'--.'."._,'-',.': :r - '2, ‘ -‘c."”\':."obW 9i" 7' '."" 'f"p‘ kl ‘.1 i H“ - . "1' ', i... i ‘ "x" “f\ | ‘ .'- , . 1.". .' u". am" ., ‘cfl , v ‘ ’,.01y'-I'- . I z .37.» .' .' A ; by.“ ;: ' 'I'I‘Ill "'Jl,‘ |¢ ‘1 x”; If} v i o "5"“ ‘5’ ),: bylf‘tr. ‘II '| I ‘ ”I 1“ '. “ I! "I '1‘II 1"". n. ' . ”:1"? . = flm $315.3 ‘;.b‘.l.n,.v I .f‘ch’ ‘3'. «1 “g a: -0 .—~. gem: .fiwsrgu - ‘ rEJé-u my: wqugfi }" *u "‘ . C I u ‘u- . ‘51" {1) .1 “q. . . :%W}‘." 3'3: .u..‘.2"‘. ‘» . 'rf‘g‘qtuuu‘.“ “n . 52““: ~21 "'*-“:.:....~ .. mum.” Cat/h :95"! "3*" “.31.: ~ ex“- : ' e‘z'z‘VJfi-t-‘Evh‘v - ~27!» v"? 1 . ‘ ",3 "‘- "A!“ '° "‘:'~,"':-:""‘ ‘ ' ., .‘ .‘_ «9.3; . ‘-. 3v ‘ .\ , I :1 "1 v a? ' ‘ 1 ‘1‘. . “L. ".31 V la} _ - 33g; 3 3:. \ an - v‘ wt». 7. Vi ., h 0v- ‘UI _“ :_- I. 4' "2:"? ' a 9,.” . Wm“ 134”" . 1J3? an}: 43.1»... "3:31,.“ .‘F'w In u '33.}; ‘4’ ‘T A '2 75 » 73-7.; ' ' f ‘ ‘4' l ‘ 3 ‘1 _,_ '.I J.‘ ~ 5' "'16... _-_ ' 1.-. . . , -- /_"'.J.‘.'.‘_1 - wrap u u _- V. ~_ m" . , L . _ n , , _ - . c. .....,.‘ 7 " ' I ~ ’ " ‘ ' ' 1‘ '—”'L" ‘3. . y ’3'...“ p.” #k“ "37"" y. 3. -,-"24§1‘i%§g7;.}¢57. 13%.. ”My. Wye A . ' -n I E .1. 6 I a ’r ~ ' ".'. ~'.‘ .- - ' : lj‘E'-LT.2.§K ‘ $6111" ‘74.,m..,f,?r “$211533" ‘ ‘ .4 "" V,» 0; -'.'l. (7”,... '. 5&1"? . - ' . J1”..- fl," ' v ,5 .-.:Qt mf‘ y AW "1.37“ «re-*«cbur ' -~'— vat" ~5- 41y} -:.'- ,- +1"- . .. . - o “Kr. 30“ '1 ‘5'9 'flv “‘2‘; {.2317}. .q. . |'-"D 1d". - .3-1 .- V ‘I‘ w z-n... q r“. can"? v.- u L: I'.1\"E’:I P. ‘ . .. ‘ 3... -3 - 33:": . - r, . :31! . L.’ _. . ...\.’1 'I.‘ , '.."'.'.I‘l Mn“ 1, A " .v ‘; it» :- n." ‘fé‘h " - I! ‘ I ’1“ . .. L.111- Z-l-"‘.\'~‘.‘... l m ‘ ”II—d ‘i '11‘ an" H. ‘ . '3 "PM ' 15mm" "h; [bl .1“ ,‘4 ___.. | ‘- .n‘sfluv [a . ,‘ ‘ L ‘ ;« ~~ . I 3“ k '3‘: {7, 0;"; L GM“? ‘31:.) H; o I q» C _q V, i n "' ‘ 1' V .\ u , - J ' "hi “5""? ~ .- NH. .~ ‘(-'.I"" 'n L. 3‘11} 'I. ‘4 .1 «; .0 III - n». ..,' .M“ . Q ”_o a... 'a ' - -‘ u I > " ’ih T"~_" _ G" ' .1 ’2‘ " .’ .‘ .I ~ , ’5‘" “ ‘%’ I. , - ' ~' ' . inn: '-' {32. “’r fifitt'hlffiu‘l {thfii‘Tftlfilue I Eofiifi 1“" ‘3 flvTfll” N-'u:|‘£+h'g‘ I IS” .séiiiiai-‘v‘rfiéi «film: it mi-v'ri‘iiim , :32“ mi "9...,“ In. - ‘ ?-?_-9 1"! J . A- ._... - ' . L- .-._ .A- ‘9‘ l ...,-..__._-- -—:i ‘1 1»_-r\.~ 1: m.) g Q P .4 I v“ - A.:-‘.~' 7“ A .A.'! ' -:.' J- if a; g a ., ‘.=‘ ' This is to certify that the dissertation entitled The effiective diffusion coefficient of a two-phase material presented by S. Sridharan has been accepted towards fulfillment of the requirements for Ph. D. degreeinm flifda Major professor 9/2434, MS U i: an Affirmative Action/Equal Opportunity Institution 0-12771 W l l // WNW /// ”WWI L/ 3 1293 00812 3070 )V1ESI.J RETURNING MATERIALS: Place in book drop to LJBRARJES remove this checkout from “ your record. FINES will be charged if book is returned after the date " ; stamped below. 11672 t m 2000 THE EFFECTIVE DIFFUSION COEFFICIENT OF A TWO-PHASE MATERIAL By S Sridharan A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Chemistry 1986 ‘thjk >5.le ABSTRACT THE EFFECTIVE DIFFUSION COEFFICIENT OF A TWO-PHASE MATERIAL By S. Sridharan This study concerns itself with calculating the effective properties of two-phase media. The theory, as developed here, applies to diffusion of a permeant species in a two-phase medium. Since other processes such as thermal and electrical conduction, dielectric phenomena and magnetic permeation are all governed by the same differential equation (the Laplace equation), the results obtained here are meaningful in those contexts as well. The system itself could be a random aggregate of the grains of the two phases which appears quite homogeneous on a macroscopic scale or it could be a particulate system where grains of one of the phases are embedded in the other phase which is continuous. The main thrust will be towards the latter case. Using a local field analysis, a formal expression is obtained for the Clausius-Mossotti(C.M.) function of the effective property in terms of formal operators that are pertinent to transport in a reference medium. Its lowest order result is then analysed for different choices of the reference medium. A multiple scattering analysis is developed for the separated grain problem (with the reference medium identical to the continuous phase) to obtain a density expansion for the C.M. function. Terms through second-order in inclusion number density are written out in terms of one-sphere scattering operators. The second order term is subsequently evaluated using a multipole expansion for the single-sphere scattering operator. When the reference medium is the same as the effective medium that describes the entire system, the resulting theory is expected to handle higher volume fractions of the inclusion-phase than a bare medium theory could handle. For this case a model system is chosen in which fictive composite spheres containing the real spheres (the inclusion phase) and the surrounding medium are embedded in the effective medium. For this case the pair-order term is evaluated only approximately. The results are compared with other effective medium approximations. It is demonstrated that the procedure of deve10ping density expansions for the C.M. function avoids divergence problems that are usually associated with virial expansions. The C.M. formula is obtained as the lowest order result of the theory. The pair—order correction to the C.M. is evaluated exactly and compared with previous findings. The approximate effective medium theory seems to be a substan- tial improvement over the bare medium theory. It compares well with Bruggeman’s formula that applies to aggregate grain systems. Finally the study shows the difficulties of developing analytical theories for systems where the constituent phases are present in almost equal proportions. TO MY PARENTS ii ACKNOWLEDGMENTS I am very grateful to Prof. Robert I Cukier for his guidance and constant encouragement throughout the course of this work. He showed a lot of patience in pushing me through my inertia during dull stages. I thank Prof. Katherine Hunt for reading the thesis in great detail, and pointing out the mistakes and typo- graphical errors. I am grateful to the Dow chemical company and the Monsanto chemical company for giving me fellowships for periods of nine and three months. Finally I thank Prof. Robert I Cukier for his timely purchase of a word-processor which I used in typing my thesis. Though it played games with me in the begining, it proved to be very co-operative towards the end. iii LIST OF LIST OF CHAPTER CHAPTER CHAPTER CHAPTER 4.4 CHAPTER CHAPTER TABLE OF CONTENTS TABLES FIGURES I INTRODUCTION II HISTORICAL BACKGROUND III DIFFUSION EQUATION AND EFFECTIVE DIFFUSION COEFFICIENTS IV SPHERE SUSPENSIONS Pair order theory Evaluation of Pair order theory: Dm = D2 case Coated metal spheres Random coil polymers V PAIR ORDER EFFECTIVE MEDIUM THEORY VI DISCUSSION APPENDIX A Multipole expansion of ;(r,r’) APPENDIX B Evaluation of gj’ j 2 3 APPENDIX C An expression for yn(1/R) REFERENCES iv vi 10 20 27 27 30 38 41 43 54 58 6O 66 68 LIST OF TABLES Table 1 List of some typical material properties 2 Table 2 Second order contribution to the virial expansion of the Clausius-Mossotti function for different models 37 Table 3 Reduced polarizability a and second order contribution to De for polymer coils 44 Figure Figure Figure Figure Figure Figure LIST OF FIGURES Plot of upper and lower bounds on 6e for (a) ellez = 5 and (b) allez = 50 Plot of ee(Bruggeman)/e2 vs volume fraction m for el/e2 = 60. The dotted lines are the upper and lower bounds on Ce Second order contribution to eq. (4.25) for 0 ; 2.1(5) = o ‘ (1.1) where D(r) is a spatially dependent diffusion coefficient. There is a host of natural phenomena that are governed by eq.(1.1). Some of the typical examples are given in table 1. The results of a study of any one of these properties can be used to describe the others with proper change of notation. The model (1.1) can be applied, for example, to diffusion in a two phase system containing small scale inhomogeneities. A macroscopic sample of such a material (one large compared with the scale of the heterogeneity) is characterised by a relation between the averaged flux J = (Q) and the averaged concentration 2N = <2n> of the form J(r) = -DegN(r). This relation defines the effective Table 1. List of some typical material properties. w E J D Electro- Electric Electric Electrical Dielectric statics potential field induction] constant intensity Displace- ment Magneto- Magnetic Magnetic Magnetic Magnetic statics potential field induction permeability intensity Electrical Electric Electric Current Coefficient conduction potential field density of electri- intensity cal conduc- tivity Thermal Temperature Temperature Heat flux Coefficient conduction gradient of thermal conductivity Diffusion Concen- Concen— Concen- Diffusion tration tration tration constant gradient flux 3 diffusion coefficient and is the objective of the calculation. There are two main problems to be addressed in a cal- culation of De' First, the composite material’s structure must be characterised. This structure is usually prescribed statistically. Second, the averaging procedure must be carried out over the statistical distribution. There are two disparate, principal composite structureslz an aggregate grain structure and a separated grain structure. The former is appropriate to taking grains of differing material property and compacting them to form the composite. The latter is formed by taking a matrix material and embedding inclusions in it. If the matrix material is a liquid the inclusions are assumed to be held at fixed positions. It is doubtful if the model defined by eq. (1.1) can be used for systems in which D(r) is allowed to vary on an atomic scale. For example, there does not seem to be much point in representing atoms in a molecule as spheres of fixed radius, each described by some D (See ref. 2). A 1. larger scale is contemplated; presumably a scale on the order of thousands of atoms would suffice. Several systems come under this length scale. In the context of diffusion macromolecular solutions serve as good examples. Several years ago Debye and Bueche3 studied the 4 dynamics of a single polymer coil and showed that it moved like a rigid sphere, in the sense that the solvent velocity inside the coil was the same as the coil velocity. Hence one can treat a dilute polymer solution as a system consisting of hard spheres floating in a solvent with the diffusivity inside the coil ranging from almost zero at its core to that of the solvent at its periphery. In the context of thermal conductivity, systems such as polymers filled with particulate solids and foams6 are good examples. The condition of fixed geometry for the inclusions requires some comments. It is certainly true that in systems such as ion exchange resins which are good candidates of the above model, the inclusions (in this case, the resins) are held at fixed positions. However it is not true in macro- molecular solutions. In a polymer coil solution, for example, the coils do move, though at a very slow rate. This system can be studied experimentally by adding a third permeant species, much smaller in size compared with the macromolecule, and using light scattering techniques to study the profile of its concentration fluctuation. If the species is small enough then the scale of its concentration fluctuation would be such that the macromolecular motion doesn’t affect it. It is in this sense that the inclusion geometry can be assumed to be fixed. In this work we present a multiple scattering formalism for these composite material effective properties. 5 The main thrust will be toward the separated grain systems, though the general formulations can be used, in principle, to discuss the aggregate grain materials. For simplicity, only aggregate grain materials for which D(g) is two-valued, D1 and D2, will be considered. In the aggregate grain structure D1 and D2 appear in a symmetric fashion. In the separated grain structure we consider the matrix to be characterised by D2 and the inclusions by D In the latter 1. case we also consider examples where D1 is multivalued inside an inclusion. I The multiple scattering approach is constructed from a propagator gm = gggn where gm(r) = (t‘lirDurfn1 is the diffusion Green’s function in a medium of diffusivity Dm' This propagator is "scattered" by the inclusions. By summing all possible scatterings among the inclusions, the fluxes and fields can be determined, averaged, and compared to yield De' The choice Dm = D2 is appropriate for density expansions since this corresponds to propagation in the "bare" medium. However, as our objective is to formulate a theory for high inclusion concentration, where density expansions are not useful, we use a medium characterized by a reference diffusion coefficient Dm' If DIn is chosen to be De’ the true effective diffusivity, a self-consistency condition is obtained which is solved by an iterative procedure. 6 The propagator gm consists of two parts (cf. Eq. (3.9)): a (long range) dipole propagator gm and a self part which is proportional to a spatial delta function. We will find that it is very useful to extract the self part and write the multiple scattering expansion in terms of the dipole propagator gm. The resulting formal expression has the appearance of what we shall refer to as a generalized Clausius-Mossotti (C.M.) formula in that the quantity (De-Dm)/(De+2Dm) appears naturally (cf. eqs. (3.22b) and (4.4)). Several known results are readily obtained from these expressions. For aggregate grain structures, the neglect of correlations between different spatial regions leads to the C.M. resultf when Dm = D2 and to the Bruggeman formulaf when DIn = De' Bedeaux7 obtained similar no-correlation results for the analogous problem of the effective shear viscosity coefficient in a two-phase flow system. In our treatment of the separated grain systems, it is natural to express the scattering from the inclusions in terms of a scattering operator, 't’. An inclusion geometry must be specified and we will use spherical inclusions. The no-correlation case corresponds to the dilute inclusion limit and, when Dm = D2, the C.M. expression is obtained again. 1 See chapter II 7 To explore the correlation effects several approaches to the generalized C.M. relation are proposed. The rigorous procedure is to construct a density expansion by setting Dm = D2 and collecting terms in the 't’ operator expansion in powers of inclusion density. We obtain the second order in density term in this fashion. It is given by all possible scatterings of the dipole propagator in the bare medium between a pair of inclusions characterized by the one sphere 't’ operator. The 't’ operator can be expressed as a multipole expansion, and this expansion is readily obtained by comparison with the solution of the corresponding classical electrostatic problem. In this fashion, we obtain De/DZ to second order in inclusion density and reproduce the results obtained by Jeffrey8 and by Felderhof, Ford and Cohenlo. The multipole moments of the one-sphere 't’ operator can be evaluated for any inclusion model for which one knows, or can work out, the corresponding classical electrostatic problem. We discuss two examples: First, a spherical inclusion which is itself composite, with an outer thin layer of material of poor conductivity with respect to the interior part as well as the matrix. This models the effect on, for example, the effective thermal conductivity of metallic spheres coated with a thin oxide coat in a low conductivity matrix such as a polymer. Second, we discuss inclusions which consist of many concentric layers with a series of diffusion coefficients gradually falling in value 8 from the solution value at the periphery to a much lower value as the center of the inclusion is approached. This models the change in diffusion coefficient of a molecular permeant within a random coil polymer solution. For the choice Dm = De’ the effective diffusion co- efficient itself, we will see that the scattering operator 't’ is non-zero everywhere in the system and it is therefore difficult to write it as a sum over sphere centers. Hence a model is chosen in which each sphere is coated with a shell of the medium to form a composite sphere within which 't’ is non-zero and outside of which the medium is characterised by De' This allows us to write the new 't' once again as a sum over sphere centers. The lowest order term in the multiple scattering series involving the new 't’, which we call 'te’, gives once again the CM result, when we adjust the ratio of the sphere volume to the volume of the composite sphere to be equal to the volume fraction m of the spheres. In order to evaluate correlation contributions to this lowest order result we have to consider overlapping configurations of the composite spheres in the effective medium. We will see some of the difficulties involved in this task. The plan of the rest of the thesis is as follows: In chapter II earlier works are reviewed under two headings. The first section is on the Clausius-Mossotti relation and constructing a virial series for the effective property. This is done in the context of the dielectric case. The 9 second section contains a review of effective medium theories to calculate such quantities. In chapter III the multiple scattering formalism is carried out in terms of the dipole, reference medium propagator to obtain the generalized Clausius-Mossotti expression for De' Also it is shown how to obtain the no correlation results of Clausius- Mossotti and Bruggeman. In the first section of chapter IV the theory is specialized to sphere suspensions and a formal expression is obtained for the second virial coefficient (the pair order theory). This is then evaluated in the next section to show that the results agree with those of Jeffrey8 and Felderhof, Ford and Cohenlo. In the third and final section of chapter IV we discuss the coated sphere and polymer diffusion problems that were outlined above. In chapter V the effective medium pair order theory is outlined. It is then evaluated approximately and the results compared with the results of other effective medium theories. Chapter VI contains a brief discussion on how to improve the above results both by analytical and numerical methods. II. HISTORICAL BACKGROUND From a historical point of view electrical conduction and dielectric phenomenon take precedence to diffusion as the number of papers that have been published in those areas since the middle of last century illustrate. Most of these works have been reviewed in an excellent article by Landauerl. We summarize below the works pertinent to our discussion. 1. The Clausius-Mossotti expression. In the language of dielectrics, we talk about electrical displacement D and electric field E in the place of flux g and concentration gradient Zn. The equivalent of D(g) is the dielectric constant 6(5). Defining an effective dielectric constant 6e for a composite dielectric, charac- terized by dielectric constants c and £2, we have 1 (D) = e (E). A: e~ Consider a composite which is formed by embedding grains of substance 1 individually in substance 2. For this system an excess polarization with phase 2 as the reference medium can be defined by B : [(e(r)-ez)/4n]g. If an 10 11 electric field is applied to the system, the field acting on a single grain is equal to the applied field plus the field due to the polarizations induced in the other grains. Provided the grains are far apart, the medium with the rest of the grains can be replaced by a uniformly polarized medium surrounding the grain under consideration. The above considerations then lead to an expression for Ge given by8 £8- £2 £1- £2 —— = —- O (2.1) (8+ 262 (1+ 2e2 where o is the fraction of volume occupied by 1. If 62 = 1 (denoting vacuum) then the above formula could be used for the dielectric constant of substances such as gases. It was essentially derived by Mossottilzin the year 1850 and later by Clausius13 in 1879. Maxwell in his treatise on electricity and magnetism14 gives an elegant derivation of equation (2.1) for the electrical conduction case. A number of other authors also have derived this relation in different contexts. Henceforth we will refer to eq.(2.1) as the Clausius-Mossotti (CM) relation. It is a good approximation to 6e for low volume fractions and yields the correct linear in m term in the density expansion of 66. It is derived by summing contributions from individual dipoles while neglecting the interactions among them. The presence of other grains is felt only in the replacement of the rest of the system by a uniformly polarized medium. 12 The derivation of eq.(2.1) was based on the assumption that 2 was the continuous medium. If we reversed the role of phases 1 and 2 we would get a different result. According to . Hashin and Shtrikmanl5 the two CM expressions thus obtained are the maximal and minimal bounds on 6e for a macros- copically homogeneous and isotropic medium. They used a variational technique in the context of magnetic permeability to show that these are the best possible bounds when no other statistical information is given about the system. The two bounds are plotted in Figure 1 for two values of 61/62. As can be seen the bounds get farther and farther away from each other as the ratio allez increases. In the limit 61/62 = ., the lower bound remains essentially close to zero all the way to m ~ 1 when it suddenly jumps to infinity. The upper bound, on the other hand, is linear in c connecting £1 and 62. There is thus a large uncertainty in the effective dielectric constant in the middle region. Hence these expressions are useless for cases when there is a large disparity between the two dielectric constants. Eq.(2.1), derived for material 1 embedded in material 2, nevertheless gives the correct answer (68 = 61) even if m = 1. The derivative, however, dee/d¢ at m = 1 is quite wrong unless material 2, when present in small amounts, still remains as a skin surrounding material 1. At 0 close to 1 it is better to use the other CM expression obtained by “I “I \ mm 013 0'6 . qb . b GU HI \ mm l l 0.3 0.6 <75 Figure 1. Plot of upper and lower bounds on 5e for (a) ellez = 5 and (b) allez = 50. For both cases eq.(2.1) denotes the lower bound. 14 reversing the roles of 1 and 2. One should then look for a theory that would interpolate between these regimes. Correction to the CM expression can be expressed in the form of a virial series in m for the Clausius-Mossotti function (the left hand side of eq.(2.1)): ee‘ ‘2 2 W2 = a¢+B¢ + (2'2) e 6-6 with a 3 1 2 e + 252 Levine and McQuarrie16 evaluated 3 for a gaseous substance treating the molecules as metallic spheres. In the present notation this would correspond to setting 62 = a = 1. They used a bispherical coordinate system to calculate the pair polarizability a12(r) of a pair of non-overlapping metallic spheres, which they then used in the calculation of B. Jeffreya, using a method due to Batchelor17, obtained the exact second virial coefficient in the virial expansion for 6e itself (in the context of thermal conduction). He solved the two-sphere Laplace problem by a twin spherical harmonics expansionls. Felderhof, Ford and Cohen9 used a multiple scattering expansion technique to obtain formal expressions for the virial coefficients, the coefficients in the density expansion of Ee' They also used a twin spherical harmonics expansion to evaluate the second virial 15 coefficientlo. Unlike the CM function, evaluation of the second virial coefficient for €e involves handling of divergent integrals which arise mainly because of the long range nature of the dipole propagator. (See chapter III.) This issue has arisen several times in the literature and people have used different procedures to get around it. 17 to While Jeffrey uses an argument due to Batchelor circumvent the problem, Felderhof et a1 use a cluster expansion technique and get absolutely convergent integrals. As we will see later this issue never arises when the expansion is carried out around the CM result. Fixman19 recently obtained approximate values for the second and third virial coefficients by using a variational technique. 2. Effective medium theories. Effective medium theories to calculate such macros- copic properties have been around for a long time. The most successful and widely used of these appeared in a 1935 paper by Bruggemanzo. The idea was based on the simple model of embedding the grains of the multicomponent medium in a homogeneous medium whose property is the unknown quantity sought after. It is then determined in a self consistent manner 0 Consider once again the system described in section 1. In Bruggeman’s model the grains of 1 and 2 are supposed to be embedded in a uniform medium of dielectric constant 68. 16 Considering a single grain (say 1)in the effective medium, a uniform applied field E0 causes a polarization in that grain due to the difference between 61 and ee. If it is assumed to be spherical then elementary electrostatic considerations lead to an expression for its induced dipole moment p: ~ (2.3) c'd I m m + co m z 0 where 'a’ is the radius of the spherical grain. If volume fraction m is occupied by such spheres then their polarization is 131 : -—1——£'¢E (2.4) 1 e Likewise one can write an expression for 22, the polariza- tion induced in~2 whose volume fraction is (1-¢). The effective medium condition is then equivalent to stating that the net polarization should vanish (see discussion on p. 17), 1080, P + P = 0 (2.5) By this relation we obtain for the above system c - e e - e 1 e 2 e e + 26 + e + 26 (1-¢) : 0 (2.6) 1 e 2 e 17 This approximation, henceforth referred to as the 'Bruggeman result’, treats the components on a symmetrical basis and can be easily generalized to any number of components. It is quadratic in Se and can be easily solved for 6e in terms of £1, £2 and m. Figure 2 shows a plot of ee(Bruggeman) along with the bounds for a two component mixture. As can be seen it agrees with the most obvious CM approximation at each end of a two component mixture range, i.e., with the C.M. approximation in which the predominant material is continuous. Eq.(2.6) is the single most used result in dealing with the problem of percolation as discussed in Landauer’s articlel. Unlike the CM approximation, the above result (which is also referred to as the Coherent Potential Approximation in solid state physics21’22) includes interactions among grains in an average manner. The mathematical details of how eq.(2.6) incorporates some of the interactions are fully documented elsewhere23’24. Physically, one considers a single grain 1 in a uniformly polarized medium that represents the presence of other grains. This polarized medium polarizes 1, but the polarization of 1 changes the polarization of the medium, the change in this polarization changes the polarization of 1, etc. until the polarization of 1 and the uniform medium representing the other grains are consistent on the average. The main problem with the Bruggeman’s theory is that it is difficult to extend it in a straightforward way to include correlations among inclusions 18 69/62 Figure 2. Plot of ee(Bruggeman)/e2 vs volume fraction O for 61/62: 60. The dotted lines are the upper and lower bounds on 6e 19 in a separated grain problem. It seems best suited to address the aggregate grain problem. Hashinzs, using an idea due to Kernerzs, considered the model of coating each of the inclusions with the surrounding medium and embedding the resulting composite spheres in an effective medium. When the ratio of the inner volume to the total volume of the composite sphere was taken as equal to o, he obtained the CM result. This model has an advantage that the correlation contributions can be calculated by considering a pair of such composite spheres in the effective medium. In chapter V we will derive Hashin's result as the lowest order approximation of the effective medium series and also consider correlation contributions to this result. III. DIFFUSION EQUATION AND EFFECTIVE DIFFUSION COEFFICIENTS The equations of this chapter apply to a random two phase system with both phases being treated on an equal footing. However when we calculate the correlation contribution to the effective diffusion constant we will consider only a particulate system, with one of the phases in the form of spherical inclusions. The steady state diffusion equation for diffusion of a third species is given by Y-j(£) + w(§) : 0 . (3.1) Here w(r) is a boundary term which produces the source field for diffusion. The flux 1(5) is given by 1(5) = -D(g)gn(r>. (3.2) with D(g) = Di in phase 1 (i=1,2). The flux, as defined here, is relative to solvent velocity, but is the same as the flux in laboratory-fixed frame when the diffusant concentration is small. Averaging eq. (3.2) over regions large compared to the typical size of the heterogeneity in the material yields {(5) : -. (3.3) 20 21 Note that the above averaging procedure hasn’t eliminated the r dependence of the averages meaning that the regions over which the average has been carried out are still small compared to macroscopic spatial gradients. We now define an effective diffusion constant De by {(5) E -DegN(r) = -De . (3.4) Calculation of De thus involves evaluating j(r) and gn(g), taking their averages as defined above and finally finding the ratio of g to EN. Next we write D(g)=Dm+5D(g) where Dm is a reference diffusion constant. In the problem of spherical particles in a continuous matrix it is usual to choose Dm as the diffusion constant of the continuous medium; leaving it as Dm defines a convenient parameter which we can choose at will later. In terms of this reference diffusion constant the diffusion equation can be written as Dmvzn(§) = ~2-8D(g)yn(g) + w(g). (3.5) Thus the diffusion is seen above as occurring in a uniform background medium with diffusion constant Dm' The uniform flux in the background medium is due to processes occurring in phases 1 and 2. This equation can be solved formally for the gradient of the density field: 22 2n2 2'80<2'>-2'n<2'> + Ynm(£> «3.6) with nmigl a -Idg'zm(g-g'>w + Ids’d£”§m<£-£’>-9<£’:£'>2'nL(£'> (3-13> where 3Dm8D(g) 0(g.g’) = D(r) + 2D 5(5-5’) 3 0(g)8(g-g’)o (3.14) " I Solving eq. (3.13) formally yields - -1 - gnL - (fl gun) gum (3.15) where an operator notation is used to represent the r-space integrations. Henceforth we will stick to this convention. Note that in r-space gm is translationally invariant and g is diagonal. The expression for the flux, eq. (3.2), can be rewritten using eqs. (3.12) and (3.15) as 1 6j(r) E -8D(r)gn(r) = -fl(i-§mfl) gnm. (3.16) Averaging eq. (3.16) we get _ -1 5; - -<(1-§mn) > gN , (3.13) 6‘1 : -<9(1-!__!mfl) L with gNL i (gnL>. Averaging eq. (3.12) yields gNL = 2N - (1/3Dm)8g, (3.19) and, by combining eqs. (3.18) and (3.19), we obtain fig = -§-gN + (1/3Dm)§~$g . (3.20) where 1 1 -1 ;(;,g') s <(n-gmn)‘ > (r,r’). (3.21) The kernel 5(g,r’)=§(r-r’) in a macroscopically homo- geneous material. Since Eq. (3.4) defines SDe as a local relation, it is clear that J§(g)dr relates g to EN; i.e. to obtain 8De we expand §(r-r’) in spatial gradients, the lowest-order term being §(r-g’) = [Idr§(r)]6(g-r’). Using this expansion and eqs. (3.4) and (3.20) we obtain Ids 25(2) {fl-ai—ldzé‘v} m fl (De-Dm) = (3.22a) 25 or (D - Du) 3D e - m(De+ 2D“) 1 - Jdr 5(2). (3.22b) This gives De as a function of D1, D2 and the unknown parameter Dm' The simplest results are obtained by neglecting correlations between different volume elements. This means 1 an average like <fl(i-§m0)- > is split up into products of averages, <fl)<(1-§mQ)-1>. Then eq. (3.21) becomes §(g-g’) = 1(£-g’) . (3.23) and, with eq. (3.14), we obtain the lowest order approxi- mation of eq.(3.22): 3Dm(De- Dm)/(De+ 2Dm) = <0). (3.24) Utilizing the definition of 9 (see eq. (3.14)) one can perform the average. (<...> denotes either a configuration or a volume average). We find De- Dm D1- Dm D2- Dm D + 20 = D + 2D °1 + D + 2D 32 (3'25) e m 1 m 2 m where m1 and o2 are respectively, volume fractions of phases 1 and 2 with ¢1+¢2 = 1. 26 For a dilute suspension of phase 1 in 2, the choice szDz , the diffusion constant in the supporting medium, is appropriate. This choice yields = _ d, (3.26) where o is the volume fraction of the dispersed phase. This is the C.M. formula in the context of diffusion. Felderhof et a111 derived the C.M. formula by summing an infinite number of a certain class of diagrams in their cluster expansion9 for the effective dielectric constant. If the volume fraction of phase 1 is comparable to that of phase 2 then it is appropriate to choose Dm=De in which case eq. (3.25) reduces to - D 1 ¢ + 2 e ¢ D + 2D 1 D + 2D 2 1 e 2 e D - De D : O. (3.27) This simple result, often referred to as Bruggeman’s formula, treats the two phases in a symmetric fashion. IV. SPHERE SUSPENSIONS We now turn to the problem of diffusion in a separated grain structure material. The expression for Q(g,r’) defined by eq. (3.14) will have contributions from N spheres of radius 'a’ and from the supporting medium. In the simpler case where Dm=D2, 0(r) is zero outside the spheres. We consider this case here and turn to the general case in Chapter V. 4.1 Pair order theory When szD2 eq. (3.14) becomes N N fl(§.g’) E 2 0a(g)8(g-g’) E Q 2 9(a-Ig-Bal)5(g-g’) (4.1) a=1 a=1 where (D-D) Q = 3 1 2 (4.2) D _ 2 (D1+ 202) and 6(x) is the step function defined by 0(x)=0(1) for x;0(>0). By substituting eq. (4.1) in eq. (3.21), it becomes evident that it is cumbersome to classify the various terms in the expansion of g according to the number 27 28 of distinct spheres involved. Therefore, we introduce the 27 one-sphere operator ; (r,g’) defined by a 2a = nan + nagm.ga (4.3a) and define (4.3b) lld’ u llbdz "(1' If we now replace Q(g,g’) by g in eq. (3.21) and demand that no two adjacent summation indices are the same (multiple-scattering exclusion condition) then we can easily show that the new expression is identical to the old one. Thus §(r,g’) is written as -<2 3; >-H -<1+gmog>'1 = <3) - -§m- + ... . (4.6) Now write X = Z X- and W = Z W. , where both 3. and W. - - - - - -J -J -J J J have j 't’ operators. Then eq.(4.5) yields N N N w. = 2 Z .00 Z t 'H ’t to.“ .t (407) =3 _ =a =m =a =m =a. al—l azval aj‘aj-l 1 2 J The g. are then given by g1 : (g1) 52 = -ogm- (4.8) The effective diffusion constant up to second order in sphere concentration (pair order) is obtained from eqs. (3.22b) and (4.8) as (D ' D ) co __s___a_ - (2) 31>... (D . 21m -1..{;1(.).;2(,).§§j 33} (.9) e m 3-3 where (2) _ (2) _ . . 5. - - E BEa , (4.10) and there are j t operators in eq. (4.10). The evaluation of eqs. (4.9) and (4.10) is carried out in the next section. 30 4.2 Evaluation of Pair Order Theory: sznz case The evaluation of ;a(§,r’) in the case Dm=DZ can be carried out by using classical electrostatics. To see this first formally solve eq. (4.3) for 3a: Q . (4.11) Comparing eq. (4.11) with eq. (3.16) shows that the Ea operator connects the flux 8j(g) and external field 2n (3), for one sphere. Thus 51(2) = -Idg'g (g.g’)-2n (g') . (4.12) Finally, substituting eq. (4.12) in eq. (3.8), we find §<£> - §m<£) = Ida’ldz” §<£-£’)°g(£’)£”)'§m(£”> ) <4-13> where §(r) = -gn(r) is minus the gradient of the density field that results when a nonuniform Em(r) is applied to one sphere located at the origin of the coordinate system. The left hand side of eq. (4.13) is therefore the gradient of the induced density field caused by the presence of the sphere. The propagators that appear in eqs. (4.11) and (4.13) are for Dm = D cf. eqs. (3.9) and (3.10); so we 2! just write them as g and g dropping the subscript 2. 31 In Appendix A we use the solution of the electrostatic problem of one sphere’s response to a imposed Em(g) field to obtain an expression for 2(r,r’) as a multipole expansion. We find that E t v‘5(r) o (v')‘s(r') (4.14) 1:0 I" ~ ~ ~ with the multipole moments t1 : B£+1/[(£+1)!(21+1)!!] (4.15) given in terms of the multipole polarizabilities Bl (see eq. (Alb). The result to lowest order in concentration is obtained from eqs. (4.7), (4.8), and (4.14) as N 3(2-2’) = §1(£-£’> = < Z ta(£)r’)> =1 - 4nD E 85+1 v1 (v')1<25( -R )5(r’-R )> (4 16) ‘ 2 o (1+1)!(21+1)! ~ ° ~ E ~a ~ ~a ° 1: a The above configuration average over the sphere centers Ba (a=1,2,...N) is (28(g-Ra)>8(r-r’) = 8(g-g’) = n5(g- ’) (4.17) a 2"! where n(r) is the number density of the spheres at r and n 32 is the average number density. Inserting eqs. (4.16) and (4.17) in eq. (4.9) then yields a a¢ (4.18) where ¢=(4n/3)na3 is the volume fraction of the spheres and a = (DI-D2)/(D1+2D2). This is the Clausius-Mossotti result, as expected. We now consider g2, the first pair term. From eqs. (4.7) and (4.8) > (4.19) Fourier transforming eq. (4.19) to wave vector space gives d3k’ 3 h(k'):(g.g')-H(g')-t(g'.5) (4.20) (21!) _ 52(5) = “2] In eq. (4.20) h(k) is the Fourier transform of the direct correlation function h(r)=g(r)-1. The radial distribution function g(r) is defined by the configuration average )> (4.21) n28(§-§’) z z <8(g-Ba)5(g’-B a=1 Bea B and is assumed to be for hard spheres: g(r)=9(r-2a). From eq (4.9) the contribution to De comes from the k40 limit of £2: 33 d3k’ 2 x (Q) = n] h(k')g(g.k’)-g(g’)-;(§'.9) (4.22) Using the multipole expansion eq. (4.14) for g(0,k) (the 5 space expansion of 't’ is obtained by formally replacing the 3’s by ik’s) and also using the fact that h(k) is angle independent (because of spatial isotropy), we see that the only factor that depends on angle in the above integral is g(k) ~ (1-3ig) which, when averaged over all directions, gives zero. Hence we conclude that §2(0) is zero. This result holds for other choices of DIn as well, the only restriction being that the inclusion geometry is spherical. The higher order terms §§2)(0),j;3 can be analyzed in similar fashion. All of them have the pair distribution function g(r) instead of h(r). In Appendix B we sketch some details of their evaluation. The general term can be written in the form (2) - - Xn (k-O) - X a a '2 n-3 £.+R. 2 2 n J 3+1 1 3D2a O 2 ... X 1{.H al.}(11+1){ .H [ ]E;:T } 21:1 ln-2= 1:1 1 3:1 13 -22£.-n+2 2 1 (2211+n-2) 1 n ii} n 2 3 (4.23) i X {n(1i+1) + (-)"’ i t with { } E 1 for n=3, and g“ 5 1x“. a is defined in eq. (4.18). The ak’s found in eq.(4.23) are the reduced 34 polarizabilities defined by 1(01- DZ) "‘1 = 1131+ (um—6; ° ”'2‘” The final expression for De’ eq.4.9, thus becomes 3(D - D ) o e 2 _ 2 2 ~(2) (5—:—§B_) - 3a¢ + a ¢ E xn (0) , (4.25) e 2 n-3 X where i 5 ‘——JL—- . n D2a2¢2 Numerical procedure For the Dm=D2 case the second order term in the effective diffusion coefficient is given in terms of the reduced polarizbilities a of the spherical objects. The 1 general term Xéz)(k=0)is a n-fold sum of suitably weighted “1’8' Though each sum goes up to infinity, in practice the sums are truncated at some point. For a given n, a cut-off value is chosen for the summand; all the terms larger than that cut-off are evaluated and summed. The procedure is repeated for two different cut-offs placed at constant interval from the first. The three values obtained for the sums are then fitted to an exponential decay to estimate the remainder. If it is larger than a given value the procedure is repeated. 35 It is clear from eq. (4.23) that as n increases the number of terms to be evaluated increases tremendously. This is more so for a case where D1 and D differ widely; 2 i.e. for cases where a is close to either 1 or —0.5. Also the number of Xn’s to be evaluated increases considerably for such cases. Except for these extreme cases, it was found that "n" never has to be more than about 8 to get 3 to 4 decimal accuracy for the-second order term. For the case of uniform diffusivity spheres the results obtained for the second order term agree with those 10 in the context of obtained by Felderhof et a1. dielectrics and Jeffrey8 in the context of thermal conductivity. Both of them used a twin spherical harmonics expansion to obtain the density field outside the spheres. We note that since X2(k=O)=0, the overlap contributions given in Felderhof et a1. and Jeffrey are obtained by expanding De/DZ in eq. (4.9) in o. It will then be found that the second—order term found on the right hand side of eq. (4.25) and their non-overlap term are the same. If we consider the spheres to be point scatterers for which the multipole moments are given by a 1 = 1 a = { (4.26) 0 1 = 2,3,... C 9 then the sum over n in eq.(4.25) can be carried out 36 analytically and the result is 2 g xéz’(0) = 2D2a 8*“ ] . (4.27) $2108 [ 8-2a n 3 This result has been derived independently for the case of a = -l/2 (perfectly reflecting spheres) in an earlier publicationzs. Table 2 gives the numbers for —.5 S a S 1 for these two models along with the results obtained in the next section. It is seen that the point-scatterer result is less than the uniform sphere result by almost a factor of two. Hence caution has to be exercised in using the point- scatterer model in practical situations.-We next consider two model systems in the following sections that are of more practical importance. Table 2. Second-order contributions to the virial expansion 37 of the Clausius-Mossotti function (coefficient of 0 term in eq.(4.25)) for different models. Reduced Polarizability Second Order Contribution a Point- Uniform Spheret Coated Metal Sphere scatterer -0.5 -0.091 -0.162 —0.162 -0.4 -0.047 -0.082 -0.088 -0.3 -0.020 -0.034 -0.040 —0.2 -0.006 -0.010 -0.014 -0.1 -0.001 -0.001 —0.002 0.0 0.0 0.0 0.0 0.1 0.0008 0.0013 -0.0002 0.2 0.006 0.010 0.004 0.4 0.049 0.083 0.057 0.6 0.169 0.290 0.229 0.8 0.408 0.718 0.627 0.9 0.586 1.055 0.969 1.0 0.811 1.506 1.506 *The reduced polarizability a has been defined in eq.(4.18). 1‘ The numbers in this column are identical to those reported 10 in Jeffrey8 and Felderhof et al . 38 4.3 Coated metal spheres As shown in Section 4.1, the second order term is expressed in terms of the one sphere 't’ operator. Thus, for a system of spheres which are themselves composite spheres, we just need to obtain the new polarizabilities a“ and use them in the expressions given in eqs. (4.9), (4.10), (4.14) and (4.15) to calculate De' As an important example of composite spheres, consider a system of metal spheres with a thin coating of poorly conducting material (an oxide coating). The conductivity of the coating is assumed to be very small compared to that of the surrounding medium. D1, D2 and Dm now refer to the conductivities of the metal, coating and the surrounding medium, respectively. The electrostatic problem of a composite sphere in a medium can be solved for the induced field due to the composite and hence its multipole polarizabilities can be found. They are given by 14 a .._.. M+l)+fll} + l‘p-p)£1+v(g+1ll‘aCRlzn-+1 1 {v(1+1)+pi}{i+1+vn} + 2(1+1)(v-u)(1-u)(a/R)2“+1 (4.28) Here a and R are the inner and outer radii of the composite, ple/DIn and szz/Dm. With a thinly coated metal sphere in mind (the coating itself is very poorly conducting compared to the sphere) we define the following two quantities: 39 1 = lim p[1-(a/R)] (R-a)* 0 P” a ( (4.29) 8 = lim up 9* 0 P" .. 4 In terms of 1 and 8, the “1’8 are given by a = 5 " *7 (4.30) 9. 8 4» (MIN It is interesting to analyze eq. (4.30) for different values of 5 and 1. First of all we note that though 1 can, in principle, take values from 0 to a, in reality it is never too large because of the damping faCtors in the expression for Xn. With this in mind if we consider the case 6))1 we see that eq.(4.30) goes to 1 in this limit, which is the conducting sphere limit of the uniform sphere case. Thus a metallic sphere with an extremely thin coating of moderate conductivity behaves like a metallic sphere. 0n the other hand in the limit 1))8 eq. (4.30) reduces to -£/(1+1) which is the same as for the perfectly reflecting uniform sphere case. Thus a thick coating of very low conductivity makes a non-conducting sphere out of a metallic sphere. It is convenient to write the al's in terms of aCEal = (5-1)/(5+21) for making comparisons with spheres of uniform diffusivity. If we do this the polarizabilities for a uniform sphere (see eq. (A1b)) and a composite sphere are 4O given by r 31g (1-1)a + (21+1) a = 4 (4.31) ac(1+2) - (1-1) (1+2) - (ll-1mc (uniform sphere) (coated metal shere) The results for these two cases are given in Table 2 for different values of a and ac with azac. The second order terms for the different models (coefficient of o2 term in eq.(4.25)) are also plotted in Figures 3 and 4 as functions of a=a . c It is clear from Table 2 that for the uniform sphere the sign of the second order term is the same as the sign of a. This follows from a theorem due to Hashin and Shtrikman15 which states that for an isotropic two phase system the Clausius-Mossotti value (eq. (3.26)) is the lower(upper) bound for the effective diffusivity when D1>D2 (D1t§¥?“v} =°- <3“ J: Unlike the Dm = D2 case, where 't’ could be written as a sum over discrete functions centered on each sphere, it is non- zero everywhere for this case. In order to be able to use the formalism we have developed to evaluate the pair term, we must discretize 't’. Hence we consider the usual model of embedding the grains of the system in the effective medium. As we have already pointed out in chapter 11, Bruggeman’s model is not in a form readily extendable to include pair interactions in the inclusion-matrix problem. Therefore, we 25 of coating the inclusions with an use Hashin’s procedure arbitrary thickness of the medium and embedding the resulting fictive composite spheres in a medium of diffusivity De' Then, from its definition (see equations (3.14) and (4.3)) 't’ is non-zero only inside the composite spheres. The t-operator is discretized once again and we will denote it as 'te’ hereafter. It is now calculable via a 45 46 multipole expansion for fields outside the fictive sphere just as in the bare medium case. In fact we already have such an expansion viz., eq.(4.28) with p = DI/De and v = D2/De. The multipole coefficients are now functions of the inner radius ‘a’ and outer radius 'R’ of the fictive sphere. A value for R has to be specified now. If the interac— tions between different fictive spheres can be neglected, then eq.(5.1) reduces to the equation (te> = 0; which, by the usual analysis, leads to a1 = 0 (with a ’s given by l eq.(4.28)). This can be written, using eq.(4.28), as (v - 1)(2v + p) + (p - 2)(2v+1)(a/R)3 = o (5.2) If R is chosen such that (a/R)3 = o, the volume of fraction of phase 1, then eq (5.2) yields the Clausius-Mossotti equation. It is useful to pursue similar reasoning at the pair order level since, in a C.M. effective medium, the fictive spheres, with (a/R)3 = o, do not scatter. This becomes evident by attempting to calculate the pair correla- tions in the effective medium. As long as the fictive spheres do not overlap, a multipole expansion for te can be used, with the expansion coefficients given by eq.(4.28). (2), Since all of the Xn s (n23) have factors of a? (from the end te’s) it is evident that the solution of eq.(5.1) is 47 a = 0, the C.M. result. 1 However, this is not the only result of effective medium theory. We haven’t taken into account the inter- actions between the composite spheres for separations from 2a to 2R. This involves overlapping configurations of the composite spheres and, for these configurations, multipole expansions for te cannot be used. If te can be calculated in these overlapping regions then eq.(5.1) can be evaluated in an iterative manner. The C.M. value can be used as an initial guess for De' Determining solutions of Laplace's equation for a pair of overlapping spheres which have two spherical inclusions (the real spheres of radius 'a’) is very difficult. While the geometry of the problem looks most suitable for treatment in a coordinate system such as toroidal or bispherical we have not been able to use any of them. In the absence of success with the above analysis we use an approximate way of working out the one sphere te—operator that is usable in the overlap region. The electrostatic problem of a composite sphere in an arbitrary external field can be solved for the potential in all three regions. In region 2, between 'a’ and 'R’, the potential has two pieces, one proportional to r1, and the other to 1/r1+1, for a given r1 external potential. The 1+1 contributions to the 1/r field come from induced dipoles which are located at a distance less than r from the origin 48 l and the r field is caused by dipoles that are farther away than r. However since it is the final field that we are interested in, we can rearrange the dipoles so that those 1 field are all distributed on the 1 contributing to the l/r1+ surface of the inner sphere and the r field arising solely from dipoles distributed on the outer surface of the composite sphere. Correspondingly 'te’ has two parts denoted by te1 and te2' The former is easy enough to calculate as a multipole expansion using the electrostatic solution for the potential in the second region. We have on a . - ___e_._&_ 1-1' . 11-1 gel(g.g ) - i 4nD81§1 12(21_1)!, g 8(3)0(2 ) 8(5) (5.3) with D (21+1){ D eD } a(n+1) e- 2 a = 4- - H W) (”53334 ) - (...W e 2 1 2 (5.4) te2 is not an easy quantity to calculate as an expansion. However we note that whereas te1 is 0(1), tez is 0(¢). Hence we just drop it. It is justified at low enough volume fractions. Thus with t = t the x (k=0)’s can be e e1 e,n calculated for separations from 2a to 2R. The result is 49 (2) _ Xe,n(k-0) * . . n-2 n-3 1.+1. 30 a2 ¢2 2--- 2 n a (2 +1) n 3 3+1 1 e e,1 1 _1 1 _1 i-l e,1i 1 ._1 1 1.+1 1' n-z‘ ‘ 9' j J -221.-n+2 221.+n-2 g, 1 a i n-l “ (2211+n-2) {1'[R] } {g(‘i*1’ * “’ Q 1i} n23 '(5.5) with { } = 1 for n = 3 . Thus the different quantities in eq.(5.1) are now known and a simple rearrangement gives _2___2 _ 2 2 “ ~(2) D + 2D - a¢ + ae o A Z Xe,n(0) , (5.6) e 2 n=3 where A . 332:_3; + 2(Dg’ ”1’(Dg'Dzi [%]3 (5 7, De+ 2D2 (De+ D2)(D1+ 2) ~ = 2 2 and xe,n - xe,n / 3De (Bl) This can be written in k space as 3 - 3 _ 2 dk’ dk” 53””‘“J 3 3 g(ng'-g"))g(g.g')-§(g')-g(g'.g”) (2W) (2W) — _ where g(k) is the Fourier transform of the radial distribution function (of. eq. 4.21). The expansion for t0, g(R) can be replaced by g(R)=gg(1/4HD2R). Thus, to calculate 53(0), we have to evaluate the contraction 2(1/R)ogn'+2(1/R). Its aB component is written as v v v i v v v v i (35) °‘ *‘1 ”2+1 R “2+1 “9. ”1 B R Appendix C gives an expression for Vi 2(1/R): 1 2?... (2+2) E ,w (_21_+3_)_2_! * ”gnu” r .. ) (- R1+3 { RR... (n+2) - 2(2£+3) flRR... (2) (1+2112+1)11-11 ' ‘“ l + 4(21+3)(21+1) nnRR... (1-2) + ...}. This is then dotted with yi+2(1/R) as shown in eq.(B5). From the third term onwards there is at least one 1 in the tensor that is fully contracted with a pair of Y’s. This 62 brings in a factor of V2(1/R) which makes the third and higher-order terms vanish. In the second term there are [122] : (1+2)(1+1)/2 terms when the symmetrized tensor is written out in terms of its components. Out of these (1+1) terms will have ‘a’ as one of the components of n. In the rest of the terms 1 is fully contracted with a pair of 2’s on the other side and hence these don’t contribute. The non— zero contribution (call it 3) is finally given by A (_)£+2 (2g+3)!! R R ... R R1+3 “ “1 “1+1 _ g;+2)(1+1) 2(g+1) ‘ v v 1 2(21+3) (1+2)(1+1) saHIRH2°°' RP1+1} Vu1+i'° P1 B R (BS) where we note that there are (1+1) identical contributions in the second term. . n+1 k 1+2 (21+1)!! a a 1 E :(-) (21+3)R - (1+1)-— v V - R1+3 { a 6R1+1 6R1 a } B R This can be written as 1+2 (21+12'v “ ‘ a“ a“ 1 s =(-) '° { (21+3)R R -— v - (n+1)—— v } v — R1+3 a 6R1 1 6R1 a B R (B7) 63 1 _ 1+2 (21+1)!! 1_ 1 * . ‘. . ( ) R1+3 [ aR1 R3 ] { ‘2”+3’Ra31 (JRVRB- 813) - (1+1)(3RGRB- sag) } - 121+1)!!L1+2)1 * 1 ‘!+1) _ . . - R21+6 { (1+2)RQRB + 2 (5aB Rana) } (BS) The expression for 53(k=0) reduces to x3’a3(k=0) = 4WDZB x Ida g(R) R21+6 { (1+2)RaRB + ilgll (saa- RaRB) } (39) In the case of 54(k=0) we have to evaluate a contraction of the form 1+1 1 1+1 m+1 1 m+1 1 Va‘2 R 0 Y Y R ° 3 Va R 1+2 (21+1111 “ “ 15 a“ (-) 1+3'° { (21+31RaR1 1 V1_ (1+1)——£ Va } R aR aR m+l 1 m+1 1 *2 R02 R using eq.(B7). Next we use eq.(B8) to evaluate both terms. 64 n 1 (-)1+2 135111++ (m+2)!(2m+l)!![ 9— ——l—— ] R1+3 6R1 Rm+3 “ * * “ - (n+1) * * _ { (21+3)RaR1[(m+2)R1Ra 2 (RVRB 613)] A A 1 - (1+1)[(m+2)RaR Rm+3 _.(_!L1_)_‘A_ a 2 (RaRp 5aB)] } The second term in the first square bracket vanishes and the final expression for the three-tensor contraction is 111+1):1(2g+111111+m+212 R21+2m+9 { (“+2’(m+2’RaR B + l (1+1)(m+1)(R R - 5 1} (810) 2 a B a3 Evaluation of one more term was necesary before we could generalize the results for arbitrary number of tensors in the contraction sequence. The final expression for §j(k=0) is given by 2 2 3'2 B1. 1 Xi’afi(k=0) : 4nDzflln Z ... Z 2 [ ( 1+ ] i 1 1.+1) 11 0 £j-2 1 X (1 +1 +2)!...(1. +1. +2)! -(22.1.+3j-3) 1 2 J-3 J-Z 1 1 x J_3 de g(R)R II (1 +2)! 1:2 k . { ? (1i+2)RaRB+ ( ) 2 2 (1i+1)(8aB RaRB) } (B11) 65 with { }*5 (11+2)! for j=3. The use of the dilute hard sphere radial distribution function g(R) = 9(R-2a) in eq. (B11) permits analytic evaluation of the integral and, after a few rearrangements, yields eq. (4.23). APPENDIX C An expression for Zn (l/R): In order to construct an expression for the above quantity consider the following contraction where k is a unit vector along the z axis of the coordinate system. AA n kk...(n) o gg...(n) % é—h[%] az II I :3 :3 'U 0 O (D G . (Bl) (See ref. 29, p. 422) Here 0090,: k . R and Pn(x) is the Legendre polynomial of C 9 order n . Pn(cose) has a power series expansion in 0039 (see ref. 29, p. 419). [n/2] Pn(cose) = Z (-)In —;——-i§fl:zflli————— (0039)“-2m m=0 2 m!(n-m)!(n-2m)! : LQE:%LLL { (cose)n- %%%i%%7 (cose)n-2+... } (82) where (2n-1)!! s (2n-1)(2n-3)...1 = (2n)!/2nn! 66 67 - 132211;; A “ n_ £12211. - n! {(k-R) 2‘2” 1) (k R1n } fl - 1222111; “ ELE_LL_ A“ - n! { RR...(n) - 2(2n_1) RR...(n-211 + ... } o hh...(n) , (33) the overhead line indicating symmetrized linear combinations of the term under it (to reflect the symmetric nature of the left hand side). Thus we have .(n) E I (<1P12n-112! { RR..,(n , Bifl—ll— RR...(n 211 + ... }. Rn+1 - 2(2n- 1) (B4) the general expression being [n/2] .. 1 (-1n ”Sn—'71,. {0b 1'll firm: RR...(n-2m111...(m1 (B5) LI ST OF REFERENCES 10. 11. 12. 13. 14. LIST OF REFERENCES R. Landauer, "Proceedings of the First Conferance on the Electrical and Optical Properties of Inhomogeneous Media", Ohio State University, 1977, eds. J. C. Garland and D. B. Tannes (AIP, New York, 1978) No. 40, p. 2. J. E. Sipe and J. van Kranendonk, Mol. Phys. 22, 1579 (1978). P. Debye and A. Bueche, J. Chem. Phys. 1 , 573 (1948). D. W. Sundstrom and Yu-Der Lee, J. Appl. Polym. Sci. 12, 3159 (1972). D. Hansen and R. Tomkiewicz, Polym. Eng. Sci. 12, 353 (1975). J. C. Harper and A. F. El Sahrigi, Ind. Eng. Chem. Fundam. 2, 318 (1964). D. Bedeaux, Physica A, 121, 345 (1983). D. J. Jeffrey, Proc. R. Soc. London ser. A 335, 355 (1973). B. U. Felderhof, G. W. Ford and E. G. D. Cohen, J. Stat. Phys. 22, 135 (1982). B. U. Felderhof, G. W. Ford and E. G. D. Cohen, J. Stat. Phys. 22, 649 (1982). B. U. Felderhof, G. W. Ford and E. G. D. Cohen, J. Stat. Phys. 22, 241 (1983). 0. F. Mossotti, Memorie di Matematica e di Fisica della Societa Italians delle Scienze Residente in Modena, 22, pt 2, 49 (1850). R. Clausius, "Die mechanische Behandlung der Electri- citat" (Vieweg, Braunschweig, 1879) p. 62. J. C. Maxwell, "A Treatise on Electricity and Magnetism" vol. 1 (Reprint: Dover, New York, 1954) Ch. 9. 68 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 69 Z. Hashin and S. Shtrikman, J. Appl. Phys. 22, 3125 (1962). H. B. Levine and D. A. McQuarrie, J. Chem. Phys. 22J 4181 (1968). G. K. Batchelor, J. Fluid. Mech. 22, 245 (1972). D. E. Ross, Aust. J. Phys. 21, 817 (1968). M. Fixman, J. Phys. Chem. 22, 6472 (1984). D. A. G. Bruggeman, Ann. Physik (Leipz.) 21, 636 (1935). D. W. Taylor, Phys. Rev. 122, 1017 (1967). P. Soven, Phys. Rev. 122, 809 (1967). R. J. Elliott, J. A. Krumhansl, and P. L. Leath, Rev. Mod. Phys. 12, 465 (1974). F. Yonezawa and K. Morigaki, Prog. Theor. Phys. Suppl. 22, 1 (1973). . Z. Hashin, J. Compo. Mater. 2, 284 (1968). E. H. Kerner, Proc. Phys. Soc. B, 22, 802 (1956). J. M. Ziman, "Models of Disorder" (Cambridge University Press, London, 1979) ch. 10. S. Sridharan and R. I. Cukier, J. Phys. Chem. 22, 1237 (1984). G. Arfken, "Mathematical methods for physicists" (Academic Press, New York, 1966). S. Kirkpatrick, Rev. Mod. Phys. 22, 570 (1973). W. H. Orttung, Ann. NY Acad. Sci. 222, 22 (1977). M. Fixman, J. Chem. Phys. 21, 3666 (1984). C. W. J. Beenakker and J. Ross, J. Chem. Phys. 21, 3857 (1986). D. J. Bergman, Phys. Rep. _2, 377 (1978). B. U. Felderhof, Physics 122 A, 430 (1984) and references therein. B. U. Felderhof and R. B. Jones, 2. Phys. B 22, 43 (1985). 7O 37. B. U. Felderhof and R. B. Jones, 2. Phys. B 22, 215 (1986). 38. J. R. Lebenhaft and R. Kapral, J. Stat. Phys. 22, 25 (1979). 1 HICH IGRN STRTE UNIV. LIBRARIES llflmllllllll(WINllIWIHIFIINNlHlllHI)HI 1293008128070 3