Viscoelasticity: Theories, Types and Models: Theories, Types and Models [1 ed.] 9781620819692, 9781613242032

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Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved. Viscoelasticity: Theories, Types and Models : Theories, Types and Models, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central,

Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved. Viscoelasticity: Theories, Types and Models : Theories, Types and Models, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central,

MATERIALS SCIENCE AND TECHNOLOGIES

VISCOELASTICITY

Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved.

THEORIES, TYPES AND MODELS

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MATERIALS SCIENCE AND TECHNOLOGIES

VISCOELASTICITY THEORIES, TYPES AND MODELS

JENNIFER N. PERKINS Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved.

AND

TYLER M. LACH EDITORS

Nova Science Publishers, Inc. New York

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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. For permission to use material from this book please contact us: Telephone 631-231-7269; Fax 631-231-8175 Web Site: http://www.novapublishers.com 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.

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Library of Congress Cataloging-in-Publication Data Viscoelasticity : theories, types, and models / editors, Jennifer N. Perkins and Tyler M. Lach. p. cm. Includes bibliographical references and index. ISBN 978-1-62081-969-2 (eBook) 1. Viscoelasticity. I. Perkins, Jennifer N. II. Lach, Tyler M. TA418.2.V57 2011 620.1'1232--dc22 2011010118

Published by Nova Science Publishers, Inc. † New York

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Contents vii

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Preface

1

Chapter I

Interfacial Dilational Rheology Related to Crude Oil Emulsion Hong-Bo Fang, Lei Zhang, Hua Zong, Lu Zhang, Sui Zhao and Jia-Yong Yu

Chapter II

Biopolymers in Food Emulsions: A Viscoelastic Approach Gabriel Lorenzo, Noemí Zaritzky and Alicia Califano

35

Chapter III

Continuous Viscoelastic Models in Food Rheology Anna Ptaszek, Paweá Ptaszek and Mirosáaw Grzesik

59

Chapter IV

Linear Viscoelasticity of Non-Fermented Dough: Effect of Gluten Absence Virginia Larrosa, Gabriel Lorenzo, Noemí Zaritzky and Alicia N. Califano

Chapter V

Chapter VI

Chapter VII

Interfacial Dilational Properties of Polymer Solutions Related to Enhanced Oil Recovery Zhen-Quan Li, Lei Zhang, Xin-Wang Song, Lu Zhang, Sui Zhao, and Jia-Yong Yu

93

115

Influence of Polyelectrolyte Addition on Rheological Properties of Zirconia and Starch Mixed Suspensions L. B. Garrido and A. N. Califano

137

Viscoelastic Properties and Sensory Quality of Whey Protein Concentrate Gels with Honey D. K. Yamul and C. E. Lupano

153

Chapter VIII

Influence of Viscoelasticity in Orthodontics Giampietro Farronato

Chapter IX

Hydrodynamic Characteristics of an Oscillating Viscoelastic Squeeze Film Muhannad Mustafa, Nusrat J. Chhanda and M. Mahbubur Razzaque

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167

177

vi Chapter X

Contents The Mesoscopic Constitutive Equations for Polymeric Fluids and Some Examples of Flows Alexsei Gusev, Grigory Afonin, Ilia Tretjakov and Grigory Pyshnogray

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Index

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187

203

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Preface Viscoelasticity is the property of materials that exhibit both viscous and elastic characteristics when undergoing deformation. In this book, the authors present topical research in the theories, types and models of viscoelasticity. Topics discussed include a viscoelastic approach to biopolymers in food emulsions; the influence of viscoelasticity in orthodontics; interfacial dilational rheology related to crude oil emulsion; the linear viscoelastic characteriztion of gluten-free non-fermented doughs and the viscoelasticity of heat induced gels from whey protein concentrate. (Imprint: Nova Press) Chapter I - Crude oil emulsions commonly are very complicated dispersion systems. The demulsification of crude oil emulsions is essential in the oil field industry, which can be achieved mainly by three means: mechanical, electrical, and chemical. The addition of chemical demulsifiers is the most widely used method. The stability of the emulsion is mainly determined by the mechanical strength of the film and the rheological properties of liquidliquid films. Chemical demulsification is a process in which the film thinning rate is enhanced and stability of the film is reduced by a chemical demulsifier. The interfacial shear property has been the most investigated property of liquid-liquid interfacial film. However, it seems that dilational viscoelastic parameters are much larger than the shear ones and show more direct correlation to emulsion stability. The study of dilational viscoelastic properties is useful to better understand the microcosmic properties of interfacial film and has been proven to be a powerful technique for probing the interfacial adsorption behavior. Interfacial dilational properties related to food emulsions have been reported in numerous publications. However, dilational rheological researches about crude oil emulsions are scarcely reported in literatures. In the present chapter, the developments of interfacial dilational properties of adsorption films containing surface-active fractions in crude oils and demulsifier molecules have been reviewed. The relationship between dilational data and emulsion stability has also been discussed. Chapter II - Food emulsions exhibit a great diversity of rheological characteristics, ranging from low-viscosity Newtonian liquids (e.g. milk, fruit beverages), to viscoelastic materials (e.g. salad dressings) and to plastic materials (e.g. butter). This diversity is the result of the different sorts of ingredients and processing conditions used to create each unique type of product. Biopolymeric macromolecules are usually key components in food emulsions to deal with creaming instability. Interactions between raw materials and the selected processing conditions will define each particular final product. There has been a growing emphasis on

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viii

Jennifer N. Perkins and Tyler M. Lach

understanding the colloidal basis of the rheology of food emulsions, rather than just treating them as a ‘black box’. Researchers are attempting to relate quantitatively the rheological properties of food emulsions to the characteristics, interactions and spatial distribution of each constituent in the dispersed system. Rheological measurements are appropriate tools for obtaining information about the organization of macromolecules in the medium, thus the correlation of microstructure information with rheological data is useful to understand the macroscopic behavior in terms of the microstructure organization. The relaxation of polymeric materials reveals the existence of a broad distribution of relaxation times and an effective way of examining this experimentally, is to determine the dynamic moduli G' (storage modulus) and G" (loss modulus). Focusing on the continuous phase behavior of an emulsified system, this chapter is oriented to interpret and model the viscoelastic characteristics of different oil-in-water food emulsions based on the molecular organization of hydrocolloids included in the aqueous phase. Biopolymeric systems containing xanthan-guar mixtures and gellan gum were selected as case studies. The analysis of these systems leads to a wide variety of food emulsions, ranging from fluid dressings to highly structured systems stabilized by a three-dimensional gel structure, caused by physical entanglements among polymeric chains. The effect of hydrocolloids concentration was studied using oscillatory measurements within the linear viscoelastic range. Time-Concentration Superposition principle was applied to find the master curves that describe the mechanical spectra of the viscoelastic materials. Viscoelastic behavior of the systems was satisfactorily modeled using Baumgaertel-Schausberger-Winter equation. This empirical model was used to predict the mechanical relaxation spectrum for both emulsions and continuous aqueous phases. Validation of the predicted spectra was carried out through creep compliance data for emulsion-filled gels and steady-state flow curves for emulsions containing xanthan –guar mixtures. With a precise knowledge of the viscoelastic behavior of food emulsions it is possible to control the desired rheological properties in a particular final product and its stability. Chapter III - In the viscoelasticity science usually there are performed two experiments: oscillatory, where deformation (stress) is sinusoidally changed in function of time (frequency domain), and step function, where deformation (stress) is constant in time domain. As a result of experiment conducted in frequency domain, the most often the complex viscoelastic modulus is obtained. The most often used and basic model describing viscoelastic system was proposed by Maxwell. Relaxation time is a value directly resulting from Maxwell model. A creep test is very frequently used in the studies of rheological properties. In this test the material is subject to a suitable constant in the time stress, and the resulting deformation is observed. For the sake of the mathematical modelling of the linear phenomena of creep phenomenological models are used. The most universal model describing the creep phenomenon (retardation) is the continuous Burger model. There are two time scales which are used in these models: a real time scale (the duration of the time of the experiment, the technological process, etc.) and an artificially created time scale called a relaxation or retardation scale. Even better approximation of viscoelastic properties of real materials is given, when continuous Maxwell model or continuous Burger model were applied. In such case, respectively Maxwell's and Burger elements create continuum, and it has effect on relaxation (retardation) time. In both Burger’s and Maxwell’s continuous model the subintegral function called a relaxation (retardation) spectrum or a distribution of the relaxation (retardation) times is characteristic of a given material. Knowledge of distribution

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Preface

ix

of relaxation times gives an insight of the structure of relaxation phenomena occurring in real materials. This knowledge can be used directly in modeling of technological process or be the starting point in the discussion regarding molecular interactions. Continuous rheological models have been known for a long time, but are rarely used in practice. It is so due to great difficulties with obtaining subintegral functions estimators based on the experimental data. In the case discussed, the estimator of the relaxation (retardation) spectrum is unknown. The direct use of the method of least squares in Maxwell’s or Burger’s model does not give the expected results because of the ill-posed of calculation problem. The solution for the ill-posed problem was found following the formulation of the regularisation method by Tikhonov. The scope of work was to analyse the viscoelastic phenomena in food science using continuous models and their applications in food rheology. Chapter IV - Gluten contains the protein fractions glutenin and gliadin. Interactions of gliadins and glutenins through covalent and non-covalent bonds to form gluten complexes result in viscoelastic dough that exhibits cohesive, elastic and viscous properties that combine the extremes of the two components. It has the ability to withstand stresses applied during mixing and to retain gas during fermentation and baking, enclosing the starch granules and fiber fragments. In recent years, gluten-free bakery products have an increase demand because of the improvement in celiac disease (CD) diagnosis. The only effective treatment for CD is strict adherence to a gluten-free diet throughout the patient’s lifetime, which, in time, results in clinical and mucosal recovery. Hydrocolloids and proteins are essential ingredients in gluten-free doughs for improving their rheological properties and final appearance. The rheological characterization of doughs provides important information for food technologists, allowing ingredient selection strategies to design, improve, and optimize the final product. Rheological studies become particularly useful when predictive relationships for rheological properties of foods can be developed based on the molecular architecture of the constituent species. This chapter focuses on the linear viscoelastic characterization of gluten-free nonfermented doughs and the effect of proteins (dry egg and ovoalbumin) and hydrocolloids (xanthan and guar gums) content. Small amplitude oscillatory data (storage modulus, G’, and loss modulus, G’’) were used to obtain the relaxation spectrum that was compared to the spectrum of a traditional wheat dough. The increase in gums content produced an increase in both moduli (G’ and G’’) and a more elastic dough was obtained. G’ was always larger than G’’ in the frequency measured range and the increase of the two moduli with frequency was small. The broadened Baumgaertel-Schausberger-Winter model was successfully used to predict the mechanical relaxation spectrum from dynamic oscillatory data. Gluten-free dough exhibited similar elastic modulus to those prepared with wheat flour. However, analyzing the tan(G) vs. frequency curves, commercial wheat flour dough presented distinctively higher values, which correspond to a more viscous contribution. The gluten matrix is more easily deformed under applied stress, and it is possible to correlate this behavior with large deformation experiments in which dough formulations are submitted to extensibility tests. Using the mechanical spectrum all dynamic data could be converted into the time domain by the application of BSW model. From the relaxation curves thus obtained, it could be notice the difference in the characteristic relaxation parameters of these systems. Chapter V - Polymers have been widely used in industry for many years. Among their applications, the polymers with ultra-high-molecular weight, especially partially hydrolyzed polyacrylamide and its modified products are used in improving oil recovery in the petroleum industry. It is said that some 8% (OOIP) oil is recovered by polymer flooding after water

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Jennifer N. Perkins and Tyler M. Lach

flooding. Interfacial dilational properties play crucial roles in Enhanced Oil Recovery (EOR). It is well known that the interfacial dilational viscoelasticities are very important for controlling the stability of foams and emulsions; consequently dominate the formation of oil bank. Moreover, the measurements of interfacial dilational properties have been proved to be a powerful tool to probe the structure of adsorption film. Bulk properties of polymer solutions such as apparent viscosity and rheology have been reported in many publications. Although it is very import to applications in EOR, the interfacial dilational rheological properties of polymer solutions are scarcely reported in literature. In the present chapter, the developments of interfacial dilational properties of polymer solutions employed in EOR process have been reviewed mainly based on the experimental results of our group and the interfacial interactions between polymers and surfactant molecules have also been discussed. Chapter VI - The preparation of porous ZrO2 had received much attention last years due to its extensive industrial applications (ceramic filters, catalyst supports, solid oxide fuel cells, membranes for gas filtration, biomedical materials, etc.). To produce porous ceramics starch is incorporated in the ceramic formulation acting as pore former and binder agent. The porous structure is highly dependent on the properties of the starting materials and suspension conditions, such as ZrO2 particle and starch granule sizes, volume ratio of the ceramic/starch and stability. The powder mixture must be dispersed in water to achieve optimum particle packing through colloidal processing. This requires the preparation and dispersion of highly concentrated suspensions which are fundamental steps in the colloidal processing of ceramics. Appropriate processing requires low viscosity and high solid content as possible. Thus rheological properties of concentrated aqueous suspensions of zirconia (3Y-ZrO2) containing different relative amounts and types of starch granules has been studied. Corn and potato starch types were incorporated in the starting powder composition as these starch varieties exhibit significant differences in their chemical and morphological characteristics. Shear viscosity and linear viscoelastic measurements using oscillatory techniques were performed on the highly concentrated (50 and 58.6 vol%) aqueous suspensions. The effect of the variable amount of an anionic polyelectrolyte added as dispersant, on rheological properties of the zirconia suspensions with and without starch was examined, providing information about differences in suspension structure due to modifications of the interparticle interactions. In the absence of starch, well stabilized zirconia suspension was obtained by 0.12 wt% of polyelectrolyte at pH 9 and began to flocculate at higher additions. Rheological measurements showed that a higher amount of the anionic polyelectrolyte was suitable to stabilize the mixed suspensions containing 45 and 68.7 vol% of corn starch in the composition. Rheological properties of zirconia suspensions containing potato starch resulted less dependent on polyelectrolyte addition. Chapter VII - Viscoelasticity of heat induced gels from whey protein concentrate prepared at pH 3.75, 4.2 and 7.0, with honey contents between 0 and 37.5%, was studied using dynamic rheological assays. The addition of honey produced a decrease in the solid– like behavior of gels, and an increase in the loss modulus of acidic gels. All gels behave as gel-like materials or strong gels, except those prepared with high honey content. Sensory evaluation of gels was performed through an affective test with non-trained panelists, which evaluated their appearance, taste, texture and general acceptability. Correlations were made between the perceived oral texture and the viscoelastic behavior of gels. Neutral gels were preferred over acidic gels. The most accepted gels were those that showed viscous

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Preface

xi

characteristics. Honey modifies the texture and taste of gels, enhancing their sensorial quality and acceptance. Chapter VIII - The study of viscoelastic behavior in orthodontic materials has a profound influence on the performance of the materials and the result of the treatment itself. Different materials used for orthodontic treatment exhibit a viscoelastic behaviour, archwires and brackets were mostly analyse in our studies. Chapter IX - Squeeze film theories have often been a major area of interest in fluid mechanics. In this paper, effects of surface roughness of rubber block on leakage flow rate and hydrodynamic force developed in fluid film between a cylindrical rigid surface and a cylindrical rubber surface are analyzed. The distribution of pressure in the fluid film and the porous material and the distribution of surface deformation of the porous rubber block are obtained by simultaneously solving the Reynolds equation, the Laplace equation and the three-parameter viscoelastic constitutive equation. Equations are discretized into finite difference equations and solved by Gauss-Siedel iteration. It is seen that with increasing standard deviation of surface height of rubber block, load carrying capacity increases significantly developing huge hydrodynamic force in the fluid film. Leakage flow rate decreases slightly with increasing standard deviation of surface of rubber block. Spatial distribution of surface texture of rubber block has no effect on leakage flow rate as well as hydrodynamic force during squeezing motion. The present analyses contribute to designing many engineering applications such as bearing, wet clutch and non-contacting face seal. The results obtained from the present model are compared with experimental results available in the literature and a very good agreement is found. Chapter X - Constitutive equations for melts and concentrated solutions of linear polymers are derived as consequences of dynamics of a separate macromolecule. The model is investigated for viscometric flows. It was shown that the model gives a good description of non-linear effects of simple polymer flows: viscosity anomalies, first and second normal stresses, non-steady shear stresses. A simple rheological equation of state (RES) which can be chosen as an initial approximation in formulating such a sequence of RES was obtained and studied. In this work, RES is extended to the case of allowance for the additional corrections caused by intrinsic viscosity and the delayed interaction of a macromolecule with its environment. Realization of this approach involves consequent solution of two problems: formulation of the equations of dynamics for a macromolecule and transition from the formulated equations to RES. The resulting equations can be recommended as a first approximation in constructing a sequence of RES. Two cases of steady-state flow between unlimited parallel planes under the action of a constant pressure gradient are considered and the same constitutive equations allow us to expand calculations also on the process of extension of the jet after the leaving of the die. Considering the processes of stretching, which occur at the lower temperatures, one has to take into account the possible process of crystallization of polymer.

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In: Viscoelasticity: Theories, Types and Models Editors: J. N. Perkins and T. M. Lach, pp. 1-34

ISBN 978-1-61324-203-2 © 2011 Nova Science Publishers, Inc.

Chapter I

Interfacial Dilational Rheology Related to Crude Oil Emulsion Hong-Bo Fang,1 Lei Zhang,2 Hua Zong,1 Lu Zhang,3 Sui Zhao,3 and Jia-Yong Yu3

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1

Shengli Engineering and Consulting Co., Ltd., Shengli Oilfield Company of SINOPEC, P. R. China 2. Geological Scientific Research Institute, Shengli Oilfield Company of SINOPEC, P. R. China 3 Technical Institute of Physics and Chemistry, Chinese Academy of Sciences, P. R. China

Abstract Crude oil emulsions commonly are very complicated dispersion systems. The demulsification of crude oil emulsions is essential in the oil field industry, which can be achieved mainly by three means: mechanical, electrical, and chemical. The addition of chemical demulsifiers is the most widely used method. The stability of the emulsion is mainly determined by the mechanical strength of the film and the rheological properties of liquid-liquid films. Chemical demulsification is a process in which the film thinning rate is enhanced and stability of the film is reduced by a chemical demulsifier. The interfacial shear property has been the most investigated property of liquidliquid interfacial film. However, it seems that dilational viscoelastic parameters are much larger than the shear ones and show more direct correlation to emulsion stability. The study of dilational viscoelastic properties is useful to better understand the microcosmic properties of interfacial film and has been proven to be a powerful technique for probing the interfacial adsorption behavior. Interfacial dilational properties related to food emulsions have been reported in numerous publications. However, dilational rheological researches about crude oil

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2

Hong-Bo Fang, Lei Zhang, Hua Zong et al. emulsions are scarcely reported in literatures. In the present chapter, the developments of interfacial dilational properties of adsorption films containing surface-active fractions in crude oils and demulsifier molecules have been reviewed. The relationship between dilational data and emulsion stability has also been discussed.

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1. Introduction An emulsion is a significantly stable suspension of particles of liquid of a certain size within a second, immiscible liquid. The formation of emulsions from two immiscible liquid phases is may be the most versatile property of surface-active agents for practical applications. Two immiscible, pure liquids cannot form an emulsion. For a suspension of one liquid in another to be stable enough to be classified as an emulsion, a third component must be present to stabilize the system. The third component is called the emulsifying agent and it is usually a surface-active agent, named emulsifying agent, such as surfactants, proteins, finely divided solids, crude oil fractions, and so on [Rosen, 2004]. Emulsions formed in Enhanced Oil Recovery (EOR) are usually water-in-oil (W/O) and oil-in-water (O/W) macroemulsions (>100 nm in diameter), which are kinetically stable systems with thermodynamical instability. In conventional oil recovery (high-energy process), the crude is often in contact with formation water or injection water, as in secondary recovery. In tertiary or enhanced oil recovery, surfactants and polymers are used purposely in water floods. Crude oil macroemulsions are produced when two immiscible liquid phases such as oil and water are mixed via the input of mechanical or thermal energy into the processes. Conventional crudes held under high pressures and temperatures amidst porous rocks are recovered by drilling. Emulsions in these oils form mainly through contact with formation water. As crude oil is pumped through various pipes, valves, chokes, etc., under high pressure and/or high temperature, fine water droplets are formed, producing macroemulsions [Angle, 2001]. The term stability, when applied to macroemulsions used for practical applications, usually refers to the resistance of emulsions to the coalescence of their dispersed droplets. The rate at which the droplets of a macroemulsion coalesce to form larger droplets and eventually break the emulsion has been found to depend on a number of factors: (1) the physical nature of the interfacial film, (2) the existence of an electrical or steric barrier on the droplets, (3) the viscosity of the continuous phase, (4) the size distribution of the droplets, (5) the phase volume ratio, and (6) the temperature [Rosen, 2004]. Among all these factors, the physical nature of the interfacial film plays the most important role. The formation of highly stable crude oil emulsions continues to challenge petroleum industry during crude oil production, transportation, and processing, which results in increased viscosity, organic deposition in processing equipment, and impacts adversely total oil recovery. This high stability of petroleum emulsion is principally attributed to the presence of a mechanically rigid or viscoelastic interfacial film formed by crude oil components. Crude oil components, such as asphaltenes, resins, waxes, solids, and naphtenic acids, are considered can adsorb onto and accumulate at oil-water interfaces to form elastic, mechanically strong films surrounding droplets. Moreover, asphaltenes have capacity to form stable thick network at the interface and are mainly responsible for stabilizing crude oil emulsions [Sjöblom et al., 2007].

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Interfacial Dilational Rheology Related to Crude Oil Emulsion

3

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The demulsification of crude oil emulsions is essential in the oil field industry, which can be achieved mainly by three means: mechanical, electrical, and chemical. The addition of chemical demulsifiers is the most widely used method. The stability of the emulsion is mainly determined by the mechanical strength of the film and the rheological properties of liquidliquid films. Chemical demulsification is a process in which the film thinning rate is enhanced and stability of the film is reduced by a chemical demulsifier [Angle, 2001]. Rheological properties are the main characteristics of the dynamic properties of a film. There are two rheological properties of the interfacial films: interfacial shear and dilational viscoelasticity. The interfacial shear property has been the most investigated property of liquid-liquid interface film [Malhotra et al., 1988]. However, it seems that dilational viscoelastic parameters are much larger than the shear ones and show more direct correlation to emulsion stability [Wasan et al., 1976; Clint et al., 1981; Neustadter et al., 1981; Wasan et al., 1982]. Moreover, the study of dilational viscoelastic properties is useful to better understand the microcosmic properties of interfacial film and has been proven to be a powerful technique for probing the interfacial adsorption behavior [Liggieri et al., 2005]. Interfacial dilational properties related to food emulsions have been reported in numerous publications [Benjamins et al., 1998; Murray et al., 1998; Prins et al., 1998; Bos et al., 2001a; Gurkov et al., 2001; Wustneck et al., 2001; Benjamins et al., 2009; Erni et al., 2009; Kovalchuk et al., 2009]. However, dilational rheology researches about crude oil emulsions and chemical demulsification are scarcely reported in literatures [Kim et al., 1996; Sun et al., 2002; Wang et al., 2003a; Wang et al., 2003b; Wang et al., 2004a; Dicharry et al., 2006; Hannisdal et al., 2007; Yang et al., 2007; Verruto et al., 2008; Zhang et al., 2008; Alvarez et al., 2009]. In the present paper, the developments of interfacial dilational properties of adsorption films containing surface-active fractions in crude oils and demulsifier molecules will be reviewed. The relationship between dilational data and emulsion stability will also be discussed.

2. Dilational Rheological Data: Principles and Methods 2.1. Principles The Gibbs interfacial dilational modulus is defined by the surface tension increase after a small increase in area of a surface element:

İ˙

dȖ dlnA

(1)

It gives a measure of the interfacial resistance to changes in area. Where İ is the dilational modulus, Ȗ is the interfacial tension and A is the interfacial area. When the interfacial area is subjected to periodic compressions and expansions at a given frequency, relaxation processes such as diffusion exchange between the surface layer and the bulk solution or molecular rearrangements within the layer may cause a phase difference

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Hong-Bo Fang, Lei Zhang, Hua Zong et al.

4

(measured by the phase angle ș ) between the applied area variation and the surface tension response. In that case İ is a complex number and can be decoupled into real and imaginary components İ d and İ Ș

ȦȘ d , respectively.

İ ˙ İ d ˇiȦȘd

(2)

where İ d is the dilational elasticity or storage modulus and İ Ș the dilational viscosity component or loss modulus that represents a combination of internal relaxation processes and relaxation due to transport of matter between the surface and the bulk. Phase angle ș is calculated according to

tanș =

İȘ İd

(3)

In the absence of relaxation processes affecting the surface dilational modulus, the phase angle ș is equal to zero and the surface layer behaves as a purely elastic body. Interfacial tension relaxation experiments are a reliable way to obtain surface dilational parameters. This technique uses small but fast axisymmetric area expansion or compression to slightly disturb the monolayer equilibrium, which causes an interfacial tension jump and then the interfacial tension will decay to the equilibrium again. For an instantaneous area change rising from ǻA(t ) = 0 for t İ0 to ǻA(t ) = ǻA for

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t > 0 , the values of İ are obtained as a function of the frequency by Fourier transformation (FT) of the interfacial tension decay obtained from the experiment by the following relationship:

FT'J (t ) FT ('A / A)(t )

H (Z )

³

f

0

³

f

0

'J (t ) exp(iZt )dt

['A(t ) / A] exp(iZt )dt

(4)

where Ȧ is the angular frequency. In an ideal system that is not diffusion controlled and in which only one relaxation mechanism occurs the decay curve of Ȗ vs. t can be represented by an exponential equation. For a real system a number of relaxation processes may occur and the decay curve would be expressed by the summation of a number of exponential functions:

'J

n

¦ 'J

i

exp(W i t )

i 1

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(5)

Interfacial Dilational Rheology Related to Crude Oil Emulsion

5

W i is the characteristic frequency of the ith process; 'J i is the fractional contribution which that relaxation process makes to restore the equilibrium; n is the total number of the where

relaxation processes. For an instantaneous change in area

³

f

0

'A exp(  iZ t ) dt A

'A / A iZ

(6)

With this, Eq. (4) becomes

H

iZ 'A / A

³

f

0

' J ( t )[cos Z t  i sin Z t ]dt

(7)

The real part of Eq. (7) is the dilational elasticity İ d and the imaginary part is the interfacial dilatonal viscosity component ZK d : i.e.

H d (Z )

'A / A ³

ZK d (Z )

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Z

f

0

Z

'J (t ) sin(Zt )dt (8)

'A / A ³

f

0

'J (t ) cos(Zt )dt (9)

Other parameters such as the tangent of phase angle, dilational modulus etc can all be obtained from these two parameters.

2.2. Methods All the methods of interfacial rheology are based on a perturbation of the mechanical equilibrium at the interface and the subsequent measurement of the system response. If the perturbation can be reduced to a small change of the interfacial area, this response is determined by a fundamental interfacial property, called the complex dynamic dilational interfacial modulus. The experimental techniques at low frequencies ( AD(RM)). The order of the relaxation function (D) was lower than 0.2 for all formulations indicating the pronounced elastic character of the doughs, typically observed in gel-like samples (Doublier et al., 1992; Steffe, 1996). However, D was significantly higher (PG´´ suggested an increase in attraction between particles. This response is usually exhibited by a weak gel structure [21]. Suspensions containing 0.5 wt% of polyelectrolyte (Figure 6) showed more elastic characteristics as both moduli were independent on frequency. However, G´ at the plateau remained similar to that for 0.12 wt% addition. Further increase of added polyelectolyte concentration to 1 wt% increased G´ at high frequencies to 1000 Pa. High and constant G´ in the low frequency region, suggesting Zc 1 indicates that the viscous behavior predominates in the sample. Honey increased the tan G values in all conditions assayed, indicating again that honey increases the viscous behavior of gels. Those

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gels with 0% and 10% honey prepared at the three pHs assayed, and with 27.5% and 32.5% honey at pH 4.2 and 7.0, presented tan G values below 1.0, revealing that the gel behavior was primarily elastic. On the contrary, gels with the highest honey content presented tan G values greater than 1.0, indicating viscous characteristics. Finally, acidic gels showed values of tan G higher than gels prepared at pH 7.0. This behavior reveals viscous characteristic in acidic gels, in agreement with results obtained in a previous work (Yamulet al., 2003). Time Sweep Tests. Figure 4 shows G’ as a function of time. The measurements were carried out immediately after 1.0 ml of dispersions were placed on the lower plate of the rheometerthermostated at 90ºC (Ȧ =1rad/sec, J= 5%). The variation of the G* was similar to that of G’ so that the viscoelastic kinetics can be represented in terms of the elastic modulus. The maximum value of G’ was reached by acidic gels 15 minutes prior to the beginning of the experiment. On the contrary, neutral gels reached the maximum value of G’ slower as the honey content increased, which suggests that honey delays the gelation process mainly in neutral gels. This fact can be confirmed by analyzing the results of Table 2, where it is possible to appreciate that honey increased mainly the gel time (t*) of the neutral gels, probably due to the interference of this component on the sulphydril/disulphide interchange reactions. The gel time or gel point represents the instant when the solution becomes a viscoelastic solid or, in the polymerization model of gelation, the time when an infinite network is formed (Aguilera, 1995). Thermal treatment at high temperatures would favour two opposite phenomenons: (a) rupture of hydrogen bonds and decrease of electrostatic interactions, thus, weakening intermolecular interactions, and (b) stabilization of hydrophobic interactions, which favour molecular associations in a three dimensional structure. The value of G’ results from this balance.If the first phenomenon predominates, the elastic modulus will be small and on the contrary, if the second is dominant, G’ reaches higher values (Chronakis, 1996). On the basis of the above-mentioned fact, it may be deduced that thermal treatment of WPC/honey dispersions, either acidic or alkaline (Figure 4) would enhance hydrophobic interactions between protein molecules.

Figure 4. Elastic modulus (G’) of WPC gels as a function of time (Ȧ =1rad/sec, J = 5%, temperature: 90ºC). Protein content of gels: 10%, w/w. pH of gels: (a) 3.75, (b) 4.2, (c) 7.0. Honey content of gels: (Ŷ) 0%, (x) 10%, (Ÿ) 27.5%, (ź) 32.5%, (i) 37.5%.

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Table 2. Values of gel time (t*) from WPC gels at different pHs and honey content. Protein content of gels 10% w/w. t*: LSD0.05=0.475 pH 3.75

4.2

t* 0.295 r 0.134 0.780 r 0.141 1.105 r 0.315 1.183 r 0.098 1.460 r 0.247 0.180 r 0.015 1.535 r 0.540 1.833 r 0.234 2.443 r 1.114 2.709 r 0.705 1.925 r 0.643 5.468 r 0.456 8.867 r 0.957 10.394 r 1.002 12.250 r 0.872

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7.0

Honey (%) 0 10 27.5 32.5 37.5 0 10 27.5 32.5 37.5 0 10 27.5 32.5 37.5

Figure 5. Sensory evaluation of WPC gels as a function of honey content. Protein content of gels: 10%, w/w. pH of gels: (Ŷ) pH 3.75; (Ŷ) pH 4.2; (Ŷ) pH 7.0. (a) Appearance, (b) Taste, (c) Texture, (d) General acceptability. Bars show standard deviation. Appearance LSD0.05=0.486. Taste LSD0.05=0.456. Texture LSD0.05=0.492. General acceptability LSD0.05= 0.470. Viscoelasticity: Theories, Types and Models : Theories, Types and Models, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central,

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Table 3. Texture score and tangent of the deformation angle (tan G) (Ȧ =1rad/sec, J= 5%, temperature: 90ºC) of WPC gels prepared at different pHs and honey content. Protein content of gels: 10% w/w. Texture score: LSD0.05=0.492. tan G: LSD0.05=0.113 pH 3.75 4.2 7.0

Honey (%) 0 37.5 0 37.5 0 37.5

Texture score 1.088 r 0.064 2.383 r 0.017 1.665 r 0.046 2.981 r 0.087 1.935 r 0.088 3.029 r 0.070

tan G 0.523 r 0.029 1.195 r 0.045 0.453 r 0.010 1.127 r 0.005 0.286 r 0.018 1.144 r 0.002

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Sensory Evaluation. Consumer Testing The sensory evaluation was perfomedas described in Confortiet al. (2007). Figure 5 shows the sensory attributes (appearance, taste, texture, and general acceptability) as a function of honey content. Significant differences (p< 0.05) were observed in sensorial attributes between acidic and neutral gels in almost all conditions assayed. On almost all samples, neutral gels received the highest score and acidic gels the lowest score. Honey can be considered a multipurpose ingredient in the food industry. Among many of its properties, its appearance enhancer property is frequently used in many foods. This property is due to the caramelization reactions that take place when honey is cooked. As a result of this process the final product has a dark color, which is perceived by consumers as an added value. As it was expected, honey enhanced the appearance (Figure 5a) of gels in all conditions assayed. Results of Figure 5a also show that pH 7.0 gels obtained the best appearance score. The addition of honey is expected to change also the taste of gels. Results of Figure 5b indicate that honey enhanced the opinion of the panelists about the taste of gels in all conditions assayed. It is important to observe that acidic gels without honey obtained the lowest score; but the addition of honey actually increased the score of these gels. The food industry aims to produce foods that are pleasurable to eat. Pleasure is derived from taste, smell, temperature, and vision (Heath et al., 1988). Texture is an organoleptic characteristic involved in the acceptability of a food product. We can define texture not only as a physical property of food, but also as a sensation perceived by the sense of touch, that is, the perception of the food structure when it is deformed in the mouth (Confortiet al., 2007). Texture perception in foods is a complex phenomenondue to the interaction between taste, aroma, and texture modalities (González-Tomás et al.,2008; Tournieret al.,2009). Due to some physical properties such as high viscosity, high density, higroscopicity, and stickiness, honey is well appreciated by the food industry as a bodying agent. This characteristic allows food manufacturers to modify the texture of a product in a desired way. Results of Figure 5c show that panelists were able to differentiate the texture among almost all of the samples. Results also show that honey enhanced the perceived texture of gels in all conditions assayed. Figure 5c also shows that the increase of gels score when honey increased from 0% to 20%

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was larger than the increase of gels score when honey increased from 20% to 37.5%; that is, the major effect of honey on the perceived texture can be observed between gels of 0% and 20% of honey. Flavor or visual differences ideally should have no impact on the oral perception of texture properties by trained panelists (Barden et al., 2009); however, both parameters could influence the oral perception texture of consumers or non trainednon-trained panelists. Figure 5d shows the general conclusion of the panelists. The increase in honey content enhanced the opinion of the panelists about the various sensorial attributes in all conditions assayed. Furthermore, neutral gels were preferred over acidic gels. Table 3 correlates an instrumental measure with a sensory descriptor. The tangent of the deformation angle (tan G = G’’/G’), as it was mentioned above, is an instrumental measure, and represents the viscoelasticity of a material (values of tan G> 1 indicate that the viscous behavior predominates in the sample). On the other hand, texture is quantified as the oral perception of food structure when it is processed in the mouth by the panelists. It must be taken into account that this comparison is limited by the fact that the measure of the tangent of the deformation angle was made at 87ºC, whereas the texture score was quantified at mouth temperature. At a given pH, the increase of the tan G increased the score of the oral perception of the food texture; in other words, panelists preferred those gels where viscous behavior predominates. On the other hand, the score of the oral perceived texture increased as pH increased from 3.75 to 7.0, whereas the value of tan G did not present the same behavior. These results suggest that not only the viscoelastic behavior of gels determine the oral perception of the food texture, but also, as mentioned above, there is an interaction between taste, aroma, and texture.

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Conclusion Honey modifies the viscoelastic properties of gels, decreasing the solid component and increasing the viscous behavior. Neutral gels are preferred over acidic gels, and the addition of honey enhances the sensorial quality of gels in all conditions assayed. It is possible to correlate sensory descriptors with viscoelastic instrumental measures, which can be useful to evaluate the consumer preferences. These gels could be used as dessert fillings, having the advantage of combine the high nutritional quality and functional properties of whey proteins with the delicate taste and properties of honey.

Acknowledgments This investigation was supported by PIP 1643 (CONICET). Authors are members of the Researcher Career of the ConsejoNacional de InvestigacionesCientíficas y Técnicas (CONICET).

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References Aguilera, M. (1995). Gelation of whey proteins. Food technology, 10, 83-89. Barden, L. M., Çakir, E., Leksrisompong, P. N., Ryan, K. N., Foegeding, E. A. and Drake, M. A. (2009). Effect of flavor on perceived texture of whey protein isolate gels. Journal. of Sensory Studies, 25, 447-462. Bernal, V. and Jelen, P. (1981). Thermal stability of whey proteins-A calorimetric study. Journal of Dairy Science, 68, 2847-2852. Cheftel, J. C. and Lorient, D. (1982). Les propriétésfonctionnelles des protéineslaitièresetleuramélioration. Lait, 62, 436-483. Conforti, P.; Yamul, D. andLupano, C. (2007). Properties of biscuits and whey protein concentrate gels with honey. In C. E. Lupano (Ed.), Functional properties of foods components (pp. 93-108). Kerala, India: Research Signpost. Cooney, M. J., Rosemberg, M. and Shoemaker, C. F. (1993). Rheological properties of whey proteins concentrate gels. Journal of Texture Studies, 24, 325-334. Chronakis, I. (1996). Network formation and viscoelastic properties of commercial soy protein dispersions: effect of heat treatment, pH and calcium ions. Food Research International, 29, 123-134. Dumay, E. (1988). Dénaturationthermique de la E-lactoglobuline et propriétésgélifiantes des concentrésprotéiques de lactosérum.In PropriétésFonctionnelles des Macromolecules Alimentaires(pp. 67-87). Paris, France: Technique et Documentation. Giboreau, A., Cuvelier, G. and Launay, B. (1994). Rheological behavior of three biopolymers/water systems, with emphasis on yield stress and viscoelastic properties. Journal of Texture Studies, 25, 119-137. González-Tomás, L., Bayarri, S., Taylor, A.J. andCostell, E. (2008). Rheology, flavor release and perception of low-fat dairy desserts. International. Dairy Journal, 18, 858–866. Heath, M. R. and Lucas, P. W. (1988). Oral perception of food texture. In J. R: Mitchell and J. M. V. Blanshard (Eds.), Food Structure, its creation and evaluation (pp 465-481). London, United Kingdom: Butterworth. IFT Sensory Evaluation Division. (1981). Sensory evaluation guide for testing food and beverage products. Food Technology, 50-59. Jong S.; Klok J. and Van DeVelde F. (2009). The mechanism behind microstructure formation in mixed whey protein-polysaccharide cold set gels. Food Hydrocolloids, 23, 755-764. Langton, M.; Astrom, A. and Hermansson A. (1997). Influence of the microstructure on the sensory quality of whey protein concentrate gels. Food Hydrocolloids, 11 (2), 217-230. Lethuaut, l., Brossard, C., Rousseau, F., Bousseau, B. and Genot, C. (2003). Sweetnesstexture interactions in model dairy dessert: Effect of sucrose concentration and the carrageenan type. International. Dairy Journal, 13, 631-641. Ross-Murphy, S. B (1995). Rheology of biopolymers solutions and gels. In Dickinson, E. (Ed.), New Physicochemical techniques for the characterization of complex food systems (pp 130-156). Glasgow, United Kingdom: Blackie Academic and Professional. Shimada, K. andCheftel, J. C. (1988). Texture characteristics, protein solubility, and sulphydryl group/disulfide bond contents of heat–induced gels of whey protein isolate. Journal of Agricultural and Food Chemistry,36, 1018-1025.

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Sopade, P. A, Halley, P. J. and Junming, L. L. 2004. Gelatinisation of starch in mixtures of sugars. I. dynamic rheological properties and behaviors of starch-honey systems. Journal of Food Engineering,69, 439-448. Spahn, G.; Santiago, L.; Baeza, R. and PilosoF A. (2008). Whey protein concentrate / Ȝ carrageenan systems: Effects of processing parameters on the dynamics of gelation and gel properties. Food Hydrocolloids, 22, 1504-1512. Steffe, J. F. (1996). Rheological methods in food process engineering. In Steffe, J. (Ed.), Viscoelasticity (pp. 295-349). East Lansing, USA: Freeman Press. Systat. V 5.0 for windows. SYSTAT, INC., Copyright, 1990. Tournier, C., Sulmont-Rossé, C., Sémon, E., Vignon, A., Issanchou, S. and Guichard, E. (2009). A study on texture–taste–aroma interactions: Physicochemical and cognitive mechanisms. International Dairy Journal, 19(8), 450–458. Yamul, D. K. and Lupano, C. E. (2003). Properties of gels from whey protein concentrate and honey at different pHs. Food Research International, 36, 25-33. Yamul, D. K. andLupano, C. E. (2009). Viscoelastic properties of whey protein concentrate gels with honey and wheat flour at different pHs. Journal of Texture Studies, 40, 319333.

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In: Viscoelasticity: Theories, Types and Models Editors: J. N. Perkins and T. M. Lach, pp. 167-175

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Chapter VIII

Influence of Viscoelasticity in Orthodontics Giampietro Farronato

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University of Milan, Milan, Italy

The study of viscoelastic behavior in orthodontic materials has a profound influence on the performance of the materials and the result of the treatment itself. Different materials used for orthodontic treatment exhibit a viscoelastic behaviour, archwires and brackets were mostly analyse in our studies. The art of orthodontics involves correction of the position of teeth and the relation of craniofacial structures. Teeth are moved by the use of forces and moments, which are delivered through the use of various types of wires. From the beginning of the profession, different types of wires have been introduced to provide forces to move teeth. Archwires are commonly used in orthodontic procedures in cooperative association with other components, typically brackets and buccal tubes. An archwire is disposed on the labial side of the arch, and when connected to brackets on the teeth it develops corrective forces for performing tooth movements. The elastic deformation of an orthodontic wire and the subsequent release of his elastic energy determine the correcting forces. In orthodontics there are four main types of material compositions for archwires which are Stainless Steel (SS), Nickel-Titanium (Ni-Ti), Beta-Titanium, and Cobalt-Chromium. A comprehensive orthodontic treatment is usually divided into leveling and aligning, space closure, anterior/posterior correction, and detailing/finishing. A variety of alloy wires can be used to generate the biomechanical forces associated with tooth movement. Once the wire is activated or bent, it is the unloading or deactivating forces that produce orthodontic tooth movement. With current orthodontic treatment, super elastic nickel-titanium wire is often used for the leveling and aligning phases, with beta titanium and stainless steel (SS) wires most frequently used for space closure and detailing/finishing. Nitinol is an elastic material and can return to its original shape when deformed, this specific characteristic is called shape memory. In the beginning stages of orthodontic treatment, Ni-Ti wires are frequently used to put gentle forces on the crooked teeth in order to

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align them. A variation of Ni-Ti wires are heat-activated Ni-Ti wires. A heat-activated Ni-Ti wire holds the deformation at room temperature, but when the wire reaches the temperature of a patient’s mouth, the wire will return to its original shape. Heat-activated Ni-Ti wires are useful in the beginning stages of treatment. If the teeth are extremely crooked, the wire can be cooled so it can be tied into the brackets easier. Then after a few minutes, it will reach the temperature of the patient’s mouth, displaying the elastic properties. Beta- Titanium wires were developed after Ni-Ti wires to offer an intermediate range of elasticity and strength. This kind of wire serves as a good intermediary wire between Ni-Ti and stainless steel. The best known properties of Nitinol alloys are thermal shape memory and superelasticity. Nitinol exhibits the shape memory effect when it is deformed at low temperature and subsequently heated when it reverts to its original shape. Super-elasticity refers to the ability of Ni-Ti to return to its original shape upon unloading after a deformation. The superelastic properties of Ni-Ti are based on the stress-induced martensite formation. While many metals exhibit super-elastic effects, only Ni-Ti based alloys appear to be chemically and biologically compatible with the human body. An important feature of super-elastic Nitinol alloys is that their unloading curves are flat over a wide deflection range. This allows the design of devices that apply a constant force. The orthodontic archwire was the first product to use this property. Nitinol is being used in a wide variety of applications in various fields. Many of the current applications of Nitinol have been in the field of medicine, especially orthodontic archwires. In a linear elastic material like stainless steel there is a large amount of force on the tooth for a small amount of corrective motion [1-5]. Stainless steel wires entered dentistry in 1919 and have been used for decades due to their high strength. In addition, stainless steel wires do not rust and can be adjusted in many different ways by the orthodontist without breaking. However, stainless steel wires are not very elastic, meaning that if you bend these wires too much, they will assume the new position and will not return to their original position. Cobalt chromium nickel has the advantage that it can be hardened by heat treatment after being formed. The orthodontic archwires can have different shape and they can be round, rectangular and square; the optimal use of the different shapes depends on each clinical situation. In orthodontic treatments, wires of different metallic alloys are used for alignment, leveling, correction of the molar position, space closing, finishing and retention. The desirable characteristics in the orthodontic wires are a low stiffness value, a good formability, biocompatibility, low surface friction, and the possibility to be soldered to auxiliaries. Stainless steel wires are of frequent use because of their biocompatibility, environmental stability, stiffness and resilience. Cobalt-chromium (Co-Cr) wires can be manipulated and can be subjected to heat treatment. Heat treatment of Co-Cr wires results in a wire with properties similar to those of stainless steel. Nitinol wires have a low stiffness and a poor formability. Beta-titanium wires present average stiffness and good formability. Multistranded wires have a high springback and low stiffness if compared to solid stainless steel wires. Optimal use of these orthodontic depends on the specific clinical situation. Developments in material science have presented new archwire materials as well as improvements in the properties of existing ones. The study of Krishnan characterizes and compares three orthodontic archwire alloys, stainless steel, beta titanium alloy (TMA), and a newly introduced titanium alloy (TiMolium). Stainless steel was the strongest archwire alloy

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with less friction at the archwire-bracket interface. TMA wires exhibited better load deflection characteristics with less stiffness than the other two wires. The surface of TMA appeared rough and exhibited very high values for friction at the archwire-bracket interface. TiMolium appeared to be an alpha-beta titanium alloy composed of titanium, aluminum, and vanadium and intermediate in nature for all the parameters evaluated. [6] Orthodontic forces are applied to the teeth basically by means of different types of orthodontic wires. Knowledge of the mechanical properties of such wires is very helpful to the clinician for the choice and application of optimal force systems during the different phases of orthodontic treatment. Stainless steel wires have demonstrated higher modulus of elasticity if compared to nickel-titanium and beta titanium alloys wires. B-titanium wires showed higher modulus of elasticity than nickel-titanium ones. In addition stainless steel wires were found to have higher values for springback than cobalt-chromium ones and lower values than nickel-titanium and B-titanium wires. [7] In the study conducted by Loftus frictional forces during simulated sliding tooth movement were measured with a model that was representative of a clinical condition. The model allowed tipping of the tooth and it also allowed rotation until the wire contacted opposite corners of the ligature tie or the buccal shield with self-ligating brackets and the base of the slot. Beta titanium arch wires produced higher frictional forces than nickel titanium arch wires, but no significant differences were found between each of the two and stainless steel arch wires. [8] The brackets can be of 2 main categories: the brackets that have their own system to connect to the archwire, known as self-ligating and the ones that need ligatures to be connected to the archwire. Self-ligating brackets contain special locking mechanisms to secure the archwire in the bracket slot, without the need for a ligature. The cardinal feature of a self-ligating bracket is a metal labial face to the bracket slot, which is referred to as a clip or slide. It reduces the chairside time and makes the ligating easier. The lower sliding resistance offered by self-ligating bracket reduces the resistance between the archwire and the slot. Different studies compared the frictional properties of self-ligating metal brackets with those of conventionally ligated metal brackets. The study conducted by Reicheneder showed that friction of the self-ligating brackets using wire with a dimension of 0.018 x 0.025 inches was 45-48 per cent lower than with 0.017 x 0.025 and 0.019 x 0.025 inch wires. Friction of the conventionally ligated brackets showed a 14 per cent or less reduced friction with 0.018 x 0.025 inch wire compared with 0.017 x 0.025 and 0.019 x 0.025 inch wires. The self-ligating metal brackets showed lower frictional forces with a 0.018 x 0.025 inch wire than conventionally ligated brackets, whereas conventionally ligated brackets showed lower friction with 0.017 x 0.025 and 0.019 x 0.025 inch wire. Friction values vary with different bracket/archwire combinations and the choice of a bracket system for treatment should consider the correct wire dimension to produce the desired frictional forces. [9] The lower frictional resistance produced by passive self-ligating brackets can be helpful during orthodontic sliding mechanics. The aims of a study conducted by Tecco et all were to evaluate the frictional resistance of brackets with passive ligation and to compare these values with conventional brackets. The study was performed using a specially designed apparatus that included 10 aligned brackets to compare the frictional resistance generated by conventional stainless steel brackets and two different types of self-ligating brackets (Damon SL II brackets and Time Plus brackets) coupled with stainless steel, nickel-titanium and beta-titanium archwires. All

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brackets had a 0.022-inch slot, and five different sizes of orthodontic wire alloys used. Time Plus self-ligating brackets generated significantly lower friction than both the Damon SL II self-ligating brackets and Victory brackets. However, the analysis of the various bracketarchwire combinations showed that Damon SL II brackets generated significantly lower friction than the other brackets when tested with round wires and significantly higher friction than Time Plus when tested with rectangular archwires. Beta-titanium archwires generated higher frictional resistances than the other archwires. All brackets showed higher frictional forces as the wire size increased. [10] Our first study was conducted in order to evaluate the use of Teflon, which is an antiadherent and aesthetic material in orthodontics. The aim of our first research was to evaluate in vitro the influence of Teflon coating on the resistance to sliding of orthodontic archwires [11]. Different stainless steel and nickeltitanium types of archwires with round and rectangular sections were tested in two passive self-ligating brackets (SmartClipTM, 3M Unitek; Opal®, Ultradent) and one active self-ligating bracket (Quick®, Forestadent). The results showed that Teflon coated archwires produced lower frictional values than the uncoated archwires and they might be used in those clinical situations where low friction values are required. Resistance to sliding is the resistance to motion when a solid object moves tangentially against another [12]. From the orthodontic point of view, Teflon or polytetrafluoroethylene (PTFE) is an anti-adherent and aesthetic material characterized by a completely fluoridated chain. The chain is responsible for its physical and chemical characteristics. Teflon is made through a sintering process and two forms exist: classical PTFE, not microporous (Teflon) and expanded PTFE (ePTFE), microporous (Gore-Tex) [13]. Teflon® coatings are usually applied in an atomizing process with purpose-cleaned compressed air as the transport medium for the atomized Teflon particles. The layer has a typical thickness of 20–25 ȝm and is tooth-colored. After being heat-processed in a chamber furnace the layer has a closed, finely structured surface with excellent sliding properties and adequate adherence to the substrate. Teflon has a low coefficient of friction and archwires with a Teflon coating reduce resistance to sliding. Different stainless steel and nickel-titanium archwires with round and rectangular sections with and without Teflon coating were tested. Teflon coated archwires (TP Italia, Gorle, BG, Italy) are made through an atomizing process with purpose-cleaned compressed air as the transport medium for the atomized Teflon particles. Each type of archwire was coupled with self-ligating brackets: SmartClipTM (3M Unitek), Quick® (Forestadent) and Opal® (Ultradent Products). The RS of each bracket – archwire – plate combination was tested 10 times by passing the wire through the test brackets at a rate of 10 mm per minute, with a frictional testing apparatus, mounted on the crosshead of an universal testing machine (Model LR30K Plus, Lloyd Instruments, Fareham, Hants, UK). Teflon coated archwires produced less friction than uncoated archwires under all tested conditions. There are several ways to reduce RS by improvement of the material surface of the archwire. Coating or refining the wire surface has an influence on the frictional behaviour during the orthodontics treatment. Compared with the conventional wire, the coating creates a modified surface which can influence the friction’s values , the esthetics, the corrosive properties, and the mechanical durability of the wires. From Husmann study on the frictional

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behaviour of coated guiding archwires can be stated that all coatings improve the friction characteristics of wires compared with uncoated ones and that the best effects with respect to friction are achieved with Teflon coating [14] Teflon coated ligatures also produced lower friction than elastomeric ligatures. Ligatures commonly used in orthodontics are heat-treated stainless steel or elastomeric rings. The friction arising from a ligature depends upon its coefficient of friction and De Franco studies showed that frictional forces were less with Teflon-coated stainless steel ligatures than with elastomeric ligatures. [15] In our study Teflon coated archwires produced lower frictional levels than the correspondent uncoated archwires. This reduction in adverse forces has positive implications for clinical applications and the findings suggest that coating orthodontic archwires with Teflon has the potential for decreasing resistance to sliding. Teflon coating has excellent aesthetic properties and together with their improved frictional performance lead to an excellent use of this type of archwire in the orthodontic practice. The purpose of our second research was to compare the values generated during sliding frictional tests of various types of single and combined Ni-Ti arches within the passive selfligating SmartClip brackets. The resistance to sliding of self-ligating brackets is generally lower than those of the conventional brackets. A self-ligating bracket restrains the archwire within the slot by means of a slide or a clip that covers the slot and for that reason passive self-ligating brackets have lower frictional resistance than conventional brackets. A study conducted by Franchi [16] evaluated the frictional forces generated by 4 types of passive stainless steel self-ligating brackets and conventional elastomeric ligatures during sliding mechanics. An experimental model was used to assess frictional forces produced by the self ligating brackets and conventional brackets. Results showed that significantly smaller static and kinetic forces were generated by the self ligating brackets if compared with conventional brackets and elastomeric ligatures. Read-Ward conducted an ex-vivo study comparing the static frictional resistance of three self-ligating brackets with a conventional steel-ligated bracket. The effects of archwire size, bracket/archwire angulation and the presence of human saliva were investigated. The study demonstrated that both increases in wire size and bracket/archwire angulation resulted in increased static frictional resistance for all bracket, with the presence of saliva having an inconsistent effect. Conventional brackets produced large individual variation, confirming the difficulty in standardizing the ligation force, although significantly larger frictional forces were observed. [17] Friction is defined by Tweney and Hughes as the resistance to motion when it is attempted to slide one surface over another with which it is in contact. A distinction is made between static frictional force which is the force needed to start a motion and kinetic frictional force defined as the force needed to resist the sliding motion of one solid object over another at a constant speed. [18] Since arch mechanics are a major part of the orthodontic treatment, friction influencing tooth movements is of great interest. Thanks to the reduced friction on one side sliding mechanics are facilitated, on the other side control of movements and root position could be reduced. Ehsani did a literature review in order to compare the amount of expressed frictional resistance between orthodontic self-ligating brackets and conventionally ligated brackets in vitro.

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Several electronic databases and a hand search were performed by going through the reference lists of the selected articles. A total of 73 papers were initially obtained and a wide range of methods were applied in the selected papers. Compared with conventional brackets, self-ligating brackets produce lower friction when coupled with small round archwires. [19] With passive self-ligating appliances, the contact between the archwire and the bottom of the bracket slot is not possible and rotations cannot be completely corrected with the initial archwire. The rotations would be corrected by the gradual increase in archwire size or by the “Piggyback” technique. The technique is simple: at the patient’s second appointment, a second small size archwire is inserted in the bracket slots on top of the original archwire. Using conventional brackets rotations are usually already corrected in the first stages of orthodontic treatment, as the ligature made it possible to seat the archwire to the bottom of the bracket slot. With passive self-ligating appliances, the intimate contact between the archwire and the bottom of the bracket slot is reduced and a 10° play exists between the archwire and the slot. [20] The rotations would eventually be corrected by the gradual increase in archwire size. The aim of our work was to compare the resistance to sliding between the archwires mostly utilized in the first stages of treatment and the combination of archwires, used in the piggyback archwires technique. Five different types of archwires were tested: 0.12 round Ni-Ti, 0.14 round Ni-Ti, 0.16 round Ni-Ti, 0.18 round Ni-Ti and 0.18 round stainless steel. For the piggyback archwire technique, a 0.14 round Ni-Ti archwire was coupled with the 0.12 round Ni-Ti, with the 0.14 round Ni-Ti and with the 0.16 round Ni-Ti archwires. Each type of archwire was tested with one type of passive self-ligating bracket: SmartClipTM brackets. Six stainless steel plates were built for each bracket type, to simulate different clinical scenarios in vitro and three brackets were positioned on each plate with orthodontic composite. The distance among the centres of the brackets was always 8.5 mm and stainless steel templates were built to position the brackets on the plates correctly. The resistance to sliding of each bracket – archwire – plate combination was tested ten times by passing the wire through the test brackets at a rate of 10 mm per minute, with a frictional testing apparatus, mounted on the crosshead of a universal testing machine (Model LR30K Plus, Lloyd Instruments, Fareham, UK). A comparison was made between the levels of friction of any single arch with those of the combination of two different arches, known as tandem. 0.12 round Ni-Ti presents values of friction smaller than the tandem arches. 0.14 round Ni-Ti and 0.16 round Ni-Ti arches present values of friction equal to or slightly higher than the 0.12 - 0.14 Ni-Ti arches in the situation simulating the greatest misalignment. The 0.18 round Ni-Ti arch presents values of friction greater than the tandem 0.12 – 0.14 Ni-Ti arches. 0.18 SS arch generated higher values than the tandem 0.12 - 0.14 Ni-Ti arches and also higher then the tandem 0.14 - 0.14 Ni-Ti arches. Tandem 0.14 - 0.16 Ni-Ti arches always presents the highest and not comparable values of friction, especially in those situations representing minor misalignment. Our study shows a proportional increase in friction values with increasing thickness of the arch section and it shows that there is a significant difference in the friction values between the individual arches and the tandem arches, which is an indication of high frictional level of the tandem arches.

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0.14 - 0.16 Ni-Ti arches are far superior to the values of other tandem arches. Tandem arches showed values of friction greater than individual arches without incurring in undesired disengagements confirming their usefulness in the final stage of alignment and leveling to completely correct rotations. Our third study investigated the effects of different bracket base materials on the bond strength of brackets to composite resin surfaces. The static and kinetic frictional coefficients of commercially pure titanium brackets were evaluated by Kusy in the passive and active configuration both in the dry and wet states against stainless steel, nickel-titanium, and beta-titanium arch wires, and compared to stainless steel brackets. Titanium brackets are grey in color and rougher in texture than the stainless steel brackets. The frictional measurements show that titanium brackets are comparable to stainless steel brackets. [21-22] The results suggested that clinically acceptable bond strength can be achieved by conventional brackets and that higher bond strength is achieved by custom-made Titanium Grade 5 brackets. In fact, Titanium Grade 5 features excellent oxidising properties towards the material used in the cementing adhesive, guaranteeing a strong chemical bond. Moreover, bracket adhesion towards cement could be further enhanced leveraging upon retention zone, placed on the bracket surface facing the teeth, that can be customized during the bracket design phase and that features higher resistance compared to traditional steel brackets. The purpose of the study is to compare the resistance of custom-made Titanium Grade 5 brackets to largely used “control” bracket (3M Unitek). In order to facilitate the used of dynamometer, Traditional Victory 3M brackets have been used, featuring the same mesh as Smartclip 3M brackets used in the Orthodontic Department of Milan University. 30 Victory 3M brackets have been used in correspondence to superior premolar. Transbond™ PLUS Color Change from 3M Unitek has been used as photo-polymerizing composite. Data have been collected using LR 30K PLUS Lloyd Instruments dynamometer. The study suggests that Titanium Grade 5 brackets present a stronger adherence to composite compared to traditional Victory bracket, with a 2.3 to ratio. The breaking point for Titanium Grade 5 brackets is reached in a more linear way, implying a less traumatic bracket-to-composite detach. However the detach is reached applying much stronger forces, compared to traditional brackets, in order to limit postdebonding damages on enamel surface. The use of a stronger force might cause micro damages on enamel surface. Consequently, it is preferable to keep a minimum composite level on the enamel surface, which can be easily manually removed afterwards. The study suggests that chemical bond between Titanium Grade 5, composite and the mesh provide an excessive bracket-to composite strength, strongly increasing the risk of damages on the enamel surface during debonding. Consequently, we are considering reducing the Titanium-to-composite mesh bond, removing the drop on the mesh base to reduce micromechanical retention.

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References [1] [2] [3] [4] [5] [6] [7] [8]

[9]

[10]

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[11]

[12] [13] [14] [15]

[16]

[17] [18]

Lidija Zorco, Rebeka Rudolf. Metallographic sample preparation of orthodontic Ni-Ti wire. Ames. Lagoudas D. C. Shape Memory Alloys, Modelling and Engineering Applications, Springer Science and Business Media LLC, New York. 2008. Kujala S. Biocompatibility and Biomechanical Aspects of Nitinol Shape Memory Metal Implants. Ph. D. Thesis, 2003. Ryhänen J. Biocompatibility Evaluation of Nickel-Titanium Shape Memory. Metal Alloy, Ph.D. Thesis, 1999. Stöckel D. Nitinol: A Material with Unusual Properties, Endovascular Update 1998. Krishnan V, Kumar KJ. Mechanical properties and surface characteristics of three archwire alloys. Angle Orthod. 2004 Dec;74(6):825-31. Kapila S, Sachdeva R. Mechanical properties and clinical applications of orthodontic wires. Am. J. Orthod. Dentofacial Orthop. 1989 Aug;96(2):100-9. Loftus BP, Artun J, Nicholls JI, Alonzo TA, Stoner JA. Evaluation of friction during sliding tooth movement in various bracket-arch wire combinations. Am. J. Orthod. Dentofacial Orthop. 1999 Sep;116(3):336-45. Reicheneder CA, Gedrange T, Berrisch S, Proff P, Baumert U, Faltermeier A, Muessig D. Conventionally ligated versus self-ligating metal brackets--a comparative study. Eur. J. Orthod. 2008 Dec;30(6):654-60. Tecco S, Festa F, Caputi S, Traini T, Di Iorio D, D'Attilio M. Friction of conventional and self-ligating brackets using a 10 bracket model. Angle Orthod. 2005 Nov; 75(6): 1041-5. Giampietro Farronato, Rolf Maijer, Maria Paola Carìa, Luca Esposito, Dario Alberzoni, Giorgio Cacciatore. The effect of teflon coating on the resistance to sliding of orthodontic archwires. European Journal of Orthodontics. In press. Rabinowicz E. Friction and wear of materials 2nd ed. Wiley-Interscience. New York. 1995. Pietrabissa R Biomateriali per protesi e organi artificiali. Pàtron. Bologna. 1996. Husmann P, Bourauel C, Wessinger M, Jäger A The frictional behavior of coated guiding archwires. Journal of Orofacial Orthopedics 2002, 63: 199–211. De Franco D J, Spiller R E Jr, Von Fraunhofer J A Frictional resistances using Tefloncoated ligatures with various bracket-archwire combinations. The Angle Orthodontist 1995,65: 63–72. Lorenzo Franchi, Tiziano Baccetti, Matteo Camporesi, Ersilia Barbato. Forces released during sliding mechanics with passive self-ligating brackets or nonconventional elastomeric ligatures. American Journal of Orthodontics and Dentofacial Orthopedics. 2008, 133: 87-90. Read-Ward GE, Jones SP, Davies EH. A comparison of self-ligating and conventional orthodontic bracket systems. British Journal of Orthodontics.1997, 24:309-317. Tweney CF, Hughes LEC. Chambers’s technical dictionary WandR Chambers Ltd. London 1961.

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[19] Ehsani S, Mandich MA, El-Bialy TH, Flores-Mir C. Frictional resistance in selfligating orthodontic brackets and conventionally ligated brackets. A systematic review. Angle Orthod. 2009 May; 79(3):592-601. [20] McLaughlin R P, Bennett J C, Trevisi H Systematized orthodontic treatment mechanics. Second Edition Mosby International Ltd. 2001. [21] Kusy RP, Whitley JQ, Ambrose WW, Newman JG. Evaluation of titanium brackets for orthodontic treatment: part I. The passive configuration. Am. J. Orthod. Dentofacial Orthop. 1998 Nov;114(5):558-72. [22] Kusy RP, O'grady PW Evaluation of titanium brackets for orthodontic treatment: Part II--The active configuration. Am. J. Orthod. Dentofacial Orthop. 2000 Dec;118(6):67584.

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In: Viscoelasticity: Theories, Types and Models Editors: J. N. Perkins and T. M. Lach, pp. 177-185

ISBN 978-1-61324-203-2 © 2011 Nova Science Publishers, Inc.

Chapter IX

Hydrodynamic Characteristics of an Oscillating Viscoelastic Squeeze Film Muhannad Mustafa*, Nusrat J. Chhanda and M. Mahbubur Razzaque Department of Mechanical Engineering, BUET, Bangladesh

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Abstract Squeeze film theories have often been a major area of interest in fluid mechanics. In this paper, effects of surface roughness of rubber block on leakage flow rate and hydrodynamic force developed in fluid film between a cylindrical rigid surface and a cylindrical rubber surface are analyzed. The distribution of pressure in the fluid film and the porous material and the distribution of surface deformation of the porous rubber block are obtained by simultaneously solving the Reynolds equation, the Laplace equation and the three-parameter viscoelastic constitutive equation. Equations are discretized into finite difference equations and solved by Gauss-Siedel iteration. It is seen that with increasing standard deviation of surface height of rubber block, load carrying capacity increases significantly developing huge hydrodynamic force in the fluid film. Leakage flow rate decreases slightly with increasing standard deviation of surface of rubber block. Spatial distribution of surface texture of rubber block has no effect on leakage flow rate as well as hydrodynamic force during squeezing motion. The present analyses contribute to designing many engineering applications such as bearing, wet clutch and noncontacting face seal. The results obtained from the present model are compared with experimental results available in the literature and a very good agreement is found.

* Present Address: Department of Mechanical Engineering, Auburn University, AL 36849, USA. Email: [email protected], [email protected] Viscoelasticity: Theories, Types and Models : Theories, Types and Models, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central,

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Muhannad Mustafa, Nusrat J. Chhanda and M. Mahbubur Razzaque

Nomenclature Roman Symbols as f F h hmin hn h0

h x

h k p pm a1,b0, b1 Q r ra Res T t

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vh vrm

Amplitude of oscillation (mm) Frequency (Hz) Hydrodynamic film force (N) Film thickness (mm) Minimum film thickness (mm) Net clearance (mm) Initial film thickness (mm) Arithmetic mean of film thickness (mm) Squeeze velocity (m/sec) Permeability of rubber block (m2) Fluid film pressure (Pa) Pressure in porous matrix (Pa) Coefficients in constitutive equations (sec, MPa and kPa-s respectively) Net leakage flow rate in radial direction (l/sec or cc/sec) Distance in the radial direction (mm) Radius of rubber block (mm) Squeeze Reynolds number Dimensionless Time Absolute Time (Second) Oscillating Velocity in vertical direction (m/sec) Average velocity in radial direction (m/sec)

Greek Symbols įs İ ȝ Ș

Viscoelastic deformation of rubber surface (mm) Strain Standard deviation of surface height of rubber block (mm) Viscocity of lubrecant oil (Pa.sec)

Abbreviation ACL

Auto-correlation length (mm)

Introduction The squeeze film theories find applications in the studies of thin film of fluids in synovial joints, computer hard disks and coalescence of bubbles. The analysis of squeeze film, in particular, between a porous block and an oscillating rigid plate has long been a subject of research in connection to the studies of bearings, wet-clutches, non-contacting mechanical

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Hydrodynamic Characteristics of an Oscillating Viscoelastic Squeeze Film

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face seals, gaskets, thrust washers, O-rings, printing processes, snow tires, etc. Although using porous rubber material in the bearing system reduces load carrying capacity due to penetration flow of lubricant oil across the rubber surface, it has some other attractive characteristics for some engineering applications. Often temperature rises significantly between the mating surfaces due to friction in case of bearings and wet clutches. Penetration flow of lubricant oil across the rubber surface contributes to cooling of the mating surfaces by removing lubricant oil with frictional heat. In wet clutches, the penetration flow across the surface attenuates the fluctuation of fluid-pressures in the engagement and disengagement processes of the friction plate. Moreover, soft deformable rubber materials in many engineering applications can isolate sensitive engine part by absorbing vibration which ensures a long life of the systems. Increasing hydrodynamic force in the fluid film between mating surfaces is often advantageous in case of bearings, wet clutches, snow tires etc due to good load carrying capacity. Whereas, decreasing leakage flow in non-contacting face seals is of prime importance, especially in critical applications such as, nuclear reactor where seal failure may have severe consequences. In the previous investigations, Rodhe et al. (1979), Hori and Kato (1979), Hori et al. (1981) and Yoo (1987) have demonstrated that in case of high frequency of squeezing motion, the pressure developed in the fluid film significantly depends on the viscoelastic effects of the bounding solid surface. The squeeze film between porous mating surfaces of low elastic modulus materials have been investigated as an elasto-hydrodynamic (EHL) problem by many researchers such as, Wu (1970), Murti (1974), Wada and Nishida (1985), Ikeuchi et al. (1989) and Horikawa et al. (1990). All these studies focus upon the squeezefilm performance considering that the mating surfaces were smooth; neither the effect of the viscoelastic deformation nor the effect of roughness of the mating surfaces was considered. However, the surfaces in sliding or squeezing contact are never completely smooth and therefore, the influence of surface roughness on the squeeze film characteristics should not be neglected. Kaneko et al. (2004) found that the effect of viscoelascity of porous rubber and fluid inertia on surface deformation cannot be ignored for the surfaces oscillating at or more than a frequency of 40 Hz. Lin et al. (2002) have shown that the mean squeeze-film characteristics of a long partial journal bearing operating under a time-dependent oscillating load are affected by the roughness in both transverse and longitudinal directions but they did not consider any rubber material in their investigation. Under the influence of axial vibration, surface roughness of porous rubber material may considerably increase hydrodynamic force with decreasing leakage flow which is desirable in many engineering applications. Mustafa et al. (2010) has developed a comprehensive model to investigate the combined effect of surface roughness and viscoelasticity on hydrodynamic characteristics of oscillating squeeze film. In this chapter, a review discussion has been conducted to show how standard deviation of viscoelastic rubber surface affects the hydrodynamic characteristics of an oscillating squeeze film.

Physical Model Figure 1(a) shows a schematic diagram of the simple model investigated in the present study. A cylindrical rigid plate is squeezing over a cylindrical porous rubber block attached to

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a rigid support. The porous block and the rigid plate are separated by a nominal distance, h. The rigid plate is oscillating sinusoidally in the direction perpendicular to its surface. It is assumed that the clearance gap is completely filled and the body of the porous block is impregnated with lubricant oil. The squeezing motion of the rigid plate causes hydrodynamic pressure to develop in the clearance gap and inside the porous block and to deform the porous rubber surface. v h [ 2Sfa s sin 2Sft ]

Rigid plate

vh

Lubricant oil

h[ h 0  a s {cos( 2S ft )  1}  G s ]

y

h

Lubricant Oil

Porous rubber block

L

ra

Porous rubber block

z r (a)

(b)

Figure 1. (a) Problem geometry (b) Co-ordinate systems.

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Cylindrical co-ordinates, as shown in Fig. 1(b), are used for the axisymmetric model. The axial co-ordinate through the clearance gap, y is measured upward from the porous rubber surface. For simplicity, the deformation (įs) of the porous rubber surface is transferred to the bottom surface of the rigid plate. For two-dimensional analysis, only circumferential roughness pattern as shown in Figure 2 is considered for the present work.

Figure 2. Images (200X) taken by Scanning Electron Microscope (SEM) of circumferential roughness pattern. [Images are taken in SEM lab of MME Dept., BUET]

Mathematical Model It is assumed that the porous rubber block has uniform porosity and permeability along z and r directions. It is assumed that quantitative effect of the strain on the local variations of the porosity and the permeability is not significant as local deformation of the porous material

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Hydrodynamic Characteristics of an Oscillating Viscoelastic Squeeze Film

181

is much smaller than the pore size. The pressures in the fluid film and the porous material and the surface deformation of the porous rubber block will be obtained by simultaneously solving the Reynolds equation, the Laplace equation and the three-parameter viscoelastic model. To evaluate fluid film pressure between the mating surfaces the modified Reynolds equation is used as governing equation (M. Mustafa, 2007) as follows: §x w § 3 wp · ¨ k wpm rh r K 12 ¨ ¸ ¨ h K wz wr © wr ¹ ¨ ©

z

· ¸ L¸ ¸ ¹

Governing equation for the pressure rise in the porous material is expressed as Laplace equation. As the seepage velocity is much smaller than the fluid velocity in the squeeze film, fluid inertia effect due to the seepage velocity in the porous material is assumed to be negligibly small. Laplace equation can be expressed as, w § wpm · w 2 pm ¨r ¸r wr © wr ¹ wz 2

0

Deformation of the rubber block is evaluated based on a three-element viscoelastic spring-dashpot model (M. Mustafa, 2007). Viscoelastic deformation of rubber surface, u is calculated using following simplified equation:

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H

b0 t

Pi ab  [1  (1  1 0 )e b1 ] b0 b1

du dz

Where, Pi is integral averaged local pressure along z-direction generated in the porous rubber material. The suffix i indicates each nodal points on rubber surface along r-direction of rubber block. Coefficients of the constitutive equations for viscoelastic model are obtained from the experimental data by Kaneko et al. (2004) as, a1 = 2.48×10-03 sec, b0 = 6.08 MPa and b1 =27.7856 kPa-s. Lateral deformation of rubber block is not considered in the present model as lateral deformation is much negligible in comparison to surface deformation of rubber block. Actually this assumption makes the present model simple, efficient and faster than previous model developed by Kaneko et al. (2004). Two-dimensional rough surfaces is generated by MATLAB code using “randn” function which is based on Ziggurat method for producing random variables. In the present model, local hydrodynamic pressures are computed numerically at various grid points over the rubber surface and hydrodynamic force F is obtained by the following equation where p is fluid film pressure between mating surfaces: ra

F

2S ³ prdr 0

Leakage flow is defined as the radial outward flow to the surroundings from the fluid film in the clearance gap between the mating surfaces. Leakage flow is calculated from the equation as follows:

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Muhannad Mustafa, Nusrat J. Chhanda and M. Mahbubur Razzaque

182 Q

2Sra hn vrm

where, vrm

§x ¨ k wpm   ³ ¨ h ra h 0 ¨© K wz 1

ra

z

· ¸ and hn ¸rdr L¸ ¹

(h  P )

Reynolds equation, the Laplace equation and the equation for three parameter viscoelastic model are discretized into finite difference equations and solved by Gauss-Siedel Iteration Method. In order to calculate the pressure in the porous material the cylindrical porous rubber block is divided into 48 nodes along the radial direction and 16 nodes along the z-direction. To calculate fluid film pressure, region of fluid film between the mating surfaces is also divided into 48 nodes along the radial direction. Grid independency test suggests that mesh sizes of 48 u 16 are sufficient to correctly evaluate the squeeze film characteristics. Squeeze Reynolds number Res is evaluated to be 0.221 for oscillating frequency f = 10 Hz and 0.443 for f = 20 Hz.

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Results and Discussion In the present work, oscillating squeeze film between the mating surfaces is modeled with silicone oil operated at atmospheric pressure. To operate fluid film in hydrodynamic regime minimum clearance is kept at hmin/μ t 3.92 where hmin denotes minimum film thickness and μ denotes standard deviation of surface height of rubber block. It is also assumed arbitrarily that the rigid block is oscillating over the rubber block at finite amplitude of 0.3 mm. For computation, input parameters are used as diameter of rubber block = 98 mm, thickness of rubber block = 29.2 mm, permeability of rubber, k = 2.63×10-10 m2, porosity = 35.7%, absolute viscosity of lubricant oil = 0.116 Pa.s, density of lubricant oil = 970 kg/m3, ambient temperature = 20 0C, initial film thickness =1.4 mm. Figure 3(a) shows the variation of hydrodynamic pressure developed in the fluid film between the mating surfaces at the center of rubber block with respect to dimensionless time parameter in terms of absolute time and oscillating frequency of rigid surface. Hydrodynamic pressure indicates integral averaged pressure throughout the fluid film. For the validation of numerical code, comparisons among fluid film pressure at the center of rubber block obtained from the proposed model, Kaneko et al.’s (2004) model and experiment by Kaneko et al. (2004) are also shown in Figure 3(a). The characteristic data on the porous rubber block, the lubricant oil and the coefficients in the constitutive equation employed in the numerical analysis are the same as those used in the experiment by the Kaneko et al. (2004). Good agreement is found comparing results obtained from the proposed model, Kaneko et al.’s model and experiment by Kaneko et al. for the frequency of oscillation 10 Hz. Figure 3(b) represents the variation of viscoelastic deformation at the center of rubber block with time for the frequency of oscillation 20 Hz. For the validation of three parameter viscoelastic model used in present model, deformation of rubber surface obtained from present model is compared with the result obtained from Kaneko et al.’s model and a good agreement is found.

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The qualitative agreement of present results with experimental results implies the validity of the present numerical simulation. 0.2 Viscoelastic Deformation, įs (mm)

50 Fluid film pressure P (kPa)

Present model Experiment Kaneko et al.’s model

25

0 0

1

2

-25

Present model Result from Kaneko et al. -10

k = 2.63 X 10

0.1

m

2

-9

k = 1.578 X 10 m2

0 0

1

2

3

4

-0.1

-0.2

-50

Dimensionless time, T=(f.t)

Time T=(f.t)

(a)

(b)

Variation of the hydrodynamic force with time under various standard deviations (μ) of surface height for different frequencies of oscillation is shown in Figure 4 (a). It is found that due to sinusoidal oscillation of the rigid surface over the rubber block hydrodynamic force is developed in the fluid film between the mating surfaces. This hydrodynamic force varies almost sinusoidally with time. The figure also reveals that with the increasing standard deviation of surface height, hydrodynamic force increases significantly. Similarly, Figure 4 (b) shows the sinusoidal variation of leakage flow rate with time under various standard deviations (μ) of surface height of rubber block for the frequency 20 Hz. It is found that leakage flow rate decreases slightly with the increasing standard deviation of surface height due to small clearance between the mating surfaces. When two mating surfaces move towards each other in normal direction, fluid needs finite time to squeeze out through the clearance. 0.4

Standard deviation 0.006 mm 0.100 mm 0.200 mm

150 50 -50 0

2

-150 ACL=1.6479 mm

-250

4

Leakage flow rate Q (l/s)

250 Hydrodynamic force F (N) .

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Figure 3.(a) Variation of fluid film pressure with time for the frequency of oscillation 10 Hz at the center of the rubber block (b) variation of viscoelastic deformation for the frequency of oscillation 20 Hz at the center of the rubber block when surface is smooth.

Standard deviation

0.006 mm 0.100 mm 0.200 mm

0.3 0.2 0.1 0 -0.1

0

2

4

-0.2 -0.3

ACL= 1.6479 mm

-0.4 Time T=(f.t)

(a)

Time T=(f.t)

(b)

Figure 4. (a) Variation of hydrodynamic force (b) variation of leakage flow rate with time for various standard deviation of surface height of rubber block for the frequency 20 Hz.

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Muhannad M Mustafa, Nusrat J.. Chhanda andd M. Mahbubuur Razzaque

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Besides, with w increasing standard deviiation of surfaace height, fluiid experiencess resistance too extrude throu ugh the gap beetween the maating surfaces. Due to small clearance, thiis extrusion prrocess is interrrupted and soome amount of o fluid retainns in the gap. Also due to viscoelastic v beehavior, rubbeer surface cannnot response instantly withh the oscillatinng motion of upper u rigid pllate which leaads to develop huge hydroodynamic forcce with decreaasing leakagee flow rate. Fiigure 5 showss the variationn of maximum m hydrodynam mic force andd leakage flow w rate with tim me under variious standard deviation of surface s heightt for the frequuencies 10 andd 20 Hz. In thhe figure, it is seen that hyddrodynamic foorce increases about 23% foor 10 Hz and 36% 3 for 20 H as standard deviation inccreases from 0.0062 Hz 0 mm too 0.29 mm whhere as leakage flow rate deecreases 11% % for 10 Hz and 18% forr 20 Hz. It im mplies that with w increasinng standard deeviation, leak kage flow raate decreases at higher rate r for highher frequencyy whereas, hyydrodynamic force increasees at higher ratte for higher frequency fr of oscillation. a importantt parameter to t describe the t spatial Auto-correelation lengthh (ACL) is an diistribution of surface texturre. It is reportted in author’ss previous puublication (Muustafa et al. 20010) that variiation of maxiimum hydroddynamic force and maximum leakage floow rate are faairly negligib ble with autoo-correlation length of ruubber surface. Therefore, it can be coonsidered as height h distribuution of the suurface roughnness over the rubber r surfacee has more siignificant effect on hydrodyynamic charactteristics of visscoelastic squeeeze film thann the spatial diistribution of surface s profilee.

Fiigure 5. Variatio on of maximum m hydrodynamicc force and maxximum leakage flow rate with time t under vaarious standard deviations of suurface height off rubber block for f various freqquencies.

Con nclusion n Effects of surface roughhness on hydrrodynamic forrce and leakaage flow rate are shown vaarying two-d dimensional surface s texturre characterisstics parametters such ass, standard deeviation. With h the incremeent of standaard deviation of surface heeight of rubbber surface, hyydrodynamic force in fluid film between the mating surfaces increases significantlly whereas, leeakage flow raate decreases slightly. s Spatial distributionn of surface rooughness of ruubber block

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Hydrodynamic Characteristics of an Oscillating Viscoelastic Squeeze Film

185

has no effect on both hydrodynamic force and leakage flow rate. It is inferred that due to viscoelastic behavior, surface roughness of rubber block has a significant effect on load carrying capacity but it has a negligible effect on leakage flow rate. Present model is more simple and faster than the previous model developed by Kaneko et al. (2004). Also, Kaneko’s model has some limitations as analysis of oscillating squeeze film is done without considering surface roughness of rubber block. Effect of surface roughness of rubber block on hydrodynamic characteristics of viscoelastic squeeze film cannot be ignored specially for the high frequencies and should be considered in many mechanical designs such as, seals, clutches and bearings.

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References Hori, Y. and Kato, T. (1979), A study of viscoelastic squeeze films, JSLE Trans., 24(3), 174181. Hori, Y., Kato, T. and Narumiya, H. (1981), Rubber surface squeeze film, ASME J. Lub. Tech., 103, 398-405. Horikawa, J., Kyogoku, T. and Nakahara, T. (1990), Lubrication characteristics with porous elastic materials: numerical analysis, Proc. JAST., 377-380. Ikeuchi, K., Oka, M. and Mori, H. (1989), A simulation of the squeeze film effect in a hip joint, Trans. Jpn. Soc. Mech. Eng., Ser. C, 55(510), 508-515. Ikeuchi, K., Oka, M. and Gi, K. (1989), An experimental study of deformation and squeeze film effect in a synovial joint, Trans. Jpn. Soc. Mech. Eng., Ser. C, 55(516), 2123-2130. Kaneko, S., Tanaka, T., Abe, S. and Ishikawa, T. (2004), A study on squeeze films between porous rubber surface and rigid surface: analysis based on the viscoelastic continuum model, J. Tribology, 126, 719-727. Lin, J. R., Hsu, C. H. and Chuan, L. (2002), Surface roughness effects on the oscillating squeeze-film behavior of long partial journal bearings, Computers & Structures, 80, 297303. Mustafa, M. (2007), Analysis of an Oscillating Squeeze Film Between a Rubber surface and a Rigid Surface, M.Sc Thesis, Department of Mechanical Engineering, Bangladesh University of Engineering and Technology, Dhaka, Bangladesh. Mustafa, M, Chhanda, N. J and Razzaque, M.M. (2010), A Numerical Model of Oscillating Squeeze Film between a Rubber Surface and a Rigid Surface”, J Tribol Intl, 43, 202-209. Murti, P. R. K. (1974), Squeeze-film behavior in porous circular disks, J. Lub. Tech., 74, 206209. Rodhe, S. M., Whicker, D. and Booker, J. F. (1979), Elastohydrodynamic squeeze films: effects of viscoelasticity and fluctuating loading, ASME J. Lub. Tech., 101, 74-80. Wu, H. (1970), Squeeze-film behavior for porous annular disks, ASME J. Lub. Tech., 92, 593596. Wada, S., and Nishida, S.(1985), Elastohydrodynamic lubrication of porous squeeze bearings with non-newtonian fluids, Trans. Jpn. Soc. Mech. Eng., Ser. C, 51(469), 2183-2190. Yoo, H. S. (1987), Some effects of viscoelastic matrix on the squeeze films, ASLE Trans., 30, 403-408.

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Copyright © 2011. Nova Science Publishers, Incorporated. All rights reserved. Viscoelasticity: Theories, Types and Models : Theories, Types and Models, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central,

In: Viscoelasticity: Theories, Types and Models Editors: J. N. Perkins and T. M. Lach, pp. 187-202

ISBN 978-1-61324-203-2 © 2011 Nova Science Publishers, Inc.

Chapter X

The Mesoscopic Constitutive Equations for Polymeric Fluids and Some Examples of Flows Alexsei Gusev, Grigory Afonin, Ilia Tretjakov and Grigory Pyshnogray Dept. Mathematics, Altai State Technical University, Barnaul, Altai, Russia

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Abstract Constitutive equations for melts and concentrated solutions of linear polymers are derived as consequences of dynamics of a separate macromolecule. The model is investigated for viscometric flows. It was shown that the model gives a good description of non-linear effects of simple polymer flows: viscosity anomalies, first and second normal stresses, non-steady shear stresses. A simple rheological equation of state (RES) which can be chosen as an initial approximation in formulating such a sequence of RES was obtained and studied. In this work, RES is extended to the case of allowance for the additional corrections caused by intrinsic viscosity and the delayed interaction of a macromolecule with its environment. Realization of this approach involves consequent solution of two problems: formulation of the equations of dynamics for a macromolecule and transition from the formulated equations to RES. The resulting equations can be recommended as a first approximation in constructing a sequence of RES. Two cases of steady-state flow between unlimited parallel planes under the action of a constant pressure gradient are considered and the same constitutive equations allow us to expand calculations also on the process of extension of the jet after the leaving of the die. Considering the processes of stretching, which occur at the lower temperatures, one has to take into account the possible process of crystallization of polymer.

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Introduction Rheological equation of state of linear polymers solutions and melts obtained using methods of statistical mechanics and thermodynamics for description and interpretations of nonlinear effects [4, 5]. The base of these equations of Brown dynamics with introduced non Markovian anisotropic noise have been formulated and it has been shown, that introduction of the certain kind of non Markovian fluctuations leads to occurrence at a macrolevel of intermediate scale of length which is meaningful radius of a tube as in theory De Gennes, or lengths of a chain between entanglements as at Graessley. At the description of process of polymeric film formation the account of that is necessary, that the received film is cooled and exposed to stretching. As these processes occur simultaneously at their mathematical modelling the joint decision of the equations for stresses and heat conduction is necessary. In work for a finding of the established stress at stretching rheological model has been used whose parameters are functions of temperature. At thermal calculation it was supposed, that thickness of a film is small. Also in this work the calculation of non-uniform flows of a nonlinear viscoelastic liquid between parallel planes under action of constant pressure gradient is executed. Comparison of results of calculation of a velocity with experimental data is executed. It allows drawing a conclusion, that in the considered case of Pouselle flow, the rheological model describes non parabolic profile of velocity between parallel plates that does not contradict experimental data.

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Film Casting Process In this work the problem of mathematical description of film casting process which schematic image is presented according to [1–3] in Figure 1 is considered, whence it is visible that at first polymer melt is pressed through die and after expression the received film is reeled up on a drum.

Figure 1. The schematical image of technological of film casting process[1-3]. Viscoelasticity: Theories, Types and Models : Theories, Types and Models, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central,

The Mesoscopic Constitutive Equations for Polymeric Fluids …

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As a result of film movement from the die to the drum the material cools, changes its width and thickness. During this moment the film is exposed to stretching and there is so-called « neck effect». All these processes occur simultaneously and consequently at their mathematical modelling the joint decision of the equations for stresses and heat conduction is necessary. Due to the work it has been calculated that the established stresses at stretching modified rheological model Vinogradov-Pokrovskii using which parameters are known functions of temperature. At thermal calculation it was considered that the film thickness is small enough and consequently it is possible to assume that the temperature on film thickness direction is constant. To solve the issue of film stretching and cooling after an exit from die will use the rheological equation of state [4,5]:

V ik

 pG ik  3

K0 a ; W 0 ik

d 1  N  E I aik  vij a jk  v kj a ji  aik dt W0 where

V ik - stress tensor; p – pressure; K 0 and W 0 – initial values of shear viscosity and

relaxation time;

Q ik

wvi – tensor gradients of velocity; aik - symmetric second rank wxi

tensor of anisotropy; I

aii – the first invariant anisotropy tensor aik ; J ik

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(1)

2 E J ik  3 aij a jk , 3 W0

1 Q ik  Q ki – 2

N and E – the phenomenological parameters of

model considering in the equations of dynamics of macromolecule the sizes and the form of macromolecular coil. Then in Cartesian co-ordinates the system of the equations of dynamics looks as following:

wQ x wQ y wQ z   wx wy wz

0,

wQ x wQ x wQ x · wV 11 wV 12 wV 13 § wQ x ¸ , Q x Q y  vz   wx wy wz ¸¹ wx wy wz © wt wQ y wQ y wQ y · wV 21 wV 22 wV 23 § wQ y ¸¸ , U ¨¨ Q x Q y Q z   w w w w w w w t x y z x y z © ¹

U ¨¨

§ wQ z wQ z wQ z wQ z · ¸ Q x Q y Q z wx wy wz ¸¹ © wt

U ¨¨

(2)

wV 31 wV 32 wV 33 ,   wx wy wz

where, vi – melt velocity along axes, and accordingly;

U – density. For non-isothermal case

the system (1, 2) should be added by the equation of energy conservation [5, 6]: Viscoelasticity: Theories, Types and Models : Theories, Types and Models, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central,

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190

§ wT wT wT wT · ¸  vx  vy  vz wx wy wz ¸¹ © wt

UCV ¨¨

(3)

w § wT · w § wT · w § wT · P ¸  ¨O ¸  (T  T0 ), ¨O ¸  ¨O wx © wx ¹ wy ¨© wy ¸¹ wz © wz ¹ h

where CV - a specific thermal capacity at constant volume; T – film temperature, T0 – an

O–

ambient temperature,

heat conductivity factor;

P – heat exchange factor,

relation of the area of the film section to section perimeter ( h

ab 2 a  2b

h

S – the p

b , for thin film 2

a !! b ). Last composed in (3) considers film cooling through its surface. We will notice that in (3) the stream of heat arising at nonzero gradients of speed is not considered. Let's consider the decision of a stationary problem when in motionless system of coordinates there are no sizes time-dependent. We will place the beginning of co-ordinates in the middle of the exhaust outlet, we will direct the Ox axis along film movement and we will search for the decision of system depending only from a variable (1-3)

Q x Q x x ; T

T x .

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Addressing the experimental data [7] received for various samples at the identical degree of stretching we will see that various polymeric samples have various width. It does not correspond to a mode one axial elongation. This effect can be considered assuming that in the axis direction Oz operates additional stretching pressure that leads to the following expression for stress tensor:

ªV 11 « 0 « «¬ 0

0 º 0 0 »» . 0 V 33 »¼ 0

(4)

Here ı11 – elongation pressure, ı33 – additional pressure. Thus speed of width change will differ from speed of its thickness change and tensor gradients of velocity will look as following:

Q ij

º ª «u c( x) 0 0 » » « 1D «0 c u ( x) 0 » .  » « 2 » « 1D «0 0  u c( x)» 2 ¼ ¬

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(5)

The Mesoscopic Constitutive Equations for Polymeric Fluids …

191

here D is a parameter of elongation anisotropy. In this case it is possible to find dependence between film velocity u (x) both film width and thickness. As from definition: Q 22

y

u ( x)a c( x)

a ( x) ˜



wy

(1  D ) u c( x) , if y=a(x) and U y 2 1D .

U y ( x, y )

a c( x) a( x)

wu y

2

, ɬɨ

dy dt

wu y wy da ( x) dt



(1  D ) u c( x) , obtained 2

a c( x) ˜

dx dt

u ( x) a c( x) , or

u c( x)

1  D u c( x) 2 u ( x)

Or after integration we receive the expression for film width:

a ( x)

§ u ( x) · ¸¸ a(0)¨¨ © Q0 ¹



1D 2

(6)

And similar distribution for film thickness:

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b( x )

§ u ( x) · ¸¸ b(0)¨¨ Q © 0 ¹



1D 2

(7)

In a stationary case, for the description of anisotropic stretching in one-dimensional approach from equations (1-3) taking into account (5), (7) we will receive:

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It is necessary to add these equations with boundary conditions for speed and temperature. In case u (x) these conditions look like: vx (0) v0 ; vx (l ) kv0 , where k factor of film stretching. For temperature we will consider boundary conditions: the film temperature at contact with a cooling roll coincides with the established temperature of the roll and is equal to ambient temperature: T (0)

T1; T (l ) T0 . Here T1 - is melt temperature at the exit from

the die. For initial viscosity and initial relaxation time the following expressions are used:

K 0 (T ) K 0 T1 e

'H § 1 1· ¨  ¸ R ¨© T  273 T1 ¸¹

, W 0 (T )

K 0 (T ) nk (T  273)

, here n – number of macromolecules

in unit of volume, k – constant Boltzmann, R – universal gas constant, 'H R – – energy of

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activation.

Figure 2.

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The Mesoscopic Constitutive Equations for Polymeric Fluids …

193

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Figure 4.

Figure 5.

Let's consider now the results of modelling the process. To have the full idea about the behaviour of the investigated system we will introduce dimensionless characteristic numbers into consideration: Prandtl number (approximating the ratio of momentum diffusivity (kinematic viscosity) and thermal diffusivity); Nusselt number (the ratio of convective to conductive heat transfer across the boundary); Reynolds number (gives a measure of the ratio of inertial forces to viscous forces) and Weissenberg number (the ratio of the relaxation time of the fluid and a specific process time) and let’s consider their influence on system decisions. Flow velocity, film temperature and width in various points represents the greatest interest for practice. Drawings make it visible that at small Pr numbers there is fast film cooling and the film temperature leaves on the final value equal to the environment temperature. At great Pr numbers of film cooling practically does not occur, what does not influence on the flows.

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Figure 6. Comparison with experimental data.

With number Nu growth there is also faster film cooling that influences its form (film width on distance to the due dependence). On the contrary the Re number increase leads to worse film cooling. The most essential impact is made by the parameter of stretching anisotropy. Apparently from drawing with growth D the film width increases. On the following Figure comparison with experimental data from work is shown. It is possible to notice the good consent of the theory and experiment that confirms the assumptions made. Here k - factor of film stretching. The picture makes it visible that on this dependence there is a site of constant width, and the basic change of film width occurs on distance of 20 %-30 % from an exit, it is the socalled effect of "neck" occurrence which would not be observed without film cooling. Thus, it is possible to draw a conclusion that rheological model (1, 2) at the account of heat exchange (3) consistently describes observed at formation of polymeric films effects [1– 3], and the received dependences making stress tensor from distance to an exit from the die can be used at modelling the process of crystallization for polymer melt under the influence of stretching and cooling.

The Flow between Parallel Planes We shall consider a problem of motion of the non-linear viscoelastic liquid, based on the established constitutive equations in a gap between parallel planes under the constant pressure gradient. The system of the equations consists of the continuity equation and the equation of dynamics, which are written in the Cartesian system of coordinates (2). Due to the form of the die, we can consider the flow in the channel of the land as a flow between two parallel planes with distance h. One can choose the beginning of coordinates at one of the die walls, directing axis Ɉx along the flow, axis Oy perpendicular to the planes, and axis Oz perpendicular to axes Ox and

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The Mesoscopic Constitutive Equations for Polymeric Fluids …

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Oy. Due to the symmetry of the problem, the solution, that is the distribution of velocity and orientation does not depend on co-ordinate z and the system of the equations (2) is reducing to:

(9) The system of the equations (9) describes the two-dimensional unsteady flows of polymeric medium. Further, the solution of equations can be considered to be independent on

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the variable x , and we consider that the gradient of pressure is constant, that is

wp wx

A

Then we shall have: wV y wy

0,

K w a xy · wp ¸¸  3 0 , wx W 0 wy ¹ wV y · § wV y K w a yy wp ¸¸  Vy 3 0 U ¨¨ , wy ¹ wy W 0 wy © wt w a xx w a xx w V x 1  N  E I a xx  Vy  2 a xy  wt wy wy W0 wV x § wV x Vy wy © wt

U ¨¨

w a xy wt

 Vy

w a xy wy

 a xy

wV y wy

 a yy

E 1 wV x 3 a xy a xx  a yy , W0 3 wy w a yy wt

 Vy

w a yy wy

 2 a yy

wV y wy



(10) 3

E a xx2  a xy2 , W0

w V x 1  N  E I  a xy wy W0

1  N  E I

W0

a yy

E 2 wV y 2 3 a xy2  a yy . W0 3 wy



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196

Note that it follows from equation (9), that V y  y is a linear function of y, but so as, due

to the boundary conditions Vy 0 Vy h

0 , one has Vy  y 0 , and the equation of

continuity is satisfied identically. Considering a steady-state case for which the above equations (8) can be reduced to simpler ones:

3

K 0 wa xy W 0 wy

wp , wx

3

K 0 wa yy W 0 wy

wp , wy

a xx a xy a yy

§ · W0 wV x E ¨¨ 2a xy  3 a xx2  a xy2 ¸¸, 1  N  E I © W0 wy ¹

(11)

· §§ W0 1 wV E ¨¨ ¨ a yy  ·¸ x  3 a xy a xx  a yy ¸¸, 1  N  E I © © 3 ¹ wy W0 ¹ E a xy2  a yy2 . 3 1  N  E I

We shall look for the solution of the system (11) by the method of consecutive approximation, considering terms of the first order with respect to small parameters N and

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E . Then the system of the equations for the zero approximation will become: waxy wy



1 W0 A, 3 K0

a xx

2W 0 a xy

wV x , wy

axy

1 wVx W0 , 3 wy

a yy

0.

From here, we shall find, replacing the partional derivatives with the full ones:

1 d2 W0 Vx 3 dy 2



W0 A d2 ; Vx 3 K 0 dy 2



A

K0

;V x  y 

1 A 2 y  ay  b. 2 K0

where constants a and b founding from the boundary conditions: Vx 0 Vx h so that in the zero approximation, one has the solution

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0

The Mesoscopic Constitutive Equations for Polymeric Fluids …

Vx  y 

1 A  y  h y 2 K0

197

1 A h  y y; 2 K0

dV x 1 A h  2 y ; dy 2 K 0 a xy

W A 1 A h  2 y 0 h  2 y ; W0 3 2K 0 6K 0

a xx

1 2 A2 h  2 y 2 . W0 2 6 2K 0

This is the well-know Poiseuille law for viscous liquid. Having introduced the symbol

AW 0

A

K0

, the above solution can be rewritten as

A h  y y; a xy 2W 0

Vx  y

A h  2 y ; a xx 6

A2 h  2 y 2 ; a yy 6

0.

The next step is the calculation of the amendments of the first order with respect to

and E . In this case, using (11) we are receiving:

1 2 2 A h  2 y ; 6 W0 1 § · W W 0 ¨1  N  E A 2 h  2 y 2 ¸; 1  N  E I 6 © ¹ 1 § 2 · dV axx 2W 0 ¨1  N  E A 2 h  2 y ¸ x axy  6 © ¹ dy I

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N

axx  a yy

1 2 § 1 4 2· A h  2 y ¸; 3E ¨ A 4 h  2 y  36 © 36 ¹ 1 2 a yy  E A 2 h  2 y ; 12 1 1 § 2 · dV § 1 2· axy W 0 ¨1  N  E A 2 h  2 y ¸ x ¨  E A 2 h  2 y ¸  6 12 ¹ © ¹ dy © 3 1 3  E A 3 h  2 y . 12

(12)

The last expression can be rewritten in the approximation of the first order with respect to parameters

N

and E :

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198

W0

a xy

dVx § 1 1 2 E 2 2· A h  2 y  A 2 h  2 y ¸  ¨  N  E 6˜3 12 dy © 3 ¹

1 3 3 E A h  2 y . 12

(13)

With the help of this expression, we are receiving from equation (10)

d 2Vx § 1 § N dV § 2N E · 2 1 E · 2·  A W0  ¸ A h  2 y  ¨  ¨  ¸ A 2 h  2 y ¸¸  W 0 x ¨ 2 ¨ dy © 3 © 18 36 ¹ dy © 9 3 9¹ ¹ 1 2  E A 3 3h  2 y  2 . 12 Having substituted here



1 A 3

W0

d 2V x dy 2

dV x in the zero approximation on N and E , we have dy

§1 § N A 3 § 2N E · E · 2 E 2· 2 2  ¸h  2 y  A 3 h  2 y . ¨¨  ¨  ¨ ¸ A h  2 y ¸¸  3 18 36 2 9 9 2 © © ¹ ¹ ¹ ©

or

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W0

d 2V x dy 2

§1 § N E · 2 2· ¨¨  ¨  ¸ A h  2 y ¸¸ ¹ © 3 © 18 36 ¹

1 § N 5E · 2  A  A3¨  ¸h  2 y . 3 ©9 9 ¹

Multiplying both parts of this expression by 3, we find:

W0

d Vx dy 2 2

§ N 5E · 2 A  A3¨  ¸h  2 y ©3 3 ¹  §N E · 2 1  ¨  ¸ A 2 h  2 y © 6 12 ¹ .

Or, keeping terms of the first order with respect to

N

and E ,

W0

d 2V x dy 2

§ § N 5E N E · 2· ¨¨ A  A 3 ¨    ¸h  2 y ¸¸ © 3 3 6 12 ¹ ¹; ©

W0

d 2Vx dy 2

§ § N 7E · 2· ¨¨ A  A 3 ¨  ¸h  2 y ¸¸ . ©2 4 ¹ © ¹

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The Mesoscopic Constitutive Equations for Polymeric Fluids … One can integrate equation (12), taking the boundary conditions Vx 0 Vx h

199

0,

into account, and get:

Vx  y

A A 3 § N 7E · 2 2 y h  y  ¨  ¸ y h  y h  2hy  2 y 2W 0 6W 0 © 2 4 ¹ or

Vx  y

A A 3 § N 7E · 2 2 y h  y  ¨  ¸ y h  y h  y  y 2W 0 6W 0 © 2 4 ¹ .









(15)

Note that the pressure gradient in the perpendicular direction is different from zero, as one can see from equations (10) and (13),

wp wy

EAh  2 y z 0 .

(16)

There is no flow in this direction inside the channel, but after the exit from the channel, it gives the extention of the jet. To calculate the volume of liquid, which is coming through a unit of the length of the gap in a unit of time, that is the specific volume, or the output flow volume rate Q, one has to integrate expression (13) with respect to the variable y from 0 to h and obtain: h

Q

³Vx  y dy

.

(17)

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0

A 3 A 3 § N 7E · 5 h  ¨  ¸h 12W 0 60W 0 © 2 4 ¹

Figure 7. The distribution of velocities in the channel of the die at different values of the specific volume Q.

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Figure 8. The distribution of velocities in the channel of the die at the different values of the specific volume Q.

Figure 9. The distribution of velocities in the channel of the die at the different values of relaxation time of the flowing system.

The reversion of this formula determines the pressure gradient as function of Q, which allows us to calculate, with help of equation (15), distribution of velocities in the channel of the die at different values of the specific volume Q. The plots of dependencies are depicted in Figure 7. One can see that the increase in the specific volume Q provokes the deviation of velocities from the Poiseuille distribution, corresponding to viscous liquid. This statement is demonstrated in Figure 9, where the ratio of velocity of flow to its maximum value is shown. The calculations show, that the distribution of velocities changes non-essentially when Q>10.

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The Mesoscopic Constitutive Equations for Polymeric Fluids …

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Figure 10. Comparison of calculated and measured velocity distribution.

We have also investigated the effects of the parameters of the rheological model on the

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distribution of velocity. Figure 9 demonstrates the influence of relaxation time ߬଴ on the velocities distribution in the channel, when the specific volume Q is constant. To test the applicability of the results to polymer liquid, we use the measurement of velocity distribution by [6]. The measurements were fulfilled inside the channel with crosssection 1 by 14 mm2, the height of the channel was 50 mm. The velocity distribution was measured at 10 mm from the outlet of the channel. The results of comparison of empirical data with theoretical ones, calculated specially for this case, are shown in Figure 10.

Conclusion The possibility of using modified rheological Vinogradov-Pokrovskii model is shown for the description of linear polymer melts flows in various modes of deformation. The system of the equations of dynamics is written down in one-dimensional approach, at the account heat exchange. Comparison with experimental data available in the literature for film half-width shows adequacy of the applied approach. Comparison of results of velocity calculation with experimental data is implemented for polymer fluids moving between parallel planes. It allows drawing a conclusion, that in the considered case of Pouselle flow, the rheological model describes non parabolic profile of velocity in this case plates that does not contradict experimental data. Thus, we have shown that the rheological Vinogradov-Pokrovskii model can be applied to describe one-dimensional uniform and non-uniform flows.

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202

Alexsei Gusev, Grigory Afonin, Ilia Tretjakov et al.

Acknowledgments The authors greatly appreciate Professor Vladimir Pokrovskii for his interest to this work and useful discussions and acknowledge financial support of the Russian Foundation for Basic Research (RFBR) under grant 10-01-00293.

References [1] [2]

[3] [4]

[5] [6]

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[7]

D. Silagy, Y. Demay, J.F. Agassant Stationary and stability analysis of the film casting process // J. Non-Newtonian Fluid Mech., 79 (1998), p. 563-583. T. Kajiwara, M. Yamamura, T. Asahina Relationship between neck-in phenomena and rheological properties in film casting// Journal of the Society of Rheology, Japan, 34(2006), p. 97 – 102. H. Ito, M. Doi, T. Isaki, M. Takeo, K. Yagi 2D flow analysis of film casting process // Journal of the Society of Rheology, Japan, 31 (2003), p. 149 – 155. G.V. Pyshnograi, V.N. Pokrovskii, Yu.G. Yanovsky, Yu.N. Karnet, I.F. Obraztsov Constitutive equation of non-linear viscoelastic (polymer) media in zeroth approximation by parameter of molecular theory and conclusions for shear and extension// Doklady Akademii Nauk, 39 (1994) p. 889 - 892. G.V. Pyshnograi, A.S. Gusev, V.N. Pokrovskii Constitutive equations for weakly entangled linear polymers// J. Non-Newtonian Fluid Mech., 163 (2009), p. 17-28. E. Wassner, M. Schmidt, H. Munstedt Entry flow of a low-density-polyethylene melt into a slit die: An experimental study by laser-Doppler velocimetry // J. Rheol., 43(1999), p. 1339-1353. C. W. Seay, D. G. Baird Sparse Long-chain Branching’s Effect on the Film-casting Behavior of PE// International Polymer Processing, (2009), p. 428-438.

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Index

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A acetic acid, 41 acid, 39, 45, 140, 145, 152 ACL, 178, 184 acrylate, 118, 119, 120, 122 active compound, 27 additives, 71, 90 adhesion, 173 adsorption, vii, x, 1, 2, 3, 7, 8, 11, 12, 17, 19, 20, 27, 28, 29, 30, 31, 32, 33, 115, 116, 119, 125, 127, 133, 134, 141, 145, 147, 148, 150 aesthetic, 170, 171 aggregation, 10, 11, 29 algorithm, 67, 68, 69, 98 alters, 37 amine, 20 amplitude, ix, 14, 41, 47, 62, 94, 97, 102, 107, 113, 140, 182 amylase, 92 angulation, 171 anisotropy, 189, 191, 194 aqueous phase, viii, 11, 28, 36, 37, 38, 46, 54, 116 aqueous solutions, 37, 56, 86, 92 aqueous suspension, x, 46, 138, 139 aromatic compounds, 13 atmospheric pressure, 182

B bacteria, 38, 39 bacterium, 39, 44 barriers, 5, 17, 117 Baumgaertel-Schausberger-Winter model, ix, 94, 105, 109, 110 behaviors, 6, 11, 28, 48, 119, 132, 133, 158, 165 biocompatibility, 168, 174

biopolymer, 37, 38, 43, 44, 48, 65, 67, 72, 88, 92, 101, 113 Biopolymeric macromolecules, vii, 35 biopolymers, vii, 42, 50, 65, 71, 85, 88, 92, 101, 102, 164 biotechnology, 71 bonds, 41, 95, 101, 113, 154, 157, 158

C calcium, 55, 152, 164 calcium carbonate, 152 carbohydrate(s), 47, 100, 107 carbon, 28, 33 carboxyl, 143, 145 carboxylic groups, 143, 150 carob, 55 case studies, viii, 36, 39 casein, 28, 32 casting, 138, 188, 202 catalyst, x, 137, 138 cation, 45, 56 cationic surfactants, 132, 133 celiac disease (CD), ix, 93, 94, 96, 112 celiac sprue, 94 cellulose, 37, 39, 92 charge density, 133 chemical, vii, x, 1, 3, 19, 37, 41, 71, 72, 92, 106, 116, 123, 138, 139, 143, 155, 170, 173 chemical characteristics, 170 chemical cross-links, 106 chemical demulsifiers, vii, 1, 3, 19 chemical reactions, 72 chloroform, 19 cholesterol, 38 chromium, 168, 169 clinical application, 171, 174

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204 CMC, 78, 80, 85 cobalt, 169 combined effect, 179 compilation, 64 complex numbers, 63 compliance, viii, 36, 49, 51, 53, 54, 55, 56, 60, 63, 85, 98 composite resin, 173 composites, 152 composition, x, 6, 11, 17, 19, 29, 47, 92, 95, 100, 111, 112, 113, 138, 139, 143, 147, 149, 150 compounds, 10 condensation, 41 conduction, 188, 189 consent, 194 conservation, 48, 189 consolidation, 11, 138, 151 consumers, 36, 38, 154, 162, 163 convergence, 91 COOH, 139 cooling, 46, 179, 189, 190, 192, 193, 194 copolymer, 119, 120, 122, 129 correlation, vii, viii, 1, 3, 19, 27, 29, 33, 35, 90, 178, 184 correlation coefficient, 90 covalent bond, ix, 93, 95 critical value, 42, 122 crude oil, vii, 1, 2, 3, 6, 7, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 19, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 116, 133, 135 Crude oil emulsions, vii, 1 crystalline, 38 crystallization, xi, 187, 194 crystals, 38 cylindrical rubber surface, xi, 177

D decay, 4, 14, 15, 16, 52 defects, 96 deformation, vii, viii, ix, xi, 9, 17, 27, 48, 59, 60, 61, 62, 64, 66, 91, 94, 97, 105, 146, 155, 159, 162, 163, 168, 177, 178, 179, 180, 181, 182, 183, 185, 201 dehydration, 97, 140 demulsification of crude oil emulsions, vii, 1, 3, 19 denaturation, 156, 157 derivatives, 92, 94, 196 deviation, xi, 177, 178, 183, 184, 200 dielectric permeability, 101 diet, ix, 93, 94 diffusion, 3, 4, 10, 16, 30, 32, 82, 117 diffusion exchange, 3, 117

diffusion process, 16 diffusivity, 127, 133, 193 digestive enzymes, 38 dilation, 120 dilational viscoelastic parameters, vii, 1, 3, 20 dispersion, vii, x, 1, 108, 111, 137, 138, 139, 140, 141, 142, 143, 145, 148 dispersion systems, vii, 1 displacement, 11, 51, 54 dissociation, 101 distilled water, 140 distribution, viii, xi, 2, 35, 36, 41, 49, 54, 59, 66, 69, 85, 90, 96, 98, 108, 143, 177, 184, 191, 195, 199, 200, 201 distribution function, 54, 66, 108 diversity, vii, 35, 36 dough, ix, 93, 94, 95, 96, 97, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113 drainage, 27 drawing, 120, 188, 194, 201 dressings, vii, 35, 36, 38, 40, 54 durability, 170 dynamic oscillatory data, ix, 94, 105, 107, 109 dynamic viscosity, 91, 104

E effluent, 12, 13 egg, ix, 56, 88, 89, 94, 101, 102, 105, 106, 107 elastic deformation, 167 elasticity modulus, 6, 28, 49, 60, 63 electric field, 29 electron, 142, 143 electron microscopy, 142, 143 elongation, 190, 191 emulsifying agents, 37 emulsion stability, vii, 1, 2, 3, 10, 27, 28, 29, 30, 31 emulsions, vii, x, 1, 2, 3, 19, 27, 28, 29, 30, 31, 32, 33, 35, 36, 37, 38, 39, 41, 43, 44, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 71, 115, 116, 133 enamel, 173 engineering, xi, 58, 152, 165, 177, 179 Enhanced Oil Recovery, x, 2, 34, 115, 133 entanglement network, 102 entanglements, viii, 30, 36, 54, 55, 95, 106, 107, 108, 109, 111, 188 environment, xi, 97, 187, 193 epidemiology, 94 equilibrium, 4, 5, 6, 11, 15, 29, 49, 53, 60, 64, 85, 88, 90, 98, 103, 104, 117, 118 equipment, 2, 38 ester, 145

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Index ethanol, 19 ethylene, 20, 25 ethylene oxide, 20, 25 evolution, 8, 11, 84, 85, 91 excitation, 49 exclusion, 48 exopolysaccharides, 44 experimental condition, 29, 146 experimental design, 154 explosives, 36 exponential functions, 4 extraction, 41 extrusion, 39, 184

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F fat, 38, 39, 54, 57, 101, 105, 106, 164 fatty acids, 38 fermentation, ix, 39, 45, 93, 95 fiber, ix, 93, 95, 100 fillers, 108 film formation, 29, 188 film thickness, 178, 182, 189, 191 films, vii, 1, 2, 3, 11, 12, 19, 20, 29, 30, 31, 33, 116, 119, 122, 125, 133, 185 filters, x, 137, 138 filtration, x, 137, 138 financial, 31, 55, 111, 134, 202 financial support, 31, 55, 111, 134, 202 fixation, 90 flexibility, 129 flocculation, 37, 56, 139, 141, 142, 146, 148, 150 flooding, ix, 34, 115, 116, 135 floods, 2 flour, ix, 94, 95, 96, 100, 102, 103, 105, 107, 108, 109, 110, 111, 112, 113, 165 flow curves, viii, 36, 51, 55, 139, 140 fluctuations, 188 fluid, viii, xi, 12, 34, 36, 41, 48, 52, 54, 86, 88, 95, 111, 135, 150, 177, 179, 181, 182, 183, 184, 193 fluid extract, 12 foams, x, 28, 33, 37, 71, 90, 115, 116, 133 food, vii, ix, 1, 3, 27, 32, 33, 35, 36, 37, 38, 39, 45, 46, 54, 56, 57, 58, 60, 71, 90, 92, 93, 94, 96, 97, 100, 101, 107, 112, 113, 152, 154, 155, 162, 163, 164, 165 food industry, 38, 45, 71, 154, 162 food products, 36, 39, 71, 94 force, xi, 6, 37, 56, 118, 168, 169, 171, 173, 177, 178, 179, 181, 183, 184 formation, x, 2, 6, 7, 10, 27, 28, 29, 30, 37, 42, 46, 56, 57, 88, 92, 101, 102, 106, 107, 111, 113, 115,

205

116, 118, 125, 129, 132, 133, 142, 146, 147, 157, 158, 164, 168, 194 formula, 102, 105, 106, 107, 113, 200 fragments, ix, 93, 95 free radical copolymerization, 118 friction, 64, 66, 86, 88, 90, 100, 169, 170, 171, 172, 173, 174, 179 FTIR spectroscopy, 145

G gas filtration, x, 137, 138 Gaussian chains, 48 Gauss-Siedel iteration, xi, 177 gel, viii, x, 7, 11, 12, 19, 28, 29, 30, 36, 37, 38, 41, 44, 45, 46, 50, 53, 54, 57, 82, 85, 86, 88, 91, 92, 98, 102, 103, 104, 106, 107, 109, 112, 146, 153, 154, 157, 158, 160, 161, 165 gelatinization temperature, 95 gelation, 11, 38, 41, 58, 76, 92, 154, 156, 158, 160, 165 glass transition, 11, 29, 50, 51, 99, 102 glass transition temperature, 29 gliadin, ix, 93, 94, 95 glucose, 39, 45, 112, 143 glucose oxidase, 112 Gluten-free dough, ix, 94, 105, 110 glutenin, ix, 93, 95, 96 granules, 100, 138, 139, 143, 145, 149, 150

H heavy oil, 12, 15 heptane, 6, 11 hexane, 19 high strength, 168 hip joint, 185 host, 57 hub, 43 human, 168, 171 hydrocarbons, 11 hydrocolloids, viii, ix, 36, 37, 38, 39, 41, 44, 45, 51, 52, 54, 56, 57, 58, 71, 76, 85, 88, 90, 91, 94, 96, 102, 107, 109, 110, 111 hydrogels, 91 hydrogen, 41, 95, 101, 158, 160 hydrogen bonds, 158, 160 hydrolysis, 118, 124 hydroxide, 28 hydroxyl, 37, 41, 143, 145 hydroxyl groups, 37, 41, 143, 145 hypothesis, 95

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Index

206

I

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immune system, 94 improvements, 168 impurities, 118, 133 in vitro, 170, 171, 172 industrial sectors, 37 industries, 36, 71 industry, vii, ix, 1, 2, 3, 19, 46, 115, 116, 162 inequality, 67, 69 inertia, 179, 181 infrastructure, 58 ingestion, 94 ingredients, vii, ix, 35, 37, 38, 93, 96, 101, 107, 154 integration, 49, 191 interface, 2, 3, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 19, 20, 21, 22, 23, 25, 27, 28, 30, 31, 32, 33, 34, 37, 116, 117, 118, 119, 120, 121, 124, 125, 126, 127, 128, 129, 131, 132, 133, 134, 135, 136, 169 interfacial adsorption, vii, 1, 3, 8 interfacial layer, 15, 16, 28, 122, 133 interference, 72, 160 intermolecular interactions, 23, 46, 160 intrinsic viscosity, xi, 187 inversion, 58, 99, 113 ionization, 146 ions, 55, 164 isotherms, 32 iteration, xi, 177

J joints, 178

K kerosene, 7, 8, 9, 10, 19, 124 kinetics, 39, 160

L large intestine, 38 lead, 10, 29, 30, 98, 101, 106, 129, 171 leakage, xi, 177, 178, 179, 183, 184 liberation, 101 light, 6, 7, 56, 92, 118 light scattering, 56, 92 linear function, 196 linear model, 99, 158 linear polymers, xi, 38, 187, 188, 202 lipids, 38, 71, 100, 105, 106

liquid interfaces, 31, 32, 33 liquid phase, 2 liquids, vii, 2, 27, 35, 36, 48, 53, 57, 86, 98 low temperatures, 138

M macromolecular chains, 101 macromolecular coil, 189 macromolecules, vii, 35, 48, 58, 120, 122, 192 magnitude, 10, 44, 46, 53, 132, 139, 140, 143, 145, 147, 148 matrix, ix, 38, 44, 94, 95, 102, 105, 106, 111, 138, 178, 185 mechanical properties, 48, 66, 85, 88, 112, 169 mechanical relaxation spectrum, viii, ix, 36, 54, 94, 105, 109 melt, 188, 189, 192, 194, 202 melting, 85 melts, xi, 187, 188 membranes, x, 137, 138 metabolized, 38 methacrylic acid, 151 methodology, 103 methylcellulose, 96 microgels, 28, 32, 132 microscopy, 56 microstructure, viii, 35, 38, 154, 164 mixing, ix, 39, 56, 93, 95, 101, 111, 112 model system, 72, 76 modelling, viii, 59, 188, 189, 193, 194 models, vii, viii, 39, 59, 63, 66, 71, 158 modifications, x, 138, 145 modules, 67, 82, 88, 90, 105 modulus, viii, ix, x, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 35, 43, 46, 49, 50, 53, 54, 56, 59, 60, 63, 64, 72, 76, 86, 88, 90, 92, 94, 98, 99, 100, 102, 103, 104, 105, 106, 107, 108, 109, 110, 113, 116, 117, 118, 119, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 145, 146, 147, 148, 153, 155, 156, 157, 158, 159, 160, 169, 179 molecular mass, 88, 95 molecular mobility, 48, 51, 102, 110 molecular weight, ix, 12, 13, 14, 15, 16, 17, 20, 30, 40, 88, 92, 95, 115, 116, 118, 120, 122, 123, 132, 133 molecular weight distribution, 40 molecules, vii, x, 2, 3, 7, 9, 10, 11, 15, 16, 17, 21, 23, 25, 26, 30, 33, 38, 41, 47, 71, 101, 102, 107, 108, 109, 111, 115, 116, 122, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 143, 145, 160 momentum, 48, 193

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Index monolayer, 4, 11, 15, 127, 134 monomers, 30, 120, 122, 129, 133, 134

N NaCl, 41, 101, 102, 105, 106, 107 neutral, 19, 154, 156, 158, 160, 162, 163 nickel, 167, 168, 169, 170, 173 nodes, 182 nonionic surfactants, 19 non-polar, 37, 101 nontropical sprue, 94 numerical analysis, 182, 185 nutrient, 39 nutrients, 94

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O octane, 134, 135 oil, vii, viii, ix, 1, 2, 3, 6, 7, 8, 10, 11, 12, 13, 16, 17, 19, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 36, 37, 38, 39, 41, 44, 46, 47, 50, 51, 52, 53, 55, 57, 97, 101, 105, 106, 107, 111, 115, 116, 118, 119, 122, 125, 133, 134, 135, 178, 179, 180, 182 oil field industry, vii, 1, 3, 19 oil production, 2 optimization, 39, 96 orthodontic treatment, xi, 167, 168, 169, 171, 172, 175 orthodontic treatments, 168 oscillation, 6, 29, 62, 88, 118, 155, 157, 158, 159, 178, 182, 183, 184 overlap, 17, 42, 119, 120

P pea starch, 152 permeability, 180, 182 PES, 20, 21, 23, 25 petroleum, ix, 2, 10, 12, 18, 30, 31, 32, 34, 36, 115, 116, 123, 135 pH, x, 6, 11, 28, 31, 33, 40, 45, 57, 58, 138, 140, 141, 142, 143, 149, 150, 153, 154, 156, 157, 158, 159, 160, 161, 162, 163, 164 phase transitions, 72 phosphate, 145 physical properties, 46, 154, 162 physicochemical properties, 96 polyacrylamide, ix, 115, 116, 118, 119, 122, 123, 133, 134, 135, 136 polyamines, 20

207

polybutadiene, 55 polyether, 20 polymer, ix, xi, 30, 33, 34, 36, 40, 42, 43, 45, 48, 50, 55, 65, 82, 88, 91, 95, 96, 99, 102, 107, 108, 109, 111, 112, 115, 116, 118, 119, 120, 121, 122, 124, 125, 126, 127, 128, 129, 133, 134, 135, 143, 187, 188, 194, 201, 202 polymer blends, 48 polymer chain(s), 30, 50, 95, 99, 102, 116, 119, 120, 127, 129, 133 polymer melts, 201 polymer molecule, 40, 42, 48, 119, 120, 125, 127, 129, 133 polymer solutions, x, 55, 115, 116, 124, 133, 135 polymer systems, 43 polymeric chains, viii, 36, 54 polymeric films, 194 polymeric materials, viii, 35, 48 polymeric medium, 195 polymerization, 160 polymers, ix, 2, 37, 38, 42, 43, 49, 56, 65, 88, 96, 99, 111, 112, 115, 116, 119, 124, 133, 134, 135, 139, 152 polypeptide, 101 polysaccharide, 37, 39, 44, 56, 57, 143, 164 Polysaccharides, 55, 91, 138 polyvinyl acetate, 112 porosity, 138, 180, 182 positive correlation, 110 potato, x, 91, 96, 138, 139, 140, 142, 143, 144, 145, 148, 149, 150 potato starch, corn starch, 96 precipitation, 45, 82 preparation, x, 39, 41, 81, 82, 137, 138, 154, 174 present value, 172 pressure gradient, xi, 187, 188, 194, 199, 200 probe, x, 5, 115, 116, 117 propylene, 20, 25 proteins, ix, 2, 31, 32, 37, 38, 71, 93, 94, 95, 96, 101, 107, 113, 154, 163, 164 PTFE, 5, 117, 170

Q quantification, 155 quartz, 6, 118

R radius, 188 ramp, 51 raw materials, vii, 35

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208 reactions, 156, 160, 162 real time, viii, 59, 62 recovery, ix, 2, 32, 33, 34, 46, 92, 93, 94, 100, 115, 116, 119, 125, 133, 134, 135 recovery process, 116 recrystallization, 37, 82 relaxation process, 3, 4, 5, 14, 15, 16, 17, 20, 21, 30, 65, 70, 86, 90, 117, 120, 121, 122, 133, 134, 135 relaxation processes, 3, 4, 5, 14, 15, 16, 17, 21, 70, 86, 90, 117, 120 relaxation times, viii, ix, 19, 35, 42, 48, 49, 51, 60, 64, 65, 69, 70, 71, 76, 82, 88, 90, 98, 106, 116, 129 repulsion, 141, 146, 147 RES, xi, 187 resins, 2, 10, 11, 12, 18, 19, 29, 30 resistance, 2, 3, 27, 117, 146, 155, 169, 170, 171, 172, 173, 174, 175, 184 response, 4, 5, 43, 44, 49, 52, 61, 62, 97, 117, 127, 139, 140, 146, 184 response time, 43, 52 restrictions, 68, 69 retail, 101, 102, 103, 104, 105, 106 retardation, viii, 53, 54, 55, 57, 59, 60, 61, 66, 69, 71, 82, 83, 84, 85, 91, 111, 113 rheological equation of state (RES), xi, 48, 187, 189 rheological properties, vii, viii, ix, x, 1, 3, 11, 27, 31, 32, 33, 34, 35, 36, 38, 57, 59, 71, 82, 86, 88, 90, 92, 93, 95, 96, 97, 101, 111, 112, 115, 116, 125, 132, 134, 135, 137, 138, 139, 143, 150, 152, 165, 202 rheology, vii, viii, ix, x, 3, 5, 11, 12, 27, 28, 29, 30, 31, 32, 33, 35, 36, 37, 38, 39, 58, 60, 63, 92, 102, 105, 115, 116, 117, 123, 134, 135, 139, 140, 155 rheometry, 110 room temperature, 41, 46, 168 root, 99, 171 roughness, xi, 177, 179, 180, 184, 185 Royal Society, 32, 34, 56, 58

S salinity, 11, 31 saliva, 37, 171 salts, 40, 71 SARA, 17, 18, 19 scaling, 43, 44, 48, 85, 88, 90 scaling coefficients, 85 scaling law, 43, 44 science, viii, 32, 59, 72, 168 sedimentation, 29, 40, 71 shape, 5, 48, 66, 67, 69, 72, 76, 105, 117, 143, 154, 158, 167, 168

shear, vii, xi, 1, 3, 27, 30, 40, 41, 47, 51, 52, 53, 57, 92, 97, 101, 103, 104, 107, 113, 116, 132, 139, 140, 141, 145, 149, 150, 187, 189, 202 shear deformation, 116 shear rates, 139, 140, 141, 149 side chain, 39, 40, 41, 101 silica, 19 silicon, 151 simulation(s), 39, 48, 183, 185 sintering, 170 small intestine, 38, 94 sodium, 41, 118 software, 6, 118, 158 solid oxide fuel cells, x, 137, 138 solid phase, 71 solid polymers, 112 solid state, 49 solid surfaces, 147 solubility, 10, 30, 101, 111, 164 solution, ix, xi, 3, 6, 30, 31, 41, 42, 45, 48, 57, 60, 65, 67, 68, 69, 87, 88, 90, 91, 117, 118, 119, 120, 122, 124, 125, 126, 127, 128, 129, 130, 133, 134, 139, 140, 141, 143, 146, 147, 150, 158, 160, 187, 195, 196, 197 solvents, 6 species, ix, 10, 34, 93, 97 specific adsorption, 143, 150 specific surface, 140, 145, 146, 150 Squeeze film theories, xi, 177 stability, vii, viii, x, 1, 2, 3, 10, 11, 26, 27, 28, 29, 30, 31, 33, 36, 38, 39, 45, 56, 58, 71, 101, 115, 116, 133, 137, 139, 141, 143, 146, 150, 164, 168, 202 stabilization, 27, 29, 37, 71, 140, 143, 151, 160 stabilizers, 37, 56, 85 standard deviation, xi, 161, 177, 179, 182, 183, 184 starch, ix, x, 37, 56, 70, 71, 72, 74, 76, 78, 80, 81, 82, 83, 84, 85, 86, 88, 90, 91, 92, 93, 95, 96, 100, 101, 102, 105, 106, 107, 108, 109, 110, 111, 112, 113, 137, 138, 139, 140, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 165 starch granules, ix, x, 82, 93, 95, 100, 108, 112, 138, 148 steady-state flow, viii, xi, 36, 51, 53, 55, 187 steel, 6, 118, 167, 168, 169, 170, 171, 172, 173 storage, viii, ix, 4, 7, 35, 38, 41, 42, 46, 49, 60, 82, 85, 86, 88, 92, 94, 95, 97, 100, 102, 103, 105, 107, 110, 113, 117, 145, 148, 155, 156 stress, viii, ix, 37, 41, 46, 48, 49, 51, 52, 59, 60, 61, 62, 63, 64, 65, 66, 70, 76, 85, 86, 88, 90, 94, 97, 98, 105, 111, 112, 139, 140, 164, 168, 188, 189, 190, 194 stretching, xi, 187, 188, 189, 190, 191, 192, 194 strong interaction, 126

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Index structural changes, 30, 82 structural characteristics, 108 structure, viii, ix, x, 10, 11, 19, 28, 34, 36, 38, 39, 40, 41, 42, 45, 46, 47, 52, 54, 57, 60, 62, 65, 69, 71, 72, 76, 82, 85, 86, 88, 90, 92, 95, 96, 101, 102, 107, 108, 109, 111, 115, 116, 119, 120, 122, 125, 128, 133, 134, 137, 138, 139, 143, 145, 146, 147, 150, 154, 157, 160, 162, 163 structure formation, 82 structuring, 37, 45 substrate, 170 sucrose, 56, 90, 154, 164 surface area, 148 surface friction, 168 surface layer, 3, 4, 11, 117, 120 surface tension, 3, 4, 6, 116, 117, 118, 120, 121 surfactant, x, 6, 7, 28, 30, 32, 33, 34, 115, 116, 118, 120, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136 surfactant molecules, x, 30, 115, 116, 125, 126, 127, 131, 132, 133, 134 surfactants, 2, 26, 30, 31, 32, 34, 37, 124, 125, 127, 129, 130, 135 suspensions, x, 41, 58, 137, 138, 139, 140, 141, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152 symmetry, 195 synergistic effect, 26 synthetic polymers, 72

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T target, 67, 156 target population, 156 techniques, x, 5, 56, 117, 138, 139, 164 technology, 32, 39, 57, 125, 164 teeth, 167, 169, 173 teflon, 174 TEM, 92 temperature dependence, 92 tension, 3, 4, 6, 9, 10, 11, 15, 16, 17, 20, 28, 34, 117, 118, 122 testing, 41, 49, 97, 105, 164, 170, 172 textural character, 154 texture, x, xi, 38, 39, 45, 56, 96, 153, 154, 162, 163, 164, 165, 173, 177, 184 thermal energy, 2 thermal stability, 45, 72 thermal treatment, 160 thermodynamics, 188 thinning, vii, 1, 3, 40, 52, 57, 104, 140, 145, 149 Tikhonov regularization method, 68, 69 titanium, 167, 168, 169, 170, 173, 175

209

toluene, 6, 7, 10, 11, 28, 29, 32 tooth, 167, 168, 169, 170, 171, 174 transducer, 6, 118 transition temperature, 45, 88 transport, 4, 11, 36, 117, 126, 170 treatment, ix, xi, 45, 58, 93, 94, 112, 160, 164, 167, 168, 169, 170, 172

U universal gas constant, 192

V vanadium, 169 varieties, x, 138, 139 vector, 72 velocity, 17, 178, 181, 188, 189, 190, 191, 193, 195, 200, 201 vibration, 179 viscoelastic liquids, 49 viscoelastic materials, vii, 35, 36, 48, 54, 113, 158 viscoelastic properties, vii, viii, 1, 3, 28, 29, 52, 58, 59, 65, 72, 85, 90, 95, 96, 139, 150, 152, 154, 163, 164 viscosity, vii, x, xi, 2, 4, 5, 14, 15, 16, 17, 19, 27, 28, 29, 35, 36, 37, 40, 41, 52, 53, 60, 62, 64, 66, 71, 82, 84, 85, 87, 88, 90, 97, 115, 116, 117, 118, 119, 120, 122, 123, 137, 138, 140, 141, 143, 144, 145, 146, 149, 150, 155, 157, 162, 182, 187, 189, 192, 193

W water-soluble polymers, 133 wettability, 32 wires, 167, 168, 169, 170, 173, 174

X xanthan gum, 38, 39, 40, 41, 45, 72, 74, 76, 83, 84, 85, 88, 90, 91, 92

Z zirconia, x, 138, 139, 140, 141, 143, 144, 145, 146, 147, 148, 150, 151 zirconia suspensions, x, 138, 144, 151

Viscoelasticity: Theories, Types and Models : Theories, Types and Models, Nova Science Publishers, Incorporated, 2011. ProQuest Ebook Central,