Coagulation: Kinetics, Structure Formation and Disorders : Kinetics, Structure Formation and Disorders [1 ed.] 9781621003885, 9781621003311

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Copyright © 2012. Nova Science Publishers, Incorporated. All rights reserved. Coagulation: Kinetics, Structure Formation and Disorders : Kinetics, Structure Formation and Disorders, Nova Science Publishers, Incorporated, 2012.

Copyright © 2012. Nova Science Publishers, Incorporated. All rights reserved. Coagulation: Kinetics, Structure Formation and Disorders : Kinetics, Structure Formation and Disorders, Nova Science Publishers, Incorporated,

RECENT ADVANCES IN HEMATOLOGY RESEARCH

COAGULATION

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KINETICS, STRUCTURE FORMATION AND DISORDERS

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RECENT ADVANCES IN HEMATOLOGY RESEARCH

COAGULATION KINETICS, STRUCTURE FORMATION AND DISORDERS

ANETT M. TALOYAN Copyright © 2012. Nova Science Publishers, Incorporated. All rights reserved.

AND

DAVID S. BANKIEWICZ

EDITORS

Nova Science Publishers, Inc. New York Coagulation: Kinetics, Structure Formation and Disorders : Kinetics, Structure Formation and Disorders, Nova Science Publishers, Incorporated,

Copyright © 2012 by Nova Science Publishers, Inc. All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. 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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Independent verification should be sought for any data, advice or recommendations contained in this book. In addition, no responsibility is assumed by the publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication. This publication is designed to provide accurate and authoritative information with regard to the subject matter covered herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services. If legal or any other expert assistance is required, the services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS. Additional color graphics may be available in the e-book version of this book. Library of Congress Cataloging-in-Publication Data Coagulation : kinetics, structure formation, and disorders / editors, Anett M. Taloyan and David S. Bankiewicz. p. ; cm. Includes bibliographical references and index. ISBN 978-1-62100-388-5 (eBook) I. Taloyan, Anett M. II. Bankiewicz, David S. [DNLM: 1. Blood Coagulation--physiology. 2. Blood Coagulation Disorders--physiopathology. WH 310] LC classification not assigned 616.1'57--dc23 2011032588 Published by Nova Science Publishers, Inc. † New York Coagulation: Kinetics, Structure Formation and Disorders : Kinetics, Structure Formation and Disorders, Nova Science Publishers, Incorporated,

CONTENTS Preface Chapter 1

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Chapter 2

vii A New Approach to the Theory of Brownian Coagulation and Diffusion-Limited Reactions M. S. Veshchunov Deregulation of Coagulation During Sepsis-Induced Disseminated Intravascular Coagulation John A. Samis

1

59

Chapter 3

Coagulation: Kinetic, Structure, Formation and Disorders Benjamín Rubio-Jurado, Díaz-Ruiz Rosbynei, Pedro A. Reyes, Carlos Riebeling and Arnulfo Nava,

77

Chapter 4

Floc Characteristics and the Influencing Factors Baoyu Gao and Weiying Xu

95

Chapter 5

Substrate Induced Coagulation in Aqueous and Non-Aqueous Media For the Preparation of Advanced Battery Materials Angelika Basch

Chapter 6

The Laboratory Diagnosis of the Pre-phase of Pathologic Intravascular Coagulation Thomas W. Stief

Chapter 7

Neonatal Coagulation Problem Viroj Wiwanitkit

Chapter 8

Coagulation and Wall Shear Stress in Living Donor Liver Transplantation Yoshinobu Sato and Hideki Nakatsuka and Toru Abo

Index

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111

137 153

163 199

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PREFACE This book presents topical research in the study of the kinetics, structure formation and disorders related to coagulation. Topics discussed include Brownian coagulation and diffusion-limited reactions; deregulation of coagulation during sepsis-induced disseminated intravascular coagulation; substrate induced coagulation (SIC) in aqueous and non-aqueous media for the preparation of advanced battery materials and neonatal coagulation problems. Chapter 1- An overview of the author‘s papers on the new approach to the Brownian coagulation theory and its generalization to the diffusion-limited reaction rate theory is presented. The traditional diffusion approach of the Smoluchowski theory for coagulation of colloids is critically analyzed and shown to be valid only in the particular case of coalescence of small particles with large ones, R1 R2 . It is shown that, owing to rapid diffusion mixing, coalescence of comparable size particles occurs in the kinetic regime, realized under condition of homogeneous spatial distribution of particles, in the two modes, continuum and free molecular. However, the expression for the collision frequency function in the continuum mode of the kinetic regime formally coincides with the standard expression derived in the diffusion regime for the particular case of large and small particles. Transition from the continuum to the free molecular mode can be described by the interpolation expression derived within the new analytical approach with fitting parameters that can be specified numerically, avoiding semi-empirical assumptions of the traditional models. A similar restriction arises in the traditional approach to the diffusion-limited reaction rate theory, based on generalization of the Smoluchowski theory for coagulation of colloids. In particular, it is shown that the traditional approach is applicable only to the special case of reactions (A + B C, where C does not affect the reaction) with a large reaction radius, rA RAB rB (where rA , rB are the mean inter-particle distances), and becomes inappropriate to calculation of the reaction rate in the case of a relatively small reaction radius, R AB rA , rB . In the latter, more general case particles collisions occur mainly in the kinetic regime (rather than in the diffusion one) characterized by homogeneous (at random) spatial distribution of particles. The calculated reaction rate for a small reaction radius in 3-d formally coincides with the expression derived in the traditional approach for reactions with a large reaction radius, however, notably deviates at large times from the traditional result in the plane (2-d) geometry, that has wide applications also in the membrane biology as well as in some other important areas.

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viii

Anett M. Taloyan and David S. Bankiewicz

Chapter 2 - Despite improvements in healthcare settings over the past 50 years, mortality from bacterial infections remains a significant and increasing clinical problem globally because of an increase in antibiotic resistance, the elderly, and immunocompromised patients. The processes of coagulation and fibrinolysis are normally intricately activated extravascularly only at the site of injury and damage to blood vessels. Disseminated intravascular coagulation (DIC) is a common acquired disorder of the coagulation and fibrinolysis processes resulting from a pre-existing pathology leading to fibrin formation in the intravascular compartment. The most common pathology resulting in the establishment of clinical DIC in humans is pre-existing gram negative or gram positive bacterial infection (Sepsis). Their understanding of bacterial-induced sepsis and DIC has been furthered upon study of the effects of these pathogens using in vivo models and clinical trials. During sepsisinduced DIC, bacterial and host components enhance the expression of host tissue factor which promotes fibrin formation in a process mediated by tumor necrosis factor- and interleukin-6, release/generation of compounds that promote fibrin formation, decreased bioavailability of anticoagulation factors, and deregulated fibrinolysis. While different pathogenic bacteria modulate the coagulation and fibrinolytic processes at different points, the presenting clinical symptoms in patients infected with different bacterial pathogens are often similar. Prolonged and/or severe bacterial sepsis disrupts the balance of coagulation and fibrinolytic processes such that bleeding is often the presenting clinical symptom. Diagnosis utilizes a number of laboratory tests which include: recognition of the pre-existing pathology associated with this condition and bacteria identification in combination with prolonged clot times, low platelet counts, and elevated levels of fibrin degradation products. Current therapies are generally aimed at treating the pre-existing pathology that leads to the establishment of DIC. While infusion with antibiotics, platelets, plasma and/or its components, or heparin have been used in the past to augment therapies, recent treatment regimens using human recombinant activated Protein C have shown encouraging patient outcomes for humans suffering from severe sepsis-induced DIC. Novel diagnostic biomarkers and therapies will ultimately result from a greater understanding of how bacterial factors interact with the host coagulation/fibrinolytic, inflammatory, and immune defence systems. This Review focuses on summarizing what is currently known about how bacteria deregulate the coagulation and fibrinolysis processes leading to the development and maintenance of the sepsis/DIC state in humans. Chapter 3 - Maintenance of normal blood flow requires equilibrium between procoagulant and anticoagulant factors it is named hemostasis, occasionally procoagulant activity predominates, leading to clots formation; frequently, tissue damage is the triggering factor. Hereditary factors, primary or acquired play a role in the development of thrombosis. Primary thrombophilia is associated to hereditary factors, which promote hypercoagulability because natural anticoagulants are not exerting their activity. On the other hand, acquired thrombophilia may occur associated to autoimmune diseases, cancer, surgical procedures, pregnancy, postpartum period, and obesity. Activation of the coagulation system is characterized by the co-participation of inflammatory response components, factors related to the subjacent disease, and other procoagulant factors. The study of hemostasis in the patients should include both inflammatory and autoimmune response markers. Chapter 4 - Floc characteristics, including floc size, strength, re-growth abilities and fractal structure, are important factors in the water treatment works. These factors could

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Preface

ix

essentially affect the solid/liquid separation process. Consequently, the investigations on the fundamental characteristics of flocs are meaningful and necessary for coagulation sedimentation units in water treatment works. The properties of flocs formed during coagulation were substantially dependent on coagulation conditions, such as properties of coagulants, coagulant dose, suspension pH, ionic strength, hydraulic conditions. In this chapter, the flocs formed in humic acids were taken as examples to present how the coagulation coagulant types, pH and hydraulic conditions affect the floc size, strength, regrowth abilities and fractal dimensions. Additionally, the opinions in this chapter were based on the investigation on various aluminum coagulants. The optimum conditions for coagulation, which contribute to the best floc characteristics, were also discussed. Chapter 5 - Substrate induced coagulation (SIC) is a dip-coating method capable of coating chemically different surfaces with finely dispersed nano-sized solid particles. In the first step of an SIC process the surface is condi- tioned with a thin layer of polymer or polyelectrolyte. In the second step the conditioned surface is dipped into a dispersion that is close to a point where coagulation occurs. The polymer or polyelectrolyte on the surface induces coagulation of the nano-sized material and a thin layer of the solid is formed. The SIC process can be performed in aqueous as well as non-aqueous media as long as the particles are stabilised (or destabilised) by elec- trostatic repulsive (or attractive) forces. The theory of Derjaguin, Lan- dau, Verwey and Overbeek (DLVO), describing colloidal stability as well as the zeta-potential and surface charging of dispersed particles are discussed for aqueous and non-aqueous dispersions and some experimental difficulties adressed. For example trace water can have a profound impact on the colloidal stability of non-aqueous dispersions. Dispersions in polar and non-polar media carbon black, titania and alumina, materials so far used for SIC, are discussed. Lithium cobalt oxide has the ability to intercalate lithium ions re- versibly. It is the preferred cathode material because it is easy to prepare and has a high specific capacity. Disadvantages of this material are its low electronic conductivity and high reactivity when charged (delithiated). A non-aqueous SIC process was used to prepare lithium cobalt oxide with an improved conductivity by coating with highly conductive carbon black. Furthermore, core-shell cathode materials of lithium cobalt oxide with a protective layer to reduce the materials reactivity when charged (delithi- ated), are prepared by coating with titania via an aqueous SIC process. Chapter 6 - The rapid laboratory diagnosis of the pre-phase of pathologic (disseminated) intravascular coagulation [PIC phase 0] is a major clinical challenge. PIC means multiple micro-thrombi that obstruct vital organ areas, and that can cause multi-organ failure. There is clinical need for reliable hemostasis parameters to diagnose the pre-phase of PIC to start the adequate treatment in time. The routine hemostasis tests such as prothrombin time and activated partial thromboplastin time are too blunted to diagnose the early phases of PIC; furthermore, thrombocytes might be dysfunctional, cell fragments might be counted as platelets, platelets-count or fibrinogen concentration are acute phase parameters that might increase instead of decrease, and antithrombin activity might change only slightly or it might be altered in an artificially diluted test matrix. In the present review new routine parameters for diagnosis of the early phases of PIC are presented: basal chromogenic thrombin activity (IIa-Test), thrombin (IIa) generation after plasma recalcification (RECA), IIa generation after intrinsic activation of plasma (INCA), or IIa generation after extrinsic activation of plasma (EXCA), fibrinogen function+antigen+ratio, undiluted antithrombin activity, and active

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Anett M. Taloyan and David S. Bankiewicz

endotoxin (here clinically understood as active lipopolysaccharide or active ß-glucan). A combination of these new assays allows to diagnose the pre-phase of PIC within minutes. Especially the functional ultra-specific IIa-Test (normal range 100 20%; 100%=5.5 mIU/ml) or the therefrom derived standardized RECA helps to distinguish the normal systemic intravascular coagulation [NIC] from the dynamic PIC phase 0 (120-150% of normal basal IIa) or from the PIC phase 1 (> 150% of normal basal IIa). All these simple photometric assays do not require special hemostasis machines to test the dynamic coagulation state and to differenciate between LPS- and ß-glucan- sepsis of each individual patient especially during the ―f irst six golden hours‖. Chapter 7 - Basically, successful blood stop must be composed of three basic effective components: good blood coagulation factor system, good platelet and good vascular structure. When there is an injury to the blood vessel, internal or external, the vessel firstly response by vasoconstriction. Then the coagulation cascade will start and process to finalize in fibrin network and platelet will go to accumulate at the injure size to get the complete blood stop process [1]. If there is any error in any basic component in any process, there will be a failure in blood stopping. On the other hand, there is a balance system to the coagulation process, the fibrinloytic system. If there is a defect in the balance system, the thrombotic disorder can also be detected. Thrombosis risk is multifactorial, with interaction of hereditary risk factors and acquired environmental and clinical conditions [2]. Newborns are at particular risk for thrombotic emergencies secondary to the unique properties of their hemostatic system, influences of the maternal-fetal environment, and perinatal complications and interventions [2]. The neonatal coagulation problem will be presented in this chapter. Chapter 8 - Over the last century many investigators have studied liver regeneration, thus giving rise to new biologic themes, concepts, and techniques. The experimental model of liver regeneration after a partial hepatectomy is considered to be very useful in understanding immune surveillance, metabolism, and also blood coagulation and fibirinolytic (BCF) system, since autoimmune diseases or carcinogenesis can lead to catastrophe but liver regeneration occurs without any treatment. There are many clinical difficulties associated with major hepatic surgery or living donor liver transplantation (LDLT). Overcoming these problems will require a better understanding of the basic mechanisms of liver regeneration. In this chapter, the authors described the BCF system between recipients and donor of a LDLT. BCF system has an important concern with hemodynamic changes and liver regeneration. They have proposed that the wall shear stress, a simple hemodynamic force caused by portal venous flow directed against vessel walls, regulates liver size and growth as well as atrophy due to the apoptosis of individual hepatocytes. Also the wall shear stress induces the dynamic changes of BCF system and immune system in the intra- and extra-hepatic circumstances accompanied with hepatic regeneration. The understanding of BCF system after liver transplantation is important for patient management. Especially in the early period after adult LDLT, there is liver regeneration accompanying portal hypertension because of a small-forsize graft. The BCF system in healthy donors mediate the physiologic regeneration pattern in the early period after liver resection. Conversely, recipients are exposed to the influences of excessive surgical stress and portal hypertension. Therefore, the authors investigated differences in the perioperative BCF system between donor and recipient after LDLT, primarily considering serum plasminogen-activator inhibitor-1 (PAI-1) and soluble fibinogen (SF) levels. And otherwise hemeoxygenase-1 (HO-1) is a stress-induced enzyme that catalyses the oxidation of heme to biliverdin. Hemeoxygenase-1 produces carbon monoxide

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(CO) as a byproduct of hemoglobin metabolism. HO-1 or HO-1 products might regulate TF and TM expression and prevent thrombus formation in human endothelial cells. Human HO-1 deficiency has been observed to involve the endothelial cells more severely, resulting in hemolysis and disseminated intravascular coagulation. The present study examines the relationship between CO production and hyperbilirubinemia following adult LDLT with special attention to the contribution of shear stress in retarding regeneration.

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In: Coagulation: Kinetics, Structure Formation… Editors: A.M. Taloyan et al

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Chapter 1

A NEW APPROACH TO THE THEORY OF BROWNIAN COAGULATION AND DIFFUSIONLIMITED REACTIONS M.S. Veshchunov* Nuclear Safety Institute (IBRAE), Russian Academy of Sciences, 52, B. Tulskaya, Moscow, Russia Moscow Institute of Physics and Technology (MIPT) (State University), Institutskii per., Dolgoprudny, Moscow Region, Russia

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Abstract An overview of the author‘s papers on the new approach to the Brownian coagulation theory and its generalization to the diffusion-limited reaction rate theory is presented. The traditional diffusion approach of the Smoluchowski theory for coagulation of colloids is critically analyzed and shown to be valid only in the particular case of coalescence of small particles with large ones, R1  R2 . It is shown that, owing to rapid diffusion mixing, coalescence of comparable size particles occurs in the kinetic regime, realized under condition of homogeneous spatial distribution of particles, in the two modes, continuum and free molecular. However, the expression for the collision frequency function in the continuum mode of the kinetic regime formally coincides with the standard expression derived in the diffusion regime for the particular case of large and small particles. Transition from the continuum to the free molecular mode can be described by the interpolation expression derived within the new analytical approach with fitting parameters that can be specified numerically, avoiding semi-empirical assumptions of the traditional models. A similar restriction arises in the traditional approach to the diffusion-limited reaction rate theory, based on generalization of the Smoluchowski theory for coagulation of colloids. In particular, it is shown that the traditional approach is applicable only to the special case of reactions (A + B  C, where C does not affect the reaction) with a large reaction radius, rA  R AB  rB (where rA , rB are the mean inter-particle distances), and becomes inappropriate to calculation of the reaction rate in the case of a relatively small reaction radius, R AB  rA , rB . In the latter, more general case particles collisions occur mainly in the kinetic regime (rather than in the diffusion one) characterized by homogeneous (at random) spatial distribution of particles. The calculated reaction rate for a small reaction radius in 3-d formally coincides with the expression derived *

E-mail address: [email protected]; Tel: +7(495) 955 2218, Fax: +7(495) 958 0040 (Corresponding Author)

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2

M.S. Veshchunov

in the traditional approach for reactions with a large reaction radius, however, notably deviates at large times from the traditional result in the plane (2-d) geometry, that has wide applications also in the membrane biology as well as in some other important areas.

1. Part 1. Brownian Coagulation Theory 1.1. Introduction Brownian motion refers to the continuous random movement (or diffusion) of particles suspended in a fluid. Brownian agglomeration occurs when, as a result of their random motion, particles collide and stick together. The theoretical treatment of agglomeration consists on keeping count of the number of particles as a result of collisions and determining the collision frequency function which depends on the particle sizes, concentrations and transport mechanisms in the system. Coagulation is regarded as a special case of agglomeration where there is instantaneous coalescence of particles after collision. Brownian coagulation was first calculated basing on the Brownian diffusion theory by Smoluchowski [1] and further developed by Chandrasekhar [2]. The theory was essentially based on assumption that the local coagulation rate should be equal to the diffusive current of particles. Namely, the expression for the collision frequency was obtained by solving the diffusion equation for particles around one particle that is assumed to be fixed using the relative diffusion coefficient D1  D2 for moving particles.

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In application to the case of suspending gas, it was outlined that this expression is valid only for particles that are large enough that they experience the surrounding gas as a continuum (so called ― continuum regime‖), whereas for particles with radius R much smaller than the mean free path m of the surrounding molecules the ― free molecular regime‖ (corresponding to large Knudsen numbers, Kn  m R  1 ) was considered (see, e.g. [3]). An expression for the collision frequency function in the free molecular regime was derived in the gas-kinetic approach assuming rigid elastic spheres. Fuchs proposed a semi-empirical interpolation formula for the whole particle diameter range [4]. In fact this formula interpolates two regimes corresponding to large and small ean drift particle sizes in comparison with the ―m ean free path‖ a (hereafter termed as the ―m distance‖) of the Brownian particles (rather than m !). The formula is reduced to the standard diffusion expression [1, 2] for relatively large particles, R  a , and to the free molecular regime collision frequency function in the opposite limiting case, R  a (rather than the above mentioned and widely used inequality, Kn  m R  1 ). The classical problem of Brownian coagulation was reconsidered in the author‘s papers [5-7] with the main conclusion that the usual approach to the calculation of the local coagulation rate via the particle diffusive current (in the continuum regime) is valid only in the particular case of collisions between large and small Brownian particles, R1  r  R2 (where r  n

1 / 3

is the mean inter-particle distance), and cannot be applied to a more general

case of particles of comparable sizes, R1 , R2  r . In particular, the significance and the

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A New Approach to the Theory of Brownian Coagulation …

3

1 / 3

necessity to introduce a new length scale, r  n , that is not frequently used in dispersed flows, however, plays an important role in the new approach, was revealed. In order to expose the main inconsistency of the traditional approach, the diffusion equation for the ensemble of Brownian particles is re-derived in the first order approximation for the small concentration n of comparable size (  R ) particles, n1/ 3 R  1 , with a special attention to restrictions on the system parameters that provide applicability of the diffusion approach (see Section 1.2). On this basis, in the second order approximation of n1/ 3 R  1 it is shown (Section 1.3) that coalescence of comparable size particles, R1 , R2  r , occurs in the kinetic regime

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(rather than in the diffusion regime) characterized by homogeneous spatial distribution function of the coalescing particles (rather than by their concentration profiles) practically in the whole considered range, owing to rapid diffusion mixing of particles in-between their collisions (Section 1.3.2). This range can be subdivided into two intervals of the model continuum‖ and ― free parameters, a  R and a  R , corresponding to different modes (― molecular‖) of the kinetic regime, and the ― transition‖ interval, a  R , amid the two limiting cases (Section 1.5). In the continuum mode of the kinetic regime, a  R (Section 1.5.1), the formal expression for the collision frequency of particles (of comparable sizes) coincides (in fact, fortuitously) with that derived in [1, 2] for the diffusion regime (being relevant only in the particular case of coalescence of small particles with large ones, R1  r  R2 ). This formal coincidence apparently explains why the traditional approach correctly describes numerous experimental measurements of the Brownian particles coalescence rate. In the opposite range a  R (Section 1.5.2), the standard free molecular expression for the collision frequency function is valid. It is shown that, despite the free molecular expression can be rigorously derived (Section 1.4) only in the case of very high collision frequency (when two subsequent collisions of a particle with other ones occur within one drift period), it can be properly extended to the whole range a R  1 . Since the transition interval a  R (Section 1.5.3) also belongs to the kinetic regime characterized by homogeneous spatial distribution of particles, the collision frequency can be numerically calculated in the same approach, generalizing the analytical method applied in the limiting cases a  R and a  R . On this base, new interpolation expressions are derived by fitting to the calculated points (Section 1.5.4), avoiding semi-empirical assumptions of the existing models. Self-consistency of the developed kinetic approach is verified in Section 1.5.5, using calculated values of the collision frequency in various ranges of the parameter a R . The interpolation expressions obtained in the first approximation of the random walk theory (so called simple random walks with a fixed length) is verified against calculations in the second approximation considering random walks with varying lengths (Section 1.6). Discussion on validity of the earlier approaches and frameworks of their justification is presented in Section 1.7. The main outcomes of the new approach to Brownian coagulation theory are formulated in Section 1.8.

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M.S. Veshchunov

1.2. Diffusion Relaxation in Ensemble of Brownian Particles Let us consider a continuous spatial distribution of centers of Brownian particles with the mean radius R and mass m that migrate throughout a fluid sample with the heat velocity and randomly change direction of the velocity for the relaxation time

0 .

The suspending fluid can be considered as a homogeneous medium, if the size Lm of the elementary volume

V  L3m (over which the averaging is carried out) is large enough in

comparison with the mean inter-molecular distance rm  nm1/ 3 , where nm is the fluid molecules concentration, Lm  nm1/ 3 . For this reason the minimum distance (or the length







scale) dr between two possible positions of a particle centre, r and r  dr , that can be considered in this approach, corresponds to the size Lm of the elementary volume and thus

dr  nm1/ 3 .



On the other hand, the local number concentration of particles n r  can be considered as

a macroscopic value (i.e. when its thermodynamic fluctuations are small in comparison with

n 2  n ) only if the size of the elementary volume V~  L3 , with respect  to which n r  is defined, is large enough in comparison with the local inter-particle distance,  L  n 1/ 3 r  . In its turn, the local inter-particle distance is limited by the particles size 1 / 3  r   2 R . (when particles touch each other), L  n its value,

1/ 3

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In the first order of approximation n R  1 , coalescence of migrating particles can be neglected. In the continuum approach it is assumed that an elementary drift (or jump) of a

~

3 particle in the time interval  0 occurs within the above defined elementary volume V  L ,

i.e. spatial variation (of the wave-length l ) of the particles distribution function is smooth on the scale of the elementary volume, l  L , and for this reason, the probability of having the 3



centre of one particle in the elementary volume d r at r can be related to the local





concentration of particles, Pr , t   nr , t d 3r .



 



Let us designate w r , d 3 the probability that the centre of a particle located at r





will relocate to r   in the time interval  0 . Then the probability to find this centre in the









 

position r   in  0 is nr w r , d 3rd 3 . Correspondingly, variation of the particles

 r

0 in is           nr , t   0   nr , t    w r   ,  n r   , t  w r ,  nr , t  d 3 , where the first term  on the right hand side represents the increase in concentration at r due to the jumps from the concentration

at





  



neighbor positions, and the second term represents the removal of particles from the position

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A New Approach to the Theory of Brownian Coagulation …

5

 r . For small  0  t the left hand side can be decomposed    nr , t  0   nr , t    0 nr , t  t , and the conservation equation takes the form          nr , t  1 (1.1)   w r   ,  n r   , t  w r ,  nr , t  d 3 . t 0





as



  



As above explained, the characteristic length scale l of the spatial variation of n r  is



assumed to be large in comparison with the local inter-particle distance n 1/ 3 r  , whereas



 

the function w r , rapidly decreases to zero with increase of  , varying at small distances





 

  n 1/ 3 r   l . This allows decomposing n r  

in the integral, Eq. (1.1),

       2 1 n r   , t  nr , t     n r , t      nr , t  . In the lack of external fields, r r r 2        w r , is a spatially homogeneous function, w r   ,  w r , , therefore, the kinetic





 



  

Eq. (1.1) takes the form

   B n  ,  A n  r  

n   t r

(1.2)

1       1 A r     w r ,  d 3 , B r    w r ,  d 3 , or, taking into  0 2 0       account homogeneity of w r , , represented in the form w r ,   w 0,  w  , one

   

where

      

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obtains

  A   01   w  d 3   01   0 , B r   B 0  B , (1.3)



where B 

 3  3 1 a2 2 2 2 2 w d        a   w d  is the mean-square , and  6 0  6 0





relocation distance in  0 , further termed as the mean drift distance (to distinguish from the ―m ean free path‖  , defined below in Section 1.3.2 as the mean distance between two subsequent collisions of a particle with other particles). Hence, Eq. (1.2) takes the form

n  D2 n , t

(1.4)

D  a 2 6 0 .

(1.5)

where

The particles move with the root mean-square heat velocity (in accordance with the equipartition theorem) Coagulation: Kinetics, Structure Formation and Disorders : Kinetics, Structure Formation and Disorders, Nova Science Publishers, Incorporated,

6

M.S. Veshchunov

uT 

1/ 2  1/ 2 u 2  3kT m   1.5kT  R 3  ,

(1.6)

related to the mean thermal speed

c  8kT m

 uT 3 8

1/ 2

1/ 2

,

(1.6a)

where  is the mass density of the particles, each particle changing a random direction of its drift for the relaxation time

 0 , which can be estimated from the Langevin theory of

Brownian movement [8] as

 0  mb 

mD , kT

(1.7)

where b is the particle mobility, calculated in the Stokes regime ( Re  1 ) as

b

Cc

6 R

,

(1.8)

 is the liquid viscosity, Cc is the slip correction factor for spherical particles, depending on the Knudsen number, Kn  m R , in the form [9]

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  A  Cc  1  Kn  A1  A2 exp   3   ,  Kn   

(1.9)

with A1 =1.257, A2 =0.40 and A3 =1.1 [10], or A1 =1.165, A2 =0.483 and A3 =0.997 obtained in recent experiments [11]. After substitution of Eqs. (1.5) and (1.6) in Eq. (1.7), the mean-square drift distance a can be roughly evaluated as

a  6kT m  0  uT 0 2 . 1/ 2

(1.10)

It is important to note that in derivation of Eq. (1.4) the terms of the higher order of  n

were neglected under condition that a  n n  l , where l is the characteristic length 1



scale of the spatial variation of n r  , which, as above explained, is much larger than the local inter-particle distance, l  n

1 / 3

, that must exceed the particle size, n

1 / 3

 R or

1/ 3

n R  1 . This results in the basic cut-off limit for the spatial fluctuation spectrum, l  n 1 / 3  R . Another restriction on the validity of the current approach for description of Brownian particles is an assumption  m   0 , where  m is the mean time between stochastic collisions of a particle with the surrounding fluid molecules, which is also generally valid and can be

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A New Approach to the Theory of Brownian Coagulation …

7

violated only for very small particles in the same size range as the (gas) molecules. Indeed, from Eq. (1.10)  0 is estimated as  0  a 2uT  am 6kT 1/ 2 , whereas  m is estimated from the



gas-kinetic theory as  m  nm R 2um 1  nm R 2

 m  1

m

8kT 

1/ 2

 nm R 2   m 2 Rm3 6kT  , 1

1/ 2

where mm ,  m and um are the mass, mass density and thermal velocity of the gas molecules, respectively. Therefore,

 m   0

aR2 nm  6 8

1/ 2

is valid when

Substituting an estimation of the gas molecules mean free path,

m 



mm

2Rm2 nm



1

1/ 2 m .

, where

Rm is the molecule effective radius, one eventually obtains the limitation on the Knudsen



number, Kn  m R  2 3

 m m 

3 1/ 2

1/ 2

m

aR Rm2 , which is really not very strong

(since the r.h.s. of the inequality is generally very large for Brownian particles) and becomes essential only in the limit when the diffusing particles are in the same size range as the gas molecules, R  Rm . Actually, in this range Kn  1 and thus Cc  1.7Kn , the gas viscosity can be evaluated from the gas-kinetic theory as

a  3 2D uT  Cc  RmkT 6

1/ 2

  mm kT 1/ 2 4Rm2 , therefore,

 1.5  Knm mm 

1/ 2

Rm2 R , and the above

derived restriction takes a simple form, m mm  1 , or R  Rm (if

  m ).

Therefore, a conclusion can be derived that the standard theory of Brownian particles can be used practically without additional restrictions in a wide range of the particle sizes,

Rm  R  n 1/ 3 . However, relaxation of the spatial distribution of particles can be described in the continuous approach by the diffusion equation (derived in the first order of 1/ 3

1 / 3

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approximation n R  1 ) only in the case when both the local inter-particle distance n and the mean drift distance a are small in comparison with the length scale l of spatial 1 / 3

 l and a  l . Both these conditions will be essential in application heterogeneities, n of the diffusion approximation to analysis of particles coalescences (Sections 1.3.1 and 1.5.1). It should be noted that the derived diffusion equation for an ensemble of particles, Eq. (1.4), is formally similar to the Fokker-Planck equation for the probability density  W r , t  of a sole particle migrating by random walks (see, e.g., [4]). However, this does not imply that solution of the Fokker-Planck equation can be generally applied to consideration of an ensemble of particles. Indeed, switch from the Fokker-Planck equation for the probability    density W r , t  to the diffusion equation for the particles concentration n r , t   nW r , t  , which should be considered as a macroscopic value, is correct only in the case when the size

~



of the elementary volume V  L , with respect to which n r  is defined, is large in 3



1 / 3 r   2 R , as above explained. This comparison with the inter-particle distance, L  n

condition can be considered as a restriction on the spatial heterogeneity length l  L for the particles ensemble. As a result, the formal solution of the diffusion equation for a sole particle probability density (which is free from this restriction on the spatial heterogeneity length for the particles



ensemble, l  n 1/ 3 r  ) can be properly applied to consideration of coalescence of small Coagulation: Kinetics, Structure Formation and Disorders : Kinetics, Structure Formation and Disorders, Nova Science Publishers, Incorporated,

8

M.S. Veshchunov

particles with large ones, R2  n 1/ 3  R1 (so called agglomeration), whereas in 1 / 3  R1 , R2 , this solution application to an ensemble of particles of comparable sizes, n

becomes inappropriate, as will be additionally shown in Section 1.3.1.

1.3. Coagulation Rate Equation 1/ 3

In the second order of approximation n R  1 , pair-wise collisions of particles during their Brownian migration can be taken into consideration. In this approximation collisions which occur among any combination consisting of more than two particles, can be ignored. For a continuous distribution of particles N R dR , the number of particles of radius R to R  dR per unit volume, under an assumption that collided particles of radii R1 and R2 immediately coalesce to form a new particle of radius coagulation equation takes the form



R

3 1

 R23  , the Smoluchowski 1/ 3



1/ 3 N R, t  1    N R1 , t N R2 , t  R  R13  R23   R1 , R2 dR1dR2 t 200 



 N R, t  N R1 , t  R, R1 dR1 , 0

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where  R1 , R2  is the collision frequency function [1].

(1.11)

The first term on the right hand side of Eq. (1.11) represents the increase in number density at R due to pair-wise coalescence between all particles, and the second term represents the removal of particles of radius R due to pair-wise coalescence between particles of radius R with particles of all radii.

1.3.1. Applicability of the Diffusion Approach to Particles Coagulation In the case when all particles, located in a medium of infinite extent, are of comparable sizes  R1 and their mean concentration n 1 obeys the condition n1R13  1 , the particles 1 / 3 can be considered as point objects ( R1  r1 , where r1  n1 is the mean inter-particles

distance), which in accordance with the diffusion equation, Eq. (1.4), tend to relax to a homogeneous spatial distribution. The situation critically changes in the case when a relatively large particle (with the characteristic size R2  R1 ) appears in the ensemble of small particles. The large particle 3 cannot be considered as a point object, if n1 R 2  1 . In this case the large particle should be

considered as macroscopic with respect to small ones, since its size R2 is much larger than 1 / 3 the mean inter-particle distance r1  n1 in the ensemble of small particles (cf. Section 1.2)

and, therefore, an additional (absorbing) boundary condition for diffusion of small particles Coagulation: Kinetics, Structure Formation and Disorders : Kinetics, Structure Formation and Disorders, Nova Science Publishers, Incorporated,

A New Approach to the Theory of Brownian Coagulation …

9

emerges on the large particle surfaces. For this reason, the induced by this boundary condition heterogeneity in the spatial distribution of small particles does not tend to disappear with time, as it was in the previous case of comparable size particles, and the steady state concentration profile of small particles around the large particle center,

n1 r   n1 R12   n1  n1 R12   1  R12 r  , is formed at

t  R122 D1 , where

R12  R1  R2  R2 is the ―influence sphere‖ radius [2]. The diffusion flux of small particles in

this

concentration

profile,

J dif R12   4D1R12 n1  n1 R12   4D1R2n1 ,

if

n1 R12   n1 , determines the condensation rate of small particles in the large particle trap, and, following analysis of [1, 2], the collision frequency function, taking into consideration migration of the traps, eventually takes the form

dif R1 , R2   4 D1  D2 R1  R2  .

(1.12)

For determination of the applicability range of this result, it should be noted that the characteristic size l of the zone around a large particle in which the small particles

concentration varies from the value n1 R12   n1 near the influence sphere surface to the value of the same order of magnitude as the mean value n1 attained at large distances from the centre, is comparable with R12  R 2 , i.e. l  R 2 . This value must naturally exceed the mean distance n11/ 3 between small particles in the vicinity of the large particle surface,

R2  l  n11/ 3 R12   n11/ 3 (in order to maintain the concentration profile of small

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3 particles around the large one), or n1 R2  1 . This condition naturally coincides with the

general requirement to applicability of the diffusion approximation, l  r1  n11/ 3 (see Section 1.2). Therefore, the standard diffusion approach [2] is valid only for coalescence of large 1 / 3  R2 , and thus particles with small ones (with sufficiently high concentration), R1  n1

cannot be applied to consideration of particles of comparable sizes. It is also important to note that Eq. (1.12) was derived in neglect of mutual coalescences between small particles, i.e. under an (implicit) assumption that the rate of small particles mutual coalescences,  c R1 , R1   4 D1  D1 R1  R1 n1 2  8D1R1n1 is negligible in comparison with the rate

in the trap, . It is straightforward to see that this  cd R1 , R2   4 D1  D2 R1  R2 n1  4D1R2 n1

assumption,

of

their

condensation

 c R1 , R1    cd R1 , R2  , is valid if R2  R1 , this additionally confirms the

important conclusion concerning applicability of the diffusion approach. In the opposite case R2  R1  R the limit of the point-wise particles restores, which is characterized by the particles tendency to a homogeneous spatial distribution. However, after coalescence of two point particles (taken into account in the second order of approximation

nR3  1 ), the local number of particles alters step-wise (from two to one) that induces a Coagulation: Kinetics, Structure Formation and Disorders : Kinetics, Structure Formation and Disorders, Nova Science Publishers, Incorporated,

10

M.S. Veshchunov

local heterogeneity in the particles spatial distribution on the length scale of the mean interparticle distance r  n 1 / 3 . Therefore, the uniform spatial distribution of coalescing particles 1 / 3

can be assumed only under condition that heterogeneities of such scale, n , rapidly relax by the particles diffusion during the characteristic time between two subsequent collisions of a particle,  c .

1.3.2. Diffusion Mixing Condition The condition of the particles rapid relaxation, or ―m ixing‖, takes the form  c   d , where

 d  n 2 / 3 6D is the characteristic time of the particles (diffusion) redistribution on 1 / 3

the length scale of the induced heterogeneities (i.e. of the mean inter-particle distance n ). In this case it may be assumed in calculations that the coalescing particles are randomly distributed in the space, and for this reason, the collision frequency can be correctly calculated in the kinetic approach. In the opposite case,  c   d , when diffusion becomes too slow in comparison with the collision rate, the induced spatial heterogeneities cannot be neglected anymore and the particles diffusion becomes the rate determining process. Therefore, coagulation of comparable size particles can occur either in the kinetic regime (characterized by a homogeneous spatial distribution of particles), or in the diffusion regime (characterized by concentration heterogeneities), depending on how rapid is diffusion relaxation of particles in comparison with their collision rate. It is shown in [5-7] that in fact coalescence of comparable size (i.e. point-wise) particles

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occurs in the kinetic regime practically in the whole considered range n1/ 3 R  1 . Indeed, the mixing condition,

 c   d  n 2 / 3 6D , can be equally represented in the form

n1 / 3  1 , where   6D c 1/ 2 is the characteristic distance between two subsequent collisions of a particle with other ones, that is generalization of the mean free path definition to the case of the Brownian particles (do not confuse with the mean drift distance a , which can be very small in comparison with  !). Therefore, the mixing condition has a clear physical sense that  cannot be smaller than the mean inter-particle distance n

1 / 3

; this can

be also confirmed by direct evaluation of  c  d in both limiting cases a  R and a  R (see Section 1.5.5). Correspondingly, it will be further assumed that the (comparable size) particles are randomly distributed in space and, in-between two subsequent collisions of a particle, the random distribution quickly reinstates owing to the particles diffusion relaxation (or mixing). In this case (corresponding to the kinetic regime) the spatial distribution of the particle centers  n r , t  can be considered as a homogeneous function characterized by their mean



concentration n t  , i.e. n r , t   n t  , slowly varying with time owing to the particles

coalescence. Respectively, the collision probability is also a spatially uniform function.

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A New Approach to the Theory of Brownian Coagulation …

11

Validity of this basic assumption for Brownian particles will be confirmed below in Section 1.5.5. In this regime (characterized by homogeneous particle distribution, rather than by their concentration profiles) the original multi-particle problem is rigorously reduced to consideration of two-particle collisions. This significantly simplifies the coagulation problem and justifies the phenomenological form of the pair-wise kernel  R1 , R2  in the Smoluchowski kinetic equation, Eq. (1.11).

1.4. Kinetic Regime: High Collision Frequency (  c   0 ) At first, the case of high collision frequency

 c1 , corresponding to an inequality

 c   0 , will be considered. In this case two subsequent collisions of a particle occur within the drift time  0 , that can be considered in the traditional free molecular (or ― ballistic‖) approximation. In this approximation it is assumed that the particles move straight  with their heat velocities u i and randomly change direction of the velocity with the frequency

 01 . Let us consider two particles of radii R1 and R2 migrating in a sample of unit volume. The first (― parent‖) particle of radius R1 can be surrounded by a sphere with the radius

R1  R2 . If the second particle centre falls into this exclusion zone, coalescence would occur.

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During the time step t , where

 m  t   c , the two particles relocate with respect 





to each other with the relative velocity u12  u1  u2 . As a result, the exclusion zone also





relocates to a distance u1  u2 t and the particle of radius R2 may be swept out by the parent particle of radius

 2  R1 with the probability, V   R1  R2  u1  u2 t .

Correspondingly, the probability of the two particles coalescence in t is equal to





V   R1  R2 2 u1  u2 t , where averaging of the swept volume is carried out over 



the Maxwell distribution of the two independent heat velocity vectors u1 and u 2 , resulting in

  u1  u2 

8kT  m11  m21 



6kT   R 2

3 1

 R23  .

If there are N 2 particles of radius R2 in a sample of unit volume randomly distributed in the space (as above assumed), the probability of coalescence of the parent particle with a particle of radius R2 reduces to N 2

V . In the first order of approximation t  c  1 , one

can neglect variation of N 2 in the time step t (occurred owing to coalescences of particles of radius R2 with other particles). Besides, N 2 is considered to be small enough that one can ignore collisions which occur in t with more than one particles of radius R2 . This implies

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12

M.S. Veshchunov

that only one particle can coalesce with the parent particle with the probability  2 3 3 2  N 2 V  N 2 R1  R2  u1  u2 t  N 2 R1  R2  6kT   R1  R2 t .





Furthermore, the number of collisions between particles of radii R1 and R2 in t , if there are N 1 particles of radius R1 randomly distributed in the sample of unit volume, is

N12  tN1 N 2 R1  R2 

2

6kT  R13  R23  . If there are more than two size groups of

particles in the sample, then the number of collisions between each pair of groups p and q in

6kT  R p3  Rq3  . Since all

t is N pq  tN p N q R p  Rq 2

N i are small enough,

we can ignore collisions which occur among any combination consisting of more than two particles. Therefore, for a continuous size distribution of particles, n R dR , the collision frequency function in this kinetic regime takes the standard free molecular form [4] 1/ 2

 6kT   fm R1 , R2       

R1  R2 

2

 1 1   3  3   R1 R2 

1/ 2

.

(1.13)

In the mean field approximation taking into consideration monodisperse particles size distribution, the probability of the two particles coalescence in t becomes equal to

3kT  1/ 2 8NR1/ 2t . Considering each particle in turn, the total rate of loss of particles by

coalescence is given by:

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 3kT  dn  4  dt   

1/ 2

n 2 R1 / 2  

n

c

,

(1.14)

where the additional factor of ½ is introduced to avoid counting each interaction twice, and



 c  3kT  1/ 2 4nR1/ 2



1

is the coalescence time corresponding to the characteristic

period between two subsequent collisions of a particle. This confirms that variation of the particles concentration can be neglected in the first order of approximation

t  c  1

during the time step t , as above assumed. Therefore, the applicability range of Eq. (1.13) derived under condition

2uT  aR3/ 2 2 9kT  , takes the form a R  3 2  1/ 2

0  a

R

 c   0 , where

Rm 

1/ 2





1/ 2

4nR 

3 1

, or

 16  3   m  nR3Cc , which should additionally obey the relationship 1/ 2

1/ 3 m mm  1 or R Rm   m   , derived in Section 1.2 (from the condition

 m   0 ). This can take place under condition of very high Knudsen numbers, Kn   m  

5/ 6

16nR 

3 1

.

However, as will be shown in Section 1.5.2, in a more general case a  R when a particle makes many jumps in-between its two subsequent collisions with other particles Coagulation: Kinetics, Structure Formation and Disorders : Kinetics, Structure Formation and Disorders, Nova Science Publishers, Incorporated,

A New Approach to the Theory of Brownian Coagulation …

13

(  c   0 ), the swept volume per unit time is nearly a constant (in time) value, and for this reason it coincides (in the first order approximation of R a  1 ) with the above calculated value





V t   R1  R2 2 u1  u2 , representing the ratio of the mean swept volume

during one jump (designated below in Section 1.5 as  V 0 ) to the jump period

 0 . Therefore,

the applicability range of Eq. (1.13) can be extended to this case, a  R , which corresponds to a more realistic condition,

Kn  0.5 m  

1/ 3

n

1/ 3 m

Rm  .

R Rm  0.5  m 

1/ 3

n

1/ 3 m

Rm  , or 2

1

1.5. Kinetic Regime: Low Collision Frequency (  c   0 ) In the case

 c   0 a particle makes many diffusion drifts (or jumps) between its two

subsequent collisions with other particles. In order to calculate the value of the collision frequency in this regime, a relatively large time step t   0 should be chosen. On the other hand, it should be sufficiently small in comparison with

 c , in order to ignore variation in t

of the mean number concentration of surrounding particles (owing to their mutual coalescences) during the time step,

 c  t   0 . Besides, this time step t should be

large enough in comparison with the diffusion relaxation (or mixing) time

 d  n 2 / 3 6D , in

order to sustain the main assumption of the kinetic regime on random (homogeneous) Copyright © 2012. Nova Science Publishers, Incorporated. All rights reserved.

distribution of coalescing particles,  c  t   d . Again, let us consider two particles of radii R1 and R2 randomly located in a sample of unit volume. The first (― parent‖) particle of radius R1 can be surrounded by a sphere with the radius R1  R2 . If the second particle centre is located in this exclusion zone, coalescence would occur. As shown in [1, 2], the relative displacements between two particles describing Brownian motions independently of each other and with the diffusion coefficients D1 and D2 also follow the law of Brownian motion with the diffusion coefficient D1  D2 . Indeed, since the two particles are not correlated in motion,

r1r2 

 0 , the Einstein equation for the relative

displacement of two particles gives

D12 

r1  r2 2 6t



r1 2 6t



r1r2  3t



r2 2 6t



r1 2 6t



r2 2 6t

 D1  D2 .

(1.15)

Therefore, in order to calculate the probability of collisions between the two particles, one can equivalently consider the second particle as immobile whereas the first one migrating with the effective diffusion coefficient, D12  D1  D2 .

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14

M.S. Veshchunov In this approximation it is assumed that the effective (mobile) particle jumps to the

1 distance a12 in random directions with the frequency  12   12 , those are unknown

(searched) values obeying the relationship for the particle diffusivity from the theory of 2 simple random walks, D12  a12 6 12 .

As a result of a jump, the exclusion zone also relocates to the distance a12 and opens the possibility that the second (immobile) particle located in a zone with the volume

V0  a12 R1  R2 2 ,

(1.16)

may be swept out by the mobile particle (dashed zone in Fig. 1.1). It is important to note that this result does not depend on the ratio between a and R , and therefore is valid in the whole considered range Rm  R  n 1/ 3 . The model parameters a12 and  12 will be self-consistently determined (below in Sections 1.5.2-1.5.4) by comparison of the collision frequency calculated in the simple random walk approach at high Knudsen numbers, Kn   , with that calculated in the free molecular approach (considering the original particles moving straight with their heat velocities).

V0   R1  R2 2 a12

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R1  R2

a12 Figure 1.1. Schematic representation of the swept zone for colliding particles of radii

R1 and R2 .

1.5.1. Continuum Mode ( a  R ) During the time step t   12 the mobile particle makes many jumps, k  t

12  1 ,

in random directions, however, the total swept zone volume V that determines the probability of the two particles coalescence in t , will be smaller than k V0  V0t

 12 ,

owing to strong overlapping of the swept zone segments at a12  R12 . This limit Coagulation: Kinetics, Structure Formation and Disorders : Kinetics, Structure Formation and Disorders, Nova Science Publishers, Incorporated,

A New Approach to the Theory of Brownian Coagulation …

15

corresponds to the continuum mode of the kinetic regime, realized at low Knudsen numbers and characterized by a random spatial distribution of particles (quickly reinstated in-between two subsequent collisions of a particle). Under this basic condition (validated below in Section 1.5.5), the probability to sweep a particle of radius R2 in the unit time is

V t N 2 , if there are N 2 particles of radius R2 per unit volume. Therefore, the number of coalescences V t N 1 N 2 between particles of radii R1 and R2 in the unit time, if there

are N 1 particles of radius R1 per unit volume, will be smaller than V0 N1 N 2

 12 .

In order to calculate the volume V swept in t , let us uniformly (in random) fill up the

space with auxiliary (fictitious) point immobile particles (―m arkers‖) of radius R*  0 with 3 a relatively high concentration, n*  R12 . In order to adequately resolve a fine structure

(with the characteristic length of a12  R12 ) of the swept zone, the markers concentration

n* should additionally obey the condition that the number of swept markers N *( 0 ) during one jump must be large,

N*(0)  R122 a12n*  1 , or n*  R122 a12  . In this case the 1

swept volume can be calculated as the total number N * of the swept markers divided by their concentration, V  N * n* .

In its turn, for the same reasons (concerning relative displacements of diffusing particles), calculation of the sweeping rate of randomly distributed immobile markers by a large particle of radius R12 migrating with the diffusivity D12 is equivalent to calculation of the condensation rate of the mobile markers migrating with the diffusivity D12 in the immobile

Copyright © 2012. Nova Science Publishers, Incorporated. All rights reserved.

trap of radius R12 (see Appendix). 3 Owing to n* R12  1 , this problem of the (point-wise) markers condensation in the large (macroscopic) trap can be adequately solved in the continuum approach of [2], as above explained in Section 1.3.1. In this approach the total number of swept markers in t is equal



to N*  4D12 R12n* t  4R12 is equal to



t D12 , and the swept volume per unit time in this case

V t   n*1 N* t   4 D1  D2 R1  R2   4D12 R12 , if the time step

2 is sufficiently large, t  16 R12 D12 . In particular, this implies that the ratio of

V t  12

2

to V0  a12 R12 is equal to

V t  V0 12   2a12

3R12 .

The spatial variation of the markers concentration occurs on the length scale l which is comparable with R12 (see Section 1.3.1), i.e. l  R12 . In accordance with the condition of the diffusion equation applicability, a  l (see Section 1.2), this result is valid only in the case a12  R12 .

t  16 R122 D12 is valid for 6D under an assumption  d  16 R122 D12 or

It is straightforward to see that the necessary condition the chosen time step

n1/ 3 R   96

1/ 2

t   d  n 2 / 3

 0.2 , that is in agreement with the basic requirement n1/ 3 R  1 . In

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16

M.S. Veshchunov

this case, the number of coalescences V t N 1 N 2 between particles of radii R1 and R2 in the unit time becomes equal to 4 D1  D2 R1  R2 N 1 N 2 .

Therefore, the collision frequency function in the kinetic equation, Eq. (1.11), takes the form ( con) R1, R2   4 D1  D2 R1  R2  , kin

(1.17)

or, in the limit of small Knudsen numbers, Kn  1 (see Eqs. (1.7) - (1.9)), when Cc  1 , ( con) R1 , R2    kin

2kT  1 1    R1  R2  . 3  R1 R2 

(1.17a)

More accurately, one should split the integrals in Eq. (1.11) into two parts, n 1/ 3 n 1 / 3



  dR dR 1

0 0

2



  0

0

 n 1/ 3

dR1dR2 

  dR dR 1

n 1/ 3

2

, and to use the kernel

0

(derived for particles of comparable sizes under condition n the kernel

(con) from Eq. (1.17) kin

1/ 3

R  1 ) in the first part and

 dif from Eq. (1.12) (correct for coalescence of small and large particles) in the

second part (and neglecting coalescences among large, and thus very slow, particles with

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(con) R  n 1 / 3 ). However, since the two kernels kin and  dif formally coincide, Eq. (1.11)

can be used with a good accuracy. It should be outlined that the coincidence of Eqs. (1.12) and (1.17) is fortuitous, since Eq. (1.12) was derived in the diffusion regime (by consideration of concentration profiles of small particles around large ones), whereas Eq. (1.17) was derived in the kinetic regime (by consideration of uniform spatial distribution of Brownian particles). The nature of this internal symmetry will be revealed below in Section 1.6.2.

1.5.2. Free Molecular Mode ( a  R ) In the opposite limit a12  R12 one can neglect the mean relative volume of the swept 3

zone segments intersections  R12

V0  R12 a12  1. In this approximation the mean

volume swept by the exclusion zone of radius R12 per unit time, V t , is a constant value equal to the sweeping rate during the jump period,

V0  12 . For this reason, the above

applied requirement to the time step t   0   12 for calculation of the collision rate (see Section 1.5) is not anymore necessary, i.e. it can be correctly calculated within one jump period. In accordance with the free molecular (or ballistic) approach traditionally applied to consideration of particles movement and collisions on the time scale of one jump period (see Section 1.4), it is assumed that the original particles move straight during their jump periods. * For this reason, the probability P12 of a collision in t *   12 of two original particles of

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A New Approach to the Theory of Brownian Coagulation …

17

radii R1 and R2 , migrating in a sample of unit volume, can be considered as a constant value, equal to the sweeping rate of the effective mobile particle (of radius R12 ),

P12*  V0  12  R122 a12  12 . On the other hand, this probability can be calculated in the free

molecular

approach

(Section

1.4)

P12*  R122 8kT  m11  m21   R122 c12  c22  , i.e. c12  a12 12 

c

2 1

as

 c22  ,

where c1 and c 2 are the mean thermal speeds of the two original particles defined in Eq. (1.6a). Correspondingly, for the effective diffusivity D12 , defined in Section 1.5, one 2 obtains D12  a12 6 12  a12c12 6 , or a12  6D12 c12 .

Therefore, the total swept volume V (after k  t

12  1 jumps) is equal to

k V0  V0t  12 , and the number of coalescences V t N 1 N 2 between particles of radii R1 and R2 (with the number densities N 1 and N 2 , respectively) in the unit time is equal

to

Eq. (1.11),

6kT  R13  R23  .

N1 N 2 V0  12  N1 N 2 R1  R2 

2

The

kernel

of

( fm ) , in this case is equal to V0  12 that coincides with the free molecular kin

expression for

 fm , Eq. (1.13), i.e.

2 ( fm )  V0  12  R1  R2   kin

6kT  R13  R23  .

(1.18)

Hence, this case corresponds to the free molecular mode of the kinetic regime and is

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realized at high Knudsen numbers, Kn  0.5   m

 1/ 3 nm1/ 3 R m  . 1

1.5.3. Transition Mode ( a  R ) In the transition interval a  R , which also belongs to the kinetic regime (characterized by homogeneous spatial distribution of particles), the collision rate can be calculated similarly to the two above considered limiting cases by evaluation of the mean swept volume per unit time, kin  V t  , however, in the numerical approach. In this approach the exact values of  kin can be calculated at different a12 R 12 and then approximated with an analytical expression [7]. For the numerical evaluation of the mean swept volume per init time, V t , a random migration of a particle of the radius R12 with the fixed jump distance a12 and jump 1 frequency  12   12 is numerically generated. The randomly generated data describe the

subsequent positions of the particle centre trajectory, which can be further used for calculation of the swept volume. For this calculation a similar to the above described procedure of random spatial distribution of auxiliary point immobile particles (markers) with a relatively high concentration, n* 

R



1 2 , 12 12

a

is numerically realized using the Monte

Carlo method. Each marker found in the swept volume is counted only once.

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18

M.S. Veshchunov

For each number of jumps k , several (up to 15) random trajectories and for each trajectory, several (up to 5) realizations of markers random spatial distribution are generated. Numbers of trajectories and markers distributions are increased until the calculated swept volume, averaged over these realizations, become insensitive to further increase of these numbers. The number of jumps k  t  12 is increased until the ratio of V t  12 to  V 0 attains a steady-state value, which in accordance with the above presented analytical calculations has to be equal to 2a12 3R12 in the limit a12  R12 and to 1 in the limit

a12  R12 . The minimum number of jumps necessary for attainment of the steady-state regime

smoothly

t  96 R

2 12

decreases

from

k min  96  R12 a12 

2

(corresponding

to

D12 ) at large R12 a12  1 (see Section 1.5.1) to k min  1 at small

R12 a12  1 (see Section 1.5.2). Therefore, in the transition interval a12 R12  1 the



inequality k  30 R12 a12



2

is a conservative requirement for the time step, that is

confidently valid under the necessary condition

 c  t   d  n 2 / 3 6D , or

k  n 1/ 3 a12  (imposed in Section 1.5 and confirmed below in Section 1.5.5) along with 2

the basic assumption n

1/ 3



R  1 , since in this case k  n 1/ 3 a12



2

 30R12 a12  . 2

From an obvious geometry (― scaling‖) consideration it is clear that the steady-state value of the mean swept volume per unit time depends only on the ratio a12 R12 (rather than on

a12 and R12 separately), that is confirmed by numerical calculations. Copyright © 2012. Nova Science Publishers, Incorporated. All rights reserved.

Examples of numerical calculations for dependence of the inverse (for convenience of graphical representation) ratio

V0 12  V t  ,

that is equal to

( fm ) kin kin , on the

number of jumps k for several values of the parameter R12 a12 are presented in Fig. 1.2. For large R12 a12  10 , the calculated steady-state values are in a satisfactory agreement (with the error < 1%) with the analytically predicted (in the limit R12 a12  1 ) values,

1.5R12 a12 .

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A New Approach to the Theory of Brownian Coagulation …

19

16

14

R12/a12 = 10 R12/a12 = 5 R12/a12 = 2

[(V/t)/(V0/)]-1

12

10

8 7.7

6

4 3.5

2 2x10

4

4x10

4

4

4

5

6x10

8x10

1x10

inverse

relative

mean

k

Figure

1.2.

Calculated

V t  V0

 12 

1

dependence

of

on the number of jumps

Table 1.1. Steady-state values of

the

sweeping

rate

k.

V0  12  V t    kin  kin( fm) 1 calculated in the

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simple random walk approach at different values of a12 R12  6D12 c12 R12

a12 R12 V0  12 V t

a12 R12

V0  12 V t

0.05

0.1

0.2

0.3

0.4

0.5

2/3

5/6

1.0

1.25

30.0 0

15.0 0

7.72

5.38

4.22

3.49

2.82

2.39

2.13

1.81

5/3

2.0

10/3

5.0

10.0

15.0

20.0

30.0

40.0

50.0

1.59

1.47

1.21

1.12

1.05

1.03

1.01

1.01

1.00

1.00

The steady-state values of the ratio

V0 12  V t  calculated in a wide range of the

parameter values, 0.1  a12 R12  50 , are collected in Table 1.1 and presented in Fig. 1.3 (by centers of circles). Using these data, an interpolation curve was searched by fitting to the calculated points and following the general requirements derived from the analytical consideration.

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20

M.S. Veshchunov

1.5.4. Interpolation Formulas Namely, taking into account (as above explained) that

V t  V0 12  depends only

on the ratio a12 R12 and varies from 2a12 3R 12 (at a12  R12 ) to 1 (at a12  R12 ), the calculated value should be approximated by an analytical expression as a function of the unique parameter a12 R12 . Since

( con) ( fm ) kin  V0  12 2a12 3R12  and kin  V0  12  ,



( con) this parameter is equal to a12 R12  1.5 kin

( fm ) , therefore, the interpolation kin

expression for  kin must be searched in the form

~

( con) ( con) ( con) ( fm ) , f1 a12 R12   kin f1  kin kin  kin kin

(1.19)

or in the equivalent form

~

( fm ) ( fm ) ( con) ( fm ) , f 2 a12 R12    kin f 2  kin  kin   kin  kin

(1.19a)

which can be also represented in terms of the independent set of the three parameters

R12  R1  R1 , D12  D1  D1 and c12 

c

2 1

~

 c12  

 kin  R122 c12 f 2 4D12 R12c12  , since

8kT  m11  m21  as (1.19b)

( con) ( fm )  4D12 R12 .  kin  R122 c12 and kin

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On the other hand, these forms, Eqs. (1.19) and (1.19a), can be specified more definitely by analytical consideration of the markers condensation in the trap of radius R12 (in the reformulated problem of the immobile particle and point markers migrating by random walks with the diffusivity D12 ). Indeed, in the limiting case a12  R12 , the markers diffusion is the rate determining step, whereas in the opposite case a12  R12 , when the jump distance of the markers significantly exceeds the length scale of their concentration heterogeneity l  R12 induced by the trap of radius R12 , the markers migrates in the kinetic (― free molecular‖) regime. Similarly to the classical problem of vapor molecules condensation in a large immobile trap [4], the transition regime for the markers with a12  R12 can be described by flux matching at the adsorbing sphere radius, r  R12  12 , separating zones of the different regimes of the markers migration, kinetic (inside the absorbing sphere) and diffusion (outside the sphere). The general solution of the flux matching problem can be searched in the form

~ ~ ( con)  kin   kin  , , where   a12 R12 and   12 R12 , which should additionally ~

obey the general restriction of Eq. (1.19), thus  has to be searched as a function of  . Owing to an uncertainty in determination of  12 , various semi-empirical models for condensation of point particles in a trap were proposed (see, e.g., [12]), which in the simplest

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A New Approach to the Theory of Brownian Coagulation …

21

approach can be reduced to the general form (with various sets of parameters B1 and B2 ) [13] ( con)  kin   kin

1  B1 , 1  B2   2 3B12

(1.20)

where   a12 R12 . In order to extrapolate the results of the point particles condensation problem to consideration of a polydisperse system of large particles, the traditional models use additional semi-empirical assumptions connected with an uncertainty in determination of the ― mean free path‖ a12 . In

the

new

a12 R12  1.5

approach

( con) kin



( fm ) kin

this

problem

is

resolved

using

for

the

parameter

 the analytical expressions, Eqs. (1.17) and (1.18), which allow

explicit calculation of a12 :

 D1  D2 

a12  where ci  8kT

kT 6 R

3 1

3 2

R





6D1  D2 

c

2 1

c

2 2





6D12 , c12

(1.21)

mi 1/ 2  6kT  2 Ri3  , in accordance with consideration in Section 1/ 2

1.5.2. This value of a12 is different from that obtained in the traditional approach by formal

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extension of the mean free path expression for vapor molecules in the condensation problem [14],   3D c , to the considered problem of polydisperse particles coagulation (see, e.g. [13])

12  3D12 c12 .

(1.22)

In terms of the three basic parameters R12 , D12 and c12 , Eq. (1.20) can be equivalently presented in the form

 kin  4D12 R12

2 3  B11 2 31  B112 2 ,   R c 12 12 2 3  B2 1  B112 2 3  B2 1  B112

(1.23)

where 1   kin

( con)

( fm ) kin  4D12 R12c12 , or 1  2 3 .

~

The key problem of determination of  (resolved semi-empirically in the traditional approach) can be resolved more accurately using the new numerical approach to the swept volume calculation. In this approach the exact values of  kin are calculated at different

a12 R 12 (Table 1.1) and then approximated using Eq. (1.20) or a more sophisticated expression with an extended set of fitting parameters, e.g. in the form Coagulation: Kinetics, Structure Formation and Disorders : Kinetics, Structure Formation and Disorders, Nova Science Publishers, Incorporated,

22

M.S. Veshchunov ( con)  kin   kin

1  B1  B4  2 , 1  B2   B3  2  2 3B4  3

(1.24)

that is consistent with Eq. (1.19). The values of the parameters in Eq. (1.24) can be determined using the least-squares method, that gives B1  14 .24 , B2  13 .61 , B3  9.52 , B4  3.52 and provides a relatively high accuracy with the maximum error of  1% in comparison with the calculation points (from Table 1.1), as presented in Fig. 1.3 (solid line). A somewhat reduced accuracy with the maximum error of  3% can be attained using the simplified model, Eq. (1.20), with parameters B1  0.51 , B2  0.78 , determined using the least-squares method (dashed line in Fig. 1.3). 100 Calculation points Interpolation (least-squares method): 2 parameters

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[(V/t)/(V0/)]-1

Interpolation (least-squares method): 4 parameters

10

1 0.1

1

10

100

a12/R12

Figure 1.3. Dependence of the steady-state values of the inverse relative mean sweeping rate

V t  V0

( fm )   12 1   kin  kin

1

interpolation curves, Eq. (1.22) with line), and Eq. (1.20) with

a12 R12  6D12 R12c12 , B1  14 .24 , B2  13 .61 , B3  9.52 , B4  3.52 on the parameter

using (solid

B1  0.51 , B2  0.78 (dashed line), in comparison with the calculated

points (centers of circles).

This optimal set of parameters for Eq. (1.20) is somewhat different from B1  0.5 ,

B2  0.855 , obtained by formal application of Fuchs – Sutugin‘s interpolation formula for ~ condensation problem [14] with   12 R12 and Eq. (1.22) (reformulated in terms of

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A New Approach to the Theory of Brownian Coagulation …

23

~   a12 R12  2 ) that provides an accuracy with the maximum error of  4% in comparison with the calculation points. A similar result can be obtained using Eq. (1.20) with B1  1 3 , B2  2 3 , proposed in

~

[13] by formal application of Dahneke‘s formula [15] with   12 R12 and Eq. (1.22), that eventually takes the form (in terms of   a12 R12 )

 kin   with

(con) kin

~ 1  ~ , 1  2 3 1  



(1.25)



~   B1   3 ,

(1.26)

and provides an accuracy with  3.5% maximum error for this set of parameters. Eq. (1.25) formally coincides with the other semi-empirical interpolation formulas from

~

the literature [4, 16], however, with different expressions for  . Namely, the expression for

~  proposed by Fuchs [4] has the form

2  22  ~    12  1 R12 R12

1/ 2

with

,

(1.27)





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3/ 2 i 1  1  3    1  i   1  i2   1 , 2 Ri 3  i 

(1.27a)

or, following Wright [16], with 5/ 2   i  1  1 1 2 5 3   2  1  i   1  i  1  i2   1  i2    1 , (1.27b) 2 Ri  i  5 3 15 

where i  li 2Ri and li  8 Di ci . Both these expressions, Eqs. (1.27a) and (1.27b), however, do not match with the abovederived general requirement of the theory, Eq. (1.19). For this reason, a more adequate

~

expression for  can be applied using corrected Fuchs‘ and Wright‘s formulas proposed in [13]





3/ 2 1 1  ~   1  12 3  1  122   1 ,   12   R12 3  12 

(1.28a)

and

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24

M.S. Veshchunov

 1  1 5/ 2  1 2 ~  5 3   12   2  1  12   1  12  1  122   1  122    1 , 3 15 R12  12  5  (1.28b) where 12  l12 R12 and l12  8D1  D2 

 c12  c22  . 1/ 2

1.1

1

kinkin(con)

0.9

0.8

Calculation points Interpolation (least-squares method): 4 parameters Dahneke's interpolation Corrected Fuchs' interpolation

Copyright © 2012. Nova Science Publishers, Incorporated. All rights reserved.

0.7

0.6 0.1

1

a12/R12 Figure 1.4. Verification of reformulated Dahneke‘s and corrected (in [13]) Fuchs‘ semi-empirical



 (con)

a

kin interpolation formulas for kin as a function of 12 parameter formula and the calculated points (from Table 1.1).

R 12

against the newly derived four-

Results of approximation with these semi-empirical expressions, Eqs. (1.28a) and (1.28b), being rather similar to each other (within < 1%), notably deviate from the calculation points (by 4–6%) in a wide range of the transition interval, 0.1  a12 R12  0.9 . Hence, in order to compare behavior of various interpolation expressions in this area, they are represented in the traditional form,

(con) (as a function of a12 R 12 ) in Fig. 1.4,  kin  kin

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A New Approach to the Theory of Brownian Coagulation …

25

demonstrating a relatively high error (up to  6% at a12 R12  0.3 ) in predictions of these semi-empirical models in comparison with the new formula and the calculation points. Therefore, the new approach allows calculation of the collision frequency function in the transition regime, without use of the semi-empirical assumptions (from the literature) in derivation of the interpolation expression and with a higher accuracy (with respect to the calculation points), which can be further improved in the next approximation of the random walk theory (see Section 1.6).

1.5.5. Applicability Range of the Kinetic Approach In the mean field approximation considering monodisperse particles size distribution (which delivers a reasonable asymptotic solution for the coagulation problem at large times), the probability of two neighbor particles coalescence in t can be evaluated from Eq. (1.17) for the case a  R . Indeed, the total rate of loss of particles by coalescence is given by

dn n  8DRn 2   c dt where

(1.29)

 c  8DRn 1 is the coalescence time corresponding to the characteristic time

between two subsequent collisions of a particle, during which the mean inter-particle distance

r  n 1 / 3 increases by a factor of  1.3. This confirms that variation of the mean particles concentration can be neglected in the first order of approximation t  c  1 during the time

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step t , as assumed in Section 1.5. Eq. (1.29) should be solved simultaneously with the condition of the total mass conservation

R3n  const.  R03n0 .

(1.30)

As already outlined in Section 1.3.1, owing to coalescence of two particles from the ensemble of point particles, the local number of particles alters step-wise (from two to one) that induces a local heterogeneity of the particles spatial distribution function on the length 1 / 3

. Therefore, the uniform spatial distribution scale of the mean inter-particle distance  n of coalescing particles can be considered under condition that the heterogeneities of such scale, n

1 / 3

, can rapidly relax by the particles diffusion during the characteristic time

between two subsequent collisions of a particle,  c . This condition of the particles relaxation, or mixing, takes the form  c   d , where

 d  n 2 / 3 6D is the characteristic time of the

particles (diffusion) redistribution on the length scale of the induced heterogeneities. Only in this case the probability to find a new particle in the vicinity of the coalesced particle at the moment of their subsequent collision will be still determined by the mean concentration of particles, as assumed in derivation of Eqs. (1.17), (1.18) and (1.24). Substitution of

 c  8DRn 1 and  d  n 2 / 3 6D in the relationship  c   d

yields the mixing condition in the form

n1/ 3 R  3 4 , which is practically

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26

M.S. Veshchunov

indistinguishable from the basic condition n1/ 3 R  1 , within the accuracy of the

 c   0 , required in the current approach for choosing a relatively large time step t   0 , takes the form characteristic times evaluation. The other condition,

8DRn 0  4 3Ra 2 n  4 3 a R nR3  1, that is valid owing to n1/ 3 R  1 and 2

a R  1 . This condition is apparently still valid in the transition regime, when a R  1 . It is straightforward to see that the mixing condition,  c   d , is valid also in the opposite case a  R , corresponding to the free molecular mode. Indeed, substituting



 c  3kT  1/ 2 4nR1/ 2



1

from Eq. (1.14) in this inequality, one obtains the mixing

condition in the form a R  4 36 

1/ 2

n1/ 3 R  18n1/ 3 R , which is valid practically in

the whole range of the free molecular mode applicability, a R  1 (taking into account that

n1/ 3 R  1 ). The other condition,  c   0 , is not anymore required in the case a  R as explained in Section 1.5.2, so, no additional restriction on the applicability of the current approach emerges. In the transition regime, when a  R (or   1 ), the value of  c derived from Eq. (1.25) is comparable with the continuum mode value,

 c  8DRn 1 , and thus also

obeys the mixing condition  c   d . Therefore, the kinetic regime is realized practically in the whole range of the considered approximation n1/ 3 R  1 , however, with various expressions for the collision frequency in Copyright © 2012. Nova Science Publishers, Incorporated. All rights reserved.

the different intervals of the parameter a R (corresponding to the different modes of the kinetic regime). It is straightforward to see that the mixing condition  c   d can be equivalently represented in the form   r  n

1 / 3

, where

  6D c 1/ 2 is the characteristic distance

between two subsequent collisions of a particle with other particles (or the mean free path of Brownian particles). From the above presented consideration it is seen that the limit

n1 / 3  1 , corresponding to  c   d , can be attained only in the case of very high densities, n1/ 3 R  1, that is beyond the applicability range of the basic approximation, n1/ 3 R  1 (or R r  1 ). This situation has a clear physical sense that the mean free path of particles in the diluted system cannot be less than the mean interparticle distance and is qualitatively similar to behavior of ordinary molecular gases. Indeed, the mean free path m of the gas molecules is estimated as

m  nm 2  , where nm  rm3 is the gas density and   d 2 is the 1

m  rm rm d 2  d rm d 3 , which results in the relationship m  rm  d under the ideal (diluted) gas condition, d rm  1 (analogous molecular cross-section, therefore,

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A New Approach to the Theory of Brownian Coagulation …

27

to the basic condition of the Brownian particles theory, R r  1 ). The mixing condition implies in this case that after each collision of gas molecules their random distribution quickly reinstates owing to their relocations on the length scale of the mean inter-molecular distance, rm . Since the gas molecules move straight with the mean thermal speed um in-between their collisions, the characteristic collision and mixing times can be evaluated as  c  m u m and

 mix  rm um , respectively. Therefore, the mixing condition,  c   mix , directly corresponds to the above grounded relationship,

m  rm . This allows calculation of the gas

atoms collision frequency in the kinetic approach (i.e. within the Boltzmann kinetic theory).

1.6. Next Approximation of the Random Walk Theory The above presented approach based on the simple random walk theory (with the fixed jump distance of migrating particle) allowed a relatively simple and rapid calculation of the new interpolation formulas for the coalescence rate, which correctly reduce to the analytical expressions in the two limiting cases, a  R and a  R , avoiding semi-empirical assumptions of the traditional models, however, not in completely self-consistent manner. Namely, one of the model parameters, a12 , was derived in Section 1.5.2 from comparison of

the collision frequency calculated at high Knudsen numbers, Kn   , in the simple random walk approach (neglecting variation of the jump distance) with that calculated in the free molecular approach, which takes into consideration the Maxwell distribution of the particles velocities (and thus resulting in a finite distribution of the jump distances). Nevertheless, since this parameter a12 enters in the calculated collision frequency implicitly (via D12 ), the Copyright © 2012. Nova Science Publishers, Incorporated. All rights reserved.

collision kernel, Eq. (1.20) or Eq. (1.24), represented in terms of the independent set of the three basic parameters R12 , D12 and c12 , Eq. (1.23), were apparently calculated with a reasonable accuracy (considered as the first approximation of the random walk theory). In order to overcome this inconsistency and to verify the simplified approach, the next approximation of the random walk theory taking into consideration variable length of the jumps will be studied in the current Section.

1.6.1. Brownian Particles Coagulation Evaluation of various physical parameters of a Brownian particle generally implies averaging of these parameters over an ensemble of identical particles (i.e. of the same mass m and radius R ). In particular, distribution for the speed of a particle, obtained by this averaging procedure, has a finite statistical dispersion (near the mean thermal speed) described by the Maxwell law. On the other hand, variation in time of the velocity components of a particle, moving under external stochastic forces exerted by the carrier gas molecules, calculated in the Langevin theory of Brownian movement [8] by averaging over

an ensemble of identical particles (of the same mass m ) as ui t   ui 0exp  t ~ ,

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28

M.S. Veshchunov

( i  x, y , z ), with the fixed value of the characteristic relaxation time, ~  mb  mD kT , is independent of the initial velocity ui 0  .

This important conclusion substantiates applicability of the random walk theory (at large times, t  ~ ) considering elementary drifts (or jumps) of a particle with the fixed

elementary drift time  0  ~ , but to different distances (owing to distribution for the particle

speed). As a result, the distribution for the elementary drift distance of a particle obeys a distribution law deduced from the Maxwell distribution law for its speed. For instance, this essentially simplifies the random walk theory for Brownian particles in comparison with that for the Boltzmann gas, where both the drift distance and drift time between two subsequent collisions of a molecule are stochastically varying values.   In this approximation for Brownian particles, the probability w r , d 3 of a particle (from    an ensemble of identical particles) at r to relocate to r   in  0 is equal to the probability for

 

 this particle to have the velocity u    0 , which is determined by the Maxwell distribution law,

~ u d 3u   m  w  2kT 

3/ 2

 mu 2  3 , exp   d u  2kT 

or

    ~ u    d 3u  w ~   d 3  3 . Therefore, the mean-square drift w r , d 3  w 0 0 0

 





distance





can

be

  4  m  aˆ 2   2    2 w r ,  d 3   3    2kT  0 0  0 

 



calculated 3/ 2

as

m  3kT 2 d   0 , or 2  2 kT  m 0   

 4 exp  

2

1/ 2

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 3kT  aˆ     0  uT 0 .  m  For the diffusion coefficient in the ensemble of particles, D 

(1.31)

 3 1 2   w d  (cf. 6 0 



Section 1.2), one obtains

D  aˆ 2 6 0  uT2 0 6 ,

(1.32)

that, as expected, coincides with the expression for a sole particle migrating by random walks with the root mean-square length

aˆ  uT 0 , where uT 

(1.33)

1/ 2  1/ 2 u 2  3kT m   1.5kT  R 3  is the root mean-square heat velocity.

On the other hand, the mean jump distance a during one walk, averaged over the Maxwell distribution function, can be calculated as

 a  u  0  c 0 ,

(1.34)

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A New Approach to the Theory of Brownian Coagulation …

c  8kT m

1/ 2

where

29

is the mean thermal speed.

From Eqs. (1.31)-(1.34) one obtains the relationship 2 D  aˆ 2 6 0  a 2 6 0 uT c    a 2 16 0  a 2 5 0 ,

(1.35)

1/ 2 aˆ a  uT c  3 8 .

(1.36)

where

As shown in Section 1.5, evaluation of the two-particle collision frequency (required for calculation of the coagulation kernel in the ensemble of comparable size particles under the mixing condition) can be reduced to calculation of the mean volume sweeping rate V t  by the effective particle of radius R12 migrating by random walks with diffusivity D12 , which in accordance with the above presented consideration is determined by the root mean2 square length aˆ12 of the particle jumps as D12  aˆ12 6 12 . On the other hand, the mean

swept volume  V 0 during one jump is determined by the mean jump distance a12 and is calculated as V0  a12 R1  R2  . 2

In the continuum mode ( a  R ) the mean swept volume per unit time (averaged over

relatively long time step t including many jump periods, t   12 ), which determines the collision frequency

( con)  kin  V t , is smaller in comparison with the mean swept volume

V0 12  , owing to strong overlapping of the swept zone segments, V t  4 D1  D2 R1  R2   4D12 R12 . In particular, this implies

per one jump period,

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and is equal to (using

V t

Eq. (1.35))

that

V0  12   2aˆ122

the

ratio

3a12 R12 ,

of

V t

to

or,

V0  12

is

using

equal

to

Eq. (1.36),

( con) ( fm ) kin kin  4D12 R12c12   4a12 R12

In the opposite, free molecular limit ( a  R ) one can neglect the relative volume of the swept zone segments intersections. For this reason, the mean volume swept per unit time,

V t , is a constant value equal to the mean volume swept per one jump period,

V0  12 . In its turn, the latter can be considered as a constant value, V0  12  R122 a12  12 , equal to the mean sweeping rate V * t *  R122 c12 during a

t *  12 . On the other hand, the mean swept volume V * represents the * probability of a collision in t of two original particles of radii R1 and R2 , migrating in a short time step

sample of unit volume, that can be calculated in the free molecular approach as

V * t *  R122 8kT  m11  m21   R122 c12  c22  ,

c12  a12  12 

c

2 1

i.e.

 c22  , where c1 and c 2 are the mean thermal speeds of the two

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30

M.S. Veshchunov

12  a12 c12 . Correspondingly, for the effective 2 2 D12  aˆ12 6 12  aˆ12 c12 6a12   16a12c12 , or

original particles defined in Eq. (1.34), or diffusivity

one

D12

a12  16  D c

1 12 12 ,

obtains

instead of Eq. (1.21) calculated in the simple random walk approach of

Section 1.5 (with the fixed jump distance, a12  const. ). Nevertheless, the collision kernel, same equation,

( fm ) , which is independent of a12 , reduces to the kin

2 ( fm )  V0  12  R1  R2   kin

defined in terms of the independent parameter



6kT  R13  R23  .

( con) kin



( fm ) kin

Besides, being

 4D12 R12c12 , Eqs. (1.20), (con) ( fm ) and kin at kin

(1.23) and (1.24) correctly reduce to the limiting expressions for

Kn  1 and Kn  1 , respectively. In the transition interval a  R , the collision rate can be calculated similarly to the two above considered limiting cases by evaluation of the mean swept volume per unit time,

 kin  V t , in the numerical approach. However, in contrast to the simple random walk approach of Section 1.5.3, the jump distance of the effective particle (of radius R12 and mass

m12 ) in random directions is calculated as a12  u12 12 , where the particle speed u12 is generated

f u12   consistency

as

a

random

with

the

probability

density

 m u2  2  m12  2 1 1   u12 exp   12 12  , where m12  m1  m2 2 kT   kT    of

the

c12  8kT m12 

1/ 2

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value

3





above

c

2 1

2 2

c

derived

expression

  8kT  m

1 1

for

1 2

m

the



1

function,

, providing selfthermal

speed,

 . The generated data describe

the subsequent positions of the particle centre trajectory, which can be further used for calculation of the swept volume. For this calculation the same procedure of random spatial distribution of auxiliary (fictitious) point immobile particles (markers) with a relatively high





1

concentration, n*  R12a12 , described in Section 1.5.3, is numerically realized using the Monte Carlo method. Each marker found in the swept volume is counted only once. 2

The number of jumps k  t  12 is increased until the ratio of

V t to V0  12

attains a steady-state value, which in accordance with the above presented analytical calculations has to be equal to  4 a12 R12 in the limit a12  R12 and to 1 in the limit

a12  R12 . In order to diminish statistical dispersion of the calculation results, a relatively large value of k  105 was chosen for each trajectory. Nevertheless, owing to variation of the jump distance, dispersion of the calculated value V t  V0 12  is notably larger in comparison with that calculated for jumps with the fixed length (in Section 1.5.3). For this reason, 150-200 calculations for each value of R12 a12 have been carried out, forming a smooth distribution plot near the mean values, which reliably converge to the limiting values at a12  R12 and a12  R12 .

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A New Approach to the Theory of Brownian Coagulation …

31

Preliminary results of these calculations, which will be presented in more details elsewhere [17], are summarized in Table 1.2. As shown in Fig. 1.5, the interpolation expression derived in the simple random walk approach (in Section 1.5.4) and represented in terms of the independent variables D12 , R12 and c12 in the form of Eq. (1.23) reasonably describes (within the calculation accuracy) the new set of calculation points,

2 31  B112  kin  , ( fm ) 2 3  B2 1  B112  kin

(1.37)

where 1  4D12 R12c12 and B1  0.51 , B2  0.78 . A similar result can be confirmed for the advanced four-parameter interpolation expression, Eq. (1.24), represented in terms of the independent variables

2 3  B11  3 2B4 12  kin ,   1 ( fm ) 2 3  B2 1  3 2B312  3 2B4 13  kin

(1.38)

where 1  4D12 R12c12 and B1  14 .24 , B2  13 .61 , B3  9.52 , B4  3.52 . From these results one can conclude that the new interpolation expression derived in the simple random walk approach (and represented in terms of the independent variables

D12 , R12 and c12 ) is reasonably confirmed by calculations in the next approximation of the random walk theory (with stochastically varying jump distance) and thus can be used instead of the semi-empirical formulas. 100

Interpolation: 4 parameters

[/(V0/)]-1

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Calculation points Interpolation: 2 parameters

10

1 0.01

0.1

1

10

100

a12/R12

Figure 1.5.

V t  V0

Verification

 12    kin  1

of



the

interpolation

expressions

( fm ) 1 kin

for

as a function of a12 R12  6D12 R12c12 obtained in the first approximation of the random walk theory with fixed jump distance (from Fig. 1.3) against the new set of calculation points obtained in the second approximation with variable jump distance (from Table 1.2).

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V0  12  V t    kin  kin( fm) 1 calculated in the next approximation of the random walk theory at R12  16 D12 c12 R12 , which is related to a12 R12  6D12 c12 R12 from Table 1.1 as a12 R12  a12 R12  3 8

Table 1.2. Steady-state values of different values of a12 0.046

0.057

0.092

0.169

0.184

0.23

0.47

0.594

0.92

1.57

3.395

9.21

21.93

a12 R12

0.054

0.067

0.109

0.2

0.217

0.271

0.552

0.7

1.085

1.846

4.0

10.85

25.84

V0  12 V t

27.40

22.38

13.69

7.965

6.87

6.12

3.29

2.765

2.04

1.54

1.196

1.06

1.02

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a12 R12

tail.action?docID=3021398.

A New Approach to the Theory of Brownian Coagulation …

r

33

Continuum mode

a12 Kinetic regime

Diffusion regime

R12

R

Figure 1.6. Schematic representation of various ranges of the parameter 12 : dashed zone corresponds to the range, where the collision kernel is described by the unique analytical expression,

12  4D12 R12 , derived under the general condition a12  R12 , corresponding to the continuum

mode either in the diffusion regime or in the kinetic regime.

1.6.2. Heavy Vapor Molecules Condensation As above explained, the new value of the mean drift distance calculated in the next approximation of the random walk theory

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a12  16  D12 c12 ,

(1.38)

is slightly different (by a factor of 1.2) from Eq. (1.21), calculated in the simplified approach. On the other hand, in the traditional approach this parameter is evaluated by formal extension of the mean free path expression for vapor molecules in the condensation problem and differs more significantly (see Section 1.5.4). These predictions can be directly compared by generalization of the new approach to consideration of the condensation problem in the particular case of heavy vapor molecules (suspended in the light molecule gas). Indeed, if the mass of vapor molecules is large in comparison with the mass of the carrier gas molecules, vapor molecules can be considered as Brownian particles (cf. Section 1.2). On the other hand, if the size R1 of vapor molecules is small in comparison with the size

R2 of the trap particle, R1  R2 , they can be considered as point-wise particles, i.e. their mutual collisions can be neglected and the condensation rate can be properly calculated as the collision probability of one molecule (randomly located in space with the probability equal to their mean concentration n ) with the trap. Since the Brownian particles coagulation problem is reduced under the mixing condition to the same two-particle collision problem, this allows rigorous consideration of the condensation problem using the above calculated collision kernel, Eq. (1.23), with the specific values of the parameters D12  D 1 and R12  R 2 . In particular, in the whole range of the continuum mode, a1  R2 (or equally

a12  R12 , where a12  a1 is the elementary drift distance of the vapor molecules), the condensation kernel calculated from consideration of the two-particle problem (in neglect of Coagulation: Kinetics, Structure Formation and Disorders : Kinetics, Structure Formation and Disorders, Nova Science Publishers, Incorporated,

34

M.S. Veshchunov

mutual collisions between point-wise vapor molecules, as explained above), is equal to 12  4D12 R12 (cf. Section 1.5.1). On the other hand, in the specific range of the continuum mode, r  n 1/ 3  R2 (or r  R12 ), corresponding to a dense gas of heavy

vapor molecules, the condensation kernel can be equally calculated (as 12  4D12 R12 ) from consideration of the diffusion flux (into the trap) in the multi-particle system of the vapor molecules (i.e. in the diffusion approach, cf. Section 1.3.1). Therefore, the condensation kernel derived in the diffusion approach formally coincides with that in the whole range of the continuum mode, including also the range r  R12  a 12 , corresponding to a rarified gas of vapor molecules, where diffusion approach is not anymore valid, however, the twoparticle collision problem has the same solution, see Fig. 1.5. This explains why the solution obtained in the diffusion regime can be correctly extended beyond the applicability range of this regime. In application of this kernel to consideration of the coagulation problem in the continuum fortuitous‖ coincidence limit a12  R12 , this feature apparently elucidates the reason for ― (revealed in Section 1.5.1) of the formal expressions for the coagulation kernel, derived either in the diffusion regime, that is valid for collisions between large and small particles (i.e. in the case R1  r  R2 , or r  R12 ), or in the kinetic regime (correctly described in the twoparticle approach), that is valid in the range r  R1 , R1 , or r  R12 . In accordance with the above derived conclusion on applicability of the Brownian particle consideration to heavy vapor molecules, diffusivity of Brownian particles determined by Eqs. (1.7)-(1.9) consistently reduces, as shown in [18], in the considered limit M B  M A to diffusivity of the vapor molecules calculated from the Chapman-Enskog

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theory for the binary diffusion coefficient,

3 2RT  1 DB  ~ 2 16 M A Np AB  AB 3

(1.39)

~

where M A and M B are masses of the carrier gas and vapor molecules, respectively; N is the Avogadro number, p is the gas pressure,  AB is the collision diameter for vapor-gas interactions,  AB  1 is the collision integral, which is equal to 1 in the hard spheres approximation, applied in the current approach. This additionally substantiates applicability of the coagulation kernel to the heavy vapor molecules condensation problem. However, the mean free path of the heavy vapor molecules ( M B  M A ) evaluated in various approximations of the kinetic theory of binary gases (e.g. [19, 12],

12  32 3 D12 c12

12  3D12 c12 [13] or 12  16   D12 c12 [20]), differ by numerical factors 1/ 2

from the drift distance of the Brownian particle, self-consistently calculated in the new approach (under essential condition of the fixed drift time,  0  const. ), Eq. (1.38), and thus can be applied to the collision kernel calculation only with a reduced accuracy.

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A New Approach to the Theory of Brownian Coagulation …

35

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1.7. Discussion A theoretical basis for the Fuchs semi-empirical approach [4] has been provided by Sitarski and Seinfeld [21], and by Mork et al. [22], who obtained the coagulation coefficient in the system of equisize particles (of radius R and diffusivity D ) through the solution of the Fokker-Planck equation for distribution function of Brownian particles. Namely, they calculated the steady rate of absorption of Brownian particles centers by a sink of radius 2 R and assumed the sink to be fixed in space, and the distribution function of particles centers to be governed by the Fokker-Planck equation (with diffusivity 2 D ). In this way, they have reduced the coagulation problem to consideration of condensation (in the large central particle fixed in space) of point masses (representing centers of original particles). At first glance, such an approach resolves the inconsistency of the SmoluchowskiChandrasekhar approach to calculation of the collision rate, based on consideration of the diffusion flux of surrounding particles to the central one, which is valid (as shown in Section 1.3.1) only for consideration of collisions between small and large particles. Namely, after turning to consideration of point-wise particles, which actually describe positions of the original particles centers, the problem of mutual collisions between particles (of finite size), moving to the central trap, is artificially removed (since collisions of point-wise particles can be ignored). However, as shown in Section 1.3.1, in fact the mutual collisions of the original particles can be neglected only in the case of their small radii in comparison with that of the central particle. Therefore, the inconsistency of the approach [21, 22] based on the FokkerPlanck equation, inherited from the diffusion approach of Smoluchowski-Chandrasekhar, in application to coagulation of the comparable size (or equisize) particles was not removed. Another approach to the Brownian coagulation problem was developed by Nowakowski and Sitarski [23] and by Narsimhan and Ruckenstein [24] through Monte Carlo simulations. Their approach, contrary to [21, 22], was essentially based on an assumption that the motion of only two colliding entities can be considered; however, applicability of this assumption was not grounded (and apparently was just derived from the phenomenological form of the pair-wise kernel in the Smoluchowski kinetic equation, Eq. (1.11)). As shown in the author‘s papers [5-7], the coagulation problem for comparable size particles can be really reduced to consideration of two colliding entities only in the case of rapid diffusion mixing of particles between their collisions, i.e. when the system of particles after each coalescence may be considered as spatially homogeneous (see Section 1.3.2). Therefore, the basic assumption of [23, 24] (valid for homogeneous systems) was directly opposite to the Smoluchowski-Chandrasekhar approach, based on the assumption (valid only for collisions between large and small particles) that the collision rate is controlled by the diffusive current of particles, that implies essentially inhomogeneous distribution of particles in space. Under condition that the Smoluchowski-Chandrasekhar approach was (and still is) widely accepted for consideration of collisions in the ensemble of comparable size particles (as well as for traditional consideration of diffusion-limited reactions, see Part 2), this question could not be treated as negligible or obvious, and thus required explicit analysis and justification (that was not attempted in [23, 24]). Following the current approach justification that only two colliding entities can be considered owing to rapid diffusion mixing in the ensemble of (comparable size) Brownian particles, the coagulation problem can be properly reduced to consideration of one immobile

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36

M.S. Veshchunov

trap (of radius R1  R2 ) and one migrating point-wise particle, which, in its turn, can be

 

described by the Fokker-Planck equation for distribution function of this particle, f r , v , t  . Naturally, this distribution function can be equally applied to formal consideration of an



ensemble of point-wise particles of concentration nr , t  

 



 f r , v , t dv , however, these

particles are fictitious (e.g. markers considered in Sections 1.5.1 – 1.5.3), since they are not related to real Brownian particles from the original multi-particle ensemble. Therefore, the Fokker-Planck approach can be eventually applied to consideration of the Brownian coagulation problem, however, indirectly, after reducing the multi-particle problem to consideration of two particles collision probability (justified under the mixing condition,  c   d or   r ). Since the Fokker-Planck equation is generally derived from the Langevin equation, which is also the governing equation of the Monte Carlo approach [23, 24], both methods should bring in similar results. On the other hand, the Langevin equation, derived under condition that forces can be split up into a systematic part (friction term) and a statistical part (stochastic term, or the Langevin force) that is not well grounded at short times t   0 , at large times

t   0 is equally reduced to the Einstein diffusion equation for the Brownian particle

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motion (see, e.g. [4, 25]), which can be properly described by the random walk theory applied in the current paper. For this reason, one should expect a rather good coincidence of the semianalytical predictions of the current approach (especially, in the second, more consistent approximation of the random walk theory, applied in Section 1.6) with results of numerical methods based on the Langevin equation (which, are, however, essentially more complicated and for this reason, still not completely reliable and often criticized in the literature).

CONCLUSION The new approach to the Brownian coagulation theory developed in the author‘s papers [5-7], is overviewed. The traditional diffusion approach to calculation of the collision frequency function for coagulation of Brownian particles is critically analyzed. In particular, it is shown that the diffusion theory [1, 2] is applicable only to the special case of coalescence between large and small particles, R1  r  R2 (where r  n

1 / 3

is the mean inter-

particle distance), and becomes inappropriate to calculation of the coalescence rate for particles of comparable sizes, R1 , R2  r . In the latter, more general case of comparable size particles, coalescences occur mainly in the kinetic regime (rather than in the diffusion one) characterized by random (homogeneous) spatial distribution of particles. This kinetic regime is realized in a wide range of the particles concentrations, obeying the basic assumption of the theory n1/ 3 R  1 , this allows calculation of the collision rate in various modes of the kinetic regime (continuum, free molecular and transition) within the same approach. In the continuum mode of the kinetic regime, corresponding to a  R , the calculated coalescence rate formally (and, in fact, fortuitously) coincides with the expression derived in

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A New Approach to the Theory of Brownian Coagulation …

37

[1, 2] for the diffusion regime that is relevant only in the particular case of large and small particles coalescences. This formal coincidence apparently explains a reasonable agreement of predictions of the kinetic equation derived in the traditional approach with experimental measurements of the Brownian particles coalescence rate. In the opposite range a  R , the standard free molecular expression for the collision frequency function is valid. It is shown that, despite the free molecular expression can be rigorously derived only in the case of very high collision frequency (higher than the particles jump frequency), it can be properly extended to the whole range of this kinetic mode,

a R  1 , corresponding to high Knudsen numbers, Kn  0.5 m  

1/ 3

n

1/ 3 m

Rm 

1

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(rather than Kn  1 often assumed in the literature). The transition interval a  R also belongs to the kinetic regime characterized by homogeneous spatial distribution of particles and, for this reason, can be described by the interpolation formulas derived within the new analytical approach with fitting parameters that can be specified numerically, avoiding semi-empirical assumptions of the existing models (connected with attempts of interpolating regimes of various nature, diffusion and kinetic, rather than two different modes of the kinetic regime as it is realized in the new approach). Numerical calculations of the coalescence rate at different values of the parameter a R are performed using the Monte Carlo method (by evaluation of the sweeping rate of randomly distributed immobile point markers by migrating Brownian particle) [7], generalizing the analytical approach applied to the limiting cases a  R and a  R [6], in two subsequent approximations of the random walk theory. The calculated points are approximated by analytical expressions obeying the general restraint, Eq. (1.19), derived in the current approach. The numerically calculated points were further used for verification of the available semiempirical interpolation formulas. However, the majority of these formulas do not obey the general restraint of the theory, Eq. (1.19), and for this reason, only a few of them can be used, including reformulated expressions of Dahneke and Fuchs-Sutugin and corrected (in [13]) expressions of Fuchs and Wright. Results of approximation with these semi-empirical formulas notably deviate from the calculated points near the border between the transition and continuum regimes, demonstrating a poorer agreement in comparison with the new (two- and four-parameter) interpolation formulas. The interpolation expressions obtained in the first approximation of the random walk theory (so called simple random walks with a fixed length) was verified in the second approximation considering random walks with stochastically distributed lengths. It was shown that the interpolation expression derived in the simple random walk approach represented in terms of the independent variables D12 , R12 and c12 reasonably describes (within the calculation accuracy) the new set of calculation points (obtained in the second approximation). The developed approach being applied to the problem of heavy vapor molecules condensation in a large trap, results in the improved description of the condensation rate in the transition regime.

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38

M.S. Veshchunov

2. Part 2. Diffusion-Limited Reaction Rate Theory 2.1. Introduction For many chemical processes, the reaction proceeds from a reaction complex formed by collision of two or more reactants. Each reaction rate coefficient K has a temperature dependency, which is usually given by the Arrhenius equation, K  K0 exp  Ea kT  , where the pre-exponential factor K 0 determines the collision frequency of reacting species and the exponential factor determines the number of collisions with energy greater than the activation energy E a of the complex (i.e. corresponds to the sticking probability of collisions).

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Diffusion-limited (or diffusion-controlled) reactions are reactions in which collisions of reactants (determining the pre-exponential factor K 0 ) are controlled by their diffusion migration in suspending solvent (rather than free-molecular collisions typical for molecular reactions in gas mixtures). Diffusion-limited reactions between two different species A and B (A + B  C, where C does not affect the reaction) show up in a vast number of applications including not only chemical (see e.g. [26]), but also biological (e.g. [27-29]) and ecological (e.g. [30]) processes that have been studied over many decades. This may apply also to the reaction of point defects, vacancies and interstitials (V + I  0), annihilation in crystals [31] produced by means of high-energy particles or electrons. A method for calculating the reaction rate of reaction partners migrating by threedimensional diffusion was developed in [32, 33] by generalization of the Smoluchowski theory for coagulation of colloids [1]. In this method the radius of the activated complex (or the ― reaction radius‖) corresponds to the ― influence-sphere radius‖ in the Smoluchowski theory (roughly equal to the sum of radii of two colliding Brownian particles, R12  R1  R2 ), which in the continuum approach is assumed to be large in comparison with elementary drift (or jump) distances a1 , a 2 of particles migrating by random walks,

R12  a1 ,a 2 . In the opposite limiting case, R12  a1 ,a 2 , the continuum diffusion approach is not anymore valid, therefore, the so-called ― free-molecular‖ (or ―b allistic‖) approximation can be used for colliding Brownian particles [4]. Formulating a reaction-diffusion model, a d-dimensional Euclidean space on which A and B particles at initial average densities (number of particles per unit volume) n A and n B diffuse freely, is usually considered in the continuum approach (see, e.g. [34-36]). In this approach reactant particles are represented as points or spheres undergoing spatiallycontinuous Brownian motion, with bimolecular chemical reactions, A + B  C, occurring instantly when the particles pass within specified reaction radius R AB between their centers. The continuum approach was further applied to the diffusion-limited reactions in one (1d) and two (2-d) dimensions (see, e.g. [37, 38]), the latter case has wide applications also in the membrane biology (see a review in [39]). The diffusion-limited bimolecular reactions between mobile vacancies and interstitials in strongly anisotropic crystals provided the mobile species is constrained to migrate in one plane only, may be also well approximated

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A New Approach to the Theory of Brownian Coagulation …

39

over a wide range of the reaction by a 2-d second order rate equation [40]. Another example of the 2-d model application is coalescence of intergranular voids on grain faces of irradiated metals or ceramics (e.g. in the practically important case of UO2 nuclear fuel) [41, 42]. In such approach, the same shortcomings of the Brownian coagulation theory that were critically analyzed in the author‘s papers [5-7] (see Part 1), are generally inherited in the diffusion-limited reaction rate models. Namely, the diffusion approach [1, 2] to calculation of the collision rate function (based on assumption that the local collision rate should be equal to the diffusive current of particles) is applicable only to the special case of coalescence between 1 / 3 is the mean interlarge and small Brownian particles, R1  r  R2 (where r  n

particle distance), and becomes inappropriate to calculation of the coalescence rate for particles of comparable sizes, R1 , R2  r . Correspondingly, the traditional approach to the diffusion-limited reaction rate theory based on a similar assumption that the local reaction rate should be equal to the diffusive current of particles, becomes invalid in the case when the characteristic reaction distance R AB for AB complex formation (i.e. the reaction radius), is small in comparison with the mean inter-particles distances, R AB  rA , rB , where

rA  n A1 / 3 , rB  n B1 / 3 (see Section 2.2.1). The new approach developed in [5-7] was generalized in the author‘s paper [43] to the case of diffusion-limited reaction kinetics. For the base case of continuum mode, a A , a B  R AB  rA , rB , the reaction rate calculated in the new approach in 3-d (see Section 2.3.1) formally (and in fact, fortuitously) coincides with the traditional result, valid only for reactions with a large reaction radius, rA  R AB  rB . However, for the base case

a A , a B  R AB  rA , rB in 2-d the traditional approach leads to considerable deviations of

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the reaction decay ni t  at large times t from that calculated in the new approach (see

Section 2.4), thus explicitly demonstrating inconsistency of the traditional approach. In the case R AB  a A , a B , the free molecular (or ballistic) regime is realized. This case can be also considered similarly to the Brownian particles coagulation problem in the corresponding regime, as well as the case of the transition regime, R AB  a A , a B (Section 2.3.2). The new approach was further generalized to consideration of reaction kinetics for particles migrating by random walks on discrete lattice sites (with the lattice spacing a ) in the author‘s paper [44]. Since the case of large reaction radius, R AB  a , is properly reduced to the continuum media limit, the opposite case, R AB  a , with reactions occurred when two particles occupy the same site (see, e.g. [45]), is of the most concern. It will be shown (in Section 2.5) that the traditional approach [38, 45] to consideration of this important case preserves the main deficiencies of the continuum media approach and thus results in erroneous predictions for the reaction kinetics. For this reason, new relationships for the reaction rate constants will be derived either for 3-d (in Section 2.5) or 2-d lattices (in Section 2.6). The main outcomes of the new approach to diffusion-limited reaction rate theory are formulated in Section 2.7.

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M.S. Veshchunov

2.2. Rate Equations In the approximation R r  1 , only pair-wise collisions of particles during their diffusional migration can be taken into consideration, and collisions which occur among any combination consisting of more than two particles, can be ignored. In the rate theory for a continuous distribution of particles N R dR , the number of particles of radius R to R  dR per unit volume, under an assumption that collided particles



of radii R1 and R2 immediately coalesce to form a new particle of radius R13  R23 Smoluchowski coagulation equation takes the form [1]





1/ 3

, the



1/ 3 N R, t  1    N R1 , t N R2 , t  R  R13  R23   R1 , R2 dR1dR2 t 200 



 N R, t  N R1 , t  R, R1 dR1 , 0

where  R1 , R2 

(2.1) is the collision frequency function which, being defined as the collision

frequency between two particles randomly located in the unit volume, does not depend on time explicitly. For this reason, N R, t  t should be calculated from consideration of pairwise collisions during a relatively short time step t when variation of concentration densities N R1 , t  and N R2 , t  can be neglected on the one hand, and t being long

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enough to attain the steady state value of  R 1 , R2  during this time step, on the other hand.

For the kinetics of an irreversible reaction A + B  C in the mean-field approximation, Eq. (2.1) being applied to the two-size ( R A and RB ) particle distribution function, is reduced to the form

dnA dnB    K ABn A t nB t  , dt dt

(2.2)

where n A and n B are the mean concentrations of reacting A and B atoms, respectively, and

K AB is a rate function, directly corresponding to the collision frequency function  for two

particles of different types (A and B), K AB   R1 , R2  R1  R A  R2  RB  . In accordance with the Smoluchowski rate theory, K AB is defined as the collision frequency of

two particles randomly located in the unit volume and for this reason, it should be considered as a value explicitly independent on time. In a self-consistent approach, the reaction rate dn A dt should be calculated choosing the time step dt that is short enough to neglect variation of the mean concentrations n A and n B in dt , and long enough to attain a steady state value of K AB dt   const  K AB . This is the main difference from the traditional

models for diffusion-limited reaction kinetics (despite they are often termed as the Coagulation: Kinetics, Structure Formation and Disorders : Kinetics, Structure Formation and Disorders, Nova Science Publishers, Incorporated,

A New Approach to the Theory of Brownian Coagulation …

41

Smoluchowski-type models), where, under assumption that the local reaction rate should be equal to the diffusive current of particles, the ― effective‖ reaction rate is calculated as an explicitly time-dependent function K AB t  (rather than K AB dt   const in the Smoluchowski theory). Similarly to analyses of the coagulation problem in Part 1, it will be shown below that this difference is connected with inadequate application of the diffusion approach to calculation of the effective reaction rate (as the diffusive current of particles) for particles with a relatively small reaction radius, R AB  rA , rB , that becomes especially critical in the 2-d case. Such an approach being valid in the case of small particles A diffusing into large circular traps B (so called agglomeration), rA  R AB  rB (with time-dependent K t  properly standing in the reaction rate equation), fails in the base case, R AB  rA , rB .

2.2.1. Applicability of the Diffusion Approach to Particles Collisions The diffusion equation for an ensemble of particles is derived (similarly to consideration of other relaxation processes in weakly inhomogeneous fluids, such as the heat transfer or viscous flow) in the quasi-equilibrium approximation. In this approximation the particles distribution function is considered to be in a local thermodynamic equilibrium, smoothly varying in space and in time following smooth variations of the fluid macroscopic parameters (e.g. temperature, pressure, concentration, velocity). In the case of the mass transfer problem (i.e. the diffusion equation) the varying macroscopic parameter is the number concentration of  particles, n r , t  .



Consideration of n r , t  as a macroscopic value (i.e. when its thermodynamic

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n 2

 n ) is valid only if the size  ~ 3 of the elementary volume V  L , with respect to which n r  is defined, is large enough  in comparison with the local inter-particle distance, L  n 1/ 3 r  , that in its turn must  exceed the minimum inter-particle distance equal to the particles size, n 1/ 3 r   2R . For fluctuations are small in comparison with its value,

this reason, only heterogeneities of particles spatial distribution on the length scale of

l  L  n 1 / 3  R can be adequately considered in the continuous diffusion approach, under an additional condition for the elementary drift distance, a  l (see Part 1, Section 1.2). In the case when identical particles (say, of type A with radius R A ) are distributed at random throughout a medium of infinite extent with the mean bulk concentration n A that obeys the dilution condition nA RA3  1 , the particles can be considered as point objects ( R A  rA , where rA  n A1/ 3 is the mean inter-particles distance), which, in accordance with the diffusion equation for an ensemble of point-wise particles, tend to relax with time to a homogeneous spatial distribution (cf. Part 1, Section 1.3.1). The situation critically changes in the case when a group of B-type traps with a relatively large ― influence-sphere‖, or reaction (with A-particles) radius R AB  R A and concentration

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42

M.S. Veshchunov

3 n B (obeying nB RAB  1 ) appears in the ensemble of A-particles. B-type traps cannot be 3 treated as point objects, if n A RAB  1. In this case traps should be considered as

macroscopic with respect to A-particles, since the reaction radius R AB is much larger than the mean inter-particle distance rA  n A1/ 3 , and just for this reason additional boundary conditions for diffusion of A-particles emerges on traps surfaces. The induced by these boundary conditions heterogeneities in the spatial distribution of A-particles do not tend to disappear with time, as it was in the previous case (without traps), and the steady state concentration profiles of A-particles around macroscopic trap centers,

2 n A r   n A R AB   n A  n A R AB   1  R AB r  , are attained at t  RAB D A [2]. The

diffusion flux of A-particles in this concentration profile calculated at the reaction radius, J dif  4DA RAB n A  n A RAB   4DA RABn A , if n A R AB   n A  n A , determines the accumulation rate of A-particles in a B-trap, and, following consideration in [1, 2], the collision frequency function between A and B particles, taking into consideration migration of traps with the diffusivity D B , eventually takes the form ( dif ) K AB  4DAB RAB ,

(2.3)

where D AB  D A  DB . For determination of the applicability range of this result, it should be noted that the characteristic size l of the zone around a large trap in which A-particles concentration varies from the value n A R AB   n A near the reaction surface to the value of the same order of

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magnitude as the mean value n A attained at large distances from the centre, is comparable

with R AB , i.e. l  R AB . This value must naturally exceed the mean distance nA1/ 3 RAB  between small A-particles in the vicinity of a B-trap surface,

RAB  l  nA1/ 3 RAB   nA1/ 3 (in order to maintain the concentration profile of small

3 particles around the trap), or n A R AB  1 . This condition logically coincides with the (above

mentioned) general requirement to applicability of the diffusion approximation, l  n A1/ 3 . This condition can be confirmed more strictly taking into consideration that the diffusion

flux at the reaction surface, J dif RAB   n A r

r  RAB

can be properly calculated only under assumption

 n A RAB  r   n A RAB  r , r  R AB . In this position

n A R AB  r   n A 2 R AB   n A 2 and thus the mean inter-particle distance in this zone

can be evaluated as r  n A RAB  r   n A 2RAB   n A 2 . On the other hand, it should be small enough to maintain the concentration profile in this spatial range (where the 1/ 3

1/ 3

1/ 3

diffusion flux is calculated), r  r  R AB , or n1A/ 3 RAB  1 . The same conclusion can be derived also in the Fokker-Planck approach based on consideration of the probability density of migrating Brownian particles (see Appendix).

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Therefore, the traditional diffusion approach, that stipulates that the local reaction rate should be equal to the diffusive current of A-particles into the traps (see, e.g. [38]), is valid only for reactions with the large reaction radius, RAB  rA  nA1/ 3 . From this analysis it can be seen that the intrinsic reason for steady-state heterogeneities in the small particles spatial distribution is connected with the additional boundary conditions (for these particles diffusion equation) induced by macroscopic (i.e. large scale, R AB  rA ) traps. These macroscopic boundary conditions vanish as soon as the reaction radius becomes comparable with the size of small particles ( R AB  R A  rA ), eliminating the driving force for emergence of steady-state spatial heterogeneities. Indeed, in the opposite case R AB  rA , rB , the limit of the point-wise particles restores, which is characterized by the tendency for the system of two type (A and B) particles to a homogeneous spatial distribution (or mixing) owing to their diffusion migration (in the absence of macroscopic boundaries).

2.2.2. Diffusion Mixing Condition In fact, reactions between point-wise particles induce l local heterogeneities in the particles spatial distribution on the length scale of their mean inter-particle distance, which in the case n A  n B  n is evaluated as r  n 1 / 3 . However, such kind of heterogeneities quickly disappear owing to rapid diffusion relaxation of particles on the length scale of their mean inter-particle distance r with the characteristic time

 d  r 2 6D (under simplifying

assumption D A  D B  D ), that is generally much shorter in comparison with the

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characteristic time

 c  K AB n  of particles concentration variation,  d   c , as will be 1

explicitly shown below either in the 3-d or in 2-d cases (in Sections 2.3 and 2.5, respectively). This allows consideration of a random distribution of particles attained during a time step

 d  t   c , chosen for calculation of the reaction rate in Eq. (2.2). In this case (corresponding to the kinetic regime) the spatial distributions of the particle  centers n A, B r , t  can be considered as homogeneous functions characterized by their mean



concentrations n A, B t  , i.e. n A, B r , t   n A, B t  , slowly varying with time owing to the

particles collisions (reactions). Respectively, the collision probability is also a spatially uniform function that can be properly calculated as the collision frequency of two particles of different types (A and B) randomly located in the unit volume.

In the case n A  n B , which at large times ( t  K AB n A 0  nB 0 ) unavoidably 1

turns to n A t   n B t  , or rA t   rB t  , each particle B can be surrounded by a sphere

r  rB t  , where a collision of this r obeying rA t   ~ (or a circle in 2-d) of radius ~

particle B with one of surrounding particles A (with the given concentration n A t  ) will take

r  rB t  , no other particles B can be considered in this sphere, therefore, place. Owing to ~

homogenization of the reaction system in t (after reactions in the previous time step) will be Coagulation: Kinetics, Structure Formation and Disorders : Kinetics, Structure Formation and Disorders, Nova Science Publishers, Incorporated,

44

M.S. Veshchunov

determined by relaxation (or diffusion mixing) of particles A (from this sphere) on the length 1 / 3 scale of their mean inter-particle distance rA  n A , i.e. by the diffusion time

 d  rA2 6D .

Apparently, this conclusion will not be violated in the case D A  DB , however, becomes invalid in the opposite case D A  DB . In this latter case mixing of particles A will be incomplete and thus the accuracy of the model predictions will be reduced. However, owing to stochastic character of particles movement and collisions, local heterogeneities (―m issing particles‖) induced by reactions between particles A and B will be randomly distributed in space, therefore, the mean collision frequency can be still considered as a spatially uniform function, but averaged over a larger scale. This implies that at least in the mean field approximation (i.e. in the large-scale limit) the current approach can be applied with a reasonable accuracy. Nevertheless, in further analysis D A  D B  D will be assumed for simplicity (and for possible generalization of the theory to consideration of concentration fluctuations, see next Section 2.2.3). The characteristic times of particles concentration variation are different for particles A and B,

 c( A, B )  K ABn A, B  , hence, the smaller one should be chosen in evaluation of the 1





1 1 time step, t   c  min  c( A) ,  c( B )  K AB nA .

Therefore, assuming for definiteness n A  n B (and D A  D B  D ) in further analysis, the

mixing

condition

can

 d  r 6D  t   c  K n 2 A

1 AB

be

generally

represented

in

the

form

1 A .

2.2.3. Applicability of the Reaction Rate Equation

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As explained above, K AB is defined as the collision frequency of two particles of different types (A and B) randomly located in the unit volume. This implies that the size of

~

3 the unit volume V  L , with respect to which K AB is defined, is large in comparison with

the minimum distance between particles of different type A and B, L  R AB . In this case, if there are n A particles of type A and n B particles of type B distributed at random through a sample of the unit volume, the number of collisions between A- and B-particles in the unit time (that defines the reaction rate) reduces to K AB n A n B . This definition of the reaction rate can be apparently extended to the case of spatial heterogeneities in distribution of A and B particles, if these heterogeneities are smooth on the length scale of the (adequately defined) unit volume, l  L  R AB . In this case the number of collisions in dt between A- and B-particles located in this volume is calculated as

  K ABnA r , t nB r , t d 3rdt , resulting in the local balance equations for the particles numbers     n A r , t   n B r , t    K AB n A r , t n B r , t  .

(2.4)

It is important to note that formal extension of Eq. (2.4) to consideration of heterogeneities of a small scale l  R AB , often performed in the traditional approach, is Coagulation: Kinetics, Structure Formation and Disorders : Kinetics, Structure Formation and Disorders, Nova Science Publishers, Incorporated,

A New Approach to the Theory of Brownian Coagulation …

45

beyond its applicability range (where K AB is defined, as explained above), and for this reason the obtained in this limit equation looses its original physical sense. Relaxation of spatial fluctuations in the particles distribution can be taken into consideration by the additional diffusion term in the r.h.s. of Eq. (2.4),

    n i r , t   Di ni r , t   K AB n A r , t n B r , t  , i = A, B,

(2.5)

since the diffusion term is defined on the length scale of l  n 1/ 3  Ri (as explained above), in a self-consistent manner with the local collision rate definition, l  RAB  Ri . This allows extension of the reaction rate theory applicability beyond the mean-field approximation, Eq. (2.2), however, only for fluctuations with long wavelengths, l  R AB . Available in the literature [34-36] results of analysis of Eq. (2.5) demonstrate that effect of ― renormalization‖ of K AB by concentration fluctuations (leading to a similar to Eq. (2.3) result) occurs on the length scale of the reaction radius, l  R AB , that is beyond the cut-off limit of Eq. (2.5), l  R AB . This additionally confirms the above derived conclusion that the results of the traditional approach are grounded only in the case of reactions with a large reaction radius, rA  R AB  rB , where short wave-length fluctuations with l  R AB (and

rA  l ) in the spatial distribution of A-particles around B-particles can be adequately described by Eq. (2.5). However, in the opposite case, rA , rB  R AB , such short wave-

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length fluctuations are beyond the cut-off limit of the theory, therefore, the kinetic approach based on Eq. (2.4) can be generally used as the lowest order approximation. In the next order approximation, taking into consideration long wavelength fluctuations, l  R AB , in Eq. (2.5), predictions of the kinetic approach may be violated at large times. For instance, in the case of equal initial concentrations, n A 0   n B 0  , the asymptotic

( t   ) decay rate n A, B becomes slower as compared with predictions of the mean-field theory (valid at intermediate times) [46, 47]. Therefore, further improvement of the kinetic approach can be attained in this case by additional consideration of long wave-length fluctuations, e.g. using the scaling and renormalization group methods (cf. [48, 49]), that is beyond the scope of the current paper.

2.3. Reaction Rate in 3-D Case As explained in Section 2.2.2, in order to calculate the local reaction rate in the kinetic regime, Eq. (2.4), a time step t relatively large in comparison with the diffusion relaxation (or mixing) time should be chosen,

t   d  n2 / 3 6D , in order to sustain the main

condition of the kinetic regime for random (homogeneous) distribution of reacting particles (where n  n A  n B and D A  D B  D are assumed, cf. Section 2.2.2). On the other hand, the time step should be small in comparison with

 c  K AB n  , i.e. t   c , that allows 1

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46

M.S. Veshchunov

neglecting variation of the mean concentrations n A and n B in t . Besides, some additional

condition for the time step should be valid, t  ~ , in order to attain a steady state value of

K AB t   const  K AB , where ~ will be evaluated below.

Following consideration in Part 1 (Section 1.5), let us consider two particles of types A and B located at random through a sample of unit volume. The first (― parent‖) particle of type A can be surrounded by a sphere with the reaction radius R AB . If the second particle centre is located in this exclusion zone, reaction would occur. As shown in [1, 2], the relative displacements between two particles describing diffusion motions independently of each other and with the diffusion coefficients D A and D B also follow the law of diffusion motion with the diffusion coefficient D A  D B . Therefore, in order to calculate the probability of collisions between the two particles, one can equivalently consider the second particle as immobile whereas the first one migrating with the effective diffusion coefficient D AB  D A  DB  2 D . In this approximation it is assumed that the effective (mobile) particle jumps to an elementary distance a AB in random directions with a frequency

 AB   01 , obeying the

2 relationship for the particle diffusivity from the theory of random walks, DAB  a AB 6 0 .

As a result of a jump, the exclusion zone also relocates to the distance a AB and opens the possibility that the second (immobile) particle with its centre located in a zone with the 2 volume V0  RAB a AB , may be swept out by the mobile particle (cf. Fig. 1.1). Depending

upon the ratio between R AB and a AB , particles migration can be considered in the continuum mode if R AB  a AB , or in the free molecular mode if R AB  a AB , with different results Copyright © 2012. Nova Science Publishers, Incorporated. All rights reserved.

for the collision rate.

2.3.1. Continuum Mode ( a A , a B  R AB  rA , rB ) During the time step t   0 the mobile particle makes many jumps, k  t

 0  1,

in random directions, however, the total swept zone volume V , that determines the probability of the two particles collision in t , will be smaller than k V0  V0t

0 ,

owing to strong overlapping of the swept zone segments at a AB  R AB . This limit corresponds to the continuum mode of the kinetic regime, characterized by a random spatial distribution of particles (quickly reinstated during the time step). Under this basic condition, the probability to sweep a B-particle in the unit time is reduced to V t n B , if there are

n B B-particles randomly distributed per unit volume. Therefore, the number of collisions

V t n An B

between A and B particles in the unit time, if there are n A A-particles

randomly distributed per unit volume, will be smaller than V0n AnB

0 .

In order to calculate the volume V swept in t , let us uniformly (at random) fill up the

space with auxiliary point immobile particles (―m arkers‖) of radius R*  0 with a relatively Coagulation: Kinetics, Structure Formation and Disorders : Kinetics, Structure Formation and Disorders, Nova Science Publishers, Incorporated,

A New Approach to the Theory of Brownian Coagulation …

47

3 high concentration, n*  R AB (following consideration in Part 1, Section 1.5.1). To

facilitate adequate resolution of a fine structure (with the characteristic length of a AB  R AB ) of the swept zone, the markers concentration n* should additionally obey the condition that the number of swept markers

N*(0) during one jump must be large,

2 2 a AB  . In this case the swept volume can be a ABn*  1 , or n*  RAB N*(0)  RAB 1

calculated as the total number N * of the swept markers divided by their concentration,

V  N * n* . In its turn, for the same reasons (concerning relative displacements of diffusing particles), calculation of the sweeping rate of randomly distributed immobile markers by a large particle of radius R AB migrating with the diffusivity D AB is equivalent to calculation of the condensation rate of the mobile markers migrating with the diffusivity D AB in the immobile trap of radius R AB (see Appendix). 3 Owing to n* RAB  1, this problem of the (point-wise) markers condensation in the large (macroscopic) trap can be adequately solved in the continuum approach of [1, 2], as above explained in Section 2.2.1. In this approach the total number of swept markers in t is

equal to



tDAB [4], and the swept volume per unit

V t  n N* t   4DABRAB , if the time step is sufficiently large, 2 ~ t    16RAB DAB . The spatial variation of the markers concentration occurs on the length scale l which is comparable with R AB (see Section 2.2.1), i.e. l  R AB . In accordance with the additional condition of the diffusion equation applicability, a AB  l , this result is valid only in the (considered here) case a AB  R AB . In this case, the number of collisions V t n A n B between A and B particles in the unit time becomes equal to 4 D A  DB R AB n A n B , that 1 *

time is equal to

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

N*  4DAB RABn*t 1  4RAB

results in

K AB  4D AB R AB . It

is

straightforward

 c  t   d  n

2 / 3

to

see

that

the

(2.6) first

restriction

on

the

time

step,

6D , can be applied if the mixing condition  c   d , or

n1/ 3 RAB  3 2 D DAB   3 4 , is valid, that is in agreement with n1/ 3 RAB  1 . The second restriction t  ~  16R2 D , can be applied owing to   ~ , or AB

AB

c

1/ 3

n RAB  1 4 , which is practically indistinguishable from the basic condition

n1/ 3 RAB  1 , within the accuracy of the characteristic times evaluation. Therefore, the correct expression for the reaction rate, Eq. (2.6), derived in the kinetic regime (by consideration of uniform (random) spatial distribution of reacting particles) for the

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48

M.S. Veshchunov

case of a relatively small reaction radius, R AB  rA , rB , coincides with the traditional expression derived in the diffusion regime (by consideration of concentration profiles and diffusive currents of particles) valid in the case of a large reaction radius, rA  R AB  rB , however, this coincidence is fortuitous and, probably, reflects some internal symmetry in the considered system in 3-d.

2.3.2. Free Molecular Mode ( R AB  a A , a B ) In the opposite case a A , a A  R AB one can neglect the mean relative volume of the swept zone segments intersections (cf. Part 1). In this approximation the swept volume per unit time, V t , is a constant value equal to the ratio of the swept volume per one jump to the jump period,

V0  0 , which can be calculated in the free molecular approach.

Correspondingly, the total swept volume V (after k  t

 0  1 jumps) is equal to

k V0  V0t  0 , and the number of coalescences V t n A n B between A and B particles (of masses

m A and mB , respectively) in the unit time is equal to

this case is equal to V0

 0 that coincides with the free molecular expression

( fm ) 2 , in 8kTmA1  mB1  . Therefore, the kernel of Eq. (2.1), K AB n AnB V0  0  n AnB RAB

( fm ) 2  V0  0  RAB 8kTmA1  mB1  . K AB

(2.7)

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This case is applicable for the reaction particles suspended in a fluid, when R AB is a small ( R AB  a A , a B ), but non-negligible parameter in comparison with the mean intermolecular distance of the suspending fluid, RAB  rm  nm1/ 3 , where nm is the fluid molecules concentration. This latter value determines the minimum distance dr  nm1/ 3







between two possible positions of a particle centre, r and r  dr , and thus allows definition of the swept volume (or area) for migrating particles. In the intermediate range a AB  R AB for reaction particles suspended in a fluid the so called transition regime is realized that can be described by the interpolation expression derived within the new analytical approach with fitting parameters specified numerically (Part 1, Section 1.5.4).

2.4. Reaction Rate in 2-D Case Similarly to the 3-d case, the problem of calculation of the area sweeping rate S t by an effective particle of radius R AB migrating with the diffusivity D AB  D A  D A  2 D (where D A  D B is assumed, cf. Section 2.2.2) in a plane can be properly reduced to consideration of point markers randomly distributed in the plane with the concentration

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n*  RAB  , migrating with the diffusivity D AB into an immobile trap of radius R AB . 2

The markers condensation rate can be calculated using a well-known analogy with the heatconduction problem in the cylindrical geometry [50]. As a result, the total number of swept markers

in

t

is

equal



2 , N*  4DABn*t ln 4DABt RAB

to



if

2 1 1 1 RAB 4DAB  t   c  K AB  min n A1 , nB1  K AB n A (where n A  n B is assumed) and

t obeys the diffusion mixing condition,  d  t . Contrary to the 3-d case, the sweeping 1 rate S t   n* N * t  in this case is a function of the time step even for very large t ,

however, this dependence is weak and can be neglected with the logarithmic accuracy. Indeed, an expression ln  xX  can be approximated as ln  xX   ln X  ln x  ln X in the case X  x  1 (and thus ln X  ln x  0 ). Therefore, choosing the time step as 2 RAB 4DAB  ~  t   c , that, under additional condition 2 ~ RAB 4DAB    c ~ ,

(2.8)





2 can be also represented in the form 0  ln t ~  ln  c ~  ln 4DAB~ RAB , one obtains

2  lnt ~  ln4DAB~ RAB2  . In this approximation the ln 4DABt R   ln 4DAB~ RAB sweeping rate can be calculated as S t   n 1 N t   4D ln 4D ~ R2 . The 2 AB

*

*



AB

AB

AB



number of collisions S t n A n B between A and B particles in the unit time becomes equal to  4D n n ln 4D ~ R2 , that corresponds to K  4D ln 4D ~ R2 and thus AB A B



AB

AB



AB



AB

2  4DABnA (if n A  n B is specified).  c  K n  ln4DAB~ RAB

AB

AB



1 1 AB A

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Substituting

this

expression

c

for

into

Eq. (2.8),

one

obtains

2  RABrA 4DAB , where rA  nA 1/ 2 ; this allows specification ~  ln 4DAB~ RAB 2 2 ~  rA2 4DAB (owing to rA RAB   ln rA RAB  , if rA R AB  1 ), that apparently obeys the necessary condition R 2 4D  ~   . Eventually one obtains for the 1/ 2



AB



AB

c

reaction rate in the mean-field approximation 2 , K AB  4DAB ln rA2 RAB

(2.9)

that depends on time implicitly (via rA  n A t  ), rather than explicitly, as obtained in the traditional approach. In the particular case n A  n B  n (or rA  rB  r ), ~ practically coincides with 1/ 2

 d  r 2 4DAB , thus, t self-consistently obeys the necessary condition  d  ~  t . In







2 2 this case the reaction rate is reduced to K AB  4DAB ln r 2 RAB   4DAB ln nRAB







2 (rather than K AB  4DAB ln 4DABt RAB in the traditional approach) and eventually results in solution of the reaction rate equation,

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50

M.S. Veshchunov 2   4D t , 1  ln nRAB AB n

(2.10)

2 that at large times, t  RAB 4DAB (before crossover to the asymptotic behaviour at t   , discussed in Section 2.2.3), is close to the traditional solution, 2  4DABt . n  ln 4DABt RAB However, in the case n A  n B situation critically changes. In this case the initial relationship n A 0   n B 0  at large times turns to n A t   n B t  , or rA t   rB t  , and

the solution of the reaction rate equation (at t  K AB n A 0  nB 0 ) results in the exponential drop of the concentration, n B t   exp  Ct  , (2.11) 1





where C  2DAB nA 0  nB 0 ln ~ rA RAB , and ~rA is the final value of rA t  , which

variation rA t   ~ rA , rA  rA t  at large times, when it approaches to ~rA , i.e. rA t   ~ is neglected in the expression for C in Eq. (11) with the applied logarithmic accuracy,

ln rA   ln ~ rA  rA   ln ~ rA   ln 1  rA ~ rA   ln ~ rA ).

The

obtained

solution,

Eq. (2.11), is much steeper in the current approach in comparison with that in the traditional approach n B t   exp  C1t ln t  ([51-53]), and thus the concentration decay rate n B is

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strongly underestimated at large times in the traditional approach. This additionally confirms the importance of the new approach to calculation of the reaction rate in 2-d. Consideration of the free molecular regime in 2-d can be performed by straightforward generalization of the 3-d case approach (presented in Section 2.3.2).

2.5. Reaction Rate on 3-D Discrete Lattice Particles migrations by random walks on discrete cubic lattice sites can be considered in two limits, R AB  a and R AB  a . In the case of large reaction radius, R AB  a , the problem is properly reduced to the continuum media limit considered in Section 2.3.1. In the opposite case, the reaction radius R AB is assumed to be small in comparison with the lattice spacing (corresponding to the elementary jump distance, a  a A  a B ), and reactions occur when two particles occupy the same site (see, e.g. [45]). In this case R AB is the minimum length scale of the problem and can be excluded from consideration. This situation is qualitatively different from the above considered free molecular regime (for reaction particles suspended in a fluid), in which R AB is also small ( R AB  a A , a B ), but non-negligible parameter ( RAB  rm  nm1/ 3 , where nm is the fluid molecules concentration). We start at t  0 with randomly distributed A and B particles on discrete cubic lattice sites, with mean concentrations n A and n B , respectively; n A, B a 3  1 . Each particle moves

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by jumps to nearest-neighboring sites, the average time interval between successive jumps being  A and  B , respectively; thus all particles perform independent random walks, with the associated diffusion coefficients

DA, B  a 2 6 A, B . Again, n  n A  n B

and

D A  D B  D , will be considered in further analysis (cf. Section 2.2.2). Similarly to the above considered continuum limit, reactions between A and B particles induce local heterogeneities in the particles spatial distribution on the length scale of the mean inter-particle distance rA  n 1/ 3  a . However, such kind of heterogeneities quickly disappear owing to rapid diffusion relaxation of particles on the length scale of their mean inter-particle distance rA with the characteristic time shorter in comparison with the characteristic time variation,

 d  rA2 6D , that is generally much

 c  K AB n  of particles concentration 1

 d   c . This allows consideration of a random distribution of particles attained

during a time step

 d  t   c , chosen for calculation of the reaction rate (owing to

 d  t ), and neglecting variation of the mean concentrations n A and n B in t (owing to

t   c  min  c( A) , c( B)  ).

In this case (corresponding to the kinetic regime) the spatial distributions of the particle  centers n A, B r , t  can be considered as homogeneous functions characterized by their mean



concentrations n A, B t  , i.e. n A, B r , t   n A, B t  , slowly varying with time owing to particles

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collisions (reactions). Respectively, the collision probability is also a spatially uniform function that can be properly calculated as the collision frequency of two particles of different types (A and B) randomly located in the unit volume and migrating with diffusivities D A and

D B , similarly to the continuum limit consideration. In its turn, this problem can be readily reduced to calculation of the collision probability between two particles, randomly located in the unit volume, one of which is immobile (say, particle B) and another (particle A) is mobile, migrating with the effective diffusivity D AB  D A  DB . However, there is also an important difference with the continuum limit. Indeed, in the continuum limit the probability of the two particles collision in t was calculated as the mean volume swept by the mobile particle (of radius R AB and diffusivity D AB ). Instead of this, in the discrete lattice limit the collision probability in t k is determined by the mean number of distinct sites visited by a k-step random walk of the mobile particle (so called the range of the random walk, S k ), where k  tk

 AB  tk 6DAB a 2  1 .

The mean value of S k can be calculated (in the case of a simple 3-d cubic lattice) as [5456]





Sk  0.718 k  0.729 k 1/ 2  O(1) ,

(2.12)

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52

M.S. Veshchunov

which for the chosen time step

a 2 6DAB  r 2 6DAB   d  tk   c , that

corresponds to k  1 , can be reduced to

Sk  0.718 k ,

(2.12a)

K AB  a 3Sk tk  4.3DABa .

(2.13)

and results in

A formally similar to Eq. (2.13) result was obtained in [38] (following [45]). In that approach the problem was also reduced to consideration of collisions between two particles A and B on discrete lattice sites, however, basing on additional (unjustified) assumptions. Namely, instead of consideration of rapid diffusion mixing of particles (as proposed in the current approach) that allows rigorous reduction of the multi-particle problem to consideration of two-particle collisions and direct calculation of the reaction rate constant, an additional setup (or Ansatz) for the reaction rate constant in the multi-particle system was applied in [38] that eventually resulted in a different (apparently erroneous) numerical factor in Eq. (2.13). Therefore, one can conclude that the currently developed approach can be generalized to consideration of the reaction kinetics on a 3-dimensional lattice, resulting in the new relationship for the reaction rate constant, Eq. (2.13).

2.6. Reaction Rate on 2-D Discrete Lattice

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Reaction rate for particles A and B migrating by random walks on discrete square lattice sites ( n A  nB  n  a 2 ), when the reaction radius is small in comparison with the lattice spacing, R AB  a , can be calculated in a similar to 3-d approach (presented in Section 2.5) using the logarithmic approximation (presented in Section 2.4). As a result, an equation (corresponding to Eq. (2.13) in 3-d case) for the reaction rate constant takes the form

K AB  a 2 Sk tk ,



(2.14)



where Sk   k log k  O k log k is the mean number of distinct square lattice sites 2

visited by a k-step random walk [54-57], k  tk diffusion

mixing

in

t k ,

the

 AB  tk 4DAB a 2 and, to provide

calculation

time

step

is

chosen

as

1 1 n . With the logarithmic accuracy one a 2 4DAB  n 2 4DAB  tk   c  K AB

obtains

K AB  4DAB log 1 na2  ,

(2.15)

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which depends on time implicitly (via n t  ). In the case n A  n B at large times this time dependence is weak and can be neglected with the applied logarithmic accuracy

K AB  4DAB log 1 n~a 2 ,

(2.16)

~ is the final value of n t  , which variation is small, n t   n t   n~  n~ , and where n

1 1 1 1 lognt a 2   logn~  n a 2   logn~a 2   log1  n n~  logn~a 2  , ~a 2 1  1  log1  n n~ . since logn

thus,

Similarly to the continuum limit in 2-d (considered in Section 2.4), the reaction rate





constant differs from that calculated in the traditional approach (with log DABt a 2 instead



~a of log 1 n

2

 in the denominator of Eq. (2.16)) and thus predicts much higher decay rate

n A, B at large times in comparison with the traditional approach [38, 45].

CONCLUSION The new approach of the author to the diffusion-limited reaction rate theory [43] that is developed on the base of a similar approach to consideration of Brownian coagulation, proposed in the author‘s papers [5-7] (see Part 1), is overviewed. The traditional diffusion approach to calculation of the reaction rate is critically analyzed. In particular, it is shown that the diffusion approach is applicable only to the special case of reactions with a large reaction Copyright © 2012. Nova Science Publishers, Incorporated. All rights reserved.

1 / 3 1 / 3 are the mean inter-particle radius, rA  R AB  rB (where rA  n A , rB  n B

distances), and becomes inappropriate to calculation of the reaction rate in the case R AB  rA , rB . In the latter, more general case of the small reaction radius, particles collisions occur mainly in the kinetic regime (rather than in the diffusion one) characterized by homogeneous (at random) spatial distribution of particles. Homogenization of particles distribution occurs owing to particles diffusion mixing on the length scale of the mean inter-particles distance with the characteristic diffusion time being small in comparison with the characteristic reaction time,  d   c . In

the

a A , a B  R AB

continuum mode of the kinetic regime corresponding to  rA , rB , where a A , a B are the elementary drift distances of particles

migrating by random walks, the calculated reaction rate in 3-d formally (and, in fact, fortuitously) coincides with the expression derived in the traditional approach (that is relevant only in the particular case of reactions with a large reaction radius, rA  R AB  rB ). This formal coincidence apparently explains a reasonable agreement of predictions of the kinetic equation, derived in the traditional approach, with experimental measurements for 3-d reaction systems.

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54

M.S. Veshchunov

However, in the 2-d geometry, corresponding to reactant particles migration constrained in a plane, the traditional approach notably underestimates the concentration decay rate n A, B at large times in comparison with predictions in the new approach, explicitly demonstrating inadequacy (in application to the base case R AB  rA , rB ) of the traditional approach based on stipulation that the local reaction rate can be calculated as the diffusive current of particles into traps. In the opposite case a A , a B  R AB (e.g. for reacting particles suspended in a fluid), the reaction rate can be calculated in the free molecular approach, also in direct analogy with the Brownian particles coagulation. The new approach was further generalized to consideration of reaction kinetics for particles migrating by random walks on discrete lattice sites (with the lattice spacing a ) [44]. Since the case of large reaction radius, R AB  a , is properly reduced to the continuum media limit, the opposite case, R AB  a , with reactions occurred when two particles occupy the same site, was additionally studied. It is shown that the traditional approach to consideration of this important case preserves the main deficiencies of the continuum media approach and thus results in erroneous predictions for the reaction kinetics. For this reason, the new relationships for the reaction rate constants are derived either for 3-d or 2-d lattices. In 3-d case a correct value of the numerical factor in the reaction rate constant is specified, whereas in 2-d a correct time dependence of the rate constant is predicted.

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APPENDIX In this Appendix it is shown that the sweeping rate of randomly distributed immobile point particles (markers) by a large particle of radius R AB migrating with the diffusivity D AB is equivalent to calculation of the condensation rate of the mobile markers migrating by random walks with the diffusivity D AB in the immobile trap of radius R AB . This assertion is important for derivation of the collision frequency function in the continuum mode of the kinetic regime (Sections 1.5.1 and 2.4). Simultaneously applicability limit of the diffusion approach to calculation of the coagulation (or reaction) rate (revealed in Sections 1.3.1 and 2.3) is additionally confirmed. Let us consider an ensemble of N   point particles randomly distributed in a sample of volume V   with the mean number concentration n  N V and migrating with the diffusivity D into the immobile trap particle of radius R . The probability for a point particle  from this ensemble located at t  0 at a distance of r from the trap particle to reach the trap



in t will be designated as w r , t  . Integration of this probability over all point particles determines the total number of point particles trapped in the time interval between 0 to t , 

  n t    nwr , t d 3 r  4n  wr, t r 2 dr ,

(A.1)

R

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in accordance with the Fokker-Planck approach to consideration of particles migration ([57], see also [4]). Correspondingly, the number of point particles trapped in the time interval between t

d n

and t  t is equal to



dt t  4nt  wr, t  t r 2 dr , that determines the R

condensation rate of point particles in the trap 

 n  d n dt  4n  wr, t  t r 2 dr .

(A.2)

R

It is important to note that Eq. (A.1) was derived under an implicit assumption that the number of particles in the volume 4r 2 dr is a large value, i.e. N r  n  4r 2dr  1 , or

n  4r 3 dr r   1 ; only in this case one can neglect fluctuations of the number of particles

in this volume,

N r 2

 N r , that allows subsequent calculation of the total number of

trapped particles [4]. In particular, this inequality should be valid at r  R , that gives

n  4R3 dr R  1 , whereas dr R  1 (in order to correctly perform integration), or 4 dr R  1 . Therefore, the necessary condition for correct calculation of the particles

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condensation rate is nR3  1 , that coincides with the condition of the diffusion approach applicability derived in Sections 1.3.1 and 2.3. This implies that, being applicable to condensation of small particles in a large trap, these calculations become invalid in the case of a small trap (comparable with the size of particles). If there is only one point particle randomly located in the sample (of volume V   ), it 1

3



3

can be found with the probability V d r in the elementary volume d r at each point r , therefore, the probability for this particle to reach the trap in t can be calculated as 

 0 t    V 1 wr, t 4r 2 dr .

(A.3)

R

The probability to reach the trap in t thus becomes equal to

d 0



dt t  4V t  t  wr, t r 2 dr , 1

(A.4)

R

or, from comparison of Eq. (A.4) with Eq. (A.2),

d0

dt t  nV  d n dt t . 1

(A.5)

On the other hand, this latter probability is equal to the probability to sweep in t a sole immobile point particle randomly located in the sample by the trap particle migrating with the Coagulation: Kinetics, Structure Formation and Disorders : Kinetics, Structure Formation and Disorders, Nova Science Publishers, Incorporated,

56

M.S. Veshchunov

diffusivity D . Indeed, as explained in Section 1.5, the relative displacements between two particles describing diffusion motions independently of each other and with the diffusion coefficients D1 and D2 also follow the law of diffusion motion with the diffusion coefficient

D1  D2 [1, 2]. In this approximation, the probability of sweeping of the sole point particle in t by the trap particle migrating with the diffusivity D is equal to V V , where V is the volume swept in t . Equating this probability to Eq. (A.5), one obtains 1

V t  n 1dn dt .

(A.6)

If there are N  nV immobile point particles randomly distributed in the sample, the total number of swept particles in t is reduced to

nV t  dn dt   n ,

(A.7)

with  n from Eq. (A.2). Therefore, the condensation rate  n of point particles migrating by random walks with the diffusivity D AB in the immobile trap particle of radius R AB is equal to the rate of sweeping of immobile point particles by the trap particle of radius R AB migrating with the diffusivity D AB .

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ACKNOWLEDGMENTS The author thanks Prof. L.I. Zaichik (IBRAE), Prof. V.V. Lebedev (Landau Institute for Theoretical Physics, Moscow) and Dr. V.I. Tarasov (IBRAE) for important critical remarks and valuable discussions, which allowed essential improvement of the manuscript. Dr. Y. Drossinos (JRC, Ispra) is greatly acknowledged for kind support and valuable recommendations. Mr. I.B. Azarov (IBRAE), who is co-author of the several reviewed papers, is thanked for the development of the numerical method and subsequent calculations. Prof. L.A. Bolshov (IBRAE) is thanked for his interest and support of this work.

REFERENCES [1] [2] [3] [4] [5] [6] [7]

Smoluchowski, M. Zeitschrift für Physikalische Chemie, 1917, vol. 92, 129-154. Chandrasekhar, S. Reviews of Modern Physics, 1943, vol. 15, 1-89. Seinfeld, H.; Pandis S. N. Atmospheric chemistry and physics. From air pollution to climate change; Wiley, New York, 1998. Fuchs, N. A. The Mechanics of Aerosols; Pergamon Press, New York, 1964. Veshchunov, M.S. Journal of Aerosol Science, 2010, vol. 41, 895-910. Veshchunov, M.S. Journal of Engineering Thermophysics, 2010, vol. 19/2, 62-74. Azarov, I.B.; Veshchunov, M.S. Journal of Engineering Thermophysics, 2010, vol. 19/3, 128-137.

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A New Approach to the Theory of Brownian Coagulation … [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]

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[19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38]

57

Langevin, P. Comptes Rendus Academie Seances (Paris), 1908, vol. 146, 530-534 Knudsen, M.; Weber, S. Ann. D. Phys. 1911, vol. 36, 981-984. Davies, C.N. Proceedings of the Physical Society of London, 1945, vol. 57(322), 259-270. Kim, J.H.; Mulholland, G.W.; Kukuck, S.R.; Pui, D.Y.H. Journal of Research of the National Institute of Standards and Technology, 2005, vol. 110, 31-54. Davis, E. J. Aerosol Science and Technology, 1982, vol. 2, 121- 144. Otto, E.; Fissan, H.; Park, S.H.; Lee, K.W. Journal of Aerosol Science, 1999, vol. 30, 17-34. Fuchs, N.A.; Sutugin, A.G. High-Dispersed Aerosols; In Topics in current aerosol research, Edited by Hidy, G. M. and Brock, J. R., Pergamon Press, New York, 1971, pp. 1-60. Dahneke, B. Simple kinetic theory of Brownian diffusion in vapors and aerosols. In Theory of Dispersed Multiphase Flow; Edited by Meyer, R. E., Academic Press, New York, 1983, pp. 97-133. Wright, P. G. Discussions of the Faraday Society, 1960, vol. 30, 100-112. Veshchunov, M.S.; Azarov, I.B. (to be published). Mädler, L.; Friedlander, S.K. Aerosol and Air Quality Research, 2007, vol. 7/ 3, 304342. Jeans, J. The dynamical Theory of Gases; Dover, New York, 1954. Loyalka, S. K. J. Chem. Phys. 1973, vol. 58, 354-356. Sitarski, M.; Seinfeld J. H. J. Colloid Interface Sci. 1977, vol. 61, 261-271. Mork, K. J.; Razi Naqvi, K.; Waldenstrom, S. J. Colloid Interface Sci. 1984, vol. 98, 103-111. Nowakowski, B.; Sitarski M. J. Colloid Interface Sci. 1981, vol. 83, 614-622. Narsimhan, G.; Ruckenstein E. J. Colloid Interface Sci. 1985, vol. 107, 174-193. Uhlenbeck, G.E.; Ornstein, L.S. Phys. Rev. 1930, vol. 36, 823-841. S.A. Rice, Diffusion-Limited Reactions. In: Chemical Kinetics, Edited by C.H. Bamford, C.F.H. Tipper and R.G. Compton, vol. 25, Elsevier, Amsterdam (1985). Alberty, R.A.; Hammes, G.G. J. Phys. Chem. 1958, vol. 62, 154-159. Murray, J.D. Mathematical Biology I: An Introduction; 3rd edition, Springer, New York, 2002. Murray, J.D. Mathematical Biology II: Spatial Models and Biomedical Applications; 3rd edition, Springer, New York, 2003. Case, T.J. An Illustrated Guide to Theoretical Ecology; Oxford University Press, Oxford, 2000. Corbett, J.W. Solid State Physics, 1966, Suppl. 17, 36. Collins, F.; Kimball, G. J.Colloid Sci. 1949, vol. 4, 425. Waite, T.R. Phys. Rev. 1957, vol. 107, 463. Burlatskii, S.F.; Oshanin, G.S.; Ovchinnikov, A.A. Chemical Physics, 1991, vol. 152, 13-21. Lindenberg, K.; West, B.J.; Kopelman, R. Phys. Rev. 1990, vol. A42, 890. Lindenberg, K.; Argyrakis, P.; Kopelman, R. J. Phys. Chem. 1995, vol. 99, 75427556. Torney, D.C.; McConnell, H.M. J. Phys. Chem. 1983, vol. 87, 1941. Torney, D.C.; McConnell, H.M. Proc. R. Soc. Lond. 1983, vol. A 387, 147-170.

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[54] [55] [56] [57]

M.S. Veshchunov

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In: Coagulation: Kinetics, Structure Formation... Editors: A.M. Taloyan et al

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Chapter 2

DEREGULATION OF COAGULATION DURING SEPSISINDUCED DISSEMINATED INTRAVASCULAR COAGULATION 1

John A. Samis 1,2,*

Medical Laboratory Science, Faculty of Health Sciences 2 Applied Bioscience, Faculty of Science University of Ontario Institute of Technology, Oshawa, ON, L1H7K4. Canada

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Abstract Despite improvements in healthcare settings over the past 50 years, mortality from bacterial infections remains a significant and increasing clinical problem globally because of an increase in antibiotic resistance, the elderly, and immunocompromised patients. The processes of coagulation and fibrinolysis are normally intricately activated extravascularly only at the site of injury and damage to blood vessels. Disseminated intravascular coagulation (DIC) is a common acquired disorder of the coagulation and fibrinolysis processes resulting from a pre-existing pathology leading to fibrin formation in the intravascular compartment. The most common pathology resulting in the establishment of clinical DIC in humans is pre-existing gram negative or gram positive bacterial infection (Sepsis). Our understanding of bacterial-induced sepsis and DIC has been furthered upon study of the effects of these pathogens using in vivo models and clinical trials. During sepsis-induced DIC, bacterial and host components enhance the expression of host tissue factor which promotes fibrin formation in a process mediated by tumor necrosis factor-and interleukin-6, release/generation of compounds that promote fibrin formation, decreased bioavailability of anticoagulation factors, and deregulated fibrinolysis. While different pathogenic bacteria modulate the coagulation and fibrinolytic processes at different points, the presenting clinical symptoms in patients infected with different bacterial pathogens are often similar. Prolonged and/or severe bacterial sepsis disrupts the balance of coagulation and fibrinolytic processes such that bleeding is often the presenting clinical symptom. Diagnosis utilizes a number of laboratory tests which include: recognition of the pre-existing pathology associated with this condition and bacteria identification in combination with prolonged clot times, low platelet counts, and elevated levels of fibrin degradation products. Current therapies are generally aimed at treating the pre-existing pathology that leads to the establishment of DIC. While infusion with antibiotics, platelets, plasma *

E-mail address: [email protected]; Tel: +1-905-721-8668 ext. 3760; Fax: +1-905-721-3179; Address: Faculty of Health Sciences, University of Ontario Institute of Technology, Oshawa, ON. L1H7K4. Canada. (Corresponding Author)

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and/or its components, or heparin have been used in the past to augment therapies, recent treatment regimens using human recombinant activated Protein C have shown encouraging patient outcomes for humans suffering from severe sepsis-induced DIC. Novel diagnostic biomarkers and therapies will ultimately result from a greater understanding of how bacterial factors interact with the host coagulation/fibrinolytic, inflammatory, and immune defence systems. This Review focuses on summarizing what is currently known about how bacteria deregulate the coagulation and fibrinolysis processes leading to the development and maintenance of the sepsis/DIC state in humans.

Keywords: Disseminated intravascular coagulation, coagulation factors, sepsis, infection, inflammation, bacteria, protease.

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INTRODUCTION The coagulation system and fibrin clot formation not only prevents excessive blood loss in response to injury, but helps to curtail the spread of bacteria throughout host tissues and the vasculature. Gram negative or positive bacteria promote sepsis, a systemic inflammatory condition during which the host immune response becomes over active. Although numerous conditions such as: trauma, organ failure, cancer, obstetric complications, liver dysfunction, and immune reactions may result in disseminated intravascular coagulation (DIC), bacteriainduced sepsis is currently the most common pre-existing clinical pathology that results in the establishment of this acquired coagulation disorder. The pathogenesis of sepsis-induced DIC involves activation of the coagulation system resulting in fibrin clot formation throughout the vasculature which may lead to: multiple organ failure, a decrease in the number of platelets and the levels of coagulation factors. This creates a scenario where cellular components and coagulation and fibrinolysis factors become consumed faster that they may be replenished and re-synthesized so that bleeding as often a presenting clinical symptom. The inflammatory response to bacterial infection occurs as part of the host immune response and involves fibrin clot formation in response to host generation of tissue factor (TF) which activates the coagulation system via the Extrinsic pathway. Fibrin clot formation retards the spread of the invading pathogen throughout host tissues and the vasculature. However, some pathogenic gram negative and positive bacteria have evolved biochemical strategies to subvert and manipulate this process by either: 1) enhancing fibrin clot formation to protect them from destruction by macrophages and/or neutrophils or 2) accelerating fibrin clot breakdown to promote their spreading throughout the host as well as transmission to new hosts. Our understanding of the biochemical events resulting in the establishment and maintenance of sepsis-induced DIC has been furthered through experimental and clinical identification and study of the bacterial and host factors that interact to initiate and establish this acquired coagulopathy. In the future, discovery of novel diagnostic biomarkers and therapeutic strategies for sepsis-induced DIC will ultimately result from a better appreciation of the biochemical interactions of host and bacterial cells and the molecular communication between the coagulation/fibrinolysis and inflammation pathways.

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1. Physiological Coagulation and Fibrinolysis The coagulation system is comprised of two major pathways: the Extrinsic and Intrinsic pathways (Fig 1). In response to tissue and blood vessel damage, the coagulation system is activated primarily by the Extrinsic pathway which involves the binding of activated Factor VII (FVIIa) to exposed cell surface-bound tissue factor (TF) [1]. The FVIIa-TF complex then activates Factor IX (FIX) and Factor X (FX) to FIXa [2] and FXa [3], respectively. FIXa in complex with activated Factor VIII (FVIIIa), a membrane surface, and calcium ions comprises the multi-component enzyme complex, tenase which converts FX to FXa [4]. FXa in complex with activated Factor V (FVa), a membrane surface, and calcium ions comprises the multi-component enzyme complex, prothrombinase which converts prothrombin to the clotting enzyme, thrombin (FIIa) [5]. Thrombin cleaves soluble fibrinogen to fibrin monomers which spontaneously associate to form an insoluble fibrin clot [6]. Thrombin also activates Factor XIII (FXIII) to FXIIIa which acts to cross-link and further stabilize the fibrin clot [7]. Activation of the coagulation system by the Intrinsic pathway, via negatively charged surfaces at sites of vessel/tissue damage, is thought to serve as a ‗secondary‘ means of augmenting thrombin generation over and above that derived from the Extrinsic pathway. This notion is supported by the observations that patients deficient in Factor XII (FXII) or Factor XI (FXI) do not or rarely have hemorrhagic events following injury or trauma [8, 9]. In addition to its ability to convert soluble fibrinogen to an insoluble fibrin clot, thrombin also functions as an anticoagulant protease. In this role, thrombin complexes with the endothelial protein, thrombomodulin (TM) which converts protein C to activated protein C (APC) [10]. APC inactivates Factor V (FV), Factor VIII (FVIII), and activated Factor (FVa) and Factor VIII (FVIIIa) and thereby down regulates fibrin clot formation [11, 12]. Fibrin clot formation is also attenuated through the inhibition of activated coagulation factors by tissue factor pathway inhibitor (TFPI) [13] and antithrombin (AT) [14]. The thrombin/TM complex also down regulates fibrinolysis by activating thrombin activated fibrinolysis inhibitor (TAFI) which cleaves carboxy-terminal lysine and arginine residues from fibrin [15], effectively attenuating plasminogen activation and plasmin binding to the fibrin clot [16]. The fibrin clot is not infinitely stable and is eventually removed by the fibrinolytic system to facilitate the tissue repair, remodelling, and wound healing processes [17, 18]. The fibrinolytic system is activated upon damaged vascular endothelial cell release of tissue plasminogen activator (tPA) and urokinase plasminogen activator (uPA), either of which may convert plasminogen to the fibrin degrading enzyme, plasmin [19] (Fig 2). Plasmin also inactivates the cofactors FV [20] and FVIII [21] and activates matrix metalloproteinases (MMPs) [22] which down regulates fibrin clot formation and promotes tissue remodelling and healing, respectively. Plasmin activity vs the fibrin clot is regulated primarily by the plasma inhibitor, 2-antiplasmin [23] and secondarily by 2-macroglobulin [24]. Plasmin activity vs fibrin is also regulated indirectly by the inhibitor, plasminogen activator inhibitor 1 (PAI-1) [25], which blocks the conversion of plasminogen to plasmin by tPA and uPA.

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Intrinsic Pathway

Extrinsic Pathway

Negatively charged/ Damaged Membrane

Injured Tissue/Trauma

FXIIa

Tissue Factor/FVIIa

FXIa

F Xa + F Va

FIXa + FVIIIa

Prothrombin (FII)

Thrombin (FIIa)

Fibrinogen

Fibrin

FXIIIa

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Cross-Linked Fibrin Figure 1. Blood Coagulation Pathway. A simplified blood coagulation pathway is illustrated. This process is comprised of an Intrinsic and Extrinsic pathway that lead to the generation of the protease thrombin that cleaves soluble fibrinogen to fibrin monomers which spontaneously associate to form an insoluble clot. FXIIIa cross-links and further stabilizes the fibrin clot. The abbreviations used are Factor (F) and ‗a‘ indicates an active coagulation Factor.

Plasminogen

Damaged Endothelium

t PA u PA

Plasmin Fibrin

Fibrin Degradation Products (FDPs; D-Dimer)

Figure 2. Fibrinolysis Pathway. A simplified fibrinolysis pathway is illustrated. This process is initiated by release of tissue plasminogen activator (tPA) or urokinase plasminogen activator (uPA) from endothelial cells at the site of vascular damage. Plasminogen activation by tPA or uPA generates the fibrin degrading protease plasmin. Plasmin degrades fibrin clots to fibrin degradation products (FDPs) that include D-dimer.

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2. Overview of Sepsis Sepsis is defined as the systemic host response to infection [26]. Sepsis is a common cause of human mortality in severely ill, elderly, or immunocompromised patients and is currently the most common cause of death in the noncoronary intensive care unit (ICU) [27]. This condition generally develops upon over activation of the host immune response towards bacterial infection which results in systemic inflammatory response syndrome (SIRS) that may lead to multiple organ failure [26]. In response to bacterial infection, activation of the host inflammatory system involves signalling through increased host production of inflammatory modulators tumor necrosis factor (TNF)- and interleukin (IL)-6 [28, 29]. Additionally, the coagulation system is also activated in response to bacterial infection primarily through generation of thrombin and activation of platelets [30]. Chronic or acute activation of the coagulation system by bacterial pathogens may result in disseminated intravascular coagulation (DIC, See below) in the host. Research now indicates that preexisting gram negative or gram positive bacterial infection equally result in the establishment of sepsis-induced DIC in humans [31, 32]. Although the bacterial surface components lipopolysaccharide (LPS) [33] or membrane proteins [34] may activate the host immune response and thus play a primary role in the pathogenesis of sepsis-induced DIC in the host, the specific cause(s) of sepsis by gram negative and positive bacteria is(are) currently unclear. However, it is important to note that very recent research has demonstrated that outer membrane vesicles (OMVs) derived from Escherichia coli (E. coli) are a previously unidentified cause of not only severe sepsis-induced DIC but lethality in mice arising from this condition as well [35].

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3. Overview of Disseminated Intravvascular Coagulation Disseminated intravascular coagulation (DIC) is a common acquired disorder of the coagulation and/or fibrinolysis processes that leads to systemic fibrin clot formation which may result in tissue hypoxia and multiple organ failure [36]. Chronic ongoing consumption of coagulation factors and platelets during the establishment and maintenance of this condition may result in an overall bleeding tendency in affected hosts and this is often the presenting clinical symptom [37]. Further, the levels of the anticoagulation factors protein C [38] and AT [39] are decreased during the onset and establishment of DIC.DIC becomes established in humans as a result of a range of pre-existing pathologies including (From greatest to least): sepsis, trauma, cancer, obstetric abnormalities, transfusion reactions, and snake or spider venom reactions [40]. The pathogenesis of DIC ultimately results in increased host generation of thrombin as a result of elevated exposure of host TF, decreased anticoagulation factor function, and deregulation of fibrinolysis (See below).It is important to understand that no one laboratory test may lead to the diagnosis of DIC. Rather, DIC is diagnosed using a combination of laboratory tests that include: a pre-existing pathology known to be associated with this condition, prolongation of the prothrombin time (PT) and activated partial thromboplastin time (aPTT), decreased platelet count, and elevated fibrin D-dimer [41]. Recent research has indicated that an abnormal biphasic transmittance waveform (BTW) profile is often observed with ICU patient plasmas in the aPTT assay [42]. The BTW has been demonstrated to be a novel and early indicator of impending DIC in humans and is

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currently being used to aid in the diagnosis and treatment of this condition in conjunction with the other laboratory tests described above.

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4. Deregulation of Coagulation by Bacteria Numerous gram negative and positive bacteria initiate the coagulation response via their ability to enhance the expression of TF in endothelial cells and monocytes [43, 44]. Further, the coagulation system is also activated during bacterial sepsis via the contact system and the intrinsic coagulation pathway. E.coli-induced sepsis DIC in baboons was associated with a decrease in blood pressure and the levels of high molecular weight kininogen (HMWK) [45]. The importance of activation of the contact system during sepsis is underscored by the ability of an anti-FXII antibody to block the decreased blood pressure and increase survival of baboons in response to E.coli-induced sepsis DIC [46]. Contact factors have been shown to assemble and become activated on the surface of gram negative and positive bacteria [47]. Lipopolysaccharide (LPS, endotoxin) from gram negative bacteria has been shown to activate the contact system by activating FXII [48]. Secreted bacterial proteases have also been shown to activate kininogen, FXII, or prekallikrein (PK) which results in the release of bradykinin [49-51]. As bradykinin is known to increase blood vessel permeability, this may promote entry and dissemination of bacteria into host tissues. Staphylococcus aureus (S.aureus) secretes a protein known as coagulase that binds prothrombin and causes it to behave like thrombin without hydrolysis [52]. The coagulaseprothrombin complex converts fibrinogen into fibrin. Finally, recent studies have shown bacterial outer membrane Omptin proteases inactivate host tissue factor pathway inhibitor (TFPI) [53]. Such a proteolytic effect of Omptins on TFPI would enhance the host coagulation system in response to bacterial infection and may contribute to the pathogenesis of DIC and other coagulation disorders that arise as a result of sepsis. As outlined above, it is important to note that in response to infection, fibrin formation around the invading pathogen may serve to ‗cordon off‘ or ‗quarantine‘ the bacteria thus preventing its spread to other tissues throughout the host. Bacterial-mediated activation of the host coagulation system to the active fibrin forming protease thrombin outlined above would promote pathogen survival and spreading throughout the vasculature and tissues because fibrin-encapsulated bacteria may also be protected from destruction by host neutrophils and/or macrophages [54, 55].

5. Deregulation of Fibrinolysis by Bacteria As outlined above, thrombin generation as a result of bacterial infection promotes fibrin formation. Fibrin accumulation in the vasculature may not only lead to platelet activation/aggregation and resultant thrombocytopenia [56], but multiple organ failure as well [57]. The breakdown of fibrin by plasmin into fibrin degradation products (FDPs) promotes the release of cytokines, interleukin (IL)-1 and IL-6 and PAI-1 from monocytes [58]. As PAI1 blocks plasmin formation and resultant fibrinolysis, elevated PAI-1 levels may thus promote further fibrin formation and may exacerbate this condition.

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The level of plasminogen decreases during clinical sepsis-induced DIC [59] and maintenance of plasminogen within normal ranges is prognostic for host survival [60]. In view of this, treatment of clinical meningcoccal infection with tPA has been reported to improve patient outcome [61]. The levels of PAI-1 increase during clinical sepsis-induced DIC [62] and this indirectly downregulates plasmin formation and fibrinolysis. Elevated plasma levels of PAI-1 have been reported to be a poor prognostic indicator of outcome from sepsis/DIC in humans [63]. It is thought that elevated levels of PAI-1 promotes reduced fibrin removal and this may result in tissue hypoxia and multiple organ failure. Numerous bacteria have been shown to release plasminogen activators. Streptococci release streptokinase which binds to plasminogen and via a change in shape causes it to take on plasmin-like activity without cleavage [64]. This conformation-induced activation of plasminogen by streptokinase protects the plasmin-like protease from inhibition by 2antiplasmin [65]. Similarly, staphylococci release staphylokinase which activates plasminogen to a plasmin-like protease by a conformational change in a process that is accelerated by fibrin as a cofactor [66]. However, unlike the streptokinase-plasmin complex, the staphylokinase-plasmin complex fibrinolytic activity is inhibited by 2-antiplasmin [67]. Furthermore, plasminogen binding to the bacterial surface increases activation by staphylokinase and protects staphylokinase-plasmin complex from 2-antiplasmin inactivation [68]. Finally, staphylokinase is able to subvert the innate immune response of the host by inactivating the anti-microbial properties of -defensins released from neutrophils [69]. Yersinia pestis (Y.pestis) posseses a surface protease, Pla, that binds and cleaves host plasminogen to generate plasmin [70]. Y.pestis-bound plasmin is thought to promote release of the pathogen from degraded fibrin clots to facilitate spreading of bacteria within and between hosts. Studies using animal models have indicated that infection of wild type mice with Y.pestis results in large numbers of free bacteria and low numbers of infiltrating inflammatory cells at sites of infection [71]. In striking contrast, Pla-deficient Y.pestis infection of control mice or Y.pestis infection of plasminogen-deficient mice leads to large numbers of inflammatory cells at sites of infection and enhanced host survival. Curiously, the survival benefits of Pla-deficient Y.pestis or host plasminogen-deficiency are lost in mice that are deficient in fibrin(ogen) [72]. Pla also undermines the host inflammatory response by inactivating the complement proteins C3 and C3b resulting in reduced bacterial destruction by phagocytes [73]. These studies underscore the importance of fibrinogen and plasminogen, the primary coagulation and fibrinolysis proteins, respectively, in regulating the host inflammatory response to bacterial infections. Numerous bacteria have been shown to not only bind plasminogen, but to recruit host tPA or uPA resulting in plasmin formation on the cell surface [74]. Two enzymes of the glycolysis pathway, glyceraldehyde-3 phosphate dehydrogenase (GAPDH) and enolase from several bacteria have been shown to specifically bind plasmin and/or plasminogen [75, 76]. Further, GAPDH and enolase have been shown to be localized on the bacterial surface where binding and activation of plasminogen to plasmin by tPA or uPA promotes fibrinolysis [77, 78]. GAPDH binding and activation of plasminogen has also been shown to increase bacterial adherence and invasion [79]. Enolase binding and activation of plasminogen has been shown to promote degradation of intracellular junctions that facilitates bacterial migration and

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dissemination in the host as part of the virulence mechanism of Streptococcus pneumoniae [80]. Streptococcus pyogenes (S. pyogenes) synthesizes several proteins that bind plasminogen. Plasminogen binding group A streptococcal M protein (PAM) binds plasminogen which may be activated to plasmin by host tPA or uPA as a means to increase bacterial virulence [81]. Helicobacter pylori (H. pylori) also bind plasminogen and synthesize a neutrophil activating protein (NAP). NAP has been shown to block fibrinolysis by enhancing monocyte synthesis of TF and PAI-2 [82]. Such an effect would enhance the coagulation process by stabilizing fibrin clot formation and may function to protect bacteria from destruction by host phagocytosis. Compared to wild type mice, plasminogen activator-deficient mice have been shown to accumulate more extensive fibrin clots upon treatment with LPS [83]. Conversely, PAI-1deficient mice are protected against fibrin accumulation upon LPS treatment. These results underscore the importance of fibrinolytic factors in regulating fibrin formation in response to LPS. Overall, bacterial-mediated activation of host plasminogen to the active fibrin degrading protease plasmin outlined above would promote pathogen spreading throughout the vasculature and tissues. This is because plasmin has been shown to not only degrade fibrin clots but the components of the extracellular matrix as well [84]. Plasmin may also mediate bacterial spreading in the host by virtue of its ability to activate procollagenase [85] and inactivate inhibitors of collagenase [22] which collectively would enhance the destruction of the extracellular matrix and tissue barriers in the immediate vicinity of the site of bacterial infection.

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6. Coagulation Factor and Inhibitor Therapies Treatment of sepsis-induced DIC is generally aimed towards the pre-existing pathology and has made use of: antibiotics, plasma or its component fractions, platelet concentrates, heparin, TFPI, AT, or APC. A low platelet count and decreased activity of coagulation factors are usually associated with development of sepsis-induced DIC in humans [86, 87] and in animal models [88, 89]. Platelet transfusions have been used successfully in patients with high risk of bleeding and low platelet counts [90]. Fresh frozen plasma or its fractions have been used for patients with defects in coagulation as a result of consumption and/or inactivation of coagulation factors [37, 91]. Unfractionated or low molecular weight (LMW) heparin have been successfully used for patients with evidence of activation of coagulation [92, 93]. Based on host generation of TF during the onset of sepsis-induced DIC, it may seem reasonable to use TFPI as a targeted therapeutic agent for this condition. The plasma levels of TFPI increase significantly during sepsis-induced DIC in humans [94]. Initial clinical trials with human recombinant TFPI showed promise in treating septic patients [95]; however, more recent clinical trials have failed to demonstrate a survival benefit with TFPI treatment [96]. Thus the role(s) of TFPI during this condition remains unknown. Further, given that several active coagulation proteases are generated during the onset of sepsis-induced DIC such as Factors IXa, Xa, and thrombin; it may also seem reasonable to block the activity of these activated coagulation factors using anticoagulation factor

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concentrates. The plasma levels of the main anticoagulation factor, AT, which blocks the activity of FIXa, FXa, and thrombin in a process accelerated by heparin, have been reported to decrease significantly in animal models [88] and humans [97] during this condition. While use of AT reduced mortality in a baboon model of Escherichia coli (E.coli) sepsis/DIC [98]; unfortunately, recent clinical trials centered on the treatment of septic/DIC patients with antithrombin concentrates and heparin have not demonstrated a significant survival benefit [99]. Strikingly, preinfusion of the anticoagulation factor APC has been shown to significantly decrease mortality of baboons in experimental sepsis models in response to lethal doses of E.coli in vivo [100]. Conversely, interference with the activation of protein C upon infusion of an antibody to the protein C receptor (Endothelial protein C receptor, EPCR) was able to convert a sublethal E.coli challenge to a lethal outcome in baboons [101]. Indeed, low plasma levels of protein C are prognostic for poor outcome from sepsis-induced DIC in animal models [102] and humans [103]. A large clinical trial has recently demonstrated a significant survival benefit with human recombinant APC (drotrecogin alfa) during severe sepsis [104]. It is important to note that in addition to inactivating Factors V/Va and VIII/VIIIa and down regulating the coagulation process, APC also modulates cytoprotective signaling functions on endothelial cells [105], protects vascular barrier integrity [106], and activates anti-apoptotic pathways of cell survival [107] which are just now being better understood through ongoing research. However, while the efficacy of APC in treating severe sepsis-induced DIC has shown recent promise in reducing mortality in humans, it has been recommended that sepsis patients with a high risk of bleeding should not be administered APC [104].

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CONCLUSION The molecular and cellular events related to the coagulation and fibrinolysis processes that develop as a result of sepsis-induced DIC are illustrated and summarized in Fig 3. Although healthcare facilities in industrialized nations have improved significantly over the past 50 years, human mortality from bacterially induced sepsis-DIC remains a global clinical concern. This is due to an increase in bacterial strains that are antibiotic resistant, elderly patients, and patients that are immunocompromised. Clearly, discovery of new therapies for this acquired coagulopathy will require a better understanding of not only an inventory of the bacterial factors that cause sepsis-induced DIC in humans but how and when these bacterial components interact with the coagulation/fibrinolysis, acquired and innate immune, and complement systems during the pathogenesis of this disorder.

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Pre-Existing Pathology/Bacterial Infection

Activation of Coagulation Process Systemic Fibrin Formation

Multi-Organ Failure

Loss of platelets and Coagulation factors

Bleeding

Activation of Fibrinolysis Process

Activation of Coagulation Process

Pre-Existing Pathology/Bacterial Infection

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Figure 3. Coagulation and Fibrinolysis Pathways During Sepsis-Induced DIC. The changes in the coagulation and fibrinolysis pathways as a result of sepsis and DIC are illustrated and summarized.

Our understanding of the underlying mechanisms that initiate and propagate sepsis-induced DIC in humans as a result of bacterial infection will be furthered by the ongoing use of experimental animal models and expanded clinical studies. Such research will ultimately lead to the discovery of new bacterial and host biomarkers that represent strategic biochemical targets for rational diagnostics and treatments that are based on a thorough scientific understanding of the biochemical changes involved.

Authors’ CONTRIBUTIONS JAS researched the topic, reviewed the current literature, and wrote the Review.

ACKNOWLEDGEMENTS The author was supported with Professional Development and Start Up Research Funding from The University of Ontario Institute of Technology. The author acknowledges the helpful discussions and insights of Dr. Michael E. Nesheim (Department of Biochemistry, Queen‘s University, Kingston, ON) whose mentorship and caring nature will always be treasured and fondly remembered.

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In: Coagulation: Kinetics, Structure Formation… Editors: A.M. Taloyan et al

ISBN: 978-1-62100-331-1 © 2012 Nova Science Publishers, Inc.

Chapter 3

COAGULATION: KINETIC, STRUCTURE, FORMATION AND DISORDERS Benjamín Rubio-Jurado1,2, Díaz-Ruiz Rosbynei2, Pedro A. Reyes3, Carlos Riebeling4,5 and Arnulfo Nava1,6,7,*

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1

Clinical Epidemiology Research Unit 2 Hematology Service UMAE, HE CMNO Instituto Mexicano del Seguro Social. 3 Research Direction, Instituto Nacional de Cardiología Ignacio Chávez 4 Clinical Epidemiology Research Unit, UMAE HP CMNS-XXI Instituto Mexicano del Seguro Social 5 Facultad de Medicina, UNAM 6 Research Direction Health Sciences, Universidad Autónoma de Guadalajara 7 Hospital General de Occidente SSJ

Abstract Maintenance of normal blood flow requires equilibrium between procoagulant and anticoagulant factors it is named hemostasis, occasionally procoagulant activity predominates, leading to clots formation; frequently, tissue damage is the triggering factor. Hereditary factors, primary or acquired play a role in the development of thrombosis. Primary thrombophilia is associated to hereditary factors, which promote hypercoagulability because natural anticoagulants are not exerting their activity. On the other hand, acquired thrombophilia may occur associated to autoimmune diseases, cancer, surgical procedures, pregnancy, postpartum period, and obesity. Activation of the coagulation system is characterized by the co-participation of inflammatory response components, factors related to the subjacent disease, and other procoagulant factors. The study of hemostasis in the patients should include both inflammatory and autoimmune response markers.

Keywords: Coagulation, Hemophilia, Hemostasis, Thrombophilia.

*

E-mail address: [email protected]; Address: Av. Juan Palomar y Arias (Antes Yaquis) # 658. Col. Providencia. Guadalajara, Jal. México. C.P. 44670. (Corresponding Author)

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Benjamín Rubio-Jurado, Díaz-Ruiz Rosbynei, Pedro A. Reyes et al.

1. Introduction Hemostasis is a physiological mechanism that allows resolving micro-hemorrhagic events by activating coagulation, clot formation; its resolution occurs without the presence of a thrombus, which, in most cases, has no clinical expression. Generally, a trauma is the antecedent that activates the coagulation system, accompanied by an inflammatory event, which includes parts of the innate immune response. Activation of both biological events facilitates their persistence and consequences.

2. Kinetics of Coagulation Systems and Clot Formation This system enables the free transit of blood through the vascular system under equilibrium processes between anticoagulant and procoagulant mechanisms (1).

The coagulation cascade is classically described by an extrinsic pathway and an intrinsic one (2). In prothrombosis and antithrombosis, participation of the endothelium surface receptors and fibronolytic and vasomotor factors intervene (3). At present, the following have been established: a) the extrinsic route of coagulation is a principal route b) a cellular model of coagulation is described c) the production by biotechnology of factors (FVIIIr, IXr, VIIr) is known, and d) the discovery of genetic polymorphisms (4).

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2.1. Primary Hemostasis The platelet activation is in the presence of vascular wall damage. Physiologically, 1 × 1011 platelets are produced daily, with a half-life of 10 days, and a third of these are stored in the spleen. Production is regulated by thrombopoeitin, hepatic synthesis. It acts via its receptor c-Mpl (cellular-myeloproliferative), which is found in platelets, megakaryocytes and endothelial cells (5). Platelet adhesion is regulated by families of surface receptors (glycoproteins) that interact with the cytoskeleton, influencing morphology and motility. Activated platelets increase their surface by means of pseudopods and have negatively charged receptors and phospholipids on their surface (6). The principal platelet receptor is an integrin of 230 kDa (αIIb3; glycoprotein IIb-IIIA). Other receptors are the integrins: αV3, α21, α51, α61, which binds to collagen, fibrinogen, von Willebrand factor, fibronectin and thrombospondin (7). The von Willebrand factor (vWF) is a plasma glycoprotein liberated from α granules of platelets (megakaryocytes and endothelial cells, adheres the platelets to surfaces with exposed collagen, and permanently adheres to the receptor GP IIb-IIIa receptor (7). Platelets have an ADP signaling route with two receptors P2Y1, P2Y12, playing a significant role in platelet adhesion and aggregation by means of α-granular liberation (with presence of tissue factor, thromboxane A2, ADP, calcium and serotonin)(8).

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2.2. Coagulation Is Constituted by Interacting Elements a) proenzymes: prothrombin, factors VII, IX, X, XI, XII, XIII, prekallikrein, protein C, antithrombin III b) soluble cofactors: FV, FVIII, protein S, vWF c) cellular factors: tissue factor (FIII), thrombomodulin d) structural proteins: fibrinogen.(2)

2.3. The Extrinsic Route of Coagulation

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This system is activated in the presence of tissue factors (extrinsic route), a mechanism initiated in vivo in response to trauma. The tissue factor is found in the adventitia of the vascular wall and constitutes the principal source as a thrombogenic agent (8). Leukocyte derivatives have been detected in soluble blood tissue factor or in microparticles (principally monocytes), which in healthy subjects is considered to have poor thrombotic activity, given the low concentration (9). The tissue factor appears after endothelial damage and the extracellular domain binds to circulating factor VIIa and activates FIX and FX (FIXa and FXa). The reaction is amplified by positive retroalimentation reaction (FIXa and FXa activate FVII = FVIIa). FXa and its cofactor Va are bound to activated platelets by membrane phospholipids and activate prothrombin (factor II) to generate thrombin. Thrombin cleaves with fibrinogen to form fibrin and a fibrin network (Figure 1), (3,10)

Modified from : Cir Cir 2007;75:313

Figure 1. Coagulation Cell Model.

2.4. The Intrinsic Route of Coagulation This begins with the contact factors: FXII, the kininogen of high molecularweight (Fitzgerald-Williams-Flaujeauc factor) and prekallikrein (Fletcher factor) called contact

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factors, acting on FXI (XIa), which then activates FIX (FIXa). El FIXa acts on FX and then the process continues by a common route [9] (Figure 1).(3,10,11)

2.5. The Cellular Model of Coagulation Establishes that thrombin is generated by an initiation phase (caused by the FT + VIIa complex), propagation phase (after 120 seconds of initiating the process it intervenes in the FVIIIa + IXa complex on the surface of activated platelets) and termination phase (determined by ending prothrombin consumption) (2). Thrombin activates FXIII (also called fibrin stabilizing factor, coagulation stabilizing factor) and activates thrombin-activated fibrinolysis inhibitor (TAFI).

2.6. Activation of the Coagulation System

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The cellular models of coagulation have been described traditionally in stages, including the expression of cellular surface receptors, signaling pathways and the presence of activated factor VII. These stages are: 1) Initiation: expression of the tissular factor binds and activates factor VII. The FVIIaFT complex activates FX and generates thrombin. 2) Amplification: thrombin activates platelets and these express, on their surface, negatively charged phospholipids, such as phosphatidylserine; on this surface, FVIII, FIX, and FX are activated, generating more thrombin 3) Propagation: on the platelet surface, the activated factors favor exponentially thrombin generation. 4) Clot stabilization: The action of F XIII on the fibrin mesh makes the latter insoluble, thereby determining a stable clot (3,4). Primary hemostasis is generated by the platelet, formation of a clot in response to tissue damage, which is attributed to an initial message generated by the endothelial cell (5). Primary hemostasis is described in three stages: adhesion, activation, and secretion, and it is mediated by the interaction of the platelet with receptors of the endothelial cell and the collagen. In the endothelial damage model, the platelet generates initially a rolling on the endothelial surface, binds in a transient and labile manner, through the Ibα glycoprotein receptor (Ibα GP) to the Willebrand factor; this binding favors dimerization of the GPαIIb-ß3 receptor, giving rise to a more stable intercellular binding between platelets and the endothelial-platelet cell. The adhered platelet recruits more platelets by secreting chemotactic elements (secretory stage). The cellular stage of activation of coagulation takes place on the platelet surface, which is exposed to the blood flow (cell-based model of coagulation) (1,5). Thrombin acts on fibrinogen and converts it to fibrin; besides, thrombin activates the fibrin stabilizing factor (Factor XIII, clot stabilizing factor) and activates the inhibitor of fibrinolysis, which is activated by fibrin, giving rise to the formation of a stable clot (11). Dissolution of the clot by the action of plasmin is called fibrinolysis. Plasmin acts on the soluble fibrin (the fibrin that has not interacted with the clot stabilizing factor, F XIII) (2).

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3. Kinetics

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3.1. Coagulation and Inflammatory Components Damage of the vascular endothelium activates hemostasis and clotting pathways to repair the damaged site (3,4). Over-expression of the tissue factor, generation of thrombin by means of the extrinsic clotting pathway, and platelets activation through its PAR4 receptor have been demonstrated in the in vitro model of endothelial damage through laser micropipettes or by instilling chemical substances. Activation of the coagulation system coexists with elements of the inflammatory response related to the underlying inflammatory disease and to procoagulating factors. A study with volunteers subjected to low doses of endotoxin demonstrated an increase of procoagulant proteins simultaneously to the inflammatory response. Patients with an acute thrombotic event course with elevation of diverse inflammation markers. In patients with asymptomatic peripheral artery failure, reactive protein C elevation and dimer-D associated with neurological impairment, related to the presence of underlying neurological atherothrombosis, have been reported, suggesting that inflammation and hypercoagulability could be implicated in its pathogenesis (12). The hypothesis that the presence of inflammation and hemostatic activation are associated to a greater cardiovascular risk finds support in the performance of the C reactive protein (CRP) levels, as marker of systemic inflammation, and which is found elevated in patients with angina, myocardial infarction, and sudden death. Hence, CRP is considered a biomarker of preclinical cardiovascular disease and underlying severity; low inflammation levels are present in patients at risk for cerebral and coronary atherothrombosis, as well as in apparently healthy subject. (13,14). High concentrations of CRP are related with an increase in the atherosclerosis and thrombosis risk, as it contributes to the generation of a procoagulant state with endothelial dysfunction associated to its capacity to: a) increase the expression of the tissular factor, b) favor opsonization, c) activate the C, d) increase the generation of cytokines, adhesion molecules, and e) favor chemotaxis (15). The usefulness of determining the circulating levels of fibrin, D-D, fibrinogen, and CRP during the first 24 h of a cardiovascular and cerebral thrombophilic event has been identified (16), being strong predictors of disease recurrence; their correlation suggests indirectly the interaction between the inflammation and coagulation systems (17). Other inflammation mediators such as the tumor necrosis factor (TNF) or endotoxin promote inhibition of anticlotting elements (heparin, alpha-1- antitrypsin, and thrombomodulin), elevation of procoagulants (tissular factor, adhesion molecules) or suppression of fibrinolysis (increase of the plasminogen-1 activator inhibitor, PAI-1) In sepsis, activation of C, the inflammation and activation of coagulation perpetuate the endothelial damage, generating thrombosis, ischemia, and multi-organ failure; this mechanism has been reproduced in animal models (18). Thrombin contributes to the amplification of the inflammatory process by activating: a) platelets with recruitment and activation of circulating leukocytes, and b) endothelial and smooth muscle cells that release mediators such as IL-6, RANTES, and CD40L, which in turn act upon circulating mononuclear cells (19,20).

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The expression of multiple receptors related to the innate and acquired immune response in the platelet membrane can explain the connection between the thrombotic event and the inflammatory one proposed for the genesis of atherothrombosis. These receptors are, among others, P-selectin and its ligand (PSGL-1), integrins such as αIIß3 (receptors for collagen, von Willebrand factor, fibronectin, and fibrinogen), adhesion molecules such as ICAM-2, binding molecules as JAM-C, toll-type receptors, proteases-activated receptor (PAR-1; constitutive receptor for thrombin), CD40 and its ligand, and chemokine receptors. (21,22)

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3.2. Coagulation and Interaction with Endothelial Cells and Leukocytes The platelet adhered to endothelial cells secretes interleukin 1ß (IL-1ß), which activates the endothelial cell promoting chemotaxis, adhesion, cellular migration, proteolysis, and thrombosis at the damage site (23). Lymphocytes and monocytes are activated by platelets through the constitutive receptors PSGL-1 and MAC-1; during the cell adhesion process, these receptors favor secretion of cytokines, chemokines, tissular factor, proteases, and the differentiation of monocytes to macrophages (23) Interleukin 6 (IL-6) is able to activate mononuclear, endothelial, or tumor cells, giving rise to over-expression of the tissue factor(19). The activated endothelial cells favor adhesion, rolling, and transmigration of T and B lymphocytes, monocytes, macrophages, and mastocytes that infiltrate multiple tissues through this mechanism generated at the inflammation or endothelial activation sites (24). The microvascular endothelial cells are active regulator of immunity, coagulation, and inflammation, as summarized in a recent review (25), an immune response resulting in inflammation depends strictly on a permissive microvasculature, which normally exerts the opposite function of preventing the indiscriminate influx of immune cells into a tissue. The microvascular bed constitutes the majority of the endothelial surface, covering an area approximately 50 times greater than that of large vessels combined (25-27). Major qualitative differences exist between macro- and micro-vascular endothelial cells, the later being able to generate a range of mediators, to display distinct adhesion molecule patterns, to activate unique set of genes, and form capillaries. At the site of inflammation, predominantly T CD4+ lymphocytes are identified, which recognize auto-antigens, such as the oxidized low density lipoproteins (oxidized LDL) through their interaction with the antigen presenting cells, initiating an acquired immune response (28). If the Th1 inflammatory response predominates, it is able to activate macrophages and endothelial cells, increase the production of proteases and adhesion molecules; thereby, favoring clot formation. In addition, it inhibits the proliferation of smooth muscle cells and the production of collagen (29). The platelet are cellular components of primary hemostasis, can co-aggregate to circulating leukocytes; after platelet adhesion to the endothelium, which serves as a surface to recruit leukocytes, this leukocytes‘ recruitment requires mechanisms of adhesion and signaling that culminate in immune cells infiltrating the vessel‘s wall (30). A major role is played by the interaction between the leukocytes and endothelial cells; as pointed out in a recent review (30).

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This interaction comprises several molecular events mainly involving adhesion molecules, which can be grouped into three families; a) Selectins are a family of three carbohydrate-recognizing molecules, of which, E-selectin is expressed on activated endothelium, P-selectin is expressed on platelets and the endothelium, and L-selectin is constitutively expressed on leukocytes(31), b) Integrins are heterodimers comprising of an alpha and beta chain and can recognize multiple ligands including proteins of the extracellular matrix, cell surface glycoproteins as well as complement factors and soluble components of the haemostatic and fibrinolytic cascade(32,33), c) The major integrin ligands involved in leukocyte adhesion belong to the immunoglobulin superfamily(34), and include intercelular cell adhesion molecules (ICAM) as well as the junctional adhesion molecules (JAMs) (35), that are expressed on endothelial and other cells.

3.3. Cross-Talk between Clotting and Complement System (C)

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The C system and the coagulation system belong both to the first line of defense against injurious stimuli and invaders. There is increasing evidence that the rapid activation of the coagulation cascade after trauma may be accompanied by a progressive inflammatory response (36,37). Interaction between both cascades has been proposed. In fact, both contain series of serine-proteases with evidence of some shared activators and inhibitors, such as factor (F) XIIa, which is able to activate c1q, and thereby the classical pathway of complement. In addition, the C1 esterase inhibitor acts not only as inhibitor of all three C pathways, but also of the endogenous coagulation activation path (kallikrein, FXIIa) (38). Recently, generation of C5a in the absence of C3 with thrombin acting as a potent C5 convertase has been demonstrated (39).

4. Disorders 4.1. Thrombophilia Thrombophilia is the tendency to develop thromboembolia and can be determined genetically or can be acquired. Genetic factors involve deficiency of natural anticoagulant proteins, such as: antithrombin III, protein C, and protein S; presence of mutations, such as: factor V (Factor V Leiden) and G20210A of prothrombin (40-42). Analysis of thrombosis is achieved from the point of view described by Virchow: anomalies of the vascular wall, anomalies of the blood flow, and anomalies of blood components (Virchow‘s triad)(43). More recently, pathogenesis of thrombosis has been considered as a multicausal model(44,45). Figure 2 depicts the factors currently considered as causes for a thrombotic event, involving genetic defects associated to the presence of an acquired factor, as for example obesity, a surgical-anesthetic event, chronic degenerative diseases, cancer, among others(11).

4.1.1. Acquired Factors Associated to Thrombosis The acquired factors associated to thrombosis include: pregnancy, smoking, thrombocytopenia induced by heparin, warfarin, oral anticonceptives, type 2 diabetes

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mellitus, hyperlipoproteinemia, polycytemia vera, hyperfibrinogenemia, nephrotic syndrome, vasculitis, neoplasias, surgery, thrombocytemia, sepsis, obesity, and others as shown in figure 3 (11,46,47). The risk for thrombosis increases with age; an analysis of the Framingham study revealed association of age with a prothrombotic state; an increase in the levels of fibrinogen, von Willebrand factor, and of PAI-1 has also been found (48). Recently, it has been proposed to use a multidimensional approach to study thrombophilic conditions, including diverse systems that probably constitute a continuum and involve release of histones, which favor cytotoxicity. Inhibition of the latter could become a protecting mechanism (25, 49).

Modified from :Cir Cir 2007;75:313-323

Figure 2. Multicausal Pathogenic Model.

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4.1.1.1. Procoagulant States and Their Association with Humoral Immunity The thrombophilic states that are related with autoimmunity exhibit mainly circulating antibodies, such as anticardiolipin antibodies (aCL), anti-β2glycoprotein I (aβ2GPI), antiprothrombin (aPT)(50), lupus anticoagulant, and anti-annexin V(46,47). The primary antiphospholipid syndrome (APS) is the most frequent cause of acquired thrombophilia (1520%)(51); showing frequently arterial or venous thrombosis (20-30%), and up to a third of them have recurrence in less than 4 years (52,53). The prevalence of antiphospholipid antibodies (aAP) in patients with thrombosis has been reported between 4 and 21%, in pregnant women of 2 to 9%, in women with recurrent stillbirths it has been found in 20%, and in clinically healthy subjects in 5 to 9%(54). In patients with cerebrovascular disease (CVD), aAP have been reported from 18 to 30% (55). In patients with systemic lupus erythematosus (SLE), 20 to 40% course with aAP, and of these, 50% course with APS, considered as secondary (56). Primary APS (PAPS) or secondary APS (SAPS) depicts diverse clinical manifestations, among them are arterial thrombosis, recurrent venous thrombosis, abortions, stillbirths, and thrombocytopenia (52,57). Prevalence of aAP in patients with deep vein thrombosis (DVT) without systemic autoimmune disease is of 5% (43). The association aAP and thrombosis yields an OR of 1.6 to 3.2 for anticardiolipin and of 6.8 to 11.0 for the lupus anticoagulant. In patients with

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venous thromboembolism, the prevalence of aCL and LA ranges from 3 to 17% and 3 to 14%, respectively; in blood donors, the prevalence of aCL has been reported in 2% (55).

Modified from: Cir Cir 2007;75:313-323

Figure 3. Acquired Prothrombotic States

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The presence of anti-prothrombin antibodies in patients with a diagnosis of APS is predictive of venous thrombosis with an OR of 1.66 and for arterial thrombosis with an OR of 2.15. In addition, the presence of anti-PT is associated with thrombocytopenia in primary APS with an OR of 6.7. The presence of anti-phosphatidylserine/prothrombin (anti-PS/PT) has been related with the lupus anticoagulant. The anti-PS/PT complex in patients with SLE is an independent risk factor for arterial and venous thrombosis with an OR of 9.36 and 7.33, respectively (58,59). 4.1.1.2. Concepts of Antibodies in Thrombophilia Conventional tests for the detection of these antibodies are: ELISA for anti-cardiolipin antibodies and lupus anticoagulant (LA), this is a phospholipid-dependent coagulation test, in which antibodies prolong coagulation time (57,58). Antiphospholipid antibodies (aAP) do not bind directly to phospholipids of cellular membranes, but to proteins with phospholipids affinity such as prothrombin and beta2-glycoprotein I (50,60). Beta-2-glycoprotein (ß2-GP I) is present in plasma at 10 to 300 μg/mL, it is synthesized in endothelial, placental, and brain cells, but mainly in hepatocytes. The anti-cardiolipin antibodies are in fact anti-ß2-GP I, which is a heterogeneous group of antibodies against epitopes of the protein, the main epitope is found in domain I, near lysine 43 (50, 57, 60). The association of IgM anti-ß2-GP I with thrombosis has been reported; traditionally it is considered that the association is with IgG anti-ß2-GP-I (61). It has been postulated that antigenicity is generated through an endogenous immunogenic adjuvant mechanism, consisting of identification of ß2-GP-I by the TLT 4 receptor or endocytosis of ß2-GP-1 oxidized by antigen presenting cells, followed by presentation to selfreacting T and B lymphocytes with the subsequent B cells proliferation and antibodies production (20, 62).

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ß2-GP I has scarce affinity with anionic phospholipids, formation of the antibody- ß2-GP I complex increases more than 100-fold, and this complex is able to interfere binding of the coagulation factors with the catalytic surface of phospholipids. The increase in the thrombotic risk is given by two main mechanisms:

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1) Interference with the phospholipid-dependent antithrombotic pathway, the protein C pathway. The latter is activated by thrombin bound to thrombomodulin, attaches to activated V and VIII factors to prevent subsequent thrombin formation. This cascade reaction of protein C takes place on a phospholipid surface. 2) Activation of platelets, monocytes, and endothelial cells (57,63), producing an increase in platelet activity (rolling, aggregation, and secretion) and inducing activity of the tissular factor, the main activator of coagulation. Activation of these cells is achieved through the binding to receptors, apoER‘2 in platelets, annexin A2 in monocytes, and toll-type receptors (TLR) in endothelial cells (57,64). In addition, these cells can inhibit antithrombin III, activate protein C, and bind to proteases involved in fibrinolysis, affecting inactivation of procoagulant factors and resolution of the clot (51). The presence of the anti- ß2-GPI-dependent positive lupus anticoagulant has been associated to thromboembolic complications with an Odds Ratio (OR) of 42.3, whereas the lupus anticoagulant not dependent on ß2-GPI represents a lower risk (OR 1.6)(52,60). Prothrombin is a vitamin K-dependent glycoprotein synthesized in hepatocytes, constituted of 579 amino acids with a molecular weight of 72 kDa, it is found at a concentration of 100 mg/mL in plasma (65). In 1959, prothrombin was proposed to be a cofactor of the lupus anticoagulant, its prevalence in patients with systemic lupus erythematosus (SLE) varies from 28 to 54% (64,65), the prevalence of anti-prothrombin antibodies (aPT) reported for SLE/APS is of 54%, for SLE/thrombosis it is of 5%, and for SLE of 16%; in this latter group, it has been accepted that aPT is useful to identify patients in anticardiolipin antibodies (aCL) - negative prothrombotic state(66). Antithrombin antibodies recognize exposed encrypted epitopes when prothrombin binds to phospholipids; prothrombin with conformational changes in its Nterminal end, in the presence of calcium, exposes hydrophobic sites to bind to phospholipids. These epitopes are located in fragments 1 and 2 of prothrombin (65). The mechanism by which aPT exerts LA activity is not clear; however, lengthening of the coagulation times, either by inhibition of the conversion of prothrombin to thrombin (hypoprothrombinemia) or by increasing depuration of the anti-PT-prothrombin complex from the circulation, is the accepted hypothesis (59).

4.2. Genetic Factors Associated to Thrombophilia 4.2.1. Deficiency of Antithrombin III Antithrombin III (AT-III) is a natural anticoagulant synthesized in the liver, it regulates fibrin formation by inhibiting thrombin and factors IXa, Xa, and XIa. The AT-III deficit was reported as the cause of thrombophilia in 1956 (67), it prevails in ≤ 2% of the general population. Patients are mostly heterozygous and depict AT-III values

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between 40 and 70% of the normal value; they can be of type I, with diminution of activity and of the antigenic value; or type II, in whom there is only diminution of the activity. Prevalence in patients with venous thromboembolic disease ranges from 1 to 7%. The thrombotic risk has been estimated to be 5- to 8-times higher than in the general population. This deficiency is caused by molecular defects such as deletions, insertions, or point mutations (68).

4.2.2. Deficiency of Protein C Protein C is synthesized in the liver and is vitamin K-dependent; it is activated by the thrombin-thrombomodulin complex and requires the presence of protein S to exert its anticoagulant action. It inhibits factors VIIIa and Va. The frequency of this deficiency in the general population is less than 1%, and in patients with thrombotic events it occurs from 2 to 5%. The risk of thrombosis is 9.7 times higher than in the general population. When it is homozygously present, it is clinically associated to purpura fulminans(67).

4.2.3. Deficiency of Protein S Protein S is synthesized in the liver, endothelium, and megakaryocytes. It is a cofactor of protein C. It exists in both the free form, which is functional, and fixed to fraction 4b of the complement. Protein S values decrease during pregnancy and by the use of anti-conceptives. This deficiency can be found in 0.03 to 0.13% of the general population, and in 3 to 7% of patients with thrombosis; it represents a 9-times higher risk of thrombosis (67).

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4.2.4. Mutation of Factor V Mutation of factor V (Factor V Leiden) causes a resistance to the action of activated protein C. Factor V is a cofactor of factor Xa that acts upon prothrombin to transform it into thrombin. The prevalence in the population of northern Europe is of 5 to 8%, and this mutation has been identified in 12 to 30% of patients with spontaneous deep vein thrombosis (DVT). The thrombotic risk in heterozygous patients is 3- to 10-times higher than in the general population. In homozygous patients, the incidence of thrombosis is high, up to 40 to 80% (67,69). Thrombosis has been associated to the participation of acquired factors. There are other polymorphisms that contribute to the resistance of activated protein C, such as factor V Cambridge, Arg306Thr. The incidence is very low; its study is recommended in individuals with activated protein C resistance and negative to factor V Leiden (42).

4.2.5. Mutation G20210A of Prothrombin The G20210A mutation of prothrombin, described in 1996(67), is a replacement of G by A in the 20210 nucleotide of the prothrombin gene; this represents an increase in prothrombin values. In average, there is a 30% increase of prothrombin in heterozygous and of 70% in homozygous individuals, and this causes an increase in thrombin generation (42). In the general population, a prevalence of 0.7 to 4% has been estimated; it is more frequent in the South of Europe, from 7 to 18% in patients with spontaneous DVT. In patients with a

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previous thrombotic event, the prevalence is of 5 to 18% and the thrombotic risk is 1- to 5times higher than in the general population. When it is accompanied by the use of oral anticonceptives it exhibits an RR of 16. This mutation has been described accompanied by a mutation in factor V; these patients have a 20-times higher risk than those without this combination (42, 67). In Mexico, a 2% to 39% resistance to activated protein C (APS) has been reported in patients with thrombotic event; of these, 10% have the mutation corresponding to factor V Leiden. Another mutation of factor V, HR2 haplotype, has been reported in 28% and mutation of prothrombin G20210A in 16%. The lack of adequate registries underestimates this problem, aside from the lack of reports on the causes of thrombosis in our population 70,71). Other factors with a probable hereditary mechanism are: hyperhomocysteinemia, increase of factors VIII, IX, X, XI(67), increase of thrombin-activatable fibrinolysis inhibitor (TAFI), and deficiency of factor XII. In these hereditary prothrombotic states, 6 to 18% of heterozygous patients (and up to 80% in homozygous ones) present a thrombotic event before the age of 50 and 50% of them will course with recurrence (72). Approximately 50% of these patients are detected due to a spontaneous first episode, some patients have more than one genetic defect and this combination increases the risk; besides, the association with conditions of temporal risks such as trauma or surgery increase also the risk of presenting a thrombotic event (43).

CONCLUSION

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Shared molecules are key issues in the cross-talk among coagulation, inflammatory response and autoimmunity (Table 1). Table 1 Major issue in cross-talk among coagulation, inflammatory response, and autoimmunity

Modified from: Clinic Rev Allerg Immunol. DOI 10.1007/s12016-010-8240-0

It is necessary to identify the actual magnitude of the interactions among inflammation, coagulation systems, and autoimmunity to understand better the activation of hemostasis. Coagulation: Kinetics, Structure Formation and Disorders : Kinetics, Structure Formation and Disorders, Nova Science Publishers, Incorporated,

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It is probable that the predominance of one system in particular influences the pathogenesis of the thrombotic state, which must be taken into account to establish adequate treatment.

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REFERENCES 1. Dahlbäck, B., Blood coagulation and its regulation by anticoagulant pathways: genetic pathogenesis of bleeding and thrombotic diseases. J Intern Med., 2005. 257: p. 209-223. 2. Bombeli T., S.D.R., Updates in perioperative coagulation: physiology and management of thromboembolism and haemorrhage. Br J Anaesth 2004. 93: p. 275-287. 3. Carrillo E. R, S.N.P., Carvajal R. R., Contreras D. V., Hernández A. C. , Rompiendo un paradigma: del modelo humoral al modelo celular de la coagulación. Su aplicación clínica en el enfermo grave. Med Crit Terapia Int. , 2004. XVIIII(1): p. 17-23. 4. Mackman, N., Role of Tissue Factor in hemostasis, thrombosis, and vascular development. . Arterioscler Thromb Vasc Biol. , 2004. 24: p. 1015-1022. 5. Ruggeri, Z.M., Platelets in atherothrombosis. Nature Medicine, 2002. 8(11): p. 1227-1234. 6. Varga-Szabo, D., Pleines, I., Nieswandt, B., Cell dhesion mechanisms in platelets. Arterioscler Thromb Vasc Biol. , 2008(28): p. 403-412. 7. Sanford J. Shattil and Peter J. Newman, Integrins: dynamic scaffolds for adhesion and signaling in platelets. Blood. 2004;104: 1606-1615 8. Bruce Furie., B.C.F., Thrombus formation in vivo. J Clin Invest, 2005. 115(12): p. 33553362. 9. Woollard, K.J., Geissmann, F. , Monocytes in atherosclerosis: subsets and functions. Nat Rev Cardiol., 2010. 7(2): p. 77-96. 10. Tanaka, K.A., MD, MSc; Key, Nigel S.,MD; Levy, Jerrold H., MD., Blood coagulation: Hemostasis and thrombin regulation. Anesth Analg. , 2009. 108: p. 1433-1446. 11. Orfeo T., B.S., Brummel-Ziedins K. E., Man K. , The Tissue Factor Requirement in Blood Coagulation. J Biol Chem. , 2005. 280: p. 42887-42896. 11. Rubio-Jurado, B., Salazar Páramo, M., Medrano Muñoz, F., González-Ojeda, A., Arnulfo Nava, TROMBOFILIA, AUTOINMUNIDAD Y TROMBOPROFILAXIS POSTOPERATORIA. Cir Ciruj., 2007. 75: p. 313-323. 12. Levi, M., Nieuwdorp, M., van der Poll, T., Stroes, E., Metabolic Modulation of InflammationInduced Activation of Coagulation. Sem Thromb Hemost., 2008. 43(1): p. 26-32. 13. Ridker PM, C.M., Stampfer MJ, Tracy RP, Hennekens CH. , Inflammation, aspirin, and risks of cardiovascular disease in apparently healthy men. N Engl J Med, 1997. 336: p. 973-979. 14. van Leuven, S.I., Franssen, R., Kastelein, J.J., Levi, M., Stroes, E.S.G., Tak, P.P., Systemic inflammation as a risk factor for atherothrombosis. Rheumatology, 2008. 47: p. 3–7. 15. Sepulveda JL, M.J., C-reactive protein and cardiovascular disease: a critical appraisal. Curr Opin Cardiol. , 2005. 20(5): p. 407-416 16. Tohgi H, K.S., Takahashi S, Koizumi D, Kondo R, Takahashi H, Activated coagulation/fibrinolysis system and platelet function in acute thrombotic stroke patients with increased C-reactive protein levels. . Thromb Res.2000; 100: 373–379, 2000. 100: p. 373379. 17. Di Napoli M, P.F., Inflammation, Hemostatic Markers, and Antithrombotic Agents in Relation to Long-Term Risk of New Cardiovascular Events in First-Ever Ischemic Stroke Patients. . Stroke. 2002; 33:1763-1771, 2002. 33: p. 1763-1771.

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36. Amara U, Rittirsch D, Flierl M, Bruckner U, Klos A, Gebhard F, Lambris JD, HuberLang M. Interaction between the coagulation and complement system. Adv Exp Med Biol. 2008;632:71-9. Review. PubMed PMID: 19025115; PubMed Central PMCID:PMC2713875 37. Hierholzer C, Billiar TR. Molecular mechanisms in the early phase of hemorrhagic shock. Langenbecks Arch Surg. 2001 Jul;386(4):302-8. Review. PubMed PMID: 11466573 38. Davis AE 3rd. Biological effects of C1 inhibitor. Drug News Perspect. 2004 Sep;17(7):439-46. Review. PubMed PMID: 15514703 39. Huber-Lang M, Sarma JV, Zetoune FS, Rittirsch D, Neff TA, McGuire SR, Lambris JD, Warner RL, Flierl MA, Hoesel LM, Gebhard F, Younger JG, Drouin SM, Wetsel RA, Ward PA. Generation of C5a in the absence of C3: a new complement activation pathway. Nat Med. 2006 Jun;12(6):682-7. Epub 2006 May 21. PubMed PMID: 16715088 40. Williamson D, B.K., Luddington R, Baglin C, Baglin T., Factor V Cambridge: a new mutation (Arg306-->Thr) associated with resistance to activated protein C. . Blood 1998. 91: p. 1140-1144. 41. Gutiérrez-Tous, M.R., Trombofilia, ¿cuándo, qué pruebas y a quién? . Sem Fund Esp Reumatol 2005. 6: p. 133-143. 42. De Stefano V, R.E., Paciaroni K, Leone G. , Screening for inherited thrombophilia: indications and therapeutic implications. . Haematologica 2002. 87: p. 1095-1108. 43. Kyrle Paul A., E.S., Deep vein thrombosis. . Lancet., 2005. 365: p. 1163-1174. 44. Hertzberg, M., .Genetic Testing for Thrombophilia Mutations. . Sem Thromb Hemost. , 2005. 31: p. 33-38. 45. Mannuccio., M.P., Genetic hypercoagulability: prevention suggests testing family members. Blood, 2001. 98: p. 21-22. 46. Johnson CM, M.L., Silver D. , Hypercoagulable states: a review. Vasc Endovascular Surg, 2005. 39: p. 123-133. 47. Samama MM, D.O., Mismetti P, et al. , An electronic tool for venous thromboembolism prevention in medical and surgical patients. Haematologica 2006. 91: p. 64-70. 48. Tofler GH, M.J., Levy D, Mittleman M, Sutherland P, Lipinska I, Muller JE, D'Agostino RB. , Relation of the prothrombotic state to increasing age (from the Framingham Offspring Study). Am J Cardiol 2005. 96: p. 1280-11283. 49. Xu J, L.F., Esmon CT, Inflammation, innate immunity and blood coagulation. Hamostaseologie, 2010. 30(1): p. 5-9. 50. Ortel Thomas, L., The syndrome antiphospholipid: What are we really measuring? How do we measure it? And how do we treat it? . J Thromb Thrombolysis 2006. 21: p. 79-83. 51. Pierangeli, S.S., Chen, P P., Raschi, E., Scurati, S., Grossi, C., Borghi, M A., Palomo, I., Harris, N., Meroni, P L., Antiphospholipid Antibodies and the Antiphospholipid Syndrome: Pathogenic Mechanisms. Semin Thromb Hemost 2008. 34: p. 236–250. 52. de Laat, B., Mertens, K., G de Groot, P G., Mechanisms of Disease: antiphospholipid antibodies—from clinical association to pathologic mechanism. nature clinical practice RHEUMATOLOGY, 2008. 4(4): p. 192-199. 53. Lim Wendy, C.M., Eikelboom John W. , Management of antiphospholipid antibody syndrome. . JAMA 2006; 295:1050-1057, 2006. 295: p. 1050-1057.

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54. Crowther, M.A., et al. , A comparasion of two intensities of warfarin for the prevention of recurrent thrombosis in patients with the antiphospholipid antibody syndrome. . N Eng J Med 2003; 349: 1133-1138, 2003. 349: p. 1133-1138. 55. Ishikura, K., Wada, H., Kamikura, Y., Hattori, K., Fukuzawa, T., Yamada, N., Nakamura, M., Nobori, T., Nakano T., High Prevalence of Anti-Prothrombin Antibody in Patients With Deep Vein Thrombosis. American Journal of Hematology, 2004. 76: p. 338–342. 56. Salmon, J.E., Girardi, G., Lockshin, M D., The antiphospholipid syndrome as a disorder initiated by inflammation: implications for the therapy of pregnant patients. NATURE CLINICAL PRACTICE RHEUMATOLOGY 2007. 3(3): p. 140-147. 57. de Groot PG, Derksen RH., Pathophysiology of antiphospholipid antibodies. Neth J Med., 2004. 62(8): p. 267-272. 58. Nojima J., I.Y., Suehisa E., Kuratsune H., Kanakura Y. , The presence of antiphosphatidylserine/prothrombin antibodies as risk factor for both arterial and venous thrombosis in patients with systemic lupus erythematous. . . Hematologica 2006; 91: 699-702, 2006. 91: p. 699-702. 59. Amengual, O., Atsumi, T., Koike, T., Specificities, Properties, and Clinical Significance of Antiprothrombin Antibodies. ARTHRITIS & RHEUMATISM, 2003. 48(4): p. 886– 895 60. Bas de Laat H., D.R., Urbanus RT., Roest M., de Groot PG. , ß2-glycoprotein Idependent lupus anticoagulant highly correlates with thrombosis in the antiphospholipid syndrome. Blood. 2004; 104:3598-3602 61. Salobir, B., Sabovic, M., Hojnikc, M., C ucnikc, S., Kveder, T., Anti-b2-glycoprotein I antibodies of IgM class are linked to thrombotic disorders in young women without autoimmune disease. Immunobiology 212 (2007) 193–199, 2007. 212: p. 193–199. 62. Giannakopoulos, B., Passam, F., Rahgoza, S., Krilis, S A., Current concepts on the pathogenesis of the antiphospholipid syndrome. Blood. , 2007. 109: p. 422-430. 63. de Groot PG, Derksen RH., Pathophysiology of the antiphospholipid syndrome. J Thromb Haemost 2007. 3: p. 1854–60. 64. Pierangeli, S., Chen, PP., González, E B., Antiphospholipid antibodies and the antiphospholipid syndrome:an update on treatment and pathogenic mechanisms. Curr Opin Hematol 2006. 13: p. 366–375. 65. Oku, K., Atsumi, t., Amengual, O., Koike, T., Antiprothrombin Antibody Testing: Detection and Clinical Utility. Semin Thromb Hemost 2008. 34: p. 335–339. 66. Bertolaccini, M., Gomez, S., Pareja, JF-, Theodoridou, A., Sanna, G., Hughes, GRV., Khamashta, MA., Antiphospholipid antibody tests: spreading the net., 2005. 64: p. 1639–1643. 67. Dalen, J.E., MD, MPH, Should Patients with Venous Thromboembolism Be Screened for Thrombophilia? Am J Med, 2008. 121(6): p. 458-463. 68. Perry, D.J., Antithrombin and its inherited deficiencies Blood Reviews, 1994. 8(1): p. 3755. 69. Rosendaal FR, K.T., Vandenbrouke JP, Reitsma PH. , High risk of thrombosis in patients homozygous for factor V Leiden (activated protein C resistance). . Blood 1995; 85:1504-8, 1995. 85: p. 1504-1508. 70. Majluf-Cruz, A., La realidad de la prevalencia de la trombosis. Gac Med Mex, 2003. 139(S2): p. 66-68.

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71. Ruiz-Arguelles GJ. Poblete-Naredo I., R.-N.V., Garcés-Eisele J., Lopez Martinez B., Gómez-Rangel JD. , Primary thrombophilia in Mexico IV: Frequency of the Leiden, Cambridge, Hong Kong, Liverpool and HR2 haplotype polymorphisms in the factor V gene of a group of thrombophilic Mexican Mestizo patients. Rev Invest Clin 2004. 56: p. 600-604. 72. Tripodi, A., Levels of coagulation factors and venous thromboembolism. Hematologica 2003. 88: p. 705-711.

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In: Coagulation: Kinetics, Structure Formation … Editors: A.M. Taloyan et al

ISBN: 978-1-62100-331-1 © 2012 Nova Science Publishers, Inc.

Chapter 4

FLOC CHARACTERISTICS AND THE INFLUENCING FACTORS Baoyu Gao and Weiying Xu Shandong Key Laboratory of Water Pollution Control and Resource Reuse, School of Environmental Science and Engineering, Shandong University, Ji‘ nan, China

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ABSTRACT Floc characteristics, including floc size, strength, re-growth abilities and fractal structure, are important factors in the water treatment works. These factors could essentially affect the solid/liquid separation process. Consequently, the investigations on the fundamental characteristics of flocs are meaningful and necessary for coagulation sedimentation units in water treatment works. The properties of flocs formed during coagulation were substantially dependent on coagulation conditions, such as properties of coagulants, coagulant dose, suspension pH, ionic strength, hydraulic conditions. In this chapter, the flocs formed in humic acids were taken as examples to present how the coagulation coagulant types, pH and hydraulic conditions affect the floc size, strength, re-growth abilities and fractal dimensions. Additionally, the opinions in this chapter were based on the investigation on various aluminum coagulants. The optimum conditions for coagulation, which contribute to the best floc characteristics, were also discussed.

Keywords: floc size, strength, fractal structure, shear rate

1. INTRODUCTION Natural organic matter (NOM) in rivers and lakes is a complex mixture of molecules with varied molecular weight (MW) and chemical nature. It can cause odor, taste, color and bacterial re-growth in potable water. Even worse, it is a precursor for disinfection byproducts, such as trihalomethanes (THMs) and haloaetic acids (HAAs), which are proven to be carcinogens [1]. Thus, effective removal of NOM is a major objective of modern drinking

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water treatment. At present, coagulation and flocculation has been the most common process used for the removal of NOM in water treatment. Polyaluminum chloride (PACl) is currently widely used as the coagulant to remove contaminants in water treatment, which has replaced many traditional aluminous coagulants (e.g. AlCl3 and alum) in recent years. Previous studies have shown that Al (Ⅲ) species in PACl can be driven into various hydrolyzed Al species such as monomeric and dimmer Al species (Al(OH)2+, Al(OH)2+ and Al(OH)4-, Al2(OH)24+, denoted as Alm), tridecamer (Al13O4(OH)247+, Al13 for short) and other unknown species including amorphous precipitate Al(OH)3(am) [2]. Al13 has been recognized as the most effective coagulation species due to its strong charge neutralization capability and high structure stability [3~5]. The aggregate characteristics, including floc size, strength, re-growth abilities and fractal structure, are believed to be the important factors influencing the efficiency of solid/liquid separation process in water or wastewater treatment works. There have been many studies on the properties of flocs formed during different coagulation systems. Boller and Blaser [6] claimed that small particles generally have lower removal efficiencies. Smaller particles will settle more slowly than larger particles of similar density. Additionally, floc strength is also a particularly important operational parameter in solid/liquid separation techniques. If the preformed flocs tend to be broken by slight increase in shear rate during water work unit process, the resultant small flocs would pose a challenge to the contaminant removal efficiency [7]. Floc strength depends greatly on the intensity of shear and the nature of the interaction between particles, which is closely associated with the characteristics of water sample and coagulants. It is known that bridging flocculation, by long-chain polymers, can give very strong flocs, whereas destabilization of particles by simple inorganic salts gives rather weak flocs [8]. The broken flocs are normally believed to re-grow to their previous sizes when the low shear conditions are restored. Yet, most previous reports proved that the floc breakage was irreversible to some extent and only limited re-growth occurred [9, 10]. For the coagulation of kaolin particles, alum and polyaluminum chloride have been shown to have the poorest regrowth, reaching only a third of their previous size after shear; poly (diallyldimethylammonium) chloride (polyDADMAC) showed nearly complete regrowth; and a copolymer of acrylamide and cationic monomer showed total regrowth [11]. These studies suggested that the particular coagulant used and thus the coagulation mechanism involved has a considerable impact on floc regrowth potential. Besides, fractal characteristic of flocs is another particularly critical parameter that deserves special attention. The complicated structure of floc aggregates is now well described by the fractal geometry theory [12-14], where the floc structure is simply described by the mass fractal dimension Df. Commonly, compact aggregates have higher Df, while aggregates with loose structures have lower Df values. Recently, small angle laser light scattering (SALLS) has been successfully used for the determination of Df of aggregates with a wide range of particle sizes, including tiny hematite paticles (5~13 m) [15], medium salt-humic flocs or salt-kaolin flocs (60~650 m) [16, 17] and large ferric precipitate (1000 m) [7]. SALLS has been well described in detail previously [13, 14] and the method is briefly given as follows: the scattered intensity I is a function of the magnitude of the scattering wave vector Q, which is shown as:

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97

Floc Characteristics and the Influencing Factors

Q=

4 n sin( / 2)

(1)

where, n, θ and λ are the refractive index of the medium, the scattered angle, and the wavelength of radiation in vacuum, respectively. For independently scattering aggregates, I is related to Q and the fractal dimension Df [18]:

I

Q

Df

(2)

So, on a log–log scale if there is a straight line, the slope of which is Df. This relationship is valid only when the length scale of the analysis is much larger than the size of primary particles and much smaller than the size of floc aggregates [7, 14]. This chapter aims to make a comparative investigation on the influences of coagulant type, dose, pH and shear conditions on the properties of flocs. Humic acid (HA) and natural surface water in Northern China were chosen as the study object and a series of Al-based coagulants were used as coagulants. HA accounts for about 50–90% of the total freshwater organic matter [19] and presents a yellowish or brown color in water. The high affinity of HA for complexation with various pollutants including heavy metals and pesticides causes contamination of surface water and more importantly, HA has been recognized as a significant precursor of DBPs [20, 21]. Thus, it is quite necessary and meaningful to explore the properties of HA flocs and then to provide a theoretical basis for the improvement of HA removal.

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2. MATERIALS AND METHODS 2.1. Coagulant Preparation and Characteristics Alum coagulant was prepared by directly dissolving Al2 (SO4)3·18H2O into deionized water. PACl was firstly synthesized by adding pre-determined amount of Na2CO3 and AlCl3 slowly into deionized water under intense agitation. The temperature was kept at 70.0±0.5 ℃ by recycling water bath. Al13 species was separated from the PACl by ethanol/acetone separation method. The total Al content in the pre-hydrolyzed Al13 species was determined by titrimetric method according to the national standard of China [22]. All the reagents used were of analytical grade and deionized water was used to prepare all solutions. The properties of alum, PACl and Al13 polymer are summarized in Table 1. As seen in Table 1, both the aluminum coagulants contain three kinds of Al species, including the Alm species, Al13 and the Alother species as mentioned above. However, there were distinct differences in the distribution of three kinds of Al species: the proportions of Alm and Alother in PACl were larger than those in Al13 prepared, whereas the proportion of Al13 species in the latter was 93.27 %, which was much higher than that in alum and PACl.

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2.2. Water Samples HA used in this study was commercial reagent grade solid (Shanghai, China) and the molecular weight of the HA was greater than 30 kDa. Table 1. Al species distribution of different coagulants Coagulants Alum PACl Al13

AlT (mol/L) 1.025 1.012 0.083

B 0 2.0 2.0

Al species distribution (%) Alm Al13 Aother 82.06 9.42 8.52 29.26 45.61 25.13 1.79 93.27 4.94

Preparation of stock solution: 1.0 g of HA was dissolved in 1 L of deionized water that contained 4.2 g of NaHCO3, under continuous stirring for 3.0 h, then stored in refrigerator at 4 ℃ for later use. The synthetic test water was prepared by dissolving 10.0 mL of HA stock solution in deionized water and diluting the solution to 1 L. The property of the synthetic test water used was as follows: UV254 = 0.297 ± 0.02, TOC = 4.75 ± 0.10 mg L−1, pH = 7.9 ± 0.2.

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2.3. Jar Tests and Floc On-Line Monitor Jar tests were performed at room temperature of 20 ± 1 ℃ using a conventional jar-test apparatus (ZR4-6, Zhongrun Water Industry Technology Development Co. Ltd., China) with 1.0 L cylindrical plexiglass beakers. A total volume of 500 mL of HA test water, previously adjusted to a desired pH with 0.1 mol L-1 NaOH and 0.1 mol L-1 HCl, was transferred into the beaker. Predetermined amount of coagulant was added at the start of coagulation. Thereafter, 1.5 min rapid mixing at 200 rpm was applied, followed by 10 min of slow stirring at 40 rpm. Then after 30 min of quiescent settling, the sample was collected from 2 cm below the surface for measurement. A filtered sample through a 0.45 mm glass fiber membrane was tested for UV254 absorbance at 254 nm wavelength using an UV-754 UV/VIS spectrophotometer (Precision Scientific Instrument Co. Ltd., Shanghai, China). Floc breakage tests were programmed similarly as aforementioned. However, after the slow stir phase (15 min of 40 rpm), the suspension was exposed to a high shear of 200 rpm (or 50, 75, 100, 150 and 300 rpm) for a further 5 min, and then a slow stirring at 40 rpm was reintroduced for flocs re-formation. A continuous laser diffraction instrument (Mastersizer 2000, Malvern, U.K.) was used to measure the dynamic floc size as the coagulation proceeded. Size measurements were taken every 30 s for the duration of the jar test.

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Floc Characteristics and the Influencing Factors

99

3. CHARACTERIZATION OF FLOCS FORMED BY DIFFERENT AL-BASED COAGULANTS 3.1. Floc Formation, Breakage and Re-Growth First, the coagulation efficiencies of three kinds of Al-based coagulant, i.e. alum, PACl and nano-Al13, were investigated. UV254 removal and zeta potentials were separately assessed and the results are shown in Figs. 1 and 2. 100 90

Removal efficiency (%)

80 70 60 50

Alum PACl Al13

40 30 20 10 2

4

6

8

10

Coagulant dose (mg/L as Al)

Figure 1. Humic acid (HA) removal efficiency as a function of coagulants dosages.

6

alum PACl Al13

4

0

Zeta potential (mV)

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2

-2 -4 -6 -8 -10 -12 -14 -16 -18 0

2

4

6

8

10

Coagulant dose (mg/L as Al)

Figure 2. Variations of zeta potential with the increasing coagulant dosage.

It could be observed that when the dose was less than 6 mg/L, the removal efficiency increased dramatically with dose for all the coagulants. With the dose further increased, the removal efficiency was obstructed and even decreased slightly when the sample water was pre-treated by Al13. Pre-hydrolyzed aluminum coagulants, i.e. PACl and Al13 displayed better HA removal efficiency than alum at the dose below 6.0 mg/L. Zeta potentials results as shown in Fig. 2 indicate that the zeta potentials caused by different Al coagulants are in the

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Baoyu Gao and Weiying Xu

following order: Al13 > PACl > alum. The stable structure and better charge neutralization ability of Al13 polymer enabled it to destabilize the HA molecules efficiently even at very low dose [23]. Based on the comprehensive consideration of the cost of coagulants and coagulation performance, the dosages of alum, PACl and Al13 polymer for the subsequent floc property studies were chosen as 7.0 mg L-1. Floc strength and re-growth ability were investigated in the breakage mode. The jar tests procedure was as follows: the suspension was rapidly mixed at 200 revolutions per minute (rpm) for 1.5 min, and then slowly mixed at 40 rpm for 15 min. When the floc reached the stead-state, an increased shear rate at 200 rpm was introduced for 5 min to break up the generated flocs. After the breakage phase, a slow stirring at 40 rpm was reintroduced for a further 15 min to allow flocs re-formation. Fig. 3 shows the floc breakage and re-growth profiles versus coagulation time. The 50 percentile floc size (d0.5) was used to denote the floc size.

400

alum PACl Al13

Floc size d0.5 (μm)

300

200

100

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0

0

500

1000

1500

2000

2500

Time (s)

Figure 3 The changes of floc size versus time for alum, PACl and Al13 at the same dosage of 7.0 mg/L.

It was found that floc formation and breakage varied obviously with different coagulants. Initially, floc size increased rapidly with coagulation time during the first several minutes, followed by a gentle growth, and finally reached a plateau, which has been generally accepted as a result of equilibrium between floc formation and floc breakage [24]. The floc steady-state size increased with the decrease of Al13 proportion in coagulants. That is, alum contributed to the largest floc with size of about 400 μm, and PACl produced aggregates with slightly smaller size of about 385 μm. The flocs formed by Al13 were the smallest, the size of which was around 300 μm. When the suspension was exposed to an enhanced shear at 200 rpm, a significant drop in flocs size could be observed immediately in all the cases, and then a gradual decline followed. The flocs began to re-grow again when the shear was reduced. However, the floc could not re-grow to their previous sizes at the steady phases. Also, the extent of floc breakage and re-growth varied for different coagulants. In order to characterize the floc breakage and re-formation, strength factor and recovery factor were calculated, which have previously been used to compare the relative breakage and re-growth of flocs [25]. They were calculated as follows:

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Floc Characteristics and the Influencing Factors Strength factor

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Recovery factor

(Sf) =

(Rf) =

d2 d1

d3 d 2 d1 d 2

100

(3)

100

(4)

The results are summarized in Table 2. It can be found that the shear did not break flocs to the same degree for different coagulants. Strength factors were in the following sequence: Al13 > PACl > alum. This could be attributed to the distinct floc size caused by different coagulants. The larger flocs caused by alum and PACl (around 400 μm and 385μm, respectively) were more exposed to micro scale energy-dissipating eddies that gave rise to floc breakage, while smaller HA-Al13 flocs (around 300 μm according to Fig. 3) were more likely to become entrained within these eddies rather than being broken by them. Similar views have been previously proposed by Boller and Blaser [26]. Another additional explanation could be devoted to the generated Al13 aggregates. It was previously claimed that Al13 remains stable at acidic pH, while it aggregates above pH 6.5 and precipitates form in alkaline condition [27, 28]. It is believed that the Al13 aggregates has much more branched structure and the bridging of particles onto Al13 aggregates could markedly enhance the flocs strength [24, 29]. The variation of recovery factor (Rf) for different coagulants followed the same trend with Sf. Al13 contributed to the largest Rf of 27.85, followed by that of HA-PACl flocs, followed by that for flocs formed by alum. The results could be interpreted in terms of the distinct charge neutralization of alum and the pre-hydrolyzed coagulants. During the break-up process, the broken flocs with newly exposed surfaces of aggregates may have a net negative, positive or neutral charge and it may depend on the nature of the original coagulation process [30]. Electrostatic patch mechanisms generally occurred between the polymers and negative particles when cationic Al polymers were applied in coagulation. The high positively charged Al13 could effectively neutralize the negative charge on the surface of the HA microflocs and consequently, the small flocs with new surfaces were all neutral and could re-form easily. In the case of alum and PACl, however, the negative charges on microflocs can not be neutralized completely. Thus the smaller flocs formed by higher shear may not be neutral and the charge on the surfaces of new flocs would resist the re-growth of flocs. Thus the results suggest that Al13 species is beneficial to both floc strength and re-formation after being broken. Table 2. Strength and recovery factors of flocs for different Al-based coagulants Coagulants Alum PACl Al13

Strength factor (Sf) 28.56 29.60 34.00

Recovery factor (Rf) 22.49 23.88 27.85

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Baoyu Gao and Weiying Xu

3.2. Effect of Shear Rate on Floc Size Gregory [31] states that when comparing different flocs, the size for a given shear rate indicates floc strength. However, this is the case for the given shear condition under which the flocs were formed, and it does not give an indication of how flocs will behave upon exposure to an increased shear rate, as could occur in a water treatment work when flocs are transferred from flocculators or to a higher shear treatment processes such as dissolved air flotation or high rate filtration. In order to investigate the relationship between shear rate and the resultant floc size, a series of high shear rate (50, 75, 100, 150, 200 and 300 rpm) was employed in the floc breakage phase as aforementioned. Particle size was monitored after exposure to each level of shear. The rate at which a floc suspension decays on exposure to shear is indicative of the strength of the flocs within the system. Previous studies revealed that the relation on a log-log scale between the average velocity gradient G in the flocculation and the floc size of the suspension in equilibrium [24, 32]: lg d =lg C － γ lg G

(5)

where d is the floc diameter; C is the floc strength constant that strongly depends on the method used for particle size measurement; G is the average velocity gradient and γ is the stable floc size exponent dependent upon floc break-up mode. The slope of the line (γ) gives an indication of the rate of degradation. A larger γ value is indicative of floc that is more prone to breaking into smaller size under an increasing shear force.

450 400 350

250

Floc size d0.5 (μm)

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300

200 150

100

alum, 6.0 mg/L solid line (slope=0.78, R2=0.97) PACl, 6.0 mg/L dash line (slope=0.74, R2=0.96) Al13, 6.0 mg/L dot line (slope=0.65, R2=0.94) 50 50

100

150

200

250

300 350

Rotary rate (rpm)

Figure 4. The relationship between the change in particle size and an increase in rotary rate for various Al-based coagulants at dosage of 7.0 mg/L.

In this chapter, the rpm instead of G was used and an adapted version of Equation 5 was used as shown below: Coagulation: Kinetics, Structure Formation and Disorders : Kinetics, Structure Formation and Disorders, Nova Science Publishers, Incorporated,

103

Floc Characteristics and the Influencing Factors lg d =lg C‘ － γ‘lg rpm

(6)

where d is the floc diameter, C‘ the modified floc strength constant, rpm the revolutions per minute on the jar tester and γ‘ a fitting coefficient. The results were quantificationally shown in Fig. 4. It was observed that the flocs presented continuous reduction in size with the increasing shear rate, which agreed with other researchers [33]. The floc size after 5 min of elevated shear was plotted against rpm and a straight line could be drawn through the data on a log-log scale. As mentioned above, the slope (γ‘) of this line was an indication of the floc degradation rate. The decreasing slope γ‘ for different coagulants was in the following hierarchy: alum > PACl > Al13. This come to the same conclusion as the investigations on floc strength factors, which confirmed that the flocs formed by Al13 were most resistant to exposure to elevated shear, followed by the flocs formed by PACl and then those by alum.

3.3. Floc Fractal Structure Analysis 6 2

alum, Df=2.30, R =0.9967 2

PACl. Df=2.35, R =0.9931

4

2

Log I

Al13, Df=2.43, R =0.9974 2

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0

-2

-4 -6.0

-5.5

-5.0

-4.5

-4.0

-3.5

-3.0

-2.5

-2.0

-1

Log Q (nm )

Figure 5. Relationship between the scattered light intensity (I) and the wavenumber (Q) on a log-log scale and the determination of floc fractal dimension (Df ) for various coagulants.

The structures of flocs formed by different coagulants were investigated in terms of fractal dimension (Df). The Df was derived from the relationship of scattered light intensity I and the scattering vector Q as shown in Eq. 2. To keep the power law relationship valid as shown in Eq. 2, the values of log Q for Df calculation were held in the range of -3.5 ~ -1.75 [7]. According to Fig. 5, there was good linear correlation for all of the data sets with R2 values (R-correlation coefficient) exceeding 0.99 and all of the Df were between 2.30 and 2.43. HA-Al13 floc presented the highest compact degree with the Df value of 2.43; while the fractal dimension values for HA-PACl and HA-alum were much looser with the Df value of 2.35 and 2.3, respectively. It was due to the higher positive charge of Al13 polymer as shown in Fig. 2, which

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Baoyu Gao and Weiying Xu

greatly weakened the repulsive forces between particles within the aggregates and hence, led to a high degree of compaction. Comparatively, the absorption, entrapment of contaminants by Al hydrolyzate in alum and PACl coagulations resulted in fluffy structures, which has been reported in the researches by Wu et al. [34] using a similar technique. 2.60 2.55

breakage (200 rpm) re-growth (40 rpm)

2.50

Fractal Dimension (Df)

2.45 2.40 2.35 2.30 2.25 2.20

alum PACl Al13

2.15 2.10 2.05 2.00 0

5

10

15

20

25

30

35

40

Time (mins)

100

a 90

Removal efficiency (%)

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Figure 6. Change in fractal dimension of flocs with time for different Al coagulants at dosage of 7.0 mg/L after growth (40 rpm) and breakage (200 rpm) followed by a return to the initial 40 rpm for regrowth.

80

70

60

Alum PACl Al13

50

40 4

5

6

7

8

9

pH

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Floc Characteristics and the Influencing Factors

5 4

Alum PACl Al13

b

3 2

Zeta potential (mV)

1 0 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 4

5

6

7

8

9

pH

Figure 6. Influence of pH on coagulation efficiencies and zeta potentials for alum, PACl and Al 13 at the dosage of 7.0 mg/L.

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Curves similar to those shown in Fig.5 were formed for the coagulating suspensions as they grew, broke, and subsequently re-grew (Fig. 6). The results demonstrated how the degree of floc compaction changed with the variation of shear rate conditions in the system. The most significant observation was that the Df of HA-Al flocs increased dramatically after being exposed to high shear force. This was due to the restructuring of primary flocs during exposure to high shear rate, such that flocs were broken into smaller aggregates and the primary particles were re-arranged into stable and more compact floc structures [35].

4. EFFECT OF PH ON FLOC PROPERTIES 4.1. Effect of PH on Coagulation Efficiency The sample pH was adjusted by HCl and NaOH solutions, and the coagulation experiments were performed on the jar tester at the alum, PACl and Al13 dosage of 7.0 mg/L. The variation of UV254 removal and zeta potentials with pH are shown in Figs. 6a and 6b. It can be found that the HA removal efficiency increased first with increasing pH at pH below 6.0, and as pH further increased, the coagulation efficiency decreased slightly. Charge neutralization was the dominant coagulation mechanism at acidic pH, while in alkaline condition, the HA was removed through both charge neutralization and adsorption on amorphous aluminum hydroxide. At pH lower than 6.0, Al13 and PACl contributed to obviously higher removal efficiency than alum, which could be attributed to the high charge neutralization abilities of Al13 and PACl as shown in Fig. 6b. With pH changes, the coagulation behaviors of Al13 and PACl were less volatile in comparison with alum due to the stable structures of the pre-hydrolyzed coagulants. The experiments results suggested that the optimum pH for Al-based coagulants was around 6.0.

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4.2. Effect of PH on Floc Formation, Breakage and Re-Growth Floc formation, breakage and re-growth tests were conducted at a series of pH values (4, 5, 6, 7, 8 and 9). The floc evolution was monitored continuously and the results for alum, PACl and Al13 are shown in Fig. 7. The results verified that pH essentially had remarkable influence on the particles aggregation. Different pH values led to different floc formation curves, and consequently produced various floc breakage and re-growth profiles. Strength factor (Sf) and recovery factor (Rf) are calculated and summarized in Table 3. From an overall perspective, a generic trend was found in all the cases, which demonstrated that the Sf and Rf decreased with the increasing pH. This indicated that acidic pH resulted in flocs with better strength and re-growth abilities. It is also worth noticing that Al13 led to lower Sf than alum and PACl at pH 4.0. This could be attributed to the complexation reactions between HA and Al species, which played a significant role during coagulation processes at acidic pH [36]. It was claimed that the Alm species in alum and PACl could react with HA to form octahedrally coordinated complexes. This complex was thought as the initial flocs in HA-Alm flocs formation process, which would join with larger flocs by charge neutralization [37]. Likewise, Al13 polymer could react with HA to form HA–Al13 complex, which could be regarded as the initial flocs in Al13 polymer treatment. However, the HA-Al m complex was much stronger than HA-Al13 complex [37]. Thus, flocs formed with alum and PACl were stronger at acidic pH. At alkaline pH, the adsorption and enmeshment of particles onto Al hydroxide precipitates produced much larger flocs and large aggregates were more readily to be broken as mention above. This is the reason why flocs formed at alkaline pH were weaker. Additionally, the flocs in the coagulation systems were in the following order: alum > PACl > Al13, as shown in Fig. 7. As a consequence, the floc Sf decreased with the floc size sequence for different coagulants. The higher Rf in acidic ambience could be ascribed to the predominant role of charge neutralization at pH 4.0, which facilitated the re-formation of broken aggregates fractions. Table 3. Strength and recovery factors of flocs at various pH for different Al-based coagulants pH Strength factor

Recovery factor

4

5

6

7

8

9

alum

49.49

33.99

31.35

30.38

27.21

21.66

PACl Al13 alum

40.49 37.22 39.77

37.31 38.95 32.20

33.18 37.28 27.56

32.15 36.08 25.62

29.61 34.66 22.79

24.80 28.53 17.73

PACl Al13

43.40 44.73

35.19 37.81

30.54 35.01

27.83 32.06

26.25 27.60

18.10 22.53

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Floc Characteristics and the Influencing Factors

450

alum

400

pH 4.0 pH 5.0 pH 6.0 pH 7.0 pH 8.0 pH 9.0

Floc size d50 (μm)

350 300 250 200 150 100 50 0 -50 0

500

1000

1500

2000

2500

Time (s)

450 400

PACl

pH 4.0 pH 5.0 pH 6.0 pH 7.0 pH 8.0 pH 9.0

Floc size d50 (μm)

350 300 250 200 150 100 50 0

0

500

1000

1500

2000

2500

Time (s)

400

Al13

pH 4.0 pH 5.0 pH 6.0 pH 7.0 pH 8.0 pH 9.0

350 300

Floc size d50 (μm)

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-50

250 200 150 100 50 0 0

500

1000

1500

2000

2500

Time (s)

Figure 7. Floc formation, breakage and re-growth profiles at various pH for different coagulants at dosage of 7.0 mg/L. Coagulation: Kinetics, Structure Formation and Disorders : Kinetics, Structure Formation and Disorders, Nova Science Publishers, Incorporated,

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4.3. Effect of PH on Floc Fractal Structures Floc fractal dimension formed in various pH ambiences was measured using the same method as shown in Fig. 5. The Df of HA flocs formed by alum, PACl and Al13 were presented in Fig. 8. An visible decrease in Df with increasing pH was found in all the cases. This indicated that charge neutralization at acidic pH resulted in compact aggregates, while bridging, adsorption onto amorphous precipitates were inclined to bring about loose, open and more branched structure of flocs. In addition, the aggregates compaction degree could be improved by the increasing Al13 species in the Al coagulants.

2.60 2.55 2.50

Fractal dimension (Df)

2.45 2.40 2.35 2.30 2.25 2.20 2.15

alum PACl Al13

2.10 2.05 2.00 1.95 1.90 4

5

6

7

8

9

pH

Figure 8. Variation of floc fractal dimension with pH for different Al coagulants.

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CONCLUSION This chapter mainly studied the influences of coagulant type, hydraulic conditions and solution pH on the characteristics of flocs formed by different Al coagulants, i.e. traditional alum, PACl and novel Nano-Al13 coagulants. The main conclusions are as follows: (1) The pre-hydrolyzed Al coagulants, PACl and Al13 species, presented higher coagulation efficiencies than the traditional Al coagulant alum, especially at low dosage. The stable structure of PACl and Al13 enabled them to display less volatility in coagulation efficiency with the changes of pH. (2) High Al13 species proportion in Al coagulants gave rise to aggregates with high strength and better ability to re-growth after being broken. Also, the flocs formed by pre-hydrolyzed coagulants were more compact. (3) Floc size decreased with the decreasing shear force and the enhanced shear force could improve the floc compact degree. (4) Solution pH had significant influence on floc characteristics, including floc size, strength, re-growth and fractal structure. Acidic pH was apt to produce small aggregates with high strength and better re-growth ability than alkaline pH. Additionally, acidic ambience also gave rise to compact flocs with higher fractal dimensions.

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preparation, Water Res., 19 (1985) 25-36. [4] Hu, C., Liu, H., Qu, J., Wang, D., Ru, J. Coagulation behavior of aluminum salts in eutrophic water: significance of Al13 species and pH control, Environ. Sci. Technol., 40 (2006) 325-331. [5] Gao, B., Zhang, Z., Ma, J., Cao, X. Solid–solid mixed method to prepare polyaluminum chloride, Environ. Chem., 24 (2005) 569-572 (in Chinese). [6] Boller, M., Blaser, S. Particles under stress, Water Sci. Technol. 37 (1998) 9–29. [7] Jarvis, P., Jefferson, B., Parsons, S. Breakage, Regrowth, and Fractal Nature of Natural Organic Matter Flocs, Environ. Sci. Technol. 39 (2005) 2307-2314. [8] Bolto, B.A. Soluble polymers in water purification, Prog. Polym. Sci. 20 (1995) 987– 1041. [9] Clark, M.M., Flora, J.R. Floc restructuring in varied turbulent mixing, J. Colloid Interface Sci. 147 (1991) 407– 421. [10] Francois, R.J., Van Haute, A.A. Floc strength measurements giving experimental support for a four level hydroxide floc structure. Stud. Environ. Sci. 23 (1984) 221– 234. [11] Yukselen, M. A., Gregory, J. The reversibility of floc breakage, Int. J. Miner. Process 73 (2004) 251-259. [12] Gregory, J. The role of floc density in solid-liquid separation, Filtr. Separat. 35 (1998) 367-371. [13] Tang, S. A model to describe the settling behavior of fractal aggregates, Colloids Surf. A 157 (1999) 185-192. [14] Waite, T.D., Cleaver, J.K., Beattie, J.K. Aggregation kinetics and fractal structure of çalumina assemblages, J. Colloid Interface Sci. 241 (2001) 333-339. [15] Lee, S.Y., Anthony, A., Fane, G., Waite, T.D. Impact of natural organic matter on floc size and structure effects in membrane filtration, Environ. Sci. Technol. 39 (2005) 6477-6486. [16] Li, T., Zhu, Z., Wang, D.S., Yao, C.H., Tang, H.X., Characterization of floc size, strength and structure under various coagulation mechanisms, Powder Technol. 168 (2006) 104-110. [17] Wei, J.C., Gao, B.Y., Yue, Q.Y., Wang, Y., Li, W.W., Zhu, X.B.. Comparison of coagulation behavior and floc structure characteristic of different polyferric-cationic polymer dual-coagulants in humic acid solution, Water Res. 43 (2009) 724-732. [18] Lin, M.Y., Lindsay, H.M., Weita, D.A., Ball, R.C., Klein, R. and Meakin, P. Universality in colloid aggregation, Nature 339 (1989) 360-362.

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[19] Zumstein, J., Buffle, J. Circulation of pedogenic and aquagenic organic matter in an eutrophic lake, Water Res. 23 (1989) 229–239. [20] Reckhow, D.A., Singer, P.C., Malcolm, R.L. Chlorination of humic materials: Byproduct formation and chemical interpretations, Environ. Sci. Technol. 24 (1990) 1655–1664. [21] Christman, R.F., Norwood, D.L., Millington, D.S., Johnson, J.D., Stevens, A.A. Identity and yields of major halogenated products of aquatic fulvic acid chlorination, Environ. Sci. Technol. 17 (1983) 625–628. [22] GB15892-1995, Water Treatment Chemicals-Polyaluminum Chloride, National Standards of the People’s Republic of China (in Chinese). [23] Wang, Y., Gao, B.Y., Xu, X.M., Xu, W.Y., Xu, G.Y. Characterization of floc size, strength and structure in various aluminum coagulants treatment, J. Colloid Interf. Sci. 332 (2009) 354-359. [24] Jarvis, P., Jefferson, B., Gregory, J., Parsons, S.A. A review of floc strength and breakage, Water Res. 39 (2005) 3121–3137. [25] Francois, R.J. Strength of aluminium hydroxide flocs, Water Res. 21 (1987) 1023–1030. [26] Boller, M., Blaser, S. Particles under stress, Water Sci. Technol. 37 (1998) 9-29. [27] Dubbin, W.E., Sposito, G. Copper-glyphosate sorption to microcrystalline gibbsite in the presence of soluble Keggin Al-13 polymer, Environ. Sci. Technol. 39 (2005) 25092514. [28] Furrer, G., Ludwig, C. Schindler, P.W. On the chemistry of the Keggin Al13 polymer. 1. Acid–base properties, J. Colloid Interface Sci. 149 (1992) 56-67. [29] Lin, J.L., Chin, C.J.M., Huang, C.P., Pan, J.R., Wang, D.S. Coagulation behavior of Al13 aggregates, Water Res. 42 (2008) 4281-4290. [30] Chu, Y.B., Gao, B.Y., Yue, Q.Y., Wang, Y., 2007. The effect of cycle shear and suifate on dynamic coagulation of alum coagulants. Sci. China Ser. B Chem. 37 (5), 440–445 (in Chinese). [31] Gregory, J. Monitoring floc formation and breakage. In: Proceedings of the Nano and Micro Particles in Water and Wastewater Treatment, Conference of International Water Association, September, 2003, Zurich. [32] Yeung, A.K. and Pelton, R. Micromechanics: a new approach to studying the strength and breakup of flocs, Journal of Colloid Interface Science 184 (1996) 579–585. [33] Bache, D.H., Rasool, E., Moffatt, D. and McGilligan, F.J. On the strength and character of alumino-humic flocs, Water Science Technology 40 (1999) 81–88. [34] Wu, R.M., Lee, D.J., Waite, T.D., Guan, J. Multilevel structure of sludge flocs, J. Colloid Interface Sci. 252 (2002) 383-392. [35] Spicer, P.T., Pratsinis, S.E., Raper, J., Amol, R., Bushell, G. and Meesters, G.. Effect of shear schedule on particle size, density, and structure during flocculation in stirred tanks, Powder Technology 97 (1998) 26-34. [36] Gregor, J.E., Nokes, C.J. and Fenton, E. Optimising natural organic matter removal from low turbidity waters by controlled pH adjustment of aluminium coagulation, Water Research 31 (1997) 2949–2958. [37] Hiradate, S. and Yamaguchi, N.U. Chemical species of Al reacting with soil humic acids, Journal of Inorganic Biochemistry 97 (2003) 26-31.

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In: Coagulation: Kinetics, Structure Formation... Editors: A.M. Taloyan et al

ISBN 978-1-62100-331-1 c 2012 Nova Science Publishers, Inc.

Chapter 5

S UBSTRATE I NDUCED C OAGULATION (SIC) IN AQUEOUS AND N ON -AQUEOUS M EDIA FOR THE P REPARATION OF A DVANCED B ATTERY M ATERIALS Angelika Basch∗ 1) Institute of Physics, University of Graz, Graz, Austria 2) Centre for Sustainable Energy Systems, The Australian National University, Canberra, Australia

Abstract

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Substrate induced coagulation (SIC) is a dip-coating method capable of coating chemically different surfaces with finely dispersed nano-sized solid particles. In the first step of an SIC process the surface is conditioned with a thin layer of polymer or polyelectrolyte. In the second step the conditioned surface is dipped into a dispersion that is close to a point where coagulation occurs. The polymer or polyelectrolyte on the surface induces coagulation of the nano-sized material and a thin layer of the solid is formed. The SIC process can be performed in aqueous as well as non-aqueous media as long as the particles are stabilised (or destabilised) by electrostatic repulsive (or attractive) forces. The theory of Derjaguin, Landau, Verwey and Overbeek (DLVO), describing colloidal stability as well as the zeta-potential and surface charging of dispersed particles are discussed for aqueous and non-aqueous dispersions and some experimental difficulties adressed. For example trace water can have a profound impact on the colloidal stability of non-aqueous dispersions. Dispersions in polar and non-polar media carbon black, titania and alumina, materials so far used for SIC, are discussed. Lithium cobalt oxide has the ability to intercalate lithium ions reversibly. It is the preferred cathode material because it is easy to prepare and has a high specific capacity. Disadvantages of this material are its low electronic conductivity and high reactivity when charged (delithiated). A non-aqueous SIC process was used to prepare lithium cobalt oxide with an improved conductivity by coating with highly conductive carbon black. Furthermore, core-shell cathode materials of lithium cobalt oxide with ∗

E-mail address: [email protected]

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Angelika Basch a protective layer to reduce the materials reactivity when charged (delithiated), are prepared by coating with titania via an aqueous SIC process.

Keywords: Non-aqueous dispersions, trace water, Li-ion battery, zeta-potential, substrate induced coagulation. PACS 82.47.Aa Lithium ion batteries, 68.05.-n Liquid solid interfaces, 82.7.Dd Colloids, 82.70.Uv Surfactants, 83.80.Hj Suspensions, dispersions, pastes, slurries, colloids, 82.45.-h Electrochemistry and electrophoresis.

1.

Substrate Induced Coagulation (SIC) in Aqueous and Nonaqueous Media

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Coagulation in a dispersion occurs at a microscopic level when particles collide and form duplets, triplets and so on until they form macroscopic flocs that settle rapidly. Particles need a kinetic barrier to form a stable dispersion. Substrate Induced Coagulation (SIC) is a dip coating method that lowers this barrier and is shown in Figure 1. It induces coagulation on a pretreated surface and that is able to coat chemically different surfaces with nano-sized solids [15], [63], [7], [8], [13], [12], [11]. The method is well established for aqueous systems and can therefore be used for any kind of material that is stable in water. Recently, it has been shown that SIC can be performed as well in non-aqueous media [5], [6], [3], [4].

Figure 1. Substrate Induced Coagulation. The substrate is dipped into a polyelectrolyte solution to condition the substrate surface (a.); Rinsing removes any unbound polymer (b.); the conditioned surface is then dipped in a dispersion of particles that coagulate on the surface (c.) and rinsed again to remove unbound particles (d.). With kind permission from Springer Science + Business media: [3]. In both aqueous and non-aqueous SIC the material is firstly conditioned with a thin organic layer by dipping the substrate into a solution of polymer or polyelectrolyte such

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Substrate Induced Coagulation in Aqueous and Non-aqueous ...

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as gelatin, polyvinylalcohol, polyvinylacetate or carboxymethylcellulose. In both cases the substrate has to be capable of adsorbing this conditioner. The substrate can be flat [38] or in the in the form of separated particles [4], [2]. Depending on the concentration, the materials and the coating conditions (such as dipping-time or number of rinses), the polymer layer can vary in thickness from 30 to 200 nm for a single coating. The polymer is usually transparent to visible light and can be removed by gently heating in oxygen (350 to 450 ◦ C). After a rinsing step the substrate is dipped into a dispersion of nano-sized solids that is close to a point where coagulation can occur. The adsorbed polymer or polyelectrolyte serves as a socalled coagulation initiator. The thickness of the layer of coagulated particles depends on the concentration and the stability of the dispersions. Dispersions used for SIC are usually stabilized by a suitable surfactant and electrolytes and are decribed in Section 2. in more detail. The molecular bridging of the conditioner is so effective that the layer of finely dispersed nano-sized solid particles is rinse-proof. The coverage of the substrate is dependent on the concentration of the dispersed solids, the conditioner and the coating conditions and can vary in thickness from 20-200 nm for a single coating. Multiple coatings as shown in Figure 2 offer the possibility to manufacture thicker layers. But it is also possible to form gradient layers by e.g. starting with dispersed solids of smaller particles and inreasing the size of the particles; or using different materials for each coating step as depicted in Figure 3.

Figure 2. Stepwise preparation of a Li4 Ti5O12 electrode using SIC. Reprinted from [38], with permission from Elsevier.

Figure 3. Principle of SIC composite coating process with different coating materials during the coating sequence. Reprinted from [38], with permission from Elsevier. SIC has shown promising results when used for the manufacturing of highly conductive composite cathodes [4] as well as the preparation of core-shell cathode materials [15], [2]. Both topics are described further in Section 3. The SIC process has also been applied to coat printed wiring boards with a conducting carbon black layer [9], [14] and to coat non-conductive surfaces such as teflon (PTFE) with conducting particles such as highly

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conductive carbon black [63]. Aqueous SIC has been used for the layer-by-layer preparation of electrodes or composite supercapacitors with defined thickness by multiple use of the coating process [38].

1.1.

DLVO Theory

The theory of Derjaguin, Landau, Verwey and Overbeek (DLVO) is a physical model of colloid stability and is widely used to explain colloidal stability and other phenomena arising from charging of particles [20], [61]. The DLVO theory attributes the stability of a dilute dispersion in a colloidal system to the total ( VT ) of repulsive electrostatic forces (VR ) and attractive van der Waals (or London) forces (VA ) shown in Equation 1 and in Figure 4. The potential energy of repulsion due to the solvent layers ( VS ) is negligible at distances above 10 nm. The energy minima of VT are dependent on the particle-particle distance, and the Debye length (κD ) [28]. A dispersion is stable when steric or electrostatic repulsive forces exceed the attracting van der Waals forces between particles.

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VT = VA + VR + (VS )

(1)

Figure 4. Total potential energy of interaction VT (Equation 1). The repulsive electrostatic forces (VR ) are positive and the basis of colloidal stability [52]. They are generated if surface groups dissociate in contact with a solvent e.g. water. As a result the surfaces become charged, attracting ions of the opposite charge and repel those of the same. The force VR decreases roughly exponentially with distance and approaches a finite magnitude.1 Van der Waals attraction (VA ), on the other hand, is always present but at large separations it might be too weak to significancely effect a dispersions stability. This force is attractive (negative), and of short range and decreases with the 6th power 1

When the electrolyte concentration increases (around and above 10−1 mol/l), electrostatic interaction becomes progressively less important while shorter range forces become more important [52].

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of the separation between atoms. 2 The sum of attractive and repulsive forces generates a primary minimum where the dispersion is stable. Depending on the magnitude of VA and VR a dispersion can exist in a metastable state with a potential energy barrier 3 that slows the rate of coagulation as shown in in Figure 4. When the electrostatic forces decrease, the dispersion becomes less stable until the system coagulates rapidly. Dispersed particles can also be stabilised sterically in aqueous and non-aqueous solution when they are kept far enough apart by chains of surfactant that van der Waals forces do not become strong enough for coagulation e.g. non-aqueous: [57]. The DLVO theory can be extended to non-aqueous media, as repulsive electrostatic forces exist in polar and non-polar media resulting from the overlap of electrical double layers [35]. Trace water often plays an important role in the charging mechanism in nonaqueous dispersions and is further described in Subsection 2.2. However, it has also been suggested that DLVO theory can not explain all systems in colloid science [28].

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1.2.

Surface Charging

Surface-active substances (such as surfactants) interact with the particles and the solvent. They strongly adsorb on the surface of the particles and yet are soluble in the medium. The dissociation of functional groups and/or preferential adsorption of ions leads to the formation of a double layer by adsorption of an ion of the opposite sign [33]. These substances therefore affect the stability of a dispersion, increasing or decreasing the electrostatic repulsion according to their charge. Effective non-aqueous dispersants are either acids or bases. The acidity or basicity of the particle surface are equally important [57]. The maximun or minimum of the resulting surface charge is also a function of the concentration of additives (surfactants, electrolytes, trace water, etc.). The magnitude of surface charge on oxides is determined by their acidity and the acidity of the solvent. The order of acidity is SiO2 > TiO2 > ZrO2 and methanol > ethanol > acetone > water for solvents. As a result a polar (basic) solid is easily protonated and the proton on the surface is abstracted by a more basic solvent, and therefore TiO2 , for example, is more negatively charged in water than in acetone [33]. There is also evidence of surface charging in non-polar liquids, but the properties of the medium can also be shifed by addition of an additive such as trace water in non-aqueous solvents. Water adsorbs on the surface of hydrophilic particles. There it can change the acid/base-character of the surface or dissolve ions [47] (see Subsection 2.2.) If there are no ions in non-aqueous media, the surface groups dissociate accepting or donating a proton [40]. The mechanism presented in Equation 2 is widely applicable since many particles or solvents are Br¨onsted acids or bases and therefore capable of donating or accepting protons. − − + SH+ 2 + B ←→SH + HB←→S + H2 B [33]

(2)

with SH being a surface group and HB a solvent molecule. 2 Theoretically it should have little effect beyond a distance of about 1 nm, but in practice it can reach a range of some tens of nanometres - quite comparable to the coulombic force [52]. 3 DLVO energy barrier of 15 kT (k= Bolzmann’s constant) is necessary for long term stability for particles ˚ of radius 1000 A.

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A common additive is Aerosol OT (AOT), the name commonly given to the compound dioctyl sulfosuccinate or bis-2-ethylhexyl sodium sulfosuccinate, which has the chemical formula C20 H32O7 SNa. It is an anionic surfactant with two double-branched hydrophobic tails and is soluble both in water and in most common non-aqueous solvents [49]. AOT is a commonly used stabiliser in non-aqueous solvents. for example AOT was found to be the most effective solute and therefore most effective at generating surface charges for carbon black, calcium carbonate, silica in xylene [47] and for many other non-aqueous systems.

1.3.

Zeta-Potential

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The Gouy-Chapman theory considers ions as point charges and the solvent as a structureless medium and provides an explanation for the stability of dilute dispersions of charged colloidal particles in aqueous media. The charge is assumed to be distributed uniformly on the particle surface. The surface charge influences the repulsive forces of the ions, molecules, or particles in the solution, attracting ions of the opposite charge and repelling those of the same charge. The distribution of ions near the surface of a particle is divided into two regions as shown in Figure 5.

Figure 5. The distribution of charges of a charged particle in solution. The Stern-layer is located closer to the surface and consists of ions of the opposite charge. It is surrounded by the diffuse layer that contains both positive and negative ions. The thickness of this layer is κ−1 α , where κα is the Debye-length, and depends on the electrolyte strength [16]. The surface potential arising from the Stern layer cannot be directly measured. The particle and ions form a charged unit. The potenital of this unit can be measured and is called zeta-potential. The zeta-potential arising from the surface charges determines the size of the electrical repulsive forces between the particles, and this in turn affects the way a suspension flows or settles. When a suspension of, for example, positively charged particles is subjected to an electric field, the particles will move towards the negatively charged cathode, while the surrounding double-layer ions will be drawn towards the anode. This process is called electrophoresis. The particle velocity ν is proportional to the applied field strength E. For spherical particles this relationship takes the form: ν = µE E

(3)

where µE is called the electrophoretic mobility of the particle. Two different models can be used to calculate the zeta potential from electrophoretic mobility. Smoluchowski presented Coagulation: Kinetics, Structure Formation and Disorders : Kinetics, Structure Formation and Disorders, Nova Science Publishers, Incorporated,

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an equation for a thin double layer (κα  1). The model assumes that the relationship between particle velocity and electric field has the form ν = [ζ/η]E

(4)

With η being the viscosity,  the permittivity and ζ the zeta-potential. Therefore, µE = [ζ/η]

(5)

H¨uckel calculated the zeta-potential from the electrophoretic mobilities for a very thick double layer (κα  1). µE = [2ζ/3η](1 + κα ) ≈ 2ζ/3η

(6)

The fomula for electrophoretic mobility given by Henry is

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µE = [2ζ/3η]f1(κα )

(7)

where f1 (κα ) is a varying function that increases from 1.0 at κα = 0 to 1.50 at κα = infinity. The Smoluchowski and H¨uckel equations (Equation 5 and 6) are, respectively, the upper and lower limits of the Henry formula (Equation 7) [28]. The Debye length κ−1 α of the double layer is required in order to utilise the appropriate equation for calculation of the zeta-potential from electrophoretic mobility data. It can be obtained by measuring the specific conductivity [34]. If the ion concentration in nonaqueous media is low, the zeta-potential will be high and has a similar magnitude to the zeta-potential in water [33]. 4 Zeta-potentials in non-aqueous media are sometimes difficult to obtain because many instruments are designed for aqueous systems. A general overview of methods for measuring the zeta-potential is given in [10]. The most reported methods in non-aqueous media involve electrophoresis, electro-osmosis or streaming potential [33]. Another measure of the stability is given by quantitative measurements involving the estimation of absolute rates of coagulation [43]. The electrophoretic mobilities of diluted dispersions can be measured by Phase Analysis Light Scattering (PALS) [45]. The charged particles move relative to an interference fringe pattern caused by two crossed laser beams, while one laser is offset in frequency. The time domain phase information can be used to determine the electrophoretic mobility. Some non-aqueous solvents can cause problems because they attack some of the plastic tubing used in common instruments. In this case PALS is a particulary attractive method as it requires only a glass cell for the measurement, compared to aqueous solutions a higher potential is needed for non-aqueous solvents. The zeta-potential of concentrated dispersions, such as those used in the presented SIC applications, can be determined by the electroacoustic method. During the measurement a high-frequency electric field is applied to charged particles, causing the particles to move back and forth. This generates a sound wave that is dependent on the size and the charge of the particles [53]. 4

In this case the Debye length would be very large and it is sometimes questionable whether it makes sense to speak of a Debye lenght in these systems [17].

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The Stability of Aqueous and Non-aqueous Dispersions Stability of Non-aqueous Dispersions

A non-aqueous dispersion can be stabilised sterically or, without the addidtion of a polymeric stabiliser, electrostatically. The electrostatic forces in a non-polar liquid are weak and long-ranged and were first directly measured using the surface-force apparatus between mica surfaces and surfactant solutions in non-polar media [17]. The extent of electrostatic stabilisation in non-aqueous media is strongly dependent on the relative permittivity (or dielectric constant)  of the solvent. In solvents with  ≥ 11 the surface charges and therefore electrostatic stabilisation occurs more or less as in aqueous systems. If an electrolyte dissociates, electrostatic stabilisation is possible when for 5 ≤  ≤ 11 [26]. Especially in non-polar media the capacity of the double layer is much smaller than it would be in water. Therefore less surface charge is necessary to stabilise a dispersion electrostatically [44]. The boundaries depend on the nature of the particle, the solute and -if present- the electrolyte in the solvent. In non-polar solvents it is often difficult to find a critical value of zeta-potential at which coagulation occurs suddenly between 35-45 mV is noticed but is often as vague as the definition of κD [33]. It has been reported that dispersions of large particles > 1µm can be electrostatically stabilised by quite modest zeta-potentials [35]. For large particles, adsorbed layers of nonionised long chain length make little contribution to the steric stability unless the chain length is larger than the particle size. Zeta-potentials in excess of 50 mV were found to be necessary for long-term stability (particle size less than 160 nm) in non aqueous media.

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2.2.

Trace Water in Non-aqueous Dispersions

In the following some examples are presented of how trace amounts of water were found to have a profound impact on the stability of non-aqueous carbon black, titania and alumina dispersions. These three materials have been reported in the literature to be used in an SIC process when dispersed in polar (aqueous and non-aqueous) media and are described in more detail later in the Section. In most non-aqueous dispersions water is present. It is either indroduced on the surface of the solids, in the liquid phase or by the hydration of chemicals [41]. The electric double layer has been considered to have an ionic nature in aqueous as well as non-aqueous systems [33]. However, the addition of ionic dispersants or additives containing any amount of water can have a huge impact on the surface-charging of particles in a non-aqueous dispersion. Hence the effect of charging is originating from ionic additives or water is predominant and surpasses the electronic effect in non-aqueous media. Traces of water can even lead to a sign shift in zeta-potentials compared to dry samples [47]. The magnitude and sign of the charge on the particles are often a complex function of the concentration of the surfactant and the quantity of water in the system. Trace water is reported to be adsorbed on the interface rendering the particle surface much more basic. As a result polar particles can have a positive zeta-potential because of the adsorption of cations such as Na+ in solution [40]. Carbon black (or barium sulfate) shows a negative charge when dispersed in a non-polar AOT solution (n-heptane, cyclohexane or benzene) which increases when low concentra-

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tions of water are added until it reaches a maximum which is shown in Figure 6. 5 This observation is explained by the solubilisation of a dissociated sodium ion in the AOT micells whereas the negative ion of AOT is adsorbed on the particle [34].

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Figure 6. The change of the ζ-potential for carbon black (and BaSO4 ), versus water content of Aerosol OT or NP-4 in nonpolar solvents. bottom-filled- ◦: Carbon-Benzene-Aerosol OT (14.5); •: Carbon-n-Heptane-Aerosol OT (11.3); ◦: Carbon-n-Heptane-Aerosol OT (78.8); right-filled-◦: Carbon-Cyclohexane-Aerosol OT (13.6). (square and triangle-shaped symbols indicate BaSO4 -solutions). Reprinted from [34], with permission from Elsevier. Trace amounts of water allow the surface potential of rutile ( TiO2 ) in an AOT solution of p-xylene to change sign or vary over a range of magnitudes in otherwise identical systems. Rigorously dried dispersions exhibit a negative charge while, because of an excess of sodium ions associated with the water on the surface, those with trace water are charged positively [42]. The zeta-potential in the same system as shown in Figure 7 increases to a maximum at about 100 ppm by volume of water. The zeta-potential is a function of the amount of water on the surface, increasing with the amount of water in solution. As the AOT concentration increases water might be in favor of the solution phase [44]. In a dispersion of alumina and AOT in cyclohexane, water plays a major role in determining the stability [41]. At low water concentration settling occurs more slowly, while at higher concentrations the amount of water adsorbed on alumina corresponds to a sharp increase in the stability as depicted in Figure 8a. The water adsorption isotherm is shown in Figure 8b and appears similar to vapour adsorption on solids, where higher vapour pressure leads to more adsorption until saturation pressure is reached. By using an analogy between these two isotherms in can be concluded that the adsoption of water is controlled by water concentration. 5 The Debye length obtained from conductivity data (which lead to choose the H¨uckel equation (Equation 6)) and was found to be smaller in wet samples.

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Figure 7. Variation of zeta-potential with water content for dispersions of rutile in AOT solutions in xylene. Aproximate concentrations of AOT: 1 mM ( ), 4 mM (x), and 20 mM (•). Reprinted from [44], with permission from Elsevier. Trace water was also found to have a critical enhancement on an alumina suspension stabilised by hydrophobic polymers in cyclohexane [60]. Higher concentrations of trace water lead to rapid flocculation of the alumina particles. Whenever dealing with non-aqueous dispersions, experimental scientists should bear in mind the effect trace water can have and consider drying the chemicals appropriately and/or investigating the amount of trace water present. In the systems described above the water concentration has been reported to be measured using the Karl-Fischer method [41] [42]. However, it is very difficult to remove traces of water below 10 ppm [33]. In the presented cases cyclohexane was dried using a molecular sieve 4A [41]. Rutile was outgassed for at least 3 h at 450 ◦C and 10−5 mm Hg, and all operations were carried out in a dry box [42]. Solid alumina was dried for 6 h at 200 ◦ C in a vacuum desiccator, the solvent was stored over molecular sieve 4A and the surfactant AOT was stored in a desiccator using P2 O5 as dessicant after drying [41]. AOT is hygroscopic, but has been reported to be dried by freeze-drying [34]. Adsorption mechanisms of AOT and the stabilisation mechanism in non-polar solvents is determined between the solvent solute and substrate [60].

2.3.

Carbon Black Dispersions

2.3.1. Carbon Black Dispersions in Polar Media Carbon black is conductive and therefore an interesting additive for highly conductive composite materials. Non-aqueous SIC gives also the opportunity to coat even watersensitive, weakly-conductive materials with a highly conductive layer. Either way, dispersions that are suitable for a SIC process should be in a state where coagulation occurs easily. The coagulation inducer, the adsorbed polymer on the surface of the substrate, forms molecular bridges to the particles dispersed in the solution. A suitable dispersion should be stabilised electrostatically instead of sterically. Electrostatic stabilisation occurs more easily when the relative permitivity of the solvent is high. Therefore a high dielectric-constant

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Figure 8. Effect of water addition on the suspension settling rate. Increase in surfactant concentration shifts the suspension destabilisation towards higher water concentrations (a.). Adsorption of water on alumina from the AOT/cyclohexane solution. Increase in surfactant concentration expands the domain of water concentration in which adsorption takes place (b.). ”•”: AOT 8.5x10−3 mol/l; ”N”: AOT = 26x10−3 mol/l. Reprinted from [41], with permission from Elsevier. solvent might be favourable as the conditions are closer to well-known aqueous systems [26]. With a dielectric constant (or relative permittivity) of 32.2 and a dipole moment of 4.0930 D, N-Methyl-2-pyrrolidinone (NMP) can still be regarded as a polar solvent with electrostatic stabilisation is more likely to occur [29]. The effect of the surfactant AOT was investigated in concentrated (1wt.%) aqueous as well as NMP carbon black dispersions [4]. In water the surfactant has a stabilising effect on the hydrophobic carbon black. With no surfactant present the particles settle immediately, while adding the surface-active surfactant AOT increases the stability. Carbon black dispersed in NMP shows a high stability over several weeks without any surfactant present. Here, adding AOT showed the opposite effect to the aqueous case and the dispersions coagulated more quickly (less than 2 h) in otherwise identical dispersions. Zeta-potentials were measured by PALS and the electroacoustic method for both cases. 6 6

With the small carbon black  18 nm particles and low electrolyte concentrations ( < 0.03 mol/L) in NMP

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The relation of the surfactant concentration and zeta-potential of aqueous and non-aqueous carbon black dispersions is shown in Figure 9. 0

Zeta potential (mV)

-10 -20 -30 -40 -50 -60 -70 0

5

10

15

20

25

AOT concentration (mmol/L)

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Figure 9. Zeta potential of carbon black particles suspended in NMP (  and ) and water (stable •, unstable dispersions ◦), as a function of AOT concentration. The points connected by dotted lines were measured with the Acoustosizer; points connected by full lines were measured by the PALS technique. Reprinted from [4], with permission from Elsevier. When the surfactant concentration increased, the aqueous dispersions showed an increased stability represented by a higher zeta-potential of dispersed carbon black while in NMP solutions the zeta-potential decreased. Carbon black is hydrophobic and shows little surface charge in water. Therefore the adsorption of (hydrophilic, ionic) surfactants increases the negative surface charge and in turn the dispersibility of the material. Carbon black is not solvophob in NMP. In a carbon black dispersion in NMP the solvent wets the particles and without any surface-active substance present the surface is charged 7 . While dynamic light scattering shows no evidence of the formation of micells in the investigated region (0-25 mmol/l) in NMP and conductivity measurements show no sign of increased conductivity caused by the interaction with the carbon black [1]. In NMP higher concentrations of AOT cause the zeta-potential and the stability of a carbon black dispersion to decrease. Because of the solvents, high-polarity dissociation of AOT occurs, creating a suspending medium with electrical double-layer interactions. In fact the effect of AOT on the stability of the dispersion in NMP is similar to a simple electrolyte such as LiCl shown in Figure 10. Dispersions of carbon black in NMP in the region of 30-40 mV zeta-potential (that contain 0-5 mmol/l AOT in 1wt.% carbon black solvent) were the best performing disperused here, it is appropriate to use the H¨uckel equation (6) to calculate zeta-potentials from measured mobilities. 7 Measuring the SO2− 4 concentration (1 per AOT molecule) in the supernatant in different AOT concentrations indicated no surfactant type behaviour for AOT adsorbing onto carbon black from NMP. No Langmuir type adsorption, the most common for carbon black in water was found.

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0

Zeta potential (mV)

-10 -20 -30 -40 -50 -60 -70 0

2

4

6

8

10

Electrolyte concentration (mmol/L)

Figure 10. Zeta potential of 1% carbon black suspension in 1 mmol/l AOT ( ) then diluted in LiCl solutions in NMP, plotted against LiCl ( N) concentration. Reprinted from [4], with permission from Elsevier.

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sions in a SIC process and were used to prepare the highly conductive composite electrodes described in Subsection 3.2. Carbon black dispersions used for SIC in aqueous media are usually stabilised by a surfactant such as AOT or cetyl trimethylammonium bromide (CTAB) and are destabilised by a suitable amount of electrolyte such as LiCl or sodium acetate (NaAc) [63], [9], [14]. 2.3.2. Carbon Black Dispersions in Non-polar Media Dispersions may also be interesting for SIC applications but have so far not been investigated. Unlike in NMP, studies of carbon black in alkanes show that the presence of the surfactant is necessary to create charged surfaces through adsorption/dissociation mechanisms [54], [55], [23]. Since surface-active groups ionisise less [54] in non-aqueous systems the electrostatic repulsive potential  ≤ 5 is very low [56]. A charging mechanism when dispersant (polyisobutene succinimide) was added to a low-polar media to stabilise a carbon black/dodecane dispersion is presented in Figure 11. Carbon black contains acidic oxygen groups (such as phenols and carboxylic acids). When basic dispersants (such as polyisobutene succinimide) are added protons go from acidic groups to basic sites of the adsorbed dispersant. These protons leave a negative charge on the particle when desorbing. Maintaining stability in this case requires a zeta-potential of 120 mV. Steric repulsion plays a relatively small role in stabilizing this dispersion. The two mechanisms of steric and electrostatic interparticle repulsion work together in this system [57]. Most work reported in the literature investigates (mainly dilute) carbon black disperisons in non-aqueous, non-polar media e.g. carbon black in an AOT solution of nheptane [39].

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Figure 11. Proposed mechanism for electrostatically stabilized particles. Adsorption is followed by a proton transfer (for acidic ”AH” solid in hydrocarbon solution of basic ”B” ˚ 2 . Reprinted dispersant). The area per molecule of dispersant adsorbed on surface is 125 A from [57], with permission from Elsevier.

2.4.

Titania and Alumina Dispersions

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2.4.1. Titania and Alumina Dispersions in Polar Media In aqueous solutions the surface charging of titania and alumina is dependent on its isoelectric point and is positive when pH is below and negative above. The switch in sign can be explained by proton exchange[16]. The isoelectric point of various anatase and rutile samples was found to be between 3.2-6.2 [18]. Aqueous and non-aqueous dispersions investigated for their use in SIC applications of titania are described in [6]. If titania (1wt.%) is dispersed in an a-protic, non-aqueous solvent such as NMP, protons cannot be accepted easily, the particle cannot donate them and may in fact even accept them (see Equation 8) [6]. In the presence of trace water the particle may donate protons, and the charge could switch to negative and cause the particles to be charged negatively. − TiO− + H2B+ ←→TiOH + HB←→TiOH+ 2 +B

(8)

A polar hydrophilic interaction is observed as AOT molecules adsorb via their hydrophobic tails with their charged head groups facing into the less-polar solvent. The AOT molecules aggregate in polar mircodomains and shield the head groups from the less polar solvent. This interaction drives agglomeration and increases the settling rate at high coverage. Wetting of NMP on titania was described using the powder contact angle method [64], [30]. The contact angle was found to be about 90 ◦ indicating a low dispersibility of the solid in the solvent. The surface between solid and solvent seeks minimisation resulting in low dispersibility and flocculation. In this case - unlike carbon black that was wetted by the solvent as described above - a flocculant seems neccessary to gain reasonable stability

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required in a SIC process for dispersed titania. The dispersion showed a much lower stability (about a day) using the same concentrations of solids, electrolytes and solvents as the carbon black dispersions described in [4]. No Langmuir type isotherm (but rather Henry type) of adsorbed AOT on titania was observed in the investigated region (0-25 mmol/L) AOT in 1% wt. titania. In the adsorption of AOT in the polar solvent onto titania a polar or hydrophobic interaction was found. The contact angle of alumina and NMP was found to be about 90 ◦ indicating a low dispersibility of the solid in NMP. Alumina in NMP showed a negative zeta-potential using the PALS method.8 The experiments were repeated using the same materials in a different lab resulting in positive zeta-potential. The only reasonable difference between the samples could be trace water, which was not measured, once more demonstrating the potentially huge impact of trace amounts of impurities [5]. 9 If AOT forms micells in the trace water at the solid-liquid interface this would explain the further decrease in stability. The surfactant AOT has been shown to be a highly effective dispersant for alumina in NMP, but also causes a destabilising effect because it is highly hygroscopic and water is capable of decreasing the stability of a dispersion. Conditions for alumina dispersions to be used in a non-aqueous SIC process are described in [5].

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2.4.2. Titania and Alumina Dispersions in Non-polar Media Most work on non-aqueous titania or alumina dispersions (mainly diluted) in the literature was done in non-polar media. These dispersions might be interesting for SIC applications as well but have so far not been investigated. A dilute rutile dispersion where AOT is used as stabiliser in p-xylene is described in [42]. Dry rutile powder in a dry AOT solution showed a negative surface-charge that turned positive after 15 h [44]. Another group found a positive zeta-potential for the same system [25]. Dispersions of titania in cyclohexane and p-xylene with Na, Mg, Ca, Ba and Al salts showed a positive zeta-potential [54]. The charging of alumina particles goes back to acid-base interactions [21], [22]. The surfactant AOT is reported to have a higher affinity for basic oxides (such as alumina). If the acidity of the solvent increases, the charge on the alumina particles and the electrostatic force increases too. Therefore, the electrostatic contribution can be tailored by changing the organic-liquid media [37]. Because of its acidity, linolenic acid should be a suitable dispersant for alumina in non-aqueous media. A dilute alpha-alumina dispersion in a solution of AOT in p-xylene is presented in [44], and in cyclohexane in [41]. Alpha-Al2O3 in propanol, butanol or pentanol changes the sign of the zeta-potential from negative to positive when water concentration is higher [58]. Using low-dielectric constant solvents, steric stabilisation dominates in alumina stabilised with a polymer dispersed in toluene [65]. Alumina shows a positive surface charge when dispersed in the azeotropic mixture of methyl ethyl ketone and ethanol stabilised with polyvinyl butryol, 8

With the small (alumina  13 nm, and titania  21 nm) particles and low electrolyte concentrations ( < 0.03 mol/L) in NMP used here, it is appropriate to use the H¨uckel equation (Equation 6) to calculate zetapotentials from measured mobilities. 9 AOT is also soluble in water and the critical micelle concentration (cmc) is 2.23 mmol/l at 303 K [49].

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where none of the ususal anionic dispersants were found to be efficient in dispersing [48]. Alumina in ethanol stabilised with polyacrylic acid showed a negative zeta-potential [50].

3.

Advanced Battery Materials

A lithium-ion battery consists of two separated electrodes that can intercalate Li-ions reversibly. In the most common case the cathode consists of lithium cobalt oxide ( LiCoO2 ) and the anode of graphite (or Li metal). During discharge the graphite acts as a source and the lithium cobalt oxide as a sink for lithium ions. The resulting negative charge is neutralised by electrons that go from the negative to the positive electrode. At this voltage water would electrolyse (depending on pH stable to about 1.23 V), therefore in a Li-ion battery non-aqueous electrolytes (such as lithium salt solutions of ethylene carbonate or dimethyl carbonate) are used. During the first charge/discharge process the anode develops a protective layer: the ”solid electrolyte interphase” (SEI) [62]. Because of the SEI formation, water and sodium should be avoided in a sealed battery.

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3.1.

The Cathode Material Lithium Cobalt Oxide

In 1980 a new layered compound capable of reversibly intercalating Li-ions at 4V, LiCoO2 , was reported [46]. The compound LiCoO2 has now reached energy densities in excess of 150 Wh/kg and 350 Wh/l cycle lives in excess of 1000 cycles and low self discharges 140 % of normal) might be pro-thrombotic, too few fibrinogen function (< 60 % of normal) might predispose to cerebral hemorrhage. Fibrinogen function is usually measured by addition of unphysiologically high activities of thrombin (> 10 IU/ml final assay concentration). Recently, two new fibrinogen tests have been described, the FIFTA (fibrinogen functional turbidimetric assay) (Table 2a) [20] and the FIATA (fibrinogen antigen turbidimetric assay) (Table 2b) [21]. In contrast to hitherto fibrinogen tests, the FIFTA works with a final physiologic thrombin activity of only 0.2 IU/ml, i.e. with the approximate thrombin activity in physiologically clotting normal plasma [17,22]. Many commercial functional fibrinogen tests depend on the plasma matrix: pathologically low albumin concentrations might alter the turbidimetric fibrin signal: this can result in a false fibrinogen concentration in patients with severe changes of the plasma matrix (plasmas of critically ill patients in intensive care units often have albumin concentrations of only 10-20 g/l (normal MV1SD= 444g/l)). In the FIFTA, 1 part of citrated plasma (e.g. 20 µl) is incubated with 2 parts of FIFTA-reagent, containing 0.3 IU/ml thrombin, 0.4 mg/ml polybrene®, 6 % human albumin, phosphate buffered saline (PBS). The turbidity increase at 405 nm within 2 min (37°C) or within 5 min (23°C) is measured. In the FIATA, fibrinogen and fibrinogen-like molecules are precipitated by the antibiotic vancomycin: 1 part of plasma (e.g. 10 µl) is incubated with 2 parts of PBS, the turbidity at 405 nm is determined, 2 parts of 4.4 mM vancomycin in PBS are added and the turbidity increase is measured after 2 min (23°C) or 40 s (37°C). FIFTA and FIATA are standardized with pooled normal citrated plasma (=100 % of normal). Via FIATA the fibrinogen concentration can be determined even in EDTA-plasma. The FIFTA/FIATA ratio (FI ratio) indicates the presence of normo-reactive (ratio=1.00.1), over-reactive (FI ratio > 1.1), or under-reactive (FI ratio < 0.9) fibrinogen. Over-reactive fibrinogen might occur in presence of enhancers of fibrin-polymerization such as soluble fibrin; under-reactive fibrinogen might occur in presence of dysfibrinogens or fibrinogen degradation products by plasmin. The normal range for FIFTA is 10020%, septic patients have 18866% of normal, the normal range for FIATA is 10020%, septic patients have 17966% of normal, and the normal range of FIFTA/FIATA ratio is 1.00.1, septic patients have 1.120.32.

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Thomas W. Stief Table 2a. FIFTA (fibrinogen function) 20 µl citrated plasma 40 µl 0.3 IU/ml thrombin, 0.4 mg/ml polybrene®, 6 % human albumin, PBS ∆A405nm/2 min (37°C)

Table 2b. FIATA (fibrinogen antigen) 10 µl plasma 20 µl PBS A405nm 20 µl 4.4 mM vancomycin in PBS ∆A405nm/2 min (23°C)

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Undiluted Antithrombin III Activity (AT3) AT3 is the most important physiological inhibitor of thrombin [23-25]. Decreased activity of AT3 results in increased thrombin generation. Therefore, AT3 activity must be determined routinely. Usual AT3 determinations work with a > 50fold dilution of plasma to overcome turbidity problems with in the assay nascending fibrin [26-28]. Recently, a new AT3 test has been developed, that works with nearly undiluted plasma [29]: in the thrombin reaction phase, 10 µl plasma or standards (100 % of normal pooled plasma and this plasma diluted with 0.9 % NaCl) are incubated with 2 µl thrombin reagent = 58 IU/ml (9.7 IU/ml final) thrombin, 6 % human albumin, 54 IU/ml (9 IU/ml final) unfractionated heparin, 1600 mM (267 mM final) arginine, pH 7.4, for 30s (23°C) in microwells with flat bottom (F-wells). Then 50 µl 2.5 M arginine, pH 8.6, 0.124 % Triton X 100® are added. After 3 min (23°C), the thrombin detection phase starts with addition of 20 µl 1 mM (0.24 mM final) chromogenic thrombin substrate HD-CHG-Ala-Arg-pNA in 1.25 M (1.87 M final) arginine, pH 8.7. ∆A/t at 405 nm is determined (Table 3). The normal range for this new AT3 test is 10015% (MV1SD), septic patients have 4727% of normal active AT3. An improvement of AT3 activity might correlate with survival of the patient. Table 3. Undiluted AT3 activity 10 µl plasma 2 µl 58 IU/ml thrombin, 6 % human albumin, 54 IU/ml heparin, 1600 mM arginine, pH 7.4 30s (23°C) 50 µl 2.5 M arginine, pH 8.6, 0.124 % TritonX100® 20 µl 1 mM HD-CHG-Ala-Arg-pNA, 1.25 M arginine, pH 8.7 ∆A405nm/t

Active Endotoxin = Endotoxin Reactivity Endotoxin is here clinically understood as lipopolysaccharide (LPS) from gram-negative bacteria or ß-glucan from fungi. The human organism is especially susceptible to active endotoxin [30]. Active (=free, unbound) endotoxin destroys susceptible cells in the blood

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stream (e.g. monocytes), which triggers intrinsic coagulation. Kallikrein and thrombin are formed and a PIC may arise. Table 4a. Active endotoxin (here: active LPS + active ß-glucan) 10 µl plasma 10 µl 60 mM chloramine-T® in 50 mM sodium citrate, pH 7.8 10 min (37°C) 10 µl 230 mM methionine 1 min (37°C) 30 µl Limulus-reagent (Limulus factors + chromogenic substrate) 0, 10, 30, 60 min (37°C) ∆A405nm

Table 4b. Active ß-glucan

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10 µl plasma 10 µl 60 mM chloramine-T® in 50 mM sodium citrate, pH 7.8 10 min (37°C) 10 µl 230 mM methionine, 0.63 mg/ml (4957 IU/ml) polymyxin B 1 min (37°C) 30 µl Limulus-reagent (Limulus factors + chromogenic substrate) 0, 10, 30, 60 min (37°C) ∆A405nm

Therefore, it is of great clinical importance to measure active endotoxin routinely to diagnose a possible pre-phase of sepsis and of PIC as soon as possible to start the adequate treatment in time. Active endotoxin can be measured in plasma within minutes by coagulation factors of the Limulus cascade: 10 µl plasma (with EDTA or citrate) or plasma-standards (1, 10, 100, 1000 ng/ml LPS reactivity) are incubated for 10 min (37°C) with 60 mM chloramine-T® in 50 mM sodium citrate, pH 7.8. 10 µl 230 mM methionine are added. After 1 min (37°C) 30 µl Limulus-reagent (Limulus factors + chromogenic substrate Ile-Glu-GlyArg-pNA) are added and the increase in absorbance with time (∆A/t) is measured (Table 4a). If only active ß-glucan is to be quantified, the reaction conditions are the same as for active endotoxin, with the exception that the methionine reagent contains also 0.63 mg/ml of the LPS-inhibitor polymyxin B, which inhibits LPS reactivities of up to 1000 ng/ml LPS added to normal plasma (Table 4b). Active LPS (LPS reactivity) is calculated with the formula: LPS reactivity [mA/t] = endotoxin reactivity [mA/t] minus ß-glucan reactivity [mA/t]

Reactivity examples: 10 ng/ml LPS reactivity corresponds to the LPS reactivity of normal plasma (containing 0.8 ng/ml LPS) that had been supplemented with additional 9.2 ng/ml purified LPS. 10 µg/ml ß-glucan reactivity corresponds to the ß-glucan reactivity of normal plasma (containing < 0.1 µg/ml ß-glucan) that had been supplemented with 10 µg/ml purified ß-glucan. The normal range of LPS reactivity is 10025 % (100% = 0.8 ng/ml). Pre-septic patients have about 2 ng/ml (200-300 % of normal) LPS reactivity. The LPS reactivity in patients with

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severe sepsis can increase up to a range of 100 ng/ml (mean value=20.5 ng/ml, standard deviation=19.6 ng/ml) [14,34]. The fungal pathology should never be underestimated: massive fungal infection or a combined bacterial/fungal sepsis can cause PIC [31,32]. Normal human plasma has < 0.1 µg/ml ß-glucan reactiviy. Pre-septic patients have about 0.5 µg/ml ß-glucan reactivity. The ßglucan reactivity in patients with severe sepsis can increase up to a range of 50 µg/ml (mean value = 4.7 µg/ml, standard deviation =7.0 µg/ml). Since free LPS is about 300fold as toxic as free ß-glucan [6,14], the load of active endotoxin of a patient can be calculated as: LPS [ng/ml] + ß-glucan [ng/ml]/300. Then patients with severe sepsis have 36.226.3 ng/ml active endotoxin [14]. In normal plasma, only about 1 % of the LPS molecules or about 10 % of the ß-glucan molecules are active=unbound=free. Fig. 1a

sepsis.2006.02.10erge;t6 Patients 1-16

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Figure 1. Fibrinogen function (FIFTA) in septic patients. Citrated plasma of 32 intensive care unit (ICU) patients with severe sepsis was analyzed for fibrinogen function with the FIFTA. Figure 1a= patients 1-16, Figure 1b= patients 17-32, each patient has his individual symbol. Patients with < 80 % FIFTA have pathologically decreased fibrinogen activities, e.g. by consumption of fibrinogen (PIC phase (2)) or by destruction of fibrinogen by plasmin (PIC phase (3)); patients with > 120 % FIFTA have pathologically increased fibrinogen activities, e.g. by hyper-fibrinogenemia or by enhancers of fibrin polymerization. Admission to ICU = stay of 0 days = day 1 in ICU.

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Special Antigentic Parameters for PIC Diagnosis Soluble Intercellular adhesion molecule-1 (sICAM-1) is a parameter for endothelial damage (normal range: 0.190.04 µg/ml, septic patients: 2.562.48 µg/ml) [14]. Plasminantiplasmin-complex (PAP; normal range: 10030%, septic patients: 313307%) and Ddimer (normal range: 0.050.03 µg/ml, septic patients: 4.03.6 µg/ml) indicate fibrinolysis activation [14]. Since D-dimer is a final product of the combined action thrombin, factor XIIIa, and plasmin, D-dimer tests cannot distinguish between coagulation activation and fibrinolysis activation. To obtain reliable test results for unstable blood, these antigenic parameters should be measured in 1.25 M arginine - stabilized EDTA-plasma [14]. The inhibited plasmatic fibrinolysis in severe sepsis can be monitored by determination of plasminogen activator inhibitor-1 (PAI-1) activity or of the global assay FIPA (fibrinolysis parameters assay with 10 min reaction time) that measures the global plasmatic fibrinolytic state [33]: PAI-1 is about 10fold elevated and FIPA is about 5fold depressed in the begin of severe sepsis. An improvement of fibrinolysis seems to correlate with survival of the patient: the FIPA mean value approximately doubles within a stay of about 1 week in intensive care unit [34].

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CONCLUSION Acute PIC is a terrible complication of many diseases; its high mortality is often caused by too late diagnosis and treatment [8], e.g. the first six hours of sepsis might be termed the ― golden hours‖ where diagnosis and therapy is still in time for a good prognosis [43]. Acute PIC is comparable with a toxic snake bite [30,35] and very often results into severe pro-thrombotic hemostasis alterations with insufficient (frustrane) activation of cellular fibrinolysis [42]. The IIa-Test (normal range 10020%; 100%= 5.5 mIU/ml basal IIa) or the standardized RECA helps to distinguish the normal intravascular coagulation (NIC; 10020 % basal IIa; IIa generation in RECA = 100 %  30 % of normal) from a pre-phase of PIC or from chronic PIC [36] where the basal IIa activity is 120-150% or the basal IIa activity is dynamically increasing, or the RECA values are > 130 % of normal. The blood half life of 2macroglobulin/protease complexes is only 15-60 min, especially cells of the reticuloenthelial system possess such receptors [37,38], which means that the 2-macroglobulin/thrombin concentration is not primarily dependent on the liver function.

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Fig. 2a

sepsis.2006.02.10erge.t10 Thomas W. Stief

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0 Copyright © 2012. Nova Science Publishers, Incorporated. All rights reserved.

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Figure 2. Fibrinogen antigen (FIATA) in septic patients. Citrated plasma of 32 intensive care patients with severe sepsis was analyzed for fibrinogen antigen by the FIFTA. Figure 2a= patients 1-16, Figure 2b= patients 17-32, each patient has his individual symbol.

The dynamic hemostasis state of each individual patient should be determined routinely, as indicated in exemplary figures 1-5 [14,30]. The low plasma volumes required for these new tests allows the determination of a complete state of hemostasis activation also in pediatrics [25,39-41]. The above reviewed new hemostasis assays help to diagnose the very early phase of PIC within minutes.

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The Laboratory Diagnosis of the Pre-Phase … Fig. 3a

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Figure 3. FIFTA/FIATA ratio in septic patients. The FIFTA/FIATA ratio (FI ratio) of the patient plasmas of Figure 1 and Figure 2 was calculated. Figure 3a= patients 1-16, Figure 3b= patients 17-32, each patient has his individual symbol. Patients with a FIFTA/FIATA ratio < 0.9 have under-reactive fibrinogen, e.g. fibrinogen degradation products as occurring in PIC phase (3) or dysfibrinogens; patients with a FI ratio > 1.1 have fibrinogen, whose function is over-reactive, e.g. due to enhancers of fibrin polymerization.

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Figure 4. Undiluted antithrombin III function in septic patients. Citrated plasma of 32 intensive care patients with severe sepsis was analyzed for undiluted AT3 activity. Figure 4a= patients 1-16, Figure 4b= patients 17-32, each patient has his individual symbol. Normal range = 100 15 %.

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sepsis.2006.02.10erge.t26 Patients 1-16

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Figure 5. IIa-Test in septic patients. EDTA-1.25 M arginine plasma of 32 intensive care patients with severe sepsis was analyzed for basal thrombin (IIa) activity. Figure 5a= patients 1-16, Figure 5b= patients 17-32, each patient has his individual symbol. Patient 26 (grey∆) had pancreatitis, his IIa activities in Figure 5b (at ICU days 0-3) have to be 2fold multiplied. IIa activities of 120-150% of norm and/or dynamically increasing IIa activities below 120 % of norm are characteristic for PIC phase (0). IIa activities > 150 % of norm and/or dynamically increasing IIa activities in the range 120-150 % of norm are characteristic for PIC phase (1).

REFERENCES [1] Bick RL. Disseminated intravascular coagulation current concepts of etiology, pathophysiology, diagnosis, and treatment. Hematol Oncol Clin North Am 2003; 17: 149-76.

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[2] Pasternak JJ, Hertzfeldt DN, Stanger SR, Walter KR, Werts TD, Marienau ME, Lanier WL. Disseminated intravascular coagulation after craniotomy. J Neurosurg Anesthesiol 2008; 20: 15-20. [3] Cotovio M, Monreal L, Navarro M, Segura D, Prada J, Alves A. Detection of fibrin deposits in horse tissues by immunohistochemistry. J Vet Intern Med 2007; 21: 1083-9. [4] Mosesson MW. Fibrinogen and fibrin structure and functions. J Thromb Haemost 2005; 3: 1894-904. [5] Stief TW. Specific determination of plasmatic thrombin activity. Clin Appl Thromb Hemost 2006; 12: 324-9. [6] Stief TW. Thrombin generation by exposure of blood to endotoxin: a simple model to study disseminated intravascular coagulation. Clin Appl Thromb Hemost 2006; 12: 13761. [7] Bick RL. Disseminated intravascular coagulation. Objective laboratory diagnostic criteria and guidelines for management. Clinics in Laboratory Medicine 1994; 14: 72969. [8] Urge J, Strojil J. Early diagnosis of DIC development into the overt phase. Biomed Pap Med Fac Univ Palacky Olomouc Czech Repub 2006; 150: 267-9. [9] Saba HI, Morelli GA. The pathogenesis and management of disseminated intravascular coagulation. Clin Adv Hematol Oncol. 2006; 4: 919-26. [10] Gando S, Saitoh D, Ogura H, Mayumi T, Koseki K, Ikeda T, Ishikura H, Iba T, Ueyama M, Eguchi Y, Ohtomo Y, Okamoto K, Kushimoto S, Endo S, Shimazaki S; Japanese Association for Acute Medicine Disseminated Intravascular Coagulation (JAAM DIC) Study Group. Natural history of disseminated intravascular coagulation diagnosed based on the newly established diagnostic criteria for critically ill patients: Results of a multicenter, prospective survey. Crit Care Med 2008; 36: 145-50. [11] Wada H, Hatada T. Pathophysiology and diagnostic criteria for disseminated intravascular coagulation associated with sepsis. Crit Care Med 2008; 36: 348-9. [12] Levi M. Disseminated intravascular coagulation. Crit Care Med 2007; 35: 2191-5. [13] Constantinescu AA, Berendes PB, Levin MD. Disseminated intravascular coagulation and a negative D-dimer test. Neth J Med 2007; 65: 398-400 [14] Stief TW, Ijagha O, Weiste B, Herzum I, Renz H, Max M. Analysis of hemostasis alterations in sepsis. Blood Coagul Fibrinolysis 2007; 18: 179-86. [15] Bauer KA, Rosenberg RD. Thrombin generation in acute promyelocytic leukemia. Blood 1984; 64: 791-6. [16] Stief TW. Quantification of thrombin generation by trypsin. Medical and Biological Sciences (Scientific Journals International) 2007; 1/ 2: 1-8. [17] Stief TW. The Recalcified Coagulation Activity. Clin Appl Thromb Hemost 2007 Dec 26; [Epub ahead of print] [18] Stief TW, Otto S, Renz H. The intrinsic coagulation activity assay. Blood Coagul Fibrinolysis 2006; 17: 369-78. [19] Stief TW, Wieczerzak A, Renz H. The extrinsic coagulation activity assay. Clin Appl Thromb Hemost (in press) [20] Stief TW. The fibrinogen functional turbidimetric assay. Clin Appl Thromb Hemost 2008; 14: 84-96.

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[21] Stief TW. The fibrinogen antigen turbidimetric assay (FIATA): the X2 test--the corrected chi-square comparison against the control-mean. Clin Appl Thromb Hemost 2007; 13: 73-100. [22] Stief TW. Inhibition of thrombin generation in recalcified plasma. Blood Coagul Fibrinolysis 2007; 18: 751-60. [23] Abildgaard U. Antithrombin--early prophecies and present challenges. Thromb Haemost 2007; 98: 97-104. [24] Sorg H, Hoffmann JN, Rumbaut RE, Menger MD, Lindenblatt N, Vollmar B. Efficacy of antithrombin in the prevention of microvascular thrombosis during endotoxemia: An intravital microscopic study. Thromb Res 2007; 121: 241-8. [25] St Peter SD, Little DC, Calkins CM, Holcomb GW 3rd, Snyder CL, Ostlie DJ. The initial experience of antithrombin III in the management of neonates with necrotizing enterocolitis. J Pediatr Surg 2007; 42: 704-8. [26] Blomback M, Blomback B, Olsson P, Svendsen L. The assay of antithrombin using a synthetic chromogenic substrate for thrombin. Thromb Res 1974; 5: 621-32. [27] Fareed J, Messmore HL, Walenga JM, Bermes EW, Bick RL. Laboratory evaluation of antithrombin III: a critical overview of currently available methods for antithrombin III measurements. Semin Thromb Hemost 1982; 8: 288-301. [28] Gallimore MJ, Friberger P. Chromogenic peptide substrate assays and their clinical applications. Blood Rev 1991; 5: 117-27. [29] Stief TW. Antithrombin III determination in nearly undiluted plasma. Laboratory Medicine 2008; 39: 46-8. [30] Stief TW. Coagulation Activation by Lipopolysaccharides. Clin Appl Thromb Hemost 2007 Dec 26; [Epub ahead of print] [31] Kalinski T, Jentsch-Ullrich K, Fill S, König B, Costa SD, Roessner A. Lethal candida sepsis associated with myeloperoxidase deficiency and pre-eclampsia. APMIS 2007; 115: 875-80. [32] Lai CC, Liaw SJ, Lee LN, Hsiao CH, Yu CJ, Hsueh PR. Invasive pulmonary aspergillosis: high incidence of disseminated intravascular coagulation in fatal cases. J Microbiol Immunol Infect 2007; 40: 141-7. [33] Stief TW, Fröhlich S, Renz H. Determination of the global fibrinolytic state. Blood Coagul Fibrinolysis 2007; 18: 479-87. [34] Stief TW, Ulbricht K, Renz H, Max M. Plasmatic fibrinolysis in sepsis. Hemostasis Laboratory 2008; 1: 61-75. [35] Paul V, Pudoor A, Earali J, John B, Anil Kumar CS, Anthony T. Trial of low molecular weight heparin in the treatment of viper bites. J Assoc Physicians India 2007; 55: 33842. [36] Iyoda M, Suzuki H, Ashikaga E, Nagai H, Kuroki A, Shibata T, Kitazawa K, Akizawa T. Elderly onset systemic lupus erythematosus (SLE) presenting with disseminated intravascular coagulation (DIC). Clin Rheumatol 2007 Dec 19; [Epub ahead of print] [37] Ohlsson K. Elimination of 125-I-trypsin alpha-macroglobulin complexes from blood by reticuloendothelial cells in dogs. Acta Physiol Scand 1971; 81: 269-72. [38] Borth W. Alpha 2-macroglobulin, a multifunctional binding protein with targeting characteristics. FASEB J 1992; 6: 3345-53. [39] Goldenberg NA, Manco-Johnson MJ. Pediatric hemostasis and use of plasma components. Best Pract Res Clin Haematol 2006; 19: 143-55.

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[40] Kowalik MM, Smiatacz T, Hlebowicz M, Pajuro R, Trocha H. Coagulation, coma, and outcome in bacterial meningitis--an observational study of 38 adult cases. J Infect 2007; 55: 141-8. Epub 2007 Apr 2. [41] Moloney-Harmon PA. Pediatric sepsis: the infection unto death. Crit Care Nurs Clin North Am 2005; 17: 417-29. [42] Langer F, Spath B, Haubold K, Holstein K, Marx G, Wierecky J, Brümmendorf TH, Dierlamm J, Bokemeyer C, Eifrig B. Tissue factor procoagulant activity of plasma microparticles in patients with cancer-associated disseminated intravascular coagulation. Ann Hematol 2008; 87: 451-7. [43] Stief TW. Innovative tests of plasmatic hemostasis. Laboratory Medicine 2008; 39: 22530. [44] Rivers EP, Coba V, Visbal A, Whitmill M, Amponsah D. Management of sepsis: early resuscitation. Clin Chest Med 2008; 29: 689-704.

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In: Coagulation: Kinetics, Structure Formation Editors: A.M. Taloyan et al

ISBN: 978-1-62100-331-1 © 2012 Nova Science Publishers, Inc.

Chapter 7

NEONATAL COAGULATION PROBLEM Viroj Wiwanitkit Faculty of Medicine, Chulalongkorn University, Bangkok, Thailand

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Introduction to Coagulation Defect in Neonatal Basically, successful blood stop must be composed of three basic effective components: good blood coagulation factor system, good platelet and good vascular structure. When there is an injury to the blood vessel, internal or external, the vessel firstly response by vasoconstriction. Then the coagulation cascade will start and process to finalize in fibrin network and platelet will go to accumulate at the injure size to get the complete blood stop process [1]. If there is any error in any basic component in any process, there will be a failure in blood stopping. On the other hand, there is a balance system to the coagulation process, the fibrinloytic system. If there is a defect in the balance system, the thrombotic disorder can also be detected. Thrombosis risk is multifactorial, with interaction of hereditary risk factors and acquired environmental and clinical conditions [2]. Newborns are at particular risk for thrombotic emergencies secondary to the unique properties of their hemostatic system, influences of the maternal-fetal environment, and perinatal complications and interventions [2]. The neonatal coagulation problem will be presented in this chapter.

Platelet Defect in the Neonate Platelet defect in either quantity or quality can result in unsuccessful hemostasis. Platelet defect in neonate can also be seen. Basically, neonatal platelets are less reactive than adult platelets to physiological agonists in whole blood, as determined by the activation-induced increase in the platelet surface expression of P-selectin and the glycoprotein (GP) IIb-IIIa complex and by the activation-induced decrease in the platelet surface expression of the GPIb-IX complex [3]. Michelson et al suggested that the mechanism of neonatal platelet hyporeactivity is, at least in part, a relative defect in a common signal transduction pathway [3]. For the quantitative defect, abnormal low platelet or thrombocytopenia can be seen. Basically, thrombocytopenia is generally known in its most severe form as acquired

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immunologic disease [4]. However the frequency of neonatal thrombocytopenia in all newborns (< 150 x 10(9)/L) has been estimated at 1-4%. eonatal thrombocytopenia is a common clinical problem [4-6]. The majority of episodes are early-onset thrombocytopenias due to impaired fetal megakaryocytopoiesis associated with placental insufficiency; the commonest causes of severe early-onset thrombocytopenia are immune thrombocytopenias, congenital infections, and asphyxia [5]. However, in some cases thrombocytopenia is constitutional and may or may not be associated with thrombocytopathy [4]. The affected cases are characterized by the absence of significant bleeding disorders, stable thrombocyte counts higher than 50 x 10(9)/l, platelet macrocytosis, normal platelet function, normal or never increased presence of megakaryocytes, and rarely positive immunological abnormalities [4]. In all cases, platelet life span clearly indicated a defect of production with destruction linked to the ageing population (more than 7 days) and no abnormal sequestration in the spleen or liver [4]. By contrast, about 90% of cases of severe thrombocytopenia presenting after the first few days of life are due to late-onset bacterial sepsis, necrotizing enterocolitis, or both. Although clinically stable neonates tolerate relatively low platelet counts without significant risk of hemorrhage, ill or clinically unstable neonates with profound thrombocytopenia often have a poor outcome [5]. Concerning the immune thrombocytopenia, neonatal immune thrombocytopenia due to the transplacental passage of maternal antiplatelet IgG is a transient passive disease in an otherwise healthy newborn [6]. Indeed, the allo-immune thrombocytopenias are the major cause of severe thrombocytopenia in the fetus and the neonate [7]. The frequency of this affection has been evaluated to be 1 out of 800 to 1000 live births [7]. Autoimmune thrombocytopenic purpura in pregnant women can induce moderate or severe thrombocytopenia in the foetus or the newborn whatever the mother's disease status [6]. Fetal thrombocytopenia can occur as early as 20 weeks of gestation [6]. Heparin-induced thrombocytopenia (HIT), an immune-mediated response to heparin administration, has been recognized in adults for some time, but only recently recognized in neonates and children [8]. The incidence of HIT Type II is 2-5 percent in adults on heparin products and may be as high in neonates and children. The mortality rate from HIT in newborns is unknown but should be high [8]. l. The cardinal sign of HIT is a drop in platelet count by 50 percent or platelet counts below 70,000-100,000/mm3 within five to ten days after the first exposure to heparin [8]. Treatment is immediate cessation of all heparin therapy and initiation of alternative anticoagulants, especially the direct thrombin inhibitors lepirudin and argatroban [8]. The major risk of severe thrombocytopenia is intracranial hemorrhage eading to death or neurological impairment [6]. Generally, an incidence of intracranial hemorrhage ICH in the neonatal period is 10-20%, and the autoimmune form with an incidence of only 1% [9]. Lopez-Lafuente et al suggested that cranial ecographic studies should be done in all newborn babies with immune thrombocytopenia even when no neurological disorder was seen [19]. They noted that an early diagnosis and suitable treatment might help to reduce the neurological sequelae [9]. Lopez-Lafuente et al also proposed that the neurological complications due to intraparenchymatous hemorrhage and visual sequelae were frequent [9]. Infection is also another important cause of thrombocytopenia in newborn. Bacterial sepsis is well-known to be the cause of neonatal thrombocytopenia [10-11]. The very low birth weight infant is the most high risk group. Common complications of sepsis in very low birth weight infants include anemia, thrombocytopenia, bleeding and multi-organ failure [10]. Strategies aimed at prevention, such as limiting the excessive use of broad-spectrum empiric

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antibiotics and the periodic review and continuous reinforcement of infection control policies will help decrease the mortality and morbidity associated with nosocomial infection in the very low birth weight infant [10]. There are important common infections relating to thrombocytopenia in the neonate that should be mentioned. Cytomegalovirus (CMV) is a common problematic infection and accepted as a member of TORCG agents that can be transmitted transplacentally. Basically, CMV is one of the Herpesviridae, known for their potential for latency and reactivation. The sequelae of fetal infection are diverse: chronic stage of early fetal infection with brain anomalies, symptomatic late fetal infection with hepatitis and thrombocytopenia and asymptomatic infection [12]. Neonatal adenovirus infection is another important infection-related cause of thrombocytopenia. Adenovirus infection in neonate is frequently disseminated and generally fatal, should be considered in the differential diagnosis of neonatal sepsis and pneumonia [13]. Common clinical findings includ lethargy, fever or hypothermia, anorexia, apnea, hepatomegaly, bleeding, and progressive pneumonia [13]. Thrombocytopenia, coagulopathy, and hepatitis are typical laboratory manifestations [13]. Necrotizing enterocolitis can be another important cause of neonatal thrombocytopenia. Necrotizing enterocolitis is the most common gastrointestinal emergency of the neonate, affecting 5-10% of infants [14]. Widely accepted risk factors include prematurity, enteral feeds, bacterial colonization and mucosal injury [14]. The activation of the cytokine cascade, in part by bacterial ligands, appears to play a key role in mucosal injury [14]. Severe thrombocytopenia frequently accompanies advanced necrotizing enterocoliti. In premature infants with severe thrombocytopenia, despite a paucity of evidence to support the practice, platelet transfusions are commonly used to maintain arbitrary levels of platelet counts in an effort to prevent hemorrhage. However, platelet transfusions contain a variety of bioactive factors, including platelet activating factor, which can augment systemic inflammatory processes [15]. Considering the impairment of platelet quality, it is usually accompanied with decreased platelet quantity. The good example is the dengue infection. Dengue infection is a mosquito borne infection that is endemic in many tropical countries. This infection manifests as a common triad: hemoconcentration, thrombocytopenia and high fever. It should be noted that dengue infection can be seen in the neonate [16]. In addition, the congenital dengue infection is also reported [16]. Wiwanitkit proposed that platelet CD61 might have an important role in causing hemorrhagic complication in dengue infection [17]. For the underlying of pathogenesis of thrombocytopenia, the immune mimicking theory is a widely mentioned [1617]. The principal aim in the management of the affected infants with thrombocytopenia is to prevent the deleterious consequences of severe thrombocytopenia [6]. Supportive treatment should be effectively administered. Platelet transfusion might be needed [4-6]. If there is an identified cause such as infection, the physician should also get rid of the primary root.

Vascular Defect in the Neonate Vascular defect that can lead to bleeding presentation is not common. A condition that relating to vascular that can lead to bleeding in pediatric population is vitamin C deficiency. Physiologically, bitamin C is an important vitamin for the hemostasis process. The main action of vitamin C in the hemostasis process is mainly on the vascular component. Vitamin

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C deficiency can lead to the most well-known vitamin deficiency disease, scurvy. Scurvy and periodontitis both manifest gingival bleeding but constitute separate entities [18]. Defective collagen in scurvy reflects many symptoms emanating from deficient vitamin C physiology [18]. Its prominent clinical features are lethargy; purpuric lesions, especially affecting the legs; myalgia; and, in advancing disease, bleeding from the gums with little provocation [19]. Common misdiagnoses are vasculitis, blood dyscrasias, and ulcerative gingivitis [22]. Untreated, scurvy is inevitably fatal as a result of infection or sudden death [19]. Fortunately, individuals with scurvy, even those with advanced disease, respond favorably to administration of vitamin C [19]. Recently, Ratanachu-Ek et al studied 28 cases of scurvy [20]. According to this study, they concluded that vitamin C deficiency was found in the children's intake of small amounts or no vegetables and fruits together with UHT-milk [20]. They noted that frequent manifestations of scurvy were limping and inability to walk and pain in the lower limbs and response to vitamin C treatment was dramatic [20]. Neonatal scurvy is an important neonate nutritional problem [21-22]. There are many possible unwanted complications of neonatal scurvy including epiphyseo-metaphyseal cupping [22] and spinal change [23].

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Coagulation Defect in the Neonate Coagulation defect that can lead to bleeding presentation is common. There are many congenital defects of coagulation factor system. The most well-known is hemophilia. Hemophilia is one of the oldest known hereditary bleeding disorder. There are two main types of hemophilia, A and B (Christmas Disease). Low levels or complete missing of a blood protein essential for clotting causes both types. Patients with hemophilia A lack the blood clotting protein, factor VIII, and those with hemophilia B lack factor IX [24]. The severity of hemophilia is usually related to the amount of the clotting factor in the blood [24]. About 70% of hemophilia patients have less than one percent of the normal amount and, therefore, have severe hemophilia [24]. A small increase in the blood level of the clotting factor, up to five percent of normal, results in mild hemophilia with rare bleeding except after injuries or surgery [24]. In people with hemophilia, blood does not clot as it should because it is missing or has low levels of one of these clotting factors [25]. If blood doesn‘t clot as quickly or as well as it should, then: a) heavy blood loss can occur, b) body organs and tissues can be injured, and c) these conditions can result in permanent damage or death [25]. Sometimes people with hemophilia need infusions of a clotting factor or factors to stop bleeding [25]. In the presence of a family history of haemophilia optimal management requires close cooperation between three individual specialist groups - obstetricians, haematologists and neonatologists, who each have an important role to play in ensuring a safe outcome for these infants [26]. Hemophilia A is caused by partial or complete depletion of factor VIII. In general, hemophilia is more common than hemophilia B [24-25]. Technologies in molecular biology have greatly advanced the knowledge regarding the origin of haemophilia A and the physiology of the factor VIII protein [27]. It is an X-linked recessive bleeding disorder affecting one in 10,000 males [28]. Prevalence of the haemophilia gene in the general population has increased recently due to advances in treatment, which has resulted in reproductive fitness among heamophiliacs [28]. A variety of different mutations in the factor

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VIII gene have been identified and their effects on the factor VIII protein described [26]. Basically, Factor VIII gene is very large with 26 exons [28]. Defects in this gene result in the deficiency of factor VIII molecule [28]. It has been shown that the frequency of haemophilia A is due to a high mutation rate predominantly in male germ cells [28]. A significant proportion is originating de novo in early embryogenesis from somatic mutations, a finding that has implications for genetic counseling [28]. Most recently it has been shown that FVIII clearance from the circulation is mediated by the low-density lipoprotein receptor-related protein (LRP) and cell-surface heparan sulphate proteoglycans (HSPGs) [28]. With the advent of recent advances in the molecular biology, it is possible to identify the multiple molecular defects such as point mutations, premature stop codons, deletions, and inversions etc in the FVIII gene in patients with haemophilia [29]. Nowadays the use of polymerase chain reaction (PCR)-based linkage analysis and direct mutation detection in the chorionic villus sample obtained at 10-12 weeks of gestation has significantly improved the prenatal diagnosis of haemophilia [28]. Southern blot analysis successfully detected a carrier in a hemophilia family for which no patient was available [29]. It should be noted that major organ bleeding in severe haemophilia A in the newborn period is rare, and this unusual complication is not well recognized [30]. Failure to recognize that the bleeding is due to a bleeding disorder, particularly in the absence of a family history, may lead to delay in appropriate management [30]. Haemophilia B, an X-linked recessive bleeding disorder due to factor IX (FIX) deficiency, has an incidence of about 1:30,000 live male births. The factor 9 (F9) gene was mapped on Xq27.1. Several factor IX deficiency disorders are reported [31]. The one known as hemophilia Bm variant, factor IXHilo, is the new deficiency first described in Europe [32]. Pathophysiologically, a point mutation that resulted in the substitution of a glutamine (CAG) for an arginine (CGG) at amino acid 180 is found in exon VI of the factor IX gene (G----A at nucleotide 20519) [32]. This mutation alters the carboxy terminal cleavage site for the activation peptide at Arg180-Val181 [32]. For hemophilia B, it is diagnosed from prothrombin time, activated partial thromboplastin time, and factor IX levels [33]. Similar to hemophilia A, several molecular tools are used for diagnosis of hemophilia B. The recent progress in the uses of factor IX gene probes in clinical diagnosis of hemophilia B carriers, as well as their use for analyzing the structural gene abnormalities that are responsible for the disease should be noted [34]. Generally, carrier females are usually asymptomatic and must be identified only with molecular analysis [33]. Linkage analysis of factor 9 polymorphisms is rapid and inexpensive but limited by non-informative families, recombinant events, and the high incidence of germline mutations; thus, various procedures have been used for the direct scan of factor 9 mutations [33]. A novel denaturing high performance liquid chromatographic procedure to scan the factor 9 gene is also launched [33]. However, the complicated technology is lack in the developing countries. The basic classical test is still needed in these settings. The clinical manifestations of hemophilia depend upon both age and the severity of the factor VIII or IX deficiency [35]. Laboratory tests include platelet count, prothrombin time, partial thromboplastin time, and bleeding time [35]. the transfusion is an important tool for treatment of the patients with hemophilia. The clotting factor can be prepared from the donated blood. The problem on finding of blood donor and donor selection should be mentioned. In addition, the problem of blood borne infection in the present day should be stated. Human immunodeficiency virus (HIV) and hepatitis C virus (HCV) in the blood

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supply are the important problematic blood borne transmitted diseases. The risk of pathogen transmission via clotting factor therapies has been reduced over the last two decades through the development of effective and progressively more sensitive pathogen screening and inactivation methods and the introduction of recombinant clotting factors for hemophilia, beginning with recombinant factor VIII in 1992 [36]. However, new understanding about the potential for transmission of an emerging infectious agent through blood and blood products has renewed concerns about vulnerabilities that remain in plasma-derived and some recombinant clotting therapies that still use plasma components during some stages of the manufacturing process [36]. In addition to congenital coagulation factor defect, there are also other acquired conditions in neonate. One of the well-known conditions is disseminated intravascular coagulopathy (DIC). This condition also affects the platelet. Quick action is necessary to determine the cause of bleeding, which determines how the infant will be treated [37]. DIC is usually an uncontrolled, simultaneous bleeding and clotting occurring as a secondary disorder in sick neonates [37]. The newborn infant is particularly susceptible to DIC because of several handicaps, such as physiological hypofunction of anticoagulant and fibrinolytic systems, an underdeveloped capacity in the reticuloendothelial system and a tendency to develop acidosis, hypothermia, hypoxia and shock [38]. Although some criteria have been reported for the diagnosis of DIC in adults, based on clinical and laboratory findings, these are not necessarily applicable to the diagnosis of DIC in newborn infants [38]. This is because a large blood sample is required, a long period of time is necessary for assay and difference in several coagulation and fibrinolysis factors exist between newborn infants and adults [38]. Shirahata et al indicated that the scoring system for platelet counts in diagnostic criteria of DIC in newborn infants was applicable to the diagnosis of DIC in very low birth weight infants, however, the scoring systems for fibrinogen and D-dimer were not usable for the diagnostic criteria of DIC in very low birth weight infants since fibrinogen and D-dimer concentrations in very low birth weight infants without DIC were lower than those in nonDIC neonates whose birth weights were above 1500 grams [39]. Knowledge of the complex physiologic mechanisms at work to maintain hemostasis contributes to the proper nursing care of infants at risk for DIC and better outcomes [37].

Thrombohemostatic Defect in the Neonate Thrombohemostatic disorder is an important hematological disorder in neonatology. Venous thromboembolism is an increasingly recognised problem in pediatric practice, particularly in the context of tertiary care pediatric services [40]. Several thrombohemostatic disoders are described for the neonates. Neonatal cerebrovenous sinus thrombosis is extremely rare, however it is a devastating condition and one needs to be aware of this condition to diagnose it [41]. Gestational or delivery complications or risk factors and comorbid conditions such as dehydration, sepsis, and cardiac defects are common for neonatal cerebrovenous sinus thrombosis [42]. Fitzgerald et al proposed that the presentation of neonatal SVT was often nonspecific, the diagnosis could be difficult to make, treatment beyond supportive care was rarely used, and outcomes can be severe [42]. Concerning the arterial thromboembolism, perinatal stroke has become increasingly recognized, but the incidence is probably underestimated because of variation in the presentation, evaluation, and

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diagnosis [43]. Perinatal arterial ischemic stroke in infants is associated with several independent maternal risk factors [44-45]. Maternal conditions that have been associated with perinatal stroke in the fetus include prothrombotic disorders, cocaine abuse, and placental complications such as chorioamnionitis and placental vasculopathy [44]. In many cases, the placenta is suspected to be the underlying embolic source for perinatal stroke, although data on placental pathology is often lacking [44]. During the delivery process, an infant may develop a cervical arterial dissection that leads to stroke [44]. Several conditions in the neonatal period predispose to perinatal stroke including prothrombotic disorders, congenital heart disease, meningitis, and systemic infection [44]. Based on estimates from populationbased studies of infants with seizures, perinatal stroke occurs in approximately 1 in 4000 term births [43]. Perinatal stroke may present with neonatal seizures during the first weeks of life or may be asymptomatic until months later when the infant is first noted to have pathological handedness [43]. Perinatal stroke underlies an important share of congenital hemiplegic cerebral palsy, and probably some spastic quadriplegic cerebral palsy and seizure disorders [46]. The outcome of perinatal stroke is variable and depends on severity, anatomic localization, and other factors not yet well clarified [44].

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REFERENCES [1] Goldenberg NA, Manco-Johnson MJ. Pediatric hemostasis and use of plasma components. Best Pract Res Clin Haematol. 2006; 19(1):143-55. [2] Thornburg C, Pipe S. Neonatal thromboembolic emergencies. Semin Fetal Neonatal Med. 2006 Jun;11(3):198-206. [3] Michelson AD. Platelet function in the newborn. Semin Thromb Hemost. 1998; 24(6):507-12. [4] Najean Y, Lecompte T. Hereditary thrombocytopenias in childhood. Semin Thromb Hemost. 1995; 21(3):294-304. [5] Roberts IA, Murray NA. Neonatal thrombocytopenia. Curr Hematol Rep. 2006 Mar;5(1):55-63. [6] Kaplan C. Immune thrombocytopenia in the foetus and the newborn: diagnosis and therapy. Transfus Clin Biol. 2001 Jun;8(3):311-4. [7] Kaplan C. Fetal/neonatal allo-immune thrombocytopenias: the unsolved questions. Transfus Clin Biol. 2005 Jun;12(2):131-4. [8] Martchenke J, Boshkov L. Heparin-induced thrombocytopenia in neonates. Neonatal Netw. 2005 Sep-Oct;24(5):33-7. [9] Lopez-Lafuente A, Campistol J, Toll MT, Iriondo M. Perinatal intracranial hemorrhage due to immune thrombocytopenia. Rev Neurol. 1999 Nov 16-30;29(10):917-22. [10] Trotman H, Bell Y. Neonatal sepsis in very low birthweight infants at the University Hospital of the West indies. West Indian Med J. 2006 Jun;55(3):165-9. [11] Boyer KM. Diagnosis of neonatal sepsis. Mead Johnson Symp Perinat Dev Med. 1982; (21):40-6. [12] Lagasse N, Dhooge I, Govaert P. Congenital CMV-infection and hearing loss. Acta Otorhinolaryngol Belg. 2000; 54(4):431-6. [13] Abzug MJ, Levin MJ. Neonatal adenovirus infection: four patients and review of the literature. Pediatrics. 1991 Jun;87(6):890-6.

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[14] Gibbs K, Lin J, Holzman IR. Necrotising enterocolitis: the state of the science. Indian J Pediatr. 2007 Jan;74(1):67-72. [15] Fernandes CJ, O'Donovan DJ. Platelet transfusions in infants with necrotizing enterocolitis. Curr Hematol Rep. 2006 Mar;5(1):76-81. [16] Wiwanitkit V. Dengue haemorrhagic fever in pregnancy: Appraisal on Thai cases. J Vector Borne Dis. 2006 Dec;43(4):203-5. [17] Wiwanitkit V. Platelet CD61 might have an important role in causing hemorrhagic complication in dengue infection. Clin Appl Thromb Hemost. 2005 Jan;11(1):112. [18] Touyz LZ. Vitamin C, oral scurvy and periodontal disease. S Afr Med J. 1984 May 26;65(21):838-42. [19] Pimentel L. Scurvy: historical review and current diagnostic approach. Am J Emerg Med. 2003 Jul;21(4):328-32. [20] Ratanachu-Ek S, Sukswai P, Jeerathanyasakun Y, Wongtapradit L. Scurvy in pediatric patients: a review of 28 cases. J Med Assoc Thai. 2003 Aug;86 Suppl 3:S734-40. [21] Hirsch M, Mogle P, Barkli Y. Neonatal scurvy: report of a case. Pediatr Radiol. 1976 Aug 20;4(4):251-3. [22] Sprogue PL.Epiphyseo-metaphyseal cupping following infantile scurvy. Pediatr Radiol. 1976 Feb 13;4(2):122-3. [23] Maclean AD. Spinal changes in a case of infantile scurvy. Br J Radiol. 1968 May;41(485):385-7. [24] Hemophilia. Available at http://www.medceu.com/tests/hemophilia.htm [25] Hemophilia. Available at http://www.nhlbi.nih.gov/health/dci/Diseases/hemophilia/ hemophilia_what.html [26] Chlamers EA. Haemophilia and the newborn. Blood Rev. 2004 Jun;18(2):85-92. [27] Oldenburg J, Ananyeva NM, Saenko EL. Molecular basis of haemophilia A. Haemophilia. 2004 Oct;10 Suppl 4:133-9. [28] Saxena R, Mohanty S, Choudhry VP. Prenatal diagnosis of haemophilia. Indian J Pediatr. 1998 Sep-Oct;65(5):645-9. [29] Fukuda K, Naka H, Morichika S, Shibata M, Tanaka I, Shima M, Yoshioka A. Inversions of the factor VIII gene in Japanese patients with severe hemophilia A. Int J Hematol. 2004 Apr;79(3):303-6. [30] Hamilton M, French W, Rhymes N, Collins P. Liver haemorrhage in haemophilia--a case report and review of the literature. Haemophilia. 2006 Jul;12(4):441-3. [31] Bowen DJ. Haemophilia A and haemophilia B: molecular insights. Mol Pathol. 2002; 55:127-44. [32] Huang MN, Kasper CK, Roberts HR, Stafford DW, High KA. Molecular defect in factor IXHilo, a hemophilia Bm variant: Arg----Gln at the carboxyterminal cleavage site of the activation peptide. Blood. 1989; 73:718-21. [33] Castaldo G, Nardiello P, Bellitti F, Santamaria R, Rocino A, Coppola A, di Minno G, Salvatore F. Haemophilia B: from molecular diagnosis to gene therapy. Clin Chem Lab Med. 2003 Apr;41(4):445-51. [34] Tonimoto M. Factor IX molecular defects in diagnosing hemophilia B: a review. Nippon Ketsueki Gakkai Zasshi. 1989 Jul;52(4):811-5. [35] Buchanan GR. Hemophilia. Pediatr Clin North Am. 1980 May;27(2):309-26. [36] Pipe S. Consideration in hemophilia therapy selection. Semin Hematol. 2006 Apr;43(2 Suppl 3):S23-7.

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[37] Keuhl J. Neonatal disseminated intravascular coagulation. J Perinat Neonatal Nurs. 1997 Dec;11(3):69-77. [38] Shirahata A, Shirakawa Y. New approach to the diagnosis of disseminated intravascular coagulation in childhood. Nippon Rinsho. 1993 Jan;51(1):61-6. [39] Shirahata A, Shirakawa Y, Murakami C. Diagnosis of DIC in very low birth weight infants. Semin Thromb Hemost. 1998; 24(5):467-71. [40] Chalmers EA. Epidemiology of venous thromboembolism in neonates and children. Thromb Res. 2006; 118(1):3-12. [41] Ibrahim HS. Cerebral venous sinus thrombosis in neonates. J Pak Med Assoc. 2006 Nov;56(11):535-7. [42] Fitzgerald KC, Williams LS, Garg BP, Carvalho KS, Golomb MR. Cerebral sinovenous thrombosis in the neonate. Arch Neurol. 2006 Mar;63(3):405-9. [43] Lynch JK, Nelson KB. Epidemiology of perinatal stroke. Curr Opin Pediatr. 2001 Dec;13(6):499-505. [44] Wu YW, Lynch JK, Nelson KB. Perinatal arterial stroke: understanding mechanisms and outcomes. Semin Neurol. 2005 Dec;25(4):424-34. [45] Lee J, Croen LA, Backstrand KH, Yoshida CK, Henning LH, Lindan C, Ferriero DM, Fullerton HJ, Barkovich AJ, Wu YW. Maternal and infant characteristics associated with perinatal arterial stroke in the infant. JAMA. 2005 Feb 9;293(6):723-9. [46] Nelson KB, Lynch JK. Stroke in newborn infants. Lancet Neurol. 2004 Mar;3(3):150-8.

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Chapter 8

COAGULATION AND WALL SHEAR STRESS IN LIVING DONOR LIVER TRANSPLANTATION Yoshinobu Sato and Hideki Nakatsuka and Toru Abo

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Division of Digestive and General Surgery, Niigata University Graduate School of Medicine and Dental Sciences, Japan Over the last century many investigators have studied liver regeneration, thus giving rise to new biologic themes, concepts, and techniques. The experimental model of liver regeneration after a partial hepatectomy is considered to be very useful in understanding immune surveillance, metabolism, and also blood coagulation and fibirinolytic (BCF) system, since autoimmune diseases or carcinogenesis can lead to catastrophe but liver regeneration occurs without any treatment. There are many clinical difficulties associated with major hepatic surgery or living donor liver transplantation (LDLT). Overcoming these problems will require a better understanding of the basic mechanisms of liver regeneration. In this chapter, we described the BCF system between recipients and donor of a LDLT. BCF system has an important concern with hemodynamic changes and liver regeneration. We have proposed that the wall shear stress, a simple hemodynamic force caused by portal venous flow directed against vessel walls, regulates liver size and growth as well as atrophy due to the apoptosis of individual hepatocytes. Also the wall shear stress induces the dynamic changes of BCF system and immune system in the intra- and extra-hepatic circumstances accompanied with hepatic regeneration. The understanding of BCF system after liver transplantation is important for patient management. Especially in the early period after adult LDLT, there is liver regeneration accompanying portal hypertension because of a small-for-size graft. The BCF system in healthy donors mediate the physiologic regeneration pattern in the early period after liver resection. Conversely, recipients are exposed to the influences of excessive surgical stress and portal hypertension. Therefore, we investigated differences in the perioperative BCF system between donor and recipient after LDLT, primarily considering serum plasminogen-activator inhibitor-1 (PAI-1) and soluble fibinogen (SF) levels. And otherwise hemeoxygenase-1 (HO-1) is a stress-induced enzyme that catalyses the oxidation of heme to biliverdin. Hemeoxygenase-1 produces carbon monoxide (CO) as a byproduct of hemoglobin metabolism. HO-1 or HO-1 products might regulate TF and TM expression and prevent thrombus formation in human endothelial cells. Human HO-1 deficiency has been observed to involve the endothelial cells more severely, resulting in hemolysis and disseminated intravascular coagulation. The present study examines the relationship between CO production and hyperbilirubinemia following adult LDLT with special attention to the contribution of shear stress in retarding regeneration.

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1. Introduction The concept of injury in liver regeneration after partial hepatectomy (Phx), and the reason hepatocytes that had not been directly regenerate, remain unclear. It is known that shear stress resulting from blood flow plays an important role in the mechanism of remodeling blood vessels, and portal pressure reflects shear stress (1-3). We proposed that acute portal hypertension (APH) as shear stress following Phx might not only become trigger of liver regeneration, but also of sinusoidal endothelial cell (SEC) and that surplus APH induces liver dysfunction (4). The blood coagulation and fibrinolytic (BCF) systems have important concern with liver regeneration following Phx (5). HO-1 produces carbon monoxide (CO) as a byproduct of hemoglobin metabolism (6,7). Moreover the CO generated by HO is thought to upregulate cGMP via activation of guanylyl cyclase, and thereby shears several biological actions with nitric oxide (NO), such as smooth muscle relaxation or inhibition of platelet aggregation. Suematsu et al. demonstrated that suppression of endogenous CO generation elicits a marked increase in the vascular resistance which is concurrent with an elevation of bile formation in the liver, suggesting the biological CO as a modulator of hepatobiliary function. In this chapter, we describe that the mechanisms of relationship between BCF and HO-CO systems during liver regeneration after clinical living donor liver transplantation.

2. Shear Stress Theory and Liver Regeneration

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Following Phx Fausto (8) reported that liver regeneration after Phx can be classified into a noninjured type, which is not due to stem cell proliferation, as distinguished from the injured type, induced by CCL4. The question remains as to whether any injury occurs after Phx. We have previously proposed that wall shear stress, which is a simple hemodynamic force cased by venous flow directed against vessel walls, regulates liver size and growth as well as atrophy due to the apoptosis of individual hepatocytes (4) (Fig.1). The intensity of wall shear stress on the endothelial cell layer was calculated by the formula τ =μ·dv/dl, where μ is the viscosity, v is flow velocity, and l is distance from center of bloodstream. Kamiya and Togawa (2) reported that shear stress against the blood vessel walls plays an integral part in the flow-mediated increases in blood vessel diameter. It was also recently shown that shear stress influences the expression of VCAM-1, CD44 (9), c-fos, cmyc, c-jun (10), transforming growth factor (TGF-β) (11), and nitric oxide synthase (NOS) (12) in vascular endothelial cells (Fig.1). Vascular endothelial cells correspond to the sinusoidal endothelial cells (SEC) in the liver. Similar phenomena are also observed in the liver after Phx (13-16).

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Figure 1: Schema of hypothesis of shear stress in liver regeneration following partial hepatectomy.(ref.4) Shear stress increases immediately after moderate PHx, stimulating liver regeneration and inducing physiologic liver regeneration. Shear stress decreases after PC shunting, inducing liver atrophy, while it overincreases after surplus massive hepatectomy, inducing liver failure

Interestingly, the sinusoid system contains doublebarreled structures formed by the sinusoid lumen and the space of Disse (17), whereby the hepatocytes are directly exposed to portal pressure through the sieve plates. Hepatocytes and SEC, as well as vascular endothelial cells, are thus responsible for portal pressure due to shear stress. The existence of a receptor of remodeling for shear stress on endothelial cells has been proposed but has yet to be demonstrated. However, a cell surface modulator (CSM) must exist on the hepatocytes and SEC as well as endothelial cells. Although the true character of the CSM involved in liver regeneration remains unclear, it is possible that the Ca ionic channel (18), Na ionic channel (19), or gap junction (20), which have already been reported, are all possible CSM candidates in liver regeneration. Furthermore, excessive shear stress after a massive hepatectomy may induce liver failure, and less shear stress may induce liver atrophy after portocaval (PC) shunt construction (4) (Fig.2). This hypothesis is thus considered to explain the absence of liver regeneration in hepatic cirrhosis despite the presence of portal hypertension. There are two possible explanations for this. The first is the ― sinusoidal capillarization‖ (21) of SEC in hepatic cirrhosis. The sieve plates disappear during the progression of cirrhosis as SEC becomes arterialized. Therefore, the excessive shear stress caused by portal hypertension may not directly affect hepatocytes or induce regeneration. A second possibility is severe fibrosis. Liver regeneration after PHx involves not only the regeneration of hepatocytes but also that of SEC. SEC regeneration leads to severe fibrosis. Fibrotic tissue does not have the capacity to regenerate. This phenomenon may inhibit the regeneration of hepatocytes, since the hepatocytes isolated from cirrhotic livers are capable of proliferation.

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Figure 2. Changes in portal pressure and platelet count after Phx. (ref.4) A. The portal pressure increased immediately after both 70% and 90% PHx. The portal pressure after 70% PHx peaked on postoperative day (POD) 3 and decreased to a normal level by POD 7, while that after 90% PHx was still significantly high even on POD 7 B. While the latelet count did not decrease in the 70% PHx rats, it de- creased markedly on POD 3 in the 90% PHx.

It has been reported that the fenestrae of sieve plates are influenced by the portal pressure and control portal blood flow, thus creating a gradient of portal pressure in the sinusoid (22,23). The portal pressure is the highest at the zone of transition from the end of the portal branch to the sinusoid, after which the pressure decreases, thereby allowing the portal blood flow to the end of the central venous branch. Although a pressure difference of 4 to 5cmH2O exists between the end of the portal branch and the end of the central venous branch, and the vasculature is quite short, the gradient of sinusoidal pressure may be very steep (24). We suggest that the elevated shear stress after Phx may destroy this gradient and damage either the hepatocytes or SEC. The ― flow hypothesis‖ of portal physiology was an established dogma well into the 1960s. Rous and Larimore (25) reported in the 1920s that a ligation of a portal branch leads to hepatic lobe atrophy of the ipsilateral and compensatory growth of the contralateral side. As a result, the experimental dogs subjected to such ligations did not suffer liver failure. Mann et al. (26) reported in 1931 that the liver does not regenerate after a partial hepatectomy with an Eck fistula (27). Fisher et al (28). demonstrated in 1954 that the liver could regenerate after Phx even when supplied by an arterial flow instead of a portal flow. However, the flow hypothesis was disproven in the report by Clarke et al (29). in which a massive increase in the portal flow by an arterialization of the portal vein was not found to be associated with a dramatic increase in the rate of cell division in the intact liver . Furthermore, a series of classic investigations of hepatotrophic factors also further support this point (30-38). Although the increase in the mitotic index in the intact liver with arterialization of the portal vein is not as dramatic as that caused by the 70% Phx in dogs in the report by Clarke et al., the increase in the mitotic index in a porto-aortic shunt group seemed to be greater than that in preoperative subjects or in a porto-spleno shunt group, thus corresponding to an increase in the portal flow volume. Furthermore, results of Clarke et al. resemble the finding of Bucher and Swaffield (39) in which the removal of 10% to 30% of the liver had only a very small effect on DNA synthesis (Fig. 3). These findings suggest that the wall shear stress caused by the arterialization of the portal vein in the intact liver may not be sufficient to

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induce liver regeneration. For this reason, it is difficult to say that these data completely disprove the flow hypothesis. We confirmed that the porto-renal artery fistula with portocaval shunt enlarged intact liver compared with sham operation. Hepatotrophic factors are very important for liver regeneration. However, the possibility that hepatotrophic factors are themselves directly toxic to hepatocytes after Phx is remote. For instance, although epidermal growth factor (40,41), TGF-α (42,43), and hepatocyte growth factor (44-46), which are widely considered to be ― complete mitogens,‖ actually augment the DNA synthesis of hepatocytes under serum-free conditions in primary culture, since the continuous infusion of these mitogens through the portal vein does not affect DNA synthesis in the intact rat liver in vivo (47,48). Moreover, events earlier than the mRNA expression of mitogens after PHx have also been reported (49-51). It is unlikely that these events occur without any in vivo injury after Phx.

Figure 3. Comparison of results of Clarke et al.(ref.29) (left) and Bucher and Swaffield (ref.30) (right). Results of Clarke et al. (open circles, partial hepatectomy; closed circles, aortoportal fistula; triangles, splenoportal fistula), in which the mitotic indexes of the intact liver with aortoportal fistula or splenoportal fistula were fewer than for a partial hepatectomy, resemble the findings of Bucher and Swaffield, where the removal of 10% and 30% of the liver had a only a very small effect on DNA synthesis.

Interestingly, the connection between these experimental results and the essence of liver regeneration has been demonstrated by recent progress in liver transplantation techniques. Liver allografts can regenerate after heterotopic liver transplantation with a porto-arterial anastomosis (52). Fisher et al (28). also reported that an increased hepatic blood flow due to porto-arterial anastomosis leads to liver regeneration. Child et al (53). demonstrated that the blood from the vena cava could induce liver regeneration by an anastomotic transposition between the portal vein and vena cava. These results suggest that liver regeneration is not controlled solely by the portal venous flow. The regenerative capacity of hepatocytes around periportal spaces is known to be higher than those around the pericentral spaces. Sigel et al

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(54). reported that the reversed flow from the portal vein to the hepatic vein induces the initial regeneration of hepatocytes around the pericentral spaces. This finding suggests that the predominance of periportal hepatocytes in regeneration is not predetermined. The above findings lead to a model in which liver regeneration is possible if the blood flow maintains an optimal sinusoidal pressure. Consistent with this hypothesis, the study by Isomura et al (55). showed that an increase in the portal flow induces c-myc expression in isolated perfused rat liver. Although MHC class I antigen expression has been observed on the sinusoidal lining cells (SLC) only around the periportal spaces in normal liver tissue, it has been seen on the SLC extending from the periportal spaces to pericentral spaces . This gradient of expression is accompanied by an increase in platelet-activating factor (PAF) production after stimulation by lipopolysaccharide (4) (Fig.4). These results thus suggest that an elevation of the shear stress after PHx may thus affect not only hepatocytes but SEC as well. Finally, the most intriguing findings of this study are that the gene expression of platelet-derived growth factor (PDGF) was upregulated by shear stress and PDGF-B chain promoter contains a cisacting fluid shear-stress-responsive element (56).

Figure 4. Changes in MHC class I antigen expression and platelet-activating factor(PAF) production capacity stimulated by LPS (1mg/kg) after 70%PHx. A: In the control rats, class I antigen expression was recognized on the sinusoid endothelial cells (SEC) only in the periportal spaces. B On POD 2 after 70% PHx, class I antigenexpression was seen in the SEC from the periportal spaces to the pericentral venous spaces. Ce; central venous area, Po; portal venous area. B: The gradient expression ofMHC class I is accompanied by an increase in platelet-activating factor (PAF) production after stimulation by lipopolysaccharide.

3. Concept of Immune System and Role of Shear Stress in Liver Regeneration Following Phx There have been few investigations of the role of the immune system in liver regeneration after a partial hepatectomy, especially in vivo. The appearance of autocytotoxic lymphocyte activation of B lymphocytes and increased immunoglobulin production have been reported

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(57-60). However, these data were obtained using either the spleen or blood, and thus the number of studies on local immunity in the regenerating liver is limited.

Figure 5. Interferon-γ inhibits liver regeneration by stimulating major histocompatibility complex class II antigen expression by regenerating liver.(ref.63). A and B:IFN-γaugment the MHC class II antigen expression of hepatocytes and Kupffer cells after 70%Phx. C: Liver regeneration with BrdU. Regeneration was inhibited more in groups given IL-2 or IFN-γthan in the group subjected to partial hepatectomy alone. D: Group 3.The number of Kupffer cells and class II positiveKupffer cells were markedly increased in the group given IFN-γ. Group 5. IL-2 and IFN-γ treatment groups, number of Kupffer cells and class II-positive cells were increased.

We hypothesized that the immune response against liver regeneration after Phx resembles ― acute rejection in liver transplantation.‖ Hence, we investigated the major histocompatibility complex (MHC) class I and class II antigen expression, which increased in the transplanted liver tissue undergoing rejection, in regenerating liver tissue. We observed the MHC class I and class II antigen expression to increase in the early phase in regenerating liver tissue, especially in the periportal spaces, after Phx. Moreover, FK506 or cyclosporine A (CYA), which are highly selective suppressors of helper T cells, accelerate liver regeneration following Phx by inhibiting class II antigen expression in the regenerating liver tissue (61,62). We therefore investigated what happens if autoantigenicity, i.e., MHC class II antigen expression, is augmented. We observed that interferon-γ (IFN-γ) inhibits liver regeneration by stimulating MHC class II antigen expression in the regenerating liver (63). In this study, the hepatocytes clearly expressed MHC class II antigens with MHC class II antigen-positive Kupffer cells and lymphocytes around the hepatic parenchyma (Fig. 5). Furthermore, the CD8+lymphocytes also increased in the periportal spaces. These data thus suggest that the MHC class I- and class II-dependent pathways, which are the basis of cellular immunity, function in the immunologic control of liver regeneration after Phx. IFN- γ mRNA expression

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in a normal young rat liver by reverse transcriptase-polymerase chain reaction (RT- PCR) showed a decreased expression in the early regenerating liver at 24h after Phx, which is the time of maximum liver regeneration activity after Phx. Interleukin-2 (IL-2) and IL-4 mRNA were not observed in the regenerating liver (64,65). This cytokine pattern may thus promote liver regeneration. Conversely, IFN- γ must play an important negative regulatory role for the control of liver regeneration after Phx. Mosmann and Coffman (66) have proposed a Th1–Th2 paradigm. Th1 cells may be more significant in the regenerating liver. However, in the spleen, the IL-2, IL-4, and IFN-γ mRNA expression increases while in the thymus the IL-2 and IFN-γ mRNA expression decreases, and the IL-4 mRNA expression increases, after PHx. The immunologic control of liver regeneration after Phx is thus complex and may give rise to different responses in various immune organs. It has recently been reported that the liver and intestine are major sites for the proliferation of extrathymic T cells (67,68). Abo et al (69). demonstrated that extrathymic T cells of liver show an intermediate intensity for the expression of CD3 and TcR, or ― intermediate TcR cells.‖ We also previously reported that intermediate TcR cells are activated in the liver during the early regenerative phase (days 2–4) after 70% Phx in mice (70) (Fig. 6). However, intermediate TcR cells have so far only been observed in mice.

Figure 6. Dual-colour staining of MNC in the liver. (ref.70). MNC obtained for the liver was analyzed by the dual-colour staining of CD3 and IL-2Rβ(or LFA-1) on the days indicated. Intermediate CD3+ cells were identified as cells with higher levels of IL-2Rβand LFA-1.

We recently demonstrated that rats also have intermediate CD3 cells (71), but these differ from the intermediate TcR cells of mice. Their numbers were almost one-half that in mice and almost always consisted of CD8＋T cells that were different from those in mice, which normally consisted of DNCD4ーCD8ーcells. Furthermore, the intermediate TcR cells, CD5＋B cells, which are associated with autoimmune disease, and natural killer (NK)-marker-positive T cells were activated in the regenerative liver in the early phase after Phx. It is unclear as to whether or not extrathymic T cells are related to the regulation of liver regeneration after Phx.

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However, it has been demonstrated that Vβ8＋T cells are cytotoxic against regenerative hepatocytes, but not normal hepatocytes (72,73).

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Figure 7. Schema of the ― Wash Out Theory‖by shear stress in the liver in the early phase during liver regeneration after Phx. We investigated the immune response after Phx differs in the liver, spleen, and thymus, as shown by the cytokine mRNA expression after Phx. These results suggest that the intrahepatic leukocytes washed out from the liver appear to influence systemic immunity. Namely, shear stress may affect the reciprocal interactions between local and systemic immunity.

A powerful paradigm has emerged in which leukocyte binding to the endothelium is explained by a three- or four- step process through the selectin family, integrin family, and related proteins (74-76). Meanwhile it has been reported that shear stress directly influences the mRNA expression of ICAM-1 (77), CD44, and VCAM-1 on endothelial cells. Moreover, it has also been demonstrated that increased shear stress suppresses the accumulation of leukocytes onto the endothelium (78). Antibodies immobilized on the wall of a flow chamber can also support leukocyte rolling in a shear flow. IgM mAb to Lewisx (CD15) and sialyl Lewisx (CD15s), which are carbohy- drate antigens related to selectin ligands, plus monoclonal antibody (mAb) to CD48 and CD59, are able to mediate such rolling. In contrast, IgM and IgG mAb to l- selectin (CD62L), LFA-1 (CD11a), CD43, ICAM3 (CD50), CD8, and CD45 only mediate firm adhesion. Antibodies supported rolling only within a restricted range of site densities and wall shear stress, outside of which firm adhesion or detachment occurred. As a result, the elevation of shear stress after Phx must also affect the leukocyte binding to SEC after Phx. Our studies therefore suggest the existence of two types of intrahepatic leukocytes; one type would tend to stay associated with SEC, while the other would not. For instance, RT6+T cells, CD4+T cells, and NK cells are easily dissociated from SEC, while CD8+T cells, LFA- 1+T cells, and macrophages tend to stay with or adhere to SEC in the early period after a partial hepatectomy. These findings might be compatible with a previous report (79), that is, CD8+T cells, LFA-1+T cells which strongly expressed LFA-1 molecules,

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and resident cells such as Kupffer cells which highly adhere to SEC, would thus be resistant to wall shear stress. In contrast, peripheral T cells such as RT6+T cells and CD4+T cells would tend to be dissociated from SEC. We therefore propose a schema for our ― Shear stress and wash-out theory‖ in Figs. 7. We mentioned that the immune response after Phx differs in the liver, spleen, and thymus, as shown by the cytokine mRNA expression after Phx. These results suggest that the intrahepatic leukocytes washed out from the liver appear to influence systemic immunity. Namely, shear stress may affect the reciprocal interactions between local and systemic immunity (80).

4. The Experimental and Clinical Data of Wall Shear Stress in the Liver

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Experimental Data Abo et al. previously reported that extrathymic T cells (intermediate T-cell receptor cells [TCRint cells]) are in situ generated in the parenchymal space of the liver in mice(81-84). They subsequently migrate to the sinusoidal lumen. In this study (85), we characterized how such extrathymic T cells, natural killer (NK) cells, and thymus-derived T cells (high T-cell receptor cells [TCRhigh cells]) localized in the parenchymal space or the sinusoidal lumen of mice. To this end, liver irrigation with physiological saline from the portal vein was performed and the distribution of lymphocyte subsets was compared between the liver (i.e., lymphocytes in the parenchymal space) and the irrigation solution (i.e., lymphocytes in the sinusoidal lumen). Extrathymic T cells and NK cells were found to be abundant in both the liver and sinusoidal lumen. As expected, thymus-derived T cells were abundant in the sinusoidal lumen. Mice. MaleC57BL/6(B6)andB6-nu/numicewereusedattheage of 8 to 9 weeks. For parabiosis experiments, B6.Ly5.1 mice were also used at the same age. These mice were originally obtained from Charles River Japan, Inc. (Tokyo, Japan) and were maintained at the animal facility of Niigata University (Niigata, Japan). All mice were fed under specific pathogen-free conditions. Liver Irrigation. Under ether anesthesia, mice were injected with 1.0 mL physiological saline through the portal vein to eliminate blood contained in the liver. This was done by the insertion of a 24-gauge needle. These mice were then killed by exsanguination via incision of the axillary artery and vein, and the liver was removed. Four groups of mice were classified. Mice of group 1 (control group) received only an injection of 1.0 mL saline. Mice of group 2 received irrigation of 30 mL saline from a height of 10 cm above the portal vein. Similarly, mice of groups 3 and 4 received irrigation of 30 mL saline from height of 20 cm and 30 cm, respectively, above the portal vein. Five mice were used in each protocol. In some experiments, physiological saline with 0.01% collagenase (Sigma Co., St. Louis, MO) was used for irrigation at a height of 30 cm to get a complete irrigation. Cell Preparation. Mononuclear cells were prepared from the liver and irrigation solution after the treatment. Hepatic mononuclear cells were isolated by an improved method as

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described previously. The excised liver was cut into small pieces with scissors, pressed through 200-gauge stainless steel mesh, and then suspended in Eagle‘s MEM medium supplemented with 5 mmol/L HEPES and 2% fetal calf serum. After being washed once, the cells were resuspended in 35% Percoll solution containing 100 U/mL heparin and centrifuged at 2,000 rpm for 15 minutes at room temperature. The cell pellet was resuspended in red blood cells (RBC) lysing solution (155 mmol/L NH4Cl, 10 mmol/L KHCO3, 1 mmol/L ethylenediaminetetraacetic acid–Na, and 170 mmol/L Tris, pH 7.3) then washed twice with the medium. Immunofluorescence Tests. The surface phenotype of cells was identified by using monoclonal antibodies (mAbs) in conjunction with 2- or 3-color immunofluorescence tests (86). The mAbs used here included fluorescein isothiocyanate- or phycoerythrin-conjugated reagents of anti-CD8 (53-6.7), anti-CD3(145-2C11), anti-macro- phage (Mac-1) mAbs, antiCD4 (RM4-5), anti-IL-2Rβ (TM-β1), and anti-NK1.1 (PK136) mAbs. A biotin-conjugated reagent of anti- granulocyte (Gr-1) mAb was also used. All mAbs were obtained from PharMingen Co. (San Diego, CA). A biotin-conjugated reagent was developed with phycoerythrin-conjugated streptavidin (Caltag Lab., San Francisco, CA). The fluorescencepositive cells were analyzed by a FACScan (Becton Dickinson Co., Mountain View, CA).

Figure 8. The number of cells yielded by the liver and the irrigation solution. The pressure of irrigation into the portal vein was determined by the height of solution from the liver. The mean and 1 SD were produced from 5 mice at each of the experiments.

Cell Yields Before and After Irrigation. Even after bleeding from the liver, some lymphocytes still remain in the sinusoidal lumen. They may attach to the sinusoidal walls. To eliminate them, liver irrigation of 30 mL physiological saline from the portal vein was performed at various pressures (Fig. 8). The pressure was determined by the height of saline

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above the liver. The number of cells yielded by both the irrigation solution and the liver was enumerated. Depending on the pressure for irrigation, the number of cells increased in the solution while decreasing in the liver. At the pressure of a height of 30 cm, the cell yields of both samples became almost the same and a higher pressure or a much greater volume of saline did not change this pattern. In other words, the sinusoidal lumen and the parenchymal space in the liver might contain similar numbers of lymphocytes.

A Comparison of the Phenotype of Cells Between the Liver and the Irrigation Solution.

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The data of cell yields suggested that cells isolated from the irrigation solution may contain abundant lymphocytes that were attached to the sinusoidal lumen, whereas those from the liver may contain lymphocytes from the parenchymal space. We then compared their phenotypes by immunofluorescence tests (Fig. 9). To determine the distribution of CD4＋and CD8＋cells, 2-color staining for CD4 and CD8 was conducted. In the left column, the data of B6 mice are represented. Those of B6-nu/nu mice are also represented in the right column. As shown by the data from B6 mice, the distribution patterns of CD4＋and CD8＋cells were not significantly changed.

Figure 9. Phenotypic characterization of lymphocytes in the liver and irrigation solution. In this experiment, various pressures of irrigation were applied. Two-color staining for CD4 and CD8 and that for CD3 and IL-2Rβ were conducted. The data shown are representative of 3 experiments. Numbers in the figure represent the percentages of fluorescence-positive cells in corresponding areas.

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Two-color staining for CD3 and IL-2Rβwas then conducted to identify natural killer (NK) cells (CD3ーIL-2Rβ＋), extrathymic T cells (CD3intIL-2Rβ＋), and thymus-derived T cells (CD3highIL-2Rβ—) (bottom row of Fig. 9). In control B6 mice without irrigation, the proportions of NK cells, extrathymic T cells, and thymus-derived T cells were 8.6%, 20.5%, and 23.8%, respectively. A difference of the phenotype of cells between the liver and solution was seen in CD3intIL-2Rβ＋extrathymic T cells. The proportion of extrathymic T cells was always higher in the liver than in the solution. Although the difference was small, the proportion of thymus-derived T cells tended to be high in the solution whereas that of NK cells tended to be high in the liver. The above-mentioned tendency was also seen in the data of nude mice (bottom row and the left column of Fig. 9). Namely, NK cells and extrathymic T cells were abundant in the liver, but very few were found in the irrigation solution. In the case of nude mice, CD3highIL-2Rβ—cells were completely absent. All these findings were confirmed by repeated experiments (n=5) and the data are represented in Fig. 10. Extrathymic T cells were abundant in the liver, whereas thymusderived T cells were abundant in the irrigation solution.

Figure 10. A comparison of the distribution of lymphocyte subsets between the liver and the irrigation solution. The experimental procedure was the same as that in Fig. 9. In the case of CD4/CD8 ratio and Extrathymic/Thymic T cells ratio, the control value is represented as 100%. The mean and 1 SD were produced by using 5 mice at each point of the experiments.

Abundance of NKT Cells in the Parenchymal Space of the Liver The majority of TCRint cells (or CD3int cells) express a NK marker, NK1.1, and they are therefore termed NKT cells (87-92). To identify NKT cells, 2-color staining for CD3 and

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NK1.1 was conducted (Fig. 11). Similar to the case of CD3int cells, NKT cells were abundant in the liver, but few were found in the irrigation solution. A similar tendency was also seen in NK cells. The data from athymic nude mice are also represented in the right column of Fig. 4. NK cells were abundant in the liver, but few were found in the solution. Basically, very few NKT cells are found in nude mice. A small population of NKT cells was observed in the liver but not in the solution. In the case of the liver, organ-specific macrophages (i.e., Kupffer cells) are present.17,18 Such Kupffer cells were identified as Mac-1�Gr-1� cells by 2-color staining for Mac-1 and Gr-1 (bottom row of Fig. 11). Although such Kupffer cells were present both in the liver and solution, they were not so abundant in the liver.

Figure 11. Further phenotypic characterization of lymphocytes in the liver and the irrigation solution. In this experiment, various pressures of irrigation were applied. Two-color staining for CD3 and NK1.1 and that for Mac-1 and Gr-1 were conducted. The data shown are representative of 3 experiments. Numbers in the figure represent the percentages of fluorescence-positive cells in corresponding areas.

Clinical Data in LDLT We confirmed the hypothesis of wash out theory in an experiment on perfused liver in mice (85). Intermediate TcR cells and NK1.1T cells tended to stay in the liver against perfused solution. Conversely, thymic T cells,compared with NKT cells, increased in the irrigated solution. In nude mice, these phenomena were more prominent. These dynamic immunological changes may influence the allo-immune reaction in the liver transplantation.

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Therefore, we investigated the changes in proportion of NKT cells and thymic T cells in the liver graft before and after perfusion by HTK solution in adult LDLT (93).

Changes of Thymus-Derived Cells in the Graft Liver by the Perfusion of HTK Solution in LDLT

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Thymus-derived cells of CD56-T cells in the graft before perfusion(37.6±10.3%) decreased immediately after perfusion of HTK solution(30.6±7.4%) and one hour after transplantation (29.1±8.9%) with statistical significance(Fig.12).

Figure 12. Changes of the proportion of thymus-derived cells in the graft liver by the perfusion of HTK solution in LDLT. Thymus-derived cells of CD56-T cells in the graft before perfusion (37.6±10.3%) decreased immediately after perfusion by HTK solution (30.6±7.4%) and one hour after transplantation (29.1±8.9%) with statistical significance.

Changes of NKT Cells Among CD3+T Cells in the Grafts Liver by the Perfusion of HTK Solusion in LDLT CD56+T cells and CD161+T cells among CD3+T cells in the graft liver tended to increase immediately after perfusion and decrease one hour after transplantation. Especially, CD56+T cells among CD3+T cells increased with statistical significance (36.9±9.1% vs 45.0±7.8%) (Fig.13). We demonstrated that the wall shear stress regulate and influence the liver regeneration and intra- and extra hepatic immune system after Phx in the both mice and human.

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Next attention is devoted to― on and off‖studies in liver regeneration after Phx. The existence of a receptor of remodeling for shear stress on endothelial cells has been proposed. A cell surface modulator (CSM) must exist on the hepatocytes and SEC as well as endothelial cells.

Figure 13. Changes of NKT cells among CD3+T cells in the graft liver by the perfusion of HTK solution in LDLT. CD56+T cells and CD161+T cells among CD3+T cells in the graft liver tended to increase immediately after perfusion and decrease one hour after transplantation. Especially, CD56+T cells among CD3+T cells increased with statistical significance (36.9±9.1% vs 45.0±7.8%).

5. Shear Stress and PAI-1 During Liver Regeneration Following Phx The prognosis for recovery after Phx, portal branch ligation, and liver transplantation depends on how well the liver regenerates after the operation. Thus understanding the mechanisms of liver regeneration is of fundamental importance to the prevention of postoperative liver failure and improving the outcome. The role of a variety of cytokines and growth factors has been extensively studied as triggers of liver regeneration (94, 95), but priming by hepatocytes themselves rather than cytokines or growth factors is now considered necessary for the initiation of liver regeneration, because liver regeneration fails to occur when a large dose of hepatocyte growth factor (HGF), a complete mitogen for hepatocytes, is administered (48, 96). The results of recent studies, including our own, have suggested that hemodynamic forces arising from changes in portal blood flow and pressure after partial hepatectomy or portal branch ligation may be involved in a priming mechanism for liver

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regeneration (4,97, 98); however, how hemodynamic forces affect liver cells has never been elucidated. Recent in vivo studies (99-102) have revealed that the expression of a number of genes, including immediate-early genes such as early growth response factor-1 (Egr-1), plasminogen activator inhibitor-1 (PAI-1), and phosphatase of regenerating liver-1 (PRL-1), is induced in hyperperfused lobes during the first few hours after partial hepatectomy or portal branch ligation. An in situ hybridization analysis of PAI-1 mRNA in the liver after partial hepatectomy showed that the majority of the positive cells were hepatocytes, although PAI-1 mRNA was also localized in venous endothelial cells, capsular mesothelial cells, and sinusoidal cells (100,102). Nevertheless, it remained unclear whether the induction of these genes in hepatocytes was attributable to shear stress. To investigate whether shear stress affects gene expression by hepatocytes, we exposed cultured hepatocytes to controlled levels of shear stress in a flow-loading apparatus and examined them for changes in gene expression. We focused on PAI-1 rather than other immediate-early genes, such as a transcription factor Egr-1 and growth-related tyrosine kinase PRL-1, because PAI-1 is involved in fibrinolysis and in a number of biological processes, including extracellular matrix degradation, cell migration, and angiogenesis. Flow-loading experiments. We applied controlled levels of shear stress to cultured cells by using the same parallel plate-type flow chamber as described previously (103). One side of the chamber was formed by the coverslip on which the hepatocytes were cultured, the base and walls were machined from a polymethacrylate plate, and the two flat surfaces of the coverslip and base were held ∼200 µm apart by a Silicone rubber gasket. The chamber had an entrance and an exit for the medium, and the entrance was connected to a reservoir with a Silicone tube. The medium was perfused through the chamber by a roller/tube pump (ATTO; Tokyo, Japan). The entire circuit was placed in an automated CO2 incubator, and the flowloading experiments were performed at 37°C in an atmosphere of 95% room air and 5% CO2. The intensity of shear stress (τ; in dyn/cm2) on the cell layer was calculated by the formula τ = 6µQ/a2b, where µ is the viscosity of the perfusate (in Poise), Q is flow volume (in ml/s), and a and b are cross-sectional dimensions of the flow path. Because the maximum Reynolds number corresponding to the highest flow rate used in this study was around 40, we assumed that the flow was laminar.

Flow-Induced Changes in Expression of the PAI-1 Gene in Hepatocytes Cultured hepatocytes were exposed for 3, 6, 12, 24, 48, and 72 h to flow exerting a shear stress of 10 dyn/cm2, and changes in PAI-1 mRNA levels were analyzed by real-time PCR. The PAI-1 mRNA level of primary cultures of rat hepatocytes began to increase 3 h after the start of exposure to flow, peaked at around threefold the static control level at 12 h, and then gradually decreased, remaining at around twofold the control level after exposure to flow for 72 h (Fig. 14A). A flow-induced increase in PAI-1 expression was also observed in cultured rat hepatocytes transformed with SV40 (RTH33; Fig. 14B), human hepatoma cells (C3A; Fig. 14C), and murine hepatocytes transformed with SV40 (TLR-2; Fig. 14D), although the temporal pattern of the PAI-1 response to flow varied from cell line to cell line.

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Figure 14. Effect of flow on plasminogen activator inhibitor (PAI)-1 mRNA levels in hepatocytes. A: primary cultures of rat hepatocytes. B: rat hepatocytes transformed with SV40 (RTH33). C: human hepatoma cells (C3A). D: murine hepatocytes transformed with SV40 (TLR-2). Total RNA was isolated from hepatocytes that had been exposed to flow (laminar shear stress: 10 dyn/cm2) for the periods indicated or maintained as a static control, indicated by time 0. Samples of mRNA were then analyzed using real-time PCR. Flow increased the PAI-1 mRNA level in primary cultures of rat hepatocytes, rat RT33 cells, human C3A cells, and murine TLR-2 cells. All values are means ±SD of 3 separate experiments. *P < 0.01 vs. static control.

Flow-Induced Changes in the Release of PAI-1 by Hepatocytes Rat hepatocytes (RTH33) were exposed to flow exerting a shear stress of 10 dyn/cm2 for 3, 6, 12, 24, 48, and 72 h, and the concentration of PAI-1 protein in the perfusion medium was determined by ELISA. The hepatocytes released very little PAI-1 under static conditions, but release of PAI-1 increased markedly in response to flow (Fig. 15). PAI-1 release began to increase at 3 h, peaked at �14-fold the control level at 12 h, then gradually decreased, remaining at �2-fold the control level at 72 h. The temporal patterns suggested that the increase in PAI-1 protein release was based on the increase in PAI-1 expression.

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Figure 15. Effect of flow on the release of PAI-1 protein by hepatocytes. Amounts of PAI-1 in perfusates of rat RTH33 cells cultured under static conditions or exposed to flow (shear stress: 10 dyn/cm2) for 3, 6, 12, 24, 48, or 72 h were measured by ELISA. F, Time course of the increases in PAI-1 release in response to flow; E, basal release under static conditions. Results are presented as means ±SD of 3 separate samples. *P < 0.01 vs. static control.

Shear Stress Dependency of Flow-Induced PAI-1 Expression To determine whether the flow-induced PAI-1 expression was dependent on shear stress or shear rate, rat hepatocytes (RTH33) were subjected to flow by two perfusates having different viscosities. PAI-1 mRNA levels increased as the shear rate increased, but at the same shear rate they increased even more when the viscosity or shear stress was higher, and the data yielded two separate curves (Fig. 16). Induction of PAI-1 expression has been shown to occur in hepatocytes in the regenerating liver a few hours after partial hepatectomy. PAI-1 induction has also been observed in vitro in hepatoma cells stimulated with glucocorticoids, transforming growth factor-β, epidermal growth factor, and interleukin-1, and thus systemic or local release of such chemical mediators after partial hepatectomy may account for the PAI-1 induction. Partial hepatectomy increases portal pressure in the remaining lobes, and the compression or stretching tension generated by the increased portal pressure may be involved in the induction of PAI-1. A recent study demonstrated that stretching tension stimulates endothelial cells to increase PAI1 production. The results of our study suggest that the rise in shear stress generated by the increase in portal blood flow is capable of increasing PAI-1 expression in hepatocytes. However, further study is needed to clarify the extent of the role played by chemical

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mediators and hemody- namic forces in the PAI-1 induction after partial hepatectomy and the role of PAI-1 in the initiation of liver regeneration.

Figure 16. Shear stress dependency of flow-inducedPAI-1expression. RatRTH33 cells were exposed to flow by two perfusates having different viscosities at various shear rates for 6 h, and changes in PAI-1 mRNA levels were determined by real-time PCR. The low-viscosity perfusate consisted of me- dium alone, and the high-viscosity perfusate consisted of medium plus 5% dextran (mol wt 162,000; Sigma). The low- and high-viscosity perfusates had viscosities of 0.0095 and 0.0378 Poise, respectively, specific gravities of 1.005 and 1.025, respectively, and osmolarities of 289 and 292 mosmol/l, respec- tively. The PAI-1 mRNA level increased with the shear rate, but at the same shear rate, the increase was always greater at the higher viscosity or at the higher shear stress.

The shear stress-dependent PAI-1 activation in hepatocytes occurred at the transcription level, not at the posttranscription level. Deletion analysis of the rat PAI-1 promoter revealed that Sp1 and Ets-1 binding sites function as the cis-element for shear stress responsiveness. EMSA and ChIP assays showed that Sp1 and Ets-1 are actually involved in the shear stressdependent activation of PAI-1 transcription. Cooperative inter- actions between Sp1 and Ets-1 have also been shown to play a critical role in regulating the transcription of genes, such as the genes encoding Fas ligand and the platelet-derived growth factor A-chain, in vascular smooth muscle cells. Nevertheless, how shear stress activates Sp1 and Ets-1, including whether shear stress increases their gene expression or activates them through posttranscriptional modifications, such as by glycosylation and phosphorylation, has not yet been determined. Recently Ando et al. (104) have demonstrated the interesting research about the CSM. The caveola is a membrane domain that compartmentalizers signal transduction at the cell surface. Normally in endothelial cells, groups of caveolae are found clustered along stress fibers or at the lateral margins in all regions of the cell. Subset of these clusters appear

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to contain the signaling machinery for initiating Ca2+ wave formation. He reported that induction of cell migration, either by wounding a cell monolayer or by exposing cells to laminar shear stress, causes caveolae to move to the trailing edge of the cell. The hepatocytes have amount of caveolae. Above findings suggest that caveolae might be a CSM for trigger of regeneration after Phx. Above findings, we investigated about the BCF systems in LDLT with paying attention to PAI-1.

6. Blood Coagulation and Fibrinolytic Systems During Liver Regeneration in LDLT To understand the BCF systems after liver transplantation is important for patient management. Especially in the early period after adult LDLT, there is liver regeneration accompanying portal hypertension because of a small-for-size graft. The BCF systems in healthy donors mediate the physiologic regeneration pattern in the early period after liver resection. Conversely, recipients are exposed to the influences of excessive surgical stress and portal hypertension. Therefore, we investigated differences in the perioperative BCF systems between donor and recipient after adult LDLT, primarily considering serum PAI-1 and soluble fibinogen (SF) levels.

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Patients and Methods Eight recipients and donors underwent LDLT between August 2003 and August 2006. We investigated changes in SF, thrombin- antithrombin complexes, protein C, fibrin degradation product, D-dimer, plasmin-alpha 2-plasmin inhibitor complex, platelet count, and PAI-1 on days 1, 3, 7, and 11 after adult LDLT aided by Mitsubishi Tanabe Pharma Corporation. The donors were three women and five men with mean age of 35.6 years (age range, 26-64 years).Recipients were 4 men and 4 women, with mean age of 53.6 years (age range, 14-61 years). Two patientshad fuluminant hepatitis, four had hepatitis C liver cirrhosis, one had Wilson disease, and one had primary biliary cirrhosis.

Results The donor coagulation system, analyzed by TAT, SF, and PT were suppressed from day 3 to 7 and recovered thereafter. Conversely, the donor fibrinolytic system of analyzed by FDP-P, D-dimer, and PIC was activated from day 3 to 7 after LDLT. Both BCF systems were unstable compared with those of the donor. Recipient TAT increased from day 1 to 3, and SF decreased over the same period, distinct from the donor findings. Recipient SF tended to recover after transplantation (Fig 17-23). The recipient fibrinolytic system was the same as the donor system except for PAI-1, which was remarkably increased on day 1 after transplantation in the recipients (Fig 17-23). We summarized BCF state following LDLT in the both donor and recipient.

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Figure 17. Changes in PAI-1 in the early period during liver regeneration after LDLT in the donor and recipient after LDLT. Serum PAI-1 was remarkably increased on day 1 after transplantation in the recipients.

Figure 18. Changes in SF in the early period during liver regeneration after LDLT in the donor and recipient.

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Figure 19. Changes in TAT in the early period during liver regeneration after LDLT in the donor and recipient.

Figure 20. Changes in Protein C in the early period during liver regeneration after LDLT in the donor and recipient.

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Figure 21. Changes in AT-III in the early period during liver regeneration after LDLT in the donor and recipient.

Figure 22. Changes in D-dimer and FDP-P in the early period during liver regeneration after LDLT in the donor and recipient.

BCF state of donor inclined to the coagulation on the day1 to day3 and after that it leaned to the fibrinolysis by the day7, and recovered to normal state 7 days after LDLT. Preoperative BCF state of recipients were DIC. Both coagulation and fibrinolysis were activated by the day3 and after that DIC state recovered immediately by the day7 after LDLT. In adult LDLT, there are several problems such as small- for-size syndrome, preoperative DIC state, malnutrition, and immunodeficiency. This situation may benefit from a marker of surgical stress in the early period after partial liver transplantation. Furthermore, recipients after adult LDLT show liver regeneration. In that sense, under- standing of the BCF system

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between donor and recipient may be important for management after adult LDLT. In consideration of both donor and recipient data, the recipient is believed to have DIC in the early period after adult LDLT, and SF may be a useful marker for improvement in the BCF system. We have previously reported that PAI-1 has an important role in liver regeneration influenced by wall shear stress. Therefore, the elevation of recipient PAI-1 on day 1 after transplantation may be a marker of injury from shear stress associated with excessive portal hypertension after adult LDLT.

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Figure 23. Changes in PIC and α2-PI in the early period during liver regeneration after LDLT in the donor and recipient.

Table 1. Summary of blood coagulation and fibrinolytic changes in the early period during liver regeneration after LDLT in the donor and recipient

Thus, wall shear stress play an important role in BCF system in liver surgery. Recently Ali et al.(105) reported that induction of the cytoprotective enzyme heme Oxygenase-1 by statins is enhanced in vascular endothelium exposed to laminar shear stress and impaired by

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disturbed flow. Next we described the relationship between HO-1 and bilirubin metabolism in LDLT.

7. Heme Oxygenase-1 and Bilirubin Metabolism in Clinical LDLT A small-for-size graft is the greatest problem in adult LDLT. We have reported that excessive portal hypertension, reflecting wall shear stress, after LDLT produces posttransplant liver dysfunction including hyperbilirubinemia (106). Reduction of excessive portal hypertension prevents liver injury or hyperbilirubinemia after LDLT. The hemedegrading enzyme HO-1 in Kupffer cells plays a key role in bilirubin metabolism. HO, a heme-oxidizing enzyme, generates biliverdin and CO. Moreover, CO generated by HO controls bile canaliculus function by changing intracellular calcium concentrations presumably through a mechanism involving cytochrome P450 reactions. Concentrations of CO in patients may be detected as CO-hemoglobin (COHb). Hyperbilirubinemia occurs following LDLT with small-for-size grafts. In this study, therefore, we investigated the relation between COHb and hyperbilirubinemia in small-for-size LDLT graft.

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Materials and Methods Adult patients (n =16, 17 to 68 years old) who underwent living-related donor liver transplantation between March 1999 and May 2001, were divided into two groups: Group 1 includes ratios of graft volume/recipient body weight (GV/RBW) ≧1.0, and Group 2:GV/RBW＜1.0. We analysed the relation between the GV/RBW ratio and the posttransplant serum total bilirubin (T.Bil), direct bilirubin/total bilirubin (D/T) ratios, COHb, and arterial ketone body ratio (AKBR). The correlation between COHb and total bilirubin was also estimated.

RESULTS Changes in Serum Total Bilirubin Following Adult LDLT Serum T.Bil levels peaked at 1 to 3 weeks and decreased thereafter. Three patients showed total bilirubin levels that exceeded 25 mg/dL which required treatment by plasma pheresis. Hyperbilirubinemia after adult LDLT was higher and more sustained among group 2 than group 1 patients (P