Mechanical Design, Materials and Manufacturing 2018060478, 2019000032, 9781536147926, 9781536147919

Citation preview

MECHANICAL ENGINEERING THEORY AND APPLICATIONS

MECHANICAL DESIGN, MATERIALS AND MANUFACTURING

No part of this digital document may be reproduced, stored in a retrieval system or transmitted in any form or by any means. The publisher has taken reasonable care in the preparation of this digital document, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained herein. This digital document is sold with the clear understanding that the publisher is not engaged in rendering legal, medical or any other professional services.

MECHANICAL ENGINEERING THEORY AND APPLICATIONS Additional books and e-books in this series can be found on Nova’s website under the Series tab.

MECHANICAL ENGINEERING THEORY AND APPLICATIONS

MECHANICAL DESIGN, MATERIALS AND MANUFACTURING

SANDIP A. KALE EDITOR

Copyright © 2019 by Nova Science Publishers, Inc. All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. We have partnered with Copyright Clearance Center to make it easy for you to obtain permissions to reuse content from this publication. Simply navigate to this publication’s page on Nova’s website and locate the “Get Permission” button below the title description. This button is linked directly to the title’s permission page on copyright.com. Alternatively, you can visit copyright.com and search by title, ISBN, or ISSN. For further questions about using the service on copyright.com, please contact: Copyright Clearance Center Phone: +1-(978) 750-8400 Fax: +1-(978) 750-4470 E-mail: [email protected].

NOTICE TO THE READER The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained in this book. The Publisher shall not be liable for any special, consequential, or exemplary damages resulting, in whole or in part, from the readers’ use of, or reliance upon, this material. Any parts of this book based on government reports are so indicated and copyright is claimed for those parts to the extent applicable to compilations of such works. Independent verification should be sought for any data, advice or recommendations contained in this book. In addition, no responsibility is assumed by the publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication. This publication is designed to provide accurate and authoritative information with regard to the subject matter covered herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services. If legal or any other expert assistance is required, the services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS. Additional color graphics may be available in the e-book version of this book.

Library of Congress Cataloging-in-Publication Data Names: Kale, Sandip A., editor. Title: Mechanical design, materials and manufacturing / editor: Sandip A. Kale (Mechanical Engineering Department, Trinity College of Engineering and Research, Pune, India; Technology Research and Innovation Centre, Pune, India). Description: Hauppauge, New York: Nova Science Publishers, Inc., [2019] | Series: Mechanical engineering theory and applications | Includes bibliographical references and index. Identifiers: LCCN 2018060478 (print) | LCCN 2019000032 (ebook) | ISBN 9781536147926 (ebook) | ISBN 9781536147919 (hardcover) | ISBN 9781536147926 (Ebook) Subjects: LCSH: Machinery. | Materials.Classification: LCC TJ7 (ebook) | LCC TJ7 .M378 2019 (print) | DDC 621.8--dc23 LC record available at https://lccn.loc.gov/2018060478

Published by Nova Science Publishers, Inc. † New York

CONTENTS Preface Chapter 1

Chapter 2

Chapter 3

Chapter 4

Chapter 5

vii Finite Element Analysis of Three DoF Strain Gauge Force Transducer Ankur Jaiswal, H. P. Jawale and Atul Sane The Basic Principle of Calculation and Analysis of the Defective Structure of Solids V. I. Talanin, I. E. Talanin, V. V. Zhdanova, D. I. Yakymchuk and A. V. Rybalko

1

17

Advanced Methods Used in Molecular Dynamics Simulation of Macromolecules Hiqmet Kamberaj

57

Study on Error in Sigmoidal Function Generation of 4R Mechanism Ankur Jaiswal and H. P. Jawale

135

Identifying Transfer Vertex from the Adjacency Matrix for Epicyclic Gear Trains Mallu Chengal Reddy, Rudraraju Manish and Yendluri Daseswara Rao

153

vi Chapter 6

Chapter 7

Chapter 8

Chapter 9

Contents 3-D Simulation Studies on Strengthening of Beam-Column Joint Khan Mohammad Firoz, Garg Aman and H. D. Chalak

165

Nanomaterials, Ceramic Bulk and Bioceramics: Synthesis, Properties and Applications H. H. M. Darweesh

175

Simulation of Low Cost Automation and Life Cycle Cost Analysis for a Special Purpose Machine Pattanayak Satyajit and Hauchhum Lalhmingsanga

263

Estimation of Axial Force in Incremental Sheet Metal Forming Bohra Murtaza and Y. V. D. Rao

291

About the Editor

303

Index

305

PREFACE Design and manufacturing processes are the important aspects of all industrial sectors. Though, the developments in the field of electronics and digital industries are significant, the importance of the basic mechanical industry remains always on the top side. The mechanical design, materials and manufacturing are noteworthy areas of research at any time. The purpose of this book is to present some advanced research studies on mechanical design, materials and manufacturing. First five chapters discuss some important design methods, tools and techniques for various mechanical engineering applications. The first chapter presents an analysis of a novel force transducer which has a special shape that allows strategic placement of the strain gauges and senses axial forces by ignoring the moments. The strain developed in all the three directions of load application is estimated by finite element analysis. The combination of physical research with the tools and means of modern information technologies leads to the creation of a new method for verifying the validity of theoretical constructions. In the field of physics and material science of silicon, this approach makes it possible to use it for the development of the foundations of defect structure management. The second chapter explains the basic principle of calculation and analysis of the defective structure of solids. The theoretical models of the physical

viii

Sandip A. Kale

process of solid state physics, material science, programming and the creation of material and devices based on it are combined in this work. The third most interesting chapter presents advanced methods used in molecular dynamics simulation of macromolecules. These methods include generalized ensemble methods, metadynamics method, umbrella sampling methods, transition path sampling methods, accelerated molecular dynamics method, conformational flooding method. This work also proposes new possible improvement of these methods, which could result in further enhancement of conformation sampling. Chapter Four explores an extended method of mathematical modelling of Freudenstein-Chebyshev approximation theory for sigmoidal function applied to four and five precision points. The sigmoidal function has been analyzed based on the structural error between the generated function and desired function. The generated sigmoidal function is analyzed and the structural parameters have been obtained. Chapter Five presents an algorithm to find the transfer vertex of a given epicyclic gear train (EGT). The algorithm used searches for gear pair and finding the transfer vertex through corresponding turning pairs. It also identifies cases where more than one transfer vertex, are possible. This algorithm reduces the number of EGTs to be checked for rotational isomorphism after displacement isomorphism. The use of composite fibers is one of the interesting research areas in all industrial fields. In the sixth chapter, an analytical study (using ABAQUS/CAE) on the strengthening of beam-column joint under seismic conditions using carbon fibre reinforced polymer (CFRP) sheets has been carried out to draw some important results. Chapter Seven discusses the hottest and most interesting topic covers recent information about preparation, properties and applications of nanomaterials, ceramics and bioceramics. It almost incorporates into all applied science branches as physics, chemistry, materials sciences, biology, agriculture, medicine, tissue engineering, bones, caffolds, dentists, cement and concrete and all types of building materials particularly bricks, ceramic and nanoceramics or in general advanced ceramics industries, biomaterials and many others.

Preface

ix

The manufacturing industry is always focusing to reduce the production cost without compromising the quality. The simulation of low cost automation and life cycle cost analysis for special purpose machine is a major area of research in industry. Chapter Eight presents a methodology to design, develop and simulate a twin spindle turning special purpose machine based on the data collected from hydraulic, pneumatic, electro pneumatic which will serve as low cost automation. The design is based on low cost automation principles rather than pneumatic and programmable logic controller principles. The last chapter is about the estimation of axial force in incremental sheet metal forming. I believe the use of this book for the researchers and students engaged in the field of mechanical engineering and material science. I am thankful to authors for contributing in this book and reviewers for valuable reviews and suggestions. Sandip A. Kale Editor

In: Mechanical Design, Materials … ISBN: 978-1-53614-791-9 Editor: Sandip A. Kale © 2019 Nova Science Publishers, Inc.

Chapter 1

FINITE ELEMENT ANALYSIS OF THREE DOF STRAIN GAUGE FORCE TRANSDUCER Ankur Jaiswal, H. P. Jawale* and Atul Sane Department of Mechanical Engineering, Visvesvaraya National Institute of Technology, Nagpur, India

ABSTRACT Strain Gauge Force transducers are the instruments deployed to measure the magnitude of force applied on the body. This chapter presents an analysis of a novel force transducer which has a special shape that allows strategic placement of the strain gauges. The arrangement facilitates to sense only axial forces and ignore the moments, simplifying the measurements and calculations. The geometry of the transducer is conceptualized with three beams on a frame like structure, such that each of the frame independently acts as a single degree of freedom cantilever beam. Due to the property of isotropic material and single body structure, it is having a proportional force conversion. The strain developed in an individual frame is estimated by Finite Element Analysis of a CAD model of the transducer. The behaviour of the transducer for uni-axial and tri-axial loadings is presented. 

Corresponding Authors Emails: [email protected]; [email protected].

2

Ankur Jaiswal, H. P. Jawale and Atul Sane

Keywords: force transducers, finite element analysis, strain gauges

1. INTRODUCTION Strain gauge based force and torque sensors are widely used in measurement technology due to their stability at room temperature and linear static characteristics. In such applications strain gauges are mounted on the primary sensing elements of a desired cross-section. The applied force or torque gets converted into displacement making the strain gauge to change the resistance which is proportional to the applied force or torque. There are different configurations of primary sensing elements that are devised and calibrated to give desired sensitivity. However, inherently these configurations also have moment components in supplement to the force component. This makes the conversion of resistance into the readable electrical signals complex. Additionally, due to the cross sensitivity, analysis becomes tedious. The performance of the strain gauge is a function of all other elements like the position of the gauge, gauge factor, input voltage applied, bridge connection and other circuitry to estimate the strain in terms of voltage. Thus the location of dipicted strain gauges directly affects the performance. Measurement of externally applied forces can be better estimated with the accurate positioning of the strain gauge on the transducer at the place of maximum strain. The six-axis force sensor with four T-shaped bars and wrist force sensor is seen to be analyzed using the FEM and analytical method. The optimized design is used for high measurement sensitivities application (Chao and Chen, 1997, Liu and Tzo, 2002). This chapter elaborates a new frame/truss type of sensor body design. The unified criteria are based on static and dynamic stiffness of the sensor and sensitivity for strain gauge. The results have been compared with the previous design (Bayo and Stubbe, 1989) on the basis of rigidity, force sensitivity and flexibility parameters.

Finite Element Analysis of Three DoF Strain Gauge …

3

The static and kinematic equations are developed through a compliance matrix form of the model. The developed model is analyzing to the two- and three-dimensional sensor applications for regular polygons and polyhedrons (Svinin and Uchiyama, 1995). The six-axis force sensor is found to be developed with the combination of two same three-axis force sensors is idealized, describing the theoretical analysis, providing the characteristic matrix connecting the load and sensor output vectors (Kaneko, 1993) Stewart platform is seen to be used for force sensing, where derived closed-form solution of forward kinematics is applied for analysis of force and torque 6-axis transducers by obtaining force and torque acting on the upper plate from six leg forces of the parallel platform (Kang, 2001). The theoretical and Finite Element Method (FEM) analysis for six-component force moment sensor (Kang, 2001)is presented. The comparative analysis of rated strains in FEM analysis, analytical equations and experiments (Kim, et al., 1999)is dealt in. Further, compact force– torque sensor is designed and developed for Stewart platform structure for light weight capabilities of sensing elements applications (Hirose and Yoneda, 1990). This chapter presents the parallel loads sharing principle of six-dimensional heavy force/torque. Analyzing the results for multidimensional time-varying heavy load conditions of dynamic measurement problems and comparing the FEM analysis (Liu, et al., 2011), Further, design and kinematic analysis of compliant parallel mechanism (CPM) for six-component Force/Torque (F/T) sensor is seen to be worked out (Liang, et al., 2013), The Design of Experiments (DOE) technique is used to analyzed and compare the Finite Element Analysis (FEA) results. The design and optimization of a novel six-axis force/torque sensor based on strain gauges for space robot is elaborated in (Sun, et al., 2015). The geometry parameters of elastic body are optimized by response surface methodology (RSM) and validating the results by FEM. The literature presented work on force and the torque sensors which were to be mounted on the robotic manipulator, being carried with atm. The complexity is involved due to which the accuracy is limited A novel approach of mounting sensors in stationary platform is developed along with

4

Ankur Jaiswal, H. P. Jawale and Atul Sane

calibration and testing of three axis force Sensor is presented in (Deshpande, et al., 2016). Present work in this chapter is extension of the three axis force sensor analysis sited above. The difficulty found with this work is tapping of unequal strains due to improper locations for depicting strain gauges. Thus, it was essential to estimate strains exactly for depiction of gauges. The estimations of stresses and strains for exiting compact transducer is the motive for work in this chapter. Also it has led to modification of transducer design followed by comparative analysis as presented herewith.

2. STRAIN GAUGE BASED FORCE SENSING Force Transducers are devices which provide an observer the numerical value corresponding to the applied force or torque on the body, which ideally should be equal to the true value. The output of a transducer is dependent upon the predetermined relationship between the measured value and the input parameter being measured. Force Transducers usually use an elastic load bearing element or their combination. Due to application of force there is a deflection in the member and that deflection is measured by the strain gauges which convert it into a measurable output. The output of a strain gauge is in the form of an electric signal with current varying according to the magnitude of applied force (Stefanescu, 2011). In recent days, there are many methods which are used to measure torque or force applied to any object, among those methods Strain gauges are the most widely used. The Strain gauge is capable of distinguishing tensile and compressive strains by a positive or negative sign. Thus it’s possible to detect expansion as well as contraction of the objects on which strain gauges are attached. The strain gauges are strategically mounted on the force sensing element which experiences a change in resistance which is induced by the deformation of the sensor. The Sensitivity of the strain gauge is decided by a fundamental factor known as the gauge factor. It is the ratio of the relative change in electrical resistance R with respect to the mechanical strain ε.

Finite Element Analysis of Three DoF Strain Gauge … R

L

GF 

where, ρ: L: A: R: ε:

5

A

(1)

R / R R / R  L / L 

(2)

Resistivity/Conductivity constant Length of the sensor Cross sectional area of sensor Electrical resistance of the sensor Mechanical Strain

The strain gauge cell is based on an elastic element to which a number of electrical resistance strain gauges are bonded. The strain produce due to application of force is dependent on the geometric shape as well as the modulus of elasticity of that element. Each strain gauge responds to the local strain at its location, and the measurement of force is determined from a combination of these individual measurements of strain (Yurish, 2014) (Yurish, 2014). In actual life scenarios forces are usually not unidirectional. Mostly applied forces are multi-directional and dynamic in nature, thus making it difficult for a single axis sensor to detect applied load. This situation is very well addressed by multi-axis Force transducers which convert input force/torque signals to voltage signals; serve to perceive external force/torque information of different dimensions. They are highly applicable in scenarios including Robotic control and manipulation, aerospace and industries (Chen, et al., 2015).

3. TRANSDUCER DESIGN This chapter discusses a novel three DOF force transducer which has a special shape that allows strategic placement of the strain gauges in Figure

6

Ankur Jaiswal, H. P. Jawale and Atul Sane

1. Because of such arrangements they only sense axial forces and ignore the moments. This makes measurements and calculations very simple. The simplest primary sensing elements of any force sensing transducer is generally similar to a cantilever configuration. Combining three cantilevers for each of the axis of Cartesian coordinate system in a single configuration can function as a three DOF transducer for force sensing. This design felicitates to study the application and results of uniaxial, biaxial and tri-axial force loading with the means of pulleys and dead weights directly on the transducer.

Figure 1. A CAD model describing the Force Transducer (green), Pulleys and dead weight for loading (brown).

(a)

(b)

Figure 2. CAD model depicting the complete assembly of FT1 and FT2.

Finite Element Analysis of Three DoF Strain Gauge …

7

There is also a new proposed design for the same transducer having some alterations to the frame and the members so that the applied force on the same region is more easily distributed in Figure 2. This helps in better stress dispersion.

3.1 Converting of Cartesian Coordinates The spherical coordinates of a point (r, θ, φ) can be obtained from its Cartesian coordinates in Figure 3 (x, y, z) by the equations given below. r

2 2 2 x y z

    a cos  

   y   a cos  2 2 2  r x y z 

(3)

y

 x z

(4)

  a tan 

Figure 3. Spherical Co-ordinate System.

(5)

8

Ankur Jaiswal, H. P. Jawale and Atul Sane

Conversely, the Cartesian coordinates may be retrieved from the spherical coordinates (radius r, inclination θ, azimuth φ), where r ∈ [0, ∞], θ ∈ [0, π], φ ∈ [0, 2π], by:

x  r sin  sin 

(6)

y  r cos

(7)

z  r sin  cos 

(8)

4. FINITE ELEMENT ANALYSIS 4.1 Material Properties The material considered for design and analysis is Structural steel with following in Table 1. Table 1. Material Properties of Structural Steel Young’s Modulus Poisson's Ratio Density

200 GPa 0.3 7850 kg/ m3

4.2 Meshing A ten node quadratic tetrahedron element C3D10 is used for meshing both the types of transducers. The Force Transducer (FT1) type model contains 6686 elements and FT2 type model contains 10100 elements. Figure 4 (a) and 4 (b) shows the meshed model for FT1 and FT2 type of transducers respectively.

Finite Element Analysis of Three DoF Strain Gauge …

9

4.3 Loading Conditions The base of the transducer is made fixed support in order to replicate actual real life scenario of the transducer firmly placed on a solid base. This improvises the results obtained by the FEM analysis on ABAQUS. Loading is applied on the topmost surface of the transducers as shown in Figure 5. The applied loads implies to a resultant force which is in range of 0.2 to 2 N. The loads are applied in the simulation packages to find out the components of this force in Cartesian coordinate system. These components (Fx, Fy and Fz) were calculated using equation (6-8).

(a) FT1

(b) FT2

Figure 4. Final Meshing of the geometry for the analysis of FT1 and FT2.

(a) FT1 Figure 5. Loading conditions on FT1 and FT2.

(b) FT2

10

Ankur Jaiswal, H. P. Jawale and Atul Sane

5. RESULTS The stresses and strains in the members of the transducer were calculated by applying ten different sets of loading conditions ranging from 0.2 N to 2 N in x, y and z directions. The calculated strains for FT1 and FT2 are shown in Table 2 and Table 3 respectively. The loads were applied on the hemispherical region on the top of the transducer and the base link of the transducer was fixed to replicate test conditions as shown in Figure 5. Figure 6 (a) and 6 (b) shows the maximum principal strain contour on the different links of transducer FT1 and FT2 respectively, at a load of 0.2 N on all the three directions. Figure 6 (c) and 6 (d) shows the maximum principal strain contour on the different links of transducer FT1 and FT2 respectively, at a load of 1 N on all the three directions Figure 6 (e) and 6 (f) shows the maximum principal strain contour on the different links of transducer FT1 and FT2 respectively, at a load of 2 N on all the three directions. Table 2. Stress values from ABAQUS for Maximum Principal Elastic Strain in FT1 load (N) 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 1.513 3.026 4.551 6.068 7.587 9.106 10.625 12.144 13.663 15.182

Maximum Principal Strain (*e^-5) for FT1 y z xyz 0.459 0.594 0.829 0.918 1.189 1.645 1.381 1.788 2.474 1.844 2.38 3.29 2.32 2.983 4.113 2.767 3.584 4.954 3.229 4.171 5.755 3.681 4.77 6.569 4.143 5.366 7.41 4.61 5.959 8.218

Finite Element Analysis of Three DoF Strain Gauge …

(a)

(b)

(c)

(d)

(e)

Figure 6. (a-f) Principal Strain distribution for TF1 and TF2.

(f)

11

12

Ankur Jaiswal, H. P. Jawale and Atul Sane

Table 3. Stress values from ABAQUS for Maximum Principal Elastic Strain in FT2 load (N) 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 1.551 3.101 4.664 6.215 7.766 9.316 10.87 12.43 13.98 15.53

Maximum Principal Strain (*e^-5) for FT2 y z xyz 0.59 0.619 1.194 1.181 1.238 2.371 1.777 1.862 3.565 2.368 2.481 4.745 2.958 3.1 5.93 3.549 3.719 7.121 4.14 4.338 8.299 4.735 4.962 9.485 5.326 5.581 10.6 5.917 6.2 11.84

6. OBSERVATIONS 1. When load is applied in any one direction (uni-axial loading), maximum strain is obtained in the member corresponding to the respective direction. Also about 5-10% of the reaming strain is observed to be distributed to other two directional members. 2. Under tri-axial loading conditions maximum strain occurs in the bottom-most member, moderate in the middle member and least in the uppermost member. 3. The transducer exhibits a linear relationship for loading and induced strain in both FT1 and FT2 type of transducers as shown in Figure 7. This is the most favourable condition for this transducer as a primary sensing element for force measurement. Due to this property the hysteresis errors would be eliminated. However, there is a minute difference in the calculated strain between the FT1 and FT2 transducers.

Finite Element Analysis of Three DoF Strain Gauge …

13

4. The difference between induced strain in FT1 and FT2 also increases linearly as shown in Figure 8, and it is maximum in case of tri-axial loading scenario due to cross sensitivity of the members. 5. Under the same loading conditions, it is observed that the middle member has the least magnitude of the strain induced.

-

Maximum Principal Strain (*e5)

16 14 12 10 8

Force Force Force Force Force Force Force Force

in X- direction (FT1) in X- direction (FT2) in Y- direction (FT1) in Y- direction (FT2) in Z- direction (FT1) in Z- direction (FT2) in XYZ- direction (FT1) in XYZ- direction (FT2)

6 4 2 0 0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Load (N)

Figure 7. Maximum Principal Strain V/s Applied Load. 4

X direction Y direction Z direction XYZ diection

Strain Difference

3.5 3 2.5 2 1.5 1 0.5 0 0.2

0.4

0.6

0.8

1

1.2

Load (N) Figure 8. Strain difference curve between TF1 and TF2.

1.4

1.6

1.8

2

14

Ankur Jaiswal, H. P. Jawale and Atul Sane

CONCLUSION Estimation of strains in a compact force transducer for application of platform type sensor is presented herewith. The observations have led to modifications in the transducer profile so that the strains are concentrated at a specific location. The comparative strain estimations are also presented which has facilitated in designing and manufacturing of the transducer. The strain profile is used for finalizing the location of gauge pasting. It helped in successful theoretical estimators of strains in terms of output voltage of the connecting bridge circuit and further calibrations. The future scope lies in the demonstration of the effectiveness of this platform type force sensor automated assembly task.

REFERENCES Bayo, Eduardo, and JR Stubbe. (1989), Six‐Axis Force Sensor Evaluation and a New Type of Optimal Frame Truss Design for Robotic Applications, Journal of Field Robotics 6 (2), 191-208. Chao, Lu-Ping, and Kuen-Tzong Chen. (1997), Shape Optimal Design and Force Sensitivity Evaluation of Six-Axis Force Sensors, Sensors and Actuators A: Physical 63 (2), 105-12. Chen, Danfeng, Aiguo Song, and Ang Li. (2015), Design and Calibration of a Six-Axis Force/Torque Sensor with Large Measurement Range Used for the Space Manipulator, Procedia Engineering 99 (1), 116470. Deshpande, MS, HP Jawale, and HT Thorat. “Development, Calibration and Testing of Three Axis Force Sensor.” Paper presented at the Mechanical and Aerospace Engineering (ICMAE), 2016 7th International Conference on, 2016. Hirose, Shigeo, and Kan Yoneda. “Development of Optical Six-Axial Force Sensor and Its Signal Calibration Considering Nonlinear

Finite Element Analysis of Three DoF Strain Gauge …

15

Interference.” Paper presented at the Robotics and Automation, 1990. Proceedings., 1990 IEEE International Conference on, 1990. Kaneko, Makoto. “A New Design of Six-Axis Force Sensors.” Paper presented at the Robotics and Automation, 1993. Proceedings., 1993 IEEE International Conference on, 1993. Kang, Chul-Goo. (2001), Closed-Form Force Sensing of a 6-Axis Force Transducer Based on the Stewart Platform, Sensors and Actuators A: Physical 90 (1-2), 31-37. Kim, Gab-Soon, Dae-Im Kang, and Se-Hun Rhee. (1999), Design and Fabrication of a Six-Component Force/Moment Sensor, Sensors and Actuators A: Physical 77 (3), 209-20. Liang, Qiaokang, Dan Zhang, Yaonan Wang, and Yunjian Ge. (2013), Design and Analysis of a Novel Six-Component F/T Sensor Based on Cpm for Passive Compliant Assembly, Measurement Science Review 13 (5), 253-64. Liu, Sheng A, and Hung L Tzo. (2002), A Novel Six-Component Force Sensor of Good Measurement Isotropy and Sensitivities, Sensors and Actuators A: Physical 100 (2-3), 223-30. Liu, Wei, Ying-jun Li, Zhen-yuan Jia, Jun Zhang, and Min Qian. (2011), Research on Parallel Load Sharing Principle of Piezoelectric SixDimensional Heavy Force/Torque Sensor, Mechanical Systems and Signal Processing 25 (1), 331-43. Stefanescu, Dan Mihai. (2011), Handbook of Force Transducers: Principles and Components, Springer Science & Business Media. Sun, Yongjun, Yiwei Liu, Tian Zou, Minghe Jin, and Hong Liu. (2015), Design and Optimization of a Novel Six-Axis Force/Torque Sensor for Space Robot, Measurement 65 (135-48. Svinin, Mikhail M., and Masaru Uchiyama. (1995), Optimal Geometric Structures of Force/Torque Sensors, The International journal of robotics research 14 (6), 560-73. Yurish, Sergey. (2014), Modern Sensors, Transducers and Sensor Networks, Lulu. com.

In: Mechanical Design, Materials … ISBN: 978-1-53614-791-9 Editor: Sandip A. Kale © 2019 Nova Science Publishers, Inc.

Chapter 2

THE BASIC PRINCIPLE OF CALCULATION AND ANALYSIS OF THE DEFECTIVE STRUCTURE OF SOLIDS V. I. Talanin, I. E. Talanin, V. V. Zhdanova, D. I. Yakymchuk and A. V. Rybalko Department of Computer Science and Software Engineering, Institute of Economics and Information Technologies, Zaporozhye, Ukraine

ABSTRACT To describe the defective structure of semiconductor silicon, a triad is created: physical plus mathematical models - computational algorithm program. This solution can be used both in studying the properties of crystals and in industrial production. Such a solution is the basic principle in the study of structural imperfections in any solid. A structural diagram of the diffusion model for the formation of structural imperfections in a crystal and a computational algorithm is presented.



Corresponding Author Email: [email protected].

18

V. I. Talanin, I. E. Talanin, V. V. Zhdanova et al.

Keywords: grown-in microdefects, defective computational algorithm, defect engineering

structure,

model,

1. INTRODUCTION Physics, and in particular solid-state physics, is the basis for application in various fields of human activity. Modern solid-state physics can not be imagined without a wide application of mathematical modeling. The mathematical model, with one or another degree of approximation to the real physical process, is a computational algorithm for the use of computer technology. This method of connecting simulation with computer technology allows you to quickly and thoroughly explore physical objects in any practical and imaginary situations. For its implementation, it is necessary to create a triad adequate to this physical process: physical plus mathematical model - computational algorithm – program. A prerequisite is the development at first of a physical model of the physical process under study, which should be adequate to the experimental results of the research. Then such a triad can serve as a key for solving various applied engineering problems in this or that sphere of human activity. The problems of solid state physics are closely intertwined with the problems of materials science and solid state chemistry. Material science is a science that studies the structure and properties of materials, which establishes a relationship between their composition, structure and behavior of materials, depending on the environmental impact. The chemistry of the solid state overlaps the issues considered by both these sections of knowledge, but it particularly touches on the issues of synthesizing new materials. The structure and composition of a solid determine its properties. In turn, the properties of a solid determine the possibilities of engineering applications of various materials.

The Basic Principle of Calculation and Analysis ...

19

Semiconductor silicon, due to the requirements imposed on it, is one of the most highly pure materials of our time. At the same time, it is also the most studied material both experimentally and theoretically. Since the properties of semiconductor silicon depend critically on its structural perfection, the main task is the development on the basis of experimental data of a theoretical description of the formation and development of structural imperfections during crystal growth and the production of devices. The solution of this problem was sought with the help of the classical theory of a solid (Born & Huang, 1954) and the classical theory of the nucleation and subsequent growth of particles of the second phase (Cristian, 1965). From these theories, a physical mechanism was developed for defects formation, called grown-in microdefects, which are formed during crystal growth (Voronkov, 1982). Numerous variations of mathematical models, which were built on this physical mechanism, have received a general name of the model of the dynamics of point defects (PDs) (Kulkarni, 2005). This model describe the formation of dislocation loops and microvoids. It is assumed that the formation of impurity imperfections is associated with subsequent thermal treatments of the crystals (Kulkarni, et al., 2004). In contrast to these statements, we developed a model for hightemperature impurity precipitation, which was based on the methods and approaches of classical theory the nucleation and subsequent growth of second-phase particles (Talanin & Talanin, 2016a). In contrast to the models of the dynamics of PDs in this model, the entire variety of the defect structure of silicon is described on the basis of the formation of impurity precipitates during crystal cooling. To confirm the results of the high-temperature impurity precipitation model, the solid-state model A. A. Vlasov was used. This model is an alternative to the classical theory of solids (Talanin & Talanin, 2016b). The general physical model of formation and development of semiconductor silicon defect structure has been called the diffusion model. It includes mathematical models of a high-temperature model for the precipitates formation, growth and coalescence, models for formation of

20

V. I. Talanin, I. E. Talanin, V. V. Zhdanova et al.

secondary microdefects. The interrelation of these mathematical models with parameters of crystal growth makes it possible to develop the algorithm for defect structure calculating. This opens up wide opportunities for solving the most important problem of the technology of ultrapure semiconductor materials - the control of the crystals defective structure.

2. HIGH TEMPERATURE IMPURITY PRECIPITATION Despite numerous experimental proofs of the existence of impurity precipitates in grown dislocation-free silicon single crystals (Talanin & Talanin, 2006), it was assumed in the theoretical description of the defect formation process that the precipitation process begins only when the already grown crystals are heat treated. This assumption was made on the basis of the theory of nucleation and growth of particles of the second phase in solids, in which the distribution function of nuclei of various sizes

𝑓0 (𝑅)~𝑒𝑥𝑝 {−

∆𝐺(𝑅) } 𝑘𝑇

where 𝑘 is a Boltzmann constant; ∆𝐺 is a minimum energy that must be expended to create an nucleus of a given size 𝑅; 𝑇 is the temperature. Then the formation of a critical nucleus at high temperature becomes impossible and the formation and growth of the nucleus occurs only with an increase in temperature. This led to the impossibility of a theoretical description of precipitation upon cooling of the crystal during its growth (Cristian, 1965). In our reasoning, we proceeded from the firmly established fact that the formation and development of the growth microdefect structure of silicon is due to the thermal conditions (Talanin & Talanin, 2016a). Numerous researches have made it possible to establish that micropores and also loops are formed in divers crystal fields and depending on 𝑉𝑔 ⁄𝐺𝑎 = 𝜀 (where 𝑉𝑔 is a crystal growth rate, 𝐺𝑎 is an axial temperature gradient at the crystal center, and 𝜀 is some constant) (Talanin & Talanin,

The Basic Principle of Calculation and Analysis ...

21

2016a). There was an idea to introduce in the theoretical calculations the growth parameter and determine the defective structure of silicon. Thus, it was necessary to solve two problems: to determine the temperature distribution along the length of the ingot during its cooling and to solve the problem of the rapid recombination process of intrinsic point defects (IPDs) near the front of crystallization. The first problem was solved very simply. It is established that during the cooling of the crystal, depending on the thermal growth parameters, the temperature distribution varies according to the law 1⁄𝑇 = 1⁄𝑇𝑚 + 𝐺(𝑟)𝑧⁄𝑇𝑚2 where 𝑧 is the distance from the crystallization front, 𝑇𝑚 is the melting point (Talanin & Talanin, 2016a). In this formula, the radial inhomogeneity of the temperature field is taken into account in accordance with formula 𝐺(𝑟) = 𝐺𝑎 + (𝐺𝑒 − 𝐺𝑎 )(𝑟⁄𝑅𝑆 )2 (where 𝑅𝑆 is the radius of the crystal; 𝐺𝑒 is an axial temperature gradient at the crystal edge; 𝑟 is a current coordinate in range from 0 to 𝑅𝑆 (Talanin & Talanin, 2016a). Hence, it is very simple to obtain a temperature dependence on time, taking into account the growth parameters of the crystal

𝑇(𝑡) =

𝑇𝑚2 𝑇𝑚 + 𝑉𝑔 𝐺𝑎 𝑡

The solution of the second problem required to abandon the assumption of rapid recombination of IPDs near crystallization front. This gave, depending on the value of the growth parameter, the homogeneous formation of microvoids or interstitial dislocation loops and to construct the simplest mathematical models (Voronkov, 1982; Kulkarni, 2005). However, this assumption was not correct, since it completely excluded any admixture from defect formation processes (Kulkarni, et al., 2004). But even in the purest unalloyed dislocation-free silicon single crystals obtained by the method of crucible-free zone melting, the concentration of

22

V. I. Talanin, I. E. Talanin, V. V. Zhdanova et al.

residual impurities of oxygen and carbon is one to three orders of magnitude higher than the concentration of IPDs near the crystallization front. Moreover, the distribution of intuitively understandable and experimentally established low-temperature recombination acts to processes occurring at high temperatures was erroneous. In (Talanin & Talanin, 2007a), we showed that the process of recombination of IPDs in dislocation-free single crystals of silicon near the crystallization front is practically impossible due to the presence of a recombination barrier. This means that the interaction “IPD-impurity” has the advantage over interaction “IPD – IPD.” This result and taking into account the influence of the thermal growth conditions of silicon single crystals in the form of the 𝑇(𝑡) dependence make it possible to theoretically describe the conditions for nucleation, growth, and coalescence of precipitates from the crystallization temperature to room temperature. The theoretical description was based on an experimentally established physical model: (i) formation near the front of crystallization the impurity complexes; (ii) precipitates formation, growth and coalescence during cooling from crystallization to room temperatures; (iii) formation at 𝑇 ≤ 1423 К micropores or loops depending on the constant 𝑉𝑔 ⁄𝐺𝑎 . Along with the developed mathematical models for all structural defects, the physical mechanism is a diffusion model for the grown-in microdefects formation (Talanin & Talanin 2016a). The central place in the diffusion model is occupied by the hightemperature impurity precipitation model (Talanin & Talanin 2010b; Talanin et al., 2007b). The precipitation process occurs throughout the entire cooling range of the crystal, while microvoids and dislocation loops are formed in a narrow temperature range at 𝑇 ≤ 𝑇𝑚 − 300𝐾. Consideration of the elastic interaction of atoms of oxygen, carbon, vacancies and intrinsic interstitial silicon atoms in undoped silicon single crystals led to the understanding that near the crystallization front there are nucleation processes (Talanin et al., 2007b).

The Basic Principle of Calculation and Analysis ...

23

In diffusion model a combined modeling method based on the solution of differential equations for small clusters and the Fokker-Planck equation for large-size clusters in the cooling temperature range 1683-1373 K was used (Talanin & Talanin, 2016a). The model of dissociative diffusion (migration of impurities) was used to calculate the formation of precipitates in the high-temperature region T ~ 1683–1403 K (Talanin et al., 2007b). When the dimensions of the nuclei are small, this approximation is used. The edge of the reaction formation front of the complex (that is, the formation of the oxygen-vacancy complexes and the carbon-interstitial silicon complexes) is located at a distance of ~ 3·10-4 mm from the crystallization front (Talanin et al., 2007b). Recombination of IPDs is absent and then their excessive concentration arises. This interval is a diffusion layer on which this excessive concentration is applied. The calculation of the defect structure at the stage of precipitates formation and growth in the model described in (Talanin et al., 2007b) agrees well with the experimental results available in the literature (Talanin & Talanin, 2006). This is true in regard to the temperatures of formation of growning microdefects; the experiments on quenching of the crystals; and the concentrations of precipitates, which were determined from the results of electron microscopy investigations (~1013–1014 cm–3) (Talanin & Talanin, 2006; Talanin & Talanin, 2010b). Consider a system consisting of oxygen and carbon atoms, vacancies, and intrinsic interstitial silicon atoms. It will allow us to describe the kinetics of nucleation and growth of new phase particles in a supersaturated solid impurity solution. A comparative analysis of the joint evolution of oxygen and carbon deposition and the optimization of the computational algorithm for the numerical solution of equations requires the carrying out of a dimensional analysis of kinetic equations and conservation laws using characteristic time constants and critical sizes of defects (Talanin & Talanin 2010b). Such an analysis, in particular, made it possible to obtain a system of equations where the mass balance of PDs is described by diffusion equations for silicon interstitials, vacancies and impurities:

24

V. I. Talanin, I. E. Talanin, V. V. Zhdanova et al. Co  2Co CoSiO2  Do  t z 2 t Cc  2Cc CcSiC  Dc  t z 2 t Ci  2Ci CiSiO2 CiSiC  Di 2   t z t t SiO2 2 Cv  Cv Cv CvSiC  Dv   t z 2 t t

(1)

where Cv , Ci , Cс , Cо is the concentration of vacancies, intrinsic interstitials of silicon, carbon and oxygen in the crystal, respectively. The equations can be used in dimensionless form ~ f SiO2



I SiO2 ~

 O ~ f SiC t I SiC  0 ~  tc c

where  

(2)

t is the dimensionless time. The time constants in (2) are to

0 0 given by tO  (nOcr ,0 )2 / g SiO , where the critical growth ; tc  (nccr ,0 )2 / g SiC 2

rates of the precipitates are defined as SiO2 0 0 SiC g SiO  NO0O exp( Gact / kT ); g SiC  NO0c exp(Gact / kT ) . 2

The precipitates normalized sizes are determined in (2):

~O  nO / nOcr ,0 ;~c  nc / nccr ,0 , where nOcr , nccr  are normalizing precipitates critical sizes. The quantities

NO0  4 (rOcr ,0 )2  SiO2 COeq ; Nc0  4 (rccr ,0 )2  SiCCceq

The Basic Principle of Calculation and Analysis ...

25

are the numbers of particles. In the system of equations (2), the normalization to the initial concentrations of the nucleation centers was made:

f SiO ~ ~ f f SiO2  0 2 ; f SiC  SiC 0 f SiO2 f SiC

(3)

The particles fluxes on right hand sides of system of equations (2) are described as BSiO ~ ~ B ASiO2  ( g~SiO2  d SiO2 )nOcr ,0  ~ 2 ; ASiC  ( g~SiC  d SiC )nOcr ,0  ~SiC ; O O

(4)

in which the following notation is used for the normalized kinetic coefficients: BSiO ~ ~ B ASiO2  ( g~SiO2  d SiO2 )nOcr ,0  ~ 2 ; ASiC  ( g~SiC  d SiC )nOcr ,0  ~SiC ; O O

BSiO2 

~ g~SiO2  d SiO2 2

; BSiC

~ g~SiC  d SiC  2

(5)

(6)

The normalized rates of growth and dissolution of the precipitates in (5) and (6) look like

g SiO d SiO ~ g ~ d g~SiO2  0 2 ; g~SiC  SiC ; d SiO2  0 2 ; d SiC  SiC 0 0 g SiO2 g SiC g SiO2 g SiC

(7)

Critical size of precipitates

rOcr 

2uV p kT ln( S0 Si i Sv v )  6uV p

(8)

26

V. I. Talanin, I. E. Talanin, V. V. Zhdanova et al.

rCcr 

2uV p

(9)

kT ln( Sс Si i Sv v )  6uV p

where So  Co

Coeg , Sc 

Cc

Cceg

, Si 

Ci

Cieg

, Sv 

Cv

Cveg

Are the supersaturations of the oxygen, carbon, silicon interstitials, and vacancies;  is a density of a surface energy of interface between the precipitate and the matrix;  is a silicon shear modulus;  and  are the relative linear and volume misfit strains of the precipitate and the matrix, respectively;

 i and  v are the fractions of the intrinsic interstitial silicon

atoms and vacancies per impurity atom attached to the precipitate, respectively; V p is the molecular volume of the precipitate; and

1  3 u  (1   i x   v x)1  ( ) (Talanin & Talanin 2010b). 1  The critical dimensions of the microdefects and the characteristic time constants associated with them indicate the scale of the changes in the distribution function over the size of grown-in microdefects with time. These parameters are important in the analysis of the evolution of growth microdefects. The processes of nucleation pass near the crystallization front (Talanin & Talanin 2010b). The dissociative diffusion model analyzes phenomena in the diffusion region of the crystallization front, and the Fokker-Planck equation determines the distribution functions of the critical size of oxygen and carbon precipitates. Both these models complement each other (Talanin et al., 2007b). A growing undoped silicon crystal can be represented as a multicomponent multiphase system in which the role of components is played by oxygen and silicon atoms, intrinsic interstitial silicon atoms, and vacancies, whereas oxygen and carbon precipitates and agglomerates of IPDs (microvoids and interstitial dislocation loops) should be treated as a new phase. The kinetics of the precipitation process includes three stages:

The Basic Principle of Calculation and Analysis ...

27

formation the nuclei of new phase, and clusters growth, and stage of coalescence (Talanin & Talanin 2004). This process is a first-order phase transition. The growth of clusters is accompanied by a significant decrease of the supersaturation solid solution. Their number in the second stage does not change. This process is described by a system of equations (Talanin & Talanin 2011): dN 0  k0 NN 0  g1 N1 , dt dN i   N i ki N  gi   gi 1 N i 1  ki 1 NN i 1 , dt dN   N  ki N i   g i N i , dt i 0 i 1

(10)

where N i is a volume average concentration of nucleation centers that attach i particles, N is a monomer concentration, ki N is the rate of attachment of a monomer for a nucleation center, and g i is the rate of detachment of monomer for nucleation center. At the initial time, in system are monomers and centers of nucleation. The precipitates growth is limited by monomer diffusion. The kinetic coefficients are given by ki  4Ri D , where Ri is a radius of free particle attachment and D is a free particle diffusion coefficient (Talanin & Talanin 2011). The system (10) obeys the law of conservation of nucleation centers

N c   Ni (t ) and the law of conservation of the total number of i 1

particles, including both monomers and particles involved in precipitates, i.e., N 0  N t  

 iN i 1

i

, where N 0 is the monomer concentration at

the initial instant of time. Therefore, the average number of particles at the nucleation centers can be represented in the form

28

V. I. Talanin, I. E. Talanin, V. V. Zhdanova et al.

 iN i N i 0

i 0

i

i



N 0  N t  Nc

(11)

For i »1, we can use the Fokker–Planck equation. In accordance with

the balance g i   k i  1N ECE i  1

CE i   k i N E , the Fokker–

Planck equation takes the form, C i, t )  2  N t   N E  k i C i, t   N t   N E  2 k i C i, t  t i 2i

(12)

where N E is a equilibrium concentration of monomers (Talanin & Talanin 2011). The mathematical expectation i(t):

di   k0 N  N E it   m , dt

(13)

where k0  4Ri D , m is a initial precipitate size, and

 is a parameter

of cluster geometry. The change in monomer concentration during the decomposition of the solid solution (Talanin & Talanin 2011):

dN t    k0 Nc1 N t   N E   N 0  mNc N t  dt

(14)

The kinetics of decrease in the monomer concentration as a result of the decomposition of the solid solution at can be written in the form m  0 can be written in the form 1   N t   N E   exp  Nc 1   N 0  N E  k0t 1  N 0  N E  





(15)

The Basic Principle of Calculation and Analysis ...

29

Average sediment radius at the growth stage:

Rt   3

3bit  4

(16)

Depending on the temperature growth conditions, the duration of the growth phase varies from 𝑇 ≈ 𝑇𝑚 − 20𝐾 for small-sized single crystals of silicon to 𝑇 ≤ 𝑇𝑚 − 300𝐾 for large crystals (Talanin & Talanin 2011). In coalescence occurs a dissolution of small-sized particles and the growth of coarse-grained particles. At this stage, new particles are not formed. The condition providing for changeover to the coalescence stage is the ratio u t  

Rt   1 , where Rcr t  is a precipitate critical radius. Rcr t 

Precipitate grows at Rt   Rcr t  and dissolves at Rt   Rcr t  . If the

supersaturation of the solute tends to zero, then the solution is possible (Talanin & Talanin 2011). The system of equations describing the nonisothermal coalescence has the form

f R, t    f R, t dR / dt   0, f0 R   f R,0  t R

(16a)



QTc   cc t     f R, t R3dR

(17)

0

L Tc t   Tc 0  c p t



  f R, t   f R R dR 3

0

(18)

0

dR 2D 2c0 Tc  R / Rk  1 при D  R  dt kTc R 2

(19)

30

V. I. Talanin, I. E. Talanin, V. V. Zhdanova et al.

dR 2D 3c0 Tc  R / Rk  1 при D  R  dt kTc R

(20)

where expression (16) is the equation of continuity in the space of sizes for the size distribution function of precipitates, expression (17) is the mass balance equation, expression (18) is the equation accounting for the amount of released heat, f 0 R  is the initial size distribution function of

precipitates, Tc 0 is the initial temperature of the solid solution,

  4 / 3 is the volume per atom in the precipitates of the new phase, QTc  is the total amount of the material in the precipitates and the

solution, cc t  is the solute concentration in the solid solution, L is the heat of the phase transition per atom of the precipitated phase, c p is the heat capacity at constant pressure per unit mass of the solid solution, and

 е is the density of the solid solution. Equations (16) - (20) form a complete system of equations describing the process of nonisothermal decomposition of the solid solution in the coalescence stage. The solution of equations (16) - (20) is carried out by the method developed in (Talanin & Talanin 2011). The simultaneous nucleation and growth of new phase particles (oxygen and carbon precipitates) during the cooling of silicon crystals after growth leads to a strong interaction between the evolution processes of these two subsystems of grown-in microdefects Absorption of increasing vacancy oxygen precipitates leads to the emission of silicon atoms into the interstitial. The intrinsic interstitial silicon atoms, in turn, interact with growing carbon precipitates, which in the process of their growth supply vacancies for growing oxygen precipitates. Such interaction leads to an accelerated transition of the subsystems of oxygen and carbon precipitates to the coalescence stage, in comparison with the independent evolution of these two subsystems (Talanin & Talanin, 2011).

The Basic Principle of Calculation and Analysis ...

31

The kinetic model of growth and coalescence of oxygen and carbon precipitates together with the kinetic models of their formation (Talanin & Talanin, 2010b; Talanin et al., 2007b) represent a unified model of the precipitation process in dislocation-free single crystals of silicon. The mathematical apparatus of this model makes it possible, in the future, to take into account and analyze the interaction of IPDs not only with background impurities of oxygen and carbon, but also with other impurities (for example, transition metals, nitrogen, hydrogen, etc.) “impurity-impurity” (Talanin & Talanin 2011). The obtained results on high-temperature impurity precipitation required confirmation with the help of another representation, different from the classical ones. To this end, we used the solid state model of Vlasov.

3. PRECIPITATION OF THE IMPURITY IN ACCORDANCE WITH THE MODEL OF THE SOLID STATE OF VLASOV In 1945, the Soviet scientist Vlasov suggested that the physical properties of gases, liquids and solids can be described with the help of the proposed earlier to describe the peculiar properties of the plasma of the kinetic equation (Vlasov, 1945). For that time such an idea seemed to be non-trivial and was subjected to severe criticism of leading Soviet theoretical physicists (Ginzburg et al., 1946). On the basis of his approach, Vlasov put the introduction of a single distribution function that depends on all the coordinates and their derivatives up to any order (non-local statistical mechanics) (Vlasov, 1978). In the classical theory of the crystal, the main conclusions about the dynamics of the system are made on the basis of the mechanical theory of atomic vibrations in the vicinity of equilibrium positions (Born & Huang, 1954). The equations of the oscillations make it possible to obtain only the time functions, while for the description of the wave processes in the original equations, explicit derivatives with respect to the spatial coordinates must be explicitly contained. According to Vlasov, the theory

32

V. I. Talanin, I. E. Talanin, V. V. Zhdanova et al.

of crystal growth cannot be created on the basis of a method that uses classical or quantum Hamiltonians with model interaction potentials between atoms (Vlasov, 1978). During phase transformations, the electron configurations of atoms and the functional form of the interaction potentials change. Field equations give a temporal evolution of the interactions of particle systems, the change in shape of which with time is a fundamental property of the kinetics of crystal growth. Consequently, the basis for the kinetics of phase transitions should be equations with a selfconsistent field (Vlasov, 1978). To describe the stationary properties of a crystal, the concept of the particle distribution density 𝜌(𝑟) = ∫ 𝑓(𝑟, 𝑣)𝑑𝑣 is used. Vlasov showed that the non-linear model of the crystal is based on nonlinear equations that make it possible to calculate the molecular potential and density of the particle location under thermal equilibrium (Vlasov, 1950): ∞

𝑈(𝑟) = 𝜆𝑘𝑇 ∫ 𝐾1,2 (𝑟)𝑒𝑥𝑝 (− −∞

𝜌(𝑟) = 𝜆𝑘𝑇𝑒𝑥𝑝 (−

𝐾1,2 (𝑟) 𝑘𝑇

𝐾1,2 (𝑟) ) 𝑑𝑟 𝑘𝑇

)

(21)

where 𝑘 is the Boltzmann constant; 𝐾1,2 is the potential of pair interaction; 𝜆 is some characteristic number. The initial equations represent the 𝜕

equations for two particles under steady-state conditions (𝜕𝑡 = 0) (Vlasov, 1950). In order to distinguish the types of interactions, one usually considers the systems of Vlasov equations (the Vlasov-Poisson, VlasovMaxwell, Vlasov-Einstein, and Vlasov-Yang-Mills equations). In applications of the mathematical theory of non-linear equations to physical problems, the main mathematical problem arises, which consists in finding the characteristic numbers. By characteristic numbers, we mean here the values of a parameter 𝜆 for which equations of the type (21) have solutions that are different from the trivial ones. By trivial solutions are meant solutions of equations of the type (21), corresponding to the case of uniform density (Vlasov, 1950). The characteristic number 𝜆 is determined

The Basic Principle of Calculation and Analysis ...

33

(Vlasov, 1950) from the main criterion for the existence of the crystalline state, and the crystallization condition can be written as follows: 4𝜋𝑁 ∞ ∗ ∫ 𝐾1,2 (𝜌)𝜌2 𝑑𝜌 𝑘𝑇𝑚 0

=1

(22)

∗ where 𝑁 is the number of particles; 𝐾1,2 = −𝐾1,2 (Vlasov, 1978). For the crystal case, Vlasov showed that the state space is determined by the temperature changes (Vlasov, 1950). The density of complexes distribution we defined as a function of crystal cooling temperature

𝜌(𝑇) = 𝜆𝑘𝑇𝑒𝑥𝑝 (−

𝑈1𝑚𝑖𝑛,2𝑚𝑖𝑛 𝑘𝑇

)

(23)

where 𝑈1𝑚𝑖𝑛 and 𝑈2𝑚𝑖𝑛 are the minima of interatomic potentials in siliconoxygen and silicon-carbon complexes, respectively. Vlasov showed that the spatial periodic distribution is one of the particular states of particle motion (Vlasov, 1950). This solution was carried out by Vlasov for an ideal (defect-free) crystal. However, real crystals always contain structural imperfections. For defective (real) crystals, we considered the possibility of pair formation of an IPD-impurity with subsequent impurity precipitation (Talanin & Talanin, 2016b). The potential of interatomic interaction is the basis for describing the interaction between the atoms of matter. Analytic consideration of the potentials of the interaction of two atoms is extremely difficult, since after quantum-mechanical calculations they are represented as functions with a large number of parameters. In this connection, model potentials with a small number of parameters are used (Mazhukin et al., 2014; Mattoni et al., 2007). To estimate the formation parameters of silicon-carbon and silicon-oxygen complexes, we presented interaction in form of Mee-Lennard-Jones potential. To determine the characteristic numbers of silicon-oxygen (𝜆1 ) and silicon-carbon (𝜆2 ) complexes, we use (22) for the number of particles in 𝐾1,2

the complex 𝑁 = 2 and 𝐾(𝜌) = 𝑘𝑇 (1 − 𝑒 − 𝑘𝑇 ). The calculation gave the

34

V. I. Talanin, I. E. Talanin, V. V. Zhdanova et al.

following values 𝜆1 = 4.482·108 eV-1 and 𝜆2 = 1.099·109 eV-1. In this computational experiment, it was assumed that the concentration of nucleation centers for carbon and oxygen complexes is ~ 1012 cm-3. We determined the evolution of the density of the complex distribution as a function of the crystal cooling temperature. The calculations of complexation with the Vlasov solid state model show that during the growth of dislocation-free silicon single crystals, silicon impurity complexes form near the crystallization front, which coalesce during the cooling of the crystal, leading to the formation of carbon precipitates (SiC) and oxygen (SiO2) (Talanin & Talanin, 2016b). Two theories of nucleation of particles of the second phase (the classical theory of nucleation and the Vlasov model for solids) lead to similar results. This allows us to say that:  

Vlasov's solid body model can be used to describe the formation of a defective structure of real crystals; Vlasov's approach to the description of substances may have a universal character.

The model of high-temperature precipitation, built on classical concepts, suggests that nucleation is possible both during cooling of the crystal during growth and during its heating (Talanin & Talanin, 2013). The Vlasov solid state model unequivocally indicates that the formation of complexes does not occur during low-temperature crystal treatments (up to ~11000 С) (Talanin et al., 2017a). The appearance of donor and acceptor centers in silicon during thermal treatments is directly related to the original defect structure. Precipitates of impurities that were created during crystal growth are in it during the coalescence stage. Therefore, the heating of the crystals promotes the dissolution of some and the coarsening of other precipitates. For the formation of electrical centers, it is precisely soluble precipitates that are responsible. The formation of silicon-impurity complexes is the determining stage in the formation of the defect structure. Depending on the thermal growth

The Basic Principle of Calculation and Analysis ...

35

conditions, interstitial dislocation loops or microvoids form during further cooling of the crystal. High-temperature impurity precipitation is associated with the subsequent transformation of growth microdefects in the production process of silicon devices and is the foundation for the creation and development of a defective structure of semiconductor silicon.

4. FORMATION OF SECONDARY STRUCTURAL IMPERFECTIONS Under certain thermal conditions for the growth of crystals, depending on the growth parameter V

G

, microvoids or interstitial dislocation loops

are formed in the temperature range of cooling 1403 ... 1223 K. The author of (Voronkov, 1982) made the assumption of rapid recombination of IPDs near the crystallization front. Remaining after the recombination, the IPDs, depending on the temperature growth conditions, form either microvoids or dislocation loops. Based on this were developed various interpretations of model the dynamics of PDs (Kulkarni, 2005; Kulkarni et al., 2004; Sinno, 1999; Brown et al., 2001). All of these variations papers never allowed impurity precipitation during crystal cooling (Kulkarni et al., 2004). The model of the dynamics of IPDs, in the general case, includes three approximations: rigorous, simplified, and discrete-continuum approaches (Kulkarni, 2005). The most rigorous model requires the solution of integrodifferential equations for the concentration fields of PDs, and the distribution of growth microdefects in this model is a function of the coordinates, the time, and the time of evolution of the size distribution of microdefects (Kulkarni et al., 2004; Sinno, 1999). In the simplified (lumped) model, the average radius of the defects is approximated by the square root of their average surface area (Kulkarni, 2005). This approximation is taken into account in the additional variable, which is proportional to the total surface area of the defects. The lumped model is effective in calculating the two-dimensional distribution of growth microdefects (Kulkarni, 2005). Both models use the classical nucleation

36

V. I. Talanin, I. E. Talanin, V. V. Zhdanova et al.

theory and suggest the calculation of the formation of stable nuclei and the kinetics of diffusion-limited growth of defects. The discrete-continuum approximation (discrete-continuous model) involves a comprehensive approach: the solution of discrete equations for the smallest defects and the solution of the Fokker–Planck equations for large-sized defects (Brown et al., 2001). We calculated the formation of microvoids and interstitial dislocation loops in accordance with the strict approximation of the model of PD dynamics, provided that there is no high-temperature recombination of IPDs. We confirm the homogeneous character of microvoid formation process (Talanin & Talanin, 2010a). Microvoids are formed on the already existing precipitates. The data from the computational experiment for determining the concentration of vacancy microvoids correlate well with the experimentally observed results (104-105 cm-3) (Kulkarni et al., 2004). For dislocation loops, where the experiments give a value of ~ 106-107 cm-3 (Brown et al., 2001), the discrepancy reaches three orders of magnitude. This may be due to the fact that, in contrast to microvoids, which are formed only by the action of the coagulation mechanism, the formation of dislocation loops occurs due to the action of both the coagulation mechanism and the mechanism of prismatic extrusion (deformation mechanism). The results of calculations show that the main contribution to the formation of dislocation loops is made by the mechanism of prismatic extrusion, when the formation of interstitial dislocation loops is associated with the removal of stresses around the growing precipitate. We know that the kinetics of high-temperature precipitation encompasses three stages: the inception of a new phase, the growth of sediment and the stage of fusion. The elastic interaction between IPDs and impurity atoms causes the formation of precipitates. The transfer of excess (scarce) matter from the sediment or vice versa is caused by elastic deformations and associated mechanical stresses. During the growth of the precipitate, its coherence with the matrix is lost. The formation and displacement of dislocation loops is due to the structural relaxation of precipitates.

The Basic Principle of Calculation and Analysis ...

37

Consider the simplest precipitate in the form of a sphere. The formation of a circular interstitial dislocation loop of nonconformity occurs due to the growth of precipitate. The total energy of deformation of the system decreases. The matrix material in the volume of the crystal is displaced by precipitate. The dislocation loop around the precipitate is formed by intrinsic interstitial atoms. On the precipitate itself, a contour of misfit dislocation is formed. The critical size of the precipitates corresponds to the critical size of the dislocation loops (Talanin & Talanin, 2012a). In silicon precipitate produces a stress field caused by mismatch between the lattice parameters of precipitate a1  and surrounding matrix

a2  . Then, the precipitate intrinsic deformation can be defined as 

a1  a2 a1

(24)

The intrinsic deformation of precipitate in the volume of matrix determined as

  xx  xy       xy  yy   zx  zy

 xz    yz   pr  ,  zz 

(25)

where the diagonal terms constitute a dilatation mismatch (between the precipitates and matrix lattices); the other terms are shear components;

  pr  is the Kronecker delta. The elastic fields of precipitate (stress  ij

and deformation  ij ) and the field of total displacement are calculated accounting for intrinsic deformation (25) and regions of precipitate

 

localization   pr . The elastic fields of precipitate are calculated according to the known scheme using elastic moduli, elastic Green function or its Fourier transform.

38

V. I. Talanin, I. E. Talanin, V. V. Zhdanova et al. Let us consider the easiest model of a spherical precipitate with

uniaxial intrinsic deformation, i.e.,  ii   ,  ij  0i  j; i, j,  x, y, z  . The elastic strain energy of a spheroidal defect rises according to the cube

 

law as the precipitate radius R pr increases (Kolesnikova et al., 2007):

E pr 

32    J   2  R3pr , 45  1   

(26)

where J is a shear modulus;  is a Poisson's ratio. Starting from a Rcrit radius, the elastic strain mechanism begins to function. This mechanism give a prismatic interstitial dislocation loop (Kolesnikova et al., 2007). The energy criterion for this mechanism is E initial  E final condition, where

Einitial, E final constitute elastic energy of the system with precipitate before and after relaxation (Kolesnikova et al., 2007). Assume that a dislocation loop of mismatch has equatorial location on the spheroidal precipitate RD  Rpr , and intrinsic energy of a prismatic loop is equal (Kolesnikova et al., 2007)

Eloop

J  b 2  RD  2  1   

 2  RD    ln  2 ,   f  

(27)

where f is the radius of the core loop; b is the magnitude of the Burgers vector. A critical value of the precipitate radius corresponds to a value at which the loop is formed on the precipitate (Kolesnikova et al., 2007)

Rcrit  where

3b  1.08Rcrit   ln , 8 1     b 

 is a constant contribution of the dislocation core.

(28)

The Basic Principle of Calculation and Analysis ...

39

The dislocation loops with R  Rcrit will increase. The dislocation loops with R  Rcrit will dissolve. The supersaturation of silicon by intrinsic interstitial atoms and the dissolution of small dislocation loops lead to the growth of large dislocation loops. In this case, the crystal growth

ratio

is

V

G

 crit

(where

0.12

mm2/K·min

 crit 

0.3 mm2/K·min (Kulkarni, 2005)). When oversaturation of vacancies ( V  crit ) occurs, the interstitial dislocation loops start to dissolve. G

Increase in the radius of interstitial dislocation loop can be defined by the formula depending on the crystal cooling time (Burton & Speight, 1985): 2 R(t )  Rcrit  j  D(t )  t ,

(29)

where D(t ) is a diffusion coefficient of silicon interstitials; t is a time of cooling; j is factor of proportionality. The crystal cooling time is defined from: T (t ) 

Tm2 (Kulkarni et al., 2004; Talanin & Talanin, 3013), Tm  U  t

where U  Vg  G is the cooling rate of the crystal. The concentration of dislocation loops should be considered as a function of the concentration of precipitates. At the stage of precipitates growth and coalescence the dislocation loops formation is determined by the deformation mechanism. Then, the loop concentration depends on the crystal cooling time (Brown et al., 2001):

N (t ) 

M (t ) , 2 1  D(t )  t 2  Rcrit

(30)

where M (t ) is the concentration of precipitates. The kinetic model of the formation and growth of interstitial dislocation loops formed in dislocation-free single crystals of silicon as a

40

V. I. Talanin, I. E. Talanin, V. V. Zhdanova et al.

result of the structural relaxation of precipitates of background impurities of oxygen and carbon shows that high-temperature precipitation of oxygen and carbon is a fundamental process in the formation of a defective silicon structure during crystal cooling after growing. The removal of the elastic deformation caused by the growing precipitate occurs through the formation and movement of dislocation loops. Within the framework of the proposed model, the obtained results were compared with experimental data. A result confirms the reliability of the proposed model and allows it to be used to correctly take into account the influence of the parameters of precipitates on the formation of interstitial dislocation loops.

5. CONTROLLING THE DEFECTIVE STRUCTURE OF CRYSTALS The results obtained radically change the physical picture of our ideas of defect formation in crystals. The entire process is responsible for hightemperature precipitation, which occurs from the time the crystal is grown until its end. Microvoids and dislocation loops are derived from the emerging precipitate structure, their formation is determined by the specific growth conditions of semiconductor silicon. The formed defect structure, in turn, is the basis for the transformation processes that occur in the crystal as a result of technological treatments when creating various devices. The computational experiments carried out made it possible to describe the process of defect structure formation. For a theoretical description of the defect structure of semiconductor silicon, we used several mathematical models with different levels of complexity of the computational experiment. In the framework of the diffusion model, they can be represented in the form of Figure 1 Each of them to some extent describes the details of the general picture of physical phenomena occurring in the original system (the formation of structural imperfections during the growth of semiconductor silicon).

The Basic Principle of Calculation and Analysis ...

41

The general requirement is one - the correspondence between the results of computational experiments and the results obtained experimentally.

Figure 1. Scheme diffusion model of defect formation.

The application of methods of mathematical modeling makes it possible to improve the technological process and accelerate the development of new technologies for growing dislocation-free single crystals of silicon with a specified minimized content of growth microdefects. In the engineering of growth microdefects, the problem of controlling the composition and state of the ensemble of PDs in crystals is put forward, requiring deep penetration into the problem of their interaction. The absence of a single algorithm for solving defect formation problems in solids leads researchers to the necessity of their phenomenological analysis for each type of crystals separately. At the same time, many phenomena occurring in different crystals are of a general nature, for example, the interaction of PDs among themselves; their interaction with dislocations; formation and transformation of dislocations, etc. Semiconductor silicon, being an extremely pure and perfect in

42

V. I. Talanin, I. E. Talanin, V. V. Zhdanova et al.

structure material, can itself be considered as an initial model for constructing theoretical models of defect formation in other semiconductor materials and metals. The obtained mathematical models and the proposed methods for their solution in silicon make it possible to formulate and solve many problems in the kinetics of diffusion processes in solids. The primacy of the processes of high-temperature precipitation is a fundamental feature that determines the general kinetics of defect formation in highly perfect crystals, both semiconductors and metals. The specificity of growing and distinguishing the crystal structure of various dislocation-free single crystals must necessarily be taken into account in the physical model and mathematical constructions of the diffusion model. The mathematical apparatus of the diffusion model with the help of modern information technologies is used as the basis for a software package. After developing mathematical models of each of the stages of this process, an algorithm was developed to calculate the formation of a defective crystal structure during growth (Talanin et al., 2017b) and on the way of creating the instrument (Talanin et al., 2012b). The algorithm is based on the dominant role of thermal growth conditions on defect formation in a crystal. From the scheme of the algorithm (Figure 2), a special role of high-temperature precipitation is seen in the formation of the defective crystal structure. Changing the thermal growth conditions, one can create a given defective crystal structure. And, conversely, the defect structure can determine the temperature conditions for crystal growth. Therefore, the software complex constructed on the basis of this algorithm will have the functions of both a research tool and devices for controlling the process of growing silicon crystals. To create such software products, a research (calculation) technique is needed that would link each point of the crystal to its real structure. The method of virtual investigation of a defective structure should be verified by experimental researches.

The Basic Principle of Calculation and Analysis ...

43

Figure 2. Algorithm for calculating of formation of a defective crystal structure during growth.

The method that determines and calculates the growth defect structure is based on a parameter that connects the main thermal conditions of crystal growth Vg

Ga

V  Ccrit . The condition g

Ga

 Ccrit in real crystals

agrees to the V-shaped distribution of precipitates. In Figure 3 shows a standard picture of the distribution of growth microdefects in dislocationfree single crystals of silicon grown by the Czochralski method with variable growth rate. The crystal diameter was 50 mm, the growth rate varied from 𝑉𝑔𝑚𝑖𝑛 = 0.5 mm/min to 𝑉𝑔𝑚𝑎𝑥 = 3.0 mm/min. Three areas of defect formation can be identified: region 1 (above the V-shaped precipitates distribution), region 2 (V-shaped precipitates distribution region) and region 3 (below the V-shaped precipitates distribution). In small-sized silicon crystals in region 1, growth microdefects (precipitates) of interstitial and vacancy types are formed at comparable concentrations; in region 2 within the V-shaped distribution the same defects as in region 1 are formed, and outside the V-shaped distribution interstitial dislocation loops and predominantly interstitial precipitates are formed. In crystals with a diameter of more than 80 mm in

44

V. I. Talanin, I. E. Talanin, V. V. Zhdanova et al.

regions 1, 2 (inside the V-shaped distribution) microvoids begin to form. In region 3, dislocation loops and precipitates of predominantly interstitial type are formed.

Figure 3. The standard picture of the distribution of grown-in microdefects in dislocation-free single crystals of silicon grown by the Czochralski method with a variable growth rate.

With increasing crystal diameter, the thermal growth conditions of the crystal change and the picture of defect formation in the longitudinal section of the crystal undergoes changes that occur in the rather rapid disappearance of regions 1 and 3. For example, in modern large-sized single crystals of silicon (diameter 150…300 mm) grown by the Czochralski method, the picture of defect formation corresponds only to region 2 in Figure 3 (Ammon et al., 1999).

The Basic Principle of Calculation and Analysis ... When Vg

Ga

45

 Ccrit in small crystals in region 1 the concentration of

vacancies is comparable with the concentration of intrinsic interstitial silicon atoms (high growth rates of the crystal). In small crystals in region 3 ( Vg

Ga

 Ccrit ), the concentration of vacancies is small compared with the

concentration of intrinsic interstitials of silicon (low crystal growth rates). For large-sizes crystals ( Vg

Ga

 Ccrit ), supersaturation in vacancies is due to

the presence of the V-shaped precipitate distribution, which leads to an impoverishment of impurity atoms within the V-shaped distribution of precipitates and the creation of conditions for the homogeneous formation of microvoids. The growth parameter V g

Ga

controls a system of interacting

PDs during crystal cooling. The first step of the technique is to specify a certain diameter of the crystal. Since Vg

Ga

 Ccrit it is determined theoretically and experimentally

in a certain range of values (0.06 mm2/K·min ≤ C crit ≤ 0.3 mm2/K·min), then for calculation we select a certain value C crit . In the second step of the procedure, the value of the axial temperature gradient at the center of the crystal (𝐺𝑎 ) is chosen, as well as the values of the minimum (𝑉𝑔𝑚𝑖𝑛 ) and maximum (𝑉𝑔𝑚𝑎𝑥 ) crystal growth rate. For each crystal diameter, these values can be determined from an analysis of the experimental and theoretical data. The third step is the choice of the axial temperature gradient at the 𝐺 edge of the crystal (𝐺𝑒 ) in the range of values 𝑒⁄𝐺 = 1.0 ... 2.5. The 𝑎 parabolic radial distribution of the axial temperature gradient 𝐺(𝑟) = 𝐺𝑎 + 2

(𝐺𝑒 − 𝐺𝑎 ) ∙ (𝑟⁄𝑅 ) , where 𝑅𝑠 is the radius of the crystal; 𝑟 is the current 𝑠 coordinate in the range from 0 to 𝑅𝑠 . In Figure 3, some value 𝑉𝑔 = 𝑐𝑜𝑛𝑠𝑡 is given as an example. In this case, we fall into region 2. Based on the analysis of the defect structure of this region within the V-shaped distribution of precipitates, the formation

46

V. I. Talanin, I. E. Talanin, V. V. Zhdanova et al.

of precipitates is calculated, and the formation of precipitates and dislocation loops is done outside the V-shaped precipitates distribution. The technique is fairly simple and can easily be implemented as a software product that can become a convenient virtual experimental device when investigating various properties of dislocation-free single crystals of silicon. The second advantage lies in the fact that the analysis and calculation of the defect formation process is determined only by the thermal growth conditions. The technique of analysis and calculation of the formation of grown-in microdefects in dislocation-free single crystals of silicon is easy to implement on a personal computer in technological and research practice. Theoretical research of the real structure of crystals depending on their thermal growth conditions with the help of an original virtual technique for analyzing and calculating the formation of grown-in microdefects is a new experimental technique. In addition, this technique allows you to replace the experimental researches of the structure with adequate theoretical researches, so the software product developed on the basis of the proposed method will be a new virtual experimental device. The development of a software package for the analysis and calculation of the formation of grown-in microdefects in dislocation-free silicon single crystals became possible after the creation of a precipitation model during the cooling of the crystal after growth, which, in conjunction with the kinetic models of the formation and growth of interstitial dislocation loops and microvoids, makes it possible to theoretically describe the processes of formation and transformation of grown-in microdefects in dislocation-free silicon single crystals of any diameter, obtained by methods floating zone method and the Czochralski method. Since the adequacy of the developed triad “mathematical model algorithm - program” of the original physical model is certified by carrying out computational experiments and their comparison with the results of physical researches, thereat replacing the experimental studies of the structure with adequate theoretical researches using the software package volume, the complex is a new virtual experimental device. Since the

The Basic Principle of Calculation and Analysis ...

47

method of application of the software is original, it represents a new technique for the experiment. At present, the computational algorithm is implemented by us on the basis of programming languages C++, C # and JavaScript (Talanin et al., 2018). These software products allow you to get information in tabular, graphical and animation forms. Software systems require conventional computer technology, are simple and convenient to use.

6. THE POSSIBILITY OF APPLYING THE MODEL OF HIGHTEMPERATURE PRECIPITATION FOR OTHER SOLIDS When creating a model of high-temperature precipitation in dislocation-free silicon single crystals, an elastic interaction was made between impurity atoms and intrinsic point defects. It is possible to take into account the elastic interaction “impurity-impurity”, as well as the Coulomb interaction. However, even without these approximations, the high-temperature precipitation model is in good agreement with the experimental data (Talanin and Talanin, 2016a; Talanin and Talanin, 2006). The phenomenon of high-temperature precipitation accompanies the growth of crystals and other substances. The similarity of the structure of dislocation-free single crystals of silicon and germanium makes it possible, as a test of this assumption, to choose germanium crystals. The creation of a high-temperature precipitation model was impossible without calculating the recombination parameters. The authors of (Cowern et al., 2013) found two separate forms of their existence in the study of point defects in germanium. The first (low-temperature) form is a widely known point defect. The second form of the defect has a structure similar to that of the amorphous pocket. This second form was called the morph (Cowern et al., 2013). This high-temperature shape has a configurational entropy value of ~30𝑘. It exists both for an interstitial defect and for a

48

V. I. Talanin, I. E. Talanin, V. V. Zhdanova et al.

vacancy. We will estimate the recombination parameters for high and low temperature regions in germanium High temperature area. The temperature dependence of the configuration entropy for the defect model considered above can be represented in the following form (Hodge, 1997): 𝑆𝑐 (𝑇) = 𝑆∞ (1 − 𝑇𝑘 ⁄𝑇)

(31)

where 𝑆∞ is the limiting value of the configurational entropy 𝑆𝑐 (at 𝑇 → 𝑇𝑚 ); 𝑇𝑚 is the melting temperature and 𝑇𝑘 is the characteristic temperature. It is assumed that the characteristic temperature 𝑇𝑘 is the minimum temperature at which structural imperfections arise in dislocation-free germanium single crystals. This temperature can be estimated to be 𝑇𝑘 = 573𝐾 as the minimum temperature of formation of thermal donors in germanium (Clauws, 2007). If we take 𝑆∞ = −30𝑘 (Cowern et al., 2013), then we obtain 𝑆𝑐 (𝑇) = −30𝑘(1 − 573⁄𝑇). Since the contribution of the enthalpy term ∆𝐻 is negligible, the free energy of the recombination barrier ∆𝐺 = −𝑇 ∙ ∆𝑆. The temperature dependence of the height of the recombination barrier is controlled by the entropy of formation of point defects ∆𝐺(𝑇) = −𝑇 ∙ [−𝑆𝑐 (𝑇)] = 𝑇 ∙ 𝑆𝑐 (𝑇)

(32)

Approximate estimation at a temperature 𝑇 = 𝑇𝑚 leads to the free energy of the recombination barrier ∆𝐺(1211𝐾) ≈ 1.65 𝑒𝑉. The experimental results on self-diffusion in germanium show that the main contribution is made by vacancies, and the contribution of intrinsic interstitial atoms is insignificant (Hüger et al., 2008; Śpiewak et al., 2007). The diffusion coefficient obeys the Arrhenius dependence over a wide range of temperatures 𝐷(𝑇) = 13.6𝑒𝑥𝑝(−3.094𝑒𝑉/𝑘𝑇) (Werner et al., 1985). Approximate estimation at a temperature 𝑇 = 𝑇𝑚 leads to 𝐷(1211𝐾) ≈ 1.81·10-12 cm2·s-1.

The Basic Principle of Calculation and Analysis ...

49

The recombination time at high temperatures (𝜏1 ) can be evaluated from the expression 𝜏1 = Ω⁄4𝜋 ∙ 𝐷(𝑇) ∙ 𝑟0 · 𝑒𝑥𝑝(− ∆𝐺(𝑇)⁄𝑘𝑇)

(33)

where Ω is the volume of the crystal lattice and 𝑟0 = 3 ∙ 10−8cm is the recombination radius. Approximate estimation at a temperature 𝑇 = 𝑇𝑚 leads to 𝜏1 ≈ 243 s. The recombination factor 𝑘𝐼𝑉 (𝑇) is described by the theory of diffusion-limited reactions together with the kinetic activation barrier (Vanhellemont et al., 2008). At high temperatures, the recombination factor can be written in the following form 𝑘𝐼𝑉 (𝑇) = 4𝜋 ∙ 𝑟0 ∙ 𝐷(𝑇) ∙ 𝑒𝑥𝑝(−∆𝐺(𝑇)⁄𝑘𝑇)⁄Ω ∙ 𝑐𝑠

(34)

where 𝑐𝑠 = 5·1022 cm–3 is the atomic density. The estimation at a temperature 𝑇 = 𝑇𝑚 leads to the recombination factor 𝑘𝐼𝑉 (1211𝐾) ≈ 8.22·10-26 cm·s-1. Lemke and Sudkamp (Lemke and Sudkamp, 1999) introduced the following criterion for “rapid recombination”: 𝑘𝐼𝑉 (1211𝐾) ∙ 𝐶𝑉𝑚 ≥ 20 𝑠 −1 where 𝐶𝑉𝑚 = 1.3·1015 cm–3 is the vacancy concentration at 𝑇 = 𝑇𝑚 (Clauws, 2007). The calculation shows that the “fast recombination” criterion under the conditions of Voronkov's model is not satisfied. Low temperature area. The estimation of the recombination time at low temperature (𝜏2 ) is carried out by the formula 𝜏2 (𝑇) = 𝜏∞ ∙ 𝑒𝑥𝑝(−∆𝐺(𝑇)⁄𝑇𝑆𝑐 (𝑇)). The quantity 𝜏∞ = 661s is determined under the condition 𝑇 = 𝑇𝑚 . Then 𝜏2 (𝑇) = 661 ∙ 𝑒𝑥𝑝(−∆𝐺(𝑇)⁄𝑇𝑆𝑐 (𝑇))

(35)

The estimate at 𝑇 = 583 K gives the value 𝜏2 → 0 (when estimating without taking into account the vibrational entropy). Consequently, under conditions of low-temperature studies, recombination of intrinsic point

50

V. I. Talanin, I. E. Talanin, V. V. Zhdanova et al.

defects proceeds at a sufficiently high rate. The theoretical calculations carried out by us confirm the model of the entropy barrier in germanium, the essence of which is that the decrease in the barrier is due to a decrease in the configuration entropy with decreasing temperature (Gosele et al., 1980; Gosele et al., 1983). In essence, this model is as follows: the dependence of the barrier height on the temperature is determined by the configuration of intrinsic point defects at high temperatures. At high temperatures, the intrinsic interstitials of silicon and vacancies are stretched into several atomic volumes (11 atoms occupy 10 cells). around the point defect there is a disordered region extending isotropically up to the atoms of the second coordination sphere (Gosele et al., 1980; Gosele et al., 1983). Recombination can occur only in the case of simultaneous compression of both defects in the vicinity of one atomic volume. Since extended defect configurations have a greater number of microstates than a point defect, this compression reduces entropy. As the temperature is lowered, the barrier decreases significantly, disappearing at low temperatures, and defects easily recombine. This is due to a change in the configuration of intrinsic point defects (Gosele et al., 1983). Thermodynamic calculations show that the process of aggregation of point defects in germanium predominates over the process of recombination between intrinsic point defects. The contribution of the recombination process at high temperatures to the aggregation process is negligible. Consequently, vacancies and intrinsic interstitial atoms coexist in thermal equilibrium. Therefore, both types of intrinsic point defects participate in the aggregation process simultaneously. The decay of the supersaturated solid solution of point defects is caused by cooling by two mechanisms: vacancy and interstitial, leading to the formation of impurityvacancy and impurity-interstitial agglomerates. This means that defect formation processes in silicon and germanium occur in an identical manner and can be described using the model of high-temperature precipitation.

The Basic Principle of Calculation and Analysis ...

51

CONCLUSION Theoretical description of problems of modern physics should be accompanied by the development, with the help of information technologies, of software products that allow obtaining and analyzing the main results of research. The combination of physical research with the tools and means of modern information technologies leads to the creation of a new method for verifying the validity of theoretical constructions. In the field of physics and material science of silicon, this approach makes it possible to use it for the development of the foundations of defect structure management. Actually, the theoretical models of the physical process of solid state physics, material science, programming and the creation of material and devices based on it are combined in one research. This solution to the problem of defect formation in semiconductor silicon can be considered as a standard for other crystalline materials. This solution is based on model of high-temperature impurity precipitation. Defective structure is transformed under the influence of technological treatments and determines the parameters and properties of the instruments. For other solids, the proposed model can meet certain difficulties. For example, for dislocation-free single-crystal germanium, many parameters are missing, which are determined from experimental studies. The same situation is characteristic of other solids. The solution to this problem requires further research to determine the microscopic and macroscopic parameters of solids. The set of defects determines the properties of real crystals and structures, and the additional introduction of non-equilibrium point defects during technological actions can significantly change both the thermodynamic state of crystals and structures, and their most important physical properties. The management of atomic processes at the interfaces, defect-impurity reactions in materials and structures becomes one of the main directions in modern materials science and the basis of the new direction in materials science of semiconductors - defect engineering.

52

V. I. Talanin, I. E. Talanin, V. V. Zhdanova et al.

Technology engineering defects are a necessary and inseparable part of the technology for the production of semiconductor silicon and devices based on it. The use of software products makes it possible to proceed to real defect engineering. These software products can be used: -

to simulate the engineering of defects in crystals of solids during their growth; to simulate the engineering of defects in crystals of solids during the manufacture of devices; for use as a tool for studying the processes of defect formation in solids.

In turn, defect engineering allows you to control the properties and quality of the materials used.

REFERENCES Ammon von W., Dornberger, E. and Hansson P. O. (1999). “Bulk properties of very large diameter silicon single crystal.” J. Cryst. Growth. 198-199: 390-398. Born M. and Huang K. (1954). “Dynamical theory of crystal lattices.” Oxford Clarendon Press. Brown R. A., Zhihong W. and Mori T. (2001). “Engineering analysis of microdefect formation during silicon crystal growth.” J. Cryst. Growth. 225: 97-109. Burton B. and Speight M. V. (1986). “The coarsening and annihilation kinetics of dislocation loops.” Phil. Mag. A. 53: 385-402. Clauws P. (2007). “Oxygen in Germanium.” In: Germanium-Based Technologies from Materials to Devices, edited Cor Claeys and Eddy Simoen, 97-130. Amsterdam: Elsevier.

The Basic Principle of Calculation and Analysis ...

53

Cowern N.E.B., Simdyankin S., Ahn C., Bennett N.S., Goss J.P., Hartmann J.M., Pakfar A., Hamm S., Valentin J., Napolitani E., Salvador D., Bruno E., Mirabella S. (2013). “Extended point defects in crystalline materials: Ge and Si.” Phys. Rev. Letters. 110: 155501. Cristian J. W. (1965). “The theory of transformations in metals and alloys.” London: Pergamon Press. Ginzburg V. L., Landau L. D., Leontovich M. A. and Fok V. A. (1946). “About inconsistency of works by A. A. Vlasov on general theory of plasma and physics of solid body.” J. Exp. Theor. Phys. 16: 246-252. Gosele U., Frank W., and Seeger A., (1980). “About the secret of selfinterstitial atoms in silicon.” J. Appl. Phys. 23: 361-367. Gosele U., Frank W., and Seeger A., (1983). “An entropy barrier against vacancy-interstitial recombination in silicon.” Solid State Commun., 45: 31-33. Hodge M.I. “Adam-Gibbs formulation of enthalpy relaxation near the glass transition”. (1997). J. Res. Natl. Inst. Stand. Technol. 102: 195-202. Hüger E., Tietze U., Lott D., Bracht H., Bougeard D., Haller E.E., and Schmidt H. (2008). “Self-diffusion in germanium isotope multilayers at low temperatures.” Applied Physics Letters 93: 162104. Kolesnikova A. L., Romanov A. E., and Chaldyshev V. V. (2007). “Elastic-energy relaxation in heterostructures with strained nanoinclusions.” Physics of the Solid State. 49: 667-674. Kulkarni M. S., Voronkov V. V. and Falster R. (2004). “Quantification of defect dynamics in unsteady-state and steady-state Czochralski growth of monocrystalline silicon.” J. Electrochem. Soc. 151: G663-G669. Kulkarni M. S. (2005). “A selective review of the quantification of defect dynamics in growing Czochralski silicon crystals.” Ind. Eng. Chem. Res. 44: 6246-6263. Lemke H., and Sudkamp W., (1999). “Analytical approximations for the distributions of intrinsic point defects in grown silicon crystal.” Phys. Stat. Sol. (a). 176: 843-865. Mattoni A., Ippolito M. and Colombo L. (2007). “Atomistic modeling of brittleness in covalent materials.” Phys. Rev. B, 76: 224103-224111.

54

V. I. Talanin, I. E. Talanin, V. V. Zhdanova et al.

Mazhukin V. I., Shapranov A. V. and Rudenko A. V. (2014). “Comparative analysis of interatomic interaction potentials for crystalline silicon.” Mathematica Montisnigri XXX: 56-75. Sinno T. (1999). “Modeling microdefect formation in Czochralski silicon.” J. Electrochem. Soc. 146: 2300-2312. Śpiewak P., Kurzydłowski K.J., Sueoka K., Romandic I., and Vanhellemont J., (2008). “First Principles Calculations of the Formation Energy of the Neutral Vacancy in Germanium.” Solid State Phenomena 131-133: 241-246. Talanin V. I. and Talanin I. E., (2004). “Mechanism of formation and physical classification of the grown-in microdefects in semiconductor silicon.” Defect & Diffusion Forum 230-232: 177-198. Talanin V. I. and Talanin I. E., (2006). “Formation of grown-in microdefects in dislocation-free silicon monocrystals.” In: New Research on Semiconductors, edited by Thomas B. Elliot, 31-68. New York: Nova Sci. Publ., Inc. Talanin V. I. and Talanin I. E., (2007a). “On the recombination of intrinsic point defects in dislocation-free silicon single crystals.” Physics of the Solid State 49: 467-471. Talanin V. I., Talanin I. E., and Voronin A. A. (2007b). “About formation of grown-in microdefects in dislocation-free silicon single crystals.” Can. J. Phys. 85: 1459-1471. Talanin V. I. and Talanin I. E., (2010a). “Kinetics of formation of vacancy microvoids and interstitial dislocation loops in dislocation-free silicon single crystals.” Physics of the Solid State. 52: 1880-1886. Talanin V. I. and Talanin I. E., (2010b). “Kinetics of high-temperature precipitation in dislocation-free silicon single crystals.” Physics of the Solid State 52: 2063-2069. Talanin V. I. and Talanin I. E., (2011). “Kinetic model of growth and coalescence of oxygen and carbon precipitates during cooling of asgrown silicon crystals.” Physics of the Solid State 53: 119-126. Talanin V. I. and Talanin I. E., (2012a). “A kinetic model of the formation and growth of interstitial dislocation loops.” J. Cryst. Growth. 346: 4549.

The Basic Principle of Calculation and Analysis ...

55

Talanin V. I., Talanin I. E., and Ustimenko N. Ph. (2012b). “Structural scheme of information system for defect engineering of dislocationfree silicon single crystals.” Science and Technology 2: 130-134. Talanin V. I. and Talanin I. E., (2013). “Diffusion model of the formation of growth microdefects as applied to the description of defect formation in heat-treated silicon single crystals.” Physics of the Solid State 55: 282-287. Talanin V. I. and Talanin I. E., (2016a). “Diffusion model of the formation of growth microdefects: a new approach to defect formation in crystals (review).” Physics of the Solid State 58: 427-437. Talanin V. I. and Talanin I. E., (2016b). “Complex formation in semiconductor silicon within the framework of the Vlasov model of a solid state.” Physics of the Solid State 58: 2050-2054. Talanin V. I., Talanin I. E., and Lashko V. I. (2017a). “Description of the real monocrystalline structure on the basis of the Vlasov model for solids.” In: New research on silicon – structure, properties, technology edited by V. I. Talanin, 5-9. Rijeka: INTECH Publ. Talanin V. I., Talanin I. E., and Lashko V. I. (2017b). “Algorithm for calculating the initial defect structure of semiconductor silicon” Engineering Physics 2(4): 60-71. Talanin V. I., Talanin I. E., and Lashko V. I. (2018). “The new software for research and the modelling of grown-in microdefects in dislocation-free silicon single crystals.” Engineering and Applied Sciences 3: 1-5. Vanhellemont J., Spiewak P., Sueoka K., Romandic I., and Simoen E., (2008). “Intrinsic point defect properties and engineering in silicon and germanium Czochralski crystal growth.” In: The 5th International Symposium on Advanced Science and Technology of Silicon Materials (JSPS Si Symposium), Nov. 10-14, 2008, Kona, Hawaii, USA. Vlasov A. A. (1945). “On the theory of the solid state.” Journal of Physics 9: 130-139. Vlasov A. A. (1961). “Many-Particle Theory and Its Application to Plasma”. New York, Gordon and Breach. Vlasov A. A. (1978). “Nonlocal Statistical Mechanics”. Moscow, Nauka.

56

V. I. Talanin, I. E. Talanin, V. V. Zhdanova et al.

Voronkov V. V. (1982). “Mechanism of swirl defects formation in silicon” J. Cryst. Growth. 59: 625-642. Werner M., Mehrer H., and Hochheimer H.D., (1985). “Effect of hydrostatic pressure, temperature, and doping on self-diffusion in germanium.” Phys. Rev. B 32: 3930.

In: Mechanical Design Materials Editor: Sandip A. Kale

ISBN: 978-1-53614-791-9 c 2019 Nova Science Publishers, Inc.

Chapter 3

A DVANCED M ETHODS U SED IN M OLECULAR DYNAMICS S IMULATION OF M ACROMOLECULES Hiqmet Kamberaj∗ Department of Computer Engineering Faculty of Engineering International Balkan University Skopje, Republic of Macedonia

Abstract Molecular dynamics (MD) simulations at atomic level have widely been used in studying macromolecular systems, such as protein, DNA and their complexes, mainly because the classical statistical mechanic’s laws can explain different phenomena occurring at specified experimental conditions. In this study, we will present the most advanced methods used in the MD simulation of macromolecular systems. Furthermore, a discussion of applications of these methods and perspective on developing new approaches will be introduced. This study aims to review the methods that are developed to enhance the conformation sampling of molecular simulations, in particular, for observing rare events in complex molecular systems. In the summary, we also present a discussion and perspective ∗ Corresponding Author Email:

[email protected]

58

Hiqmet Kamberaj on the methods described in this chapter and propose the new possible improvement of these approaches, which could result in further enhancement of conformation sampling.

Keywords: molecular dynamics simulation, enhanced sampling, rare events, conformation transitions.

1.

Introduction

MD approach at atomic resolution is often used to study complex biomolecular systems (M. Karplus and J.A. McCammon, 2002), mainly because the classical statistical mechanic’s laws can explain different phenomena occurring at specified experimental conditions (van Gunsteren et al., 2006). In particular, MD is used to study the internal fluctuations (A. Amadei et al., 1993; M. Karplus and J.A. McCammon, 2002), protein folding dynamics (Rogal and Bolhuis, 2008), transition path sampling (P. G. Bolhuis et al., 2002), proteinDNA, protein-protein and protein-ligand complexes, and free energy calculations (S. L. Seyler and O. Beckstein, 2014). However, standard MD simulation has limited time and size scale, which makes it difficult to study typical phenomena of macromolecular systems, such as slow conformation motions (Clarage et al., 1995; Palmer, 1982). Therefore, it has been argued elsewhere (G. Ciccotti and E. Vanden-Eijnden, 2015; M. K. Transtrum et al., 2015) that these limitations may be avoided by employing new statistical and computational approaches to be studied efficiently. There have been different efforts in developing new approaches for enhancement of conformation sampling of simulations using MD technique, as discussed elsewhere (van Gunsteren et al., 2006). These MD approaches are used in many applications for lowering conformation transition barriers by increasing the rate of rare events occurrence by introducing a bias that can be rigorously removed a posterior, or even without the bias term. Different approached have been used to introduce bias during MD simulation, such as by changing the shape of potential energy surface until a (quasi) flat landscape is obtained (A. Piela et al., 1989; Hamelberg et al., 2004b; Hünenberger and van Gunsteren, 1997; Laio and Parrinello, 2002a), using soft-core potential interactions (Hünenberger and van Gunsteren, 1997), conformational flooding (Grub-

Advanced Methods Used in Molecular Dynamics ...

59

müller, 1995), (geometrical) constraints (Wells et al., 2005), or using Tsallis dynamics (Andricioaei and Straub, 1997). Other approaches include parallel tempering, such as replica-exchange (Berg and Neuhaus, 1992; Earl and Deem, 2005; Frantz et al., 1990; Marinari and Parisi, 1992; Wang and Swendsen, 1986), multi-canonical algorithms (Y. Okamoto, 2004), and swarm-like dynamics (H. Kamberaj, 2015, 2018; Huber and van Gunsteren, 1998; K. K. Burusco et al., 2015). For gaining an increase in both time a size scale of the systems, the coarse-grained models have also shown a great interest, for instance, by decreasing the number of interacting particles (I. Bahar and R. L. Jernigan, 1997; Irbäck et al., 2000; McCammon et al., 1980; Oldziej et al., 2004; Smith and Hall, 2001a,b; Tozzini, 2005; Tozzini and McCammon, 2005; Tozzini et al., 2006; Y. Ueda et al., 1978), or reducing the dimensional space to only essential degrees of freedom (Kamberaj, 2011; Lange and Grubmüller, 2006; Stepanova, 2007). Recently (Dror et al., 2011; Friedrichs et al., 2009), using computer engineering, longer MD simulation runs have been reported scaling from hundreds of microseconds to milliseconds timescale. Besides, development of multiple time step integration numerical schemes, such as reference system propagator algorithm (RESPA), have provided other approaches for extending the time scales of MD simulations (M.E. Tuckerman et al., 1992; Minary et al., 2004; Tuckerman and Martyna, 2000). In this chapter, we will describe in details some of these methods which are most often used to improve the sampling of configuration space in the MD simulations. We aim to critically review these methods and provide a discussion and perspective of the approaches introduced here. Also, we will further discuss possible improvements of some these methods, which could yield an increase of sampling efficiency of MD simulation in studying more complex phenomena of macromolecular systems.

2.

Multiple Time Step Integrator

MD simulations of complex molecular systems, such as biomolecules characterized by multiple time scales, show some disadvantage due to the small time steps used to ensure the stability of numerical integration of the fast motions. Hence, too many time steps are needed for observation of slow conformation transitions, which practically requires a large number of force computa-

60

Hiqmet Kamberaj

tions. For these reasons, the Reference System Propagator Algorithm (RESPA) method is introduced to reduce computational efforts for simulations of such system (Tuckerman and Berne, 1991a,b; Tuckerman et al., 1990, 1991). The time-reversible forms of the RESPA methods have also been developed, named r-RESPA, which have shown to be very stable concerning the order and stability of numerical integrators (M.E. Tuckerman et al., 1992). The r-RESPA, which will be discussed below in more details, uses Trotter factorization of the classical Liouville propagation operator (Creutz and Goksch, 1989; H.D. Raedt and B.D. Raedt, 1983; Takahashi and Imada, 1984). Following the discussion in literature (M.E. Tuckerman et al., 1992) (see also Ref. (M.P. Allen and D.J. Tildesley, 1989)), for a system with f degrees of freedom the Liouville operator, L, is defined as  f  ∂ ∂ + p˙ j (1) iL = {· · · , H} = ∑ x˙ j ∂x j ∂p j j=1 where Cartesian coordinates are used with (x j , p j ) ≡ Γ the position and conjugate momenta of the system, p˙ j gives the force along the jth direction, and {· · ·} represents the Poisson bracket of the system. L is a linear Hermitian operator of square integrable function on the phase space of Γ. The time propagation operator as a function of L is defined by U(t) = exp (iLt) which is a unitary: U(−t) = U −1 (t). The position and conjugate momenta state point of the system at a given time t is defined as Γ(t) = U(t)Γ(0), which allows determining one time step propagation as the following: Γ(∆t) = exp (iL∆t)Γ(0) where ∆t = t/P is the size of a time step. Here, t is the total evolution time and P are the number of integration points. By splitting the Liouville operator into n different terms, like the following: n

iL =

∑ iLk

k=1

Advanced Methods Used in Molecular Dynamics ...

61

and use the Trotter factorization scheme (H.F. Trotter, 1959), then the propagator becomes (" # n−1

∑ Uk (∆t/2)

U(t) =

Un (∆t)

(2)

k=1

×

"

n−1

∑ Un−k(∆t/2) k=1

where Uk (h) = exp (iLk h). Denoting "

#)P

n−1

∑ Uk(∆t/2)

G(∆t) =

k=1

×

"

+ O(t 3 /P2 )

#

Un (∆t)

n−1

∑ Un−k(∆t/2) k=1

#

As shown in Ref. (M.E. Tuckerman et al., 1992), G(∆t)G(−∆t) = 1, therefore, G(∆t) generates time-reversible dynamics. The multiple time step integrator is based on splitting the system into the fast and slow degrees of freedom. Equivalently, decomposing the forces entering into the equations of motion into long-range forces, Fl (r) and short-range forces Fs (r) (M.E. Tuckerman et al., 1992): F(r) = Fs (r) + Fl (r) The short-range forces in the system are related to the slow degrees of freedom, and thus, they determine the multiple time step of the integrator δt. On the other hand, the long-range forces are related to the fast degrees of freedom, and thus, they determine the most extended time step of the integrator ∆t. The relationship is established as ∆t (3) δt = NMT S where NMT S is the number of multiple steps. Here, the short-range forces are calculated every time step δt, and long-range forces are calculated after every NMT S time steps (i.e., every time step ∆t). Hence, the degrees of freedom are advanced using ∆t as a time step. In the r-RESPA implementation, this procedure

62

Hiqmet Kamberaj

decreases the number of calls for forces calculations, which reduces, in turn, the overall computational time. The basic idea of r-RESPA implementation, as discussed elsewhere (M.E. Tuckerman et al., 1992; Minary et al., 2004; Tuckerman and Martyna, 2000), is on determining a reference system force Fs (r) for short range interactions. Then, Eq. (1) can be written in the following form:   ∂ ∂ ∂ + Fs (x j ) + Fl (x j ) iL = ∑ x˙ j ∂x j ∂p j ∂p j j=1 f

f

= iLs + ∑ Fl (x j ) j=1

(4)

∂ ∂p j

and the propagator operator is factorized as f



∆t ∂ G(∆t) = ∏ exp Fl (x j ) 2 ∂p j j=1



(5)

× exp (iLs ∆t)   f ∂ ∆t Fl (x j ) × ∏ exp 2 ∂p j j=1 where the operator exp (iLs ∆t) propagates the state vector using the short range forces with a shorter time step δt (see Eq. 3). Here, this operator is factorized using the Trotter formula (M.E. Tuckerman et al., 1992): "   f δt ∂ exp (iLs ∆t) = ∏ exp Fs (x j ) (6) 2 ∂p j j=1   f ∂ × ∏ exp δtFs (x j )x˙ j ∂x j j=1  #NMT S f δt ∂ × ∏ exp Fs (x j ) 2 ∂p j j=1 Here, NMT S is usually chosen a priory to guarantee the stability of numerical integrator (M.E. Tuckerman et al., 1992). Usually, when the operator G(∆t) is

Advanced Methods Used in Molecular Dynamics ...

63

applied to an initial state (r(0), p(0)), it gives a solution for both position and velocity similar to Verlet numerical integrator (M.E. Tuckerman et al., 1992). Following the discussions in Refs. (M.E. Tuckerman et al., 1992; M. Tuckerman and M. Parrinello, 1994; W.B. Street et al., 1978), for a LennardJones type of fluid, exists only the translational relaxation time characteristic. In that case, the integration time step can easily be chosen. On the other hand, for biomolecules, indeed there exists more than one time-scale. For example, in addition to the translational and rotational relaxation times, there exists the time characterizing intra-molecular motion, such as bond stretching, angle bending, and dihedral angle motion. Furthermore, the inter-molecular motion, including van der Waals and electrostatic interactions, is of the typical timescale of one or more orders in magnitude larger than intra-molecular motion. In such cases, the system is characterized by stiff nonlinear differential equations, which require the use of a small enough time step to observe fast motion, if treated using one time-scale. Other systems that are characterized by more than one timescales are those consisting of high-frequency oscillators interacting with a bath of slow motion (Tuckerman et al., 1990), and the systems consisting of large mass particles (slow degrees of freedom) interacting with lighter ones (fast particles) (Tuckerman et al., 1991). The method is also used to treat systems coupled to a Nosé heat bath (Nosé, 1984; S. Nosé, 1984a; W.G. Hoover, 1985) used to keep temperature and/or pressure fixed during MD simulations. Here, heat bath includes extra fast degrees of freedom into the system, treated using multiple time stepping algorithms (M.E. Tuckerman et al., 1992), typically, two or more time steps. The method has been used by many molecular simulation software codes in performing simulations of complex systems, for example, in CHARMM program (B. R. Brooks et al., 2009). However, the approach is limited by the so-called resonance phenomena, which restricts the use of time steps higher than ∆t < 8 fs by r-RESPA in MD simulations of biomolecular systems (Bishop et al., 1997; Ma et al., 2003; Schlick et al., 1998). It must be noted that not just time-reversible integrators, but also multiple time step symplectic integrators (Skeel et al., 1997) show numerical instability limiting the use of large time steps (Wolfram, 2002). According to Ref. (Schlick et al., 1998), the resonance phenomena is the re-

64

Hiqmet Kamberaj

sult of using the perturbation techniques to derive the numerical integrators. To overcome these problems, numeric methods have been introduced to increase the time steps in molecular dynamics simulations. For example, nonsymplectic Langevin Molly (LM) integration method (Izaguirre et al., 2001) and the so-called LN integrator, which combines the force separation approach with Langevin dynamics (Barth and Schlick, 1998a,b). These methods allow using more substantial time steps in MD simulations using stochastic approaches to increase the numerical stability of integration. A stable version of r-RESPA integrator has also been introduced, named the Targeted Mollified Impulse method (Ma and Izaguirre, 2003), which includes the Langevin dynamics to improve the accuracy of multiple time stepping integrator. In Ref. (Minary et al., 2004), authors discuss a reversible, resonance-free integrator which allows for using time steps of the order up to 100 fs or even larger depending on the time length correlations studied. This integrator uses non-Hamiltonian dynamics, which are shown to sample a canonical distribution of physical configuration space (Minary et al., 2004) (q1 , q2 , · · · , q3N ) ≡ ((x1 , y1 , z1 ), · · · , (xN , yN , zN )) Here, we have written the equations of motion governing dynamics by modifying those given in Ref. (Minary et al., 2004) as the following: q˙i =

pi , mi

(7) (s)

p˙i = Fi − λi pi − P1 pi   (i) (i) 2 L M Q1 (ξ1, j ) (i) (i) η˙ (i) = − ∑  ξ2, j − ∑ ξk, j  kB T j=1 k=2 (i) (i) (i) (i) (i) (i) ξ˙ 1, j = −ξ1, j ξ2, j − λb ξ1, j − λi ξ1, j

j = 1, · · · , L (i)

(i) ξ˙ k, j =

Gk, j (i) Qk

(i)

(i)

− ξk+1, j ξk, j

j = 1, · · · , L; k = 2, · · · , M − 1

Advanced Methods Used in Molecular Dynamics ...

65

(i)

(i) ξ˙ M, j =

GM, j (i)

QM, j

,

j = 1, · · · , L

(s)

s˙i,k = Pi,k , k = 1, 2, · · · , M Γi,k (s) (s) (s) P˙i,k = − Pi,k+1 Pi,k , k = 1, 2, · · · , M − 1 Wi,k Γi,M (s) , P˙i,M = Wi,M for i = 1, 2, · · · f ( f = 3N), where 1 λi = 2K(p, ξ) (i)

λb = 2K(p, ξ) =



pi Fi mi



L−1 1 − 2K(p, ξ) L p2i L − 1 + mi L

L

(i)

∑ Q1

(8) L

(i) (i) (i) 2 ξ2, j (ξ1, j )

∑ Q1 j=1

(i)

(ξ1, j )2

!

(9) (10)

j=1

which ensures that maximum total kinetic energy accumulated in each degree of freedom is LkB T . In Eq. 7, M is the Nosé-Hoover chain length of ther(i) mostats, ξk, j (k = 1, · · · , M and j = 1, · · · , L) are the thermostat velocities associated with Lagrangian multiplier along the i degrees of freedom and η(i) is the corresponding thermostat coordinate, which is used to control the accumulated kinetic energy fluctuations. Fi is Newton’s force on the i degrees of freedom. (i) The thermostat forces Gk, j are defined as (i)

(i)

(i)

Gk, j = Qk−1(ξk−1, j )2 − kB T,

(k = 2, 3, · · · , M)

(11)

for ( j = 1, · · · , L), were L is an adjustable parameter. Here, kB is the Boltz(i) mann constant and Qk determine fictitious thermostat masses optimized in Refs. (G. J. Martyna et al., 1992; S. Nosé, 1984b): (i)

Qk = kB T τ,

k = 1, 2, · · · , M

(12)

66

Hiqmet Kamberaj (i)

where τ is a time scale associated with the thermostat. λi and λb are the Lagrangian multipliers which are determined such that equations of motion have to satisfy the following constraint: 2K(p, ξ) = LkB T (s)

In Eqs. 7, si,k and Pi,k are, respectively, the thermostat coordinates and their associated velocities (k = 1, · · · , M) at temperature T for ith degrees of freedom of the real system. Thermostat forces Γi,k are defined as p2i − kB T mi  2 (s) Γi,k = Wi,k−1 Pi,k−1 − kB T,

Γi,1 =

(13) k = 2, · · · , M

Wi,k are thermostat masses determined by Eq. 12, as in Refs. (G. J. Martyna et al., 1992; S. Nosé, 1984b). Eqs. 7 can be numerically solved using the Liouville operator formalism and Trotter factorization schemes as suggested elsewhere (Minary et al., 2004) (and the references therein). The classical Liouville operator can be expressed as: f  ∂ ∂ + q˙d (14) iL = ∑ v˙d ∂v ∂q d d d=1 M

+

L

k=1 j=1 M

+

(s)

∑ P˙d,k k=1

∂

(d)

∑ ∑ ξ˙ k, j

(d) ∂ξk, j

∂ (s) ∂Pd,k

+ η˙ (d)

+ s˙d,k

∂ ∂η(d) 

∂  ∂sd,k

which can then be decomposed for every degree of freedom d as " # f

iL =

f

∑ iL(d) = ∑

d=1

d=1

(d)

Nd

(d)

(d)

iL1 + ∑ iL2,n + iLNHC n=1

where (d)

iL1 = vd

M ∂ ∂ ∂ + η˙ (d) (d) + ∑ s˙d,k ∂qd ∂sd,k ∂η k=1

(15)

Advanced Methods Used in Molecular Dynamics ... ! (n) F ∂ (d) (n) (n) (s) (n) d iL2,n = − λd vd − Pd,1 vd (n) (n) md ∂vd (n)

+

Γd,1

L

∂

Wd,1 ∂P(s) j=1 d,1 M L G(d) ∂ k, j (d) iLNHC = (d) (d) k=2 j=1 Qk, j ∂ξk, j

∑∑ L

∂

(n) (d)

− ∑ λd ξ1, j

(d)

∂ξ1, j

M−1 L

−

67

n = 1, 2, · · · , Nd

(d) (d)

∑ ∑ ξk, j ξk+1, j

k=1 j=1

∂ (d)

∂ξk, j

∂

(d) (d)

− ∑ λb ξ1, j

(d)

∂ξ1, j

j=1 M

M−1 Γd,k ∂ ∂ (s) (s) − Pd,k Pd,k+1 (s) ∑ (s) ∂P k=2 Wd,k ∂P k=1

+∑

d,k

k

where Nd is the number of parts that the force on every degree of freedom can be split, that is Nd

Fd =

(n)

∑ Fd

n=1

for each degree of freedom d (d = 1, 2, · · · , f ). Here, vd is the velocity of the d-the degree of freedom, vd ≡ q˙d = pd /md . Note that it is assumed that the force’s strength is decreasing with n. Introducing the multiple time step parameters (Minary et al., 2004): δt =

∆t , NMT S

Nd

NMT S = ∏ sn

(16)

n=1

n−1

sNd = 1,

wn = ∏ sk ,

w1 = 1

k=1

Using the Trotter factorization scheme for classical Liouville operator, as suggested in Ref. (Minary et al., 2004), then the approximation of true evolution can be written as:  (t)  h is1 −2 ˜ ˜ (t) iL˜ N δt d Γ(∆t) ≈ e · · · eiL2 δt eiL1 δt (17)

68

Hiqmet Kamberaj  s −2 iL˜ 2 δt 2 iL˜ Nd δt ×e · · ·e Γ(0)

where (d)

˜

δt

eiLk δt = eiLNHC 2 (d) iL2,1 δt2

×e

(18) (d) iL1 δt

e

(d) ∑kn=1 iL2,n wn δt2

e

(d) iLNHC δt2

×e

for each degree of freedom d (d = 1, 2, · · · , f ). Thus, the weak or long-range forces correspond to large values of n, and hence are calculated less often, but they are weighted with larger wn to equalize they time step with that of shortrange forces, where sn =

wn+1 wn

gives the ratio of strengths between (n + 1) and n forces. It can be seen that the number of n force evaluations is NMT S /wn . The error in one time step is O(∆t 3 ), and for the entire trajectory of length t, it is O(t∆t 2) (Minary et al., 2004). Analytical solutions can be obtained for each of the exponential factorized parts of the classical Liouville operator using the following relations:   ∂ f (x) = f (x + a), (19) exp a ∂x   ∂ exp ax f (x) = f (ea x) ∂x where a is a constant. Furthermore, Nosé-Hoover part iLNHC of the classical operator can also be decomposed using Trotter factorization schemes as suggested in Ref. (G. J. Martyna et al., 1996). The method, using an unmodified version of Eq. 7 as in Ref. (Minary et al., 2004), has been implemented in the PINY-MD software (M. E. Tuckerman et al., 2000) and it is applied for different test systems, including a protein studied in vacuo using CHARMM22 force field (Jr et al., 1998). The results published in Ref. (Minary et al., 2004) have shown that large time steps of ∆ = 100 f s provided perfect agreement with other methods using much smaller

Advanced Methods Used in Molecular Dynamics ...

69

time steps. Efforts should be made to also include the solvent as a part of the system and check the efficiency of the method in the simulation of large macromolecular systems in the solvent. Future work should also focus on the comparison of the efficiency of sampling conformation equilibrium space of such complex systems using other methods discussed below or combining this method with other enhanced sampling techniques.

3.

Generalized Ensemble Methods

It has been suggested (Hansmann and Okamoto, 1993) that generalizedensemble can be used for a better sampling of configurations characterized by lower energies in computer simulations. This class of methods includes approaches, such as multicanonical sampling (Berg and Neuhaus, 1991, 1992), the broad histogram method (de Oliveira, 1998; de Oliveira et al., 1996), WangLandau algorithm (Wang and Landau, 2001), Tsallis weights methods (Tsallis, 1988), and parallel tempering or replica exchange method (Geyer, 1992; K. Hukushima and K. Nemoto, 1996; Penna, 1995). These methods are often used to study the dynamics of biomolecular systems (Hansmann and Okamoto, 1999). All of the above mentioned generalized-ensemble approaches have the same starting point, that is, the replacement of canonical Boltzmann-like weights at temperature T exp (−β∆E) with non-Boltzmann weights, which allows the system escaping from the local minimum states. Here, ∆E represents the energy barrier height and β is the inverse temperature of the simulation, β = 1/kB T . In the canonical ensemble (characterized by fixed N, V , and T ), each state point, (r, p), in the phase space is associated with a Boltzmann weight, which is defined in terms of the Hamiltonian function H(r, p): WB(r, p, β) = exp (−βH(r, p))

(20)

Since momentum p and coordinates r are independent, we can integrate according to the momentum space Eq. 20, and re-write the Boltzmann factor as a

70

Hiqmet Kamberaj

function of the instantaneous value, E, of the potential energy function U(r): WB(E, β) = exp (−βE)

(21)

The probability distribution function of a canonical ensemble is proportional to the product of WB (E, β) and the density of states Ω(E): P(E, β) ∝ Ω(E)WB(E, β) Here, Ω(E) is a monotonically increasing function of the energy E. Since WB (E, β) is a monotonically decreasing function of E, then P(E, β) has a Gaussian shape distribution with a maximum around average energy E for a fixed inverse temperature β. In a typical MD simulation, due to sampling problems, accurate calculation of Ω(E) is not possible, especially, at low temperatures and complex systems, which can be trapped at some local minimum energy state. Here, we will discuss how these weights are chosen for those methods which are most often used in molecular dynamics simulations.

3.1.

Multicanonical Sampling Method

The main aim of the multicanonical ensemble (the so-called MUCA) is to multiply the states with a non-Boltzmann multicanonical factor, Wmu (E), which yields a uniform probability energy distribution, Pmu (E) (Berg and Neuhaus, 1991, 1992): Pmu (E) ∝ Ω(E)Wmu(E) ≡ constant

(22)

Since probability is uniform (i.e., flat), the multicanonical ensemble achieves free random walks is in the potential energy space. In this way the system is able to escape faster any local energy minimum state, hence enhancing the configuration phase space sampling in an MD simulation. From Eq. 22, we can calculate the non-Boltzmann weight as Wmu(E) ≡ exp (−βEmu (E, β0 )) ∝

1 Ω(E)

(23)

where Emu (E, β0 ) is the multicanonical potential energy function given by Emu (E, β0 ) = kB T0 ln Ω(E) =

1 S(E) kB β0

(24)

Advanced Methods Used in Molecular Dynamics ...

71

where S(E) = kB ln Ω(E) is the entropy function of the microcanonical ensemble and β0 is the multicanonical inverse temperature. The density of states is practically unknown a priory, therefore, the nonBoltzmann’s weights Wmu (E) are determined, in general, using short MD simulation runs (Berg and Neuhaus, 1991, 1992), and this is one of the limitations of standard multicanonical ensemble approach, which can be overcome by combining MUCA with other methods as discussed in the following sections. The implementation of the MUCA in MD simulation is conveniently introduced by modifying equations of motion with new forces, F˜i , acting on particles as (Bartels and Karplus, 1998; Hansmann et al., 1996; Nakajima et al., 1997): ∂Emu (E, β0 ) F˜i = − ∂qi ∂Emu (E, β0 ) = Fi , ∂E

(25) i = 1, 2, · · · , f

(26)

where Fi is the Newton force acting on the i degree of freedom. Eqs. 7 describing the dynamics of a system in the multicanonical ensemble are re-written as pi mi β(E) (s) p˙i = Fi − λi pi − P1 pi β0   (i) (i) 2 L M Q (ξ ) 1 1, j (i) (i) η˙ (i) = − ∑  ξ2, j − ∑ ξk, j  k T B j=1 k=2 q˙i =

(i) (i) (i) (i) (i) (i) ξ˙ 1, j = −ξ1, j ξ2, j − λb ξ1, j − λi ξ1, j

j = 1, · · · , L (i)

(i) ξ˙ k, j =

Gk, j (i) Qk

(i)

(i)

− ξk+1, j ξk, j

j = 1, · · · , L; k = 2, · · · , M − 1 (i)

(i) ξ˙ M, j =

GM, j (i)

QM, j j = 1, · · · , L

(27)

72

Hiqmet Kamberaj (s)

s˙i,k = Pi,k , k = 1, 2, · · · , M Γi,k (s) (s) (s) − Pi,k+1 Pi,k , k = 1, 2, · · · , M − 1 P˙i,k = Wi,k Γi,M (s) , P˙i,M = Wi,M where β defines the simulation inverse temperature, such as   1 ∂S(E) β(E0 ) = kB ∂E E0 The multicanonical weighting factor is usually determined by short trial MD simulation runs at high temperature T0 using a canonical ensemble (Berg and Celik, 1992; Okamoto and Hansmann, 1995), as described by Eqs. 7. From these trial runs, we can then determine ( (1) Emu (E, β0 ) = E (1) Wmu (E, β0 ) = WB(E, β0 ) = exp(−β0 E) A maximum value of energy Emax is determined as an average of potential energy function at temperature T0 : Emax = hEiT0 Then, for E ≤ Emax , a flat energy distribution is achieved, and for E > Emax, we obtain the canonical ensemble distribution at T0 . At every MD time step, t, the probability distribution weighting factor is given by:   W (t) (E, β0 ) = exp −β0 E (t) (E, β0 ) (t)

Then, a histogram N (t) (E) is accumulated for distribution Pmu (E) of potential (t) energy. Denoting by Emin the minimum energy value obtained until the t time step. For the (t + 1) time step, the multicanonical potential energy is obtained

Advanced Methods Used in Molecular Dynamics ...

73

as  E, E ≥ Emax   
(t)  1 (t)  (t)  ln N (E) −c , E (E, β ) + mu  0  β0   (t) (t+1) Emin ≤ E < Emax Emu (E, β0 ) =  (t)   β(t+1) (Emin )   (t)   E − Emin   β0    (t+1) (t) (t) +Emu (Emin , β0 ), E < Emin

(28)

where c(t) are used to ensure the continuity of energy function at E = Emax , determined as  1  (t) c(t) = ln N (Emax) β0

The MD simulation continues until a reasonably flat potential energy function is obtained, which is determined by comparing the values of energy for all E < Emax and requiring to be of the same order of magnitude. After this (t) convergence is reached, Emin should be equal to the global minimum potential energy function value. Note that during MD simulation, a polynomial or sometimes a cubic spline function (Y. Sugita and Y. Okamoto, 2000) is used to fit the histograms each MD simulation time step (Nakajima et al., 1997). Long MD simulation in a multicanonical ensemble is performed, after the optimal weighting factor is obtained. Then, the ensemble average of any physical quantity, A , is determined using the Weighted Histogram Analysis Method (WHAM) (Gallicchio et al., 2005), which is described in details, for the general case, in Section 3.10..

3.2.

The Wang-Landau Multicanonical Method

In the Wang-Landau method (WLM), a random walk in the energy space with probability proportional to the density of states Ω(E): P(E) ∝

1 Ω(E)

(29)

generates a flat energy distribution (Wang and Landau, 2001). To achieve this the estimated density of states is modified systematically until a flat distribution

74

Hiqmet Kamberaj

is produced in the energy space. In this procedure, simultaneously, the density of states converges to the true value, by controlling a so-called modification factor at each iteration step. At the end of the simulation, this modification factor becomes very close to one, representing a random walk with the true density of states (Wang and Landau, 2001). Initially, the density of states, Ω(E), is unknown, therefore, it is set to one: Ω(E) = 1,

∀E

Then, a sampling of energy space is performed with a probability given by Eq. 29. In general, if E1 and E2 are two energy states, then the transition probability from state E1 to E2 is   Ω(E1 ) ,1 (30) P(E1 → E2 ) = min Ω(E2 ) Every time an energy state E is visited, we multiply existing density of states by the factor γ > 1 (Wang and Landau, 2001) Ω(E) → Ω(E)γ

(31)

or in algorithmic scale ln(Ω(E)) → ln (Ω(E)) + ln (γ)

(32)

If the move is rejected, then E remains unchanged, and we modify the current Ω(E) with the same factor γ. In the first publication (Wang and Landau, 2001), the suggested initial value of factor γ is γ = γ0 = e1 = 2.71828 which allows faster convergence of Ω(E) to the true density of states even for a very large system. On the other hand, as discussed in Ref. (Wang and Landau, 2001), if γ0 is too small, then the convergence is extremely slow. However, values of γ0 being too large will produce high statistical errors. During the simulation a histogram H(E) is accumulated, representing counts for every visited energy bin, E, by the system. After histogram becomes flat in the sampled energy range, we say Ω(E) has converged to the true value

Advanced Methods Used in Molecular Dynamics ...

75

with an accuracy proportional to the factor ln(γ). Then, the modification factor is decreased according to (Wang and Landau, 2001) √ γ1 = γ0 At this moment, the histogram is reset, and the random walk sampling restarts. Now, the density of states is multiplied by a smaller value of factor γ1 at each step. The algorithm continues in this way, and each time that the histogram becomes flat, the modification factor is decreased as (Wang and Landau, 2001) √ γi+1 = γi The algorithm stops when
γfinal = exp 10−8 ≈ 1.00000001

It can be seen that the accuracy is controlled by estimating the density of states and the length of a simulation by a factor γ. In addition to γfinal , the accuracy of estimating Ω(E) depends on the complexity and size of the system, criterion of the flat histogram, and algorithm’s implementation (Wang and Landau, 2001). In MD simulations WLM is implemented by modifying equations of motion for the multicanonical method, as in Refs. (Hansmann et al., 1996; Nakajima et al., 1997). Here, we purpose to modify equations of motion, given by Eqs. 27, as the following pi mi β (s) p˙i = Fi − λi pi − P1 pi β0   (i) (i) 2 L M Q1 (ξ1, j ) (i) (i) η˙ (i) = − ∑  ξ2, j − ∑ ξk, j  kB T j=1 k=2 q˙i =

(33)

(i) (i) (i) (i) (i) (i) ξ˙ 1, j = −ξ1, j ξ2, j − λb ξ1, j − λi ξ1, j

j = 1, · · · , L

(i) ξ˙ k, j =

(i) Gk, j (i) Qk

(i)

(34) (i)

− ξk+1, j ξk, j

76

Hiqmet Kamberaj j = 1, · · · , L; k = 2, · · · , M − 1 (i)

(i) ξ˙ M, j =

GM, j (i)

QM, j j = 1, · · · , L (s)

s˙i,k = Pi,k , k = 1, 2, · · · , M Γi,k (s) (s) (s) − Pi,k+1 Pi,k , k = 1, 2, · · · , M − 1 P˙i,k = Wi,k Γi,M (s) P˙i,M = , Wi,M where d lnΩ(E) (35) dE with β and β0 being, respectively, the inverse simulation and multicanonical temperatures. Note that the Wang-Landau application in MD simulation consists in calculation of density of states Ω(E) from Eq. 31 or Eq. 32, then using Eq. 35 to run MD simulation (T. Nagasima et al., 2007). In order to calculate accurately Ω(E) and hence β0 , a histogram bin of energy distribution is estimated, and the bin width will determine the accuracy of calculation of Ω(E) and β0 since it defines the ruggedness of energy distribution (T. Nagasima et al., 2007). To smooth the ruggedness, the energy distribution is approximated by a Gaussian distribution, and then, WHAM can be used to estimate Ω(E) and multicanonical inverse temperature. The method has found application to Ising spin lattice systems (Landau et al., 2004; Wang and Landau, 2001). It has been used to study the conformation transitions of proteins using confined lattice models (Pattanasiri et al., 2012), protein folding (Wüst and Landau, 2012), and optimizing temperature distribution in replica exchange method (H. Kamberaj and A. van der Vaart, 2009). β0 =

3.3.

Tsallis Statistics Molecular Dynamics Method

In the Tsallis statistics molecular dynamics (TSMD) approach (Tsallis, 1988), the principle of maximum generalized entropy is employed to obtain the gener-

Advanced Methods Used in Molecular Dynamics ...

77

alized statistical mechanic’s formalism. The probability weights can be determined as (Tsallis, 1988) q q − 1 WT (E, β) = [1 + (q − 1)β(E − E0 )] −

q is an adjustable parameter taking real values and E0 is the system’s ground energy. Note that WT (E, β) > 0. Besides, for q → 1, the Boltzmann’s weight can be obtained, and for q > 1, probability distribution has longer tails. The long tails of the Tsallis distribution have inspired construction of generalized distributions which will enhance the excursion towards regions with higher energy by decreasing the magnitude of the force close to these regions. This increases the rate of barrier crossing and hence allows the system escaping the local minimum energy states (A. Karolak and A. van der Vaart, 2012; Andricioaei and Straub, 1997; H. Kamberaj and A. van der Vaart, 2007; J. Kim and J. E. Straub, 2009; Sugita and Okamoto, 1999). The aim of Tsallis statistical ensemble is to weight each state by a weighting factor, WT (E, β) (Sugita and Okamoto, 1999): PT (E, β) ∝ Ω(E)WT(E, β)

(36)

The implementation of Tsallis statistics in MD simulations is obtained by defining the Tsallis weights as the following (Sugita and Okamoto, 1999) WT (E, β) = exp (−βUeff ) where Ueff is an effective potential defined as Ueff(E, β) =

q ln (1 + β(q − 1)(E − E0 )) β(q − 1)

(37)

In the new generalized ensemble, MD simulations use the new potential function Ueff, which replaces the old one E. The new forces that drive Newton’s equations of motion are written as (Sugita and Okamoto, 1999) ∂Ueff (E, β) F˜ i = − ∂qi ∂Ueff (E, β) Fi = ∂E

78

Hiqmet Kamberaj =

1 Fi 1 + β(q − 1)(E − E0 )

Fi is the Newton force on particle i (i = 1, 2, · · · , N). Then, the equations of motion describing a generalized canonical ensemble according to Tsallis statistics can be given as the following: q˙i =

pi mi

(38)

1 (s) Fi − λi pi − P1 pi 1 + β(q − 1)(E − E0 )   (i) (i) 2 L M Q (ξ ) 1 1, j (i) (i) ξ2, j − ∑ ξk, j  η˙ (i) = − ∑  k T B k=2 j=1 p˙i =

(i) (i) (i) (i) (i) (i) ξ˙ 1, j = −ξ1, j ξ2, j − λb ξ1, j − λi ξ1, j

j = 1, · · · , L (i)

(i) ξ˙ k, j =

Gk, j (i) Qk

(i)

(i)

− ξk+1, j ξk, j

j = 1, · · · , L; k = 2, · · · , M − 1 (i)

(i) ξ˙ M, j =

GM, j (i)

QM, j j = 1, · · · , L (s)

s˙i,k = Pi,k , k = 1, 2, · · · , M Γi,k (s) (s) (s) P˙i,k = − Pi,k+1 Pi,k , k = 1, 2, · · · , M − 1 Wi,k Γi,M (s) P˙i,M = , Wi,M TSMD has successfully been employed to different molecular systems, such as simulation of atomic clusters (Andricioaei and Straub, 1996, 1997), protein folding (I. Fukuda and H. Nakamura, 2002; S. Jang et al., 2008; U. H. E. Hansmann and Y. Okamoto, 1997; Y. Pak and S. Wang, 1999), and molecular docking (Y. Pak and S. Wang, 2000). The approach has also been

Advanced Methods Used in Molecular Dynamics ...

79

implemented with replica exchange method by replacing Boltzmann’s weights with Tsallis weighting factors for each replica (J. Kim and J. E. Straub, 2009; S. Jang et al., 2003; T. W. Whitfield et al., 2002).

3.4.

Swarm Particle-Like Molecular Dynamics Method

As we mentioned above, Eqs. 7 can be used to describe the Nosé-Hoover dynamics (Hoover, 1985; S. Nosé, 1984b) of a system of N atoms coupled to a chain of thermostats (G. J. Martyna et al., 1992). Recently (H. Kamberaj, 2015) a new approach was introduced based on the swarm particle social intelligence, which is tested to improve the conformational sampling (H. Kamberaj, 2015, 2018). In this approach, in addition to the Newtonian forces, a random force is exerted on each particle (H. Kamberaj, 2015). This is similar to Langevin dynamics (Schlick, 2010). In particular, the MD equations of motion given by Eqs. 7 can be modified following Ref. (H. Kamberaj, 2015) as: pi mi

pLbest q˙Lbest = i δ U(q) < U qLbest , i mi Gbest

p q˙Gbest = i δ U(q) < U qGbest , i mi q˙i =

(39)

(s)

p˙i = Fi − λi pi − P1 pi m

+ ∑ Pi j γ1 u1 (cLbest − c j ) + γ2 u2 (cGbest − cj) j j j=1

p˙Lbest i Gbest p˙i

= −γ1 u1 (qLbest − qi ) i

= −γ2 u2 (qGbest − qi ) i   (i) (i) 2 L M Q (ξ ) 1 1, j (i) (i) η˙ (i) = − ∑  ξ2, j − ∑ ξk, j  k T B j=1 k=2

(i) (i) (i) (i) (i) (i) ξ˙ 1, j = −ξ1, j ξ2, j − λb ξ1, j − λi ξ1, j

j = 1, · · · , L

(40)


80

Hiqmet Kamberaj (i)

(i) ξ˙ k, j =

Gk, j (i) Qk

(i)

(i)

− ξk+1, j ξk, j

j = 1, · · · , L; k = 2, · · · , M − 1 (i)

(i) ξ˙ M, j =

GM, j (i)

QM, j j = 1, · · · , L (s)

s˙i,k = Pi,k , k = 1, 2, · · · , M Γi,k (s) (s) (s) P˙i,k = − Pi,k+1 Pi,k , k = 1, 2, · · · , M − 1 Wi,k Γi,M (s) P˙i,M = , Wi,M where the vector c = (c1 , c2 , · · · , cm )T characterizes the essential degrees of freedom in the system. Projection operator, P, transforms the real coordinates q to the so-called collective coordinates c according to: f

c j = ∑ Pi j qi i=1

In Eq. 39, {cLbest }mj=1 and {cGbest }mj=1 are defined as (H. Kamberaj, 2018): j j f

cLbest = ∑ Pi j qLbest j i i=1 f

cGbest = ∑ Pi j qGbest i j i=1

which are updated every time step. Here, qLbest is configuration vector with the lowest value of the potential energy of the system and qGbest is configuration vector of the final state of the system. In Eq. 39 ui (i = 1, 2) denotes a uniformly distributed random number in (0, 1), and γ1 and γ2 are adjustable parameters. In Eq. 39, the δ function is given as:


1, if U(q) < U qLbest Lbest δ U(q) < U q = 0, otherwise

81

Advanced Methods Used in Molecular Dynamics ... and Gbest

δ U(q) < U q





=



1, 0,

if U(q) < U qGbest otherwise




The augmented dynamical system, which is given by Eq. (39) , sample an equilibrium canonical distribution with conserved total energy given by: f

p2i i=1 2mi

Eext = ∑

f

∑

+

i=1 f

∑

+

i=1

|

pLbest i 2mi


2 !

pGbest i 2mi


2 !

δ U(q) < U qLbest

1 2

f

{z

M

+∑ ∑





}

h
2
2 i Lbest Gbest u γ q − q + u γ q − q j 2 2 j ∑ 11 j j

j=1

{z

Ubias

f





δ U(q) < U qGbest Etot,kin

+U(q) + |

(41)

Qi,k ξ2i,k

i=1 k=1

|

2 {z

Ethermo

+ kB T si,k

!

}

}

where f is the total number of degrees of freedom of the system ( f = 3N). These equations represent an extended phase space of the augmented dynamical system with real variables:
Gbest Gbest

(qi , pi ) , qLbest , pLbest , qi , pi , i = 1, 2, · · · , f i i and thermostats variables: (si,k , ξi,k ),

i = 1, 2, · · · , f ; k = 1, 2, · · · , M

In Eq. 41, Etot,kin is the total kinetic energy of augmented system, Ubias is the total potential energy including bias term, and Ethermo is the thermostat energy.

82

Hiqmet Kamberaj

WHAM is used to recover the equilibrium canonical distribution of the real system (H. Kamberaj, 2015, 2018). The augmented dynamical system is shown to sample metastable (H. Kamberaj, 2018) and rare transition events (Hummer and Szabo, 2010), and to enhance the conformation sampling (Andricioaei and Straub, 1997; J. Kim and J. E. Straub, 2009). In Eq. (39), the first bias term steers the system towards the state with the lowest energy, which has been visited at any instant time t and hence enhancing the local basin sampling. Besides, the second bias term indicates the “information" about configuration with the lowest energy ever visited, and hence enhancing the barrier crossing rate.

3.5.

Replica Exchange Method

Another class of methods that use the generalized distributions for sampling the conformation phase space is also the so-called temperature Replica Exchange Method (REM) (Earl and Deem, 2005; Falcioni and Deem, 1999; Neal, 1996; Sugita and Okamoto, 1999; Wang and Swendsen, 1986). REM is often used to solve the problems of quasi-ergodicity in simulations of (bio)molecular systems. In REM, replicas representing the system are simulated independently at different temperatures (Wang and Swendsen, 1986). In particular, consider a system of N atoms each with a mass mi , position vector ri = (xi , yi , zi ), and conjugated momentum pi = (pxi , pyi , pzi ). In standard REM the generalized ensemble corresponds to L independent replications of the original system coupled to L thermostats at different temperatures. Using Nosé-Hoover dynamics (Hoover, 1985; S. Nosé, 1984b), each replica is in equilibrium with a chain of thermostats (G. J. Martyna et al., 1992) and the equations of motion are given here as the following for each replica α: q˙i,α =

pi,α mi

(42) (α,s)

p˙i,α = Fi,α − λi,α pi,α − P1 pi,α   (i,α) (i,α) 2 L M Q (ξ ) 1 1, j (i,α) (i,α) η˙ (i,α) = − ∑  ξ2, j − ∑ ξk, j  k T B α j=1 k=2 (i,α) (i,α) (i,α) (i,α) (i,α) (i,α) ξ˙ 1, j = −ξ1, j ξ2, j − λb ξ1, j − λi,α ξ1, j

j = 1, · · · , L

Advanced Methods Used in Molecular Dynamics ...

83

(i,α)

(i,α) ξ˙ k, j =

Gk, j

(i,α)

(i,α)

− ξk+1, j ξk, j

(i,α) Qk

j = 1, · · · , L; k = 2, · · · , M − 1 (i,α)

(i,α) ξ˙ M, j =

(α) s˙i,k

GM, j

(i,α)

QM, j

j = 1, · · · , L (α,s)

= Pi,k , (α)

(α,s) P˙i,k =

(α,s) P˙i,M =

Γi,k

(α) Wi,k (α) Γi,M (α) Wi,M

k = 1, 2, · · · , M (α,s) (α,s)

− Pi,k+1 Pi,k ,

k = 1, 2, · · · , M − 1

Two neighboring thermostats (e.g., i and j) swap at regular interval of times their configurations (replicas) with probability, Pacc , which preserves the detailed balance (Sugita and Okamoto, 1999; Wang and Swendsen, 1986):  Pacc = min 1, exp(−(β j − βi )(Ei − E j )) (43)

where Ei and E j correspond to the total energies of replicas i and j, respectively. In REM, high-temperature replicas are able to cross more often energy barrier. On the other hand, low-temperature replicas sample more often potential energy basins. It has been suggested (H. Fukunishi et al., 2002) that the number of replicas scales as the square root of system’s degrees of freedom. Note that increasing the number of replicas requires longer simulation runtime, which is necessary to optimize the rate of round trips between the two extreme temperatures. Omitting the solvent degrees of freedom through the use of implicit or hybrid explicit/implicit solvent models (A.E. Garcia and J.N. Onuchic, 2003; A. Okur et al., 2006; D. Bashford and D.A. Case, 2000; R. Zhou, 2003; R. Zhou and B.J. Berne, 2002) have increased the efficiency of REM. Another approach includes the use of separate heat baths for the solute and solvent (X. Cheng et al., 2005). It has been argued (A.E. Garcia and J.N. Onuchic, 2003; P. Liu

84

Hiqmet Kamberaj

et al., 2005; R. Zhou, 2003; R. Zhou and B.J. Berne, 2002) that reductions in system size may not accurately describe the structure and dynamics of the system. Other approaches, similar to REM, are also proposed. For instance, the Hamiltonian Replica Exchange Method (HREM) (H. Fukunishi et al., 2002). An HREM with biasing the backbone dihedral potentials yielded a reduction in the number of replicas (K. Srinivasaraghavan and M. Zacharias, 2007). In resolution HREM approach, which uses implicit solvent models only (E. Lyman et al., 2006; P. Liu and G.A. Voth, 2007), in addition to different temperature couplings, the replicas exchange a subset of configuration coordinates from a coarse-grained model (E. Lyman et al., 2006). Use of temperature scaling for the solvent-solvent and solvent-protein interactions in REM has also shown to reduce the number of replicas (P. Liu et al., 2005), which has further been improved by using the Tsallis biasing potential (H. Kamberaj and A. van der Vaart, 2007). Efforts have also been made to optimize the distribution of temperatures among the replicas as in Refs. (A. Kone and D.A. Kofke, 2005; Berg, 2004; C. Predescu et al., 2004; D.A. Kofke, 2002; Escobedo and MartinezVeracoechea, 2007; Gront and Kolinski, 2007; H.G. Katzgraber et al., 2006; Li et al., 2007; Nadler and Hansmann, 2007; N. Rathore et al., 2005; Predescu et al., 2005; Sabo et al., 2008; S. Trebst et al., 2004, 2006; Sugita and Okamoto, 1999). Some of these methods (Escobedo and Martinez-Veracoechea, 2007; H.G. Katzgraber et al., 2006; Nadler and Hansmann, 2007; S. Trebst et al., 2004, 2006) have particularly been important in studying the protein folding/unfolding transitions, which represent a difficult case study in standard REM because of the low rate of accepted swaps between replicas across the transition temperature (Huang et al., 2007; S. Trebst et al., 2006). To further increase the efficiency of REM, other approaches to REM have also been proposed (H. Kamberaj and A. van der Vaart, 2009), which aims to obtain a flat generalized probability distribution function in temperature space using the Wang-Landau algorithm (F. Wang and D.P. Landau, 2001a,b). The method addresses two problems of REM: it increases the probability of swapping, and it decreases the bottleneck for exchange at the transition temperature. Note that a WHAM is used for analyzing the data from all replicas (see Section 3.10.).

Advanced Methods Used in Molecular Dynamics ...

3.6.

85

Swarm Particle-Like Replica Exchange Method

More recently (H. Kamberaj, 2015), a combination of replica exchange method with Swarm Particle-like Molecular Dynamics (SPMD) is introduced. SPMD showed to improve conformation sampling when applied to LennardJones atomic cluster systems (H. Kamberaj, 2015) and protein folding problems (H. Kamberaj, 2018) when combined with replica exchange approach. Here, the equations of motion given in Ref. (H. Kamberaj, 2018) are modified as the following: pi,α mi

pLbest i,α δ U(qα ) < U qLbest , q˙Lbest = α i,α mi pGbest

i,α q˙Gbest = δ U(qα ) < U qGbest , i,α α mi q˙i,α =

(α,s)

p˙i,α = Fi,α − λi,α pi,α − P1 pi,α  m (α) α,Lbest + ∑ Pi j γ1 u1 (c j −cj ) j=1

(α)

+ γ2 u2 (cGbest −cj ) j



Lbest p˙Lbest i,α = −γ1 u1 (qi,α − qi,α )

p˙Gbest = −γ2 u2 (qGbest i,α i,α − qi,α )   (i,α) (i,α) 2 L M Q (ξ ) 1 1, j (i,α) (i,α) η˙ (i,α) = − ∑  ξ2, j − ∑ ξk, j  k T B α j=1 k=2 (i,α) (i,α) (i,α) (i,α) (i,α) (i,α) ξ˙ 1, j = −ξ1, j ξ2, j − λb ξ1, j − λi,α ξ1, j

j = 1, · · · , L (i,α)

(i,α) ξ˙ k, j =

Gk, j

(i,α) Qk

(i,α)

(i,α)

− ξk+1, j ξk, j

j = 1, · · · , L; k = 2, · · · , M − 1

(44)

(45)

86

Hiqmet Kamberaj (i,α)

(i,α) ξ˙ M, j =

(α) s˙i,k

GM, j

(i,α)

QM, j

j = 1, · · · , L (α,s)

= Pi,k ,

(α,s) P˙i,k =

(α,s) P˙i,M =

k = 1, 2, · · · , M

(α) Γi,k (α,s) (α,s) − Pi,k+1 Pi,k , (α) Wi,k (α) Γi,M , (α) Wi,M

k = 1, 2, · · · , M − 1

where all the variables have the same meaning as in Eq. 39 for the replica α and {cGbest }mj=1 is related to the global best coordinates qGbest corresponding to j configuration with the lowest energy among all replicas through the projection operator P: f

cGbest j

= ∑ Pi j qGbest i i=1

It has been shown elsewhere (H. Kamberaj, 2018) that the Eqs. 44 preserve the detailed balance condition. Following Ref. (H. Kamberaj, 2018), assuming that a Markovian chain of states is formed, the probability of obtaining a trajectory in the configuration space of the replica k can be written as: T −1

Pk (XkT ) = exp (−βk E(xk,0 )) ∏ π(xk,t → xk,t+1 )

(46)

t=0

with βk being the inverse temperature of the thermostat k. In Eq. 46 E(xk,t ) is the total energy obtained for the configuration xk,t . Here, XkT represent T replicas of the system: XkT = {xk,0 → xk,1 → · · · → xk,T −1 } The initial configurations of each replica are obtained from a canonically distributed with an initial unbiased energy of the system for replica k E(xk,0 ): ρinit (xk,0 ) = exp (−βk E(xk,0 ))

Advanced Methods Used in Molecular Dynamics ...

87

In Eq. 46, π(xk,t → xk,t+1 ) is the propagation probability at each time step, which depend on the details of deterministic or stochastic dynamics. In general, the Markovian transition probability π(xk,t → xk,t+1 ) can have any distribution that conserves the Boltzmann distribution. Here, π(xk,t → xk,t+1 ) represents the action characterized by augmented system given in Eq. (44), which produces a Boltzmann distribution in the extended phase space of variables. In the general case of the Newtonian dynamics, we can write: p(xt → xt+1 ) = δ(xt+1 − Φ∆t (xt )) where δ is the delta function and Φ∆t (xt ) is the discrete flow map of one time step ∆t propagation operator. In this case, a trajectory can be generated using an initial state sampled from some canonical distribution and then propagating in time using usual Hamiltonian dynamics. Note that for Hamiltonian dynamics is easy to find a time-reversible discrete flow map. On the other hand, when dynamics are governed by Eq. (44), the structure is not symplectic, but still, it is time reversible. It is important to note that WHAM can be used for analyzing the data from all replicas in the case of REM:SPMD simulation as presented in Refs. (H. Kamberaj, 2015, 2018) (see Section 3.10.).

3.7.

Replica Exchange Multicanonical Method

To overcome the problems of standard REM (e.g., the large number of replicas and high computational demands) and of MUCA (e.g., difficulties on determining the weighting factor), a new method has been proposed called replica exchange multicanonical (REMUCA) (Y. Sugita and Y. Okamoto, 2000). In REMUCA, the first step is to perform a short replica exchange method simulation (e.g., of L replicas) to calculate the multicanonical weighting factor. Then, a standard multicanonical simulation run is performed using this weighting factor. The multiple-histogram re-weighting technique (Ferrenberg and Swendsen, 1989; Kumar et al., 1992) can be used to calculate the energy density of states as described in the next section (see 3.10.). After we obtain the energy density of states, the multicanonical weighting factor is obtained from Eq. 23 and Eq. 24. Note that the multicanonical energy Emu (E, β0 ) obtained in this way is determined in the range between hEiT1 and hEiTL , where T1 and TL denote the lowest and highest temperature, respectively,

88

Hiqmet Kamberaj

used in the replica exchange simulation. While outside this interval the potential energy of multicanonical simulation is determined through extrapolation:    ∂Emu (E, β0 )   (E − E1 )   ∂E  E 1     +Emu (E1 , β0 ), E < E1 (0) E (47) Emu (E) = mu (E, β0 ), E 1 ≤ E ≤ EL   ∂E (E, β )  mu 0  (E − EL )    ∂E  EL  +Emu (EL , β0 ), E > EL

In multicanonical MD simulations, Eq. 25 (or Eq. 27) is used to govern New(0) ton’s dynamics where Emu (E, β0 ) is replaced by Emu (E). Then, after the simulation run, the trajectories are analyzed using WHAM for a single run, as described in the next section. Eq. 47 can also be written as  T0 T0  (E − E1 ) + T0 S(E1 ) = E + constant,    T T 1 1    E < E1 = hEiT1  (0) Emu (E) = T0 S(E), E1 ≤ E ≤ EL (48)   T0 T0   (E − EL ) + T0 S(EL ) = E + constant,   TL   TL E > EL = hEiTL and the dynamical equations of motion are defined here as: q˙i,α =

pi,α mi

 β1   Fi,α , E < E1 = hEiT1   β   β0 (E) (α,s) α p˙i,α = −λi,α pi,α − P1 pi,α + Fi,α , E1 ≤ E ≤ EL  β0    β   L Fi,α , E > EL = hEiTL β0   (i,α) (i,α) L Q1 (ξ1, j )2 (i,α) M (i,α) (i,α)  η˙ =−∑ ξ2, j − ∑ ξk, j  k T B α j=1 k=2

(49)

Advanced Methods Used in Molecular Dynamics ...

89

(i,α) (i,α) (i,α) (i,α) (i,α) (i,α) ξ˙ 1, j = −ξ1, j ξ2, j − λb ξ1, j − λi,α ξ1, j

j = 1, · · · , L (i,α)

(i,α) ξ˙ k, j =

Gk, j

(i,α)

(i,α)

− ξk+1, j ξk, j

(i,α) Qk

j = 1, · · · , L; k = 2, · · · , M − 1 (i,α)

(i,α) ξ˙ M, j =

(α)

GM, j

(i,α)

QM, j

j = 1, · · · , L (α,s)

s˙i,k = Pi,k , (α)

(α,s) P˙i,k =

(α,s) P˙i,M =

Γi,k

(α) Wi,k (α) Γi,M (α) Wi,M

k = 1, 2, · · · , M (α,s) (α,s)

− Pi,k+1 Pi,k ,

k = 1, 2, · · · , M − 1

From Eq. 23, the Boltzmann’s factor depends on temperature T and energy E, and hence, scaling potential energy (and so the force) by a constant κ is similar to scaling the temperature by 1/κ (Hansmann et al., 1996; R. Yamamoto (0) and W. Kob, 2000). Therefore, Emu given by Eq. 47 (or Eq. 48) generates a canonical ensemble distribution at T = T1 , multicanonical ensemble distribution for E1 ≤ E ≤ EL , and canonical ensemble distribution simulation at T = TL for E > EL .

3.8.

Multicanonical Replica Exchange Method

The replica exchange method in multicanonical simulation, REMUCA, can also be introduced as a multicanonical replica exchange method (MUCAREM) (Y. Sugita and Y. Okamoto, 2000). In MUCAREM, the final MD simulation run is a replica exchange with fewer replicas, say L , in contrast to REMUCA, where the final run is a standard multicanonical MD simulation. Since the degree of energy probability distribution overlapping in a multicanonical simulation is higher compared to canonical one, a fewer number of replicas

90

Hiqmet Kamberaj

are needed for the final simulation run to guarantee an optimal distribution of replicas among thermostats. In MUCAREM, similar to REMUCA, short replica exchange MD simulations are performed with L replicas and L thermostats, covering a temperature range from T1 to TL . During this short simulation run, we can estimate the energy density of states Ω(E) for all range of energy using WHAM techniques. After we define the density of states Ω(E), we can chose L pairs of thermostats (m) (m) (m) (m) with temperatures (TL , TH ), for m = 1, 2, · · · , L , where TL < TH . In practice, the temperatures are arranged such that ensure sufficient overlapping be(1) (L ) tween neighboring pairs. Here, we have TL = T1 and TH = TL , and   E (m) = hEi (m) , L TL (50)  EH(m) = hEi (m) , m = 1, 2, · · · , L T H

Then, we chose L thermostats at temperatures T1 , T2 , · · · , TL and assign to each the multicanonical potential as    ∂Emu (E, Tm)   (E − E (m) )   (m)  ∂E  E L     +Emu (EL(m), Tm ), E < EL(m)  (m) (m) (m) Emu (E) = (51) Emu (E, Tm), E L ≤ E ≤ EH      ∂E (E, T ) mu m   (E − E (m) )   (m) ∂E  EH    (m) (m) +Emu (EH , Tm ), E > EH

where the multicanonical potential energy, Emu (E, T ), is obtained for the entire (m) interval of energy. Also, this choice of Emu (E) generates a canonical distribu(m) (m) (m) (m) tion at T = TL for E < EL , a multicanonical distribution for EL ≤ E ≤ EH , (m) (m) and a canonical distribution simulation run at T = TH for E > EH . In final step of MUCAREM, the production run is defined as a replica exchange simulation with L different thermostats at temperatures T1 , T2 , · · · , TL (1) (2) (L ) and multicanonical potential energies, Emu (E), Emu (E), · · · , Emu (E). The transition probability of swapping two replicas of neighboring temperatures is given by  1, ∆≤0 (i) ( j) w(xm | xm+1 ) = (52) exp(−∆), ∆ > 0

Advanced Methods Used in Molecular Dynamics ...

91

where h i (m+1) (m+1) ∆ = βm+1 Emu (E(q(i))) − Emu (E(q( j))) h i (m) (m) − βm Emu (E(q(i))) − Emu (E(q( j)))

(53)

Here, the multicanonical potential energies, E (m) (E(q( j))) and E (m+1)(E(q(i))), have to be calculated since E (m) (E) has different values for m (Y. Sugita et al., 2000). Using the multiple-histogram reweighting method, the canonical distribution can be obtained (Ferrenberg and Swendsen, 1989; Kumar et al., 1992), as presented in Section 3.10..

3.9.

Tsallis Replica Exchange Methods

Tsallis’s weight factor for a configuration q at inverse temperature β` has the following general form (J. Kim and J. E. Straub, 2009): q h i ` (0) 1 − q ` W`(q) = 1 − β` (1 − q` )(U(q) −U` )

(54)

(0)

where U` is a reference minimum value of the potential energy U(q) at Tsallis entropy parameter q` of replica ` associated with thermostat held at inverse temperature β` . Note that in the limit when q` → 1, the Boltzmann’s weight of standard MD simulation can be obtained:   WB(q) ∼ exp −β` (U −U (0) )

There may exist two approaches of applying Tsallis-like replica exchange method. In the first implementation (named q-REM) L replicas are initially setup running at different values of q` for ` = 1, 2, · · · , L and equal temperature T as proposed (S. Jang et al., 2003). While in a second implementation, the Tsallis-like dynamics is incorporated with REM in the form of the generalized ensemble (TSREM) (T. W. Whitfield et al., 2002). In TSREM implementation, L replicas are associated with L thermostats at different inverse temperatures β` and Tsallis entropy parameter q` (` = 1, 2, · · · , L). Usually, in both implementations the reference replica samples the phase space using MD simulation with

92

Hiqmet Kamberaj

original potential energy function, that is, it has q` = 1 and β` = β0 , where β0 is the required inverse temperature. On the other hand, the other replicas sample using biased potential energy function associated with effective potential energy that is given by Eq. 37 (i.e., q` > 1 for all replicas ` > 1). In Tsallis-like REM a swapping between two neighboring replicas 1 and 2 has an acceptance probability given by Pacc(1 ↔ 2) = min [1, exp (−∆12 )]

(55)

where ∆12 = β2 (Ueff,2 (q1 ; q2 , T2 ) −Ueff,2 (q2 ; q2 , T2 ))

(56)

+ β1 (Ueff,1 (q2 ; q1 , T1 ) −Ueff,1 (q1 ; q1 , T1 ))

where Ueff,1 and Ueff,2 are the effective Tsallis potential energy of replica 1 and 2, respectively. Close to the barrier regions, the magnitude of the force, because of lowering the barriers, is reduced for q larger than one (Andricioaei and Straub, 1997; Plastino and Anteneodo, 1997). Therefore, in these regions resistance on the particles decreases and the barrier crossing rates increases. However, the largest value of q has to be carefully determined after some preliminary test runs depending on the system. It is empirically suggested (H. Kamberaj and A. van der Vaart, 2007; U. H. E. Hansmann and Y. Okamoto, 1999) an upper value as q = 1 + 1/ f , where f is the number of degrees of freedom. A more general approach for optimization of q values has been suggested in Ref. (van Giessen and Straub, 2005). Based on this approach, expression of ∆12 in Eq. 56 is written as 1 ∆12 = kB

Zq1 

q2

 1 1 − dz T2(z) T1(z)

where T is the effective Tsallis temperature given as   ∂(βUeff) −1 kB T (q; q, T ) = ∂U U

(57)

Advanced Methods Used in Molecular Dynamics ...

93

The performance of either q-REM or TSREM will directly depend on the rate of accepted replica attempted swaps, defined by the average Pacc of acceptance probability of each swapping attempt: Pacc(1 ↔ 2) =

Z

dq1 dq2 P1 (q1 )P2 (q2 ) min[exp (−∆12 )]

where Pi (qi ) is the probability that replica i has a configuration qi or energy U(qi ), which can be written Pi (qi ) ≡ Pi (U) = Ω(U)WT,i(U)

(58)

where Ω(U) represents the energy density of states and WT,i (U) Tsallis weighting factor of replica i. Eq. 58 can further be simplified as (van Giessen and Straub, 2005): Pacc(1 ↔ 2) =

Z

dUdU 0 θ(∆12 )P1 (U)P2 (U 0 )

+

Z

dUdU 0 θ(−∆12)P1 (U 0)P2 (U)

(59)

where θ(∆) denotes the Heaviside step function:  1, x > 0 θ(x) = 0, otherwise

and ∆i (U,U 0) = −∆i (U 0 ,U), from which we can obtain (van Giessen and Straub, 2005): Pacc(1 ↔ 2) = 2

Z

dUdU 0 P1 (U)P2 (U 0)θ(∆12)

(60)

The optimal Pacc(1 ↔ 2) is obtained by maximizing the overlapping integrals between two neighboring Tsallis probability distribution functions. For that, we can approximate the function of Pi (U) by local expansion around the stationary point U0 as (van Giessen and Straub, 2005): lnPi (U) = ln Pi (U0 ) −

1 (U −U0 )2 + · · · 2σq

(61)

where U0 is the energy where both Tsallis effective temperature and statistical temperature, TS (U) are equal: TS (U0 ) = T (U0 )

94

Hiqmet Kamberaj

where 

TS (U0 ) =

∂S ∂U

−1 U0

where S(U) is the microcanonical entropy: S(U) = kB lnΩ(U) In Eq. (61) σq denotes the width of Tsallis probability density function at Gaussian approximation given by (van Giessen and Straub, 2005):  0  TS (U) Ti 0 (U) −1 σq (U0) = − (62) TS2 (U) Ti 2 (U) U0 where TS0 (U) =

∂TS ∂Ti , T 0 (U) = . ∂U i ∂U

If TS (U) is assumed to be linear function of U around U0 , then TS0 (U) is a constant. Furthermore, the equivalence between the microcanonical and canonical ensembles indicates that this constant is 1/CV (T0 ), thus TS0 (U0 ) = 1/CV (T0 ) Then, we obtain (van Giessen and Straub, 2005) σq (U0) = where

σ0 1−κ

(63)

κ = (qi − 1)CV (T0 ), σ0 = T02CV (T0 ) where T0 = TS (U0 ) and σ0 is the Gaussian width of the canonical probability density function at temperature T0 . It can be seen from Eq. 63 that in the limit of qi → 1, which is the limit of Boltzmann distribution, σq → σ0 , corresponding to a Gaussian distribution. Moreover, if 1 < qi < 1 + qc , where qc = TS0 (U0 ) =

1 CV (T0 )

Advanced Methods Used in Molecular Dynamics ...

95

then σq > σ0 (Eq. 63), and hence Tsallis probability density function, Pi (U), has a broader distribution than canonical function at T0 . Whereas, for qi < 1, Pi (U) becomes narrower compare to canonical probability density function. However, in both cases, Tsallis distribution has its maximum at stable point U0 as canonical distribution at T0 . For qi = 1 + qc , the Tsallis effective temperature, T (U), it is tangential to canonical temperature TS (U) function at U0, and Pi (U) is locally flat around U0 , indicating only marginal stability. Thus, the choice qi = 1 + qc generates the most delocalized Tsallis distribution for standard Tsallis MD simulation run. For qi > qc, the local minimum of Pi (U), namely U0 , is an unstable crossing point (van Giessen and Straub, 2005). The WHAM is used to estimate the averages of unbiased system quantities at required temperature T0 (H. Kamberaj and A. van der Vaart, 2007). The configuration probability density for each replica k (k = 1, 2, · · · , K) at inverse temperature β` (` = 1, 2, · · · , L) is written as P(U; qk, β` ) =

1 q Zk`k

(64)

h i qk (0) 1−qk × Ω(U) 1 + β` (qk − 1)(U −U )

where U is unbiased potential energy, Ω(U) is density of states, Zk` is configuration partition function: Z h iql /(1−qk) Zk` = dUΩ(U) 1 + β` (qk − 1)(U −U (0) ) The canonical distribution at the required inverse temperature β0 is given by   P(U; β0) = f k`P(U; qk, β` ) exp β`Ukbias − β0U (65)

where Ukbias is the bias potential energy function of replica k and q

fk` =

Zk`k Z0

or    −1 P(U; qk, β`) = f k` P(U; β0) exp − β`Ukbias − β0U

Further details of the method are presented in Section 3.10..

(66)

96

Hiqmet Kamberaj

3.10. The Weighted Histogram Analysis Method The WHAM is often used to analyze the data from replica exchange molecular dynamics simulation. This is considered an efficient technique of data processing since it combines all the data from replicas. In WHAM, it assumed that K copies of the same system (namely the replicas) are in equilibrium with L thermostats at inverse temperature β` (` = 1, 2, · · · , L). In addition, to each replica unbiased potential energy, U`(qk ) (` = 1, 2, · · · , L; k = 1, 2, · · · , K) a biasing potential energy term is added ∆U`(qk ). Then, a histogram of M bins is created for the unbiased potential energy combining all of the replicas, with Um (m = 1, 2, · · · , M) being the energy at the center of the bin. Thus, for each replica k and histogram unbiased potential energy bin m we count the number of independent snapshots, namely Hkm. The probability of observing the system at energy bin m and thermostat ` is defined as (Gallicchio et al., 2005): P`m = Z`−1C`m Ωm e−β0Um

(67)

where Ωm = Ω(Um) is the density of states at the energy bin m and the constant C`m determines both the effect of temperature and biasing potential in probability distribution as: C`m = exp (− (β` − β0 )Um) × exp (−β` ∆U`)

(68)

In Eq. 67, Ωm e−β0Um gives the unbiased probability of the bin m at the target temperature and Z` is the partition function at β` . Note that ∑M m=1 P`m = 1 must be satisfied. Combining Eq. 67 and Eq. 68 we obtain: bias −F ) `

P`m = Ωm e−β` (Um

(69)

where Umbias gives the value of biased potential energy at the center of bin m and F` is the Helmholtz free energy, which has to be estimated, given as F` = −(1/β` ) lnZ` Let nk` be the number of saved snapshots from replica k visiting thermostat `, then the accumulated probability density for energy bin m can be determined as: K

Pm = Ωm ∑

L

nk`

bias m −F` )

∑ Nk e−β (U

k=1 `=1

`

(70)

Advanced Methods Used in Molecular Dynamics ...

97

where Nk is the total number of saved snapshots from the replica k. Pm can also be approximated as (Chodera et al., 2007): K

Pm ≈

Hkm k=1 Nk

∑

(71)

Using the last two equations, we obtain K

∑ Hkm k=1

Ωm =

K

L

(72) bias −F ) `

∑ ∑ nk`e−β` (Um

k=1 `=1

F` = −

M bias 1 ln ∑ Ωm e−β`Um β` m=1

To take into account any possible correlations between configurations saved from simulations, the histogram bin statistical inefficiency for each energy bin m from replica k, gkm, can be introduced (Chodera et al., 2007), which determines the effective number of snapshots from replica k with unbiased potential energy eff , and the effective number of snapshots from replica k in falling in bin m, Hkm equilibrium with thermostat `, neff k` : eff Hkm =

Hkm ; gkm

neff k` =

nk` gkm

ˆ m is given as: Then, the estimated value of the density of states Ω K

ˆm= Ω

eff ∑ Hkm

k=1 K

L

(73) bias −F ) `

−β` (Um ∑ ∑ neff k` e

k=1 `=1

F` = −

M 1 ˆ m e−β`Umbias ln ∑ Ω β` m=1

ˆ m depends on F` , and F` also depends on Ω ˆ m . Therefore, From Eq. 73, Ω ˆ m are usually determined iteratively from Eqs. 73, starting from some F` and Ω

98

Hiqmet Kamberaj

arbitrary choice F` = 0 (` = 1, 2, · · · , L) and continuing until a convergence is ˆ m is given by (Chodera et al., 2007): reached. The statistical error σ2Ωˆ of Ω m

σ2Ωˆ = m

ˆm Ω K

L

(74) bias −F ) `

−β` (Um ∑ ∑ neff k` e

k=1 `=1

The estimated average value of any physical quantity A of the system at the target inverse temperature β0 is computed by summing the weighted values from all configurations: K

Nk

∑ ∑ Wkn(β0 )Akn ˆ 0) = A(β

k=1 n=1 K Nk

(75)

∑ ∑ Wkn(β0 )

k=1 n=1

In Eq. (75) Wkn(β0 ) are the weights given by ˆ m −β U Ω e 0 m H km m=1 M

Wkn(β0 ) =

∑

The chain rule of error propagation is used to obtain the statistical error of ˆ 0 ) (Chodera et al., 2007): A(β 2  2   σX σY2 σ2XY hXi 2 σAˆ = + −2 (76) hY i (hXi)2 (hY i)2 hXihY i where hXi =

1 Nk ∑ Wn(β0)An Nk n=1

(77)

hY i =

1 Nk ∑ Wn(β0) Nk n=1

(78)

Nk gX ∑ (Wn(β0)An − hXi)2 Nk (Nk − 1) n=1

(79)

σ2X = σY2 =

Nk gY ∑ (Wn(β0) − hY i)2 Nk (Nk − 1) n=1

(80)

Advanced Methods Used in Molecular Dynamics ... σ2XY =

Nk gXY (Wn (β0 )An − hXi) ∑ Nk (Nk − 1) n=1

99 (81)

× (Wn (β0 ) − hY i)

Here, gX(Y,XY ) are the statistical inefficiencies determined from (auto)correlation functions of replica exchange simulations. If ∆U` = 0 (` = 1, 2, · · · , L), the standard WHAM of replica exchange simulations is obtained, discussed already in the literature. (Chodera et al., 2007)

4.

Metadynamics Method

Metadynamics method has been developed by Alessandra Laio & Michele Parrinello (Laio and Parrinello, 2002b). The method consists on finding a limited number of essential collective coordinates, ci (i = 1, 2, · · · , m), upon which the free energy depends on F(c). At any moment of time t, the free energy surface is explored based on the dynamical equations of motion determined by the forces acting on the system: fit = −

∂F ∂cti

In metadynamics method, a bias potential function Ubias(c) is constructed that is added to the Hamiltonian function of system. This bias potential is written as a sum of Gaussian distributions, which are added at any time t of the trajectory in subspace expanded by collective coordinate as generated during the MD simulation. The main effect of the bias potential is to not allow the system visiting configurations that are already explored. The mathematical form of the bias potential is (Barducci et al., 2011; Laio and Parrinello, 2002b) ! Z t m 0 2 (c (t) − c (t )) i i Ubias(c,t) = dt 0ω exp − ∑ 2σ2i 0 i=1 Here, ω gives the rate of energy change on time: ω=

W τ

100

Hiqmet Kamberaj

where W and τ are the Gaussian height and the time interval of deposition, respectively (Barducci et al., 2011). Here, σi is the Gaussian distribution width of collective coordinate i. W , τ and σ are adjustable parameters to optimize the algorithm (Laio and Parrinello, 2002b). The main benefit of using metadynamics method is being able to escape from the local minimum free energy metastable states, and hence it increases the rate of sampling rare events. Besides, the metadynamics method allows sampling of new reaction pathways after the system escapes local minimum states (Barducci et al., 2011). In the metadynamics method, there is no need for a priory knowledge of the exact topology of the free energy landscape. After a certain long time, metadynamics technique will eventually give a bias potential Ubias (Laio et al., 2005): Ubias(c,t → ∞) = −F(c) +C where C is an integration constant, F(c) is the underlying free energy of system, defined as Z  1 drδ(c − c(r))e−βU(r) F(c) = − ln β where r is the vector of coordinates, β = 1/kB T and U(r) is the potential energy function. This formula has been tested for simplified models (Laio et al., 2005) and for other complex systems (Gervasio et al., 2005; Laio and Parrinello, 2002b). A formal proof of this expression is shown in Ref. (Bussi et al., 2006). The free energy surface can be obtained up to an uncertainty, which is inversely proportional to the inverse temperature β and intrinsic diffusion coefficient D of the system in collective coordinates subspace (Gervasio et al., 2005; Laio et al., 2005):  1/2 ω ε∝ Dβ In practical applications of metadynamics method, ε is estimated by comparing different independent simulation runs (Angioletti-Uberti et al., 2010; Barducci et al., 2006; Provasi and Filizola, 2010) or using block averaging (Pfaendtner et al., 2009).

Advanced Methods Used in Molecular Dynamics ...

101

The main advantage of using metadynamics method include parallelization, which is an intrinsic property of metadynamics. For instance, one can run multiple interacting copies of metadynamics simulations for reconstruction of a free energy surface, where every simulation contributes to the time-dependent potential (Raiteri et al., 2006). The implementation of method yields an algorithm that scales very well linearly with the number of processors, independent on the type of processor. However, there are two disadvantages of the metadynamics simulations. The first, in single metadynamics simulation the convergence of bias potential Ubias is not reached to a constant value, but it oscillates about a constant value, making the criteria for stopping the simulation too tricky in practice (Barducci et al., 2011). The second disadvantage is related to the identification of collective coordinates for describing complex topological free energy is very difficult. The well-tempered metadynamics method (Barducci et al., 2008) provides a solution to the first problem of standard metadynamics. In this method, the rate of bias accumulation decreases over the course of the simulation, which is made possible using this expression for the bias potential:   ωN(c,t) 0 Ubias(c,t) = kB ∆T ln 1 + kB ∆T where N(c,t) is a histogram accumulated during the simulation for collective variables c and ∆T a free parameter with dimensions of temperature. The deriva0 tive Ubias (c,t) with respect to time t is   ωδc,c(t) U(c,t) 0 ˙ Ubias(c,t) = = ω exp − δc,c(t) ωN(c,t) kB ∆T 1+ kB ∆T The new approach can easily be related to standard metadynamics by taking δc,c(t) to be a Gaussian function. This is practically implemented by defining the height of a Gaussian W as   Ubias(c,t) W = ωτ exp − kB ∆T There are two main characteristics of the well-tempered metadynamics compare to standard metadynamics (Barducci et al., 2011). The first, rate of bias

102

Hiqmet Kamberaj

accumulation decreases with simulation time as 1/t and the deviations from the equilibrium dynamics are small. Secondly, a convergence of the bias potential is reached up to a constant value, C, though a complete compensation of the free energy surface is not obtained: ∆T F(c) +C Ubias(c,t → ∞) = − T + ∆T In long run simulation, the collective coordinates probability distribution becomes:   F(c) P(c) ∝ exp − kB (T + ∆T ) where for ∆T → 0, standard MD simulations are recovered, and for ∆T → ∞ we have standard metadynamics. In contrast to standard metadynamics, in welltempered metadynamics, the bias potential converges to a finite value during one run. For all other values of ∆T the extent of free energy surface exploration is determined by adjusting ∆T (Barducci et al., 2011).

5.

Umbrella Sampling Methods

Umbrella sampling method is developed in Refs. (G.M. Torrie and J. P. Valleau, 1977; Torrie and Valleau, 1974). This technique adds a bias term to the potential energy function applied to the system, which ensures efficient sampling along the path of a reaction coordinate. The bias term can be added in a single simulation run or in multiple copies of simulations (often called windows) with overlapping distributions. The umbrella sampling method aims to connect regions of phase space that are separated by high energy barriers, which is a reason for naming it the umbrella sampling. The bias potential as a function of the reaction coordinate, let say q, has the following form: Ubias(q) = −F(q) where F(q) is not usually known a priory. The umbrella sampling aims to calculate the F(q) by employing two main types of bias potentials, such as harmonic biases consisting of a set of windows centered at different points along q and adaptive bias modeled to match −F(q) in only one window enveloping the whole range of q.

Advanced Methods Used in Molecular Dynamics ...

5.1.

103

Harmonic Bias Potential

In this approach, the entire range of values of q is divided into a subset of small size windows, Nw . Then, a bias potential function is employed in every window allowing the system to fluctuate around a reference point qref i centered at the window i of the form (Kästner, 2011):
2 ki q − qref (82) Ui,bias(q) = i 2 Free energy curves calculated in this way are combined using WHAM technique (Ferrenberg and Swendsen, 1989; Kumar et al., 1992). From Eq. (82), the bias potential Ubias is characterized by ki , which can also be adjusted depending on the window, number of windows, Nw , and qref i . In general, qref are chosen uniformly distributed in all range of q. In praci tice (Frenkel and Smit, 2002), there is a compromise between the statistical errors and computational time required. For instance, increasing the number of windows results in a smaller statistical error, but longer computational time is needed. However, the advantage is that MD simulations in each window are completely independent, and hence they can run in parallel by producing multiple MD simulation copies, which takes into account the advantage of the parallelism of computer architecture available. This approach has already been used in atomistic simulations of protein folding (Nymeyer et al., 2004). Combination of the umbrella sampling with replica exchange have also been suggested (Auer and Frenkel, 2004) to improve the conformational sampling. Strength, ki , of the bias potential, has to be chosen before simulation runs, such that the bias potential allows steering the system across potential energy barriers. On the other hand, too large ki will cause very narrow probability distributions. Often, if the probability distributions have too large gaps between the windows, then additional windows could also be inserted. Overlapping between distributions at each window has to be sufficiently large for WHAM and it could also be advantageous in umbrella integration (Kästner and Thiel, 2006). Large values of ki can also lead to instability of numerical integration of the equations of motion unless small time steps are used. In addition, for too large values of ki only the configurations with high energies will be sampled (Straatsma and McCammon, 1991). The statistical error can also be derived analytically as a function of ki (Kästner and Thiel, 2006). Location of the next sampled window (qref i+1 ) can be chosen

104

Hiqmet Kamberaj

as (Woolf and Grossfield, 2002) ref qref i+1 = qi + wi

where wi is the window width. Alternatively, the experimental data can also be used to determine optimized values of bias parameters (Mills and Andricioaei, 2008).

5.2.

Adaptive Bias Umbrella Sampling

On the other hand, this method aims to screen the entire interval of interest for the reaction coordinate q in a single simulation run (Bartels and Karplus, 1997, 1998; Hooft et al., 1992; Laio and Parrinello, 2002a; Mezei, 1987), by choosing a bias potential of the following form: Ubias(q) = −F(q) Adding this bias term to the potential yields an exactly flat energy surface, and hence the resulting probability distribution is uniform along q. Usually, the simulations start with an initial guess for Ubias(q), because F(q) is not known a priory. Then, iteratively, Ubias(q) is improved until a complete uniform distribution is obtained in q space.

6.

Transition Path Sampling Methods

In typical chemical reactions in solutions, the difficulty of computer simulations is in understanding the rare events occurring in complex systems when moving from one basin of attraction to another on a multimodal potential energy landscape. In particular, determining the transition state of these processes as a function of order parameters, which have also to be defined, is notoriously difficult problem (D. J. Wales, 2015), which will allow sampling using the standard MD simulations starting from this initial state (Anderson, 1973; Chandler, 1978; Hänggi et al., 1990; Keck, 1962). The disadvantage of this approach is that the transition state is not always known a priory. Furthermore, because of the high dimensionality of the problem phase space, the energy landscape has a complex topology with many transition states, and hence the reaction coordinate may not be represented accurately by the order parameter (Dellago et al., 1998).

Advanced Methods Used in Molecular Dynamics ...

105

On the other hand, transition path sampling (Bolhuis et al., 2000; Dellago et al., 1998), as an alternative method, does not require previous knowledge of transition states, but it relies on the calculation of isomorphic reversible work from reactive flux correlation functions. In this approach, L + 1 copies of the trajectories in phase space characterize a path in space-time: XL = {x0 → x1 → · · · → xL } Here, xt (t = 0, 1, · · · , L) represent points in a 2D-dimensional phase space. A relationship between the sequence time t and physical simulation run time exists depending on the transition path (Dellago et al., 1998), which is represented by 2D(L + 1) coordinates. If we assume visited states form a Markovian chain, then the probability of simulating the trajectories is given by L−1

P (XL) exp (−βE(x0)) ∏ p(xt → xt+1 )

(83)

t=0

where β is the inverse simulation temperature and E(xt ) is the total energy at configuration xt . Here, the initial configuration (at t = 0) is generated from a canonical distribution (Dellago et al., 1998): ρinit (x0 ) = exp (−βE(x0 )) In Eq. (83), p(xt → xt+1 ) gives the transition probability for each time step, which is determined based on the natural dynamics. Usually, any Markovian transition probability p(xt → xt+1 ) is such that it should obey to the Boltzmann distribution and is normalized (Dellago et al., 1998). Typically, two approaches are proposed as a choice for p(xt → xt+1 ): the Markovian action and Langevin action (Dellago et al., 1998). If the natural dynamics are governed by the Newton’s equations of motion for the Hamiltonian systems, then p(xt → xt+1 ) = δ(xt+1 − Φ∆t (xt )) where δ is the delta function and Φ∆t (xt ) is the discrete flow map of one-time step propagation.

106

Hiqmet Kamberaj

A transition path sampling can be performed by applying the constraints hA (x0 ) and hB (xL ) at the endpoints in path probability as the following: −1 PAB (XL ) ≡ ZAB (L) hA (x0 )P (XL)hB (xL )

ZAB (L) =

Z

(84)

d L x hA (x0 )P (X; L)hB(xL )

where hA (x0 ) and hB (xL ) are indicator functions defined as  1 if x ∈ A, B hA,B (x) = 0 if x ∈ / A, B Here, hA (x0 ) constraints the trajectory path to start in the region A (i.e., reactant) and hB (xL ) constraints the trajectory path to stop in the region B (i.e., product). Typically, L steps are used to take the system from states A to the state B, with action defined by Eq. (84). This approach has been used to probe the dynamics of folding pathways for the C-terminal β-hairpin of protein G-B1 using MD simulation at room temperature of protein in explicit solvent (Bolhuis, 2003). It can be suggested for time propagation of the system to be governed by the swarm particle-like dynamics given in the following form (i = 1, 2, · · · , f ): x˙i =

pi mi

(85) (s)

p˙i = Fi − λi pi − P1 pi

+ γ1 u1 (xLbest − xi ) + γ2 u2 (xGbest − xi ) i i   (i) (i) 2 L M Q (ξ ) 1 1, j (i) (i) η˙ (i) = − ∑  ξ2, j − ∑ ξk, j  k T B j=1 k=2 (i) (i) (i) (i) (i) (i) ξ˙ 1, j = −ξ1, j ξ2, j − λb ξ1, j − λi ξ1, j

j = 1, · · · , L (i)

(i) ξ˙ k, j =

Gk, j (i) Qk

(i)

(i)

− ξk+1, j ξk, j

j = 1, · · · , L; k = 2, · · · , M − 1

Advanced Methods Used in Molecular Dynamics ...

107

(i)

(i) ξ˙ M, j =

GM, j (i)

QM, j

j = 1, · · · , L (s)

s˙i,k = Pi,k , k = 1, 2, · · · , M Γi,k (s) (s) (s) P˙i,k = − Pi,k+1 Pi,k , k = 1, 2, · · · , M − 1 Wi,k Γi,M (s) , P˙i,M = Wi,M where xGbest represents the coordinates of final state B (product), i.e., xL . Then, we can generate different trajectories starting in the region A (reactant) and biased towards the end in final region B. As advantage this method does not postulate a priory knowledge of the transition state.

7.

Accelerated Molecular Dynamics Method

Accelerated molecular dynamics approach is proposed as a robust method to bias the potential energy function to efficiently enhance the barrier crossing during the simulations (Hamelberg et al., 2004a), based on previously introduced methods (Grubmüller, 1995; Rahman and Tully, 2002; Voter, 1997). The method has been described in details elsewhere (Hamelberg et al., 2004a, 2007). In this approach, a reference boost potential energy U0 term is defined with a value slightly lower in magnitude than the local potential energy minimum (Hamelberg et al., 2004a), then each step of simulations the potential energy U(r) is modified by a continuous non-negative bias potential ∆U(r) as (Hamelberg et al., 2004a, 2007) Ubias(r) = U(r) + ∆U(r) where the bias term is given as  2  (U0 −U(r)) ∆U(r) =  (U0 −U(r)) + α 0

(86)

If U(r) < U0 Otherwise

108

Hiqmet Kamberaj

where α is used to adjust the depth of potential energy minimum and modulate local smoothness of the energy basins of Ubias. In this approach, the bias term ∆U(r) raises the potential surfaces near the minimum states and leaves unaffected surface points near the barriers. Another form of the bias term ∆U(r) has also been proposed (de Oliveira et al., 2008), such as Ubias(r) = U(r) − ∆U(r) where the bias term is given as  2  (U(r) −U0 ) ∆U(r) =  (U(r) −U0 ) + α 0

(87)

If U(r) ≥ U0 Otherwise

In this new MD simulation approach, transitions are accelerated by lowering the barriers instead. With increasing α the modified landscape becomes rougher, and it moves closer to the original potential. It is interesting to note that taking into account the relationship between average potential energy surface roughness and the diffusivity, the method allows acquiring approximately the kinetics of original potential energy landscape (Doshi and Hamelberg, 2011; Hamelberg et al., 2005; Xin et al., 2010). The method has successfully been used to study the sampling of slow diffusive conformation transitions of torsion angles for biomolecules in timescales longer than milliseconds (Hamelberg et al., 2007). The approach is also tested in the ab initio molecular dynamics simulations (Bucher et al., 2011). The method is efficiently used to increase accuracy and convergence of free energy computations in condensed-phase systems when combined with thermodynamic integration simulations (de Oliveira et al., 2008). The approach is also used as replica exchange by varying the degree of acceleration among the replicas for gas-phase model systems (Fajer et al., 2008). The approach is used to study the waters contribution to the energetic roughness from peptide dynamics (Johnson et al., 2010). Besides, the method has also shown to retrieve the kinetic rate constant when applied in simulations of the helix to beta strand transition of alanine dipeptide in explicit solvent (de Oliveira et al., 2007). Recently (Pierce et al., 2012), the approach has been used to access conformation changes in

Advanced Methods Used in Molecular Dynamics ...

109

time scales of milliseconds for bovine pancreatic trypsin inhibitor protein emphasizing one of the method’s advantage for not needing prior knowledge of free energy landscape or reaction coordinate.

8.

Conformational Flooding Method

Helmut Grubmüller (Grubmüller, 1995) introduced a new approach examining conformation transitions in complex macromolecular systems at the atomistic level. In this method, first, the so-called conformation space (Ansari et al., 1987) is defined for the system characterized by the Hamiltonian H as a restricted region of the configuration phase space, in which system spends a long time. Typically, this determines the time needed for the system to sample enough phase space for the correct determination of the statistical averages (Grubmüller, 1995). In this confined space, the free energy landscape is determined in order to k find an effective Hamiltonian Heff , where k indicates one of the subspaces in configuration space. Here, it is assumed that this configuration space is made up by regions of low free energy F, which are separated from each other by high energy barriers of order ∆F. The free energy landscape of the subspace is expressed in terms of the so-called collective coordinates (A. Amadei et al., 1993; Frauenfelder et al., 1989; Go and Scheraga, 1969; Grubmüller, 1994), characterized by the vector c = (c1 , c2 , · · · , cm )T where m is the number of essential degrees of freedom of the configuration subspace. This describes a coarse-graining of the configuration space, leaving out 3N − m degrees of freedom, with N being the total number of atoms. According to (Grubmüller, 1995), in the subspace describe by the collective coordinates, ˜ the conformation space density ρ(c) is defined as ˜ ρ(c) =

Z

dx0 ρ(x0)δ c − c(x0 )




(88)

where x is a 3N-dimensional Cartesian vector of the N particle positions, and ρ(x) is the configuration space density. Hence, the free energy landscape can be

110

Hiqmet Kamberaj

evaluated as (Grubmüller, 1995): 1 ˜ F(c = − ln ρ(c) β ˜ Calculation of the ρ(c) requires knowledge of ρ(x), which is an integration on a 3N configuration space difficult to be evaluated in practice, since the system has to be ergodic (Grubmüller, 1995). However, in time scales covered by MD simulations (typically of order a few hundred nanoseconds), systems (e.g. biomolecular systems) are non-ergodic at all time scales (see for example (Clarage et al., 1995) or (Grubmüller, 1995) and the references therein). In (Grubmüller, 1995), the configuration density of the subspace k, ρk (x) is approximated as   1 T −1 −1 ρk (x) = Z exp − (x − x¯ ) C (x − x¯ ) (89) 2 In Eq. (89), the partition function, Z, is given by   Z 1 T −1 Z = dx exp − (x − x¯ ) C (x − x¯ ) 2 Here C is the covariance matrix, which is a ℜ3N×3N symmetric and positive matrix, defined as (Grubmüller, 1995) (and the references therein): ¯ C = h(x − x)(x − x¯ )T ik where x¯ = hxik Here, h· · ·ik denotes ensemble average in the configuration subspace k. The matrix C is calculated based on MD trajectories, which then is diagonalized: C = ET Λ−1 E

(90)

where E and Λ are the matrix of eigenvectors and the diagonal matrix of eigenvalues, respectively. Projection of the Cartesian coordinates fluctuations along the space spanned by eigenvectors E is ¯ q = ET (x − x)

Advanced Methods Used in Molecular Dynamics ... Equation 89 can be simplified as   1 T −1 ρk (x) = Z exp − q Λq 2

111

(91)

Coarse-graining of the configuration subspace k allows the definition of m essential collective coordinates (A. Amadei et al., 1993) c = (q1 , q2 , · · · , qm)T with the largest eigenvalues, which characterize the low-frequency fluctuation modes, and the remaining 3N − m eigenvalues, which characterize the highfrequency fluctuation modes, are assumed not to influence the conformation transitions (Grubmüller, 1995). The justification of this choice is based on the fact that the first m eigenmodes are anharmonic and with high amplitude, whereas the other 3N − m eigenmodes are essentially harmonic and with small amplitudes, and hence only m coordinates will dominate the collective motions in biomolecular systems (see for example discussion in Ref. (Grubmüller, 1995) and the references therein.) Thus, the conformation subspace density is defined as   1 T −1 ˜ ρ˜ k (c) = Zc exp − c Λc c 2 where Λc is a reduced matrix of m diagonal elements and Z˜c is the corresponding subspace partition function. Then, the effective Hamiltonian becomes, k Heff (c) =

1 T c Λc c 2β

(92)

k is fundamenThis coarse-grained model for the configuration subspace Heff tal in designing the so-called flooding potential Vfl (c) chosen as a multivariate Gaussian in order to fulfill the criteria discussed in Ref. (Grubmüller, 1995):   1 T Vfl(c) = Efl exp − c Λfl c (93) 2

where Efl characterizes the strength of Vfl (c) and Λfl determines the shape of Vfl (c) in conformation space, which is chosen to be Λfl = Λc /γ2

112

Hiqmet Kamberaj

where γ is a proportionality constant specified as a function of Vfl (Grubmüller, 1995): γ = (βEfl )1/2 The method is applied to probe conformation transitions in argon clusters and simplified protein model (Grubmüller, 1995). Other examples used to demonstrate the application of flooding to accelerate conformation transitions and chemical reactions are also examined (Lange et al., 2006).

9.

Discussion and Perspectives

Both industry and academic research are often using the molecular dynamics technique and its variants to a wide range of problems and systems, from inorganic and organic fluids to macromolecular. Yet, there are several issues identified in applying molecular dynamics simulations to these systems as we probed in this work, such as time and size scale, and rare events. The recent advances in parallel supercomputing have made possible to brace larger spatial scales, but increasing simulation timescales remains still a challenge. For instance, simulations spanning up to microseconds in the lifetime of a macromolecule need to cover billions of numerical integration time steps, which is a challenge from the computation point of view. This is because in simulations of biomolecules, at each time step, only a relatively small amount of computation can be run on parallel among a large number of processors. Hence, indeed, billions of simulation time steps can only be executed in a considerable amount of time. In nowadays, for molecular dynamics method development scientists, in particular, a significant challenge is to effectively employ the computers of near future to perform simulations of systems with millions of atoms (Hardy et al., 2011; Phillips et al., 2014; Stone et al., 2011, 2013, 2014). Another approach is exploitation of hardware parallel supercomputers for MD simulations with processors that can execute traditional MD codes orders of magnitude faster, such as Anton supercomputer (Scarpazza et al., 2013). In long MD simulation runs up to milliseconds timescale using fully atomistic physical models, force field accuracy will determine the overall accuracy achieved. Very recent studies (Piana et al., 2014) (and the references therein) have shown that prediction of native-structures and folding rated can be more

Advanced Methods Used in Molecular Dynamics ...

113

robust concerning error compare to the potential energy function. Moreover, the numerical integrator used in MD simulations should guarantee energy conservation and stable trajectories in long time scale simulations (Gray et al., 1994; Kamberaj, 2005). Large computer storage is also needed to save, analyze, and better computer graphics to handle a large amount of data produced. Coarse-grained models of macromolecular systems have probed problems of biologically relevant size and timescale during simulations when combined with computer power (Tozzini and McCammon, 2005). However, the coarsegrained models of proteins remain demanding, because of the challenges in building useful energy potential functions representing all the physics of interactions (Kamberaj, 2011; Lange and Grubmüller, 2006; Stepanova, 2007). The most tested coarse-grained model is the bead-spring model of polymers. Solvent effects are added using Brownian dynamics (Ermak and McCammon, 1978) or Stokesian dynamics (Phung et al., 1996). In this study, we attempted to give a big picture of the methods used to enhance sampling in molecular dynamics simulations and thus being able to simulate rare events for complex molecular systems. Also, we presented the advantages and limitations of each method. Our final message from this study is that probing relevant time and size scales of (bio)physical and chemical phenomena in macromolecular systems may need new statistical models of data processing and computational theoretical models to allow studying them efficiently (G. Ciccotti and E. Vanden-Eijnden, 2015; M. K. Transtrum et al., 2015).

Acknowledgments The author thanks International Balkan University for the support.

Conflict of Interest The author declares that there is no conflict of interest regarding the publication of this chapter.

114

Hiqmet Kamberaj

References A. Amadei, A. B. M. Linssen, and H. J. C. Berendsen. Essential dynamics of proteins. Proteins: Struct., Funct., Genet., 17:412–425, 1993. A.E. Garcia and J.N. Onuchic. Folding a protein in a computer: An atomic description of the folding/unfolding of protein A. Proc. Natl. Acad. Sci. USA, 100:13898–13903, 2003. A. Karolak and A. van der Vaart. Importance of local interactions for the stability of inhibitory helix 1 of Ets-1 in the apo state. Biophys. Chem., 165-166 (3):74–8, 2012. A. Kone and D.A. Kofke. Selection of temperature intervals for paralleltempering simulations. J. Chem. Phys., 122:206101, 2005. J. B. Anderson. Statistical theories of chemical reactions. Distributions in the transition region. J. Chem. Phys., 58:4684, 1973. I. Andricioaei and J. E. Straub. Generalized simulated annealing algorithms using Tsallis statistics: Application to conformational optimization of a tetrapeptide. Phys. Rev. E, 53:R3055–R3058, Apr 1996. doi: 10. 1103/PhysRevE.53.R3055. URL https://link.aps.org/doi/10.1103/ PhysRevE.53.R3055. I. Andricioaei and J. E. Straub. On Monte Carlo and molecular dynamics methods inspired by Tsallis statistics: Methodology, optimization, and application to atomic clusters. J. Chem. Phys., 107(21):9117–9124, 1997. doi: 10.1063/1.475203. URL https://doi.org/10.1063/1.475203. S. Angioletti-Uberti, M. Ceriotti, P. D. Lee, and M. W. Finnis. Solid-liquid interface free energy through metadynamics simulations. Phys. Rev. B, 81: 125416, 2010. A. Ansari, J. Berendzen, D. Braunstein, B. R. Cowen, H. Frauenfelder, M. K. Hong, I. E. T. Iben, J. B. Johnson, P. Ormos, T. B. Sauke, R. Scholl, A. Schulte, P. J. Steinbach, J. Vittitow, and R. D. Young. Rebinding and relaxation in the myoglobin pocket. Biophys. Chem., 26:337–355, 1987.

Advanced Methods Used in Molecular Dynamics ...

115

A. Okur, L. Wickstrom, M. Layten, R. Geney, K. Song, V. Hornak, and C.J. Simmerling. Improved efficiency of replica exchange simulations through use of hybrid explicit/implicit salvation model. J. Chem. Theory and Comput., 2:420–433, 2006. A. Piela, J. Kostrowicki, and H. A. Scheraga. On the multiple-minima problem in the conformational analysis of molecules: Deformation of the potential energy hypersurface by the diffusion equation method. J. Phys. Chem., 93: 3339–3346, 1989. S. Auer and D. Frenkel. Numerical prediction of absolute crystallisation rates in hard-sphere colloids. J. Chem. Phys., 120(6):3015–3029, 2004. A. Barducci, R. Chelli, P. Procacci, V. Schettino, F. L. Gervasio, and M. Parrinello. Metadynamics simulation of prion protein: Beta-structure stability and the early stages of misfolding. J. Am. Chem. Soc., 128:2705–2710, 2006. A. Barducci, G. Bussi, and M. Parrinello. Well-tempered metadynamics: A smoothly converging and tune-able free energy method. Phys. Rev. Lett., 100:020603, 2008. A. Barducci, M. Bonomi, and M. Parrinello. Metadynamics. Advanced Review, 1:826–843, 2011. C. Bartels and M. Karplus. Multidimensional adaptive umbrella sampling: Applications to main chain and side chain peptide conformations. J. Comput. Chem., 18:1450–1462, 1997. C. Bartels and M. Karplus. Probability distributions for complex systems: Adaptive umbrella sampling of the potential energy. J. Phys. Chem. B, 102: 865–880, 1998. E. Barth and T. Schlick. Overcoming stability limitations in biomolecular dynamics. I. Combining force splitting via extrapolation with Langevin dynamics in LN. J. Chem. Phys., 109(5):1617–1632, 1998a. E. Barth and T. Schlick. Extrapolation versus impulse in multiple-time stepping schemes. II. Linear analysis and applications to Newtonian and Langevin dynamics. J. Chem. Phys., 109(5):1633–1642, 1998b.

116

Hiqmet Kamberaj

B. A. Berg and T. Celik. New approach to spin-glass simulations. Phys. Rev. Lett., 69:2292–2295, Oct 1992. doi: 10.1103/PhysRevLett.69.2292. URL https://link.aps.org/doi/10.1103/PhysRevLett.69.2292. B. A. Berg and T. Neuhaus. Multicanonical algorithms for first order phase transitions. Phys. Lett. B, 267(2):249 – 253, 1991. ISSN 0370-2693. doi: https://doi.org/10.1016/0370-2693(91)91256-U. URL http://www. sciencedirect.com/science/article/pii/037026939191256U. B. A. Berg and T. Neuhaus. Multicanonical ensemble: A new approach to simulate first-order phase transitions. Phys. Rev. Lett., 68:9–12, Jan 1992. doi: 10.1103/PhysRevLett.68.9. URL https://link.aps.org/doi/10.1103/ PhysRevLett.68.9. B.A. Berg. Markov chain Monte Carlo simulations and their statistical analysis. World Scientific, Singapore, 2004. T. C. Bishop, R. D. Skeel, and K. Schulten. Difficulties with multiple time stepping and fast multipole algorithm in molecular dynamics. J. Comput. Chem., 18(14):1785–1791, 1997. P. G. Bolhuis. Transition-path sampling of β-hairpin folding. Proc. Natl. Acad. Sci. USA, 100(21):12129–12134, 2003. P. G. Bolhuis, C. Dellago, P. L. Geissler, and D. Chandler. Transition-path sampling: Throwing ropes over mountains in the dark. J. Phys.: Condens. Matter, 12:A147–A152, 2000. B. R. Brooks, C. L. Brooks, A. D. MacKerell, L. Nilsson, R. J. Petrella, B. Roux, Y. Won, G. Archontis, C. Bartels, S. Boresch, A. Caflisch, L. Caves, Q. Cui, A. R. Dinner, M. Feig, S. Fischer, J. Gao, M. Hodoscek, W. Im, K. Kuczera, T. Lazaridis, J. Ma, V. Ovchinnikov, E. Paci, R. W. Pastor, C. B. Post, J. Z. Pu, M. Schaefer, B. Tidor, R. M. Venable, H. L. Woodcock, X. Wu, W. Yang, D. M. York, and M. Karplus. CHARMM: The biomolecular simulation program. J. Comput. Chem., 30(10):1545– 1614, 2009.

Advanced Methods Used in Molecular Dynamics ...

117

D. Bucher, L. C. T. Pierce, J. A. McCammon, and P. R. L. Markwick. On the use of accelerated molecular dynamics to enhance configurational sampling in ab initio simulations. J. Chem. Theory Comput., 7:890–897, 2011. G. Bussi, A. Laio, and M. Parrinello. Equilibrium free energies from non equilibrium metadynamics. Phys. Rev. Lett., 96:090601, 2006. D. Chandler. Statistical mechanics of isomerization dynamics in liquids and the transition state approximation. J. Chem. Phys., 68(6):2959–2970, 1978. doi: 10.1063/1.436049. URL https://aip.scitation.org/doi/abs/10. 1063/1.436049. J. D. Chodera, W. C. Swope, J. W. Pitera, C. Seok, and K. A. Dill. Use of the weighted histogram analysis method for the analysis of simulated and parallel tempering simulations. J. Chem. Theory Comput., 3:26–41, 2007. J. B. Clarage, T. Romo, B. .K. Andrews, B. M. Pettitt, and G. N. Jr. Phillips. A sampling problem in molecular dynamics simulations of macromolecules. Proc. Natl. Acad. Sci. USA, 92:3288–3292, 1995. C. Predescu, M. Predescu, and C. Ciobanu. The incomplete beta function law for parallel tempering sampling of classical canonical systems. J. Chem. Phys., 120(9):4119–4128, 2004. M. Creutz and A. Goksch. Higher-order hybrid Monte Carlo algorithms. Phys. Rev. Lett., 63:9–12, Jul 1989. doi: 10.1103/PhysRevLett.63.9. URL https: //link.aps.org/doi/10.1103/PhysRevLett.63.9. D.A. Kofke. On the acceptance probability of replica-exchange Monte Carlo trials. J. Chem. Phys., 117(15):6911–6914, 2002. With erratum in Ref. ?. D. Bashford and D.A. Case. Generalized Born model of macromolecular salvation effects. Annu. Rev. Phys. Chem., 51:129–152, 2000. C. A. F. de Oliveira, D. Hamelberg, and J. A. McCammon. Estimating kinetic rates from accelerated molecular dynamics simulations: Alanine dipeptide in explicit solvent as a case study. J. Chem. Phys., 127:175105, 2007.

118

Hiqmet Kamberaj

C. A. F. de Oliveira, D. Hamelberg, and J. A. McCammon. Coupling accelerated molecular dynamics methods with thermodynamic integration simulations. J. Chem. Theory Comput., 4:1516–1525, 2008. P. M. C. de Oliveira. Broad Histogram Simulation: Microcanonical Ising Dynamics. Int. J. Mod. Phys. C, 9:497–503, 1998. P. M. C. de Oliveira, T. J. P. Penna, and H. J. Herrmann. Broad histogram method. Braz. J. Phys., 26:677–683, 1996. C. Dellago, P. G. Bolhuis, F. S. Csajka, and D. Chandler. Transition path sampling and the calculation of rate constants. J. Chem. Phys., 108(5):1964– 1977, 1998. D. J. Wales. Perspective: Insight into reaction coordinates and dynamics from the potential energy landscape. J. Chem. Phys., 142:130901–12, 2015. U. Doshi and D. Hamelberg. Extracting realistic kinetics of rare activated processes from accelerated molecular dynamics using Kramer’s theory. J. Chem. Theory Comput., 7:575–581, 2011. R. O. Dror, D. H. Arlow, P. Maragakis, T. J. Mildorf, A. C. Pan, H. Xu, and D. W. Borhani. Activation mechanism of the β2-adrenergic receptor. Proc. Natl. Acad. Sci. USA, 108:18684–18689, 2011. D. J. Earl and M. W. Deem. Parallel tempering: Theory, applications, and new perspectives. Phys. Chem. Chem. Phys., 7:3910, 2005. E. Lyman, F. M. Ytreberg, and D. M. Zuckerman. Resolution exchange simulation. Phys. Rev. Lett., 96:028105, Jan 2006. doi: 10.1103/PhysRevLett.96. 028105. URL https://link.aps.org/doi/10.1103/PhysRevLett.96. 028105. D. L. Ermak and J. A. McCammon. Brownian dynamics with hydrodynamic interactions. J. Chem. Phys., 64:1352, 1978. F.A. Escobedo and F.J. Martinez-Veracoechea. Optimized expanded ensembles for simulations involving molecular insertions and deletions. i. closed systems. J. Chem. Phys., 127:174103, 2007.

Advanced Methods Used in Molecular Dynamics ...

119

M. Fajer, D. Hamelberg, and J. A. McCammon. Replica-exchange accelerated molecular dynamics (REXAMD) applied to thermodynamic integration. J. Chem. Theory Comput., 4:1565–1569, 2008. M. Falcioni and M. W. Deem. A biased Monte Carlo scheme for zeolite structure solution. J. Chem. Phys., 110(3):1754, 1999. A. M. Ferrenberg and R. H. Swendsen. Optimized Monte Carlo data analysis. Phys. Rev. Lett., 63:1195–1198, 1989. D. D. Frantz, D. L. Freeman, and J. D. Doll. Reducing quasi-ergodic behavior in Monte Carlo simulations by J-walking: Applications to atomic clusters. J. Chem. Phys., 93(4):2769–2784, 1990. doi: 10.1063/1.458863. URL https: //doi.org/10.1063/1.458863. H. Frauenfelder, P. J. Steinbach, and R. D. Young. Conformational relaxation in proteins. Chem. Scr., 23A:145–150, 1989. D. Frenkel and B. Smit. Understanding Molecular Simulations: From Algorithms to Applications. Academic Press, San Diego, CA, second edition, 2002. M. S. Friedrichs, P. Eastman, V. Vaidyanathan, M. Houston, S. Legrand, A. L. Beberg, D. L. Ensign, C. M. Bruns, and V. S. Pande. Accelerating molecular dynamics simulation on graphics processing units. J. Comput. Chem., 30: 864–72, 2009. F. Wang and D.P. Landau. Efficient, multiple-range random walk algorithm to calculate the density of states. Phys. Rev. Lett., 86(10):2050–2053, 2001a. F. Wang and D.P. Landau. Determining the density of states for classical statistical models: A random walk algorithm to produce a flat histogram. Phys. Rev. E, 64:056101–16, 2001b. E. Gallicchio, M. Andrec, A. K. Felts, and R. M. Levy. Temperature weighted histogram analysis method, replica exchange, and transition paths. J. Phys. Chem. B, 109:6722–6731, 2005.

120

Hiqmet Kamberaj

G. Ciccotti and E. Vanden-Eijnden. The trees and the forest. Aims and objectives of molecular dynamics simulations. Eur. Phys. J. Special Topics, 224: 2515–2518, 2015. F. L. Gervasio, A. Laio, and M. Parrinello. Flexible docking in solution using metadynamics. J. Am. Chem. Soc., 127:2600–2607, 2005. G. J. Geyer. Practical Markov Chain Monte Carlo. Stat. Sci., 7(4):473–483, 1992. G. J. Martyna, M. L. Klein, and M. E. Tuckerman. Nosé-Hoover chains: The canonical ensemble via continuous dynamics. J. Chem. Phys., 97(4):2635– 2643, 1992. doi: 10.1063/1.463940. URL https://doi.org/10.1063/1. 463940. G. J. Martyna, M. E. Tuckerman, D. J. Tobias, and M. L. Klein. Explicit reversible integrators for extended systems dynamics. Mol. Phys., 87(5):1117– 1157, 1996. G.M. Torrie and J. P. Valleau. Nonphysical sampling distributions in MonteCarlo free-energy estimation: Umbrella sampling. J. Comput. Phys., 23(2): 187–199, 1977. N. Go and H. A. Scheraga. Analysis of the contribution of internal vibrations to the statistical weights of equilibrium conformations of macromolecules. J. Chem. Phys., 51(11):4751–4767, 1969. doi: 10.1063/1.1671863. URL https://doi.org/10.1063/1.1671863. S. K. Gray, D. W. Noid, and B. G. Sumpter. Symplectic integrators for large scale molecular dynamics simulations: A comparison of several explicit methods. J. Chem. Phys., 101(5):4062–4072, 1994. D. Gront and A. Kolinski. Efficient scheme for optimization of parallel tempering Monte Carlo method. J. Phys.: Condens. Matter, 19:036225, 2007. H. Grubmüller. PhD thesis, Technische Universität München, 1994. H. Grubmüller. Predicting slow structural transitions in macromolecular systems: Conformational flooding. Phys. Rev. E, 52:2893–2906, 1995.

Advanced Methods Used in Molecular Dynamics ...

121

D. Hamelberg, J. Mongan, and J. A. McCammon. Accelerated molecular dynamics: A promising and efficient simulation method for biomolecules. J. Chem. Phys., 120:11919, 2004a. D. Hamelberg, J. Mongan, and J.A. McCammon. Accelerated molecular dynamics: A promising and efficient simulation method for biomolecules. J. Chem. Phys., 120(24):11919–11929, 2004b. D. Hamelberg, T. Shen, and J. A. McCammon. Relating kinetic rates and local energetic roughness by accelerated molecular-dynamics simulations. J. Chem. Phys., 122:241103, 2005. D. Hamelberg, C. A. F. de Oliveira, and J. A. McCammon. Sampling of slow diffusive conformational transitions with accelerated molecular dynamics. J. Chem. Phys., 127(15):155102, 2007. P. Hänggi, P. Talkner, and M. Borkovec. Reaction-rate theory: Fifty years after Kramers. Rev. Mod. Phys., 62:251, 1990. U. H. E. Hansmann and Y. Okamoto. In Annual reviews in computational physics VI. (World Scientific, Singapore, 1999. U. H. E. Hansmann and Y. Okamoto. Prediction of peptide conformation by multicanonical algorithm: New approach to the multiple-minima problem. J. Comput. Chem., 14:1333–1338, 1993. U. H. E. Hansmann, Y. Okamoto, and F. Eisenmenger. Molecular dynamics, Langevin and hydrid Monte Carlo simulations in a multicanonical ensemble. Chem. Phys. Lett., 259:321–330, 1996. D. .J. Hardy, J. .E. Stone, K. L. Vandivort, D. Gohara, C. Rodrigues, and K. Schulten. Fast molecular electrostatics algorithms on GPUs. In In Wen mei Hwu, editor, GPU Computing Gems, pages 43–58. Morgan Kaufmann Publishers, 2011. H.D. Raedt and B.D. Raedt. Applications of the generalized Trotter formula. Phys. Rev. A, 28(6):3575, 1983. H.F. Trotter. On the product of semi-groups of operators. Proc. Am. Math. Soc., 10:545–551, 1959.

122

Hiqmet Kamberaj

H. Fukunishi, O. Watanabe, and S. Takada. On the Hamiltonian replica exchange method for efficient sampling of biomolecular systems: Aplacation to protein structure prediction. J. Chem. Phys., 116:9058–9067, 2002. H.G. Katzgraber, S. Trebst, D.A. Huse, and M. Troyer. Feedback-optimized parallel tempering Monte Carlo. J. Stat. Mech., page P03018, 2006. H. Kamberaj. Conformational sampling enhancement of replica exchange molecular dynamics simulations using swarm particle intelligence. J. Chem. Phys., 143:124105–8, 2015. H. Kamberaj. Faster protein folding using enhanced conformational sampling of molecular dynamics simulation. J. Mol. Graph. Model., 81:32–49, 2018. H. Kamberaj and A. van der Vaart. Multiple scaling replica exchange for the conformational sampling of biomolecules in explicit water. J. Chem. Phys., 127:234102–234109, 2007. H. Kamberaj and A. van der Vaart. An optimised replica exchange method for molecular dynamics simulations. J. Chem. Phys., 130:074904, 2009. R. W. .W. Hooft, B. P. van Eijck, and J. Kroon. An adaptive umbrella sampling procedure in conformational analysis using molecular dynamics and its application to glycol. J. Chem. Phys., 97:6690–6694, 1992. W.G. Hoover. Canonical dynamics: Equilibrium phase-space distributions. Phys. Rev. A, 31:1695–1697, 1985. X. Huang, M. Hagen, B. Kim, R.A. Friesner, R. Zhou, and B.J. Berne. Replica exchange with solute tempering: efficiency in large scale systems. J. Phys. Chem. B, 111:5405–5410, 2007. T. Huber and W. F. van Gunsteren. SWARM-MD: Searching conformational space by cooperative molecular dynamics. J. Phys. Chem. A, 102:5937–5943, 1998. G. Hummer and A. Szabo. Free energy profiles from single-molecule pulling experiments. Proc. Natl. Acad. Sci. USA, 107(50):21441–21446, 2010.

Advanced Methods Used in Molecular Dynamics ...

123

P. H. Hünenberger and W. F. van Gunsteren. Computer Simulation of Biomolecular Systems, Theoretical and Experimental Applications. Kluwer, Dordrecht, The Netherlands, 1997. I. Bahar and R. L. Jernigan. Inter-residue potentials in globular proteins and the dominance of highly specific hydrophilic interactions at close separations. J. Mol. Biol., 266:195–214, 1997. I. Fukuda and H. Nakamura. Tsallis dynamics using the Nosé-Hoover approach. Phys. Rev. E, 65:026105, 2002. A. Irbäck, F. Sjunnesson, and S. Wallin. Three-helix-bundle protein in a Ramachandran model. Proc. Natl. Acad. Sci. U.S.A., 97:13614–13618, 2000. J. A. Izaguirre, D. P. Catarello, J. M. Wozniak, and R. D. Skeel. Langevin stabilization of molecular dynamics. J. Chem. Phys., 114(5):2090–2098, 2001. J. Kim and J. E. Straub. Optimal replica exchange method combined with Tsallis weight sampling. J. Chem. Phys., 130:144114–11, 2009. Q. Johnson, U. Doshi, T. Chen, and D. Hamelberg. Water’s contribution to the energetic roughness from peptide dynamics. J. Chem. Theory Comput., 6: 2591–2597, 2010. A. D. MacKerell Jr, D. Bashford, M. Bellot, R. L. Dunbrack Jr., J. D. Evanseck, M. J. Field, S. Fisher, J. Gao, H. Guo, S. Ha, D. Joseph-McCarthy, L. Kuchnir, K. Kuczera, F. T. K. Lau, C. Mattos, S. Michnick, T. Ngo, D. T. Nguyen, B. Prodhom, W. E. Reiher, B. Roux, M. Schlenkrich, J. C. Smith, R. Stote, J. Straub, M. Watanabe, J. Wiorkiewicz-Kuczera, D. Yin, and M. Karplus. All-atom empirical potential for molecular modelling and dynamics studies of proteins. J. Phys. Chem. B, 102(18):3586–3616, 1998. H. Kamberaj. A theoretical model for the collective motion of proteins by means of principal component analysis. Cent. Eur. J. Phys., 9(1):96–109, 2011. H. Kamberaj. PhD thesis, Manchester Metropolitan University, Manchester, UK, May 2005. J. Kästner. Umbrella sampling. Advanced Review, 00:1–11, 2011.

124

Hiqmet Kamberaj

J. Kästner and W. Thiel. Analysis of the statistical error in umbrella sampling simulations by umbrella integration. J. Chem. Phys., 124:234106(1–7), 2006. J. C. Keck. Statistical investigation of dissociation cross-sections for diatoms. Discuss. Faraday Soc., 33:173–182, 1962. K. Hukushima and K. Nemoto. Exchange Monte Carlo method and application to spin glass simulations. J. Phys. Soc. Jpn., 65:1604–1608, 1996. K. K. Burusco, N. J. Bruce, I. Alibay, and R. A. Bryce. Free energy calculations using a swarm-enhanced sampling molecular dynamics approach. Chem. Phys. Chem., 16:3233–3241, 2015. K. Srinivasaraghavan and M. Zacharias. Enhanced sampling of peptide and protein conformations using replica exchange simulations with a peptide backbone biasing potential. Proteins: Struct., Funct., Bioinf., 66:697–706, 2007. S. Kumar, J. M. Rosenberg, D. Bouzida, R. H. Swendsen, and P. A. Kollman. The weighted histogram analysis method for free-energy calculations on biomolecules. I. The method. J. Comput. Chem., 13(8):1011–1021, 1992. A. Laio and M. Parrinello. Escaping free-energy minima. Proc. Natl. Acad. Sci. USA, 99:12562–12566, 2002a. A. Laio and M. Parrinello. Escaping free-energy minima. Proc. Natl. Acad. Sci. USA, 99(20):12562–12566, 2002b. A. Laio, A. F. L. Rodriguez, F. Gervasio, M. Ceccarelli, and M. Parrinello. Assessing the accuracy of metadynamics. J. Phys. Chem. B, 109:6714–6721, 2005. D. P. Landau, S.-H. Tsai, and M. Exler. A new approach to Monte Carlo simulations in statistical physics: Wang-Landau sampling. Am. J. Phys., 72(10): 1294–1302, 2004. O. F. Lange and H. Grubmüller. Can principal components yield a dimension reduced description of protein dynamics on long time scales? J. Phys. Chem. B, 110:22842–22852, 2006.

Advanced Methods Used in Molecular Dynamics ...

125

O. F. Lange, L. V. Schäfer, and H. Grubmüller. Flooding in GROMACS: accelerated barrier crossing in molecular dynamics. J. Comput. Chem., 27(14): 1693–1702, 2006. X. Li, C.P. O’Brien, G. Collier, N.A. Vellore, F. Wang, and R.A. Latour. An improved replica-exchange sampling method: Temperature intervals with global energy reassignment. J. Chem. Phys., 127:164116, 2007. Q. Ma and J. A. Izaguirre. Targeted mollified impulse: A multi scale stochastic integrator for long molecular dynamics simulations. Multiscale Model. Simul., 2(1):1–21, 2003. Q. Ma, J. A. Izaguirre, and R. D. Skeel. Verlet-I/r-RESPA/Impulse is limited by nonlinear instabilities. SIAM J. Sci. Comput., 24:1951, 2003. E. Marinari and G. Parisi. Simulated tempering: a new Monte Carlo scheme. Europhys. Lett., 19:451, 1992. J. A. McCammon, S. H. Northrup, M. Karplus, and R. M. Levy. Helix-coil transitions in a simple polypeptide model. Biopolymers, 19:2033–2045, 1980. M.E. Tuckerman, B.J. Berne, and G.J. Martyna. Reversible multiple time scale molecular dynamics. J. Chem. Phys., 97(3):1990, 1992. M. E. Tuckerman, D. A. Yarne, S. O. Samuelson, A. L. Hughes, and G. J. Martyna. Exploiting multiple levels of parallelism in molecular dynamics based calculations via modern techniques and software paradigms on distributed memory computers. Comput. Phys. Commun., 128:333, 2000. M. Mezei. Adaptive umbrella sampling: Self-consistent determination of the non-Boltzmann bias. J. Comput. Phys., 68:237–248, 1987. M. Mills and I. Andricioaei. An experimentally guided umbrella sampling protocol for biomolecules. J. Chem. Phys., 129:114101(1–13), 2008. P. Minary, M. E. Tuckerman, and G. J. Martyna. Long time molecular dynamics for enhanced conformational sampling in biomolecular systems. Phys. Rev. Lett., 93(150201-4), 2004.

126

Hiqmet Kamberaj

M. Karplus and J.A. McCammon. Molecular dynamics simulations of biomolecules. Nat. Struct. Biol., 9(9):646–652, 2002. With corrigenda in Nat. Struct. Biol. 9(10),788 (2002). M. K. Transtrum, B. B. Machta, K. S. Brown, B. C. Daniels, C. R. Myers, and J. P. Sethna. Perspective: Sloppiness and emergent theories in physics, biology, and beyond. J. Chem. Phys., 143:010901–13, 2015. M.P. Allen and D.J. Tildesley. Computer simulation of liquids. Oxford University Press, 1989. M. Tuckerman and M. Parrinello. Integrating the Car-Parrinello equations. I. Basic integration techniques. J. Chem. Phys., 101(2):1302, 1994. W. Nadler and U.H.E. Hansmann. Generalized ensemble and tempering simulations: A unified view. Phys. Rev. E, 75:026109, 2007. N. Nakajima, H. Nakamura, and A. Kidera. Multicanonical ensemble generated by molecular dynamics simulation for enhanced conformational sampling of peptides. J. Phys. Chem. B, 101:817–824, 1997. R. M. Neal. Sampling from multimodal distributions using tempered transitions. Statistics and Computing, 6(4):353–366, 1996. S. Nosé. A molecular dynamics method for simulations in the canonical ensemble. Mol. Phys., 52:255, 1984. N. Rathore, M. Chopra, and J.J. de Pablo. Optimal allocation of replicas in parallel tempering simulations. J. Chem. Phys., 122:024111, 2005. H. Nymeyer, S. Gnanakaran, and A. E. Garcia. Atomistic simulations of protein folding. using the replica exchange algorithm. Methods in Enzymology, 383: 119–149, 2004. Y. Okamoto and U. H. E. Hansmann. Thermodynamics of helix-coil transitions studied by multicanonical algorithms. J. Phys. Chem., 99:11276–11287, 1995.

Advanced Methods Used in Molecular Dynamics ...

127

S. Oldziej, A. Liwo, C. Czaplewski, J. Pillardy, and H. A. Scheraga. Optimization of the UNRES force field by hierarchical design of the potential-energy landscape. 2. Off-lattice tests of the method with single proteins. J. Phys. Chem. B, 108:16934–16949, 2004. R. Palmer. Broken ergodicity. Adv. Phys., 32:669–735, 1982. B. Pattanasiri, Y. W. Li, D. P. Landau, T. Wüst, and W. Triampo. Conformation transitions of a confined lattice protein: A Wang-Landau study. J. Phys.: Conf. Series, 402:012048, 2012. T. J. P. Penna. Traveling salesman problem and Tsallis statistics. Phys. Rev. E, 51:R1, 1995. J. Pfaendtner, D. Brabduardi, M. Parrinello, T. D. Pollard, and G. A. Voth. Nucleotide-dependent conformational states of actin. Proc. Natl. Acad. Sci. USA, 106:12723–12728, 2009. P. G. Bolhuis, D. Chandler, C. Dellago, and P. L. Geissler. Transition path sampling: Throwing ropes over rough mountain passes, in the dark. Annu. Rev. Phys. Chem., 53:291–318, 2002. J. C. Phillips, J. .E. Stone, K. .L. Vandivort, T. .G. Armstrong, J. M. Wozniak, M. Wilde, and K. Schulten. Petascale Tcl with NAMD, VMD, and Swift/T. In SC’14 workshop on High Performance Technical Computing in Dynamic Languages. IEEE Press, 2014. T. N. Phung, J. F. Brady, and G. Bossis. Stokesian dynamics simulation of brownian suspensions. J. Fluid Mech., 313:181–207, 1996. S. Piana, J. L. Klepeis, and D. E. Shaw. Assessing the accuracy of physical models used in protein-folding simulations: quantitative evidence from long molecular dynamics simulations. Curr. Opin. Struct. Biol., 24:98–105, 2014. L. C. Pierce, R. S. Ferrer, C. .A. F. de Oliveira, J. A. McCammon, and R. .C. Walker. Routine access to millisecond time scale events with accelerated molecular dynamics. J. Chem. Theory Comput., 8(9):2997–3002, 2012. A. R. Plastino and C. Anteneodo. A dynamical thermostatting approach to nonextensive canonical ensembles. Ann. Phys., 255(2):250–269, 1997.

128

Hiqmet Kamberaj

P. Liu and G.A. Voth. Smart resolution replica exchange: An efficient algorithm for exploring complex energy landscapes. J. Chem. Phys., 126(4):045106–6, 2007. P. Liu, B. Kim, R.A. Friesner, and B.J. Berne. Replica exchange with solute tempering: A method for sampling biological systems in explicit water. Proc. Natl. Acad. Sci. USA, 103(39):13749–13754, 2005. C. Predescu, M. Predescu, and C.V. Ciobanu. On the efficiency of exchange in parallel tempering Monte Carlo simulations. J. Phys. Chem. B, 109:4189– 4196, 2005. D. Provasi and M. Filizola. Putative active states of a prototype G-proteincoupled receptor from biased molecular dynamics. Biophys. J., 98:2347– 2355, 2010. J. A. Rahman and J. C. Tully. Puddle-skimming: An efficient sampling of multidimensional configuration space. J. Chem. Phys., 116:8750, 2002. P. Raiteri, A. Laio, F. L. Gervasio, C. Micheletti, and M. Parrinello. Efficient reconstruction of complex free energy landscapes by multiple walkers metadynamics. J. Phys. Chem. B., 110:3533–3539, 2006. J. Rogal and P. G. Bolhuis. Multiple state transition path sampling. J. Chem. Phys., 129:224107, 2008. R. Yamamoto and W. Kob. Replica-exchange molecular dynamics simulation for supercooled liquids. Phys. Rev. E, 61:5473–5476, 2000. R. Zhou. Free energy landscape of protein folding in water: explicit vs. implicit solvent. Proteins: Struct., Funct., Bioinf., 53(2):148–161, 2003. R. Zhou and B.J. Berne. Can a continuum solvent model reproduce the free energy landscape of a β-hairpin folding in water? Proc. Natl. Acad. Sci. USA, 99:12777–12782, 2002. D. Sabo, M. Meuwly, D.L. Freeman, and J.D. Doll. A constant entropy increase model for the selection of parallel tempering ensembles. J. Chem. Phys., 128: 174109, 2008.

Advanced Methods Used in Molecular Dynamics ...

129

D. P. Scarpazza, D. J. Ierardi, A. K. Lerer, K. M. Mackenzie, A. C. Pan, J. A. Bank, E. Chow, R. O. Dror, J. P. Grossman, D. Killebrew, M. A. Moraes, C. Predescu, J. K. Salmon, and D. E. Shaw. Extending the generality of molecular dynamics simulations on a special-purpose machine. In Proceedings of the 27th IEEE International Parallel and Distributed processing Symposium, pages 933–945, Boston, MA, USA, 2013. IEEE Computer Society. T. Schlick. Molecular Modeling and Simulation. Springer, New York, NY, 2nd edition, 2010. T. Schlick, M. Mandziuk, R. D. Skeel, and K. Srinivas. Nonlinear resonance artefacts in molecular dynamics simulations. J. Comput. Phys., 140(1):1–29, 1998. S. Jang, S. Shin, and Y. Pak. Replica-exchange method using the generalized effective potential. Phys. Rev. Lett., 91:058305, 2003. S. Jang, E. Kim, and Y. Pak. All-atom level direct folding simulation of a ββα miniprotein. J. Chem. Phys., 128:105102, 2008. R. D. Skeel, G. Zhang, and T. Schlick. A family of symplectic integrators: stability, accuracy, and molecular dynamics applications. SIAM J. Sci. Comput., 18(1):203–222, 1997. S. L. Seyler and O. Beckstein. Sampling large conformational transitions: adenylate kinase as a testing ground. Mol. Sim., 40(10-11):855–877, 2014. A. V. Smith and C. K. Hall. Alpha-helix formation: discontinuous molecular dynamics on an intermediate-resolution protein model. Proteins, 44(3):344– 60, 2001a. A. V. Smith and C. K. Hall. Assembly of a tetrameric alpha-helical bundle: Computer simulations on an intermediate-resolution protein model. Proteins, 44:376, 2001b. S. Nosé. A unified formulation of the constant temperature molecular dynamics methods. J. Chem. Phys., 81:511, 1984a. S. Nosé. A molecular dynamics method for simulation in the canonical ensemble. Mol. Phys., 52:255, 1984b.

130

Hiqmet Kamberaj

M. Stepanova. Dynamics of essential collective motions in proteins: Theory. Phys. Rev. E, 76(5):051918, 2007. J. .E. Stone, D. .J. Hardy, B. Isralewitz, and K. Schulten. GPU algorithms for molecular modelling. In D. A. Bader and J. Kurzak, editors, Scientific Computing with Multicore and Accelerators, pages 351–371. Chapman & Hall, 2011. J. E. Stone, K. L. Vandivort, and K. Schulten. GPU-accelerated analysis and visualisation on petascale supercomputing platforms. In Proceedings of the 8th International Workshop on Ultrascale Visualization, pages 6:1–6:8, New York, NY, USA, 2013. UltraVis’13. J. E. Stone, R. McGreevy, B. Isralewitz, and K. Schulten. GPU-accelerated analysis and visualisation of large structures solved by molecular dynamics flexible-fitting. Faraday Discuss., 169:265–283, 2014. doi: 10.1039/ C4FD00005F. URL http://dx.doi.org/10.1039/C4FD00005F. T. P. Straatsma and J. A. McCammon. Multiconfiguration thermodynamic integration. J. Chem. Phys., 95:1175–1188, 1991. S. Trebst, D.A. Huse, and M. Troyer. Optimizing the ensemble for equilibrium in broad-histogram Monte Carlo. Phys. Rev. E, 70:046701, 2004. S. Trebst, M. Troyer, and U.H.E. Hansmann. Optimized parallel tempering simulations of proteins. J. Chem. Phys., 124:174903, 2006. Y. Sugita and Y. Okamoto. Replica-exchange molecular dynamics method for protein folding. Chem. Phys. Lett., 314:141–151, 1999. M. Takahashi and M. Imada. Monte Carlo calculation of quantum systems. II. higher order correction. J. Phys. Soc. Jpn., 53:3765–3769, 1984. T. Nagasima, A. R. Kinjo, T. Mitsui, and K. Nishikawa. Wang-Landau molecular dynamics technique to search for low-energy conformational space of proteins. Phys. Rev. E, 75:066706, 2007. G. M. Torrie and J. P. Valleau. Monte Carlo free energy estimates using nonBoltzmann sampling: Application to the sub-critical Lennard-Jones fluid. Chem. Phys. Lett.., 28:578–581, 1974.

Advanced Methods Used in Molecular Dynamics ...

131

V. Tozzini. Coarse-grained models for proteins. Curr. Opin. Struc. Biol., 15: 144, 2005. V. Tozzini and J. McCammon. A coarse grained model for the dynamics of the early stages of the binding mechanism of hiv-1 protease. Chem. Phys. Lett., 413:123–128, 2005. V. Tozzini, W. Rocchia, and J. A. McCammon. Mapping all-atom models onto one-bead coarse-grained models: General properties and applications to a minimal polypeptide model. J. Chem. Theory Comput., 2:667–673, 2006. C. Tsallis. Possible generalization of Boltzmann-Gibbs statistics. J. Stat. Phys., 52:479–487, 1988. M. .E. Tuckerman and B. J. Berne. Molecular dynamics algorithm for multiple time scales: Systems with long range forces. J. Chem. Phys., 94:6811, 1991a. M. .E. Tuckerman and B. J. Berne. Molecular dynamics in systems with multiple time scales: Systems with stiff and soft degrees of freedom and with short and long range forces. J. Chem. Phys., 95:8362, 1991b. M. .E. Tuckerman and G. J. Martyna. Understanding modern molecular dynamics: Techniques and Applications. J. Phys. Chem. B, 104:159–178, 2000. M. .E. Tuckerman, G. J. Martyna, and B. J. Berne. Molecular dynamics algorithm for condensed systems with multiple time scales. J. Chem. Phys., 93: 1287, 1990. M. .E. Tuckerman, B. J. Berne, and A. Rossi. Molecular dynamics algorithm for multiple time scales: Systems with disparate masses. J. Chem. Phys., 94: 1465, 1991. T. W. Whitfield, L. Bu, and J. E. Straub. Generalized parallel sampling. Physica A: Statistical Mechanics and its Applications, 305:157–171, 2002. U. H. E. Hansmann and Y. Okamoto. Tackling the protein folding problem by a generalized-ensemble approach with Tsallis statistics. Braz. J. Phys., 29: 187, 1999.

132

Hiqmet Kamberaj

U. H. E. Hansmann and Y. Okamoto. Generalized-ensemble Monte Carlo method for systems with rough energy landscape. Phys. Rev. E, 56(2):2228– 2233, 1997. A. E. van Giessen and J. E. Straub. Monte Carlo simulations of polyalanine using a reduced model and statistics-based interaction potentials. J. Chem. Phys., 122:024904, 2005. W. van Gunsteren, D. Bakowies, R. Baron, I. Chandrasekhar, M. Christen, X. Daura, P. Gee, D. P. Geerke, A. Glättli, P. H. Hünenberger, M. A. Kastenholz, C. Oostenbrink, M. Schenk, D. Trzesniak, N. F. A. can der Vegt, and H. B. Yu. Biomolecular modeling: Goals, problems, perspectives. Angew. Chem. Int. Ed., 45(25):4064–4092, 2006. A. F. Voter. Hyperdynamics: Accelerated molecular dynamics of infrequent events. Phys. Rev. Lett., 78:3908, 1997. F. Wang and D. P. Landau. Efficient, multiple-range random walk algorithm to calculate the density of states. Phys. Rev. Lett., 86:2050, 2001. J. S. Wang and R. H. Swendsen. Replica Monte Carlo simulation of spin glasses. Phys. Rev. Lett., 57:2607–2609, 1986. W.B. Street, D.J. Tildesley, and G. Saville. Multiple timestep methods in molecular dynamics. Mol. Phys., 35:639, 1978. S. Wells, S. Menor, B. Hespenheide, and M. Thorpe. Constrained geometric simulation of diffusive motion in proteins. Phys. Biol., 2(4):S127–S136, 2005. W.G. Hoover. Canonical dynamics: Equilibrium phase-space distributions. Phys. Rev. A, 31:1695, 1985. S. Wolfram. A new kind of science. Wolfram Media, Inc., Champaign, IL, 2002. T. B. Woolf and A. Grossfield. Interaction of tryptophan analogs with POPC lipid bilayers investigated by molecular dynamics calculations. Langmuir, 18:198–210, 2002.

Advanced Methods Used in Molecular Dynamics ...

133

T. Wüst and D. P. Landau. Optimized Wang-Landau sampling of lattice polymers: Ground state search and folding thermodynamics of HP model proteins. J. Chem. Phys., 137(6):064903, 2012. X. Cheng, G. Cui, V. Hornak, and C. Simmerling. Modified replica exchange simulation methods for local structure refinement. J. Phys. Chem. B, 109: 8220–8230, 2005. Y. Xin, U. Doshi, and D. Hamelberg. Examining the limits of time reweighing and kramer’s rate theory to obtain correct kinetics from accelerated molecular dynamics. J. Chem. Phys., 132:224101, 2010. Y. Okamoto. Designing generalized statistical ensembles for numerical simulations of biopolymers. J. Mol. Graphics Modell., 22:425–439, 2004. Y. Pak and S. Wang. Folding of a 16-residue helical peptide using molecular dynamics simulation with Tsallis effective potential. J. Chem. Phys., 111: 4359, 1999. Y. Pak and S. Wang. Application of a molecular dynamics simulation method with a generalized effective potential to the flexible molecular docking problems. J. Phys. Chem. B, 104:354–359, 2000. Y. Sugita and Y. Okamoto. Replica-exchange multicanonical algorithm and multicanonical replica-exchange method for simulating systems with rough energy landscape. Chem. Phys. Lett., 329:261–270, 2000. Y. Sugita, A. Kitao, and Y. Okamoto. Multidimensional replica-exchange method for free-energy calculations. J. Chem. Phys., 113:6042–6051, 2000. Y. Ueda, H. Taketomi, and N. Go. Studies on protein folding, unfolding, and fluctuations by computer simulation. II. A. three-dimensional lattice model of lysozyme. Biopolymers, 17:1531–1548, 1978.

In: Mechanical Design, Materials … ISBN: 978-1-53614-791-9 Editor: Sandip A. Kale © 2019 Nova Science Publishers, Inc.

Chapter 4

STUDY ON ERROR IN SIGMOIDAL FUNCTION GENERATION OF 4R MECHANISM Ankur Jaiswal and H. P. Jawale* Department of Mechanical Engineering, Visvesvaraya National Institute of Technology, Nagpur, Maharashtra, India

ABSTRACT This chapter presents an extended Freudenstein-Chebyshev approximation method for sigmoidal function generation applied to four and five precision points. The mathematical model for sigmoidal function generation is presented and the structural error parameters are obtained. The structural error between the generated function and desired function is estimated. The least-square method is used for minimizing the structural error. Comparison of the results of the least square method and analytical method is carried out. The proposed methodology is demonstrated for designing a four bar function generation planar mechanism.



Corresponding Author Email: [email protected].

136

Ankur Jaiswal and H. P. Jawale

Keywords: kinematics synthesis, optimization, function generation, precision points

1. INTRODUCTION A mechanism is an assemblage of rigid bodies connected together for the purpose of obtaining the desired motion. Dimensional synthesis is a method of determining the geometric characteristics of the links. The fourbar linkage has a long history in both the theoretical kinematics as well as plenty applications for spcial motion and path generating tasks. Various approaches of mechanisms synthesis are found in literature for path generation, function generation and rigid body guidance. The function generation is a approach to obtain desired characteristic at output link of the menchaism. The approaches are many times extended to estimate accuracy and optimize the performance. Traditionally, the objective function of mechanism synthesis is specified using the Freudenstein equation and the solved by the Chebyshev approximation theory. Within the constrints in obtaining the solutions, the desired accuracy is possible to obtain at limited precision points. In literature, solution for three four and five precision points function generation task has been dealt successfully using various approaches like freudenstein-chebyshev method, Loop-closure equation, (George and Arthur, 1984, Hartenberg and Denavit, 1964, Norton and Wang, 2004, Rao, 1979, Saxena and Ananthasuresh, 2003, Sun, 1982, Todorov, 2002). Also, the least square method, Galerkin’s method, non-linear programming approach, continuation method and sequential quadratic programming (SQP) method is used to optimize the structural error and compare the past results of the mechanism (Akcali and Dittrich, 1989, Cossalter, et al., 1992, Lin, 1998, Shariati and Norouzi, 2011, Wilde, 1982).

Study on Error in Sigmoidal Function Generation of 4R …

137

Analytical treatment on function generation in spherical four bar mechanisms (Alizade and Kilit, 2005) and five accuracy points is dealt in (Rose and Sandor, 1973, Sheu, et al., 2008) comparing the continuation method (Huang, et al., 2009). Comparison of the error using heuristic optimization techniques and gradient based method is presented (Mirmahdi and Norouzi, 2013). A new approach for dimensional synthesis of planar mechanisms with mixed exact - approximate points for path and function generation task is found to be used (Cervantes-Sánchez, et al., 2009, Diab and Smaili, 2008) for rigid links and for adjustable links is (Soong and Chang, 2011, Zhou, 2009) structural error is analyzed. The interpolation approximation, least squares approximation and chebyshev approximation for solving the function generation five point synthesis problem (Alizade and Gezgin, 2011), approximate precision-position synthesis (Freudenstein, 2010), and classical chebyshev synthesis is extended to obtain error minimization (Ceccarelli and Vinciguerra, 2000). The gauss elimination and homotopy continuation method (Wu, 2005) applied to slider crank mechanism, sum of square method (Simionescu and Beale, 2002) is also seen. Optimization of the path synthesis using adjustable approaches (Peng and Sodhi, 2010). The work on four and five precision point synthesis and therein optimization of error is required to be carried out with newer techniques, extending scope for the analysis to sigmoidal function generation. The present script aims to extend the freudenstein-chebyshev method for four and five point synthesis and optimization using least square method. The objective function is part of exponential function. A sigmoidal function is a mathematical functionhaving an “S” shape (sigmoid curve) and the limit are 0 to1 (Costarelli and Spigler, 2014). The objective function has been analyzed based on the structural error between the generated function and desired function.

F ( x) 

1 1  e x

138

Ankur Jaiswal and H. P. Jawale

2. KINEMATIC SYNTHESIS 2.1. Four Precision Point Synthesis The Chebyshev equation to determine the accuracy points of the mechanism can be given as

x( j ) 

x 0  x n 1 x n 1  x 0 cos(2 j  1)  . 2 2 2n

(1)

Freudenstein method (Hartenberg and Denavit, 1964) is an analytical method for synthesizing four-bar linkage in Figure 1. For three-position synthesis, except the three link length ratio, all other parameters were chosen arbitrarily. In the case of four-position synthesis, four parameters are required for satisfying four positions. One more parameter in addition to the three parameters taken in three point synthesis is chosen (Jaiswal and Jawale, 2017, Jaiswal and Jawale, 2017).

D1 cos   D2 cos  D3  cos(  )

(2)

Equation (2) i.e., Freudenstein equation is used to synthesize a four bar mechanism for three accuracy points where: D1 

L1 L D2  1 L4 , L2 ,

D3 

L12  L22  L23  L24 2 L2 L4

Study on Error in Sigmoidal Function Generation of 4R …

139

A B

L3 L2

L4

ψ

φ L1 O2

O4 Figure 1. Four bar mechanism.

Equation (2) is modified for four accuracy points by substituting –

 j   i  ( ) j where, ѱi = initial output angle; Substituting cos i  s1 Using values of D1, D2 and D3 on re-arranging R1 sin( j  ( ) j )  R2 cos( j  ( ) j )  R3 cos  j  R4  R5 sin( ) j  cos( ) j

(3)

where: 1

L (1  s12 ) 2 R1  2 L1s1 ,

R2 

L2 L1 ,

R3 

L2 L4 s1

R4 

,

(4)

L L L L 2 L1 L4 s1 2 1

2 2

2 3

2 4

,

140

Ankur Jaiswal and H. P. Jawale 1

(1  s12 ) 2 R5  s1

Equation (3) is the required design equation for synthesis of four bar linkage up to four accuracy points. In equation (3) and (4), there are five variables (R1, R2, R3, R4 and R5). For four accuracy points, there are only four equations [29] and similarly in five points case.

R1  R2 R5

(5)

R j  Aj  λB j

(6)

Here, Aj and Bj are selected so as to satisfy the following two sets of four linear equations. A1 sin( j  ( ) j )  A2 cos( j  ( ) j )  A3 cos  j  A4   cos( ) j

(7)

B1 sin( j  ( ) j )  B2 cos( j  ( ) j )  B3 cos  j  B4  sin( ) j

(8)

where, j = 1, 2, 3, 4 and Solving for Aj and Bj by using matrix method



( A  B )  ( A2  B1 ) 2  4 A1 B2  1 2 2 B2



1 2

(9)

2.2. Five Precision Point Synthesis The three link ratios, and two angles, as design variables in case of five point synthesis. Let the crank and follower angles for five accuracy points be φj and ψj (j = 1, 2, 3, 4, 5), then,

Study on Error in Sigmoidal Function Generation of 4R …

 j  i    j

and

141

 j   i  ( ) j

Substituting the above in the Freudenstein equation (2), R1 cos{ 1 j   1 j }  R2 sin{ 1 j   1 j }  R3 cos 1 j

 R4  1 j  R5  R6 sin  1 j  cos 1 j

(10)

where:

R1 

s1 [1 

R2  R3 

R4  R5 

R6 

1  s . 1  s  2 1

2 2

K 2 .s1 .s 2



s1 1  s  1  s 2 1

K 2 s1 s 2 K 1 s1 K 2 s 2 (11)



K 1 . 1  s12 K 2 .s 2

2 2

,

 ,

 (11)

K3 K 2 s2 ,

1  s  2 1

s2

Equation (10) is the required design equation for synthesis of four bar linkage up to five accuracy points. Using compatibility condition for linearization in equation (10) and (11) and applying gauss elimination method is used to solve the non-linear equations. Derive the linear equation (12) and (13) is to determine the design parameters of the mechanism.

142

Ankur Jaiswal and H. P. Jawale



R1 R4  R3 R2 R1 R3  R2 R4

A1 cos(1q   1q )  A2 sin(1q   1q )  A3 cos( )1q  A4 sin( )1q

 A5  cos 1q  0

(12)

B1 cos(1q   1q )  B2 sin(1q   1q )  B3 cos( )1q  B4 sin( )1q

 B5  sin  1q  0

(13)

where, q = 1,2,3,4 and 5.

3. OPTIMIZATION OF THE STRUCTURAL ERROR 3.1. Modifying the Design Equation for Four and Five Presicion Points Here, least-square technique is used as the optimization approach for mechanism synthesis. Now, on applying the least square technique to minimize the errors [29], the overall error can be written as - rearranging the equation (3) D1 cos(q  ( )q )  D2 sin( )q  D3 cos q  D4  D5 sin(q  ( )q )  cos   q  0

D1 

(14)

L2 L1 1

(1  s12 ) 2 D2  s1

D3 

L2 L4 s1

D4 

L L L L 2 L1 L4 s1 2 1

(15) 2 2

2 3

2 4

Study on Error in Sigmoidal Function Generation of 4R …

D5

143

1  s  

1 2 2 1

L1 s1

Rearranging the above equation and replacing R1-R6 by D1-D6 respectively. The design equation (10) can be written as: D1 cos{ 1q   1q }  D2 sin{ 1q   1q }  D3 cos 1q

 D4 sin  1q  D5  D6 sin  1q  cos 1q  0

where: D1 

D2  D4 

D5  D6 

(16)

1  s . 1  s 

s1 [1 

2 1

K 2 s1 s 2



s1 1  s12  1  s 22 K 2 s1 s 2



K1 . 1  s

2 1

2 2

,

 ,

(17)



K 2 .s 2

K3 K 2 s2 ,

1  s  2 1

s2

3.2. Optimization for Four and Five Presicion Points From equations (15) and (17) – D5 = D1D2and Assuming, D5  1 (for four points) D6 

D1 D4  D2 D3 D1 D3  D2 D4 (for five points)

(18)

(19)

144

Ankur Jaiswal and H. P. Jawale



E    Aq    Bq  2

2



(20)

For minimizing overall error, Σ (ƐAq)2& (ƐBq)2 are minimized [30]. By derivative method first derivative must be equal to zero, such that

  Aq 2   Bq 2 0 0 An , Bn

(21)

 Aq  A1 sin( j  ( ) j )  A2 cos( j  ( ) j )  A3 cos  j  A4  cos( ) j

(22)

 Bq  B1 sin( j  ( ) j )  B2 cos( j  ( ) j )  B3 cos  j  B4  sin( ) j

Aq  A1 cos(1q   1q )  A2 sin(1q   1q )  A3 cos( )1q  A4 sin( )1q  A5  cos( )1q

Bq  B1 cos(1q   1q )  B2 sin(1q   1q )  B3 cos( )1q  B4 sin( )1q  B5  sin( )1q

(23)

(24)

(25)

The equations (22) & (23) and (24) & (25) determine the optimal design parameters for four and five accuracy points respectively [29, 30].

4. RESULTS AND DISCUSSION The following Table 3.1 shows a comparative study of the four and five point synthesis of sigmoidal function and their optimization using least square technique. The results of synthesis for sigmoidal functions at four and five accuracy points are summarized in Table 3.1. The structural errors for sigmoidal functions are shown in Figure 2 and 3.

Study on Error in Sigmoidal Function Generation of 4R …

145

Figure 2. Plot of error for sigmoidal function generator with four precision points.

Table 3.1. Plot of error for sigmoidal function generator with four and five accuracy points Design Variables

Interval of ‘x’ Range of φ , deg Range of ψ , deg Initial crank angle, φi, deg Link length ratio’s Follower angle, ψj,deg Maximum error Maximum output error %

-----

The results of four point synthesis by Sigmoidal function [1/(1+e-x)] Analytical Least Square method Optimization 0x1 0x1 90 90 60 60 15 9

The results of five point synthesis by Sigmoidal function [1/(1+e-x)] Analytical Least Square method Optimization 0x1 0x1 90 90 60 60 73.1478 71.9893

L2/L1 L3/L1 L4/L1 --

3.0336 9.7791 7.2282 51.2460

2.9389 6.3517 8.8720 46.2122

-1.0389 1.6085 -1.7275 61.6060

-1.7078 1.6123 -1.0215 60.3716

---

2.8614e-04 0.0391

2.8033e-04 0.0383

9.2931e-06 0.0013

5.4204e-06 7.4145e-04

146

Ankur Jaiswal and H. P. Jawale

Figure 3. Plot of error for sigmoidal function generator with five precision points.

Present work depicts error for sigmoidal function by analytical method 0.0391% and 0.0013%. After optimization by least square method the results are 0.0383% and 7.4145e-04%. The maximum output error with four and five accuracy points synthesis 2.8614e-04 and 9.2931e-06 by an analytical method. After optimization the position of the accuracy points are different and determine the new design parameter of the mechanism and maximum output error with four and five accuracy points are 2.8033e-04 and 5.4204e-06. The error of mechanism is dependent on the position of the accuracy point. The results obtained with the least square method for synthesis of five point mechanism is optimal. Figures 4 and 5 show the mapping of independent variable and input angles of the syntheze mechanism. The precision points of the sigmoidal function generating mechanism for four and five point case shown in Figure 4 and 5. The deviation of the precision points using least square method is 0.0001 to 0.01. Evaluation of the four and five point synthesis after optimization results is more accurate, it provides minimum error and mechanism behaves constrained motion to the other mathematical functions.

Study on Error in Sigmoidal Function Generation of 4R …

147

100 90

Precision Points

Input Angle (  )

80 70 60 50 40 30 20 10

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Independent Variable (x)

Figure 4. Independent variable Vs Input angle for four accuracy points. 90 80

Precision Points

Input Angle (  )

70 60 50 40 30 20 10 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Independent Variable (x)

Figure 5. Independent variable Vs Input angle for five accuracy points.

CONCLUSION The work reported in this chapter describes a methodology to design a four bar planar mechanism to generate sigmoidal function for four and five point synthesis. The different technique for four and five point synthesis of planar mechanisms has been discussed in the chapter. The proposed method is simple and easy to implement in the four bar mechanism. The

148

Ankur Jaiswal and H. P. Jawale

function generation of four-bar linkage can be used in automobiles, especially for control systems. The results obtained by mathematical analysis have following silent features: 1. The sigmoidal function generation by proposed method is found to generate minimum error and mechanism behaves with predictable motion than other function generation i.e., trignometric and hyperbolic functions (Hartenberg and Denavit, 1964, Jaiswal and Jawale, 2017). 2. The least square method is developed to optimize the structural error between the desired function and mechanically developed function, further used to optimize the generated design parameter by four and five point synthesis. 3. The developed methodology increases the global accuracy in the mechanisms by minimizing the maximum errors to the order of 104 to 10-5. In future work, the approach can be extended to cover five and six link mechanisms and evolutionary optimization problems.

REFERENCES Akcali, I., and Dittrich, G. (1989). Function Generation by Galerkin's Method. Mechanism and Machine Theory, 24 1: 39-43. Alizade, R., and Gezgin, E. (2011). Synthesis of Function Generating Spherical Four Bar Mechanism for the Six Independent Parameters. Mechanism and Machine Theory, 46 9: 1316-26. Alizade, R. I., and Kilit, Ö. (2005). Analytical Synthesis of Function Generating Spherical Four-Bar Mechanism for the Five Precision Points. Mechanism and Machine Theory, 40 7: 863-78.

Study on Error in Sigmoidal Function Generation of 4R …

149

Ceccarelli, M., and Vinciguerra, A. (2000). Approximate Four-Bar CircleTracing Mechanisms: Classical and New Synthesis. Mechanism and Machine Theory, 35 11: 1579-99. Cervantes-Sánchez, J. J., Medellín-Castillo, H. I., Rico-Martínez, J. M., and González-Galván, E. J. (2009). Some Improvements on the Exact Kinematic Synthesis of Spherical 4r Function Generators. Mechanism and Machine Theory, 44 1: 103-21. Cossalter, V., Doria, A., Pasini, M., and Scattolo, C. (1992). A Simple Numerical Approach for Optimum Synthesis of a Class of Planar Mechanisms. Mechanism and Machine Theory, 27 3: 357-66. Costarelli, D., and Spigler, R. (2014). Sigmoidal Functions Approximation and Applications. Roma Tre” University, Rome, Italy, Diab, N., and Smaili, A. (2008). Optimum Exact/Approximate Point Synthesis of Planar Mechanisms. Mechanism and Machine Theory, 43 12: 1610-24. Freudenstein, F. (2010). Approximate Synthesis of Four-Bar Linkages* [1, 2]. Resonance, George, N., and Arthur, G. Advanced Mechanism Design: Analysis and Synthesis. Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1984. Hartenberg, R. S., and Denavit, J. Kinematic Synthesis of Linkages. McGraw-Hill, 1964. Huang, X., He, G., Liao, Q., Wei, S., and Tan, X. “Solving a Planar FourBar Linkages Design Problem.” Paper presented at the Information and Automation, 2009. ICIA'09. International Conference on, 2009. Jaiswal, A., and Jawale, H. (2017). Comparative Study of Four-Bar Hyperbolic Function Generation Mechanism with Four and Five Accuracy Points. Archive of Applied Mechanics, 87 12: 2037-54. Jaiswal, A., and Jawale, H. “Comparative Study of Structural Error in Four Bar Mechanism for Hyperbolic Functions.” Paper presented at the Advances in Mechanical, Industrial, Automation and Management Systems (AMIAMS), 2017 International Conference on, 2017. Lin, C.-C. (1998). Complete Solution of the Five-Position Synthesis for Spherical Four-Bar Mechanisms. Journal of marine science and Technology, 6 1: 17-27.

150

Ankur Jaiswal and H. P. Jawale

Mirmahdi, S., and Norouzi, M. (2013). On the Comparative Optimal Analysis and Synthesis of Four-Bar Function Generating Mechanism Using Different Heuristic Methods. Meccanica, 48 8: 1995-2006. Norton, R. L., and Wang, S. S.-L. Design of Machinery: An Introduction to the Synthesis and Analysis of Mechanisms and Machines. McGrawHill Higher Education, 2004. Peng, C., and Sodhi, R. S. (2010). Optimal Synthesis of Adjustable Mechanisms Generating Multi-Phase Approximate Paths. Mechanism and Machine Theory, 45 7: 989-96. Rao, A. (1979). Optimum Design of Four-Bar Function Generators with Minimum Variance Criterion. Journal of optimization theory and applications, 29 1: 147-53. Rose, R. S., and Sandor, G. N. (1973). Direct Analytic Synthesis of FourBar Function Generators with Optimal Structural Error. Journal of Engineering for Industry, 95 2: 563-71. Saxena, A., and Ananthasuresh, G. (2003). A Computational Approach to the Number of Synthesis of Linkages. Transactions-american society of mechanical engineers journal of mechanical design, 125 1: 110-18. Shariati, M., and Norouzi, M. (2011). Optimal Synthesis of Function Generator of Four-Bar Linkages Based on Distribution of Precision Points. Meccanica, 46 5: 1007-21. Sheu, J.-B., Hu, S.-L., and Lee, J.-J. (2008). Kinematic Synthesis of a Four-Link Mechanism with Rolling Contacts for Motion and Function Generation. Mathematical and Computer Modelling, 48 5-6: 805-17. Simionescu, P., and Beale, D. (2002). Optimum Synthesis of the Four-Bar Function Generator in Its Symmetric Embodiment: The Ackermann Steering Linkage. Mechanism and Machine Theory, 37 12: 1487-504. Soong, R.-C., and Chang, S.-B. (2011). Synthesis of Function-Generation Mechanisms Using Variable Length Driving Links. Mechanism and Machine Theory, 46 11: 1696-706. Sun, W. (1982). Optimum Design Method for Four-Bar Function Generators. Journal of Optimization Theory and Applications, 38 2: 287-93.

Study on Error in Sigmoidal Function Generation of 4R …

151

Todorov, T. S. (2002). Synthesis of Four-Bar Mechanisms by Freudenstein–Chebyshev. Mechanism and Machine Theory, 37 12: 1505-12. Wilde, D. (1982). Error Linearization in the Least-Squares Design of Function Generating Mechanisms. Journal of Mechanical Design, 104 4: 881-84. Wu, T.-M. (2005). Non-Linear Solution of Function Generation of Planar Four-Link Mechanisms by Homotopy Continuation Method. J. Appl. Sci, 5 4: 724-28. Zhou, H. (2009). Synthesis of Adjustable Function Generation Linkages Using the Optimal Pivot Adjustment. Mechanism and Machine Theory, 44 5: 983-90.

In: Mechanical Design, Materials … ISBN: 978-1-53614-791-9 Editor: Sandip A. Kale © 2019 Nova Science Publishers, Inc.

Chapter 5

IDENTIFYING TRANSFER VERTEX FROM THE ADJACENCY MATRIX FOR EPICYCLIC GEAR TRAINS Mallu Chengal Reddy1,, Rudraraju Manish1 and Yendluri Daseswara Rao1 1

Mechanical Department, BITS-PILANI, Hyderabad, India

ABSTRACT In this chapter, an algorithm to find the transfer vertex of a given epicyclic gear train is given. The algorithm used searches for a gear pair and finding the transfer vertex through corresponding turning pairs. It also identifies cases where more than one transfer vertex, are possible. This algorithm reduces the number of epicyclic gear trains to be checked for rotational isomorphism after displacement isomorphism.

Keywords: epicyclic gear train, isomorphism, adjacency matrix, transfer vertex 

Corresponding Author Email:[email protected].

154

Mallu Ch. Reddy, Rudraraju Manish and Yendluri D. Rao

1. INTRODUCTION The assumptions normally used while synthesizing Epicyclic Gear Trains (EGT) are the mechanism should be planar and should have only binary joints. Also, should obey the general degree of freedom equation, have unlimited rotatability for all links. Each gear should have a turning pair on its axis and each link should have at least one turning pair to maintain constant center distance between each gear pair. There are three ways to represent an epicyclic gear train   

Functional representation, Graphical representation, and Rotation graph

Functional Representation of a gearbox is shown in Figure 1. In a graphical representation, the mechanism is illustrated using a graph (Buchsbaum, F. and Fruedenstein, F. 1970).

Figure 1. Functional Representation of a gearbox.

Identifying Transfer Vertex from the Adjacency Matrix …

155

The links are represented as vertices and joints as edges, connecting the vertices. A graphical representation of the gearbox is shown in Figure 2. In graphical representation, a turning pair is represented by a thin edge, whereas a gear pair is shown as a thick edge. Graphs shown with thin and thick edges are called bicolored graph and unlabeled when its edges are not labeled (Buchsbaum, F. and Fruedenstein, F. 1970), (Dobrjanskyj, L. and Fruedenstein, F. 1967). To draw a rotation graph of epicyclic gear trains, graphical representation of an EGT is divided into Fundamental Circuits (FC) consisting of one gear pair and associated turning pairs.

Figure 2. A graphical representation of the mechanism.

156

Mallu Ch. Reddy, Rudraraju Manish and Yendluri D. Rao

Figure 3. Three fundamental circuits.

For example, if the EGT in Figure 2b is considered, the three FCs are shown in Figure 3. In a rotation graph of EGTs, the turning pair edges are removed and each geared pair is labeled with the symbol of the associated transfer vertex of the fundamental circuit (Tsai L. 1987). For example, for the graphs of EGTs in Figure 2(a) and 2(b), rotation graphs are as shown in Figures 4(a) and 4(b). The fundamental rules followed (Buchsbaum, F. and Fruedenstein, F. 1970), (Ravisankar, R. and Mruthyunjaya, T. S.) while drawing a graph are that, for n-links, epicyclic gear train having one degree of freedom, there are n-vertices, n-1 turning pairs and n-2 gear pair edges. The subgraph formed by deleting all the geared edges is a tree and a geared edge added to the tree will form a fundamental circuit having one geared edge and several turning edges. The number of fundamental circuits will be equal to the geared edges and a fundamental circuit containing only turning edges, is not permitted as it violates the rotatability rule of an EGT. All vertices must at least be connected to one turning edge. The level of a turning pair identifies its location of its axis. For example, for the graph of EGT shown in figure 2(b), the levels are shown in Figure 3. For any fundamental circuit, the differential degree of freedom should be the number of vertices minus two, with a minimum of one. Two turning pair edges at the same level must intersect at a common vertex (Ravisankar R. and Mruthyunjaya, T. S.). All edges on one side of the transfer vertex are at the same level and all edges on the other side are at a different level.

Identifying Transfer Vertex from the Adjacency Matrix …

157

Figure 4. Rotation Graphs of Figure 2(a) and 2(b).

1.1. Linkage Adjacency Matrix Linkage adjacency matrix (Uicker, J. J. Jr. and Raicu A. 1975) is the matrix, which represents the connectivity of links in the EGT. For an n-link EGT, the adjacency matrix is a square matrix of order n obeying the following rules (Fruedenstein, F. 1971), (Ravisankar, R. and Mruthyunjaya, T. S.). A(i, j) = 1 (if link i and link j are connected by a turning pair) = 2 (if link i and link j are connected by a gear pair) = 0 (otherwise, including the case i = j) The adjacency matrix for the graph of EGT in figure 2 is given below. Adjacency matrix of Figure 2(a):

158

Mallu Ch. Reddy, Rudraraju Manish and Yendluri D. Rao

0 1  1  1 2

1 1 1 2 0 2 2 0 2 0 0 1  2 0 0 0 0 1 0 0

Adjacency matrix of Figure 2(b):

0 1  1  2 1

1 1 2 1 0 2 1 0 2 0 0 0  1 0 0 2 0 0 2 0

1.2. Hamming Procedure In the adjacency matrix, a hamming matrix is generated using the Boolean algebra. To generate the hamming matrix, there are few rules to be followed: from the elements in the adjacency matrix, the hamming distance is measured between two links, which act as elements of hamming matrix. hij = ∑aik+ ajk if aik ≠ ajk hii = 0 also hij = hji and (aik + ajk) = 0 if aik = ajk Hamming matrix and hamming value of each link (last column) of graphs in Figure 2(a) and 2(b) are as follows,

Identifying Transfer Vertex from the Adjacency Matrix …

159

 0 10 9 8 6 33 10 0 7 6 8 31    9 7 0 1 7 24    8 6 1 0 6 21  6 8 7 6 0 27 

0 9  8  8 4

9 8 8 4 29 0 5 9 5 28 5 0 8 4 25  9 8 0 8 33 5 4 8 0 21

From the elements of hamming matrix, hamming value of a link is calculated which is the sum of the elements of the specific link’s row. The sum of hamming values of all links gives hamming number of graphs or hamming value of the chain/graph. Hamming string is obtained by concatenating the hamming number and the hamming values all the links. The hamming string acts as an invariant and can be considered as a parameter to check for structural isomorphism. Identical hamming strings are considered as structurally isomorphic graphs. In doing this, we write a computer code, which computes the hamming matrix, generates the hamming string, and eliminates the matrices, which have identical hamming strings. The hamming strings of the graphs are as follows:

136

33 31 27 24 21

136

33 29 28 25 21

From the hamming strings of the graphs, we can conclude that they are not displacement isomers to one another.

1.3. Rotational Analysis After the structural isomers are eliminated, the remaining graphs are tested for rotation analysis. In this, a computer code is run to identify fundamental circuits and find out the transfer vertex of the graph. If a

160

Mallu Ch. Reddy, Rudraraju Manish and Yendluri D. Rao

graph has a single transfer vertex, those graphs are directly taken to be rotationally non-isomorphic. If a graph has more than one transfer vertex possible, those graphs need to be manually iterated to find the transfer vertex. After finding out the transfer vertex, rotation graphs are drawn and the corresponding rotation matrices are generated. These rotation graphs are now matched with the earlier adjacency matrices. On finding a match, the graph is said to be rotationally isomorphic and is eliminated. Since the rotation matrices that do not find a match, rotation-hamming strings are generated and the string is iterated for a match to be eliminated as a rotational isomer. The graphs, which are unique, are said to be rotationally non-isomorphic.

1.4. Example Graph in Figure 2(a) on rotation analysis shows that, it is a single transfer vertex i.e., vertex 3 and can be said to be rotationally nonisomorphic, which is not the case with Figure 2(b). Hence, Figure 2(b) needs to be manually checked. The fundamental circuits in graph 2 are,

Corresponding to the rotational graph and fundamental circuit, 0 9 7 7 1 24 9 0 6 10 8 33   7 6 8 0 6 27    7 10 8 0 6 31 1 8 6 6 0 21

136

33 31 27 24 21

Identifying Transfer Vertex from the Adjacency Matrix …

161

The rotation hamming matrix: Hamming string with hamming value column. We observe that, the hamming string is identical to the hamming string of the graph in Figure 2 (a). Hence, we conclude that graph in Figure 2(b) is rotationally not isomorphic to graph in Figure 2(a).

2. INTELLIGENT METHODS The first graph of the given EGT is drawn and linkage adjacency matrix is written using the rules mentioned above. Search for the gearing pair in the adjacency matrix, in each FC. If a gear pair, denoted by weightage 2, is in (i, j) position in the adjacency matrix in a FC, search in a row i and row j for a set of turning pairs with weightage 1 having a common vertex. The common vertex (link) is the Transfer Vertex (TV) for that gear pair. In case there is no such common vertex (link), check for other connecting pairs. If the number of turning pairs are more than two in a FC, then the number of TVs are possible. In such cases where more than one TV is possible, identify all the FCs in the EGT. Then assign levels to each of the turning pairs without violating the fundamental rules of EGTs (Ravisankar, R. and Mruthyunjaya, T. S.), (Tsai L. 1987). There by the TV in that particular FC is identified.

2.1. Example For example, in the adjacency matrix of Figure 2(a), one gear pair is at position (1, 5). Then the search is for tuning pairs in rows 1 and 5. We find that column 3 in each row has turning pair. That is, elements in the adjacency matrix (1, 3) and (5, 3) are turning pairs (weightage 1). Therefore, in the FC 1-5-3 as shown in Figure 5, it can be concluded that vertex 3 is the TV for the Gear pair 1-5.

162

Mallu Ch. Reddy, Rudraraju Manish and Yendluri D. Rao

Considering another example, in the adjacency matrix of Figure 2(b), gear pair with weightage 2 is at the element (4, 5). So, rows 4 and 5 are searched for all turning pairs with an entry of 1 in adjacency matrix.

Figure 5. In the FC 1-5-3.

Figure 6. Fundamental Circuit of Four Members.

However, in this case there is no common vertex among the turning pairs in both the rows. That is, in the adjacency matrix, elements (4, 2) and (5, 1) are 1, but no common vertex like (5, 1)-(4, 1) or (4, 2)-(5, 2). However, since (2, 1) element is 1 (a turning pair), then this FC has more than one TV as shown in Figure 6. That is, both the vertices 1 and 2 are possible TVs.

Identifying Transfer Vertex from the Adjacency Matrix …

163

The actual TV is ascertained by assigning the levels to the turning pairs in the FC. As levels for all the pairs in the EGT are assigned, the TV for the case where ambiguity is there is identified. Graph with levels assigned is shown in Figure 7. The rotation graphs for this EGT are as shown in Figure 4b.

Figure 7. Graph with levels assigned.

CONCLUSION Structural isomorphism and rotational isomorphism are the two stages in epicyclic gear trains synthesis. For rotational isomorphism transfer vertex for each of the fundamental circuits in the epicyclic gear train is essential. In this work, a simplified algorithm is given to find transfer vertex of a given epicyclic gear train. This reduces the computational time and effort largely. This algorithm also reduces the demand for manual effort in rotational isomorphism test.

164

Mallu Ch. Reddy, Rudraraju Manish and Yendluri D. Rao

REFERENCES Buchsbaum, F. and Fruedenstein, F. (1970), Synthesis of Kinematic Structure of Geared Kinematic Chains and Other Mechanisms, Journal of Mechanisms, 5, 357 - 392. Cheng-Ho Hsu and Jin-Jhu Hsu (1996), An efficient methodology for the structural synthesis of geared kinematic chains, Mech. Mach. Theory, 32(8), 957 - 973. Dobrjanskyj, L. and Fruedenstein, F. (1967), Some Application of Graph Theory to the Structural Analysis of Mechanisms, ASME Journal of Engineering for Industry, 89(B), 153 - 158. Fruedenstein, F. (1971), An Application of Boolean Algebra to the Motion of Epicyclic drives, ASME Journal of Engineering for Industry, 93(B), 176 - 182. Jae UK Kim and Byung man Kwak (1989), Application of edge permutation group to structural synthesis of epicyclic gear trains, Mech, Much. Theory, 25(5), 563 - 574. Jae Kun Shin and Sundar Krishnamurthy (1992), Standard code technique in the enumeration of epicyclic gear trains, Mech. Mack. Theory, 28(3), 347 - 355. Rao YVD and Rao AC (2008), Generation of Epicyclic Gear Trains of One Degree of Freedom, ASME J. Mech. Des., 130(5). Ravisankar, R. and Mruthyunjaya, T. S. Computerized Synthesis of the structure of Geared Kinematic Chains, Mechanisms and Machine Theory, 20(5), 367 - 387. Tsai L. (1987), An Application of the Linkage Characteristic Polynomial to the Topological Synthesis of Epicyclic Gear Trains, ASME. J. Mech. Trans. and Automation, 109(3), 329 - 336. Uicker, J. J., Jr. and Raicu, A. (1975) A Method for the Identification and Recognition of Equivalence of Kinematic Chains, Mechanisms and Machine Theory, 10,375 - 383.

In: Mechanical Design, Materials … ISBN: 978-1-53614-791-9 Editor: Sandip A. Kale © 2019 Nova Science Publishers, Inc.

Chapter 6

3-D SIMULATION STUDIES ON STRENGTHENING OF BEAM-COLUMN JOINT Khan Mohammad Firoz, Garg Aman and H. D. Chalak Department of Civil Engineering, NIT Kurukshetra, India

ABSTRACT The beam-column joint is one of the most critical elements of a structure. A slight rupture or loss in the strength of any beam column joint in the building may lead to a decrease in its load carrying capacity, which may further result in its progressive collapse. So it is important to analyse for stresses and deflection in the vicinity of the beam-column joint. However, the behavior of beam-column joint when subjected to lateral loading is different as compared to when it is subjected to dead load and live load. Over the years, the research in earthquake engineering has been enhanced continuously leading to modification of codal provisions for the design of structures. To incorporate these changes, the structural members have to be reinforced adequately, and beam-column joints need to be strengthened accordingly. In this research work, an analytical study has been carried out with ABAQUS/CAE on the strengthening of a beam-column joint under seismic conditions using 

Corresponding Author Email: [email protected].

166

Khan Mohammad Firoz, Garg Aman and H. D. Chalak Carbon Fiber Reinforced Polymer (CFRP) sheets. A comparison analysis is done for two different models created in ABAQUS/CAE namely Model a: Beam column joint without the CFRP strengthening. Model B: Beam-Column joint strengthened with CFRP sheets. Both the members are subjected to the same seismic loading. Model A is subjected to a seismic load of max amplitude of 30Mpa and Model B is subjected to two different seismic loads, one with a maximum amplitude of 30 MPa and the second one with a maximum amplitude of 60 MPa and later on the behavior of beam-column joints of both models is observed in terms of displacement and stress.

Keywords: beam-column joint, seismic analysis, strengthened beamcolumn joint

1. INTRODUCTION In RC building, a portion of columns that are common to beams at their intersection is called beam-column joints. The complete knowledge of it is of utmost importance if the building lies in the high seismic zone. Over the past some decades, intensive research work has been reported on reinforced concrete (RCC) beam-column joint. Improper design of reinforced concrete structures gives rise to critical issues involving technical as well as social aspects because these structures are designed for gravity loads only, not for lateral loads. During an earthquake, columns having a minimum cross-sectional area and longitudinal reinforcement will not be able to meet the requirement of shear demand and flexure. Secondly, construction of a strong beam-weak column under seismic loads may lead to the formation of local hinges in the column. This associated failure mode is characterized by catastrophic and brittle structural failure. In interior joints where rebar is improperly anchored, bond failure in longitudinal reinforcement has also been observed. The strength of reinforced concrete beam-column joint plays a significant role in the performance of the RC frame structure especially for the cases when subjected to seismic loads. Inadequately detailed reinforced

3-D Simulation Studies on Strengthening of Beam-Column Joint 167 Concrete beam-column joints, especially exterior joints will be the most critical element as they may fail due to excess amount of shear stresses. Some researchers reported the work related to the strengthening of the beam-column joint. In most of the cases, ferrocement composites were used to strengthen the joint which led to the improvement of the joint in terms of stiffness, energy dissipation and shear stress (Li et al., 2015). Use of lightweight concrete was also done and tested, which led to joint getting light weighted but it was useful for a certain limit of intensity of load (Decker et al., 2015). Analysis of precast pinned beam column joint under cyclic loading with the help of new expression (formula) was carried out where the expressions were limited by geometrical configurations (Georgia et al., 2017). An analytical study on a new moment resisting connection of beam to the precast concrete column was carried out using ABAQUS. The performance of that proposed connection was found similar to that of monolithic connection (Bahrami.S et al., 2017). Various researches were still being done in order to increase the efficiency of the joint. In the present chapter, a study has been carried out on the reinforced beam column joint analytically using ABAQUS (v.6.14). Following two models are modelled for the analysis, and the comparison between the results has been reported.

2. MATERIAL PROPERTIES The metal properties used in the simulation are as follows (Naveeena.N and Ranjitham.M, 2016). Longitudinal Rebar and stirrups used in the column and beam are circular in profile. The materials, as well as model properties, are given below in Table 1 and 2. During analysis, concrete damage plasticity model is introduced to predict the exact behavior of the structure. From Figure 1 and 2 the location and orientation of the CFRP sheet put on the joint can be determined. The interaction between beam and column has been defined as a tie connection in ABAQUS, and the connection

168

Khan Mohammad Firoz, Garg Aman and H. D. Chalak

between the CFRP sheet and beam-column joint in Model B is also defined as tie connection. The interaction between reinforcement and concrete is defined as an ideal, i.e., friction between the reinforcement and concrete is infinite. Table 1. Material properties Material

Modulus of elasticity (MPa)

Concrete Steel

22360 200000

CFRP sheet

(Ex) – 2.3x105 (Ey) – 1.79x105 (Ez) – 1.79x105 (Gxy) – 1.179x105 (Gxz) – 1.179x105 (Gyz) – 6.88x105

Poisson Ratio 0.15 0.3

Density (tons/mm3) 2.2 E-9 7.85E-9

(Vxy) – 0.22 (Vxz) – 0.22 (Vyz) – 0.30

1.6E-9

Strength (MPa) 20 (28 days) 415 (Yield Strength) 3500 (Tensile strength)

Table 2. Model properties Element

Length (mm)

Beam Column

600 800

Figure 1. Model A.

Cross section (mm × mm) 200x150 200x150

Main reinforcement

Shear stirrups

8 bars of 12 mm φ 8 bars of 12 mm φ

Seven stirrups of 8 mm φ 6 stirrups of 8 mm φ

Figure 2. Model B.

3-D Simulation Studies on Strengthening of Beam-Column Joint 169 Following finite elements are used for modelling: 1. C3D8R (explicit 8-node linear brick, reduced integration, enhanced hourglass control): For beam, column. 2. S3: (A 3-node triangular general-purpose shell, finite membrane strains) for CFRP sheet 3. T3D2 (Explicit 2-node linear 3-D truss): for longitudinal reinforcement and shear stirrup.

3. MODEL VALIDATION The present model is validated with the results obtained by N. Naveeena.N et al. (2016). Two different cyclic loads of 30 kN and 60kN have been applied at the tip of the beam of unstrengthened and strengthened beam column joint respectively. Strengthening of the joint is done by the wrapping of CFRP sheet up to 300mm on each side of the joint region. The results obtained are presented in the form of table and graphs. From the Figure 3 and 4, we can see that the stresses in the vicinity of beam-column joint of Model B are comparatively less as compared to the stresses in model A. The values by which the stresses differ are presented in table 3 below. It is also noticed that the stresses generated in the corresponding beam and column are comparatively less in model B.

Figure 3. Stress variation (Model A).

170

Khan Mohammad Firoz, Garg Aman and H. D. Chalak

Figure 4. Stress variation when subjected to 60 MPa seismic loads (Model B)

Figure 5. Displacement variation (Model A).

Figure 6. Displacement Variation when subjected to 60 MPa seismic (Model B).

3-D Simulation Studies on Strengthening of Beam-Column Joint 171 Table 3. Validation of Results Beam-Column joint type

Cyclic Load (kN)

Shear Stress ANSYS ABAQUS (Naveeena.N and Ranjitham.M, 2016)

30

Deflection ANSYS ABAQUS (Naveeena.N and Ranjitham.M, 2016) 6.2073mm 8.663mm

Unstrengthened joint (Model-A) strengthened joint with CFRP wrapping up to 300mm on each side of the joint region (Model-B)

60

5.9673mm

19.926 MPa

6.169mm

23.225 MPa

24.74 MPa 25.99 MPa

From Figure 5 and 6, we can easily compare the displacements occurring at the vicinity of beam-column joint of both the Model A and Model B. The displacement occurring in Model B is clearly very less as compared to that of Model A. Given in Table 3 are the values of both stresses and displacement of Model A and Model B.

4. CONVERGENCE STUDY Convergence study has been carried out on unstrengthened reinforced concrete beam-column joint subjected to 30KN cyclic loading at the tip of the beam. Here the graph is plotted between the size of mesh and deformation occur in the joint. Mesh size is on the x-axis and deflection on the y-axis. The graph clearly shows that on converging the size of mesh, the value of deflection is getting more refined and accurate. However, on increasing the size of the mesh, the deflection value is decreasing, and after 50*50 mesh size, it is becoming constant. In the further model, the mesh size of 50*50 is taken.

172

Khan Mohammad Firoz, Garg Aman and H. D. Chalak

Figure 7. Graph showing convergence of deflection with mesh size.

5. RESULTS AND DISCUSSIONS Comparative results are shown as follows in the form of table and graphs. Figure 4 shows the variation of deflection and Figure 5 shows the variation of shear stress with loading for the unstrengthened and strengthened model. From the Table 4, it is clear that Model B outperforms Model a regarding deflection and stress generated. Given below is the graph comparison of the analysis of both models. We can see from the graphs plotted; Model B can give better results than Model A even at the load double to that in the case of Model A.

Figure 8. Variation of deflection with load in different model.

3-D Simulation Studies on Strengthening of Beam-Column Joint 173

Figure 9. Variation of Shear Stress with Loading for Different Model.

Table 4. Comparison of Unstrengthened and Strengthened Joint Parameter Deflection Shear stress

Model-A at 30kN 8.663mm 24.74 N/mm2

Model-B at 30kN 2.768mm 13.57 N/mm2

Model-B at 60kN 5.941mm 22.74 N/mm2

CONCLUSION The following observation and conclusions can be drawn based on the analytical results of the study. 

 

By doing CFRP strengthening on the beam-column joint, the joint becomes more effective is withstanding the stresses and deflections. This clearly states that strengthening by CFRP sheet can be considered as one definite way of improving the working of joint. The load carrying capacity of strengthened Model-B is 45.15% more than the unstrengthened Model-A. Deflection in Model B is reduced by 68.04% when compared to the Model A.

174

Khan Mohammad Firoz, Garg Aman and H. D. Chalak 

 

On comparing the results, we can confirm that the load carrying capacity of the strengthened Model-B is comparatively more than that of the unstrengthened Model A. CFRP strengthened Model-B shows better ductility and strength as compared to unstrengthened Model-A. This work can be further carried out by strengthening these joint of a building frame, which can further lead to a comparison of stresses generated in the corresponding beams and columns meeting at a joint

REFERENCES Bahrami. S, Madhkhan M, Shirmohammadi. F, Nazemi. N, (2017), Behavior of two new moments resisting precast beam to column connections subjected to lateral loading, Eng. Struct., volume 132:808821. Decker. CL, Issa MA, Meyer KF, (2015), Seismic investigation of interior reinforced concrete sand-lightweight concrete beam-column joints, ACI Struct J. volume 112(3):287-297. Georgia DK, Yasin. MF, Loannis LP, Ioannis NP, Spyros GT, (2017), Numerical investigation of the resistance of precast RC pinned beamto-column joint under shear loading, Earthq. Eng. Struct. Dyn. Volume 46(9):1511-1529. Li. B, Lam. ESS, Wu B, Wang YY, (2015), the Seismic behavior of reinforced concrete exterior beam-column joints strengthened by ferrocement composites, Earthq. Struct. Volume 9(1):233-256. Naveeena. N, Ranjitham.M M., Numerical Study on Retrofitting of BeamColumn Joint Strengthened with CFRP, (2016), Int. Res. J. Eng. Tech., volume 03(01):914-920.

In: Mechanical Design, Materials … ISBN: 978-1-53614-791-9 Editor: Sandip A. Kale © 2019 Nova Science Publishers, Inc.

Chapter 7

NANOMATERIALS, CERAMIC BULK AND BIOCERAMICS: SYNTHESIS, PROPERTIES AND APPLICATIONS H. H. M. Darweesh Department of Refractories, Ceramics and Building Materials, National Research Centre, Cairo, Egypt

ABSTRACT The overwhelming demand and interest of nanotechnology is one of the most exciting disciplines and it almost incorporates into all applied science branches as: physics, chemistry, materials sciences, biology, agriculture, medicine, tissue engineering, bones caffolds, dentists, cement and concrete, and all types of building materials particularly bricks, ceramics and nanoceramics or in general advanced ceramics industries, biomaterials, and many others. The author interests with using the nanomaterials or accurately nanoparticles to prepare the ceramic batches containing ultra fine and nano raw materials as nano-SiO2 and nanoAl2O3 to indicate the importance of nano particles, to improve the 

Corresponding Author Email: [email protected].

176

H. H. M. Darweesh physical, chemical, mechanical properties and thermal properties as: thermal shock, thermal expansion, firing resistance, and also durability of the resulting bioproducts against aggressive environments. The bioceramics can help people who are suffering from the deficiency in their body bones as a result of accidents. These bioceramic products resulting from biomaterials can compensate people for their lost or broken bones, as arms, legs, fingers or even toes, and hence they feel happy.

Keywords: nanomaterials, nanoparticles, bioceramics, firing, sintering, expansion, shrinkage

ABBREVIATIONS C: CaO S: SiO2 F: Fe2O3 H: H2O C3S: 3CaO.SiO2: C2S: 2CaO.SiO2: C3A: 3CaO. Al2O3: C4AF: 4CaO. Al2O3. Fe2O3: C2AS: 2CaO. Al2O3. SiO2 CAS2:CaO. Al2O3. 2 SiO2 β-CS: CaO. SiO2 2C2S.CaCO3: 4CaO. 2 SiO2. 2 CaCO3 Al3 (Mg, Fe)2.(AS. SiO5. O13): 2MgO. 2Al2O3. 5SiO2 A3S2: 3Al2O3. 2SiO2 3 MgO. 4 SiO2. H2O W.A. B.D. A.P. B.S. T.F.

A: Al2O3 CH: Ca (OH)2 Tricalcium silicate Dicalcium silicate Tricalcium aluminate tetracalcium aluminate ferrite Dicalcium aluminate silicate (Gehlenite) calcium aluminate disilicate (Anorthite) Calcium silicate (Wollastonite) (Spurrite) (Cordierite) (Mullite) montmorillonite water absorption bulk density apparent porosity bending strength thermal features

Nanomaterials, Ceramic Bulk and Bioceramics

177

1. NANOMATERIALS 1.1. Introduction Recently, nanomaterials are the backbone and basis of nanotrchnology. Nanoscience is the most recent branch of the sciences, including vast or nearly all specialties of researches. It is growing and increasing continuously all over the world in the last two decades. It has the potential to occur a revolution in the ways by which the materials and/or units could be produced successfully. This already has a significant commercial effect that will increase in the future. Nanomaterials are of a great interest due to the fact that unique optical, rheological and electromagnetic properties, and many others could be arised. These features are of a great importance in nano-or-bioceramics, medicine, bones, etc. (Figure 1).

Figure 1. A carbon nanotube as a nanomaterial.

Nanotechnology differs from a field to a field and from a place to a place and it is essentially used as a “catch all” to describe very small things (Nalwa, 2000, Nalwa, 2011, Shackelford, 2004). Nanotechnology is commonly defined as the understanding, control, and restructuring of matters on the order of nanometers, i.e., less than 100 nm to make materials with essentially new characteristics and functions (Poole, 2003, Drexler et al. 1991). Nanotechnology comprises two main approaches: the

178

H. H. M. Darweesh

“top-down” approach, in which larger structures are minimized in size to the nanoscale in a condition that it must maintain their original characteristics without an atomic-level control, e.g., miniaturization in the domain of electronics or deconstructed from larger structures into their smaller, composite parts and the “bottom-up” approach that it can also be called “molecular nanotechnology” or “molecular manufacturing,” introduced by Drexler et al. (Drexler et al., 2003), in which materials are engineered from atoms or moleculars through a process of assembly or self-assembly(Figure 2). Most contemporary technologies depend on the “top-down” approach. Molecular nanotechnology holds a great promise for break throughs in materials and manufacturing, electronics, medicine and healthcare, energy, biotechnology, information technology, and national security (Askeland and Phule, 2006, Koch, 2002, Smith and Hashemi, 2005, Ashby and Jones, 2006, Sobole and Ferrada-Gutiérrez, 2005, Darweesh, 1992, Tseng and Nalwa, 2009, Ashby, 2005).

Figure 2. The “Top-Down” and “Bottom-Up” approaches in nanotechnology.

Nanomaterials, Ceramic Bulk and Bioceramics

179

Recently, the idea of creating the situations under which atoms or molecules could be self assembly into beneficial structures was driven by the reduction of its energy. The most important advantage of the selfassembly is that the system concentrates to a definite configuration without the need to a further control. Typically, the aggregates formed by the selfassembly seemed to be bonded by relatively weak bonds with binding energies only a few times higher than the thermal energy per atom. The most important example of the self-assembly is the production of carbon nanotubes. These nanostructures are composed of carbon atoms that assemble into cylinders of about 1.4 nm in diameter (Figure 3). The use of carbon nanotubes to make simple gears evolved by bonding ligands onto the external surfaces of carbon nanotubes is to produce “gear teeth” (Figure 4).

Figure 3. A single-wall carbon nanotube (as wrapped sheets of graphene) of 1.4 nm diameter.

Figure 4. Nanogears made of carbon nanotubes with added teeth.

180

H. H. M. Darweesh

Figure 5. Self assembly of nanoclusters.

The efficiency of these gears depends on placing those gears teeth just right in atomically precise positions. Also, the self assembling molecules form micelles are very important. Micelles are aggregates of amphiphilic molecules. Molecules with one end are soluble in water and the other end that rejects it. These aggregates form spontaneously at a size that depends on the concentration of the amphiphilic molecules in solution. The center of the micelles acts as a chamber for chemical reactions (Darweesh, 1992; Tseng and Nalwa, 2009; Ashby, 2005). Thus, it dictates the size of the nanoparticles created (Figure 5).

1.2. Ultrafine Particles Ultrafine particles are those particles that are having a grain size distribution of nanometers (10-9 m). These are considered as nanostructured materials (Nalwa, 2000; Drexler, 1991; Tseng and Nalwa, 2009), i.e., any material comprised grains as clusters below 100 nm and/or layers of the same dimensions had been seen as a nanostructure unit (Nalwa, 2000; Poole and Owens, 2003; Ashby, 2005). The importance of those materials is due to its too small size as: particles, grains or phases. Its high surfaceto-volume ratio, these materials shows unique mechanical, optical, electrical and magnetic characteristics (Poole and Owens, 2003; Nalwa, 2000; Drexler, 1991; Ashby et al., 2009; Tseng and Nalwa, 2009). The

Nanomaterials, Ceramic Bulk and Bioceramics

181

features of nanostructured materials mainly depend on the following microstructural items: i. ii. iii. iv.

Grain size distribution (< 100 nm). Chemical analysis and its ingredients. Existence of interfaces. Reactions between constituents dormant.

These features evidently and can easily determine the marvelous characteristics of nanostructured materials.

1.3. Nanomaterials Nanomaterial: It distinguishes between the external dimensions and its internal structure of the whole object. Thereby, it can be >100 nm in all three dimensions that are still being considered a nanomaterial if it has structural features within the nanoscale range, i.e., between 1-100 nm (Askeland and Phule, 2006; Koch, 2002; Smith and Hashemi, 2005; Ashby and Jones, 2006; Sobolev and Ferrada-Gutiérrez, 2005; Darweesh, 1992; Tseng and Nalwa, 2009; Ashby, 2005). Sometimes, nanomaterials occur naturally. Engineered nanomaterials are very important. To obtain useful advantages of extra fineness of particles, they are designed to be applied in several commercial scopes as sunscreens, sporting tools, pain-resistant clothes, cells, ceramics, bones, dentists or teeth, and others (Nalwa, 2000; Ashby, 2009; Ashby, 2005). The ratio of a fineness-to-a volume of nanomaterials is > its conventional forms, which in turn, can lead to a greater chemical reactivity that affected positively on their strength. At the nanoscale, quantum effects can become much more important to identify many of materials properties leading to optical, electrical and magnetic behaviors. Now, nanomaterials can be used commercially in stain-resistant and wrinkle-free textiles, cosmetics, sunscreens, electronics, paints, varnishes, windows, sporting machines, bicycles and auto cars.

182

H. H. M. Darweesh

Nanoscale TiO2 and nano scale SiO2 are being used as a filler in several products as bioceramics as: bones, teeth or dental fillings (Nalwa, 2000; Poole and Owens, 2003; Sobolev and Ferrada-Gutiérrez, 2005; Tseng and Nalwa, 2009; Ashby et al., 2009; Ashby et al., 2007; Callister, 2007).

1.4. History and Advances of Nanomaterials Nanomaterials started at once after the big bang when nanostructures were formed in the early meteorites. Later, the nature evolved a lot of nanostructures as: seashells and skeletons. Nanoscale smoke particles were formed during the use of fire by early humans. However, the scientific history of nanomaterials started much later. Nanostructured catalysts have also been studied for over the last seventy years. By the early 1940’s, the precipitated waste of silica fume nanoparticles were being manufactured and could be sold as substitutes for ultrafine carbon black for rubber reinforcements, and also for cement industry. Nanosized amorphous SiO2 could be applied in many consumer products as: automobile tires, optical fibers and catalyst supports. During the period of 1960s to 1970’s, metallic nanopowders for magnetic recording tapes were developed and improved. In 1976, nanocrystals produced now by the most popular inert-gas evaporation technique (Poole and Owens, 2003; Drexler et al., 1991; Sobolev and Ferrada-Gutiérrez, 2005; Tseng and Nalwa, 2009). Today, the menopause engineering has expanded very rapidly to develop a number of organic and/or inorganic structural and functional materials. This permits to impact mechanical, catalytic, electric, magnetic, optical and electronic functions. The production of nanophase or clusterassembled materials is often based on the devoloping of small clusters separately, which in turn are fused into a bulk-like material or on their embedding into compact liquid or solid matrix materials, i.e., Si-nanophase which differs from normal Si in physical and electronic properties that could be applied to macroscopic semiconductors to develop new devices. When a normal glass is adapted with colloids and/or semiconductors, it

Nanomaterials, Ceramic Bulk and Bioceramics

183

becomes a high performance optical medium (Sobolev, K. and FerradaGutiérrez, 2005; Darweesh, 1992; Tseng and Nalwa, 2009; Ashby, 2005; Ashby et al., 2009; Ashby et al., 2007; Callister, 2007).

1.5. Classification of Nanomaterials Nanomaterials have very small size particles which are having 1-3 dimensions ≤ 100 nm or more. Nanomaterials can be nanoscales in one dimension, e.g., surface films, two dimensions, e.g., fibers and three dimensions, e.g., particles. They often exist in single, fused, aggregated and/or agglomerated forms in spherical, tubular, and irregular shapes. Conventional kinds of nanomaterials comprise nanotubes, dendrimers and fullerenes. Nanomaterials have displayed different physical and chemical characteristics from normal chemicals, i.e., nanosilver, carbon nanotube, fullerene, photocatalyst, nanocarbon, nano-SiO2, nano-Fe2O3 and nanoAl2O3 (Koch, 2002; Sobolev and Ferrada-Gutiérrez, 2005; Tseng and Nalwa, 2009; Ashby, 2005; Ashby et al., 2009; Ashby et al., 2007; Callister, 2007).

Figure 6. I: (0D: Spheres and clusters), II: (1D: Nanofibers, wires and rods), III: (2D: Films, plates and network), V: (3D: Nanomaterials).

Nanomaterials are materials that characterized by an ultra fine grain size (< 50 nm) or by a dimension limited to 50 nm. Nanomaterials can be created with various dimensions. Nanostructured materials could be

184

H. H. M. Darweesh

classified into:- Zero dimension (0D) as: clusters and/or cluster assemblies and filaments; one dimension (1D) as: nanofibers, wires, multi-layers and rods; two dimensions (2D) as: ultrafine-grained over-layers or buried layers, films, plates, and networks and three dimensions (3D) nanostructures as: nanomaterials, nanophase materials consisting of equiaxed nanometer sized grains (Figure 6).

1.6. Importance of Nanomaterials In recent years, nanomaterials have taken an overwelming interest due to the following reasons: 



 



Nanoceramics are flexible at high firing temperatures if compared with those containing coarse size particles (Nalwa, 2000; Darweesh, 1992; Sobolev and Ferrada-Gutiérrez, 2005; Tseng and Nalwa, 2009; Ashby, 2005; Ashby et al., 2009; Ashby et al., 2007; Callister, 2007). Structured nanosemiconductors did not display different lineal optical characteristics. Always, they are used as solar cells window layers. Nanosized minerals ashes often use to produce gas-tight materials as: dense parts of porous linings. The very small or nanoparticles have special atomic structures with distict electronic states, which give rise to special properties. Nanomagnetic composites are used for mechanical strength devolve as: magetoliquids, a high density storage knowledge and magneto refrigeration. Structured nanomineral clusters and colloids of mono- or crystalized mineral compositions have a particular action in catalytic applications using as precursors for recent hetero-geneous catalyses or cortex-catalyses. They tend to give essential advantages regarding to activation, selectivity and lifetime in

Nanomaterials, Ceramic Bulk and Bioceramics





185

chemical or electrocatalylic transformations or fuel cells (Nalwa, 2000; Tseng and Nalwa, 2009; Ashby, 2005; Ashby, 2009). Structured nanomineral-oxide thin films are having a continual attention for inquiring gas sensors as: NO3, CO, CO2, CH4 and some aromatic hydrocarbons. Structured nanomineral-oxide, MnO2 could be applied in re-chargeable batteries for cars or consumer goods. Crystalline nono-SiO2 sheets for highly transparent contacts in solar cells and also structured nano-TiO2 porous films for its high transmission and significant fineness increasing going to a strong absorption in dyesensitysed solar cells (Poole and Owens, 2003; Drexler et al., 1991; Askeland and Phule, 2006 Koch, 2002; Tseng and Nalwa, 2009). Composites of polymers with high levels of inorganics with a high dielectric constant are interesting materials for structured photonic band gaps.

A variety of size-related effects can be incorporated by controlling the sizes of ingredients (Nalwa, 2000; Sobolev and Ferrada-Gutiérrez, 2005; Ashby, 2005; Ashby et al., 2009; Ashby et al., 2007; Callister, 2007), e.g., the mechanical strength of nanostructured minerals and nanoceramics are get be better than that of the conventional materials which is mainly due to the ultrafine microstructured (Nalwa, 2000; Sobolev and FerradaGutiérrez, 2005; Ashby, 2005; Ashby et al., 2009; Ashby et al., 2007; Callister, 2007). Nanostructured materials as the manufacturing of devices with large magetoresistance enfluences are of magnetic applications (Nalwa, 2000). Furthermore, nanostructured minerals and nanoceramics are tended to be candidiates for new catalytic applications (Nalwa, 2000; Poole and Owens, 2003). The growth of semiconductors and nanoclusters is on the circle of intense research works particles (Nalwa, 2000; Sobolev and FerradaGutiérrez, 2005). It is important because it is leading to electrical, optical, optoelectronic, magnetic and magneto-optical properties, e.g., quantum dots can be modified to emit and absorb a desired wavelength of light by varying the particle diameters (Nalwa, 2000).

186

H. H. M. Darweesh

2. CERAMICS The modern term ceramic was derived from an ancient Greek word which referred to a burning or firing process. More recently, a ceramic product was defined as an article made from clay with or without the addition of either materials, shaped in the plastic state, dried and then fired to give it the required strength and durability. Also, it may be defined as one composed of inorganic, but non-metallic material (Jackson, 1969; Kingery et al., 1975; Rayan and Radford, 1987). On this basis, ceramic products are usually subdivided into four sections as follows:1) Structural products as bricks, roofing tiles, pipes, wall and floor tiles. 2) Refractories, i.e., ceramics used mainly for their resistance to high temperatures as kiln linings, steel and glass plants, and many other products used at elevated temperatures. 3) Electrical and special ceramics: Modern ceramic materials, some of which show exceptional qualities of refractoriness, hardness, strength, chemical stability and specific electrical and inorganic properties. 4) Whitewares as: Table and decorative wares, wall tiles and sanitary ware products. Nowadays, ceramics include a vast range of products, many of which may be contain no clay whatsoever, so that a simple and definite definition is impossible. Firing is still an essential feature in the manufacture of the vast majority of ceramic articles, but there are few exceptions where the product is classified as ceramic even in its nature, i.e., without firing (kingery et al., 1975; Rayan and Radford, 1987). In the developing countries, the average standard of living has been improved to a great extent throughout the last two decades and is still improving continuously with time. In Egypt, the policy of construction has been greatly increased in order to face the overpopulation problem. So, there is an increasing

Nanomaterials, Ceramic Bulk and Bioceramics

187

demand for ceramic tiles. Accordingly, new ceramic producing factories were built and some others are still under construction so as to meet the overwhelming demand on ceramic tiles. Though the world production of ceramic floor and wall tiles is approximately 100 million square meter/year, it cannot reply the world needs (Drews, 1983; Rayan, 1978). Nanoceramics are composed of raw nanoparticles which can be classified as inorganic, heat-resistant, nonmetallic solids compounds. In the last decade, researchers have shown that nanoceramics have a lot of functions than known materials (Khalil, 2012). Nanoceramics were found out during early 1980s. It is fabricated by sol-gel techniques, i.e., mixing raw nanomaterials in a gel solution to produce nanoparticles. Recently, they involved by sintering through a pressure and a heat due to many advances in fields of electronics (Njindam et al.,2018; Martín-Márquez et al., 2008; Nuttawat et al.,2013; Kim and Hwang, 2016; Ke et al., 2016). Increase the various solid wastes in the countries particularly those do not resolve easily in nature by weathering for long periods as glass bottle wastes that created ground pollution. The recycling of these wastes is an important environmental and economical solution. The glass is composed of high amounts of SiO2 (70–74) % and a reasonable amount of alkaline earth oxides: Na2O (12–16) %, CaO (5–11) % and MgO (1–3) % (Njindam et al., 2018; Martín-Márquez et al., 2008). In Egypt, there are huge quantities of waste glass bottles of several shapes and sizes. This evidently causes an environmental problem and acts as a heavy burden on the environment. Therefore, it is very necessary to solve this serious problem through scientific researches that could use these priceless waste materials in traditional ceramics like as the wall and/or floor tiles, table wares or porcelain stoneware tiles. These may be used for out and/or indoor applications. The inert component glass waste can be used as a fluxing agent in ceramic industry. Clay is a product which is coming from the decomposition of granite rocks. It is essentially composed of Al2O3 and SiO2 which represent the major constituents of clay (Nuttawat et al., 2013; Kim and Hwang, 2016; Ke et al., 2016). Clay is known to provide the plasticity of the whole mixture. Granite often decomposes into feldspar that is composed of

188

H. H. M. Darweesh

Al2O3, SiO2, and a flux of an alkaline nature e.g., Na, K, Li and Ca. The fluxing agents as feldspar, talc, pyrophyllite, etc. could produce the liquid or glassy phase during sintering while SiO2 favours the dimension control of the product during firing (Nuttawat et al., 2013; Kim and Hwang, 2016; Ke et al., 2016). The nanoceramics have unique properties due to its smaller size particles and molecular structure. To construct these nanostructures using a “three-step process” to develop these complex structures, the nanoceramic Greer is currently used Al2O3, or aluminum oxide, and its maximum compression is about 1 micron from a thickness of 50 nanometers. After its compression, it can revert to its original shape without any damage on the structure and was coated with Al2O3, different molecules could yield a different result (Sonjida, 2011; Vorrada, 2009; Darweesh, 2015).

2.1. Synthesis For making a nanoceramic, it begins first with the sol-gel process. It involves a chemical solution, made of nanoparticles in the phase, usually a gel or polymer, made of molecules immersed in a solvent. The sol-gel is then mixed to produce an oxide material which is generally a type of ceramic. The rest of the excess liquid solvent is removed with a drying method by evaporation. The desired particles are then heated through densification to return a solid product (Jackson, 1969; Darweesh, 2016). This method could also be applied to produce a nanocomposite by heating the gel on a thin film to form a layer of nanoceramic on top of the film (Sridhar, 1996).

2.2. Sintering Sintering is a process of consolidating nanoceramic powders using high temperatures to press the powder into a solid material. High temperature sintering results in a rough material that deteriorates the

Nanomaterials, Ceramic Bulk and Bioceramics

189

properties of ceramics and requires a lot of time to obtain an end product. Microwave sintering has been developed to overcome the time and energy problems of the previous technique. In microwave sintering, radiation is produced from a magnetron to vibrate the powder and thus heating them. This method allows for heat to be instantly transferred across the entire volume of material instead of a graduate temperature gradient (Jackson, 1969; Darweesh, 2016). They first start to place the nanopowder in an insulation box that is composed of low insulation boards. They place it in an insulation box in order to increase temperature because nanoceramics don't absorb microwave energy well in the room temperature. Inside the boxes are suspectors that absorb microwaves at room temperature to act as an initial heat source and initialize the sintering process. When the box is placed in the microwave furnace, the microwaves heat up the suspectors, to about 600 °C. Then, the nanoceramics start to absorb the microwaves and bind the nanoparticles together.

2.3. History In the early 1980s, Roy Komareni developed the first way to synthesize nanoparticles, specifically nanoceramics (Julie, 2002). He used a process called sol-gel and enabled researchers to test the properties of nanoceramics. This process was later replaced by sintering in the early 2000s and continued to advance microwave sintering. (Nissan, 2014). Seashells are very tough and yet are very similar to ceramics. They discovered that the durability of seashells come from their microarchitecture (Kimm, 2013). In 2012, California Institute of Technology succeeded in replicating the skeletal structure of sea creatures like sea sponge using ceramics (Sridhar, 1996). They developed the nanoarchitecture that is known as nanotruss. It can be used for materials other than ceramics to take on the properties of that materials, while retaining its durable build. Their largest scale of nanoceramic is a 1mm cube and they are still currently working on.

190

H. H. M. Darweesh

Nanoceramics have a large variety of functions and can be applied in many fields including energy supply, and storage, communication, transportation systems, construction, electrical and medical technologies. There are also physical applications such as nanoceramics in nanotruss architecture. Nanoceramic holds very flexible and durable properties making the material very light and yet be as strong as current building materials like concrete or steel. Medical technology has been using nanoceramics for bone repair and developing materials that can enter the body without conflict (Singer and Singer, 1971). Ceramic floor and wall tiles are building materials which are designated for use as floor and wall coverings both indoors and outdoors regardless of shape or size. They undergo some processing such as milling, screening, blending and wetting. Tiles are shaped by pressing methods at room temperature (ASTM; Mostafa and Darweesh, 1992; Njindam et al., 2016). The possibility to utilize the glass waste (10-40 %) as a flux with clay to produce porcelain stoneware tiles was studied (Ashby et al., 2009). The firing shrinkage inceased with glass content only up to 30 %, but more than that the product started to melt and spread. It is increased also with firing temperature. The water absorption decreased with glass waste content up to 30%, while the bulk density increased. This reflected positively on the mechanical strength.

2.4. Classification of Ceramics The ceramic industries could be classified commonly into the following main categories owing to the similarities in service, manufacturing method and raw materials used (Ke et al, 2016; Sonjida et al., 2011):

2.4.1. Heavy Clay Wares Heavy clay wares or structural ceramics are the products used extensively by the building trades and in civil engineering projects. They include bricks of all types, commons, facing and engineering bricks,

Nanomaterials, Ceramic Bulk and Bioceramics

191

hollow or perforated bricks, clay roofing tiles, draining and sewer pipes. The used raw materials vary widely from easily worked highly plastic clays to hard coal measure shales (Ke et al, 2016; Sonjida et al., 2011).

2.4.2. Refractories Refractory products are those used in high temperature applications, in kilns and furnaces of all types, in steel works and in other industries, in glass works and other power stations. The raw materials used include fire clays for lower temperature applications and oxides such as alumina (Al2O3) and magnesia (MgO) for the higher temperature ones. The manufacturing methods include extrusion and moulding, whereas dry pressing techniques account for the great majority of the non-clay refractories (Njindam et al., 2018; Sonjida, 2011). 2.4.3. Special Ceramics Special or technical ceramics include many products which have little or no resemblance to the conventional ceramics because they are always made from pure synthetic raw materials to confirm technical specifications, for example: electronics, aeronautics, nuclear engineering and electrical power (ASTM, 1980; Mostafa and Darweesh, 1992). 2.4.4. Pottery (Whitewares) The term pottery or white wares includes normal domestic tablewares, wall tiles, sanitary wares and porcelain tills. The tableware also includes earth wares, bone china and porcelains, other types include stonewares and a number of proprietary brands. Ceramic products can be divided into those used at normal and high temperatures. These divisions may be subdivided into those products which have a porous body after firing and those with non-porous ones (Chen-Chi and Jackie, 1999; ASTM, 1980) as shown in Figure 7. Each product must possess their properties in use such as: beauty, mechanical strength, temperature resistance or any other property. The properties of the finished ceramic article will broadly depend on:

192 i. ii.

H. H. M. Darweesh Types, purity and relative amounts of materials from which it is made. Methods and conditions used in its manufacture.

But, the two cannot be divorced since the materials used will dictate to a considerable extent, the most suitable manufacturing processes which are available. The composition of the traditional ceramic units lies, in general in definite limits. They are controlled by some factors: 1-Shape of the ware which needed to the inclusion of a sufficient plastic material (clay). After shaping, the clay also holds the ware together during drying. 2-Avoiding of cracking during firing due to the incorporation of the non-clay materials which do not shrink during firing as: flint, quartz and tiles grog. 3-The quantity of fluxes so that a sufficient glassy material formed to hold it together when fired and so it does not warp (Njindam et al., 2018; Darweesh, 2015).

Figure 7. The Classification of Ceramic products.

Nanomaterials, Ceramic Bulk and Bioceramics

193

Figure 8. Triaxial diagram showing areas of commercial wares.

Now, the traditional bodies are essentially made up of clay, flint or quartz, and feldspar that can be represented by a tri-axial diagram as shown in Figure 8. The most natural method is by means of the batch composition. But, it was shown from the raw materials analysis that the traditional ones, namely clays and feldspars, are of available composition and properties (Darweesh, 2016; Kimm, 2014). Clay is an essential ingredient in the majority of white wares as well as ceramic tiles. It possesses a number of characteristics which the ceramic technologist must consider when choosing the most suitable clay for a particular application (Njindam et al., 2018; Martín-Márquez et al., 2008; Nuttawat et al., 2013; Kim et al., 2016; Ke et al., 2016; Sonjida et al., 2011).

2.5. Properties of Ceramics 2.5.1. Plasticity When a dry clay is mixed with water, there are two water contents which could be recognized easily and simply by working the mass of the

194

H. H. M. Darweesh

clay/water body in the hands. Above the higher value of water content, the mass becomes sticky and difficult to handle. Below the lower one, it tends to crack and crumble. Accordingly, between these two values (upper and lower plastic limits), the clay can usually be worked without any difficulty. Therefore, the plasticity can be expressed as workability which enables a material to be deformed continuously under a force exceeds certain maximum values. Plasticity enables a material to be changed in shape without rupturing by the application of an external force and to retain that shape when the force is removed or reduced below a certain value, but there is no a quantitative definition or units for plasticity (Njindam et al., 2018; Vorrada et al., 2009; ASTM, 1980). Many trials were carried out to find a relationship between the stress applied and the resultant measured strain. The stress may be considered as a force per unit area and the strain as the resulting deformation of the body. Basically, there are three types of stress which may be applied as tensile, compressive or shear forces. The property of a body to return to its original size and shape after having been stretched, compressed or deformed, is simply that stress is proportional to strain (Rayan and Radford, 1987) as follows,

K

Stress Strain

(1)

where, k is a constant and is called as “Modulus of elasticity.” When a body is subjected to a simple tension (or compression), K is known as “Young’s modulus” and the stress = force per unit area (F/A). Strain =Extension per unit length

S

dI I

(2)

Nanomaterials, Ceramic Bulk and Bioceramics

195

2.5.2. Factors Affecting Plasticity Fineness: By separating clay samples into a number of size fractions, Kalnin (kalnin, 1967; Kalnin, 1968) was able to relate measured plasticity indexes to fineness expressed as surface area. His results indicated that the relation was:

log S  AP  log B

(3)

Where, S is the surface area of the clay sample (expressed as cm2/1000 g), P is the measured plasticity index, A and B are constants. The plotting of log S against B gave straight line graphs with intercept on the log S axis giving the value of log B. In all samples, B was found to be 1.8 x 106. Thus for P to have any value, S related to be greater than 1.8 x 106 cm2/100g (Rayan and Radford, 1987; Drews, 1983; Mostafa and Darweesh, 1992). As the material becomes finer, not only does plasticity increase, but also the amount of water needed to produce optimum plasticity increases, and plasticity is exhibited over a wider range of water content. Figure 9 indicates this behavior for various size fractions of clay.

Figure 9. Plasticity index as a function of water content for five size fractions of clay sample.

196

H. H. M. Darweesh

2.5.3. Determination of Plasticity There are many test methods can be used for measuring plasticity, but its most are not accepted a definite or accurate test for plasticity. Tests which depended mainly on the moisture content are the optimum (Rayan and Radford, 1987; Chiang, 1997). So, the most used tests are: 2.5.3.1. Pfefferkorn Test A standard weight is released from a fixed height and falls on to a cylindrical sample of the material under test. The cylinder is deformed. The deformation ratio is defined as: Deformation ratio 

Original length of the cylinder Deformed length of the cylinder

(4)

Pfefferkorn number is calculated as the moisture content at which the deformation ratio lies in the range of 3:1. The higher the Pfefferkorn number is the more plastic material. The test is repeated several times and then the mean value is considered (Rayan and Radford, 1987; Drews, 1983; Rayan, 1978; Khalil, 2012). 2.5.3.2. Atterberg Plasticity Test Two moisture contents are measured corresponding to the plastic and liquid limits. The plastic limit is the moisture content below which the material stops to behave plastically and become crumbly. While the liquid limit is the moisture content above the material acts as a fluid. The material becomes as a paste if it is being in contact with water and placing it in a standard cup, a standard groove is cut in the sample and the cup is given a number of standard bumps to close the groove. The liquid limit can be defined as the moisture content at which about 25 bumps are needed to close the gap (Rayan and Radford, 1987; Drews, 1983; Rayan, 1978; Khalil, 2012). The plasticity index is the numerical difference between the moisture content of the liquid and plastic limits. This method is looking for the range of a moisture content over which the material can behave plastically,

Nanomaterials, Ceramic Bulk and Bioceramics

197

where the greater the range is the higher plasticity. Figure 10 shows the BCR compression plastometer by which the plasticity is measured. The motor moves the lower plate upwards at a constant speed and when the cam scale reads zero deformation, the upper plate just come into contact with the top of the sample. As the cam turns further and the lower plate rises, the sample is compressed between the upper and lower plates (Drews, 1983; Chiang, 1997).

Figure 10. A diagrammatic representation of the BCR compression plastometer.

Readings on the dial gauge caused by flexing of the beam (proportional to the stress causing deformation) are taken at 10 % and 50 % deformation as registered on the cam scale (proportional to the strain). The plasticity index P can be deduced by the following relation:

P  1.8 

Dial reading at 10 % deformation Dial reading at 50 % deformation

(5)

where, 1.8 is a factor used to permit for a change in the cross-sectional area of the cylinder during deformation and is being equal to:

Cross-sectional area of sample at 50 % deformation Cross-sectional area of sample at 10 % deformation

198

H. H. M. Darweesh

Care must be taken to ensure that the samples are of the correct length and have parallel ends. Deformation of the sample cylinder must therefore be avoided when unmoulding it from metal die. Often sex samples are tested and the mean value is considered. Shrinkage of the shaped ceramic articles takes place during drying and firing stages of manufacture. The extent of these contraction is very important where not only will the size of the finished product depend on their values, but also if they are too large, or if they are occurring too rapidly, cracking or distortion of the article is likely. So, the ceramist must recognize the wet-to-dry, dry-to-fired and wet-to-fired processes. Each may be expressed on either a linear or a volume basis depending on the method of measuring (Rayan and Radford, 1987; Drews, 1983; Rayan, 1978; Khalil, 2012) as the wet-to-Dry contraction. In most pottery bodies, the clay content is primarily responsible to determine the extent of the contraction though the ratio, size and size distribution, particle shape of the non-clay component also play a vital role. The fineness and plate-like character of the clay crystallites lead not only to the desirable properties of the high plasticity and dry strength, but also to high drying contraction. It is well known that a small drying shrinkage is often desired in pottery products, where it permits the formed product to shrink away from the plaster mould and facilitates its removal from the mould. However, the excess shrinkage, specially anisotropic shrinkage, i.e., different shrinkage values in different directions due usually to a particle alignment that taking place during pugging, plasticmaking or slip casting, is responsible for a considerable rate of manufacture losses. High drying shrinkage is also resulting in a loss of dimensional accuracy and variations, which may give problems in decoration, e.g., a lithograph is applied to the circumference of a plate, a high dimensional accuracy and the moisture content of the body may be kept low as possible (Rayan, 1978; Khalil, 2012), ASTM, 1980). When a plastic body dries, water is removed from the surface and replaced by water from the interior. As water content is reduced, particles are drawn closer together. The particles eventually will touch each other and though water still exists in the voids between particles, no further shrinkage can take place on the removal of this water (Figure 11). If a

Nanomaterials, Ceramic Bulk and Bioceramics

199

piece of plastic clay or body lets to dry, values of the volume and the corresponding moisture content are given at intervals, a plot of volume as a function of moisture content is as shown in Figure 12. On drying, the shrinkage takes place as water is removed. During this stage, water still separate the particles, but the particles are moving closer together if water is removed. The point at which there is no further removal of water is known as the critical moisture content “CMC.” Its value often varies from clay to clay and also from a ceramic body to another. This depends essentially on clay to non-plastic ratio, particle size, distribution and particle shape. So, this value is very important where drying from water content above it will result in a shrinkage involving a risk of distortion or cracking, and if below it, no shrinkage is involved.

Figure 11. Shrinkage of a plastic body during drying.

Figure 12. The volume as a function of moisture content for a plastic body during drying.

200

H. H. M. Darweesh

At moisture content higher than the CMC, the rate at which water is lost will be constant under constant conditions. Since, drying involves only evaporation of water from the surfaces of samples which is replaced by water flowing from the interior of the surface. This is known as “Constant rate period” (Rayan and Radford, 1987; ASTM, 1980; Konta, 1972). The interval between the CMC and the dryness is known as “Falling rate period” (Figure 13). Clay particles are a plate-like in nature and when mixed with water to form a paste or slip, the particles are free to orientate themselves when a force is applied. This orientation occurs to present the least resistance to the applied force, e.g., in a pug, the particles will align themselves in the direction of the force causing extrusion. The extruded column of the clay will contain particles oriented so that their major axes are parallel to the length of the column (Figure 14). Similar orientations occur during plastic making and slip casting processes.

Figure 13. A plot of water removal rate as a function of moisture content for a plastic body during drying.

Figure 14. Preferable orientation of the produced particles resulting from anisotropic drying shrinkage.

Nanomaterials, Ceramic Bulk and Bioceramics

201

2.5.3.3. Test Methods a) Clays About 0ne kg of the clay is mixed with water to give a slurry which is sieved through 120 mesh (China clay) or 80 mesh sieve (plastic clay). The resulting suspension is poured on to a plaster of Paris batt and then let to dry to the plastic condition. When sufficiently dry, the clay is removed from the batt and thoroughly wedged by the hand till all air inclusions have been removed, and a freshly cut surface (cut with a cheese wire) has a homogeneous appearance. The test samples made for measuring its shrinkage may be either slabs or, a hand-moulded in a plaster of Paris moulds, or rods of a circular cross –section of about ½ in diameter. Rods are generally extruded with the help of wad box, and may be used for shrinkage and modulus of rupture. When the mechanism is operated, a plastic clay is forced through a tapered section and ultimately through a ½ in diameter orifice to produce rods. Soon after extrusion, the rods are placed in the grooves of a wooden pallet. The rods are then let to dry in air for 24 hours after which drying is completed in a drier at 110 ºC. The distance between the marks is then measured on each of the five rods and the mean dry length is obtained (kingery, 1975; (Rayan, 1978, Todor, 1972). The linear wet-to-dry shrinkage is reported on the wet basis, i.e., the % linear drying shrinkage (wet basis) is given by:

Linear drying shrinkage, %=

Wet length - Dry length (Average) 100 Wet length

The moisture content of the rods at the beginning of drying should also be quoted with these results. The rods should be then fired through the production kiln in such a position as to receive an average firing temperature, and a pyroscope (cone or ring) should be fired with the samples. Again, the distance between marks is measured on the rods, and the average fired length is calculated. It is suggested that the linear dry-tofired shrinkage is reported on the dry basis, i.e., the % linear firing shrinkage (dry basis) is given by:

202

H. H. M. Darweesh Linear frying shrinkage, %=

Dry length - Fired length (Average) 100 Dry length

The pyroscope reading should be reported together with the result of firing shrinkage (kingery, 1975; Khalil, 2012; ASTM, 1980; Mostafa and Darweesh, 1992). b) Ceramic Body Concerning the shrinkage measurements on the body, the production of a casting slip or a plastic body should be used as the starting material, where the same procedures as for clay are used. c) Volumetric Contraction The more accurate measurement of both drying and firing shrinkages could be obtained on the basis of volumes by using Doulton Densometer (Figure 15). Two weights are often required during drying of sample, the weight of the sample (w1) and the down thrust force (w2), which is equal to the up thrust of mercury + the sample weight when the sample is immersed in mercury and held immersed by the cradle.

Figure 15. Doulton Densometer for bulk density and volume shrinkage.

Nanomaterials, Ceramic Bulk and Bioceramics

203

An egg-shaped sample is moulded from the plastic clay or body using a special plastic mould. Then, it is weighed to an accuracy of 0.01 g on the top-pan balance. A sample beaker containing 1 g of mercury is placed on the balance pan, the cradle is lowered into the mercury using the coarse hand wheel adjustment. The bridge is then locked in a place with the locking screw. Using the micrometer adjustment, the needle pointer is adjusted to just touch the surface of the mercury. The balance is then turned back to zero. The cradle is then raised using the coarse adjustment and the sample placed on the surface of the mercury under the cradle. The cradle is lowered using the coarse adjustment till the sample is immersed. The bridge is locked and the pointer is once again adjusted to touch the surface of the mercury using the micrometer adjustment. The balance is read to 0.01 g. this weight represents w2 the up thrust due to mercury displaced + the weight of the sample (Rayan and Radford, 1987; Drews, 1983; Rayan, 1978; Khalil, 2012). Several readings of w1 and w2 at intervals are taken, while the sample is drying. The sample is fully dried in an oven and the measurements of w1 and w2 are determined. The moisture contents of the sample at different stages of drying can be obtained by subtracting the dry weight of the sample from the various values of w1 that obtained during the drying process. The volumes of the samples corresponding to these moisture contents can be calculated as w2/D, where D is the density of mercury at the room temperature. So, w2 is measured to an accuracy of 0.01 g, the volume is obtained to an accuracy of 0.001 cm3. The volume of a sample is then plotted against to a moisture content. d) Adsorbed Water Plasticity is associated with adsorbed water films around each clay particle or micelle. The electrical charges on clay particles enable them to adsorb polar micelles like water at their surfaces. These adsorbed films appear to act as lubricants, and are responsible for the plastic behavior. If water in a clay paste is replaced by a liquid of lower polarity, e.g., alcohol, the plasticity is much reduced since adsorption of the liquid is lessened.

204

H. H. M. Darweesh

Non-polar liquids as benzene give almost no plasticity when mixed with clays (Jackson, 1969; kingery et al., 1975; Rayan and Radford, 1987). e) Ionic Effects All clay particles carry a negative charge, and hence have the ability to attract and adsorb cations at their surfaces. These cations are not firmly held and on treatment with suitable electrolyte solutions one type of cations can be replaced by another, g. a clay having adsorbed Ca2+ ions at its surface can have these replaced by Na+ ions on treatment with Na2CO3,

Ca-Clay + Na 2CO3  Na-Clay + CaCO3

(6)

The Na-clay produced has very different physical properties than those of Ca-clay. In a flocculated clay or body, the particles attract to each other giving open packed particle arrangement requiring such water to produce an optimum plasticity. Therefore, flocculation is used to increase the workability of industrial tableware units. Such treatment suffers from a disadvantage that unfired strength is reduced due to open particle packing and reduced its bulk density (Rayan and Radford, 1987; Drews, 1983; Rayan, 1978; Khalil, 2012). f) Hand Wedging (Pugging) This process increases the workability of the material. This is mainly attributed to the improvements which can take place in the body texture which is resulting from the elimination of air inclusions and reduction of moisture distribution throughout the material. g) Ageing of Souring It is the practice of storing plastic clay or body for a time period prior to its use, which commonly employed in the heavy clayware production, but less so in white wares. The improvements in workability are essentially due to two factors:

Nanomaterials, Ceramic Bulk and Bioceramics

205

1) The penetration of water into hard aggregates to wet not previously wetted outer surfaces. This increases the effective surface area of the material. 2) Some flocculation of the material caused by dilute acids formed by the decomposition of organic matter by a bacterial action. When the clay dries from the plastic state or the dried unfired ware, its strength has been developed. The dry strength of the clay depends on the nature of the clay and the size of the clay particles. As a general rule, the more plastic clays are the strongest. The construction usually takes place as the clay dries due to the loss of water of plasticity. A little shrinkage is desirable and excessive construction leads to problems such as warping and cracking. A further construction always takes place during the firing of clay article due to the melting of fluxes which are present in varying amounts in the different clays (kingery, 1975; Mostafa and Darweesh, 1992). Darweesh, 2015 studied the dry and firing shrinkage of some ceramic composites containing 0, 5, 10, 15, 20 and 25 wt. % cement kiln dust waste (CKD) at 0, 1000, 1050, 1100 and 1150ºC to throw light on the behavior of the samples on heating according to ASTM-Specification, 1980. Table 1and Figure 16 illustrate the dry and firing shrinkage as a function of CKD content. He found that the dry and firing shrinkage slightly increased with the increase of CKD content up to 25 wt. %, and also with the increase of firing temperature (kingery, 1975; 1978; Khalil, 2012; ASTM, 1980; Mostafa and H. H. M. Darweesh, 1992; Chiang, 1997). Table 1. The dry and firing shrinkage of ceramic products with various CKD content at different firing temperatures Firing temperature, ºC

0 1000 1050 1100 1150

0 0 0.022 0.106 0.313 0.473

CKD content, wt % 5 10 15 0 0 0 0.043 0.105 0.165 0.152 0.286 0.482 0.406 0.654 1.108 .518 0.654 1.312

20 0 0.189 0.535 1.185 1.342

25 0 0.251 0.603 1.223 1.363

206

H. H. M. Darweesh

Figure 16. Dry and firing shrinkage of ceramic products with various CKD content.

h) Fired Color The unfired color of the clay becomes another color after firing. For example: many pottery clays in the dry state are brown or dark grey and become almost white color by firing. A slip is a suspension of clay in water. When 9 parts by weight of dry constituents are mixed with 10 parts by weight of water, the resulting slip will have the consistency of a stick cream. If we now add a small amount of two common alkaline materials (CaCO3 and Na3SiO3), the slip will become quite fluid and it will be possible to add a further 17-18 parts by weight of the dry materials before the slip starts to thicken again without adding any more water. The process of rendering a thick slip fluid in this manner is known as deflocculation and the materials used to bring it about are called deflocculents. Deflocculation of a pottery body is dependent on the clays present and is very important in the production of a slip for the making process known as casting. This method is used for hollowware, tea ports, jugs, etc., and in the manufacture of sanitary wares (Jackson, 1969; Mostafa and Darweesh, 1992).

Nanomaterials, Ceramic Bulk and Bioceramics

207

Clay and Loams are sedimentary species; clay; silt and sand with low alumina content from 16-23% and high proportion of iron compounds, are used in the manufacture of colored ceramic tiles. They are usually divided into two catigories: not-easily fusible pottery clays and loams, and common pottery ones. It is subdivided by konta (Konta, 1981) into: 1) Pottery clays and/or loams containing 3.5-16% Fe2O3 and a slight to nearly zero amount of CaO. 2) Pottery clays and loams, calcareous clays (or lime clays), containing sometimes up to 40% calcite, which is suitable for the production of stone tiles at low fusion temperatures (12001300ºC).

2.6. Raw Materials The chemical composition of raw materials used in the ceramic industry is shown in Table 2 and are frequently classified to: Table 2. The chemical composition of ceramic raw materials, wt. % Materials Oxides SO2 Al2O3 Fe2O3 CaO MgO MnO Na2O K2O TiO2 SO3 P2O5 Cr LOI

Clay 53.47 26.78 3.99 0.60 1.38 0.03 1.15 1.18 1.12 ---0.51 ---9,72

Felspar 75.37 13.62 0.41 0.63 ---0.03 3.44 5.84 0.05 0.02 ---0.02 0.67

Quartz 93.63 3.64 0.08 0.18 ---0.02 0.17 0.14 0.16 0.14 ---0.06 1.78

Limestone 0.08 0.03 0.04 56.84 0.10 ---0.12 0.05 0.01 0.02 ---0.08 42.63

208

H. H. M. Darweesh

2.6.1. Fillers This is a rather loose term used to describe non-plastic materials used in body compositions, which are not fluxes. In practice, the filler of white ware bodies is always silica, either in the form of flint of sand though few bodies often used alumina. Fillers serve to open up the body facilitating the drying process, reducing the drying shrinkage and plasticity. Moreover, they are controlling the thermal expansion and contributing the whiteness of the fired bodies. Fillers are often representing a large ratio of the total body composition. So, the particle size distribution is very important in the determination of the particle packing in the body. This is in turn affects drying and firing shrinkage. Therefore, the fineness of the filler affects the thermal expansion of the fired body. Fillers are high melting, chemically resistant inorganic materials whose main function is to reduce a ceramic body's tendency to warp or distort when fired to high temperatures. They also play an important role in the determinaton of thermal expansion. Quartz and/or flint are the most common fillers used in white wares and other ceramic bodies. Though Al2O3 is an important ingredient in many refractory and technical ceramic bodies, it is used in a few tableware bodies (Jackson, 1969; kingery, 1975; Rayan and Radford, 1987; Mostafa and Darweesh, 1992; Grim, 1962). 2.6.2. Quartz Sand Quartz or silica is the most abundant oxide in the earth's crust. It is estimated that silica occupies approximately 19 wt. % of the outer 25 miles of the earth. White wares often contain siiica in the form of silicates, i.e., silica can be combined with other elements to form the silicate minerals, but a large proportion occurs as free silica, mostly in the form of quartz. Since, this is the mineral form which is stable under the normal atmospheric conditions. Earthen wares for an example are containing 7276 % silica. In many countries like UK, flint is the popular form of the silica component of white wares, though the continental countries have used quartz sand for many years. Recently, however and mainly for economic purposes, many British manufacturers have taken to the continental practice and replaced flint with sand in their formulations.

Nanomaterials, Ceramic Bulk and Bioceramics

209

Sandstones are common rocks, but the sandstone pure enough to use in white ware bodies are infrequent. Many are containing iron compounds, which stain them brown or red colors (kingery, 1975; Mostafa and Darweesh, 1992; Chiang, 1981).

2.6.3. Flints Flints contain small amounts of water, minute air spaces and interstitial amorphous SiO2 due to its mode of formation from the skeletons of sponge-like organisms which dissolved in sea water and were later deposited in chalky deposits. The fine microstructure of flints makes it more reactive than quartz, and in particular, it converts during firing more readily to cristobalite (another crystalline form of silica) which is profoundly affects the thermal expansion of the body (kingery, 1975; ASTM, 1980). Flint occurs as hard nodules in the middle and upper parts of the chalk deposits often arranged in layers along bedding planes separating the beds rather than evenly distributed throughout the chalk. Therefore, flints were formed by precipitation from silica in solution. This is confirmed by the fact that the white skin of flint gradually varies in composition from CaCO3 to silica. In order to prepare flints for use, flints pebbled from the sea shore or from chalk deposits are calcined up to 1100ºC to make it friable, and to facilitate its grinding to the required fineness (45-50%less than 10 µ). During calcinations, flints easily shatter due to expansion of air and water pockets within the structure. This changes its color from black to white as an organic material is burnt off. The specific gravity of flint is 2.62 reduced to 2.50 after calcination due to the exfoliation of the flint on heating and to a lesser extent to the formation of cristobalite (spec. gr. 2.3). About 18% of quartz is converted to cristobalite during calcination process of flint. The calcined flint is wet ground to the required size and used in a slop form in the body mix. Grinding to the correct fineness is very important since it affects the extent of conversion of quartz to cristobalite during firing of the body (kingery, 1975; ASTM, 1980; Darweesh, 1992). At all, flints of interest to the potter are divided to:

210

H. H. M. Darweesh

2.6.3.1. Chalk Flint These are generally large lumps with a thick chalk crust. 2.6.3.2. Wash Mill Flints These are a by-product of the cement industry and by far most important source of free silica for earthenware and tiles. 2.6.3.3. Beach Flints These are smooth flint pebbles that occur on a number of beaches at the southern coasts of England, and at the northern coasts of Belgium and France, where the surface chalk is completely washed away (Rayan and Radford, 1981; Sonjida et al., 2011; Vorrada et al., 2009; Darweesh, 2015). The firing temperature of all pottery bodies is several hundred degrees below the melting point of the filler. SiO2 existed in a variety of crystal forms which differ in density and thermal expansion characteristics from one another. The three most important crystal forms of SiO2 are quartz, cristobalite and tridymite. Quartz exists in two forms differing slightly in density. Below 573ºC, α -quartz, the denser form, is the stable one, while above this temperature, β-quartz is stable. The change between the two occurs rapidly as the temperature changes through 573ºC and since only a comparatively small displacement in the relative positions of the SO4 tetrahedra is removed. This type of change between high and low temperature modifications of silica is referred to as a displacive change. When quartz is subjected to temperature above approximately 1000ºC, bonds between atoms are broken and a new crystal structure results. The mineral cristobalite has such lower density than quartz. Since the change involves a major reconstruction of the crystal. Cristobalite also exists in three modifications (Khalil, 2012; ASTM, 1980; Darweesh, 1992). 

Inversion and conversion of silica

The conversion of SiO2 to cristobalite is slow depending on the degree of temperature, soaking of firing and the presence of mineralizers the

Nanomaterials, Ceramic Bulk and Bioceramics

211

thermal reactions. Thermal reactions representing (kingery, 1975; ASTM, 1980; Konta, 1981) of the conversion of SiO2 at different temperatures are as follows,

 -Quartz

5730 C

 -Quartz

(7)

8700 C

 -Quartz  2 -Tridymite

(8)

14100 C

2 -Tridymite   -Cristobalite  -Cristobalite

2 -Tridymite 1 -Tridymite

220 2800 C

1630 C

1170 C

 -Cristobalite

(9)

(10)

1 -Tridymite

(11)

 -Tridymite

(12)

Flint is composed of extremely small crystals of quartz. The surface of each minute crystal is available for conversion. Consequently, more cristobalite is formed in a body, whereas the flint filler is more than in the same body in which the ground sand is substituted for flint. Therefore, flint is preferred in the earthenware industry. Since, the contraction which accompanies the change as the cristobalite cools is an important factor in preventing crazing (kingery, 1975; Khalil, 2012; ASTM, 1980; Mostafa and Darweesh, 1992). They are added and usually used in ceramic tiles or any ceramic masses as fillers. It is a common accessory constituent in clays used for the production of tiles (Todor, 1972). Quartz relics act as a source of flaws in ceramic products. To get over it, the grain size of the quartz should be greatly reduced. In this case, it participates readily in the liquid or glassy phase formation.

212

H. H. M. Darweesh

2.6.4. Fluxes A flux is a material which reduces the fusion temperature of the materials or mixtures to which it is added. It is not strictly possible to divide material to fluxes and refractory materials, since whether or not a material acts as a flux depends not only on the material itself, but also on the material to which it is added. It is good mention that as the alkali content increases in a fluxing material, its fluxing action increases. The F2O3 content of the fluxing material must be in its lowest content, or it will impart its color to the fired ceramic body. Also, the grain size of the fluxes is of great and vital importance (kingery, 1975; Rayan and Radford, 1987; ASTM, 1980; Mostafa and Darweesh, 1992). Fluxes melt when the ware is fired. On cooling, the glass solidifies and provides the bond that holds the whole mass together.That is responsible for the strength of the pottery ware. Cornish stone, feldspar and nepheline syenite are fluxes (kingery, 1975; Grim, 1962; Schuller and Jager, 1979). 2.6.4.1. Feldspars It is essentially a mixture of two feldspars namely, orthoclase-potash feldspar, KAlSi3O8 or K2O. Al2O3. 6SiO2 and albite-soda feldspar, NaAlSi3O8 or Na2O. Al2O3. 6SiO2. Small amounts of mica and quartz are usually present. The feldspar is often extracted by a normal quarrying or mining with visual control over the extracted material at the quarry face. Briefly, the method starts by crushing the rock to less than one inch by means of jaw and gyratory crushers which followed by wet milling to pass a 30 mesh sieve. The three stage froth flotation process which follows, first removes the mica, garnet and other similar materials. Then, the feldspar is frothed from the quartz. Finally, the separated feldspar undergoes a further flotation to reduce the quartz as much as possible. The concentrated feldspar and quartz which are separated by this method are then dried and stored. Though feldspar contains about 5% quartz, quartz contains 6-8% feldspar (kingery, 1975; Mostafa and Darweesh, 1992; Schuller and Jager, 1979).

Nanomaterials, Ceramic Bulk and Bioceramics

213

Table 3. Comparative analyses of some commercial fluxes, wt. % Materials Oxides SiO2 Al2O3 Fe2O3 CaO MgO K2O Na2O TiO2 LOI

Feldspar 67-70 10-18 0.1-0.3 0.3-0.5 0.03-0.15 9.0-11.0 2.5-3.3 0.02-0.07 04-0.7

Cornish stone 72-74 14-16 0.1-0.3 1.5-1.9 0.05-0.20 4.0-4.5 3.0-4.0 0.05-0.15 1.0-2.0

Neph. Syen. 1 61.0 32.0 0.07 0.6 0.10 4.5 10.0 ---0.6

Neph. Syen. 2 56.0 25.0 0.08 0.75 ---9.2 8.1 0.1 0.75

The temperature at which fluxes start to melt is largely depending on the total alkali content and the particle size of the material. Mica, particularly in the extremely finely divided form in which it occurs in the clays, is one of the first of glass-formers to melt. The outer fluxes all resemble one another at the lower temperature, e.g., pressed discs of Cornish stone, feldspar and nepheline syenite when fired up to 1100ºC have very much the same appearance-hard and with sharp well defined edges. As the temperature is raised, the expected variation becomes apparent with the nepheline syenite showing the greatest tendency to flow. So, increasing the amount of flux in a body will not necessarily reduce the fired porosity, particularly at relatively low temperatures (Grim, 1962; Schuller and Jager, 1979). Schuller and Jager, 1979 showed that the viscosity of feldspar melts increased with the rise in K2O/Na2O ratio. Schuller, 1964found that soda feldspar seems to increase the reactivity of the melt and promotes the attack on the aggregates of primary mullite (3A2O3.2SiO2 or A3S2) at a higher temperature. Table 3 illustrates the chemical analysis of some commercial fluxes. 2.6.4.2. Nepheline Syenite It contains a high alkali ratio (9-10%) and therefore, it is considered as a powerful flux. So, it can be used in a body composition of glazes and porcelain enamels have increased considerably over the last 20 years. It is

214

H. H. M. Darweesh

claimed that the substitution of nepheline syenite instead of feldspar in the ceramic body reduces the firing temperatures resulting in saving time and fuel. Also, it increases the firing ranges. There are two main sources for nepheline syenite in Canada known as “Lakefield variety” and in Norway known as “North Cape variety.” The second variety has a total alkali content and K2O/Na2O ratio greater than those of the first. Therefore, the Norway variety has a higher fluxing action than the Canadian (Jackson, 1969; kingery, 1975; Rayan and Radford, 1987, Drews, 1983; Rayan, 1978). 2.6.4.3. Cornish Stone It is clear from Table 3 that Cornish stone is not a powerful flux as either feldspar or nepheline syenite, where its total content is considerably less, but it was however very widely used as a flux. In this country is being the only native deposit of fluxing material commercially viable. Therefore, its use has declined recently, and feldspars and nepheline syenite are being imported to replace it. In fact, the Cornish stone is effectively a mixture of kaolin and feldspar. Generally, it is graded on the basis of its color. There are three Cornish stone varieties known as “Hard purple,” “Mild purple “and “Dry white.” All varieties contain feldspar, quartz, clay minerals, mica, fluorspar (CaF2), and other minor impurities. The purple varieties are the richest in feldspar, while the white variety is the poorest. Hence, the hard purple is the strongest flux and the dry white is the weakest. The color is mainly due to the impurity content exist in the fluorspar of the stone. During firing, as the feldspar and fluxing ability increases, the fluorspar content increases too. This often accompanied with the emission of fluorine. This is usually detrimental to both environment and human health (Jackson, 1969; kingery et al., 1975; Rayan and Radford, 1987; Drews, 1983; Rayan, 1978; Darweesh, 1992). 2.6.4.4. Talc It is a magnesium silicate and a member of montmorillonite group having the formula of 3MgO. 4SiO2. H2O. It is a very cheap source of magnesia which is acting as a flux. The presence of Talc in bodies that

Nanomaterials, Ceramic Bulk and Bioceramics

215

fired at high temperatures results in the formation of cordierite. This imparts low thermal expansion and therefore good thermal shock resistance. This makes it to be more suitable for the production of table wares. The only disadvantage is the very narrow range of firing which makes it easily distort on firing. Furthermore, according to its lower expansion, they become more difficult to match with the glaze of a suitable expansion or decorate to an acceptable standard. It is believed that limebearing talc in particular gives bodies of low shrinkage the high strength and good crazing resistance. They are successfully used in the manufacture of both table wares and wall tiles (Jackson, 1969; kingery, 1975; Rayan and Radford, 1987; Drews, 1983; Rayan, 1978; ASTM, 1980).

2.6.5. Clyas Clay minerals are often formed from the decomposition of igneous rocks as granite which is composed mainly of roughly equal proportions of potash mica (K2O. 3 Al2O3. 6SiO2. 2H2O), quartz (SiO2) and potash feldspar (K2O. Al2O3. 6SiO2). Kaolinitic clay was the decomposition or kaolinization of feldspar in the presence of air and water (Jackson, 1969; kingery, 1975; Singer and Singer, 1971; ASTM, 1980; Mostafa and Darweesh, 1992) as follows: K2O.3 Al2O3 .6SiO2  H 2O  Al2O3 .2SiO2 .H 2O  K 2O  4SiO2

(13)

Kaolinite is a crystalline material in which the crystals are being flat and extremely small hexagonal in shape. The crystal size may vary from 5 µ to 1 µ 0r 10-4 meters, which in turn responsible for their extreme properties (Jackson, 1969; kingery, 1975; Rayan, 191987; Drews, 1983; Rayan, 1978). Clays provide the plasticity which simplifies the manufacture of clay wares and reduces handling losses. When clay is formed from the parent rock, it may have been deposited at its origin site. Hence, it is known as “Primary or residual clay,” but if it is transported by means of water or winds or any other means and deposited into a distance

216

H. H. M. Darweesh

away is known as “Secondary or sedimentary clay” (ASTM, 1980; Mostafa and Darweesh, 1992). At 100-200ºC, the volume of certain argillaceous minerals shrinks as a result of the water loss which causes a dimensional change. At 200-300ºC, the oxidation of certain organic materials present in argillaceous minerals begins. The degree of oxidation depends on the nature of the organic materials, the amount of oxygen available in the furnace and the readiness with which this can penetrate into the mass of the argillaceous material in order to promote the oxidation. In general, the degree of oxidation increases with rising temperatures (ASTM- Specification, 1980; Mostafa and Darweesh, 1992). The main clay mineral for potters is kaolinite (Al2 Si2 O5 (OH)4 or Al2O3. 2SiO2. 2H2O or AS2H2), which is seen in the SEM to crystallize into minute hexagonal plates. Kaolinite is a decomposition product of feldspar 450-500ºC to form metakaolin (Al2O3. 2SiO2 or AS2) and water vapor (Khalil, 2012; Mostafa and Darweesh, 1992). 450500 C Al2O3 .2SiO2 .H 2O   Al2O3 .2SiO2  H 2O 0

(14)

At this moment, the ceramic units suffered from a slight expansion, where it becomes enlarged than before. At 980ºC, some heat was released followed by the total destruction of the whole structure to form mullite (3Al2O3. 2SiO2 or A3S2) and quartz minerals. 980 C Al2O3 .2SiO2   3 Al2O3 .2SiO2  SiO2 0

(15)

2.6.6. Limestone It is an important ingredient in wall and floor tile compositions due to its capacity to redduce the moisture expansion of the resulting ceramic units. This is a very critical charater specially in floor tiles (Kingry, 1975, Mostafa and Darweesh, 1992). Limestone is often used at a range of 10-12 wt. % addition in tile products to lower the quantity of melting component

Nanomaterials, Ceramic Bulk and Bioceramics

217

during firing.This is the feature of the body which is prone to moisture expansion which is more complecated. Accordingly, it cannot be considered as a filler or flux. Inside the kiln, limestome starts to break down into CaO and CO2↑ as shown in the following equation: 700900 C CaCO3   CaO  CO2 0

(16)

The formed CaO tends to react with the surrounding ingredients to form anorthite (CaAl2Si2O8) instead of going to the melting minerals. This reduces in the melting condition and then prevents the firing shrinkage. Limestone or ground chalk or CaCO3 is commonly known as “Whiting” could also be used as an important glaze material (Kashlyak and Kareev, 1978; Kisel, 1980). Tiles (Rayan and Radford, 1987; Danto, 1979) are generally classified according to their degree of water absorption (W.A.) into three main groups: i. ii. iii. 

Low water absorption (W. A. < 3%). Medium water absorption (W. A. = 3-6%). High water absorption (W. A. > 10%). Floor tiles are belonging to the 1st and 2nd groups with W. A. only up to 6% (Rayan and Radford, 1987; Drews, 1983; Rayan, 1978). There are two different types of floor tiles:

a) Floor Tiles They are dense bodies, fully vitrified, colored frequently by the color of the natural clays used in the batch or by colors, and can be obtained in various colors as well as different textures. The extenal surfaces of floor tiles are fine and soft. In some cases, surfaces are self-glazed. These are characterized by their high resistance to abrasion and weather stains.

218

H. H. M. Darweesh

b) Quarry Tiles The quarry tiles are thicker and extremaly bigger if compared with floor tiles and are having a rough surface appearance with W. A. between 2-5%. These tiles are used in floor of public buildings, silos and wall linings. c) Decorative Floor Tiles These are ready dense unglazed tiles. There is a wide range of bright colors in these colored types. The floor tiles production all over the world is based on the available local raw materials and it is usually established near the sites of these raw materials. Since, it is considered that the transportation of raw materials over large distances is uneconomic. In addition, the simple shape of the floor tiles and the conventient size of each tile in the production, make it possible to mechanization and automization of the industry (kingery, 1975; Kashlyak and Kareev, 1978). Many attempts were made in the world, each to use local raw materials in the production of floor tiles (Danto, 1979; Vauequelinet et al., 1980; Fiori and Fabbri, 1980; Darweesh and El-Din, 2000). Accordingly, it was found that tiles with water absorption of 7-9% can be prepared by replacing the expensive refractory clay type with local clay mixtures (Danto, 1979). On the other hand, the use of granite or kaolinized granite for the manufacture of red tiles by rapid firing was done giving a firing temperature 40ºC lower than the best comparable clay body one (Schuller, 1964; Kashlyak and Kareev, 1978). In some compositions, the firing temperature was found to reach 1000ºC (Bacanac and Babic, 1984; Belous, et al. 1987). Belous et al., 1989 [57] studied the influence of chemical and structure composition on the characteristics of floor tiles. They found that the presence of ZrO2 had no a significant effect, but the bending strength was increased by increasing the Al2O3 content and was lowered by more SiO2 content. The compressive strength reached a maximum value 3100 MPa in presence of 95% Al2O3. Ceramics, in general, are distinguished by their high mechanical strength that enables them more convenient for many applications. So, Watchman, 1969 had classified ceramics into different categories as follows:

Nanomaterials, Ceramic Bulk and Bioceramics 1) 2) 3) 4) 5) 6)

219

Single crystal types. Glass fibers and film types. Polycrystalline aggregates in a glassy matrix types. Polycrystalline glass-free ceramic types. Devitrified ceramic types. Composites types.

Traditional whiteware ceramics specially floor tiles fall into the category of polycrystalline aggregate in a glassy matrix, e.g., multiphase body with phases having too different coefficients of thermal expansion which may tend to develop some defects. So, for bodies made of polycrystalline aggregate in a glassy matrix, the physicomechanical properties, i.e., the ceramic parameters, are highly affected by stresses arised in the melting condition and corroded relics of quartz grains and mullite crystals (Warshow and Seider, 1967). Quartz, which is one of the ingradients in the floor tile composition, is characterized by a relatively higher thermal expansion than other components, that reaches a value of 26.2 x 10-6 between room temperature and 1000ºC (Genin, 1958). So, quartz controls the total thermal expansion of tile units. Therefore, the overall cracking is resulting from expansion changes owing to phase conversions of SiO2 in the form of quartz. In addition, the difference in expansion between the glassy phase and SiO2 was proved in the work done by Genin, 1958. Kalnin, 1967 and Kalnin et al., 1968. They found that both young's and their modulus as well as the strength of white wares were increased with the total amount of mullite which was also claimed by Mortel, 1977 in case of the fast fired bodies and by Giamlem and Lyng, 1974 with the addition of anorthothite as a flux to rise the bending strength of the produced articles. Schamberg et al., 1984 stated that the bending strength of the tiles can be increased with more than 20 N/mm2 by preventing the formation of gehlenite compound by the addition of about 8% barium compound in the tile composition and firing at 800-1200ºC.

220

H. H. M. Darweesh

On another side, strengthening of glassy phase of tile bulk was by rising its Al2O3 and lessening its K2O contents (El-Alfi et al, 2004; Darweesh et al., 2011; Darweesh and M. G. El-Meligy, 2014; Aggarwal, et al. 1980), whereas the MgO, K2O and Na2O are reasons that controlling the maturing temperature, liquid phase and crystallization which can develop the different properties. In recent trends, to save raw materials and to reach good mechanical as well as ceramic properties of floor tiles and moreover to use both industrial byproducts and wastes, the industrial waste materials containing oxides suitable for tile composition were introduced in the tile mix composition. The most important waste materials among this field of application are both FA and cement kiln dust. Aggarwal et al., 1980 found that FA from power stations and aluminium industry waste can be used in the manufacture of wall tile up to 42 wt.%. The bending strength of ceramic tiles was increased and their water absorption decreased by the addition of fly ashes and slags from coal-fired power plants. The sintering effect of slags was due to the increased amount of FeO and that of ashes to the amount of alkalies in the composition (Khrundzhe and Babushkin, 1983; Ibrahim et al., 1980; Pinalova et al., 1984). Pinalova et al., 1984 used the tailing from a coal benification for the manufacturing of ceramic materials. A typical waste 16-20 mm diameter contained 7-16% moisture and 75-85 wt. % ash was examined. The waste was added to the ceramic batch and ground to a particle size 0.5 mm, mixed with water and plasticizers, and then pressed or extruded at 20-30 kg/cm2. The products were dried, fired at 800-1110ºC, and then optionally glazed at 980ºC to give tiles and facing materials. On the way, to increase the capacity of tile firing furnaces and decrease the energy consumption in tiles manufacture, silica flour was used in the ceramic batch composition to produce tiles glazed and fired by glost firing at 1050-1250ºC for 20-60 minutes only (Pitskhelauri et al., 1988). Also, tiles had 1.0-1.5% shrinkage and 13.8-14.5 MPa bending strength were produces at the Slavyanshoe plant (ASTM, 1980) by using the waste slurry obtained during the production of WG. This waste has a high SiO2 content (66.67%) and alkalies (Na2O, 2.5-3.0%) but a low content of Al2O3 (0.5-0.6%). Also, to

Nanomaterials, Ceramic Bulk and Bioceramics

221

decrease volume and weight with retained strength and decrease the firing temperature of the tiles (ASTM, 1980), the raw mixture of tiles contains wastes from asbestos cement production was used.

2.7. Methods of Investigation 2.7.1. Densification Parameters White ware products are known as “Vitreous” in which water absorption is less than 1% or “Porous” in which water absorption is more than 1% and may be reached to 18% as in wall tiles. Open pores are voids which allow the ingress of a penetrating fluid from the surface of the article. The voids are created due to the migration of gases during the drying and firing. Sealed pores are often formed during firing when bubbles of gases are frozen into the glassy matrix or when open pores are sealed by the molten material. Some clays and bodies are bloated when over fired due to the melting action of the fluxes together with the evolution of gases from such impurities as calcium sulphate, CaSO4 (kingery, 1975; Rayan and Radford, 1987; Drews, 1984; Rayan, 1978; ASTM, 1980; Mostafa and H. H. M. Darweesh, 1992). The density of a ceramic material or as “apparent volume,” since the terms are also appropriate to porosity and water absorption evaluations. The density of a material has been defined as the relationship between its mass (weight) and its volume (Rayan and Radford, 1987; Rayan, 1983; Rayan, 1978) or, Density 

Mass Volume

(17)

For a vitreous object, there is only one weight and one volume involved, but for a porous solid, there are three common volume expressions as follows:

222

H. H. M. Darweesh 1) Apparent volume: This is the envelope volume of the porous solid which is usually called “bulk volume.” The material often comprises both open and sealed pores. It can be determined by: A physical method, a mercury displacement method, the difference between the soaked (S) and immersed (I) weights, since the water is used, the numerical value (S - I) g, gives the apparent volume (Rayan, 1978; Khalil, 2012; Njindam, 2018). 2) True volume: This refers only to the volume of the solid component. It can be determined by the “density bottle method,” where the material is crushed to be in the powder form and then all pores are disappeared. 3) Apparent solid volume: This lies between the two previous volumes, where it can be determined by the difference between the dry weight (D) and the immersed weight (I) of the material. At all, S - I, S - D and D – I are the volume of the open pores + sealed pores + solid. and therefore by subtraction, respectively (kingery, 1975; Rayan, 1978; Ke et al., 2016; Sonjida et al., 2011; Vorrada et al., 2009; Kimm, 2014; Kimm, 2013; Nissan, 2014). There are three expressions of density corresponding to the three previous volumes discussed above as follows, Appearent (bulk ) Density 

True Density 

Weight D  Appearent volume S  I

Weight True volume

Appearent solid Density 

Weight D  Appearent solid volume D  I

(18)

(19) (20)

where, D, S and I are the dry, soaked and immersed weights of the material, respectively.

Nanomaterials, Ceramic Bulk and Bioceramics

223

This property is measured by comparing the volume of the pores to the weight of the material itself. There two widely known expressions for porosity are “Apparent porosity or AP” and “Water absorption or WA.” The AP is a ratio of open pore volume to the total volume, whereas the WA is a ratio of the open pore volume to the weight of the test material. The 1 cm diameter/1 cm thickness disc-shaped specimens were prepared under 20-22 KN loading using a suitable piston. At first, the prepared specimens are left to dry on air for 24 hours and then at 105ºC for another 24 hours. The dried specimens are then fired at different firing temperatures using soaking time of 1-2 hours. The used furnace must leave to cool slowly over night. The fired ceramic bodies after cooling are then subjected to densification parameters (ASTM, 1980; Darweesh et al., 2012), in terms of water absorption, %; bulk density, g/cm3 and apparent porosity, % which could be calculated from the following relationships,

W . A, % 

W1  W2 100 W3

W3 W1  W2

(22)

W1  W3 100 W1  W2

(23)

B.D, g / cm3 

A.P, % 

(21)

where, W1, W2 and W3 are the saturated weight in air, suspended weight and the dry weight, respectively. Darweesh et al., 2012 studied and discussed W.A, B.D and A.P of some ceramic batches containing various contents of CKD waste up to 25 wt. % (Figures 17-19). Table 4 shows the chemical analysis of the starting raw materials used in this work. They reported that the W.A and A.P decreased with firing temperature up to 1100ºC, but only decreased with CKD content up to 10 wt. %, and then decreased with more addition of CKD content as shown in Figures 17 and 18, respectively, while the bulk density increased as shown in Figure 19.

224

H. H. M. Darweesh Table 4. The chemical composition of raw materials, wt. %

Materials Oxides

T-Clay (TC)

Feldspar (F)

Sand (S)

Limestone (L)

Homra (H)

L.O.I SiO2 Al2O3 Fe2O3 CaO MgO MnO K2O Na2O TiO2 SO3 P2O5 Cl-

9.72 53.47 26.78 3.99 0.60 1.38 0.03 1.18 1.15 1.12 --0.51 ---

0.67 75.37 13.62 0.41 0.53 --0.03 5.84 3.44 0.05 0.02 --0.02

1.78 93.63 3.64 0.08 0.18 --0.02 0.14 0.17 0.16 0.14 --0.06

42.63 0.08 0.03 0.04 56.84 0.10 --0.05 0.12 0.01 0.02 --0.08

58.22 28.25 8.16 0.79 0.46 --1.46 1.32 1.34 -------

Leached C. Dust (LCD) 24.51 12.84 1.86 1.53 52.51 1.84 --1.65 0.83 --2.43 -----

Figure 17. Water absorption of ceramic products with different CKD contents up to 25 wt. %.

Nanomaterials, Ceramic Bulk and Bioceramics

225

Figure 18. Apparent porosity of ceramic products with various LCD contents up to 25 wt. %.

Figure 19. Bulk density of ceramic products with various LCD contents up to 25 wt.%.

226

H. H. M. Darweesh

2.7.2. Mechanical Properties Rod-shaped specimens of the dimensions 1 x 1 x 7 cm3 are prepared using water as a binder under 20-22 KN loading by a suitable piston. At first, the prepared specimens are left to dry on air for 24 hours, and then at 105ºC for another 24 hours. The dried specimens are then fired at different firing temperatures using soaking time of 1-2 hours. The used furnace must leave to cool slowly over night. The fired rod-shaped specimens after cooling are then subjected to flexural or bending strength measurements (Khalil, 2012; Mostafa and Darweesh, 1992) by three point adjustments system (Figure 20), where, S is the span (cm), W and T are width and thickness of the sample (cm), respectively. The beam load was applied perpendicular to the axis of the sample. The F.S. or B.S. (Mukhamedzhanovet al., 1990) could be calculated from the following relation,

F .S , MPa 

3PL 2bd 10.2

(24)

where, δ: flexural strength, MPa, P: the load of rupture, kg, L: span or the distance between the two lower beams (5 cm), b: width of sample, cm and d: thickness of sample, cm.

Figure 20. Schematic diagram of the bending strength, B: beam, S: span, W: width and T: thickness.

Nanomaterials, Ceramic Bulk and Bioceramics

227

Figure 21. Flexural or bending strength of ceramic products with different contents of CKD waste, %.

Darweesh et al., 1992 studied B.S. or F.S. of some ceramic batches incorporating different contents of CKD waste (Figure 21). They found that the B.S. improved and enhanced with firing temperature up to 1100ºC, but only increased up to 10 wt. % CKD content and then decreased.

2.8. Body Glaze Generally, the glaze is a glass layer covered the ceramic body. It is a vitreous material which is a super-cooled liquid below the point at which it might have crystallized. It is a liquid of very high viscosity due to the following facts: 1) A glass rod ultimately deformed and bends when suspended between two supports. 2) On heating, the glass softens and liquifies over a temperature range of 1000-1150ºC, but it does not recorded a limited melting point.

228

H. H. M. Darweesh

Figure 22. Structure of crystalline silicate and silicate glass.

Normal solids are crystalline and have a definite arrangement of atoms and molecules. This can be shown by X-ray diffraction patterns (XRD). Glazes however, have no a definite structure, the atoms are oriented in a random network. This may be appreciated by comparing the structure of crystalline silicate minerals with glazes (Rayan and C. Radford, 1987; Mukhamedzhanov et al., 1991; Keijiro et al., 1991; Moroz et al., 1991).The unit of structure is the silicon tetrahedron. This shape is dictated by the size of the atoms, on a comparative scale oxygen = 1.32 and silicon = 0.39. An example of the structure of a crystalline silicate and a silicate glass is shown in Figure 22, which clearly indicates the difference in their structures. They have a random three-dimentional network, but no units repeat itselves at fixed and regular intervals. On heating, the glaze always softens over a given temperature range due to the varying amounts of energy required to detach different parts of the network, which are not structurally equivalent. The interstices within the network may be filled with atoms of other elements which modify the physical properties as color, brilliance and so on. Glazes could be classified to:

Nanomaterials, Ceramic Bulk and Bioceramics

229

a) Lead and Leadless Glazes i. Lead glazes have a brilliant appearance and can be used to glost firing temperature of 1150ºC above which the lead tends to volatilize. ii. Leadless glazes have been developed and improved. Now, they have a reasonable brilliancy and firing range. However, glazes that mature below 1000ºC often have poor craze resistance. This is mainly due to the fact that high proportions of soda and potash required to give the glaze a low melting point, and also confer a relatively high expansion, but the glaze must have a lower thermal expansion compared to the ceramic body. b) Fritted and Raw Glazes A fritted glaze often has one or more material used in its formulation, which have been subjected to the fritting operation, i.e., the heating together of components to form a glass which is subsequently ground to a given particle size. The main reasons for fritting are: i.

ii.

iii.

When soluble compounds fritted with other materials in the right ratios, it become insoluble. This means that glaze components should be insoluble, otherwise they tend to immigrate into the pores of glass wares. When the glaze fired, it may then have a starved appearance. Even, if a vitreous ware is used with a glaze of slightly soluble components, segregation will take place as the drying process progresses. Though lead compounds are available to ceramic industry, they are generally insoluble in water, but they may have a considerable solubility in dilute acids. For an example, if the compounds are soluble in dilute HCl acid, they become toxic, where the gastric juices are highly acidic. If glazes contain high ratios of clay materials, it may crack on drying. The chemically bound water content tends to release during the glost firing. This may lead to a poor glost finishing.

230 iv.

H. H. M. Darweesh If any compound evolved gases on heating like limestone (CaCO3) used in a glaze with a low maturing temperature, is fritted so that the used glaze is not likely to suffer from pin-holing and bubbling.

For a common practice, it often used two frits, i.e., Lead and Borax frits or its oxides (PbO and B2O3). When these oxides are fritted together, the lead does not generally reach satisfactory limits of insolubility. However, lead borosilicate can be produced satisfactory if the PbO:B2O3 ratio is carefully selected (Rayan and Radford, 1987; Mostafa and Darweesh, 1992; Mukhamedzhanov et al., 1991). The glaze could be classified also due to the type of ware on which the glazes applied as for Earthern wares, Sanitarywares and Porcelain glazes. There is a wide range of glazes available within each group. This classification has been broadened to include firing and maturing temperatures noticing that in the following examples, glaze molecular formulae have been given, where the oxides have been categorized as basic, amphoteric, acidic. The sum of the basic oxides is one, where the temperatures are referred to the maturing range (Jackson, 1969; Kingery, 1975; Rayan and Radford, 1987; Njindam et al., 2018; ASTM, 1980). 1- Majolica (900-1050ºC) [0.7 PbO +0.3 CaO] +0.15 Al2O3 + [2.0 SiO2 + 0.3 B2O3] 2- Earthenware (1000-1150 C-Lead glaze [0.4 PbO+0.3 CaO+0.3(Na,K)2O] 0.25 Al2O3 [2.5 SiO2+0.5 B2O3] 3- Leadless glaze [0.55 CaO 0.3 (Na,K)2O] 0.30 Al2O3 [3.0 SiO2 + 0.8 B2O3] 4- Sanitaryware (1200-1250ºC) [0.6 CaO+0.2 (Na,K)2O+0.2 ZnO] 0.35 Al2O3 [3.0 SiO2] 5- Hard paste porcelain (1400ºC) [0.68 CaO+0.20 MgO+0.12 ZnO] 1.0 Al2O3 [10.0 SiO2]

Nanomaterials, Ceramic Bulk and Bioceramics

231

There is a wide range of commercially coloring agents or stains which are used extensively in the ceramic industry. White wares may be decorated using coloring oxides in different ways as follows: - Colored bodies. - Colored slips (engobes). - On glaze.- Under-glaze. - In glaze. Colored bodies and slips have a few percent of the desired color added to the bulk mix during preparation in the slip house. The major disadvantage of using colored bodies is that shading of wares may take place. A ceramic color seldom consists simply of a coloring oxide or compound. This fact is understood by considering the following cobalt blue color known as “Royal blue” or “Maz blue” (Jackson, 1969; Rayan, 1978; ASTM, 1980). Cobalt oxide 45%, Whiting 10%, Flint 18%, Alumina 5%, Feldspar 22%. The color may be analyzed in terms of the function of its components. Cobalt is the basic coloring compound giving the intense blue color. CaCO3 acts as a modifying agent that modified the blue color to give new blue tint and stability. Flint and Al2O3 act as a diluents, i.e., as filler giving a slightly faint but more stable color. Feldspar which uses as a flux, is always assisting in the sintering of color mixes through the calcination process noticing that the underglazed colors need about 5% flux, but on glaze colors need about 70%. Ceramic colors are usually prepared to fix and carefully control specifications. Even so, it is difficult to produce decorative wares of consistent quality on a commercial scale. It must always remember that the final color depends mainly on the composition of the glaze, firing temperature and kiln type. These are very important particularly in sanitary ware industry, where the finished product can be sold to an exact standard color, while the vitreous sanitary ware is generally colored with opaque pastel glazes which have to match other colored fittings, in the cloak or path rooms manufactured from quite different materials. The metamerism phenomena must be taken into consideration which is two color samples may appear to match, under a given illumination, even though their spectrophotometric curves are not the

232

H. H. M. Darweesh

same (Figure 23). So, the sanitaryware colored glazes are formulated from a color and compatible base glaze to an exact specification. If a different illuminant is used, the samples may no longer match where the perceived color depends essentially on the illuminant and the reflection curve at the surface (Drews, 1983; Danto, 1979, Vauequelinet et al., 1980; Fiori and Fabbri, 1980; Darweesh and El-Din, 2000). The color specification by CIE system gives the chromaticity values or coordinates x, y and Y where these values may be used in conjunction with the CIE chromaticity diagram (Figure 24).

Figure 23. The metamerism curves of ceramic products.

Figure 24. The CIE chromaticity diagram.

Nanomaterials, Ceramic Bulk and Bioceramics

233

2.9. Thermal Expansion The difference between the contraction of the ceramic body and the glaze during cooling often causes some faults. The thermal behavior or expansion of ceramic products during firing can be measured by using Ortom Automatic Dilatometer (Figure 25). In this type of instruments, the temperature can be controlled in the range of 0.2-10ºC/min. and the rate of heating is 3-5ºC/min. (Darweesh, 2016) investigated the thermal behavior or expansion of some ceramic batches as 0, 5, 10, 15, 20 and 25% cement kiln dust waste (CKD) using Orton Automatic Dilatometer to throw light on the behavior of the samples on heating.

Figure 25. Ortom Automatic Dilatometer for thermal expansion.

Table 5 and Figure 26 illustrate the coefficient of linear thermal expansion versus CKD content and its effect on the thermal expansion of ceramic units incorporating it. The coefficient of linear thermal expansion decreased with the increase of CKD content up to 25 wt. %.The equipment (Figure 27) in its fully automated form is a direct reading dilatometer which plots the percentage of expansion against temperature. Specimens between from 55 to 80 mm can be accommodated. The silica correction is added automatically and after the initial setting up the instrument requires no supervision. The Dilatometer is also available in a hand-operated form and the following test procedures relate to this less sophisticated

234

H. H. M. Darweesh

instrument. The test piece with parallel ends is preheated to 900ºC, cooled and then pushed along the silica tube using thrust rod till it contacts the silica stopping disc. The dial gauge is firmly clamped into position and the needle adjusted so that a slight pressure is exerted on the thrust rod. The outer ring on the dial gauge is moved to zero reading. The thrust rod should move quite freely within the silica tube. The base of the apparatus should be gently tapped till the dial gauge gives a consistent zero reading. Figure 28 shows the typical thermal expansion of earthenware body and low sol glaze. At 500ºC, the thermal expansion is 0.38% and 0.32% for body and glaze, respectively. The difference of expansion is 0.06%. Table 5. The coefficient of linear thermal expansion as a function of CKD content and its effect on the thermal expansion of ceramic products CKD, wt. % 0 5 10 15 20 25

Coefficient of Linear thermal Expansion, α/K 8.731 x 10-6 8.492 x 10-6 8.049 x 10-6 7.145 x 10-6 6.382 x 10-6 6.120 x10-6

Figure 26. The coefficient of linear thermal expansion as a function of CKD content and its effect on the thermal expansion of ceramic products.

Nanomaterials, Ceramic Bulk and Bioceramics

235

Figure 27. Thermal expansion apparatus (A: Slate base, B: Silica tube, C: Stopping disc, D: Fused silica rod, E: Invar extension rod, F: Dial gauge, G: Furnace, H: Test specimen.

Figure 28. Thermal expansions of earthen ware body and glaze.

2.10. Applications Some investigators (Mukhamedzhanov et al., 1991; Moroz et al., 1991) used mining tailings and metallurgical wastes in the production of ceramic floor tiles by dewatering, grinding, pressing and firing at 1000-1100ºC. Mining tailings consist of argillaceous shale and sandstone, where the argillaceous shale serves as a binder, while sandstone as a flux substituted for pegmatite. Ceramic tiles of a low porosity and good mechanical properties were obtained from mixtures containing argillaceous shale 70-

236

H. H. M. Darweesh

76, sandstone 15-20, chamotte 3-5 and bentonite 1-5 wt. % (Keijiro et al., 1991). The CKD waste is the main source of air pollution in many countries, which resemble the present day problem, be a matter of great importance due to the precipitation of huge amounts of it behind the kilns and its characteristics such as: light weight and fine grains so that it can be transported easily by air storms to many agriculture and population areas. Keijiro et al., 1991 manufactured high strength ceramic tiles from slurry containing cement and mineral materials, e.g., feldspar, siliceous stone and SiO2 by moulding the slurry followed by drying and firing. The cement is selected from Portland and/or white Portland cements. Abdel-Fattah and Nour, 1981; Abdel-Fattah et al., 1982; Abdel-Fattah and Eh-Didamony, 1981) studied the CKD supplied by the factory. The dust was fired between 1000-1250ºC, then two calcines (70:30) and (50:50) were prepared by blending the raw dust and kaolin. It was stated that it consisted mainly of dolomitic limestone, minor amount of Na2O, K2O, SiO2, 2 C2S.CaCO3 and 2 C2S.CaSO4. The formed phases in the 1st calcined mix, was found to be mainly of gehlenite (C2AS), but in the 2nd calcined mix, it was mainly of β-C2S. Using of CaO sources in ceramics fields and tile production is recommended as a whiting and a flux (Darweesh, 2001). The cement kiln dust (CKD) is a mixture of finely divided particles and partially calcined raw materials with condensed volatile salts (Anlagenbau et al., 1995; Saltyvskaya, 1985). So, the use of dust along with kaolin in the form of calcined mix proved also to be used successfully in replacing feldspar in porcelain bodies. Darweesh, 2010 also studied and utilized the alkaline wastes obtained from cement kilns to produce porcelaineous bodies. Darweesh, 2010 exploited the CKD waste from cement factories as a source of CaO in ceramic industry by 25% to produce wall and floor tiles. W.A, B.D. and A.P. as well as B.S. and T.F. were researched (Figures 2931). The addition of the waste improved and enhanced all properties of the prepared units. Darweesh, 2001; Darweesh and khalil, 2001 demonstrated that adding 5 up to 10% waste to the aumina cement improved the specific

Nanomaterials, Ceramic Bulk and Bioceramics

237

characters of the cement hydrated up to 28 days. More than 10% waste, adversely reflected on these properties (Figures 32-34).

Figure 29. Water absorption, bulk density and apparent porosity of the ceramic bodied of the base batch and the batches containing CKD waste.

Figure 30. Compressive and bending strengths as well as abrasion resistance of the base batch and the batches containing CKD waste.

238

H. H. M. Darweesh

Figure 31. Firing shrinkage and linear thermak expansion of the base batch and the batches containing CKD waste.

Figure 32. Chemically-combined water content of the various alumina cement pastes containing cement kiln dust waste hydrated up to 28 days.

Nanomaterials, Ceramic Bulk and Bioceramics

239

Figure 33. Free lime content of the various alumina cement pastes containing cement kiln dust waste hydrated up to 28 days.

Figure 34. The compressive strength of the various alumina cement pastes containing cement kiln dust waste hydrated up to 28 days.

3. BIOCERAMICS 3.1. Introduction There is an advanced branch of ceramics known as “Bioceramics” including the ceramic products which can be used mainly in the human

240

H. H. M. Darweesh

bodies as bioactive bone scaffolds. Ca2 (PO4)2 was used in bioceramics which recently have received a great attention as bone-scafolds substitutes due to its good biocompatility and osteoconductive properties if compared to other materials. Hydroxyapatite (HA) has a similar composition to the major inorganic constituents of the skeletal system of vertebrates. Using of ceramic units in the human body, i.e., to exchange the broken or expired bones with a ceramic material that can function the remaining years of the patient’s life is very important. Moreover, the overwhelming demand for spare parts of bones for human beings took a particular interest nowadays. The cronical bone is the solid part of bones. Trabecular bones like scaffolds or a honey-comb (Figure 35). Spaces among bones are filled with fluid bone narrow cells which are making the blood and some fat cells. Figure 36 shows a simple structure of the bone. Bones have different shapes, sizes and structure tissues. The common bioactive ceramic materials are Al2O3 and ZrO2 due to its excellent biocompatibility. The main advantages of Al2O2 are its higher hardness and wear resistance, while ZrO2 has a higher strength and fracture toughness, in addition to the lower Young’s modulus (Taha, 2011; Mohamed, 2012; Kokubo, 1991; Iarry, 1991; Klein,1994; Kong et al., 1999; Susan, 2007).

Figure 35. The outside (cortical) and Inside (trabecular) of a bone.

Nanomaterials, Ceramic Bulk and Bioceramics

Figure 36. A simple structure of a bone.

Figure 37. Chemical structure of Gelatin.

Figure 38. Chemical structure of Agarose.

241

242

H. H. M. Darweesh

The sol-gel produces porous Al2O3 and ZrO2 toughened alumina (ZTA) substrates that can be prepared and impregnated with a synthetic body fluid (SBF) and dicalcium phosphate (DCP) solutions for different periods to prepare the bioactive composites. The resulting porous ceramic substrates are then characterized by measuring their physical properties in terms of B.D and A.P. The phase composition and the microstructure of the prepared scaffolds or porous ceramic substrates can be determined by using XRD and SEM techniques (Mohamed, 2012; El-Hady et al., 2015; Hench and West, 1990). El-Hady et al., 2015; Hench and West, 1990 used a certain polymer based on the gelatin from bovine skin (Figure 37) and agarose (Figure 38) to prepare composite scaffolds using the sol-gel technique. They also used ciprofloxacin (Figure 39) to avoid bacterial infections (Daniel and ValletRegi, 2010; Balamurugan et al., 2008; Barba-Izquierdo et al., 2000).

Figure 39. Chemical structure of ciprofloxacin.

3.2. SOL-GEL Techniques and Applications The sol–gel process is often utilized to prepare glasses and ceramics (Figure 40). The process is based on inorganic polymerization reactions of metal alkoxide precursors. These precursors undergo hydrolysis and condensation reactions if dissolved in a solvent to form soluble metal hydroxides (Mohamed, 2012; Saravanapavanand Hench, 2003; Phulé and Wood, 2001; Julian et al., 2006; Balamurugan et al., 2006; Newport et al., 2007; Gupta et al., 2005).

Nanomaterials, Ceramic Bulk and Bioceramics

243

Figure 40. The Sol-gel processing could be used to produce nanostructured layers and/or coatings as well as nanoporous membranes. 1- Hydrolysis. 2: Condensation. 3: Gelation, 4: Evaporation and Drying. 5: Nanoporous membranes.

The main characters of a multi-component sol–gel process are a homogeneous solution that formed before polymerization at lower temperatures, but also phase separation, crystallization and chemical decomposition can be eliminated (Saravanapavan and Hench, 2003). Briefly, the advantages of the sol-gel processes include lower processing temperatures, high levels of purity, control of concentrations, and the capacity to synthesize multicomponents in various forms (Phulé and Wood, 2001). In addition, the sol–gel derived bioactive glasses tend to have more simple compositions and enhanced bioactivity as well as resorbability due to the mesoporous texture (2–50 nm) inherent to the sol– gel process (Julian et al, 2006). Enhanced bioactivity is due to their residual hydroxyl ions, micro pores, and a large specific surface area (Balamurugan, et al., 2006; Newport et al., 2007). The dynamics of the sol–gel processes depend on various physicochemical properties of the

244

H. H. M. Darweesh

sol–gel composition, water to precursor ratio, the type of catalyst, choice of precursors, pH value, temperature and solvent (Gupta et al., 2005). Now, calcium silicate could be analysed by the sol–gel process. It is composed mainly of SiO2 (50-53%), and CaO (48-50%) by the chemical reactions between tetraethyl orthosilicate and calcium nitrate tetrahydrate. Tetraethyl orthosilicate was used as metal alkoxide precursors of the corresponding oxide SiO2. Metal alkoxides represent the precursors of choice for gel processing of oxide ceramics and glasses (Phulé and Wood, 2001). Metal salts are in the form of NO3- because they are less thermally stable if compared to other anions. This is due to that it is easy to decompose. Acetate salts are highly basic leading to a rapid gelation in silicate systems (Saravanapavanand Hench, 2003). CaCO3 did not use for biomedical applications, where CO32- ions incorporated in the hydrated layer slow down the growing of the apatitic (Siriphannon et al., 2002; Drouet et al., 2007). Moreover, when used the salt of calcium nitrate tetrahydrate,a problem of differential hydrolysis and polycondensation of various alkoxides arised (Phulé and Wood, 2001). Hence, Ca2+ in the form of calcium nitrate tetrahydrate could be incorporated.This explains the cause to select calcium nitrate tetrahydrate as a starting material as well as the percentage of them to prepare wollastonite. So, HNO3 acid must use to catalyze hydrolysis during sol-gel process. Where, the use of acids as catalysts allows the hydrolysis of all compounds outcomes due to the partial charge distribution in hydrolyzed alkoxides (Saravanapavanand and Hench,2003; Meiszterics and Sink´o,2007; Crayston, 2003).The choice for using HNO3 was depending on using it by several investigators (Saravanapavan and Hench,2003; Phulé and Wood, 2001; Balamurugan et al., 2006; Meiszterics and Sink´o, 2007; Alemany et al., 2005; Lukito et al., 2005; Olmo et al., 2003; Barba et al., 2006; Mila and Vallet- Regm, 2001; Wei-Hong et al., 2006; Xia and Chang, 2006; Brinker and Scherer,1990). In the present study the procedure was carried out also without catalyst and this leads to delay in hydrolysis and condensation of TEOS and gelling time to produce powder. The powder was obtained after 7 days instead of 10 hr in case of the use of HNO 3 as a catalyst. Meiszterics and Sink´o, 2007 reported that gelling time of the

Nanomaterials, Ceramic Bulk and Bioceramics

245

powder be based on the kind of the catalyst and the porosity of the gel (Saravanapavanand Hench, 2003).On the other hand, Brinker and Scherer, 1990 considered that the gelation is essentially depends on the ratio of H2O/TEOS. At lower ratios, the gelation time became quicker. Water is added as an initiator and it can be generated by condensation reactions, e.g., esters by a condensation of carboxylic acids and alcohols (Iarry, 1991). In some researches, ethyl alcohol could be used as a solvent, where hydrolysis takes place by water into a non-aqueous solution as alcohols, whilist Saravanapavan and Hench, 2003 reported that CaO ratio is < the theoretical value. This is attributed to the variation in the leaching of cations during aging and drying stages when the process was carried out in presence of the pore liquor (ethyl alcohol). In the present study ethyl alcohol did not used. After a transparent green sol was obtained, The sol was held at 60°C (aged) to reach to a viscosity close the gel point, where transparent gel was formed (in the present procedure two hours at 60°C, the gelation began. This temperature was in agreement with other research (Saravanapavan and Hench, 2003; Balamurugan et al., 2006; Crayston, 2003; Lao et al., 2006; Balamurugan et al., 2005; Liu and Miao2004; Wang et al., 2008; Lao et al., 2007), and it was chosen according to observation of temperature at which the solution become completely miscible or in other word when the solution become completely miscible the temperature already reach to 60°C. So, we kept the solution at this temperature to obtain transparent gel system. Hydrolysis and condensation continue giving an increase in viscosity (Chen et al., 1999; Saravanapavan and Hench, 2003). Minimizing of gelling time and rising of Ca2+ ions are due to a lack of ionic charges on sol-gel by the salt (Saravanapavanand Hench, 2003; Wang et al., 2003). Gupta et al., 2005 reported that gelation increases the viscosity, shrinkage, weight loss, but decreases the pore size distribution. Scherer, 1999 also reported that after the gelation, the chemical reactions still proceeded, increased the rigidity and shrinkage. Gel was dried slowly up to 110°C to remove water and also adsorbed moisture from pores. Heat will densify the gel by removing of water.Then, discs transformed from transparent to translucent opaque. The drying

246

H. H. M. Darweesh

process yields a powder-like product. Drying of the gel was achieved by the gradual increase of temperature 110°C in order to avoid cracking. Viitala et al., 2002 reported that drying process is very important to obtain homogeneous and free crack products. Crayston, 2003 reported that drying determines the nature of final products. Scherer, 1999 reported that on drying, a capillary pressure develops a higher shrinkage and so, the modulus increases and shrinkage stops, where the slow drying reduces gradients and damage. Saravanapavan and Hench, 2003 noted that at lower Ca2+ ions, the gels were white in colour, whereas the other compositions ranged from light yellow to yellow due to the increase of NO3-1 ions in the gels. When wet gels were exposed to the atmosphere, the discoloration was stronger and a light yellow was obtained. After drying, the gel networks were converted to white powder. The dried gels were calcined at 500, 600 and 800˚C for two hours soaking with a heating rate of 5°C/min. Objects are to eliminate the organic content, and to achieve nitrate removal and further densification. Calcination temperature 800°C gives no weight loss. Therefore, it completes elimination of nitrate and the calculated amount of wollastonite was obtained.The holding time was two hours calcination time that provides a sufficient time to create additional nucleation sites in glass powders. Wollastonite powder was calcined at 500°C in some researches (Siriphannon et al., 2002; Viitala et al., 2002; Binnaz and Hazar, 2007). Other researches calcined wollastonite at 600°C (Alemany et al., 2005; Hayashi et al., 1999), whereas other researches calcined wollastonite at 700°C (Wang et al., 2008; Padilla et al., 2005; Long et al., 2006), in addition to some researches (Hayashi et al, 1999; Lin et al, 2005; Longet al, 2006) that calcined it at 800°C. The holding time used in this study for calcination was used by some researches (Siriphannon et al., 2002, 120, 126, Viitala et al., 2002, Padilla et al., 2005; Long et al., 2006; Lin et al., 2005; Longet al., 2006); Haiyan and Chang, 2004). Saravanapavan and Hench, 2003 noted that there is an initial endothermic process around 100°C which can be attributed to loss of residual water and ethanol. This weight loss over 450°C is due to the loss

Nanomaterials, Ceramic Bulk and Bioceramics

247

of organics. The second endothermic peak occurs at 55°C with a weight loss of 35 % which is attributed to the loss of nitrates. Alemany et al., 2005 found that the TG curve indicated that a weight loss was observed at 0150°C and 150-600°C. Then, it became stable to 1350°C. DTA curve showed that four endothermic peaks at 150, 570, 560 and 1200°C, respectively. To achieve NO3- elimination and more densification, Lao et al., 2006 heated the gel powder at 700°C for 24 hours soaking. Liu and Miao, 2004 reported that at 800°C, the bioglass powder still kept the amorphous crystals, but on further increase of temperature, crystallization occurs. Padilla et al., 2005 stated that materials fired at 700°C and 800°C, the XRD showed typical amorphous materials. Meiszterics and Sink´o, 2007 showed that according to the TG curves, heating of Ca silicate gel systems should be carried out ≥ 600°C. After calcinations step, powders were ground and sieved to obtain 63 µm particulates. This particle size was chosen by other researches (Olmo et al., 2003; Hayashi et al., 1999; Long et al., 2006; Haiyan et al, 2004), where the range of particle size was from 32 to 63 µm. Sieved powder that pressed into a cylinder with 15 mm in diameter and under different compaction loads to obtain the optimum load. A pressure of 60 KN was chosen on the basis of some experiments which will be discussed later. To obtain a hard unit, cylindrical specimens of CaSiO3 were sintered for 3 hours in an electric furnace at 900-1250°C, where samples were let to cool to room temperature. Lin et al., 2004 and 2005 confirmed that the increase of holding time from 1-3 hours increased mechanical properties and remained constant with further increase. So, 3 hrs were the optimum condition to achieve the best mechanical strength of CaSiO3 ceramics. Also, other investigators (Mila and Vallet- Regm, 2001; Xia and Chang, 2006; Hayashi et al, 1999; Haiyan and Chang, 2004; MeseguerOlmo et al, 2008; Lin, 2004; Wu et al, 2007; Kokubo et al, 2003; Wan et al, 2008; Arstila et al, 2007; Liu, and Ding, 2002) chose 3 hrs as holding time and this was the basis for our choice, while the sintering temperatures were chosen on the basis that there is no weight loss in the produced powder after 800°C. In addition, from the CaO/SiO2 phase diagram, the

248

H. H. M. Darweesh

phase transition temperature of β-CaSiO3 and α-CaSiO3 is 870°C (from amorphous to β-CaSiO3) and 1125°C (from β-CaSiO3 to α-CaSiO3; Hayashi et al., 1999), respectively. Furthermore, the XRD analysis was done to measure crystalline phases present at these temperature and to be sure that wollastonite is the major phase without any traces of impurities. Finally these range of temperature (900-1300°C) was the range which used by other researches (Balamurugan, 2006; Crayston, 2003; Lukito et al., 2005; Lin et al., 2005; Long et al., 2006; Haiyan and Chang, 2004; Meseguer-Olmo et al,, 2008; Lin, 2004; Wu et al., 2007; Kokubo et al., 2003). Xia and Chang, 2006 stated that the DSC showed a sharp exothermic peak at 832°C, which is belongs to β-CaSiO3, and a broad exothermic peak at 1142°C, which is related to α-CaSiO3. Saravanapavan and Hench, 2003 reported that gel-glasses produced are confirmed to be amorphous even after stabilization at 600°C. Arstila et al., 2007 stated that wollastonite type glasses crystallized at 900°C. Liuand Ding, 2002 stated that the sintering did not < 1300°C due to CaO–SiO2 binary phase diagram, the tridymite and metastable phases of quartz are easy to precipitate from the wollastonite that melts at 1436°C. The lower sintering temperature in sol-gel process was explained by Phulé and Wood, 2001. The conversion of a sol into a crystalline ceramic or glass can be often accomplished at lower temperatures than those in traditional units. Crayston, 2003 attributed the Lower temperature synthesis to the homogeneity and the smaller particle size, nucleation and growing of crystalline phases of dried powder/gel units that can occur at lower temperatures. While Saravanapavanand Hench, 2003 suggested that the excess free energy of a gel to a glass of the same composition, a gel might be changed to glass at lower temperatures, i.e., below the liquid phase temperature of the same composition. Bioactive glasses and glass-ceramics represent wollastonite in the study which suggested some biomaterials for a bone tissue regeneration. The use of Al2O3, ZrO2 and TiO2 could widen their application fields which represented a bioinert material with the bioactivity of wollastonite as implant and dental implants. Materials which used as additives to improve mechanical properties and used it in load-bearing bone substitution were

Nanomaterials, Ceramic Bulk and Bioceramics

249

chosen due to its high wear resistance, their excellent scratch resistance, good frictional properties and fracture toughness (Nalwa, 2000; Tseng and Nalwa, 2009; Ashby, 2009). The high stability of oxide ceramics as low corrosion and low ion release is due to the high heat of formation of their molecules, in addition to its biocompatibility with the physiological environment, that make these materials reliable for a variety of applications as biomedical applications including knee-prostheses and dental implants. Hence, ceramic oxides of Al2O3, TiO2 and ZrO2 have been introduced as reinforcing agents (Kokubo et al., 2003; Wan et al., 2008; Arstila et al., 2007; Liu, and Ding, 2002; Wei-hong et al., 2006). Sol-gel helps the formation of nano-particles, nano-films, and nanoporous membranes (Figure 40). A precursor solution in a solvent, e.g., alkoxides and cations can form a colloidal suspension (SOL-GEL) due to some polymerization reactions, which by adding a surfactant the dispersed particles could be kept in a suspension used to extract the particles for more processings to be a substrate. Solvent evaporation creates dense or nano-porous films. Sol-gel techniques can develop many new materials as paints, nano and/or bioceramics (Wang et al., 2008; Lao et al., 2007; Binnaz and Hazar, 2007; Padilla et al, 2005; Long et al., 2006; Long et al., 2006; Meseguer-Olmo et al., 2008; Wu et al., 2007; Wan et al., 2008; Arstila et al., 2007; Peng et al., 2004).

CONCLUSION 

It is well known recently that nanotechnology is one of the most exciting disciplines, and it incorporates physics, chemistry, materials sciences, biology, cement and building materials, ceramic and bioceramics industries, biomaterials, medicine and many others. In this chapter, the author interests with using the nanomaterials to prepare the ceramic batches containing ultrafine and nono-raw materials to indicate the importance of

250

H. H. M. Darweesh





nanomaterials and/or nanoparticles for improving the physical and chemomechanical properties of the resulting bioproducts. It could be concluded that the use of nanoparticles of ceramic materials as clay, limestone, quartz, feldspar and many others achieved better properties particularly the mechanical strength like tensile and bending strengths than those of the traditional particles. The crystals of the resulting ceramic products are sharp and well defined. The morphology or external appearance of the fired units is well developed. The nano - and/or biomaterials are recently used to prepare the ceramic batches containing ultrafine or nanoparticles to produce different shapes and sizes of bone scaffolds to place them inside the human body to lighten the pains of patients subjected to traffic accidents and lose a part or more of their body bones. In the future, the scientists must interest to increase and look for new and more effective materials suitable to produce artificial bone scaffolds.

REFERENCES Abdel-Fattah, W. A. & Nour, F. A., Utilization of electrostatic precipitator dust in borcelain, XIII- Silicon, for silicate Industry and Science Conferences, Budapest, June, 1981. Abdel-Fattah, W. I. & Ei-Didamony, H., Thermal investigation on electrostatic precipitator kiln dust,” Thermochimica, Acta, 51, 1981, 297-306. Abdel-Fattah, W. I.; El-Didamony, H. & Taha, A. S., Synthesis and thermal characteristics of Ca rich phases,” TIZ-Fachberichte, 106 (5), 1982, 351-354. Aggarwal, P. S.; Lele, R. V. & Sen, S. K., Utilization of fly ash for making wall tiles, Central Glass and Ceramic Res. Inst. Calcotta, 700 012 India, Cent. Glass Ceram. Res. Inst. Bull., 27 (1-2), 1980.

Nanomaterials, Ceramic Bulk and Bioceramics

251

Alemany, M. I.; Velasquez, P.; de la Casa-Lillo, M. A. & De Aza, P. N., Effect of materials processing methods on the in vitro bioactivity of wollastonite glass-ceramic materials, J. Non-Crystalline Solids 351, 2005, 1716–1726. Anlagenbau, A., Ceramic tiles, GmbH, Ger. Offen. DE 3 313, 594 (CI. CO 4B33/34), Oct., 18, 1984. Chem. Abst. 102:49975 k, 1995. Arstila, H.; Vedel, E.; Hupa, L. & Hupa, M., Factors affecting crystallization of bioactive glasses,” Journal of the European Ceramic Society 27, 2007, 1543–1546. Ashby, M. F. & Jones, D. R. H., Engineering materials 1, 3rd ed., Elsevier Butterworth Heinemann, ISBN 0-7506-6380-2, 2006. Ashby, M. F. Ferreira, P. J. & Schodek, D. L. “Nanomaterials, Nanotechnology and Design,” Text Book, Elsevier Ltd, ISBN: 978-07506-8149-0, 2009. Ashby, M. F., Materials selection in mechanical design, 3rd ed., Butterworth-Heinemann, Elsevier, 2005. Ashby, M. F., Shercliff, H. R. & Cebon, D. Materials: engineering, science, processing and design, Butterworth Heinemann, ISBN-13: 978-0-7506-8391-3, 2007. Askeland, D. R. & Phule, P. P., The science of engineering materials, 5th ed., Thomson Publishing, 2006. ASTM-Specification, C326-32-76, “Standard Test Method for shrinkage of ceramic white ware clays after drying and firing,” Part, 17: 1980, 266267. ASTM-Specification, C373-72, “Standard Test Method for water absorption, bulk density, apparent porosity and specific gravity of whiteware products,” Part, 17: 1980, 308-309. ASTM-Specification, C674-71, Standard Test Method for flexural properties of ceramic white ware Products,” Part, 17: 1980, 668-671. Bacanac, M. & Babic, M., Materials for ceramic tiles with a backing temperature of 1080C and use of Trokyte, Technika, Belgrade, 39 (4), 1984, 1-5. Balamurugan, A.; Balossier, G.; Laurent-Maquin, D.; Pina, S.; Rebelo, A. H. S.; Faure, J. & Gerreira, J. M. F., An in vitro biological and

252

H. H. M. Darweesh

antimactobial study on a sol-gewl derived silver-incorporated bioglass system,” Dental Materials, 24, 2008, 1343-1351. Balamurugan, A.; Sockalingum, G.; Michel, J.; Fauré, J.; Banchet, V.; Wortham, L.; Bouthors, S.; Laurent-Maquin, D. & Balossier, G., Synthesis and characterization of sol gel derived bioactive glass for biomedical applications, Materials Letters 60, 2006, 3752–3757. Barba, I.; Conde, F.; Olmo, N.; Lizarbe, M. A.; García, M. A. & ValletRegí, M., Vitreous SiO2–CaO coatings on Ti6Al4V alloys: Reactivity in simulated body fluid versus osteoblast cell culture, Acta Biomaterialia, 2, 4, 2006, 445-455. Barba-Izquierdo, I.; Salinas, A. J. & Vallet-Regi, M., “In vitro calcium phosphate layer formation on sol-gel glasses of the CaO-SiO2, J. Biomed. Mater. Res., 47, 2000, 243-250. Belous, K. P.; Kushel, M. I.; Evplov, Yu. N. & Ohkrimenco, G. M., “Effects of technological factors on the strength of ceramic materials,” Inst. Probl. Prockn., Kiev, USSR, 2, 1989, 42-49. Binnaz, A. Hazar, Y., Preparation and in vitro bioactivity of CaSiO3 powders, Ceramics International, 33, 2007, 687–692. Brinker, C. J. & Scherer, G. W., The physics and chemistry of sol-gel processing, Academic Press Inc., San Diego, CA, USA, 1990. Budnikov, P. P., The technology of Ceramics and Refractories, Edward Arnold Publishers Ltd., London, 1964. Callister, W. D. Jr., Materials science and engineering, an introduction, 7th ed., Wiley, ISBN 0-471-73696-1, 2007. Chen, Q.; Miyaji, F.; Kokubo, T. & Nakamura, T., Apatite formation on PDMS-modified CaO-SiO2-TiO2 hybrids prepared by sol-gel process, Biomaterials 20, 1999, 1127-1132. Chen-Chi, W. & Jackie, Y., “Sol−Gel Synthesis and Hydrothermal Processing of Anatase and Rutile Titania Nanocrystals.” Chemistry of materials, 11 (11): 1999, 3113–20. Chiang, Y. M.; Birnie, D. P. & Kingery, W. G., “Physical Ceramics Principals for Ceramic Science and Engineering,” 3rd Edn., John Wiley and Sons, Lehigh Press. Inc., USA, 1997.

Nanomaterials, Ceramic Bulk and Bioceramics

253

Crayston, J. A, Sol-gel, Comprehensive Coordination Chemistry II 1, ISBN 0-08-0443230, 2003, 711–730. Drexler, K. E., Peterson, C. & Pergamit, G., Unbounding the future: the nanotechnology revolution William Morrow, New York, 1991. Daniel, A. & Vallet-Regi, M., Sol-gel silica based biomaterials and bone tissue regeneration, Acta Biomaterialia, 6, 2010, 2874-2888. Drouet, C. R. C.; Sfihi, H. & Barroug, A., Physico-chemical properties of nanocrystalline apatites: Implications for biominerals and biomaterials, Materials Science and Engineering C 27, 2007, 198–205. Darweesh, H. H. M., Utilization of Cement Kiln Dust in Ceramics to minimize Environmental Pollution, M. Sc. Thesis, Inst. Environmental Studies and Researches, Ain Shams Univ., 1992. Darweesh, H. H. M., “Ceramic Wall and Floor Tiles Containing Local Waste of Cement Kiln Dust - Part I: DensificationParameters,” American Journal of Environmental Engineering and Science, 2, 5, 2015, 35-43. (http://www.aascit.org/journal/ajees), ISSN: 2381-1153. Darweesh, H. H. M., “Ceramic Wall and Floor Tiles Containing Local Waste of Cement Kiln Dust- Part II: Dry and Firing Shrinkage as well as Mechanical Properties,” American Journal of Civil Engineering and Architecture, 4, 2, 2016, 44-49. DOI: 10.12691/ajcea-4-2-1, (http:// pubs.sciepub.com/ajcea/4/2/1). Darweesh, H. H. M. & El-Din, M. N., Characterization of some Egyptian Granites and Marbles as dimensional stones, Bull. NRC, Egypt, 25, 4, 2000, 323-339. Darweesh, H. H. M.; Awad, H. M. &Tawfik, A., Red Bricks from Dakhla Formation Clay - Tushka area-Incorporated with some Ind. Wastes or byproducts” Industrial Ceramics, Italy, Vol. 31(3) 2011, 201-207. Darweesh, H. H. M. & El-Meligy, M. G., Non-conventional light-weight Clay bricks from homra and kraft pulp wastes, J. Chemistry and Materials Research, 1(4) 2014, 1-7. Darweesh, H. H. M.; Wahsh, M. M. S. & Negim, E. M., Densification and Thermomechanical Properties of Conventional Ceramic Composites Containing Two Different Industrial Byproducts,” Amer.-Eurasian Journal of Scientific Research, 7 (3), 2012, 123-130.

254

H. H. M. Darweesh

Darweesh, H. H. M., Building materials from siliceous clay and low grade dolomite rocks, Ceramics International, 27, 2001, 45-50. Darweesh, H. H. M., “Utilization of Cement Kiln Dust Waste as a source of CaO in Ceramic Industry,” Sil. Industrials (Ceramic Science and Technology), 66 (3-4) 2001, 47-52. Darweesh, H. H. M. & Khalil, N., Effect of cement bypass dust waste on the physico-mechanical properties of alumina cement, Ind. J. Chem. Techn. India, 7, 2001, 287-291. Danto Co. Ltd. “Decorative tiles,” Jpn. Kokai, 79/70, 250, Jan. 14, 1979. Drews, M., “Wall and floor tiles,” Ceramic Monographs 2.4.1. Verlag Schmid GmbH, 1983. El-Alfi, E. A.; Radwan, A. M. & Darweesh, H. H. M., Effect of sand as non-plastic material on ceramic properties of clay bricks, Inter Ceram (Intern. Cer. Review), 53 (5) 2004, 330-333. El-Hady, B. I.; El-Kady, A. M. & Hassan, M. M. A., Preparation and characterization of biocomposites for localized bone treatment based on Agarose and Gelatin, Ph. D. Thesis (Biophysics), University College of Women, Ain Shams University, 2015. Fiori, C. & Fabbri, B., Granite containing bodies for the production of Stoneware tiles, Res. Inst. Ceram. Techn. National Res. Council, Faenza, Italy, Inter Ceram 32 (1) 1980, 21-22. Genin, G., Influence of Quartz on the strength of porcelain, Stekelo. Keram., 15, 1958, 35-41. Giomlem, K. & Lyng, S., Anorthosite as a ceramic raw material-Part II; Anorthosite for tile manufacture, Trans. Brit. Ceram. Soc. 73, 1974, 139146. Grim, R. E., Clay Mineralogy, McGraw Hill, NewYork, 1962. Gupta, R.; Mozumdar, S. & Chaudhury, N. K., “Fluorescence spectroscopic studies to characterize the internal environment of tetraethyl-orthosilicate derived sol–gel bulk and thin films with aging,” Biosensors and Bioelectronics 20, Issue 7, 2005, 1358-1365. Haiyan, L. & Chang, J., Fabrication and characterization of bioactive wollastonite /PHBV composite scaffolds, Biomaterials, 25, 2004, 5473-5480.

Nanomaterials, Ceramic Bulk and Bioceramics

255

Hayashi, S.; Nakagawa, Z.; Yasumori, A. & Okada, K., Effects of H2O in EtOH-H2O disperse medium on the electrophoretic deposition of CaSiO3 Fine Powder,” Journal of the European Ceramic Society, 19, 1999, 75-79. Hench, L. L. & West, J. K., The sol-gel process, Chemical Review, 1990, 90-133. Iarry, H. L., Bioceramics from concept to clinical, Amer. Cer. Soc., 74, 1991, 1487-1491. Ibrahim, D. M. M; Abdel-Fattah, W. I. & Nour, F., Air-cooled blast furnace slag as a fluxing material in the Ceramic Industry, Ber. Dt. Keram. Ges. 57, 9-10, 1980, 187-194. Jackson, G., “Introduction to white wares,” Elsevier Publishing Co., 1st Edn., 1969. Joachim, S.; Manfred, S. & Peter, T., Ceramic composition for lightcolored gehlinite-free glazed earthenware tiles, VEB kombinat Fliesen and Sanitaerkeramic “Kurt Buerger,” Ger. (East), DD 205, 883, CI CO 4B33 /00, Jpn. Jan, 11, 1984. Julian, R. J.; Lisa, M. E. & Larry, L. H., Optimising bioactive glass scaffolds for bone tissue engineering, Biomaterials, 27, 7, 2006, 964973. Julie, C. D., “R & D 2002: Nano Ceramics.” MIT Technology Review. Retrieved Dec. 1, 2002. Kalnin, I. L., Strength & elasticity of Whitewares-Part I; Relation between Flexural strength and elasticity, Ceram. Bull., 46, 1174, 1967, Kalnin, I. L.; Cucka, P.; Warshow, S. I. & Seider, R., “Strength and elasticity of Whitewares,” Ceram. Bull., 47, 1968, 498-503, Kashlyak, L. L. & Kareev, Yu. B., Production of tiles for floor and Façade tiles in the Leningrad Ceramic Products Plant, Keram from St. 8, 10, 1978. Ke, S.; Wang, Y.; Pan, Z.; Ning, C. & Zheng, S., Recycling of polished tile waste as a main raw material in porcelain tiles, J. Cleaner Prod. 115, 2016, 238–244. Keijiro, S.; Takashi, o.; Yoshifumi, K.; Yoshiyuki, O; Masaru, T. & Kozo, M., Manufacture of ceramics containing Portland cement, Sumitomo

256

H. H. M. Darweesh

Cement Co. Lmt., Jpn. Kokai, Tokyo Koho JP, o3, 215, 350, 1991, [91, 215, 350] (C1. CO 4B35/16). Chem. Abst. 115: 285800p. Khalil, A., “Advanced Sintering of Nano-Ceramic Materials.” Ceramic Materials - Progress in Modern Ceramics, 2012. Khrundzhe, A. V. & Babushkin, V. I., State Reginal electric power plant wastes used for the manufacture of ceramic tiles, USSR, Steklo, Keram. 3, 1983, 6-8. Kim, K, and Hwang, J., Characterization of ceramic tiles containing LCD waste glass, Ceram. Int. 42, 2016, 7626–7631. Kimm, F., “Ceramics Don't Have to Be Brittle.” Cal Tech. Retrieved 11, 2014. Kimm, F., “Made-to-Order Materials.” Cal Tech. Retrieved 5, 2013. Kingery, W. D.; Bowen, H. K. H. K. & Uhlmann, D. R, Introduction to Ceramics, 2nd edn., John Wiley and Sons, New York, London, Sydney, Toronto, 361, 594, 1975. Kisel, I. I.; Strodyreva, G. V. & Demidovisch, V. B., Possibilities of using local clay in manufacturing floor tiles, Steklo, Sitally Silik, 1980. Klein, I. Sol-gel optics: Processing and applications, Springer Verlag, ISBN 0792394240, 1994. Koch, C. C., Nano structured materials: Processing properties and applications, Noyes Publications, 2002, ISBN 0-7155-1451-4. Kokubo, T., Bioactive glass ceramics, Properties and applications,” Biomaterials, 12, 1991, 155-163. Kokubo, T.; Kim, H. & Kawashita, M., Handbook of Advanced Ceramics, 2003, 385-416. Chapter 14 - 14.1 Ceramics for Biomedical Applications. Kong, Y. M.; Kim, S. & Kim, H. E., Reinforcement of hydroxyapatite bioceramics by addition of ZrO2 coated with Al2O3, Amer. Cer. Soc., 82, 1999, 2963-2968. Konta, J., Properties of ceramic raw materials, Ceramic Monograph 1.1.4. Verlag Schmid G.m.b.H, 1981. Lao, J. Nedelec, J. M.; Moretto, Ph. & Jallot, E., Biological activity of a SiO2–CaO–P2O5 sol–gel glass highlighted by PIXE–RBS methods,

Nanomaterials, Ceramic Bulk and Bioceramics

257

Nuclear Instruments and Methods in Physics Research B 261, 2007, 488–493. Lao, J.; Nedelec, J. M.; Moretto, Ph. & Jallot, E, Micro-PIXE characterization of interactions between a sol–gel derived bioactive glass and biological fluids, Nuclear Instruments and Methods in Physics Research B 245, 2006, 511–518. Lin, K., Preparation of macroporous calcium silicate ceramics, Materials Letters 58, 2004, 2109–2113. Lin, K.; Zhai, W.; Ni, S.; Chang, J.; Zeng, y. & Qian, W., Study of the mechanical property and in vitro biocompatibility of CaSiO3 ceramics, Ceramics International, 31, 2005, 323–326. Liu, J. and Miao, X., Sol–gel derived bioglass as a coating material for porous alumina scaffold, Ceramics International 30, 2004, 1781–1785. Liu, X. & Ding, Ch., Characterization of plasma sprayed wollastonite powder and coatings, Surface and Coatings Technology, 153 (2-3) 2002, 173-177. Long, L. H.; Chen, L. D. & Chang, J., Low temperature fabrication and characterizations of β-CaSiO3 ceramics” Ceramics International, 32, 2006, 457–460. Long, L. H.; Chen, L. D.; Bai, S. Q.; Chang, J. & Lin, K. L., Preparation of dense β-CaSiO3 ceramic with high mechanical strength and HAp formation ability in simulated body fluid, J. European Ceramic Society 26, 2006, 1701–1706. Lukito, D.; Xue, J. M. and Wang, J., In vitro bioactivity assessment of 70 (wt.) %SiO2–30 (wt.) %CaO bioactive glasses in simulated body fluid, Materials Letters 59, 2005, 3267 – 3271. Martín-Márquez, J.; Rincón, J. Ma & Romero, M., Effect of firing temperature on sintering of porcelain stoneware tiles, Ceram. Int. 34, 2008, 1867–1873. Meiszterics, A. & Sinkó, K., Sol–gel derived calcium silicate ceramics, Colloids and Surfaces A: Physicochem. Eng. Aspects, 2007. Meseguer-Olmo, L.; Bernabeu-Esclapez, A.; Ros-Martinez, E.; Sa´nchezSalcedo, S.; Padilla, S.; Martìn, A. I.; Vallet-Regì, M.; Clavel-Sainz, M.; Lopez-Prats, F. & Meseguer-Ortiz, C. L., In vitro behaviour of

258

H. H. M. Darweesh

adult mesenchymal stem cells seeded on a bioactive glass ceramic in the SiO2–CaO–P2O5 system, Acta Biomaterialia, 4, 2008, 1104-1113. Mila, R. A. & Vallet- Regm, M., Static and dynamic in vitro study of a solgel glass bioactivity, Biomaterials 22, 2001, 2301-2306. Mohamed, E. M., Inorganic –organic hybrids based on PVA and Silica with added Titania, M. Sc. Thesis (Biophysics), University College of Women, Ain Shams University, 2012. Moroz, B. I.; Gumenyuk, A. E.; Mel’nekov, V. M. &Trubachov, V. I. “Bodies for the manufacture of ceramic tiles,” USSR, Stroit. Mater. Konstr., 2, 17, 1990, 2, 17. Chem. Abst. 114: 10710y, 1991. Mortel, H., Porcelains for fast firing, Inter Ceram, 3, 1977, 65-69. Mostafa, M. Z. & Darweesh, H. H. M., “Mechanical properties of ceramic floor tiles containing local waste of cement kiln dust,” The 8th SIMCER Intern. Symposium on Ceramics, Rimini (Italy)-Exhibition Centre, Nov. 10-12, 1992. Mukhamedzhanov, M. T.; Palvanov, V. P.; Irkahodzhaeva, V. P. & Toirova, M. Kh., The use of mining tailings in the manufacture of ceramic tiles, Inst. Sredazuiprotsvtmet, Almalyk, USSR, Kompleksen. Ispol’z. Miner. Syr’ya, 11, 1990, 82-84. Chem. Abst. 114: 212495n, 1991. Nalwa, H. S. Ed., Encyclopedia of Nanoscience and Nanotechnology, Volumes 1-25, American Scientific Publishers, Los Angeles, CA (2004/2011). Nalwa, H. S. Ed., Handbook of Nanostructured Materials and Nanotechnology, Volumes 1-5, Academic Press, San Diego, CA, 2000. Newport, R. J.; Skipper, L. J.; FitzGerald, V.; Pickup, D. M.; Smith, M. E. & Jones, J. R. “In vitro changes in the structure of a bioactive calcia– silica sol–gel glass explored using isotopic substitution in neutron diffraction,” Journal of Non-Crystalline Solids 353, 2007, 1854–1859. Nissan, B., “Nanoceramics in Biomedical Applications.” MRS Bulletin 29 (1): 2014, 28–32. Njindam, O. R.; Njoya, D.; JMache, R.; Mouafon, M. Messan, A. & Njopwouo, D., “Effect of glass powder on the technological properties and microstructure of clay mixture for porcelain stoneware tiles

Nanomaterials, Ceramic Bulk and Bioceramics

259

manufacture,” Construction and Building Materials 170, 512–519, www.elsevier.com/locate/conbuildmat, 2018. Nuttawat, K.; Jaimasith, M. & Thiemsorn, W., Fabrication of ceramic floor tiles from industrial wastes, Suranaree J. Sci. Technol. 21, 2013, 65– 77. Olmo, N.; Ana, I. M.; Antonio, J. S.; Turnay, J.; Vallet-Reg, M. & Lizarbe, M. A., Bioactive sol–gel glasses with and without a hydroxycarbonate apatite layer as substrates for osteoblast cell adhesion and proliferation, Biomaterials 24, 2003, 3383–3393. Padilla, S.; Román, J.; Carenas, A. & Vallet-Regì, M., The influence of the phosphorus content on the bioactivity of sol–gel glass ceramics, Biomaterials, 26, 2005, 475–483. Peng, F.; Liang, K.; Hu, A. &Shao, H., “Nano-crystal glass-ceramics obtained by crystallization of vitrified coal fly ash,” Fuel 83, Issues 1415, 2004, 1973-1977. Phulé, P. P. & Wood, T. E., Ceramics and Glasses, Sol-Gel Synthesis of” Encyclopedia of Materials: Science and Technology, ISBN: 0-080431526, 2001, 1090-1095. Pinalova, D.; kukushiva, M. & Belchev, K., Wastes from Coal benefication for effective manufacture of ceramic products, Stroit. Mater. Silik., 25, 6, 1984, 14-15. Pitskhelauri, I. A.; Lomidze, N. M.; Chkheidze, K. S.; Sakandlidze, A. A. & Zambakhidze, N. P., Raw mixtures for producing ceramic wall articles,” Tbilisi State Academy of Arts, USSR, Su 1, 357, 392 (C1./ CO 4B33/00), Dec. 7, 1987. Chem. Abst. 108: 100003v, 1988. Poole, C. P. Jr., & Owens, F. J., Introduction to nano technology, WileyInterscience, ISBN 0-471-07935-9, 2003. Rayan, R. W. & Radford, C., “White wares production, testing and quality control,” 1st Edn. The Institute of Ceramics, Pergamon Press. Oxford, New York, Beijing, Sedney, Frankfurt, Sao Paulo, Tokyo, Toronto, 1987. Rayan, W., “Properties of ceramic raw materials,” 2nd Edn. Pergamon Press, 1978.

260

H. H. M. Darweesh

Saltyvskaya, I. M., Use of chemical industry byproducts for the manufacture of ceramic tiles, (Khar’k. Inzh.-Ekon. Inst. Khakov, USSR). Steklo Keram, 9, 19-20. Chem. Abst. 102: 30662j, 1985. Saravanapavan, P. & Hench, L. L., Mesoporous calcium silicate glasses. I. Synthesis, J. Non-Crystalline Solids 318, 2003, 1-13. Scherer, G. W., Structure and properties of gel,” Cem. Concr. Res., 29, 8, 1999, 1149-1157. Schuller, K. H. & Jager, H., The chemistry and properties of feldspar and their effects in porcelain, Ber. Dt. Keram. Ges., 56, 29, 1979. Schuller, K. H., Reactions between Mullite and glassy phase in Porcelains, Trans. Brit Ceram.Soc., 63, 1964, 103-110. Shackelford, J. F., Introduction to materials science for engineers, 6th ed., Prentice Hall, 2004. Singer, F. & Singer, S. S., “Industrial Ceramics,” 2nd Edn. Leipziger, Druchkaus, GDR, 1971. Siriphannon, P.; Kameshima, Y.; Yasumori, A.; Okada, K. & Hayashi, S., Formation of hydroxyapatite on CaSiO3 powders in simulated body fluid, J. European Ceramic Society 22, 2002, 511–520. Smith, W. F. & Hashemi, J., Foundations of materials science and engineering, 4th ed., McGraw Hill, 2005. Sobolev, K. & Ferrada-Gutiérrez, M., How nanotechnology can change the concrete, world: Part 1, Am Ceram Soc Bull, 84 (10), 2005, 14–17. Sonjida, M; Ahsan, M.; Hamid, A. & Ahmed, S., Effect of waste glass powder on physico-mechanical properties of ceramic tiles, Bangladesh J. Sci. Res. 24, 2011, 169–180. Sridhar, K., “Some significant advances in sol-gel processing of dense structural ceramics.” Sol-Gel Science and Technology 6 (2): 1996, 127132. Susan, H., Basic biomaterials, 5th Edn., 88 ISBN 0071260412, 2007. Taha, R. T. A., Bioceramic composites suitable for bone graft, M. Sc. Thesis (Biophysics), University College of Women, Ain Shams University, 2011. Todor, D. N., Thermal analysis of minerals, English Edn., Abacus Press, 1972.

Nanomaterials, Ceramic Bulk and Bioceramics

261

Tseng, T. Y. & Nalwa, H. S., Handbook of Nanoceramics and Their Based Nanodevices, Volume 1-5, American Scientific Publishers, Los Angeles, CA, 2009. Vauequelinet, M; Gault, C & Boch, B., Reheology of clay mixture ceramic bodies, Ind. Ceram. 744, 1980, 736-740. Viitala, R.; Jokinen, M.; Peltola, T.; Gunnelius, K. & Rosenholm, J. B., Surface properties of in vitro bioactive and non-bioactive sol–gel derived materials, Biomaterials 23, 2002, 3073–3086. Vorrada, L.; Panyachai, T.; Kaewsimork, K. & Siritai, C., Effects of recycled glass substitution on the physical and mechanical properties of clay bricks, Waste Manage. (Oxford) 29, 2009, 2717–2721. Wan, X.; Hu, A.; Li, M.; Chang, Ch. & Mao, D., Performances of CaSiO3 ceramic sintered by Spark plasma sintering, Materials Characterization 59, 2008, 256-260. Wang, E.; Chow, K.; Kwan, V.; Chin, T.; Wong, C. & Bocarsly,A., Fast and long term optical sensors for pH based on sol–gels, Analytica Chimica Acta, 495 (1-2) 2003, 45-50. Wang, H.; Sockalingum, G.; Michel, J.; Fauré, J.; Banchet, V.; Wortham, L.; Bouthors, S.; Laurent-Maquin, D. & Balossier, G. Synthesis and microwave dielectric properties of CaSiO3 nanopowder by the sol–gel process,” Ceramics International 34, 2008, 1405-1408. Warshow, S. I. & Seider, R., Comparison of strength of triaxial porcelains containing alumina and silica, J. Amer. Ceram; Soc., 50, 1967, 337342. Watchman, J. P., Mechanical properties of ceramics-An introductory Survey, Ceram Bull., 46, 1967, 756-762. Wei-hong, Z.; Jin-shu, C.; Jian, Q.; Xian-chun, L. & Jian, L. I., Crystallization and properties of some CaO-A12O3-SiO2system glassceramics with Y2O3 addition, Trans. Nonferrous Met. SOC. China, 16, 2006, 105-108. Wu, Ch.; Ramaswamy, Y.; Kwik, D. & Zreiqat, H., The effect of strontium incorporation into CaSiO3 ceramics on their physical and biological properties” Biomaterials 28, 2007, 3171–3181.

262

H. H. M. Darweesh

Xia, W. and Chang, J., Well-ordered mesoporous bioactive glasses (MBG): A promising bioactive drug delivery system, J. Controlle Release, 110 (3) 2006, 522-530.

In: Mechanical Design, Materials … ISBN: 978-1-53614-791-9 Editor: Sandip A. Kale © 2019 Nova Science Publishers, Inc.

Chapter 8

SIMULATION OF LOW COST AUTOMATION AND LIFE CYCLE COST ANALYSIS FOR A SPECIAL PURPOSE MACHINE Pattanayak Satyajit and Hauchhum Lalhmingsanga Department of Mechanical Engineering, National Institute of Technology Mizoram, Aizawl, India

ABSTRACT The need for simulation of low cost automation and life cycle cost analysis for a special kind of machine is a promising research area in industries. This machine is capable of machining (undercut, head diameter, fillet, chamfering) operation of engine valve, used in automobile industry. The objective of this work is to reduce rejection, lower the cost, increase productivity and quality of performance. In this chapter, a methodology to design, develop and simulate a twin spindle turning Special Purpose Machine is presented based on the data collected from hydraulic, pneumatic, electro pneumatic which will serve as low 

Corresponding Author Email: [email protected]

264

Pattanayak Satyajit and Hauchhum Lalhmingsanga cost automation. The electro pneumatic circuit design presented here is simulated using commercially available Fluid SIM software. Special purpose machine tool development based on special product manufacturing (engine valve) requirements is also presented in this chapter. The design is based on Low Cost Automation principles rather than pneumatic and Programmable Logic Controller principles. A 3D model of the machine is also presented using Pro-E software to represent the implemented components after the design and development process.

Keywords: low cost automation, special purpose machine, productivity, cost reduction, life cycle cost, simulation

NOMENCLATURE Ph S L LS W TC Cp BP

Parts per hour Single part time hours Cost of labour (USD) Lot size Wage rate dollar Tool cost dollar Cost per part dollar Break-even point

ABBREVIATIONS SPM LCA LCC PLC CPU NC DCV

Special Purpose Machine Low Cost Automation Life Cycle Cost Programmable Logic Controller Central Processing Unit Numerical Control Directional Control Valve

Simulation of Low Cost Automation and Life Cycle Cost …

265

1. INTRODUCTION Low Cost Automation (LCA), is the introduction of simple pneumatic, hydraulic, mechanical and electrical devices into the existing production machinery, with a view of improving their productivity (Sulivan, 1989; Albarlos, 1998). This will also enable the operation of this equipment by semi-skilled and unskilled labour, with a little training. This will involve the use of standardised parts and devices to mechanise or automate machines, processes and systems (Esposito, 1994; Freund et al., 2000). In the context of globalisation and liberalisation, quality improvement and cost reduction are two major steps to increase the productivity. Moreover, making any heavy investment is not possible considering many factors like economic recession, slump in demand, risk and lack of funds (Erbe, 1995,1996,2000,2003; Inagaki, 2000; Lay, 2002). One of the very practical, safe, economical and rewarding strategies is the application of Low Cost Automation. LCA should not be regarded in terms of a specified maximum capital outlay, but as an approach to automation using equipment and control devices that are, in general, both technically and economically, within the scope of the company concerned (Legierski et al.,1998; Hohwieler et al., 2002). The lower level technologies can be made highly productive by automation at low cost and in simple form (Ollero, 2002). Life Cycle Cost (LCC) accounts for the total costs from starting point of equipment and projects to their disposal which is derived analytically through an estimation of the total costs for their life time (Carvalho, 1986; Chuang et al. 1992;1993). Usually Engineers used LCC based analysis to justify the selection of equipment and process. This approach helps in choosing the most cost-effective approach with the aim of achieving minimum long-term cost. This is because the initial purchase price is usually much less than the cost of maintenance operation and disposal (Barringer et al., 1996). The total costs anticipated to be incurred in the design, production, development, maintenance, operation, support and disposition over the life span of a major system constitute its life cycle cost. A minimum total of LCC correspond to the optimal balance of the

266

Pattanayak Satyajit and Hauchhum Lalhmingsanga

cost elements (Swider, 2002; Swider et al., 2003;2004). Automation demands replacement of conventional Radial Drilling Machines by Special Purpose Horizontal Multi Spindle Drilling Machine by application of LCA (Gajmal et al., 2014). The machining areas in cellular manufacturing have been proven to be an economic, efficient and lean approach by bringing flexibility. After investigating existing analytical methods for measuring work, the LCA is useful for identifying and quantifying the different manual tasks of cellular manufacturing line (Soloman, 1996; Seifermann et al., 2014). Three machines viz. conventional lathe machine, NC lathe machine and special purpose machine which performs machining works like turning head diameter, chamfering, fillet and undercut or grooving are taken into consideration. In these three machines; select one machine, which is greater as per LCC principle. Hence, economic analysis should be done in these three machines by selecting one machine which is best as per LCC principle and our required objectives like increasing productivity, number of quality products, production cost reduction, reduction of time and avoiding skilful labour can be fulfilled.

2. METHODOLOGY 2.1. Key Main Product Features of Automation Designs of low cost automation in special purpose turning machines have some key features. Following are the main components of Low Cost Automation: 1) 2) 3) 4) 5)

Chute on/off Loader on/off Pusher on/off Ejector on/off Collet on/off

Simulation of Low Cost Automation and Life Cycle Cost …

267

1) Chute is nothing but manually conveying systems of material as shown in Figure 1. Chute is fitted in a machine at certain angle. So, the part is moving down through gravitational forces. Here, chutes door is closed which is operated by pneumatic cylinder. When the cylinder is retracting at that time, the part is moving down to loader plate.

Figure 1. Chute assembly.

2) Loader is loading the product, which comes from the chute as shown in Figure 2. It has one cylinder which is moving down and up by using pneumatic cylinder. When the cylinder is extending at that time, loader is down and set the same line of spindle axis.

Figure 2. Loader assembly.

3) Pusher as shown in Figure 3 is pushing the product from loader to spindle collet. Pusher is also fitted with the same line of spindle axis. So, pusher pushes the product easily in collet in the spindle.

268

Pattanayak Satyajit and Hauchhum Lalhmingsanga

Figure 3. Pusher.

4) Collet is a standard product as shown in Figure 4. The collet end side has thread which holds the spring. Another spring is attached in pneumatic cylinder rod. When cylinder is retracted, spring is pulled to the collet at the same time and collet comes on to hollow spindle and fixed the work part.

Figure 4. Collet.

5) After the machining process, the spindle collet cylinder given in Figure 5 moves forward and pushes the collet. Hence, the part of the collet is loose and at the same time, ejector move forward and eject the part from collet. Ejector, spindle and collet assembly is given in Figure 6.

Figure 5. Spindle collet assembly.

Simulation of Low Cost Automation and Life Cycle Cost …

269

Figure 6. Ejector, spindle and collet assembly.

Figure 7. Assembly of Special Purpose Machine.

2.2. Purpose of Special Purpose Turning Machine The Special Purpose Turning Machine, having two spindle head unit, is used for carrying out the turning head diameter, undercut, chamfering head and neck operations during manufacturing of engine valves. Assembly of Special Purpose Turning Machine is shown in Figure 7. The engine valve manufacturing involves a sequence of operations starting from casting to final finishing, out of which turning head diameter, undercut, chamfering and fillet form four intermittent operations. These operations are required for turning head diameter as per dimension, undercut operation for holding the spring, chamfering operation for good fittings. The Special Purpose Turning Machine will have the combined advantages of a CNC Machine and Special Purpose Machine. This machine is built only for the purpose of turning head diameter, undercut

270

Pattanayak Satyajit and Hauchhum Lalhmingsanga

and chamfering generation in valve manufacturing with computerized controlled.

2.3. Circuit Design 2.3.1. Sequence of Operation A-A+B+C+D-C-B-D+E+EA- Open Chute door cylinder B- Upward Loader Cylinder C- Retraction Pusher Cylinder D- Close Collet Cylinder E- Retraction Ejector Cylinder

2.3.2. Position Step Diagram

Figure 8. Position step diagram.

A+ Close Chute Door Cylinder B+ Downward Loader Cylinder C+ Forward Pusher Cylinder D+ Open Collet Cylinder E+ Forward Ejector Cylinder

Simulation of Low Cost Automation and Life Cycle Cost …

271

2.3.3. Flow Diagram of Step Sequence Operation

Figure 9. Flow diagram of step sequence operation.

2.3.4. Drawing of Pneumatic Circuit The typical pneumatic circuit is shown in Figure 10. This pneumatic circuits have five cylinders, ten flow control valves, eleven 5/2 directional control valves, ten roller type 3/2 DCV for limit switches, one AND valve, One timer etc. This circuit is very complicated and require a big space for assembling.

272

Pattanayak Satyajit and Hauchhum Lalhmingsanga The main disadvantages of this circuit are noted below: • • • • •

Very complicated Large space is required Difficult in assembling and dissembling Difficulty in maintenance. One labour is always required when process is going on, for operating the start button at every cycle.

For these reasons, it is necessary to design electro pneumatic circuit design.

Figure 10. Pneumatic circuit.

Simulation of Low Cost Automation and Life Cycle Cost …

273

3. RESULTS AND DISCUSSION 3.1. Circuit Simulation As per the sequence of operation in Figure 11, the electro pneumatic circuit simulation as per the required sequence of operation in various steps is presented. Dark colour shows air flow to the cylinder from compressor and light colour shows return air from cylinder to exhaust as shown in Figure 12.

Figure 11. Electro pneumatic circuit design.

Figure 12. Simulation of home position of electro pneumatic circuit.

274

Pattanayak Satyajit and Hauchhum Lalhmingsanga

Figure 13. Simulation of second step.

Step 1: Home position of the circuit, (A+ B- C- D+ E-). Figure 12 Shows home position of the circuit. Step 2: The opening of chute cylinder door indicate Y1 solenoid is actuated and cylinder retraction position is (A- B- C- D+ E-). Figure 13 shows that chute cylinder A is retracted. Step 3: Chute Cylinder door is closed as shown Figure 14. When piston is at A- position, at the same time sensor is sensed and Y2 solenoid is actuated. Hence, air is going from port 1 to port 2 through blank end of cylinder and cylinder is forwarded, (A+ B- C- D+ E-).

Figure 14. Simulation of third step.

Simulation of Low Cost Automation and Life Cycle Cost …

275

Figure 15. Simulation of fourth step.

Step 4: A loader cylinder is downward (B+) as shown in Figure 15. When chute cylinder is at A+ position and at the same time sensor is sensed, then Y3 solenoid is actuated. Hence, air goes from port 1 to port 4 through blank end of loader cylinder. So, cylinder is moving forward, (A+ B+ C- D+ E-). Step 5: Fifth step shows a pusher cylinder push the product in collets as given in Figure 16. When loader cylinder is coming to B+ position and at the same time sensor is sensed, then Y3 solenoid is actuated. Air goes from port 1 to port 4 through blank end of pusher cylinder. Hence, pusher cylinder is moving forward (A+ B+ C+ D+ E-).

Figure 16. Simulation of fifth step.

276

Pattanayak Satyajit and Hauchhum Lalhmingsanga

Figure 17. Simulation of sixth step.

Step 6: A collet is clamped to the part by gripper cylinder as shown in Figure 17. When pusher cylinder piston comes at C+ position, Y2 solenoid is actuated at the same time. Air goes from port 1 to port 4 through rod end area of gripper cylinder. So, gripper cylinder is retracted. Here, we can see that the pipe line from valve 4 to cylinder is different to another loader, pusher and ejector cylinder as our requirement is to open collet at home position. It is important to design the port 2 connected with blank end area of gripper cylinder and port 4 connected with rod end area of gripper cylinder, (A+ B+ C+ D- E-). Step 7: Pusher cylinder is retracted as shown in Figure 18. When gripper cylinder is at D- position and Y6 solenoid is actuated at the same time. Air goes from port 1 to port 2 through rod end area of pusher cylinder, (A+ B+C- D- E-). Step 8: In this step, loader cylinder is retracted as shown in Figure 19. When pusher cylinder is at C- position and Y4 solenoid is actuated at the same time. Air goes compressor from port 1 to port 2 (V2) through rod end area of loader cylinder. Then, the loader cylinder is retracted, (A+ B- C- D- E-). Step 9: The gripper cylinder unclamped the part as shown in Figure 20. When loader cylinder is at B- position and timer is there at the same time, then the time will be set as per requirement of machining operation, thereby actuating Y8 solenoid. Air goes to compressor from port 1 to port 2

Simulation of Low Cost Automation and Life Cycle Cost …

277

(V4) and to blank end area of gripper cylinder. therefore, gripper cylinder moves forward, (A+ B- C- D+ E-). Step 10: Ejector cylinder moves forward as shown in Figure 21. When gripper cylinder is at D+ position and Y9 solenoid is actuated at the same time. Air goes to compressor from port 1 to port 4 (V5) and to blank end area of ejector cylinder. Therefore, ejector cylinder is ejecting the part from collet, (A+ B- C- D+ E+).

Figure 18. Simulation of seventh step.

Figure 19. Simulation of eighth step.

278

Pattanayak Satyajit and Hauchhum Lalhmingsanga

Figure 20. Simulation of ninth step.

Figure 21. Simulation of tenth step.

Figure 22. Simulation of eleventh step.

Simulation of Low Cost Automation and Life Cycle Cost …

279

Step 11: Ejector cylinder is retracted as shown in Figure 22. When ejector cylinder is at E+ position and Y10 solenoid is actuated at the same time. Air goes to compressor from port 1 to port 2 (V5) and to rod end area of the ejection cylinder. Therefore, ejector cylinder is retracted, (A+ B- CD+ E-) which means it return back to home position. This completes one cycle of the feeding to ejection automation process.

3.2. Programmable Logic Controller Ladder Programming 1) The sequence is A-A+B+C+D-C-B-D+E+E-. This can be performed by using Programmable Logic Controller (PLC) programming. Performance of this sequence is shown in Figure 23. In the programmed, one side is connected with 24V and the other side is connected with 0V. Here, only one input is given, i.e., START button. When the start button is started, it gives an output i.e., CR2 activated which means Y2 solenoid in 5/2 DCV1 is actuated (A-). 2) When A- is actuated which means chute door is open, timer is activated and set at 01 s. After 01 s, CR1 output is activated which means Y1 solenoid in 5/2 DCV1 is actuated (A+). 3) When A+ is actuated at the same time, the timer is set at 01 s. After 01 s, CR3 is activated which means Y3 solenoid in 5/2 DCV2 is actuated (B+). 4) After setting the timer at 01 s, CR5 is activated which means Y5 solenoid in 5/2 DCV3 is actuated (C+). Then, cylinder pusher is extended. 5) When the pusher is extended, the sensor sensed to activate CR8 which means Y8 solenoid in 5/2 DCV4 is actuated (D-). Then, Gripper cylinder is clamping the part. Then, one timer is set again at 01 s. After 01 s, CR6 is activated which means Y6 solenoid in 5/2 DCV3 is actuated. After which the pusher is retracted.

280

Pattanayak Satyajit and Hauchhum Lalhmingsanga

Figure 23. PLC ladder diagram.

Simulation of Low Cost Automation and Life Cycle Cost …

281

6) When the pusher is retracted and at the same time, loader is also retracted. After 01 s, CR4 is activated which means Y4 solenoid in 5/2 DCV2 is actuated (B-). 7) When loader is retracted, the required machining work begin. After completing machining work, ejector cylinder is activated. Then, CR7 is activated which means Y7 solenoid in 5/2 DCV4 is actuated (D+). Before ejecting the part, the gripper cylinder should unclamp the part from collet. 8) After unclamping the part, the ejector cylinder is extended and eject the part from collet. Then, the ejector cylinder is retracting and one cycle is completed. Then, CR9 is activated which means Y9 solenoid in 5/2 DCV5 is actuated (E+). The next round has one timer set at 01 s. After 01 s, CR10 is activated which means Y10 solenoid in 5/2 DCV 5 is actuated (E-). This complete one cycle of automation by using PLC programming.

3.3. Life Cycle Cost Analysis 3.3.1. Economic Analysis The tool designer must furnish management with an idea of how much the tooling will cost and how much the production method saves over a specific run. This information is generally furnished in the estimated cost of the tool and projected savings over alternate methods. The estimate also includes any special conditions that may justify the cost of tooling, such as close tolerances or high-volume production. For a valid estimate, the tool designer must accurately estimate the cost and productivity of the design in terms of materials, labour, and the number of parts per hour the tool will produce. Edward G., 2004 had determined calculation of economic analysis.

282

Pattanayak Satyajit and Hauchhum Lalhmingsanga

3.3.1.1. Economic Analysis of Conventional Lathe Machine Machining Operation: a) b) c) d)

Turning Head Diameter Chamfering Fillet Undercut/Grooving

First step: Calculation of parts per hour The first step in estimating is calculating the number of parts per hour the machine will produce simplest method is to divide 1 hour by the singlepart time or the time takes to load machine and unload each part. It can be expressed as given in Equation 1: 1

𝑃ℎ𝑐𝑜𝑛𝑣 = 𝑠

(1)

Second step: Calculation of labour cost Labour cost is the most expensive factor in manufacturing. If labour expenses can be reduced, so can overall production costs. The cost of labour can be calculated as given in Equation 2: 𝐿𝑆

𝐿𝐶𝑜𝑛𝑣 = 𝑃ℎ × 𝑤

(2)

Third step: Calculating the cost per part For accuracy, it is a must to calculate how much the design is worth in terms of total production and cost per part. Equation 3 gives the formula for finding this value: 𝐶𝑝 (𝐶𝑜𝑛𝑣) =

𝑇𝐶+𝐿 𝐿𝑆

3.3.1.2. Economic Analysis of NC Turning Machine Machining Operation: a) Turning Head Diameter

(3)

Simulation of Low Cost Automation and Life Cycle Cost …

283

b) Chamfering c) Fillet d) Undercut/Grooving First step: Calculation of parts per hour The first step in estimating is calculating the number of parts per hour the machine will produce with simplest method is to divide 1 hour by the single-part time, or the time taken to load machine and unload each part. It can be expressed as given in Equation 4: 1

𝑃ℎ𝑁𝐶 = 𝑠

(4)

Second step: Calculation of labour cost Labour is the single most expensive factor in manufacturing. If labour expenses can be reduced, so can overall production costs. The cost of labour can be calculated as in Equation 5: 𝐿𝑁𝐶 =

𝐿𝑆 𝑃ℎ

×𝑤

(5)

Third step: Calculating the cost per part For accuracy, it is very important to calculate how much the design is worth in terms of total production and cost per part. Equation 6 gives the formula for finding this value as follows: 𝐶𝑝 (𝑁𝐶) =

𝑇𝐶+𝐿 𝐿𝑆

(6)

Fourth step: Calculating the Break-Even Point The break-even point is the minimum number of parts a tool must produce to pay for itself. Any number less than this minimum results in a loss of money; any number more results in a profit. It is logical to assume that the lower the break-even point, the higher the profit potential. The formula to find the break-even point is given in Equation 7:

284

Pattanayak Satyajit and Hauchhum Lalhmingsanga 𝐵𝑃 = (𝐶

𝑇𝐶 𝑝(𝐶𝑜𝑛𝑣) −𝐶𝑝(𝑁𝐶)

(7)

Fifth Step: Calculating total savings To determine the most economical production method, it is required to compare production alternatives. The following formula given in Equation 8 assumes that both alternatives are being considered which require special tooling to produce the part is: 𝑇𝑆 = 𝐿𝑆 × {𝐶𝑝(𝐶𝑜𝑛𝑣) − 𝐶𝑝(𝑁𝐶) }

(8)

3.3.1.3. Economic Analysis of Special Purpose Machine Machining Operation: a) b) c) d)

Turning Head Diameter Chamfering Fillet Undercut/Grooving

First step: Calculation of parts per hour The first step in estimating is calculating the number of parts per hour the machine will produce by simplest method is to divide 1 hour by the single-part time, or the time taken to load machine and unload each part. It can be expressed as given in Equation 9: 1

𝑃ℎ𝑆𝑃𝑀 = 𝑠

(9)

Second step: Calculation of labour cost Labour is the single most expensive factor in manufacturing. If labour expenses can be reduced, so can overall production costs. The labour cost can be calculated as per Equation 10: 𝐿𝑆

𝐿𝑆𝑃𝑀 = 𝑃ℎ × 𝑤

(10)

Simulation of Low Cost Automation and Life Cycle Cost …

285

Third step: Calculating the cost per part For accuracy, it is important to calculate how much the design is worth in terms of total production and cost per part. The formula for finding this value is given Equation 11: 𝐶𝑝 (𝑆𝑃𝑀) =

𝑇𝐶+𝐿 𝐿𝑆

(11)

Fourth step: Calculating the Break-Even Point The break-even point is the minimum number of parts a tool must produce to pay for itself. Any number less than this minimum results in a loss of money; any number more results in a profit. It is logical to assume that the lower the break-even point, the higher the profit potential. The following formula given in Equation 12 is used to find the break-even point: 𝐵𝑃 =

𝑇𝐶 (𝐶𝑝(𝑁𝐶) −𝐶𝑝(𝑆𝑃𝑀)

(12)

Fifth Step: Calculating total savings To determine the most economical production method, it is required to compare production alternatives. The following formula, which assumes that both alternatives which are being considered required special tooling to produce the part is given in Equation 13: 𝑇𝑆 = 𝐿𝑆 × {𝐶𝑝(𝑁𝐶) − 𝐶𝑝(𝑆𝑃𝑀) }

(13)

3.3.2. Comparative Analysis of These Three Machines A comparative analysis of these three machines are shown in Table 1. By comparing each method, the tooling requirements in terms of costs versus savings can be obtained. Then, the method that returns the most for each dollar spent can be selected. When preparing this comparison, it is important to weigh all the economic factors in relation to expenses and productivity.

286

Pattanayak Satyajit and Hauchhum Lalhmingsanga Table 1. Comparison work sheet of three machines

COMPARISON WORK SHEET Economic & Productivity Conventional Factors Lathe Machine Lot Size (Ls) 80 Tool Cost (Tc) $312.77 Parts Per Hour (Ph) 7.49≠8 Labour/Hour(W) $1.50 Labour/Lot (L) $16.02 Cost Per Part (Cp) $4.10 Quality of The No. of Part/Lot 8 Maintenance Cost of The $223.41 Machine Per Annum Electricity Charge $16.37 Coolant Tank Size 0

NC Lathe Machine 80 $312.77 29.81≠30 $0.744 $1.99 $3.93 32 $670.24

Special Purpose Turning Machine 80 $312.77 148.14 $0.5585 $0.3015 $3.91 76 $446.82

$27.37 63 Litres

$22.34 63 Litres

Figure 24. Comparison graph.

From Table 1, it can be seen that the first machine is not suitable, when cost and quality are the only factor. The second machine also has slow production rate and low labour cost. But number of quality of the product

Simulation of Low Cost Automation and Life Cycle Cost …

287

is less and electricity charge is higher than other machines. This machine’s total cost of savings is $13.6. When quality is the only factor, the second machine is not suitable. The third machine produces the parts at a higher production rate and a lower labour cost. The number of quality product is high and electric charge is minimum as compared to second machine. The savings are again offset, this time by the tool cost. If the production run were greater, this method would have been the least expensive. Our requirements are high volume production rate, the number of quality product, less rejection and reduction of cost of production rate. In this machine skilled labour is not required. So, the third machine is selected for production (Graphically shown in Figure 24).

CONCLUSION This work is carried out based on the industry requirements to achieve low cost in production, maintaining quality with reduced rejections. LCA techniques, use of simulation software and PLC programming, have been considered to achieve the objective of this work. At first, design has been carried out by using the pneumatic circuit for the required operation sequence, but this pneumatic design technique has some disadvantages such as large space requirement, high maintenance cost and labour cost. To overcome these disadvantages, it is decided to adopt electro pneumatic circuit design, because this is very easy to implement and also nullifies the disadvantages of the pneumatic circuit. The electro pneumatic design is simulated using FLUID-SIM software and PLC programming and the required operation sequence is presented by simulation. Life Cycle Cost principle is used for the economic analysis and comparison of three machines, namely conventional lathe, NC lathe and the Special Purpose turning Machine had been investigated. It can be concluded that the Special Purpose Turning Machine will be profitable for long run mass production.

288

Pattanayak Satyajit and Hauchhum Lalhmingsanga

REFERENCES Albarlos, P. (1998). Trends in Low Cost Automation, In: Proceedings of the 5th IFAC Symp. On Low Cost Automation, Elsevier Science Ltd., Oxford. Barringer H. P., Barringer P. E, and Weber P. D. (1996). Life Cycle Cost Tutorial. 5th International Conference on Process Plant Reliability, Gulf Publishing Company and Hydrocarbon Processing, Marriott Houston Westside Houston, Texas, 1-58. Carvalho, A. J. P.(1986). SYSSIM- interactive systems for real time control simulation, IFAC Proceedings, 19 (13), pp.215-218. Chuang, S. H., Du, C. Y., Chao, C. H., Lee, C. E., and Tsai, B. Y. (1992). Software Development of Pneumatic Loop Design and Simulation, PLDS 2.0, Technical Report, NCHU. Chuang, S. H., Du, C. Y., Chao, C. H., Lee, C. E., and Tsai, B. Y. (1993). Software Development of Pneumatic Loop Design and Simulation, PLDS 2.1, Technical Report, NCHU. Edward G. (2004). Jigs and Fixture Design, Hoffman Published. Erbe, H. H. (1995). Refitting conventional machine tools to numerical control. In: Preprints, 4th IFAC Symp. on Low Cost Automation, Buenos Aires, Argentina, pp. 16-20. Erbe, H.-H. (1996). Technology and Human Skills in Manufacturing, In: Balanced Automation Systems IT, Chapman & Hall, London, pp 483 490. Erbe, H. H. (2000). Technologies for Cost Effective Automation in manufacturing (Low Cost Automation). Center for Human-Machine Systems, Technische Universitat, Berlin, and Germany, IFAC Professional Brief, Berlin, 1-32. Erbe, H. H. (2003). Manufacturing With Low Cost Automation IFAC Automatic Systems for Building the Infrastructure in Developing Countries, Elsevier IFAC Publications, Istanbul, Republic of Turkey, 95-100.

Simulation of Low Cost Automation and Life Cycle Cost …

289

Esposito, A. (1994). Fluid Power with Applications, Prentice-Hall, Englewood Cliffs, NJ, 3rd ed, 1-589. Freund, E., A. Hypki, F. Heinze and R. Bauer (2001). COSIMIR PLC - 3D simulation of PLC programs. In: Proceedings of the 6th IFAC Symposium on Cost Oriented Automation, Berlin. Gajmal. S. S. and Bhatwadekar. S. G., (2014). Low Cost Automation (LCA): A Case Study, IJAGET, 2 (12), 172-181. Hohwieler, E. and Berger, R. (2002). New Electronic Service for the Production Stor, In: Proceedings of the 6'h IFAC Symp. On Cost Oriented Automation, Elsevier Science Ltd. Oxford, pp. 237-241. Inagaki T. (2000). Automation and the cost of authority. Institute of Information Science, center for Taka, University of Tsukuba, Tsukuba, Japan, 8305-8573. Lay, G. (2002). Is High Automation a Dead End? Cutbacks in Production Over engineering in the German Industry, In: Proceedings of the 6thIFACSymp. Cost Oriented Automation, Elsevier Science Ltd. Oxford, pp. 225-230. Legierski, T., and Wyrwa, J., (1998). PLC Programming, Jaska Skalmierski’s Publishing Company, Gliwice (in polish). Ollero, A. (2002). Low Cost Automation in field Robotics In: Proceeding of the 6th IFAC Symp. Cost Oriented Automation, Elsevier Science Ltd. Oxford. Seifermann. S., Böllhoff. J., Metternich. J., and Bellaghnach. A. (2014). Evaluation of Work Measurement Concepts for a Cellular Manufacturing Reference Line to enable Low Cost Automation for Lean Machining, Procedia CIRP 17, 588 – 593. Soloman, S. (1996). Affordable Automation, McGraw-Hill, New York. Sullivan, J. A. (1989). Fluid Power Theory and Application, Prentice-Hall, Englewood Cliffs, NJ, 3rd ed. Swider, J. (2002). Control and Automation of Technological Process and Mechatronics Systems, Silesian University Publishing Company, Gliwice. pp. 546. Swider, J. and Wszoek, G. (2003). The methodical collection of labouratory and project tasks of technological process control.

290

Pattanayak Satyajit and Hauchhum Lalhmingsanga

Pneumatic and electropneumatic systems with logical PLC control, Silesian University Publishing Company, Gliwice, pp. 270. Swider, J. and Wszoek, G. (2004). Analysis of complex mechanical systems based on the block diagrams and the matrix hybrid graphs method, Journal of Materials Processing Technology, Elsevier, the Netherlands, 157-158.

In: Mechanical Design, Materials … ISBN: 978-1-53614-791-9 Editor: Sandip A. Kale © 2019 Nova Science Publishers, Inc.

Chapter 9

ESTIMATION OF AXIAL FORCE IN INCREMENTAL SHEET METAL FORMING Bohra Murtaza and Y. V. D. Rao Department of Mechanical Engineering, Birla Institute of Technology and Science Pilani, Hyderabad, India

ABSTRACT In sheet metal incremental forming, the sheet blank edge rigidly clamped deforms using a spherical tool, assisted by a CNC controlled incremental tool movement. Limitation in Single Point Incremental Forming (SPIF) is that it is limited to the production of parts with wall angle between 70-80 degree. The formability of a metal sheet differs according to the direction of the tool movement because of the plane anisotropy. The formability of a sheet metal is better accounted for using Forming Limit Diagrams. A considerable amount of knowledge about the forming forces is required for the preservation of tooling and machinery. A dynamical model is to be developed and using numerical techniques and FEM simulations, an approximated value of axial force will be computed. Based on these results a decision is to be made regarding a servo-controlled motor for the robot used for base rotation. 

Corresponding Author Email: [email protected].

292

Bohra Murtaza and Y. V. D. Rao

Keywords: incremental sheet forming, CNC technology, servo controlled, forming limit diagrams, axial force

1. INTRODUCTION Metal Forming is a manufacturing process which involves plastic deformation of metal to change the shape of the sheet. In metal forming a tool applies stress on the sheet which exceeds the yield stress limit and the metal takes a shape determined by the shape of the die. Stresses applied to plastically deform the metal sheet can be of any type like Compressive, Tensile, and Shear. Some of the examples are rolling, extrusion, die forming, forging, stretching, deep drawing, spinning, etc. Our main focus on this paper is Single Point Incremental Forming (SPIF). The incremental forming process is originated from stretch forming and metal spinning process. The SPIF have the combined advantage of both Stretching and Spinning. In SPIF a single point tool is made to move on a metal sheet. Recently, research has shown that it is also possible to produce low-cost, small-batch, polymer sheet components by means of single point incremental forming at room temperature (Martins et al., 2009) (Yonan et al., 2014). The tool generally moves in a spiral way either inwards or outwards in the plane of the sheet taking small incremental steps in the downward direction forming a 3-D dimensional shape from a 2-D sheet. In SPIF the stresses are localized near the point of contact of the tool with sheet surface. This increases the formability of the process also the quality of the metal sheet after forming is better than the conventionally formed sheets, therefore, ISF is primarily used in Aerospace and Automobile industry where high-quality surface finish and precision is required. SPIF uses modern CNC technology to produce complex 3D shapes although it produces complex geometries it is not suitable for mass production because it is a relatively slow process. However, the limitation in SPIF is, it can produce parts with maximum wall angle between 70-80 degree (Tisza, 2012) (Crina, 2011). To overcome this problem another

Estimation of Axial Force in Incremental Sheet Metal Forming

293

technique known as Multi-Stage Single Point Incremental Forming (MSPIF) is used. MSPIF was proposed to overcome the abovementioned difficulty of conventional SPIF related to the production of complex sheet metal parts with vertical walls (Skjødt et al., 2010). MSPIF is donemainly by alternating or sometimes repeating the tool movement from upwards to downwards, and the wall angle is increased gradually in consecutive stages. The number of intermediate stages is kept minimum to avoid significant surface wear and to limit the overall forming time. The risk of wrinkling is said to increase with the decrease in the intermediate number of stages (Hirt et al., 2004) (Bambach et al., 2004).

2. THE BASIC SETUP FOR SPIF In SPIF, the sheet blank’s edges are usually rigidly clamped and the sheet is then deformed usually by a hemispherical tool which moves as per the numerical computer program feed, following specific contours which result in the required shape of the sheet blank. Incremental sheet forming can be performed on any universal milling machine which is having at least 3-axis CNC control system. Thus, the basic elements of incremental forming processes are the material to be formed, a sheet blank holder clamping the blank, universal forming tool and the forming machine with the CNC control.

Figure 1. Basic Setup for SPIF showing all the Perimeters.

294

Bohra Murtaza and Y. V. D. Rao

A basic setup of SPIF is shown in Figure 1, here 𝜓 = Wall Angle, and Δz = Incremental step height in the downward direction. It’s known from the theory of spinning that the thickness of the part changes following the sine-law. tf = tO𝑆𝑖𝑛(𝛼) = tO𝑆𝑖𝑛(90 − 𝛼)= tO𝐶𝑜𝑠(𝜓), here tO = Initial thickness of sheet, tf = final thickness of sheet

3. TYPES OF ASYMMETRIC INCREMENTAL SHEET METAL FORMING TECHNIQUE 







Single point incremental forming - It uses a single point tool to form the elements into desired 3-D shapes as shown in Figure 2 (a). Multi-point Incremental sheet forming - It uses two single point tools one main which is in contact with the metal sheet on one side and another supporting tool which is in contact on the other side of the metal sheet. Instead of supporting tool partial die or fully developed dies are also used as a supporting structure, as shown in Figure 2 (b). ISF using Hydraulic Fluid - In this type of incremental forming one side of the metal sheet is in contact with the single point tool and the other side is supported by pressurized fluid, as shown in Figure 2 (c). Hybrid Incremental Forming - This type of ISF uses both stretches forming as well as the single point incremental forming. First, the metal sheet is stretched over a die and then a single point tool is moved from giving it a perfect shape and for making grooves. This type of technique is primarily used in making body panels for Automobiles as shown in Figure 2 (d). (Khare and Pandagale, 2014) (Tisza, 2012).

Estimation of Axial Force in Incremental Sheet Metal Forming

(a) Single point incremental forming

(b) Multi-point Incremental sheet forming

(c) ISF using Hydraulic Fluid

(d) Hybrid Incremental Forming

Figure 2. Types of sheet metal forming techniques.

295

296

Bohra Murtaza and Y. V. D. Rao

3.1 Clamping Clamping is done to fix the metal sheet onto the blank holder. There are two types of clamping methods used. In the first one, Solid or Rigid clamping the sheet is fixed on the blank holder using nuts and bolts. And in the second case, Ball clamping the sheet is clamped or fixed by forcing free rotating balls along its perimeter. The ball clamping method is more effective than solid clamping as it provides a relatively freer flow of the sheet during the ISF process (Khare and Pandagale, 2014). There is a lot of scope for research in ball clamping method.

3.2 Types of Tools Used in ISF  

Hemispherical Tool - With this tool, the contrasting point of the tool has a hemispherical shape. Ballpoint tool - In this tool at contacting tip a hardened steel ball is placed which is free to rotate just like the nib of a ballpoint pen.

Experiments show that spherical tool gives more depth and provides better surface roughness when compared to a hemispherical tool, this is because of its smooth rolling action over the blank surface without rubbing. Kim and Park, 2002 found that the formability of SPIF process is improved when a ball tool of a particular size is used with a small feed rate and a little friction. The feed rate is the speed the forming tool with which moves around the mill bed. Decreasing the feed rate also increases the time of the process, therefore it should be wisely decided according to the shape of the part and surface finish required. The formability also differs according to the direction of the tool movement because of the plane anisotropy.

Effect of Lubrication Lubrication is very important in incremental sheet forming. It is used to decrease the wear of tool and sheet. Also, lubrication helps in getting a

Estimation of Axial Force in Incremental Sheet Metal Forming

297

better surface finish and helps in dissipating heat generated at the contact area between tool and metal sheet. It is observed that spherical tool without lubrication gives better formability because the friction at the tool and sheet interface increases the tool pressure lowering state of stress in the sheet (Khare and Pandagale, 2014).

3.3 Forming Limit Diagram All the metals have a maximum limit up to which it can be stretched or deformed plastically. If a particular metal is stretched beyond its Ultimate tensile strength necking will occur which will ultimately lead to tearing or fracture of the part. Due to this reason, formability of a particular metal part is restricted to a certain limit. To account for the formability of a Sheet Metal, Forming Limit Diagrams (FLD) are widely used as an appropriate solution. The amount of strain that a sheet metal can tolerate just before localized necking is called as limit strain. Based on this principle, FLDs represent limiting major and minor available principal strains in the plane of the deformed sheet that can be achieved. Due to this FLDs becomes a valuable tool for analyzing sheet metal forming. FLDs were initially introduced by Keeler (Keeler, 1968) and Goodwin (Goodwin, 1968) popularly known as Keeler-Goodwin diagram. FLDs represent comprehensively sheet metal formability and has been used widely as one of the criteria for optimizing Forming processes, in the designing of dies and selection of tools for forming. To get FLDs basic strategy used is that a grid of circles is etched on the surface of a sheet metal. The sheet metal is then stretched and subjected to deformation. The circles get deformed into elliptic shapes Figure 3. The strain along two principal directions could be expressed as the percentage change in length of the major and minor axes. Negative minor strain results when minor axis contracts i.e., if Minor axis length is less than original circle’s radius. The strains measured near necks or fracture are called failure or limiting strain. A plot of the major strain

298

Bohra Murtaza and Y. V. D. Rao

versus minor strain is then made. This plot is called Keeler-Goodwin forming limit diagram Figure 4. This plot gives limiting strains corresponding to safe deformations. The FLD is thus a plot of the combinations of major and minor strains which lead to fracture. Any combination of strains which lies in the region above the limiting curves in the Keeler-Goodwin diagram represents failure, while those in the region below the curves represent safe deformations. Such diagrams indicate both of the principal strains ε1 and ε2 at diffuse or localized instability in the plane-stress state for different strain paths. Factors which influence the property of FLDs are strain hardening exponent, anisotropy constant (Sowerby and Duncan, 1971), tool geometry, friction effect, strain rate sensitivity (Ghosh and Hecker, 1974), grain size (Stachowicz, 1989) and strain path changes(Graf and Hosford, 1994) all these factors affect the property of FLDs.

Figure 3. Geometrical transformation of circle to ellipse due to stretching of sheet.

Forming limit analysis plays a major role in SPIF as it helps in deciding or choosing a particular metal for sheet, type of tool and design of the part. FLDs in ISF can also be useful in separating safe and unsafe areas. Extensive research has shown that FLD for ISF differs a lot from FLDs of conventional forming (Tisza, 2012) although the method to obtain

Estimation of Axial Force in Incremental Sheet Metal Forming

299

FLD remains same. Figure 5 shows that the FLD for SPIF lies above the FLD obtained for conventional forming. This shows the higher formability of SPIF compare to conventional forming. The main reason for the higher formability of SPIF is because the strain here is localized near the forming tool.

Figure 4. FLD for conventional sheet metal forming.

Figure 5. FLC for conventional and incremental forming (Tisza, 2012).

300

Bohra Murtaza and Y. V. D. Rao

4. FORCE PREDICTION IN SPIF DEDUCED FROM EXPERIMENTAL AND FINITE ELEMENT ANALYSIS During ISF it is absolutely necessary to have a considerable amount of knowledge about forming forces for the preservation of tooling and machinery. Also, it is primitively important, especially when using a robotic arm as a forming platform, because the end effector of a robotic arm is not a stiff structure and due to forming forces it deflects. This causes undesirable deviations in the tool path which affects the end results inducing errors in the geometry of the achieved parts. One of the solutions to this problem is by including a compensation for the deflection to be expected in the tool path. In the literature (Aerens et al., 2009), a study was done to establish formulae allowing to predict the forces occurring during the SPIF process. During experiments, forces were measured in steady state and truncated cones were formed. A model was generated partially obtained through experiments and regression techniques and partially through numerical techniques using FEM simulations allowing to compute an approximated value of axial force Fz as a function of tensile strength Rm of material, sheet thickness t, wall angle ψ, tool diameter dt, and scallop height Δh. Fz= 0.0716Rm t1.57 dt0.41 Δh0.09 ψCos(ψ), here Fz is expressed in N, Rm in N/mm2, t in mm, dt in mm, Δh in mm, and ψ in deg.

CONCLUSION Forming limit analysis plays a major role in SPIF as it helps in deciding or choosing a particular metal for sheet, type of tool and design of the part. For the preservation of tooling and machinery, a good knowledge of forming forces is essential. In addition, it is of utmost importance when

Estimation of Axial Force in Incremental Sheet Metal Forming

301

using a robotic arm as a forming platform. This is so because the end effector of a robot is not a stiff structure and gets deflected under forming forces. Using experimental results, a model is developed. With the help of regression techniques and numerical techniques using FEM simulations an approximated value of axial force Fz as a function of Tensile strength Rm of material, sheet thickness t, wall angle ψ, tool diameter dt, and scallop height Δh is computed. Using the results, a decision is to be made regarding a servo controlled motor to give the base rotation on which blank is clamped.

REFERENCES Aerens R, Eyckens P, Bael A Van and Duflou JR. Force prediction for single point incremental forming deduced from experimental and FEM observations. 2009, Int J Adv Manuf Technol doi: 10.1007/s00170009-2160-2. Bambach M, Hirt G, and Ames J, Modelling of optimization strategies in the incremental CNC sheet metal forming process. In Materials processing and design: modeling, simulation and applications, 2004. Proceedings of the Eighth International Conference on Numerical methods in industrial forming processes (Numiform 2004), AIP Conference Proceedings, Vol. 712, Columbus, Ohio, USA, 13–17 June 2004, pp. 1969–1974 (American Institute of Physics, Melville, New York). Crina, Radu, Determination of the Maximum Forming Angle of some Carbon Steel Metal Sheets, 2011, Journal of Engineering Studies and Research – Volume 17 No. 3. Ghosh AK and Hecker SS, Stretching limits in sheet metals: in-plane versus out-of-plane deformation. 1974, Metall. Trans. A 5A (1974) 1607– 1616. Goodwin GM, Application of strain analysis to sheet metal forming problems. 1968, Metall. Ital. 60 (1968) 767–771.

302

Bohra Murtaza and Y. V. D. Rao

Graf A and Hosford W, The influence of strain-path changes on forming limit diagrams of Al 6111 T4, Int. J. Mech. Sci. 36 (1994) 897–910. Hirt, G., Ames, J., Bambach, M., and Kopp, R. Forming strategies and process modelling for CNC incremental sheet forming. 2004, CIRP Ann. Mfg. Technol., 52(1), 203–206. Keeler SP, Circular grid systems: A valuable aid for evaluating sheet metal formability, 1968SAE Technical paper 680092 (1968) 633–640. Khare Unmesh and Pandagale Martand, A Review of Fundamentals and Advancement in Incremental Sheet Metal forming. 2014, International Conference on Advances in Engineering & Technology (ICAET-2014). IOSR Journal of Mechanical and Civil Engineering (IOSR-JMCE) eISSN: 2278-1684, p-ISSN: 2320-334X PP 42-46. Kim YH and Park JJ Effect of process parameters on formability in incremental forming of sheet metal. 2002, J Mat Process Technol 130:42–46. Martins, PAF, Kwiatkowski, L, Franzen, V, Tekkaya, AE and Kleiner, M Single point incremental forming of polymers, CIRP Annals Manufacturing Technology 58 (2009) 229–232. Paul SK Theoretical analysis of strain- and stress-based forming limit diagrams.2013, J Strain Analysis, 48(3) 177–188. Skjødt, M, Silva, MB, Martins, PAF, and Bay, N. (2010), Strategies and limits in multi-stage single-point incremental forming. Journal of Strain Analysis for Engineering Design, 45(1), 33-44. DOI: 10.1243/03093247JSA574. Sowerby R and Duncan JL, Failure in sheet metal in biaxial tension. 1971, Int. J. Mech. Sci. 13 (1971) 217–229. Stachowicz F, Effects of microstructure on the mechanical properties and limit strains in uniaxial and biaxial stretching. 1989, J. Mech. Work. Technol. 19 (1989) 305–317. Tisza, M, General overview of sheet incremental forming, 2012, JAMME, Issue 1 Volume 55. Yonan, S Alkas, Silva, MB, Martins, PAF and Tekkaya, AE, Plastic flow and failure in single point incremental forming of PVC sheets, (2014), eXPRESS Polymer Letters Vol.8, No.5 301–311.

ABOUT THE EDITOR

Sandip A. Kale is working as Associate Professor in Mechanical Engineering Department at Trinity College of Engineering and Research, Pune affiliated to Savitribai Phule University Pune (Previous Pune University), India. Additionally, he is handling the responsibility of Dean, Research and PhD Coordinator in the institute. He is also driving Technology Research and Innovation Centre, Pune, India. He has completed his PhD and Master’s Degree from Pune University. He has more than eighteen years industrial, academic and research experience. He has published more than fifty research articles in various conferences, books and journals. He has worked as Guest Editor for publishing many journal special issues. He has filed eleven patents. He has worked as

304

About the Editor

Conference Chair for five international conferences and one national conference. He is working as Editorial Board member and reviewer for many international journals.

INDEX A adjacency matrix, v, 153, 157, 158, 161, 162 algorithm, viii, 17, 18, 20, 23, 41, 42, 46, 47, 59, 69, 75, 84, 100, 101, 116, 119, 121, 126, 128, 131, 132, 133, 153, 163 aluminum oxide, 188 amplitude, 111, 166 anisotropy, 291, 296, 298 Arrhenius dependence, 48 atoms, 22, 23, 26, 30, 32, 33, 36, 37, 39, 45, 47, 48, 50, 53, 79, 82, 109, 112, 178, 179, 210, 228 automation, ix, 263, 264, 265, 266, 279, 281 automobiles, 148, 292 axial force, vi, vii, ix, 1, 6, 14, 291, 292, 300, 301

B barriers, 58, 92, 102, 103, 108, 109 beam-column joint, vi, viii, 165, 166, 167, 168, 169, 171, 173, 174 beams, 1, 166, 174, 226 bedding, 209

bending, 63, 176, 218, 219, 220, 226, 227, 237, 250 binary joints, 154 binding energies, 179 bioceramics, vi, viii, 175, 176, 177, 182, 239, 249, 255, 257 biomaterials, viii, 175, 248, 249, 250, 252, 253, 255, 257, 259, 260, 261, 262 biomedical applications, 244, 249, 252 biomolecules, 59, 63, 108, 112, 121, 122, 124, 125, 126 biopolymers, 133 biotechnology, 178 Boltzmann constant, 20, 32, 65 Boltzmann distribution, 87, 94, 105 Boolean algebra, 158, 164 break-even point, 283, 285

C carbon nanotube, 177, 179, 183 carboxylic acids, 245 casting, 198, 200, 202, 206, 269 ceramic, viii, 175, 186, 187, 188, 190, 191, 192, 193, 198, 199, 205, 206, 207, 208,

306

Index

211, 212, 214, 216, 219, 220, 221, 223, 224, 225, 227, 229, 231, 232, 233, 234, 235, 236, 237, 239, 240, 242, 248, 249, 250, 251, 252, 254, 255, 256, 257, 258, 259, 260, 261, 262 ceramic bulk, vi, 175 ceramic materials, 186, 220, 240, 250, 251, 252 Chebyshev approximation theory, viii, 136 chute, 266, 267, 270, 274, 275, 279 clay, 186, 187, 190, 191, 192, 193, 195, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 214, 215, 216, 218, 224, 229, 250, 253, 254, 256, 259, 261 CNC technology, 292 coal, 191, 220, 260 coatings, 243, 252, 257 collet, 266, 267, 268, 269, 270, 276, 277, 281 computational algorithm, 17, 18, 23, 47 computer simulations, 69, 104 computer technology, 18, 47 configuration, 6, 48, 50, 59, 64, 70, 80, 82, 84, 86, 91, 93, 95, 105, 109, 110, 111, 128, 179 convergence, 73, 74, 98, 101, 102, 108 cooling, 19, 20, 21, 22, 23, 30, 33, 34, 35, 39, 40, 45, 46, 50, 54, 212, 223, 226, 233 correlation function, 99, 105 cost, ix, 263, 264, 265, 266, 281, 282, 283, 284, 285, 286, 287, 289, 292 cost reduction, 264, 265, 266 Coulomb interaction, 47 crystal growth, 19, 20, 32, 34, 39, 42, 43, 45, 52, 55 crystal structure, 42, 43, 210 crystalline, 33, 51, 53, 54, 209, 215, 228, 248 crystallites, 198 crystallization, 21, 22, 23, 26, 33, 34, 35, 220, 243, 247, 251, 260

crystals, 17, 19, 20, 22, 23, 29, 30, 33, 34, 35, 40, 41, 42, 43, 45, 46, 47, 51, 52, 53, 54, 55, 115, 211, 215, 219, 247, 250

D decomposition, 28, 30, 187, 205, 215, 216, 243 defect engineering, 18, 51, 52, 55 defect formation, 20, 21, 40, 41, 42, 43, 44, 46, 50, 51, 52, 55 defective structure, v, vii, 17, 18, 20, 21, 34, 35, 42 defects, 19, 23, 35, 43, 50, 51, 52, 56, 219 deformation, 4, 36, 37, 38, 39, 171, 194, 196, 197, 297, 301 differential equations, 23, 35, 63 diffusion, 17, 19, 22, 23, 26, 27, 36, 39, 40, 41, 42, 48, 49, 53, 56, 100, 115 directional control valve, 264, 271 dislocation, 19, 20, 21, 22, 26, 31, 34, 35, 36, 37, 38, 39, 40, 41, 43, 44, 46, 47, 48, 51, 52, 54, 55 displacement, viii, 2, 36, 37, 153, 159, 166, 171, 210, 222 displacement isomorphism, viii, 153 distributed memory computer, 125 distribution, 11, 20, 21, 22, 26, 30, 31, 32, 33, 34, 35, 43, 44, 45, 64, 70, 72, 73, 76, 77, 81, 82, 84, 87, 89, 90, 94, 95, 100, 104, 105, 180, 181, 198, 199, 204, 208, 244, 245 distribution function, 20, 26, 30, 31 DNA, 57, 58

E ejector, 266, 268, 269, 270, 276, 277, 279, 281 elastic deformation, 36, 40

Index

307

energy, 20, 38, 53, 58, 65, 69, 70, 72, 73, 74, 76, 77, 80, 81, 82, 83, 86, 87, 88, 89, 90, 91, 92, 93, 95, 96, 97, 99, 100, 101, 102, 103, 104, 107, 108, 109, 113, 115, 118, 120, 122, 124, 125, 127, 128, 130, 132, 133, 167, 178, 179, 189, 190, 220, 228 energy consumption, 220 energy density, 87, 90, 93 energy supply, 190 engineering, vii, ix, 18, 41, 51, 52, 55, 165, 182, 190, 191, 250, 251, 252, 260, 289 entropy, 47, 48, 49, 50, 53, 71, 76, 91, 94, 128 epicyclic gear train, v, viii, 153, 154, 155, 156, 163, 164 expansion, 4, 93, 176, 208, 209, 210, 215, 216, 219, 229, 233, 234, 235, 238

forming forces, 291, 300 forming limit diagrams, 291, 292, 297, 302 four precision point synthesis, 138 fracture toughness, 240, 249 free energy, 48, 58, 96, 99, 100, 101, 102, 108, 109, 114, 115, 128, 130, 248 freedom, 1, 59, 60, 61, 63, 65, 66, 67, 68, 71, 80, 81, 83, 92, 109, 131, 154, 156 Freudenstein equation, 136, 138, 141 Freudenstein-Chebyshev approximation method, 135 friction, 168, 296, 297, 298 fuel cell, 185 function generation, 135, 136, 137, 148, 149, 150, 151 functional representation, 154

F

graphical representation, 154, 155 grown-in microdefects, 18, 19, 22, 26, 30, 44, 46, 54, 55 growth, 19, 20, 21, 22, 23, 24, 25, 26, 27, 29, 30, 31, 32, 34, 35, 36, 37, 39, 40, 41, 42, 43, 44, 45, 46, 47, 52, 53, 54, 55, 185 growth rate, 24, 43, 44, 45

finite element analysis, v, vii, 1, 2, 3, 8, 300 firing, 176, 184, 186, 188, 190, 191, 192, 198, 201, 202, 205, 206, 208, 209, 210, 214, 215, 217, 218, 219, 220, 221, 223, 226, 227, 229, 230, 231, 233, 235, 236, 238, 250, 253, 258 five precision point synthesis, 137, 140 flooding, viii, 58, 111, 112, 120 fluctuations, 58, 65, 110, 133 fluid, 63, 130, 196, 206, 221, 240, 294 force, vii, ix, 1, 2, 3, 4, 5, 7, 9, 12, 14, 59, 60, 62, 64, 65, 67, 68, 71, 77, 78, 79, 89, 92, 112, 115, 127, 194, 200, 202, 291, 292, 300, 301 force transducers, 2, 4, 15 formability, 291, 292, 296, 297, 299, 302 formation, 17, 19, 20, 21, 22, 23, 26, 31, 33, 34, 36, 37, 39, 40, 41, 42, 43, 44, 45, 46, 48, 50, 52, 54, 55, 56, 129, 166, 209, 211, 215, 219, 249, 252, 257

G

H Hamiltonian, 64, 69, 84, 87, 99, 105, 109, 111, 122 hamming matrix, 158, 159, 161 hamming number, 159 hamming procedure, 158 hamming string, 159, 160, 161 histogram, 69, 72, 73, 74, 75, 76, 87, 91, 96, 97, 101, 117, 118, 119, 124, 130 hybrid incremental forming, 294, 295

308

Index I

incremental sheet forming, 292, 296, 302 ions, 204, 243, 244, 245, 246 iron, 207, 209 ISF using hydraulic fluid, 294, 295 isomerization, 117 isomers, 159 isomorphism, 153, 159, 163 isotope, 53

J joints, 154, 155, 165, 166, 167, 174

K kinematics synthesis, 136 kinetic equations, 23 kinetic model, 31, 39, 46, 54 kinetics, 23, 26, 28, 32, 36, 42, 52, 108, 118, 133

L landscape, 58, 100, 104, 108, 109, 118, 127, 128, 132, 133 lattice parameters, 37 least-square method, 135 life cycle cost, vi, ix, 263, 264, 265, 281, 287, 288 linkage adjacency matrix, 157, 161 liquids, 31, 117, 126, 128, 204 loader, 266, 267, 270, 275, 276, 281 low cost automation, vi, ix, 263, 264, 265, 266, 288, 289 low temperatures, 50, 53, 70, 213

M acromolecular systems, 57, 58, 59, 109, 113 macromolecules, viii, 117, 120 magnetic characteristics, 180 magnitude, 1, 4, 13, 22, 36, 38, 63, 73, 77, 92, 107, 112 manufacturing, vii, ix, 14, 178, 185, 190, 191, 192, 220, 256, 264, 266, 269, 270, 282, 283, 284, 288, 292 Markov chain, 116 materials, vii, viii, 18, 19, 20, 42, 51, 52, 53, 167, 175, 177, 180, 181, 182, 183, 184, 185, 186, 187, 189, 190, 192, 206, 208, 212, 216, 218, 220, 229, 231, 236, 240, 247, 249, 250, 251, 252, 254, 255, 260, 261, 281 materials science, viii, 18, 51, 175, 249, 260 matrix, 3, 26, 36, 37, 110, 111, 140, 153, 157, 158, 159, 161, 162, 219, 221, 290 MD simulation, 57, 58, 59, 63, 64, 70, 71, 72, 73, 75, 76, 77, 88, 89, 90, 91, 95, 99, 102, 103, 104, 106, 108, 110, 112, 113 MD technique, 58 mechanical properties, 176, 235, 247, 248, 254, 261, 302 melting, 21, 48, 205, 208, 210, 216, 217, 219, 221, 227, 229 microdefects, 18, 19, 20, 22, 23, 26, 30, 35, 41, 43, 44, 46, 54, 55 model, 1, 3, 6, 8, 17, 18, 19, 22, 23, 26, 31, 32, 33, 34, 35, 36, 38, 39, 40, 41, 42, 46, 47, 48, 49, 50, 51, 54, 55, 84, 108, 111, 112, 113, 115, 117, 122, 123, 125, 128, 129, 131, 132, 133, 135, 166, 167, 168, 169, 170, 171, 172, 173, 174, 264, 291, 300, 301 modulus, 5, 26, 38, 194, 201, 219, 240, 246 moisture content, 196, 198, 199, 200, 201, 203

Index molecular dynamics (MD), v, viii, 57, 58, 59, 61, 63, 64, 65, 67, 68, 69, 70, 71, 72, 73, 75, 76, 77, 79, 81, 83, 85, 87, 88, 89, 90, 91, 93, 95, 96, 97, 99, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133 molecular nanotechnology, 178 Monte Carlo method, 120, 124

N nano-Al2O3, 175, 183 nanoceramics, viii, 175, 184, 185, 187, 188, 189, 190, 259, 261 nanomaterials, vi, viii, 175, 176, 177, 181, 182, 183, 184, 187, 249, 251 nanometers, 177, 180, 184, 188 nanoparticles, 175, 176, 180, 182, 184, 187, 188, 189, 250 nano-SiO2, 175, 183 nanostructured materials, 180, 181 nanostructures, 179, 182, 184, 188 nanotechnology, 175, 177, 178, 249, 251, 253, 259, 260

O optimization, 3, 15, 23, 92, 114, 120, 136, 137, 142, 143, 144, 145, 146, 148, 150, 301 oxygen, 22, 23, 24, 26, 30, 31, 33, 34, 40, 54, 216, 228

P peptide, 108, 115, 121, 123, 124, 126, 133 phase transformation, 32 phase transitions, 32, 116

309

physical and mechanical properties, 261 physical properties, 31, 51, 204, 228, 242 physicochemical properties, 243 plastic deformation, 292 plasticity, 167, 187, 194, 195, 196, 197, 198, 203, 204, 205, 208, 215 point defects, 19, 21, 47, 48, 50, 51, 53, 54 polymerization, 242, 243, 249 polymers, 113, 185, 302 polypeptide, 125, 131 porosity, 176, 213, 221, 223, 225, 235, 237, 245, 250 precipitation, 19, 20, 22, 26, 31, 33, 34, 35, 36, 40, 42, 46, 47, 50, 51, 54, 209, 236 precision points, viii, 135, 136, 145, 146, 148, 150 probability, 70, 72, 73, 74, 77, 83, 84, 86, 87, 89, 90, 92, 93, 94, 95, 96, 102, 103, 104, 105, 106, 117 probability density function, 94, 95 probability distribution, 70, 72, 77, 84, 89, 93, 96, 102, 103, 104 production costs, 282, 283, 284 productivity, 263, 264, 265, 266, 281, 285, 286 programmable logic controller, ix, 264, 279 programming, viii, 47, 51, 279, 281, 287 programming languages, 47 properties of ceramics, 189, 261 protein(s), 57, 58, 68, 76, 78, 84, 85, 103, 106, 109, 112, 113, 114, 115, 119, 122, 123, 124, 126, 127, 128, 129, 130, 131, 132, 133 protein folding, 58, 76, 78, 85, 103, 122, 128, 130, 131, 133 pusher, 266, 267, 268, 270, 275, 276, 279, 281

Q quality products, 266

310

Index

quartz, 192, 193, 207, 208, 209, 210, 211, 212, 214, 215, 216, 219, 248, 250, 254

R random walk, 70, 73, 74, 75, 119, 132 raw materials, 175, 190, 191, 193, 207, 218, 220, 223, 224, 236, 249, 256, 260 recombination, 21, 35, 36, 47, 48, 49, 50, 53, 54 reduction of time, 266 rheological and electromagnetic properties, 177 rotation graph, 155, 156, 157, 160, 163 rotational analysis, 159 rotational isomorphism, viii, 153, 163

S seismic analysis, 166 semiconductor(s), 17, 19, 35, 40, 42, 51, 52, 54, 55, 182, 185 servo controlled, 292, 301 shear, 26, 37, 38, 166, 167, 169, 172, 174, 194 sheet metal forming, vi, ix, 291, 295, 297, 299, 301 shrinkage, 176, 190, 198, 199, 200, 201, 202, 205, 206, 208, 215, 217, 220, 238, 245, 246, 250, 253 sigmoidal function generation, v, 135, 137, 148 silica, 182, 208, 209, 210, 220, 233, 235, 253, 259, 261 silicon, vii, 17, 19, 20, 21, 22, 23, 24, 26, 29, 30, 31, 33, 34, 37, 39, 40, 41, 42, 43, 44, 45, 46, 47, 50, 51, 52, 53, 54, 55, 56, 228 simulation(s), v, vi, viii, ix, 9, 18, 57, 58, 59, 63, 69, 70, 71, 72, 73, 74, 75, 76, 78,

83, 87, 88, 89, 90, 91, 95, 96, 99, 100, 101, 102, 103, 104, 105, 106, 108, 112, 115, 116, 118, 119, 121, 122, 123, 126, 127, 128, 129, 132, 133, 165, 167, 263, 264, 273, 274, 275, 276, 277, 278, 287, 288, 289, 301 single crystals, 20, 21, 22, 29, 31, 34, 39, 41, 43, 44, 46, 47, 48, 54, 55 single point incremental forming, 291, 292, 293, 294, 301, 302 sintering, 176, 187, 188, 189, 220, 231, 247, 248, 255, 258, 262 SiO2, 34, 175, 176, 182, 183, 185, 187, 209, 210, 213, 215, 218, 219, 220, 224, 230, 236, 244, 247, 248, 252, 257, 258 sol-gel, 187, 188, 189, 242, 243, 244, 245, 248, 252, 255, 259, 260 SOL-GEL techniques and applications, 242 solid state, viii, 18, 31, 34, 51, 55 special purpose machine, vi, ix, 263, 264, 266, 269, 284 special purpose turning machine, 266, 269, 286, 287 spindle collet, 267, 268 strain gauges, vii, 1, 2, 3, 4, 5 strengthened beam-column joint, 166

T techniques, vii, 64, 69, 90, 125, 126, 137, 187, 191, 242, 249, 287, 291, 295, 300, 301 technology(ies), vii, 2, 18, 20, 41, 42, 51, 52, 55, 178, 190, 252, 259, 265, 292 thermal expansion, 176, 208, 209, 210, 215, 219, 229, 233, 234 transducer, vii, 1, 2, 4, 5, 7, 9, 10, 12, 14 transfer vertex, v, viii, 153, 156, 160, 161, 163 tri-axial loadings, 1

Index V vacancies, 22, 23, 24, 26, 30, 39, 45, 48, 50 valve, 263, 269, 270, 271, 276 variables, 81, 86, 87, 101, 140 variations, 19, 35, 198 vector, 38, 62, 80, 82, 100, 109 velocity, 63, 67 viscosity, 213, 227, 245

W Wales, 104, 118 waste, 182, 187, 190, 205, 220, 223, 227, 233, 236, 237, 238, 239, 254, 255, 256, 258, 261

311

water absorption, 176, 190, 217, 218, 220, 221, 223, 250 water vapor, 216

X X-ray diffraction (XRD), 228, 242, 247, 248

Y Yang-Mills, 32

Z ZnO, 230