© Copyright 2001 American Chemical Society
OCTOBER 2, 2001 VOLUME 17, NUMBER 20
Letters Analysis of Concentrated Colloidal Dispersions Armik V. Khachatourian and Anders O. Wistrom* Department of Chemical and Environmental Engineering, University of California, Riverside, California 92521 Received November 20, 2000. In Final Form: July 31, 2001 A long-standing area of research in colloid science concerns the determination of electrostatic, magnetic, and elastostatic fields in dispersions comprising densely packed particles whose material properties differ markedly from that of the background. As new materials with complex microstructures are being developed, there is a corresponding need for analytic and numerical methods to predict and understand their properties. Dense systems with large numbers of close-to-touching particles require very fine discretization which causes the linear system of equations that arise to be highly ill-conditioned. A framework for solving the many-body interface problems is provided by a mathematical transformation of the boundary value integral equation. The transformation avoids the use of asymptotics and involves no uncontrolled approximations which distinguishes this work from previous methods. Pointwise analysis of local charge densities in the bulk and at interfaces is now possible.
1. Introduction
2. The Many-Body Problem
An important problem in colloid research concerns the determination of electrostatic fields in concentrated dispersions consisting of randomly distributed particles suspended in a solvent of uniform composition. Important quantities which can be obtained from the electrostatic calculations include the electrical transport properties of the composite as well as pointwise values of stress fields in the bulk and at grain boundaries. Time and space resolved structural information would complement thermodynamic information for producing as complete a picture of colloidal systems as possible. Despite methodological advances, there are virtually no accurate largescale numerical simulations of dense random dispersions in three dimensions. In this letter, we report on the analytical formulation of the surface charge density distribution of a particle suspended in a random assembly of charged particles which may be close to touching. The framework for the mathematical description of the manybody electrostatic problem that allows for pointwise evaluation of local properties is provided in a form making the numerical evaluation of concentrated many-body systems tractable.
There are a variety of theoretical approaches to the static many-body problem including methods based on classical potential theory1-3 for example4-10 and the method of images.11-14 In the former method, the charge density is represented as a Fourier series where the multipole moments (or Fourier coefficients) are the unknowns. In the case of dilute systems, where particles are well separated, the number of moments which need to be retained to resolve the charge density is relatively (1) Jackson, J. D. Classical Electrodynamics; Wiley: New York, 1975. (2) Jaswon, M. D.; Symm, G. T. Integral Equation Methods in Potential Theory and Electrostatistics; Academic Press: New York, 1977. (3) Van Bladel, J. Electromagnetic Fields; McGraw-Hill: New York, 1964. (4) Greengard, L.; Moura, M. Acta Numerica, 1994; Cambridge University Press: Cambridge, 1994. (5) Helsing, J. Proc. R. Soc. London, Ser. A 1995, 450, 343. (6) Helsing, J. J. Comput. Phys. 1996, 127, 142. (7) McPhedran, R.; McKenzie, D.; Derrick, G. Proc. R. Soc. London, Ser. A 1978, 362, 211. (8) McPhedran, R.; Perrins, W. T.; McKenzie, D. Proc. R. Soc. London, Ser. A 1979, 369, 207. (9) Sangani, A. S.; Acrivos, A. Proc. R. Soc. London, Ser. A 1982, 368, 263. (10) Sangani, A. S.; Yao, C. Phys. Fluids 1988, 31, 2426.
10.1021/la0016120 CCC: $20.00 © 2001 American Chemical Society Published on Web 09/07/2001
6032
Langmuir, Vol. 17, No. 20, 2001
Letters
modest and Rayleigh’s15 method gives excellent results. However, for concentrated systems, where particles are close to touching, the charge density becomes nearly singular and the number of degrees of freedom required grows exponentially. The linear system of Fourier coefficients also becomes highly ill-conditioned which renders the multipole method impractical for concentrated systems. The method of images represents the electric fields in terms of a series of dipoles. The method of images proceeds by reflecting the dipole source onto the other surfaces with a strength proportional to the reciprocal of the distance between the charge and the field point. As the particles approach each other, more and more reflections are required to satisfy the boundary conditions and a series solution is obtained in which the successive terms decay geometrically. For the two-body problem, the number of reflections is manageable16 but the total number of reflections required grows exponentially with the number of particles, thus rendering the method unwieldy for largescale problems. A hybrid method that combines the method of images for close-to-touching particles with a multipole representation of particles at larger separations eliminates some of the computational burden associated with each method taken separately but introduces approximations that are difficult to evaluate.17,18 In this letter, we present a formula for evaluating the surface charge density distribution of a charged particle in the presence of more than one other charged surface, for example, a charged particle suspended in a manybody system such as a concentrated dispersion. A transformation of the series solution results in a reordering of the sums to yield a finite matrix of coefficients such that the sought-after charge density distribution can be evaluated by simple matrix inversion. Importantly, the transformation is mathematically rigorous, avoids the use of asymptotics,19,20 and involves no uncontrolled approximations or model-dependent parameters which distinguishes this work from previous methods. The theoretical framework demonstrated here provides a basis for developing practical methods for producing time-resolved structural and space information of many-body systems at high concentrations. 3. Computing the Electrostatics Consider a dispersion of random spherical particles where each particle carries a surface potential
V)
1 4π
∫ σ RdS
(1)
where integrations are performed over the surface of the spheres, with charge densities σ, weighted with the relative curvature R-1, and is the effective permittivity of the background medium. A solution to eq 1 requires a (11) Honein, T.; Honein, E.; Herrmann, G. Q. Appl. Math. 1992, 3, 337. (12) Lifshitz, E. M.; Landau, L. D.; Pitaevskii, L. P. Electrodynamics of Continuous Media, 2nd ed; Pergamon Press: Oxford, 1984. (13) McPhedran, R.; McKenzie, D. Proc. R. Soc. London, Ser. A 1978, 359, 45. (14) Poladian, L.; McPhedran, R.; Milton, G. W. Proc. R. Soc. London, Ser. A 1988, 415, 185. (15) Lord Rayleigh Philos. Mag. 1892, 34. (16) Soules, J. A. Am. J. Phys. 1990, 58 (12), 1195. (17) Cheng, H.; Greengard, L. J. Comput. Phys. 1997, 136, 629. (18) Cheng, H.; Greengard, L. SIAM J. Appl. Math. 1998, 58 (1), 122. (19) Batchelor, G. K.; O’Brien, R. Proc. R. Soc. London, Ser. A 1997, 355, 313. (20) Bonnecaze, R.; Brady, J. Proc. R. Soc. London, Ser. A 1990, 430, 285.
Figure 1. Scaled charge density distribution aσ/(V) as a function of θ (in deg) (‚‚‚) and sin θ multiplied by scaled charge density distribution (aσ sin θ)/(V) as a function of θ (in deg) (/ / /). V1 ) V2 ) V, a1 ) a2 ) a, and separation distance d/a ) 2.
numerical approach whereby the surface charge densities on each particle boundary are first discretized into area elements of charge. For spherical particles, charge elements with densities σ are appropriately expressed in terms of Legendre polynomials, Pm(x). The finite size of the area elements is the only approximation in the method, and its effect can in principle be made arbitrarily small given sufficient computing power. Also, in the expression for force the surface charge is modified by the angular component, sin θ, measured relative to the axis connecting the particle centers which lessens the effect of a coarse discretization (Figure 1). An infinite dimensional linear system of equations of discrete charges gim can be cast in the form of the Fredholm equation with a singular kernel according to procedures developed in earlier work,21 ij gim ) -Γijmβj - Ωm,k gjk
(2)
ij ij where Ωm,k is the many-body kernel and Γijm is Ωm,0 . Superscripts refer to family matrices where the indices i and j run from unity to the number of particles, N. βj () Qj/(4π)) is related to the charge at infinite separation, Vj ) Qj/(4πaj). Also, for each m the subscript k runs from zero to infinity and summing over repeated indices is assumed. The unitless companion matrix to Γm () Ωm,0) is the pure traceless matrix Ωm,k whose diagonal elements are ij contain the zero. The off-diagonal elements in Ωm,k ij succession of coefficients where Ωm,k ) [(k + m)!/(k!m!)] (aim+1ajk/hijm+k+1) ) [(k + m)!/(k + m)!](zij)m+1(zji)k where zij ) (ai/hji), ai is the particle radius, and its center-to-center separation distance to aj is hij () hji) (Figure 2). Hence, the number of reflections required grows geometrically with decreasing separation and exponentially with the number of particles. The consequence is an infinite dimensional system of coefficients that requires massive computational storage capacity for concentrated many-body systems, a problem which has, until now, prevented application of the method of images to large-scale problems.
(21) Khachatourian, A. V.; Wistrom, A. O. J. Phys. A: Math. Gen. 2000, 33, 307.
Letters
Langmuir, Vol. 17, No. 20, 2001 6033
required a fundamentally different approach to formulating the coefficients of each image series without introducing uncontrolled approximations. The problem we have solved is a mathematical reformulation of the family matrices that allows us to reorder the sums in eq 3 into a closed form. To complete the reordering, we use the mathematical identity ∞
gk ) gk,0 )
∑ (gk,m - gk,m+1)
(4)
m)0
A sequence of algebraic manipulations using above identity yields ∞
gm ) -[1 +
Figure 2. The geometry of a many-body system. The charge on particle 1, σ1, is due to the self-charge in addition to contributions from particle 2 through θ1,2 and from particle 3 through θ1,3.
The magnitude of the surface charge, σ, on particle i is obtained by summing the contributions from all other surfaces connected to i by the inner product of the unit vectors aˆ i‚hˆ ij and is given by
Iijσj ) βi + Lijmgjm
(3)
where Iij ) ai2δij and Lijm ) (2m + 1)Pm(cos θij) is again a pure traceless matrix. Equation 3 is the full solution for the charge density on a conducting sphere and includes all sources of charge from the reflections from all other conducting spheres. The series of coefficients that stems from the serial reflections is easily seen to converge. For the pair interaction, this does not constitute a problem since the reflections stem only from one surface at a time. Unfortunately, the many-body problem is not so simple. Suppose that the first-order reflections are computed from each pair interaction. These images must then be reflected into all other particles, generating second-order reflections and so on, with the total number of reflections required growing faster than exponentially with the number of particles. Even if each image series is truncated after a finite number of terms, the number of coefficients that must be stored increases with at least an exponential trend which requires massive computational memory allocations even for moderately sized systems. Since more than 1000 terms are needed to resolve the pair interaction of closeto-touching particles, the problem of branching coefficients effectively hinders evaluation of gim for other than smallscale and dilute systems. Only recently has application of hybrid methods began to yield results. Methods that combine the method of images for close-to-touching particles with a multipole representation of particles at larger separations eliminate some of the computational burden associated with each method taken separately but introduce approximations that are difficult to evaluate.17,18 Nevertheless, a rigorous solution of the many-body system that allows for numerical evaluation has remained an unsolved problem until now. We found that an unambiguous solution of this problem
∑ Ωk,m]-1Γmβ k)0
(5)
∞ where ∑k)0 Ωk,m ) zij(zji)m/(1 - zij)m+1 is again a pure traceless matrix with all diagonal elements zero. Each term is now finite, and the sought-after charge density distribution can now be evaluated by simple matrix inversion and summed over m using eq 3 noting that the matrix is singular when particles are touching. The reformulation is mathematically rigorous; for example, eq 2 satisfies eq 5. An important problem in colloid research is the evaluation of electrical properties of suspensions. One example is the net electric charge of a suspension of particles given the surface potential. By definition, Q ) ∫π0 ∫2π 0 sin ω dω dφIσ ) 4π(β + L0g0), and then the net electric charge is given by
1
∞
∑
Ωm,0]-1Γ0β Q ) β - L0[1 + 4π m)0
(6)
Similarly, the local charge density distributions are readily obtained which provides for pointwise evaluation of stress fields in the bulk and at grain boundaries. 4. Conclusion The theoretical approach we have presented here makes possible a complete numerical solution to large random many-body systems. The key to success in this work is the use of mathematical transformations that were originally invented to prove formal theories in combinatorial identities and that eliminates the use of asymptotics or arbitrary truncation. Although the theoretical framework presented here will certainly be improved upon, the real importance is that the only approximation in the method is the finite size of the surface elements containing the surface charge. The procedure we have outlined involves no uncontrolled approximations, and the effects of the numerical approximations can in principle be made arbitrarily small, given sufficient computing power. This fact distinguishes this approach from other theoretical methods that have been proposed to tackle many-body problems. Some have been found to give excellent results, but thus far all have involved approximations that cannot be systematically eliminated. We expect that the ideas presented here will lead to further developments of numerical methods for examining properties of many-body systems drawing on methods and concepts from recent developments in numerical schemes for handling large-scale problems. LA0016120
