Received 2 November 1995; accepted 17 January 1996. An off-lattice random-flight simulation procedure is described which accurately predicts the obstruction ...
Solvent diffusion in ordered macrofluids: A stochastic simulation study of the obstruction effect Haukur Jo´hannesson and Bertil Halle Condensed Matter Magnetic Resonance Group, Lund University, Chemical Center, P.O. Box 124, S-22100 Lund, Sweden
~Received 2 November 1995; accepted 17 January 1996! An off-lattice random-flight simulation procedure is described which accurately predicts the obstruction factors for self-diffusion of small molecules in macrofluids. The simulation procedure, employing a continuous step length distribution that ensures rapid and accurate convergence, was validated by comparison with exact results for cylindrical and spherical obstructions on 2D and 3D lattices. The exact results were computed with Rayleigh’s multipole method, which also was used to derive a new analytical formula for the obstruction factor of parallel cylinders on a hexagonal lattice, of much higher accuracy than the commonly used approximations. Random-flight simulations were used to assess the accuracy of existing mean-field approximations for the obstruction factors of orientationally ordered nonspherical objects. Due to a near-cancellation of errors, the mean-field result accurately describes the obstruction effect on the trace of the diffusion tensor, as measured in isotropic systems, up to moderately high volume fractions. In contrast, the diffusion anisotropy, a sensitive indicator of microstructure in anisotropic fluids, is accurately predicted by mean-field theory only at low volume fractions. © 1996 American Institute of Physics. @S0021-9606~96!50316-0#
I. INTRODUCTION
Transport properties, such as the mass diffusion coefficient and the electrical conductivity, are widely used to characterize the structure of heterogeneous fluids via the effect of interactions on the thermal motion of intrinsic molecular species. Molecular self-diffusion coefficients, usually measured by NMR pulsed-gradient spin-echo techniques,1 have thus provided structural information about a wide range of complex fluids, including surfactant solutions, microemulsions, gels, and liquid crystals.2–7 Furthermore, diffusion-weighted magnetic resonance imaging is used increasingly for clinical applications.8,9 In the absence of long-range interactions, which are important only for ion diffusion,10–13 the interactions need not be explicitly incorporated in the diffusion equation ~as a systematic drift term! but appear implicitly via the boundary conditions ~excluded volume! and, possibly, via a locally modified diffusion coefficient ~near boundaries!. In the present work, we focus on the excluded volume or obstruction effect, which contains the desired structural information. By spatially restricting the diffusion trajectories of solvent molecules ~or other mobile species!, obstructions reduce the effective self-diffusion coefficient measured on length scales that greatly exceed the characteristic length of the structural heterogeneity. The obstruction factor, also known as the tortuosity factor, is defined as the ratio, D/D 0 , of the measured large-scale diffusion coefficient D and the local diffusion coefficient D 0 ~usually identified with the bulk solvent diffusion coefficient!. It depends primarily on the shape and volume fraction of the obstructing objects. At high volume fractions, the obstruction factor depends also on the spatial and ~for nonspherical shapes! orientational distribution of the obstrucJ. Chem. Phys. 104 (17), 1 May 1996
tions. To extract structural information from experimental diffusion coefficients, one must know how the obstruction factor depends on all these structural parameters. The mathematical problem of calculating the obstruction factor for diffusion amounts to a solution of the steady-state diffusion equation, i.e., Laplace’s equation, under appropriate boundary conditions. This problem is isomorphic to the analogous problems of the dielectric permittivity, magnetic permeability, electrical or thermal conductivity of composite materials.14 Unfortunately, the problem does not in any nontrivial case admit a closed-form solution valid over the full range of volume fractions. Ever since the classic studies of Maxwell16 and Rayleigh,17 the problem has therefore been addressed by a variety of numerical methods and simplifying approximations. In the case of periodic distributions of obstructions, Rayleigh applied a multipole expansion method which reduces the problem to the solution of an infinite system of algebraic equations involving image multipoles and lattice sums. Truncating the system of equations, Rayleigh obtained closed-form results, valid at low and moderate volume fractions, for the obstruction factor for infinite cylinders on a square lattice and for spheres on a simple cubic lattice.17 More recently, this method was numerically implemented to calculate the obstruction factor for spheres on cubic lattices at all volume fractions up to close packing.18 Rayleigh’s multipole method provides a systematic procedure for deriving increasingly accurate closed-form approximations to the obstruction factor. If one is satisfied with numerical results, however, a simpler approach is the direct numerical integration of Laplace’s equation on a grid by finite-difference or finite-element methods. The finite difference method has been used by Photinos and Saupe for infinite19 and truncated20 cylinders periodically arranged on
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H. Jo´hannesson and B. Halle: Solvent diffusion in ordered macrofluids
planes. This method is computationally demanding at volume fractions near close-packing, where a fine mesh is required. In the present work, we used Rayleigh’s multipole method to validate the simulation procedure ~cf. below!. The conceptually simplest and most generally applicable approach to the numerical calculation of the obstruction factor is computer simulation. While a full molecular dynamics simulation can be used,15 a stochastic simulation involving only the position variable is generally sufficient. Randomflight simulations have recently been carried to a limited extent;21–23 however, due to methodological deficiencies these results are unreliable ~cf. below!. Here we present results of off-lattice random-flight simulations with variable step length, systematically studying the effects of shape, volume fraction, and spatial distribution on the obstruction factor in two and three dimensions. Closed-form approximations to the obstruction factor, valid at low volume fractions, have been obtained by various mean-field approaches. The most familiar of these is the effective-medium approximation ~EMA!, which yields the Maxwell relation for the conductivity16 and the formally identical Clausius–Mossotti relation for the dielectric permittivity.24 In this approach, Laplace’s equation is solved for a system containing one sphere ~or obstruction of other shape! taking the surrounding spheres into account only in an average way. As expected, the EMA result for the effective diffusion coefficient in the case of spherical obstructions is of the same form as these classical relations.14 The EMA also yields closed-form results for the obstruction factor in the case of ellipsoidal obstructions.25 A related, but distinct, development is the effective cell approximation ~ECA!, first used for spherical and cylindrical polyelectrolytes.10–13 According to the ECA, the real system, consisting of some spatial distribution of obstructing objects, is replaced by an effective cell of the same geometry as the obstruction and of a size that makes the volume fraction in the cell equal to that of the real system. Unlike a Wigner– Seitz cell, the geometry of the effective cell is not rigorously related to the real system. It is therefore not obvious what type of boundary condition one should impose at the cell boundary. By invoking the principle of minimal entropy production, however, Jo¨nsson et al. developed a consistent and general formulation of the ECA.26 None of the mean-field approaches can describe the effect of the spatial distribution of obstructions on the diffusion. Thus, for example, they cannot distinguish between planar, columnar, or cubic arrangements of spheroidal obstructions, provided that the shape and volume fraction is the same in all cases. The mean-field approaches have the great virtue of producing closed-form results for simple obstruction shapes, such as spheres, cylinders, and ellipsoids. ~Shapes that cannot be described in terms of a single coordinate system, such as spherocylinders, are not so easily handled.! For spherical and cylindrical obstructions, the EMA and ECA produce identical results. As expected, these results have the correct asymptotic form at low volume fractions. In fact, they are correct to fourth order in the volume fraction, being identical to the lattice-independent contribu-
tion obtained from the multipole method.17,18 For ellipsoidal obstructions, however, the EMA ~Ref. 25! and ECA ~Ref. 26! results differ significantly. Since there is no systematic procedure for improving the accuracy of the mean-field results, it is important to test them against exact numerical results. This has so far been done only for spheres,18 where the mean-field result is quantitatively accurate up to volume fractions of ;0.3. In the present work, we assess the accuracy of the mean-field results for cylindrical and spheroidal ~and certain related shapes! obstructions. The emphasis is on the anisotropic obstruction factors in the case of orientationally ordered spheroidal obstructions ~as well as certain other nonspherical shapes!. For isotropic fluids, the shape dependence of the obstruction factor is often too weak to be of practical utility for structural investigations. This is the case, for example, for prolates of all aspect ratios and for oblates of moderate aspect ratios.25,26 The situation is different in orientationally ordered fluids, such as lyotropic nematic phases. In a uniaxial phase, the difference between the two principal components of the effective diffusion tensor, i.e., the diffusion anisotropy, depends strongly on the shape of the obstructions. This fact has been utilized in a series of electrical conductivity studies of micellar nematic phases.27,28 The structural interpretation of these data have usually been carried out in terms of the EMA relations for spheroids. However, the local electric fields, which may have a significant effect on the conductivity in these highly charged systems,10–13 have not been properly included in these studies. This problem is avoided in the water diffusion approach, recently applied to micellar nematics.29–31 The main motivation for the present study comes from the need for a quantitative analysis of such data, which might resolve the longstanding problem of separately determining the shape and orientational order of the micellar aggregates in nematic phases.32–34 The outline of the present paper is as follows: In Sec. II we describe the numerical methods used here to calculate obstruction factors; the random-flight simulation and the multipole method. These are then applied, in Sec. III, to calculate the obstruction factor for infinite cylinders ordered on hexagonal and square lattices or disordered in the plane. The multipole calculations are used here to validate the simulation protocol. In addition, we assess the accuracy of the mean-field approximation. A further check on the simulation procedure is provided in Sec. IV, where we investigate spherical obstructions ordered on simple and face-centered cubic lattices or spatially disordered. For the ordered cases, we compare the simulation results with those of the multipole method. The principal results of this work are presented in Sec. V, where we present the principal obstruction factors for oblate and prolate spheroids, disks, and spherocylinders, over a wide range of aspect ratios and, in the case of oblate spheroids, for all volume fractions up to close-packing. While the mean-field approximations adequately describe the obstruction factor in isotropic solutions at low and moderate volume fractions, they fail already at low volume fractions to quantitatively describe the diffusion anisotropy, which is the
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H. Jo´hannesson and B. Halle: Solvent diffusion in ordered macrofluids
key to the structural characterization of orientationally ordered phases.
II. METHODOLOGY A. Random-flight simulation
Consider a particle undergoing force-free diffusion in a heterogeneous system. This dynamic process is fully described by the space–time propagator p~r,t!, obeying the diffusion equation
] p ~ r,t ! 5“•Dloc~ r! •“p ~ r,t ! , ]t
~2.1!
with Dloc~r! the spatially varying local diffusion tensor. On length scales much larger than the characteristic dimensions of the heterogeneity, one can use a coarse-grained description where the system appears spatially homogeneous and the macroscopic propagator P~r,t! satisfies the diffusion equation
] P ~ r,t ! 5“•D•“ P ~ r,t ! , ]t
~2.2!
with a spatially uniform macroscopic diffusion tensor D. If the symmetry group of the system contains a threefold or higher axis, the system is uniaxial with respect to a second-rank tensor,35 i.e., D has only two distinct principal components, D i 5D zz and D' 5D xx 5D y y . If there are at least two orthogonal threefold or higher axes, as in cubic crystals and isotropic fluids, D transforms like a scalar, i.e., it is proportional to the unit tensor, D5DU. Since the propagator P~r,t! obeying Eq. ~2.2! is of the usual Gaussian form, the mean-square displacement along a principal direction a is given by36
^ R a2 ~ t ! & 5
E
dRP ~ R,t ! R a2 52D aa t.
~2.3!
Consider now a system where the microscopic heterogeneity consists of a fixed spatial distribution of impenetrable objects embedded in a continuous, locally isotropic, fluid. Such a system is described by the microscopic diffusion Eq. ~2.1!, with the local diffusion tensor given by Dloc~r!5D 0 U outside the obstructions and Dloc~r!50 inside the obstructions. The obstructions thus enter the microscopic description solely via the boundary conditions associated with the differential Eq. ~2.1!. The obstruction factor, defined as the ratio of a principal component of the macroscopic diffusion tensor with and without obstructions, can now be expressed as A aa 5
D aa ^ R a2 ~ t ! & 5 2 , D0 ^ R a~ t ! & 0
~2.4!
where the zero subscript signifies absence of obstructions. To obtain the experimentally accessible obstruction factors, we thus need to calculate the mean-square displacements along the principal directions. This can be done, to any desired accuracy, with the aid of the random-flight model.
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We thus model the diffusive motion of the particle among the obstructions as a random flight, where each step vector d is chosen from the same a priori step vector distribution f ~d!. However, attempted steps that would lead the particle into obstructed regions are rejected. This procedure simulates the reflecting boundary condition at the obstruction surface, associated with the microscopic diffusion Eq. ~2.1!. According to the central limit theorem, after a large number N of steps the random flight is equivalent to a Gaussian diffusion process,36 i.e., the mean-square displacement ^ R a2 (N) & is proportional to N. This is true for any isotropic step vector distribution f ~d! with finite second moment.37 In particular, we have in the absence of obstructions, lim ^ R a2 ~ N ! & 0 5N ^ d a2 & 0 ,
~2.5!
N→`
with
^ d a2 & 0 5
E
d d f ~ d! d a2 .
~2.6!
According to Eqs. ~2.4! and ~2.5!, the obstruction factors can be obtained from the random-flight simulation as A aa 5
1
^ d a2 & 0
lim
^ R a2 ~ N ! &
N→`
N
.
~2.7!
The freedom in choosing the step vector distribution f ~d! can be used to optimize the performance of the simulation procedure in the sense of ~i! fast evaluation of the step vector, ~ii! rapid convergence of Eq. ~2.7! with increasing N, and ~iii! faithful sampling of the obstruction geometry. To satisfy the last criterion, we considered only uniform distributions for the step vector orientation. ~A 3D lattice simulation, with only 6 allowed directions, is clearly inferior in this respect.! Furthermore, a distribution with fixed step length d was rejected since it does not fulfill both the criteria ~ii! and ~iii!; to avoid a significant d-dependence in A aa , d has to be made so small that a very large number N of steps are required to average over the microscopic heterogeneity. While fulfilling criteria ~ii! and ~iii!, a Gaussian distribution has the drawback of a comparatively slow evaluation. We have therefore used a truncated spherical distribution with the step vector uniformly distributed within a sphere of radius D. For this distribution,
^ d a2 & 0 5
H
D 2 /4
in
2D,
D /5
in
3D.
2
~2.8!
A truncated spherical distribution of step vectors d was obtained by first generating a set of three random numbers ( d x , d y , d z ), uniformly distributed over the interval @2D,D#. The step vector d5( d x , d y , d z ) was accepted provided that d 2 5 d 2x 1 d 2y 1 d 2z 2p/~9)!'0.40 for the 3D fcc lattice. Depending on obstruction shape, volume fraction, and lattice type, the 3D random-flight simulations required, for a given D, 1–5 cpu h on an IBM Risc 6000/250 workstation.
B. Multipole method
In the external space outside the obstructions, the stationary diffusion propagator p~r!, giving the number density of tagged particles, satisfies Laplace’s equation, ¹ 2 p ~ r! 50.
~2.9!
In the multipole method, the particle density p~r! at any point r can be expressed in terms of image sources located at the lattice points ~on which the obstructions are centered! and external sources ~at infinity! that maintain the stationary particle flux. The particle density outside a reference obstruction, whose center defines the origin, is expanded as FIG. 2. Obstruction factor, A, for spheres on a face-centered cubic lattice vs the step length distribution cutoff D at three volume fractions ~fcp50.7405!. The lines represent linear fits to the simulation data ~open circles, error bars within symbols!; the solid circles give the exact results obtained by the multipole method.
p ~ r! 5A 0 1
(k ~ A k r k 1B k r 2k ! cos~ k u ! ,
in 2D ~2.10a!
and
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H. Jo´hannesson and B. Halle: Solvent diffusion in ordered macrofluids
p ~ r! 5A 001
~ A lm r k 1B lm r 2k21 ! Y lm ~ u , w ! , ( l,m
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in 3D. ~2.10b!
Using the translational symmetry of the lattice and the boundary condition of vanishing particle flux across the obstruction surface, one obtains a system of linear equations that can be expressed in matrix notation as17,18 ~2.11!
~ 11M! B5C.
In the 2D case, the vector B has nonzero components B 2k11 (k50,1,2,...), while the only nonzero component of C is C 1 5Ha 2 , with H the applied macroscopic density gradient and a the cylinder radius. The elements of the matrix M are M kn 5S 2 ~ k1n21 ! a 2 ~ k2n !
SD
a 3 l
2 ~ k1n21 !
@ 2 ~ k1n ! 23 # ! , ~2.12! @ 2 ~ k1n ! 25 # ! ~ 2n22 ! !
where l is the lattice parameter and S n 5 ( m,m 8 8 (m 8 1 b m) 22n are the so-called lattice sums, with b 5i for the 2D square lattice and b 5(11i))/2 for the 2D hexagonal lattice. The obstruction factor involves only the dipole moment B 1 ,17 A5
S
D
B1 1 122 f . 12 f Ha 2
~2.13!
The matrix ~11M!21 can be developed in a power series as 12M1M22M31•••, leading to an expansion of B in powers of a/l. Using this procedure, we obtain Ha 2 /B 1 511 f 20.3058f 4 1••• ~ square lattice! ,
~2.14a!
Ha 2 /B 1 511 f 20.07542f 6 1••• ~ hexagonal lattice! . ~2.14b! Due to slow convergence near close-packing, a direct numerical matrix inversion of ~11M! is preferable at high volume fractions. This method was used for all multipole calculations presented here. The 3D case, which has been discussed in detail elsewhere,18 is completely analogous to the 2D case. III. CYLINDERS
FIG. 3. Obstruction factor, A' , vs volume fraction f of parallel cylindrical obstructions on a 2D square lattice. Shown are simulation results ~filled circles, error bars within symbols!, the exact result obtained by the multipole method ~solid curve!, the mean-field result ~3.1! ~dotted curve!, and the higher-order multipole formula ~3.2! ~dashed curve!.
by comparison with the ~exact! results from the multipole method, and ~ii! to assess the accuracy of the mean-field result. The mean-field result for circular obstructions in 2D or cylindrical obstructions in 3D, obtained directly or as the r→` limit of the EMA ~Ref. 25! or ECA ~Ref. 26! results for prolate spheroids, is A' 5
~3.1!
with f the area ~2D! or volume ~3D! fraction. This simple formula must clearly fail at high volume fractions, since A' 50 at close-packing ~where the cylinders touch!. Using the multipole method, Rayleigh obtained for cylinders on a square lattice14,17 A' 5
S
D
1 2f 12 , 12 f 11 f 20.3058f 4
~3.2!
which coincides with the mean-field result ~3.1! up to order f4. For cylinders on a hexagonal lattice, we obtain similarly A' 5
We consider first a 2D diffusion problem with circular obstructions arranged on a square or hexagonal lattice or on a disordered square lattice ~cf. Sec. II A!. For these lattice symmetries, the 2D diffusion tensor is isotropic, i.e., the obstruction factor A' is the same for any direction in the plane. The principal experimental connection here is to smallmolecule diffusion through liquid-crystalline phases built from long cylindrical amphiphilic aggregates21 or polymers, or through fibrous biomaterials.23,38–40 In these cases, A i 51 trivially and the solution to the 2D problem yields the obstruction factor A' for diffusion in the transverse plane. The results are also applicable to true 2D systems, such as the lateral diffusion of phospholipids in biological membranes with integral proteins acting as obstructions. The objective in this section is twofold; ~i! to validate the simulation method
1 , 11 f
S
D
1 2f 12 , 12 f 11 f 20.07542f 6
~3.3!
which coincides with the mean-field result ~3.1! up to order f6. Figure 3 shows the simulation and multipole results for a square lattice up to the close-packing volume fraction, fcp5p/4'0.7854. The excellent agreement with the multipole calculations demonstrates that our simulation procedure is highly accurate. It is also seen that the simple mean-field formula ~3.1! remains quantitatively accurate up to f'0.4, while Rayleigh’s higher-order result ~3.2! is useful up to f'0.7. The corresponding results for a hexagonal lattice are shown in Fig. 4. Again the simulation and multipole results agree quantitatively all the way up to close-packing at fcp5p/~2)!'0.9069. The mean-field result ~3.1! is clearly
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H. Jo´hannesson and B. Halle: Solvent diffusion in ordered macrofluids
FIG. 4. Obstruction factor, A' , vs volume fraction f of parallel cylindrical obstructions on a 2D hexagonal lattice. Shown are simulation results ~filled circles, error bars within symbols!, the exact result obtained by the multipole method ~solid curve!, the mean-field result ~3.1! ~dotted!, the higher-order multipole formula ~3.3! ~short dash!, the empirical formula ~3.4! ~dash–dot!, and the formula proposed in Ref. 38 ~long dash!.
a better approximation for a hexagonal lattice ~f,0.6! than for a square lattice ~f,0.4!. This is not unexpected; the mean-field approximation, which ignores spatial correlations, should be more accurate the higher the symmetry of the lattice. The relatively better performance of the mean-field approximation for the hexagonal lattice is also evident from the higher-order multipole approximation ~3.3!, which differs from the mean-field result only in order f7. The new multipole formula ~3.3! is seen to be quantitatively accurate up to f'0.8. It is remarkable that not even the simple mean-field formula ~3.1! is widely known. On the basis of a random-flight simulation, Celebre et al. recently proposed the following formula for the obstruction factor for cylinders on a hexagonal lattice21 1 1 A' 5 [12( f / f cp!1/2#1/21 @12f/fcp] 1/2. 2 2
~3.4!
As seen from Fig. 4, Eq. ~3.4! is greatly inferior to the simple mean-field formula ~3.1!; in fact, it does not even have the correct asymptotic form, 12f, at low volume fractions. Since the simulation results of Celebre et al. are accurately represented by Eq. ~3.4!,21 it follows that their simulation procedure is highly inaccurate. We note that simulations with a fixed step length that is not negligible compared to the cylinder radius, as used by these authors, tends to produce an artificial increase in the obstruction volume since the particle will then be reflected from the obstruction before coming into actual contact with its surface. Figure 4 also shows that the formula A' 5~12f/fcp!/~11f/fcp!, obtained by Rorschach et al. from an approximate solution of Laplace’s equation for cylinders on a hexagonal lattice,38 grossly overestimate the obstruction effect at intermediate volume fractions. Szafer et al.39 recently presented another approximate formula for the obstruction factor of parallel cylinders;
FIG. 5. Obstruction factor, A' , vs volume fraction f of parallel cylindrical obstructions on a disordered 2D square lattice. The simulation results for the disordered lattice ~filled circles, error bars within symbols! are compared with the exact multipole result for the ordered lattice ~solid curve, same as in Fig. 3!.
A' 51/(11 f 3/2). Neither of these formulas reduce to the correct asymptotic form, 12f, at low volume fractions. In Fig. 5 we compare the obstruction factors for cylinders on regular and disordered square lattices. As expected, disorder reduces the obstruction ~increases A' ! since the particle can now traverse the system via regions with a lower local volume fraction. ~The situation is analogous to a network of resistors of varying resistance.! It is intuitively clear that the effect of disorder must vanish at low f as well as at close-packing; hence, we observe a maximal effect at intermediate volume fractions. Although the mean-field treatment neglects spatial correlations, it does not become more accurate when the spatial order is destroyed ~as in a fluid!. In fact, the mean-field result ~3.1! now fails even at low volume fractions, although it might be improved by taking into account a polydispersity in the local volume fraction.26
IV. SPHERES
The obstruction factor for spheres on cubic lattices was recently investigated in detail18 and is included here mainly to validate the simulation procedure for a 3D case. The mean-field result for spherical obstructions is A5
1 . 11 f /2
~4.1!
This formula can be obtained as a special case of Maxwell’s classical EMA result for the effective conductivity14,16,25 and is also the ECA result.26 Using the multipole method, Rayleigh derived for spheres on a simple cubic lattice the approximate result7 A5
1 ~ 12 f !
S
12
D
3f , 21 f 20.3914f 10/3
~4.2!
which coincides with the mean-field result ~4.1! up to order f4.
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H. Jo´hannesson and B. Halle: Solvent diffusion in ordered macrofluids
FIG. 6. Obstruction factor, A, vs volume fraction f of spherical obstructions on a simple cubic lattice. Shown are simulation results ~filled circles, error bars within symbols!, the exact result obtained by the multipole method ~solid curve!, the mean-field result ~4.1! ~dotted curve!, and the higher-order multipole formula ~4.2! ~dash–dotted curve!.
Figure 6 compares the simulation result with the exact result obtained by the multipole method16 for spheres on a simple cubic ~SC! lattice up to the close-packing volume fraction, fcp5p/6'0.5236. The excellent agreement demonstrates that our simulation procedure yields reliable results also in 3D. The simple mean-field formula ~4.1! remains quantitatively accurate up to f'0.25, while Rayleigh’s higher-order result ~4.2! is useful up to slightly higher volume fractions. The corresponding results for a face-centered cubic ~fcc! lattice are shown in Fig. 7. Again the simulation and exact multipole results agree quantitatively all the way up to closepacking at fcp5&p/6'0.7405. As in the 2D case, the meanfield result is found to be a better approximation for the more symmetric fcc lattice. In contrast to the 2D case, the solvent medium remains macroscopically connected at close-packing so that A does not go to zero. At f5fcp simulation and exact
FIG. 7. Obstruction factor, A, vs volume fraction f of spherical obstructions on a face-centered cubic lattice. Shown are simulation results ~filled circles, error bars within symbols!, the exact result obtained by the multipole method ~solid curve!, and the mean-field result ~4.1! ~dotted curve!.
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FIG. 8. Cross sections of nonspherical obstructions. ~Top! Oblate and prolate spheroids with symmetry axis z and x, respectively. ~Bottom! Hemitoroidal disk and spherocylinder with symmetry axis z and x, respectively.
numerical results are 0.72160.005 and 0.722, respectively, for the SC lattice, and 0.61560.004 and 0.617 for the fcc lattice. We have also computed the obstruction effect for spheres on a disordered SC lattice. As in 2D, disorder reduces the obstruction ~increases A!, but in 3D the effects are barely significant ~data not shown!. V. NONSPHERICAL OBSTRUCTIONS IN 3D
Having verified that the random-flight simulation yields correct results for cylinders and spheres, we shall now use this method to calculate the obstruction factors for spheroids, disks, and spherocylinders. For oblate and prolate spheroids, only mean-field results have been obtained previously.25,26 In order to extract information about obstruction shape from experimental diffusion data, it is clearly essential to quantitatively establish the accuracy of these approximate results. To assess the extent of model dependence in such interpretations, we also study two related nonspherical shapes, for which no previous results have been reported. Figure 8 illustrates the geometry of the four types of obstructions considered. The simulated systems consist of orientationally ordered obstructions arranged on a 3D lattice obtained by uniformly expanding a fcc lattice in one ~prolates and spherocylinders! or two ~oblates and disks! dimensions. For the spheroids, the volume fraction at close-packing is the same as for spheres, fcp5&p/6'0.7405. Since the obstructions and the lattice are both uniaxial, it follows that the overall system is uniaxial and the diffusion tensor has two distinct components. The corresponding obstruction factors are denoted A i along the symmetry axis and A' in the transverse plane. We also examine the isotropically averaged obstruction factor,
^ A & 5Tr@ D/D 0 # /35 ~ A i 12A' ! /3,
~5.1!
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H. Jo´hannesson and B. Halle: Solvent diffusion in ordered macrofluids
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and the obstruction anisotropy,
a 5 ~ A i 2A' ! / ^ A & .
~5.2!
The averaged quantity ^ A & is the obstruction factor for a fcc lattice of isotropically disordered uniaxial obstructions, or for a polycrystalline sample of orientationally ordered fcc domains ~small compared to the relevant diffusion length scale!. This is not the same as the obstruction factor of an isotropic solution, which is both orientationally and positionally disordered. As seen in Secs. III and IV, positional disorder reduces the obstruction effect, although this reduction appears to be small in 3D. Using a variational principle, Hashin and Shtrikman established a rigorous upper bound for the obstruction factor for a ‘‘macroscopically homogeneous and isotropic’’ system,41 which in our notation reads 1 . ^A&< 11 f /2
~5.3!
This result holds for obstructions of any shape provided that they are isotropically disordered and distributed on a translationally invariant lattice. It will be noted that the right-hand side of Eq. ~5.3! is nothing but the mean-field result for spheres, Eq. ~4.1!. For spheres, therefore, the mean-field approximation provides an upper bound, as can be confirmed from Figs. 6 and 7. For positionally disordered systems, however, Eq. ~5.3! is not strictly valid. In an orientationally ordered fluid, such as a lyotropic nematic phase, the orientational order is never complete. To obtain the experimentally accessible obstruction anisotropy, the anisotropy a calculated here for complete orientational order ~with the symmetry axes of all obstructions parallel! should be multiplied by the second-rank orientational order parameter, S5~3^cos2 u&21!/2, with u the angle between the optic axis of the phase and the symmetry axis of the obstructing object.27,28 General bounds on the obstruction anisotropy are readily established. For oblates and disks, where D i
