J . Phys. Chem. 1994,98, 1002-1005
1002
Oil and Water Self-Diffusion in Bicontinuous Phases Michael W. Deem Department of Chemical Engineering, University of California, Berkeley, California 94720 Received: September 2, 1993; In Final Form: November 2, 1993"
W e analyze the long-time self-diffusion properties of a density field theory for oil, water, and surfactant microemulsions. Specifically, we study the reduction of oil and water self-diffusion coefficients due to the complicated geometrical arrangement of oil-rich and water-rich phases in a bicontinuous microemulsion. This reduction, the tortuosity factor, is calculated by a massively-parallel, quenched Monte Carlo calculation. The direct calculation shows diffusive behavior persists to roughly 0.01 s in this system. Analytical arguments suggest that such behavior persists a t longer times within the idealized quenched ensemble. The quenched ensemble becomes inappropriate a t long times, due to surfactant membrane fluctuations and dynamic oil-water equilibration, and these physical effects imply that diffusive behavior continues for times longer than those directly simulated.
1. Introduction We address the long-time diffusive properties of oil and water species in bicontinuous phases by a massively-parallel Monte Carlo simulation. The geometrical structure of the bicontinuous phases that often occur in oil, water, and surfactant mixtures must be known in order to define oil and water self-diffusion. A great deal of experimental information is available about such bicontinuous phases. In particular, small-angle neutron scattering (SANS) experiments provide information about the equilibrium geometries of these systems.lq2 Freeze-fracture transmission electron microscopy (TEM)3 as well as direct imaging TEM4 lead to real space images of microemulsion systems rapidly quenched to ultralow temperatures. N M R self-diffusion experiments provide numerical values of the reduced self-diffusion coefficients observed in the complicated oil and water geometries of these system^.^ A microscopically-motivated density field theory of oil, water, and surfactant microemulsions has recently been developed and a n a l y ~ e d . 6This ~ field theory predicts self-assembly, as evidenced by the variety of structures seen in Monte Carlo simulation^.^ This field theory is also in accord with several sets of measured scattering curves, within an optimized Gaussian treatment.* Furthermore, a more refined, optimized transformed Gaussian treatment shows the model supports distinct interfaces in bicontinuous phasese9 In particular, the model agrees with the bulk-contrast SANS experiments of ref 2. The model predicts the system to be microphase-separated and bicontinuous, in agreement with NMR, conduction, and freeze-fracture experiments. The surfactant system of ref 2 is thus an archetypical one in which to study oil and water self-diffusion within the context of a well-defined theoretical model. This system seems to be microphase-separated into a bicontinuous network of large, interconnected oil-rich and water-rich regions. These regions have a sharply defined boundary so that diffusion within an oilrich or water-rich region is a well-defined concept. Furthermore, membrane relaxation kinetics occurs on second time scales,lO so membrane fluctuations can be ignored for shorter times, and the oil-rich and water-rich regions can be considered to be static. While numerical analysis of the diffusion equation in various proposed models of bicontinuous phases has been performed,lI apparently there has been no analysis of a microscopicallymotivated theory fit to experimentally-measured data. There has recently been a great deal of interest in anomalous diffusion induced by random media. That is, in some systems,
* Abstract published
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( r * ( t ) ) tzv, where 0 Iv 5 I/*. The reviews of Havlin and Ben Avraham12 and Isichenkol3 indicate cases where diffusion on fractal media generate such subdiffusive scaling laws. The review of Bouchaud and Georges discusses many cases where particularly strong or highly correlated disorder induces subdiffusion in Euclidean media.14 Given the expected tortuous passageways in Euclidean space through which oil and water particles must diffuse in bicontinuous microemulsions, subdiffusion might, perhaps, be expected to occur in these systems. We show by a massivelyparallel Monte Carlo calculation that regular diffusion persists to long times in one model of bicontinuous microemulsions. Both analytical and physical arguments imply that diffusion persists to longer times than those directly simulated. We specialize an oil, water, and surfactant density field model to just one field, the difference between the local water and oil densities being 6p(r) = pw(r) - po(r). We then identify 6p(r) = po(1 - 2c[9(r)]), where po is the total fluid density. We choose the transfer function c ( x ) as a Heaviside function and specify +(r) as a Gaussian field variable. These identifications correspond to an extreme form of Gaussian clipping in the language of ref 9, which shows that a field theory model of this system indeed possesses a highly bimodal site density. These identifications also correspond to the random-wave algorithm that has often been assumed to generate representative bicontinuous microemulsions configuration^.^^ With the statistics of the 4(r) field specified, we generate typical members of the field theory ensemble, in which Monte Carlo diffusion experiments are performed. In section 2 we derive the computational approach to the calculation of oil and water self-diffusion constants. Section 3 presents the results of the calculations. In section 4 we discuss what happens at times longer than we are able to simulate, both within our idealized model and within the real microphase separated system. We conclude in section 5. 2. Simulation Methodology
The oil, water, and surfactant density field theory is fit to experimental SANS data in refs 8 and 9. In particular, for the bicontinuous phase scattering data of ref 2, the expected highly bimodal site density distribution results. Thus the oil-water density difference field
in Advance ACS Abstracts, December 15, 1993.
0022-3654/94/2098- 1002$04.50/0
0 1994 American Chemical Society
The Journal of Physical Chemistry, Vol. 98, No. 3, 1994 1003
Self-Diffusion in Bicontinuous Phases 2.0 I
I
0.0 0.00
0.05 k
0.10
(A")
Figure 1. Correlation functions x(k) for the fields 6(r) (thick line) and bp(r) (thin line). The normalization is arbitrary. The density4ensity correlation function is essentially the scattering curve measured by SANS.
an oil-water density configuration occurs with the probability
where Vis the system volume and &(k) is the spectrum of the Bhost field that generates the experimental scattering curve xpp(k).Here, the square grid is N X N X Nand V = P P , where N is the grid size and 1 is the lattice constant. The summation in eq 3 is over the unique Fourier components associated wch a finite, cubic volume in real space. Equation 3 implies each 4(k) in half of Fourier space is independently Gaussian, with the other half given by a(-k) = $*(k) since #(r) is real. Equation 2 relates the oil-water density difference to the ghost field configuration at each site. Oil-water density difference fields are generated with the correct probabilities on a three-dimensional square lattice with periodic boundary conditions by assigning Gaussian random values to the fields &k), inverting by a fast-Fourier transform to get d(r), and clipping to get 6p(r).* Once an oil-water density difference field is generated, selfdiffusion coefficients can be determined by Monte Carlo calculation.~3That is, a random walker is started at a site, whether oil or water does not matter, as the model is perfectly symmetric. At each time step, the random walker randomly tries to move in one of the six nearest-neighbor directions. If the density at the attempted site is the same as at the initial site, the move is accepted, otherwise the move is rejected. Time is incremented after each move, whether accepted or failed. Our walker is, thus "blind" in the standard t e r m i n ~ l o g y .The ~ ~ self-diffusion coefficient is defined by the relation (r2(t)) = 6Dt
t (MCS)
Figure 2. Mean square displacement for oil and water self-diffusion in a bicontinuous phase. Data from 2563,5123, and 10243grids of spacing I = 30 A are plotted. Note that these are Monte Carlo data, and the line fit to these data is not plotted. The dense sets of simulation data appear as a line on this scale, and the three sets overlap.
102 103 104 105 108 107
t
(MCS)
Figure 3. Mean square displacement for oil and water self-diffusion in a bicontinuous phase. Data from 2563,5123, and 10243grids of spacing I = 60 A are plotted. Note that these are Monte Carlo data, and the line fit to these data is not plotted. The dense sets of simulation data appear as a line on this scale, and the three sets overlap.
(4) The averages implied by eq 4 are averages over initial sites of the random walker as well as averages over the quenched oil-water configurations, with probability expressed by eq 3. The Monte Carlo calculations should not be continued beyond where (r2(t)) = L2, whereL = lNis theboxlengthofthequenched configuration. This restriction is necessary since the random walker is exploring a nonrandom medium beyond this distance when periodic boundary conditions are used. With this constraint, we find it sufficient to average over initial conditions of the random walker in eq 4 without an average over quenched field configurations. That is, our clipped-Gaussian model for the configurations, eqs 2 and 3, does not produce glassy behavior, and quenched averages correspond to annealed averages. The memory consumed by the lattices generated by eq 3 grows as N3. This limits the size of lattices that can be generated in a computer. In fact, for N = 1024,4 Gbytes (222bytes) of RAM is required to generate the final oil-water density difference configuration. The final configuration, however, is generated by clipping the $(r) lattice (a word to bit transformation), and it occupies only 128 Mbytes. The 8 Gbytes of RAM available on a Thinking Machine Corporation CM-200 (a massively-parallel computer with 64K parallel processors) is used to generate the oil-water density differenceconfigurations. While the CPU time required to generate these lattices is not large, the CM-200 is roughly 200 times as fast as an IBM RS/600 Model 350 at this task for small clusters. The Monte Carlo calculations on the quenched lattices are performed on workstations, however.
can be identified as 3. Monte Carlo Results
(2) @(r) = POU - 2c[d(r)l) where $(r) is a Gaussian field variable. A highly bimodal site density distribution implies the transfer function c ( x ) is roughly approximated by a Heaviside function. Given this is the case, the statistics of d(r) can be defined so that the experimental SANS data are reproduced.* Figure 1 presents the correlation functions of both d(r) and 6p(r). Since the field d(r) is Gaussian,
Monte Carlo diffusion simulations are performed on lattices of size N = 256,5 12, and 1024. Additionally, for each size lattice, simulations are performed for 1 = 30 and 60 A. The reciprocal lattice in eq 3 is such that wave vectors in the range 0
