Computer simulation studies of diffusion in gels: Model structures

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1 Computer simulation studies of diffusion in gels: Model structures Paulo A. Netz a) and Thomas Dorfmüller Fakultät für Chemie, Physikalische Chemie I, Universität Bielefeld, Postfach 1131, 3351 Bielefeld, Germany Received 3 June 1997; accepted 28 August 1997 We have investigated particle diffusion through different obstacle geometries by computer simulations. The model structures used in this work randomly placed point obstacles and cage-like structures were chosen with the aim of represent a broad range of geometrical structures similar to gels and in order to be compared with our previous simulations of particle diffusion through polyacrylamide gels. The diffusion behavior was studied as a function of tracer size and obstacle concentration. The isomorphism between the diffusion of finite-sized tracers and the diffusion of point tracers in the presence of expanded obstacles was applied. Only hard-sphere interactions of the tracer with the immobile obstacles were considered and the theoretical description was made in terms of theory of the obstruction effect. In the case of randomly placed point obstacles an analytical expression for the dependence of the diffusion coefficient on tracer radius and obstacle concentration, applying the model of spherical cells, could be deduced. The same description was applied numerically to the other model systems. Up to moderatly high fractions of excluded volume this description was found to be successful. For very high fractions of excluded volume higher concentrations or larger tracers the validity of Fick s second equation for describing diffusion breaks down and anomalous diffusion was found. The anomalous diffusion exponent diverges as the tracer size becomes comparable to the size of the pores. Analysis of the trajectory of tracers in the cases where an anomalous diffusion takes place shows a Levy-flight-like characteristic American Institute of Physics. S X I. INTRODUCTION Diffusion is a process that can be seen, in a macroscopic description, as the time development of a concentration distribution: For example, an initially sharp peaked concentration profile becomes smoother as time progresses. 1 From a microscopic point-of-view we can, however, focus our attention on the erratic motion of individual particles. In this approach, more akin to the methods used in molecular simulations, diffusion is described by the time evolution of the mean square displacement. Assuming conservation of mass, the validity of Fick s second law and isotropy, diffusion can be characterized by a diffusion coefficient that can be related to the mean square displacement t r2 t2dd, where d is the space dimensionality. This equation can also be described as a simple scaling relation: the mean square displacement is a linear function of time r 2 tt. Equation 1, that describes the normal or Euclidean diffusion, is the fundamental link between the microscopic and the macroscopic descriptions of diffusion. Using this relation we can study the diffusion in a system by calculating the mean square displacement of a tracer along a simulated trajectory. a Also at: Universidade Luterana do Brasil - ULBRA 9242 Canoas, Brazil. 1 2 The diffusion of solutes in gels is determined by the interaction of diffusing solute molecules with the solvent and with the polymer molecules. Although the nature of this interaction is specific i.e., the chemical nature of the gel plays a significant role it can be approximated in terms of general effects such as geometrical obstructions, hydrodynamic drag, alteration of the interspacing between solvent molecules and effects of the polymer involvement. 2 The relevant factors are therefore concentration of the polymer gel, the solute size, the geometrical arrangement of the polymer chain and also some not often considered parameters as shape, thickness and stiffness of the chains. 3,4 The theoretical description of these factors can be carried out by means of two approaches: geometrical considerations and hydrodynamics. 5 The influence of hydrodynamics is found to be secondary in many cases, so that the main contributions to the influence of the gel on diffusion come from steric hindrance. 4 This steric hindrance is usually modeled in terms of an obstruction effect: 2,6,7 The obstructions lead to a lengthening of the diffusion path. The obstruction can, in some cases, be estimated using the information about the geometry of the polymer, as in the phenomenological approach considering the spaces in a random suspension of fibres 6 which was tested against experiments 2,3,7 and simulations. 4,8 Recently it was observed that for diffusion processes in some media Fick s second law does not hold. In such cases the mean square displacement is not proportional to time. More precisely the exponent describing the time dependence of the mean square displacement is different from one J. Chem. Phys. 17 (21), 1 December /97/17(21)/9221/13/$ American Institute of Physics 9221

2 9222 P. A. Netz and T. Dorfmüller: Diffusion in gels r 2 t 2/d w. Here, d w is the anomalous diffusion exponent. Thus in the case of a hindered diffusion the exponent is fractional and we have d w 2. Several mechanisms can lead to anomalous diffusion: a broad distribution of jump times, a broad distribution of jump lengths or strong correlations in diffusive motion. 9 Obstruction produces strong correlations, because the local isotropy of space breaks down. We will show in this paper that a time-scale-dependent broad distribution of jump lengths a distribution of square displacements for fixed time interval can be detected depending on the observation time. The study of anomalous diffusion has become more important in the last years. Anomalous diffusion was experimentally detected in the diffusion of large latex spheres in polyacrylamide gels. 1 NMR studies on diffusion of water 11 have detected anomalous diffusion for water molecules, in form of a time-dependent diffusion coefficient, in the case of polyacrylamide gels in the neighborhood of a volume phase transition. In this case the results were interpreted in terms of diffusion through permeable barriers, although the calculated pore dimensions using this model were quite large. Anomalous diffusion of water molecules has been detected in some cancerous as well as normal glandular and fibrous tissues 12 and was interpreted as occurring when the water molecules experience obstruction by a hierarchy of structures in a broad range of length scales. Beside the experimental studies, and indeed earlier, the study of anomalous diffusion by computer simulation proved to be an important means to study these phenomena. 9,13 15 In these studies simplified models were used, that can, however, take the essential features of diffusion into account, especially for weakly interacting molecules. The studied systems range from percolation clusters and other fractal structures 13 to geometrical models of membranes 9,15 and gels. 14 Evidence of anomalous diffusion in somewhat more complex systems has also been found: Bizzarri and Cannistraro 16 showed with canonical-ensemble molecular dynamics simulations that the mean square displacement of protein hydration water at low hydration levels and in the proximity of the protein surface does not increase linearly with time. Although the computational and even the experimental studies of anomalous diffusion do not explain quantitatively the full diffusion behavior, they suggest that some caution must be taken when we use experimental or calculated values of diffusion coefficients. As already pointed out 17 the diffusion of solutes in colloidal systems is very difficult to study because the distribution of colloidal particles is neither completely random nor ordered which leads to a complicated diffusion path. In a previous paper 14 we have studied the anomalous diffusion in polyacrylamide gels using Monte Carlo simulation of tracer diffusion through immobile obstacles that represent realistically the main geometrical properties of this gel. The geometrical characterization of these structures is difficult and had to be made using concepts such as nearest-neighbor volume and pore distribution. In the present study we use model 3 structures that reproduce some of the main features of gels and polymer solutions. The model structures here used are sets of randomly distributed point obstacles and cage-like structures. Investigations about self-diffusion in hard-sphere suspensions can be found in the literature. 18,19 Here we consider, however, only the case of immobile point obstacles with one mobile tracer. The obstacles constitute obstructions that hinder the diffusion of tracer particles only in a geometrical way: In our simulations we consider only repulsive interactions between tracer and polymer. These interactions are modeled within the frame of the obstruction effect. Surely this is an oversimplification, but especially for disordered systems, where the role of hydrodynamics is less important 4 we expect that this approach captures the essentials of the diffusion process. The role of the inhomogeneities is of central interest in our study. On the one hand, the simulated model structures of randomly distributed points although disordered, have a rather homogeneous characteristic. On the other hand the cage-like structures are heterogeneous and more ordered. It is expected to find the structures of real gels as a broad spectrum between these two extremes. In the following section we will discuss the methods used. Simulation results are then presented for structures of randomly distributed points and cage-like structures and compared with previous simulations using more realistic gel structures. 14 Both studies take only into account the effect of immobile obstacles. In order to find an analytical expression for the reduced diffusion coefficients the cell model 4,17 is applied to the random suspension of point obstacles. Anomalous diffusion is studied for the model structures by analysing the deviations of the mean square displacement from the linear behavior with time. It is expected that in the case that the obstructions cause a strong disturbance of the diffusion path the behavior would become anomalous, as found in the simulations with the gel structures. 14 II. DESCRIPTION OF THE METHODS The motion of the diffusing tracer is simulated by means of a Monte Carlo program. The diffusing tracer interacts with the obstacles represented by hard core potentials. The results are thus interpreted in the context of the obstruction theory. The model obstacles here studied are sets of randomly placed points and cage-like structures. Obstacle coordinates and tracer trajectory are defined in a simulation box with periodic boundary conditions. The tracer carries out an off-lattice random walk and its position is recorded periodically. The moves are made by adding a vector with length of 1 Å and random orientation. The trial moves are tested using the distances between the particle in the new position and the coordinates of the obstacles. One time step is added and the new position is accepted if the minimum distance between tracer and obstacles is larger than R the radius of the tracer otherwise the old configuration is retained. The mean square displacement

3 P. A. Netz and T. Dorfmüller: Diffusion in gels 9223 r 2 t 1 t max t maxt rt trt of each tracer was calculated over trajectories with 21 7 steps in the case of random points the simulated trajectory and for the cage-like structures. If the Einstein relation is valid the diffusion coefficient can be calculated as the limiting slope of the the mean square displacement for long times D 1 6 lim d dt r2 t. 5 t In our simulations we calculate a relative diffusion coefficient (D*), where the diffusion without obstacles (D ) is taken as reference. This relative diffusion coefficient was studied as a function of size of the probe and obstacle concentration D*R,c DR,c 1 d D D dt r2 t,r,c. 6 The geometry of the free space was characterized by the nearest-neighbor volume V 2,21 n using a method discussed elsewhere. 14,22 The nearest-neighbor volume fraction normalized nearest-neighbor volume, divided by the total volume V T ) can be identified with the void nearest-neighbor probability density H v (r) of Torquato et al. 23,24,21 and represents the probability that a randomly chosen point lies in a volume region defined in such a way, that this region constitutes a layer between the distances r and rdr around the obstacles. The nearest-neighbor volume is sampled by the direct Monte Carlo method. We consider here the isomorphism between the diffusion of a tracer particle of radius R around obstacles with radius a and the diffusion of a point particle around the obstacles with radius Ra at the same number density of obstacles. 21 The excluded volume for a tracer with radius R is the integral of the neighbor volume 2 V exc R R Vn rdr. The free volume is the complement of the excluded volume V free RV T V exc R. 8 The definitions of the volume fractions are exc R V excr V T 7 and free R V freer V T. 9 According to this isomorphism we can expect that the influence of both obstacle concentration and tracer size can be expressed by a single parameter: The excluded volume fraction exc, assuming that the obstacles have a characteristic length R*, with R*R*(c) and exc exc (R,R*). The combined influence of both tracer size and concentration can also be expressed by the excluded and free volume fractions and thus D*(R,c)D*( exc ). In order to compare our results with theoretical approaches the cell model 4,25,17 was applied. The relative diffusion coefficient is considered as the result of local flows in different microscopic subsystems n(v c ) cells with volume V c or, equivalently, f (b) cells with radius b D 1 V nvc DV c dv c 1 V fbdbdb. 1 With appropriate assumptions regarding the geometry of the cells as well as the distribution of space between the obstacles, it is possible to calculate the distribution of cell sizes and, therefore, the effective diffusion coefficient for the whole system. For the case of randomly placed point obstacles, an analytical expression for D can be derived. Taking spherical cells with radius b centered in the obstacles as the elementary microscopic subsystems and assuming a frequency distribution in terms of radius equal to f (b) the diffusion coefficient can be expressed as D D Db f b db. 11 D Assuming a distribution of spaces in a random suspension of points similar to that deduced by Ogston, 6 we can calculate f (b) and an analytical expression for the diffusion coefficient as a function of radius of the tracer and obstacle concentration where D D e 2 e E 1 2, E 1 x x e u u du 12 is the exponential integral function. This is exactly the same expression for D/D as deduced by Johansson et al. 4 but with a different definition for, taken into account the spherical symmetry of cells in our case 4 3 R 3, 13 where N P /L 3 is the density of points. As we shall see in section III, this parameter is also related to the excluded volume fraction. Details about the deduction can be found in the Appendix. For the other model systems the reduced diffusion coefficient was estimated by numerical solution of equation 11 using a similar method. When the influence of obstacles is large the mean square displacement turns out to be not a linear function of time equation 3 and using equation 5 to calculate diffusion coefficients makes no longer sense. Instead of this the anomalous diffusion exponent d w is the property to be calculated as a function of concentration and tracer size d w d w (R,c). This is best done using a logarithmic plot of the mean square displacement versus time.

4 9224 P. A. Netz and T. Dorfmüller: Diffusion in gels TABLE I. Model structures using randomly distributed points as obstacles. A chosen number of points (N P ) is randomly placed in a simulation box with L8 Å with a minimum-distance restriction. The concentration units are explained in the text. Equivalent Minimum Number of points concentration Concentration distance Structure N p g/ml points/å 3 ] Å A B C D E F G H FIG. 1. Neighbor volume fraction (V n (r)/v T - solid line and its integral excluded volume fraction - dashed line as function of tracer radius for the structure A of randomly placed point obstacles. III. RESULTS AND DISCUSSION A. Random points The first model systems studied are sets of randomly placed points, acting as immobile obstacles and representing a disordered, perfectly random and homogeneous system. A number of points is placed in a simulation box with box length 8 Å with the restriction that the distance between points cannot be smaller than a fixed parameter. The diffusion is followed by the calculation of the mean square displacement of a probe particle. The influence of the obstacles on the diffusion was studied by varying the concentration of obstacles and the radius of the diffusing particle. In Table I are listed the number of points, the equivalent concentration, the concentration in points/å 3 and the minimum-distance parameter for each model structure. This last parameter seems to have no influence on the results. The equivalent concentration is defined, in order to compare with our previous simulations of polyacrylamide structures, 14 as the concentration that the set of random points would have, if each random point were an united atom. We have derived an expression for free space in a random suspension of point obstacles, using a method similar to that used by Ogston 6 for a random suspension of fibres. It can be shown that, for N P /L 3 randomly placed points per unit volume the probability distribution of empty spaces with radius r is simply dpr4r 2 e 4/3 r3 dr 14 and, therefore, the fraction of the total volume of the suspension which can accommodate a particle with radius R is equal to the probability that the distance r between a randomly chosen spherical particle and the obstacles is greater than the radius of the particle, with P rr R dpre 4/3 R This volume fraction is simply the free volume fraction free. For the calculation of the nearest-neighbor volume for the structures of randomly placed point obstacles we have taken a set of 1 6 sampling points, accumulated in a histogram with 32 channels according to the method described in the methods section. Figure 1 shows the nearest-neighbor volume and its integral, the excluded volume fraction, for structure A of Table I. The nearest-neighbor volume can be well described by equation 14. According to the analogy between diffusion of finite-sized tracers and diffusion of point tracers in the presence of expanded obstacles, this volume fraction is the same as that is excluded to a tracer of corresponding radius. 21 With help of the density of points we can calculate the mean distance between the points, i.e., is the theoretical characteristic length of this system x L 1/3 N 1/3. 16 P This characteristic length can also be interpreted as a kind of volume averaged mean pore radius. Defining a reduced radius zr/r*, 17 the nearest-neighbor volume curves for all the structures can be plotted as a master curve exc 1e 4/3 z3. 18 This result is indeed an equation identical to equation 15. The real characteristic length R* is calculated from the best fit to an equation of type 18. The small deviations between R* and x, calculated from 16 can be traced back to finite size effects. Because the nature of the distribution of points is statistical and disordered, the structure can be characterized analytically. It is important to note that this excluded volume fraction is a function of tracer size and obstacles concentration. The parameters obtained in the analysis of the nearest-neighbor volume such as the maximum of the nearest-neighbor volume curve, the characteristic length R* and the mean distance x are listed in Table II for the model structures of randomly placed points. The diffusion behavior was studied as a function of tracer size and obstacle concentration. The slopes of the

5 P. A. Netz and T. Dorfmüller: Diffusion in gels 9225 TABLE II. Characterisation of the structures of the model system of randomly distributed point obstacles. Maximum in the neighbor-volume R* x Structure Å Å Å A B C D E F G H curves of the mean square displacement against time are proportional to the diffusion coefficient. The dependence of the relative diffusion coefficient on the concentration of obstacles is shown in Figure 2 for tracers with radius 3.5 and 5.5 Å. These curves show a good qualitative agreement with experimental determinations of the dependence of the diffusion coefficient on the concentration 2 and with simulation results in two-dimensional systems. 26 It is important to note here that the curves have a positive curvature: i.e., the variation of the concentration at low concentration has a stronger influence on the diffusion coefficient than a variation in the higher-concentration region. The calculated dependence of the diffusion coefficient on the radius of the particle is a smooth curve shown in Figure 3 and has the same negative curvature as observed for experimental results on the dependence of the diffusion coefficient on tracer size. 3 We note that for small particles only a slight decrease of the diffusion coefficient can be detected. A particle size of about 6. Å is necessary in order that the diffusion coefficient becomes half of the diffusion coefficient of the unperturbed case. Results for several systems of randomly placed point obstacles can be plotted on a master curve as shown in Figure 4 as a function of the reduced FIG. 3. Dependence of the relative diffusion coefficient on tracer radius for the structure A of randomly placed point obstacles. The upper abscissa is scaled in terms of reduced radius see the text for details. radius, defined in equation 17. The statistical fluctuations for medium-sized tracers are very small. Long range diffusion can be detected for tracers with reduced radius z smaller than.9. Figure 4 also shows the analytical predictions for hardsphere diffusion from the cell model for a random suspension of point obstacles and assuming a distribution of free spaces similar to that deduced by Ogston 6 and confirmed in the analysis of the nearest-neighbor volume see above. The analytical expression equation 12 is a very good description of the diffusion behavior and is indeed a universal curve, in the sense that the diffusion coefficients for several concentrations and tracers sizes can be plotted as a master curve for almost the whole concentration and tracer size region. Because of the simple and unique dependence of the free volume on the reduced radius of the tracer we can plot the reduced diffusion coefficients as a function of the excluded volume fraction, as shown in Figure 5. A very remarkable point is that for these three dimensional systems it is possible to have a diffusion coefficient different from zero even for systems where the fraction of free volume is only 5%, because in these structures the free space has a percolating structure. This result can be compared with the simulations in two-dimensional systems of point tracers, 26,27 where it has FIG. 2. Concentration dependence of the relative diffusion coefficient for tracers with radius 3.5 Å solid line and 5.5 Å dashed line in the systems of randomly placed point obstacles. The diffusion coefficients are mean values over several structures with the same concentration. The concentration is shown in units of points/å 3 and g/ml. FIG. 4. Dependence of the relative diffusion coefficient on the reduced radius. The curve obtained by equation 12 is shown for comparison.

6 9226 P. A. Netz and T. Dorfmüller: Diffusion in gels FIG. 5. Dependence of the relative diffusion coefficient on the excluded volume fraction for the systems of randomly placed points. The theoretical predictions according to Frickes approach light dashed line, a phenomenological prediction applying the concept of tortuosity solid line as well as the predictions of equation 12 expressed as a function of exc are compared with simulated data. been found that diffusion cannot be detected if the free surface fraction is less than 6% square lattice or 5% triangular lattice. In Figure 5 the results are compared with the approach of Fricke 2,28 that relates the decrease in the diffusion coefficient to the volume fraction of obstacles D* /2 and with a phenomenological approach applying the concept of tortuosity 2,7 2 1 D*. 2 1/2 A transformation of equation 12 as a function of the excluded volume fraction is also shown in Figure 5. We see in the figure that the former two approaches works only for very small volume fractions and cannot describe the diffusion of large tracers but the last approach describes correctly the trend over the entire region. The predictions assuming a hard-core interaction are indeed an upper estimate of D: In real systems the decrease is expected to be more pronounced and we must consider other effects that cannot be reduced to the pure geometrical obstruction. But within the scope of this work we can only state that even the obstruction effect is not satisfactorily reproduced by Fricke s approach in the region of large volume fraction of obstacles. Our results are qualitatively in good agreement with the results of Kim and Torquato 21 and with BD simulation by Johansson and Löfroth 8 but the behavior in the region of low volume fraction is somewhat different. For very high volume fractions only equation 12 still shows a qualitatively good agreement but the other approaches are unsatisfactory. A very important point is that even the linear dependence between mean square displacement and time a necessary condition to define a diffusion coefficient does not hold for systems where the excluded volume fraction is large. FIG. 6. Diffusion behavior of tracers in the structure A of randomly placed point obstacles. It is shown from top to bottom the mean square displacement for tracers with radii.5, 1.5, 3.5, 5.5, 7.5, 8.5, 9. and 9.5 Å as a function of time in a double logarithmic plot a and in a transformated way b see the text for details. In b the dashed lines are linear fits to the intermediate time region. Figure 6a shows the logarithmic plot of the mean square displacement for the same system analyzed before structure A. For tracers with size 7.5 Å or larger the dependence of the mean square displacement on time is not linear. We see three different regions of diffusion behavior: in the limit of short time intervals the diffusion is normal. For intermediate times we have anomalous diffusion and for some tracers we can see a transition to normal diffusion in the limit of large times. A similar behavior was found in both experimental 1 and simulated systems. 9,14 The relative extension of these regions depends on the tracer size. A plot of ln(r 2 )/t against ln(t) Figure 6b shows these regions in a very clear way. With help of this transformation 9,14 it is easier to recognize the regions and to calculate the inclination of the curves and thus the anomalous diffusion exponent d w The anomalous diffusion exponents for the simulations of tracer diffusion through randomly distributed points show a universal dependence. Using the reduced radius we can plot all the calculated d w collapsed in a master curve, as shown in Figure 7. When the radius of the tracer approaches the mean

7 P. A. Netz and T. Dorfmüller: Diffusion in gels 9227 FIG. 7. Universal plot of the dependence of the anomalous diffusion exponent on reduced tracer size for diffusion in randomly placed point obstacles. Curves described by equations 21 heavy dashed line and 22 light dashed line are shown for comparison. distance between the points the radius of trapped tracer the anomalous diffusion exponent diverges. The anomalous diffusion becomes remarkable for tracers with reduced radius greater than.8. In Figure 7 are shown, as a guide to the eye, the curve z d w as well as the function 1 d w z 23 that was found to be a good description for anomalous diffusion in realistic polyacrylamide structures 14 but that in this case is evidently not adequate. In a similar way it is possible to find a master curve that relates the anomalous diffusion exponents to the excluded volume fraction curve not shown. Although these systems have, strictly speaking, no fractal properties, we can find anomalous diffusion. Moreover, we see that the transition between anomalous diffusion and normal or trapped diffusion is continuous. FIG. 8. Cage-like structures ordered point obstacles: a isolated cubes, b merged cubes. The nearest-neighbor volume for these structures is sampled similarly as in the case of the systems of randomly placed points and shown in Figures 9a and b. In these figures is also shown the integral of the nearest-neighbor B. Cagelike structures Cage-like structures as shown in Figures 8a and b are probably a better approximation to realistic gel structures than randomly placed point obstacles because the network character of gels is better reproduced. The ordered geometry is, however, the main unrealistic feature in this model. Moreover, these structures represent a case of high heterogeneity. In the model system of isolated cubes the 188 point obstacles are geometrically distributed forming the edges of a cube with length 16 Å, that is placed in the center of a simulation box with box length 32 Å. This corresponds, in terms of concentration, to points Å 3. In the model system of merged cubes the 386 points constitute a system of periodic linked cubes ( points Å 3 ). In Figures 8a and b the periodic structure of both systems is made clear. FIG. 9. Neighbor volume fraction (V n (r)/v T - heavy solid line and its integral excluded volume fraction - heavy dashed line as function of tracer radius for the cage-like structures of a isolated cubes, b merged cubes. It is also shown a fit to an equation of type 17 as a best fit to data light dashed line and as a fit to the corresponding system of randomly placed point obstacles with the same number of points.

8 9228 P. A. Netz and T. Dorfmüller: Diffusion in gels FIG. 1. Dependence of the relative diffusion coefficient on the excluded volume fraction a and c and on on tracer radius b and d for the cage-like structures of isolated cubes a and b and merged cubes c and d. It is shown in the Figures relating the diffusion to the excluded volume fraction the Fricke s approach, the simplest obstruction effect. Is is also shown in b and d the numerically estimated reduced diffusion coefficient using the cell model with fibres with radius. solid lines or 1. dashed lines. volume, i.e., the dependence of the excluded volume fraction on the tracer size for both structures. We show also the curves of the excluded volume fraction for randomly placed points with the same number of points and the best fit of the excluded volume fraction to equation 18. This best fit yields R*1.3 for isolated cubes and R*8.2 for merged cubes. The x for the corresponding number of points are 5.6 for isolated cubes and 4.4 for merged cubes. The excluded volume fraction is larger in the case of randomly distributed point obstacles. The difference can be regarded as a measure of the heterogeneity of the obstacle structure. In these simulation series the diffusion has also been studied as a function of tracer size. In the simulations of tracer diffusion in the system of isolated cubes we start the trajectory in the center of the cube. The calculated relative diffusion coefficients shown in Figure 1a as a function of excluded volume fraction and in Figure 1b as a function of tracer radius can be well described by Frickes approach up to an excluded volume fraction of about 3% Figure 1a. A numerical application of the cell model, obtained by numerical solution of equation 11 see the Appendix yields a better estimate to the diffusion coefficient. In this case the natural geometry of the cells is the cylindrical one. Starting with the excluded volume fraction which is known numerically as a function of tracer size we can calculate f (b) and therefore D/D. This is shown in Figure 1b. The geometry of the system of merged cubes is very similar to the system of isolated cubes, but the diffusion behavior is very different. The obstruction effect in this case is more pronounced than in the case of isolated cubes. The diffusion coefficients that were calculated from the slope of the curves of mean square displacement against time are shown as a function of excluded volume and tracer radius in Figures 1c and d, respectively. Here we see that the decrease of the diffusion coefficient is larger than in the case of the isolated cubes, but because of the large deviations from the linear dependence of the mean square displacement on time see above we must be careful on relating a value of the diffusion coefficient to the slope of the mean square displacement. The description using a numerical solution applying the cylindrical cell model, calculated in a similar way as for the isolated cubes, seems to be satisfactory for moderately sized, but not satisfactory for very large tracers. In the system of isolated cubes the diffusion of small tracers is only slightly disturbed, as one can see in Figure 11a that shows the log-log plot of the mean square dis-

9 P. A. Netz and T. Dorfmüller: Diffusion in gels 9229 FIG. 11. Double logarithmic plot of the mean-square displacement for tracers in the structures of isolated cubes a and merged cubes b. It is shown, from top to bottom, the curves for tracers with radii 4., 7., 7.5, 8., 8.25, 8.5 and 9. Å a and 2., 4., 6., 7., 7.5, 7.7, 7.8 and 8. Å b. placement of tracers as a function of time. We see the very sharp transition from normal diffusion to trapped behavior. There is no evidence for anomalous diffusion. There could be anomalous diffusion in the case of isolated cubes only if the tracers larger than the half of the box length could start in points evenly distributed over the entire configurational space. Contrasting to that we can see, for diffusion in the system of merged cubes, Figure 11b that the linear dependence between the mean square displacement and the time does not hold for large tracers. Indeed the diffusion is strongly anomalous, although the region of the anomalous diffusion is very small. Similar to the case of random points we can see the three diffusion regimes: normal, anomalous and long-time normal diffusion; especially the transition from the second to the third regime is clear. The anomalous diffusion exponent in this case grows with larger tracer radius and diverges near the radius of the trapped tracer. The intermediate regime extends over less than a decade. In this case we also see that a clearly non-fractal structure can lead to an anomalous diffusion. C. Comparison with realistic gel structures We have also simulated diffusion of hard-sphere tracers through polyacrylamide networks modelled with atomistic detail about the network formation and tracer diffusion in FIG. 12. Dependence of the relative diffusion coefficient on the excluded volume fraction a and tracer radius b for a polyacrylamide structure. It is also shown, for comparision the curves for randomly distributed point obstacles heavy dashed line, isolated cubes light dashed line and merged cubes dotted line in a as well as the numerical solution applying the cell model with fibres with radius. solid line,.5 heavy dashed line and 3. light dashed line. these structures see Refs. 14 and 22. These model structures were built on a diamond lattice, in a simulation box with periodic boundary conditions, by means of a random distribution of knots and further interconnection constructing polyacrylamide bridges between randomly chosen pairs of knots. In the case of polyacrylamide structures acting as obstacles the influence on the decrease of the diffusion coefficient, shown in Figure 12a and b, is similar to that of random points and both are more effective than the isolated cubes and comparable with the merged cubes. In the polyacrylamide structures, however, diffusion is found even for an excluded volume fraction close to 1%. Although the realistic gel structures exhibit cage-like features, due to the large distribution of pore sizes and mainly the presence of very large pores it is possible that large portions of the space become almost insensitive to the obstacles and the obstruction has effect only for very large tracers. The application of the numerical calculation using the cylindrical cell model to the polyacrylamide structures yield a good description of the diffusion coefficient, especially if we consider an effective tickness of the polymer fibres, as shown in Figure 12b. In the numerical solution of equation 11 we use the nearest-neighbor volume as structural information and the cylidrical cells with radius b are calculated centered in

10 923 P. A. Netz and T. Dorfmüller: Diffusion in gels the obstacles. The artificially high reduced diffusion coefficient calculated for small tracers using expanded obstacles is explained in terms of a negative frequency function of cells f (b) with small b in this system. An important point that emerges from our results is the comparison of the effectiveness of obstacles in acting as a barrier to the diffusion of tracers. This topic has already been studied in two-dimensional systems by Saxton 29 who found, considering the same area fraction of obstacles, that clustercluster aggregates are more effective barriers to diffusion than point obstacles. The latter are on the other hand more effective barriers than the same area fraction of compact obstacles. This higher effectiveness of the fractal obstacles for hindering the diffusion of point tracers compared with the FIG. 13. Small sample trajectory 125 sample points for tracer diffusion in our model systems. It is shown the curves for random points using tracers with radius 7. a and 8.5 Å b, for a polyacrylamide structure using tracers with radius 7. c and 3 Å d, for merged cubes using tracers with radius 7. e and 7.5 Å f and isolated cubes using tracers with radius 7. g and 7.5 Å h.

11 P. A. Netz and T. Dorfmüller: Diffusion in gels 9231 FIG. 13. Continued. same area fraction of other obstacles is in sharp contrast with the higher effectiveness of point obstacles to act as a barrier for large tracers, 26 also found in our simulations. A key quantity in the effectiveness is the homogeneity of the obstacles: more homogeneously distributed obstacles have a dominant effect for larger tracers. Here, it is important to point out that for very large tracers or excluded volume fractions the diffusion in this gel systems is found to be sometimes strongly anomalous and therefore we must be careful in identifying the inclination of the curves of mean-square displacement with a reduced diffusion coefficient. Computer simulations of tracer diffusion in twodimensional systems 9 have shown that, in presence of moderate concentration of obstacles, the diffusion is anomalous over short distances and normal over long distances, similar as we have found, in the case of large tracers, for randomly placed point obstacles and also for the cage-like system of merged cubes and for the polyacrylamide model networks. The anomalous diffusion is related to the typical trajectory of the tracers, as we see in Figures 13a h. In the case of isolated cubes the trajectory resembles the typical Brownian motion. For larger tracers diffusion takes place predominantly between the cubes: It is entropically improbable that the tracer reenters the cubes. In the cases where the anomalous diffusion is pronounced for merged cubes, randomly distributed point obstacles and gel networks the trajectory has a typical pattern that resembles a Levy-flight: 3 tight clusters linked by somewhat large jumps. For these systems the tracer is so to speak forced to carry out an induced Levyflight. This has as consequence that in this case the diffusion of a large tracer becomes strongly anomalous. This is particularly clear in the case of merged cubes, where the tracer cannot escape from the cubes geometry. IV. CONCLUSIONS We have investigated tracer diffusion through sets of obstacles with different geometry. The model obstacles were chosen in order to represent a broad range of geometrical structures present in gels and to be compared with our previous simulations of tracer diffusion in polyacrylamide gels. The randomly distributed point obstacles represent the borderline case of a homogeneous but randomly disordered system, while the cage-like structures represent a very ordered and heterogeneous structure. In the simulation of tracer diffusion through randomly distributed point obstacles in three dimensions we have investigated both the dependence of the diffusion on tracer size and obstacle concentration. We obtained results that are qualitatively similar to those of diffusion in two dimensions, but in our case we see that the free space has a percolating structure and permits a long-range diffusion even when its fraction is as small as 5%. The effect of concentration and tracer size can be expressed in terms of the excluded and free volume fractions. The reduced diffusion coefficient is as a simple function a master curve of the excluded volume fraction or, equivalently, of the reduced tracer radius. The diffusion was described in the frame of the obstruction effect, using the cell model 4,17 and the analytical expression for the distribution of spaces in the suspension. 6 The equation describing the effect of the randomly placed point obstacles on the diffusion coefficient is indeed the same as found by Johansson et al. 4,8 for the case of diffusion in a system of randomly oriented fibres, but the parameter describing the universal nature of the dependence of diffusion coefficient on tracer size and obstacle concentration is different, according to the different underlying geometry. For the cagelike structures and gel networks the application of the same procedure was not possible because of the lack of an analytical expression for the distribution of spaces. Nevertheless, a numerical procedure was applied and it is found that this description is indeed better than the usual approches related to the obstruction effect. We found that knowing the distribution of the obstacles in the space we are able to make a very good prediction of the diffusion behavior for a broad range of concentrations and probe sizes. The reverse can also be true: maybe the knowledge of the diffusion behavior for several tracers could yield information about the structure of the system. For large excluded volume fractions the presence of ob-

12 9232 P. A. Netz and T. Dorfmüller: Diffusion in gels stacles produces a strong anomalous diffusion behavior in almost all systems. Anomalous diffusion is detected for the model structures in the case that the obstructions cause a strong disturbance of the surroundings. The disturbance that leads to anomalous diffusion is not a simple function of the concentration, but the topology of the structure must be taken into account. The anomalous diffusion is pronounced if the tracer has a radius larger than 8% of the mean pore radius. For larger tracers we can divide the diffusion regime the time evolution of the mean square displacement into three regions: a normal diffusion for short times, an anomalous diffusion regime and a transition to normal diffusion for larger times provided that the tracer does not become trapped. The anomalous diffusion exponents were calculated and show a continuous dependence on the relative tracer size, with divergence when the tracer size approaches the size of the trapped tracer this divergence is thus related to the transition in the free volume from a percolating structure to a non-percolating one. Analysing the tracer trajectories we could conclude that the anomalous diffusion is related to a kind of forced Levy-flight. ACKNOWLEDGMENTS We gratefully acknowledge the Deutscher Akademischer Austauschdienst, the Fonds der Chemischen Industrie and the Graduiertenkolleg Strukturbildungsprozesse for financial support and Dr. Michael Buchner for his very valuable suggestions. APPENDIX: APPLICATION OF THE CELL MODEL TO ESTIMATE THE DIFFUSION COEFFICIENT IN THE MODEL STRUCTURES In this section we show a deduction for equation 12, similar to that of Johansson et al., 4 but adapted for spherical symmetry. According to Jönsson et al. 17 the effective diffusion coefficient in a colloidal system can be expressed as a function of the local flows in microscopic subsystems of different volume V c the cells. These cells are centered at the obstacles and are defined in such a way that V nvc dv c. A1 The effective diffusion coefficient is a volume average of the diffusion coefficients in the cells D 1 V nvc DV c dv c. A2 A key quantity in the cell model is the probability distribution of spaces of size R, that corresponds to the nearestneighbor volume distribution 23,24 and is related to the free volume fraction See Section III/ A. This quantity must be either known analytically or must be determined both for the geometry of a cell and for the whole system. For a spherical cell with radius b we have g sc R,b 3R2 SRb, A3 3 b where S(x) is a step function Sx 1 x x. A4 Considering pure obstruction effects and considering, for the spherical cells, a reduction of the diffusion coefficient in the cell analogous to the case of cylindrical cells 4,25,17 we have, for the reduced coefficient of a tracer with radius R in a cell with radius b Db 1 D 1R/b SRb. 3 A5 Equations A1 and A2 can be expressed also as a function of the cell radius b, considering a frequency distribution of cells with radius b equal to f (b). We have, for the global probability distribution of spaces of size R gr f bg sc R,bdb A6 and for the effective diffusion coefficient D D Db f b db. A7 D As we have seen in section III A the distribution of free space in the system of randomly distributed points follows the analytical expression similar to that deduced by Ogston 6 gr4r 2 e 4/3 R3. A8 The insertion of equation A3 and A8 into equation A6 leads to an integral equation that can be solved for f (b). If we call the average number of obstacles displaced by the tracer, with we have 4 3 R 3, R e R 2 f b db, A9 R 3 b whose solution is f b3 2 R 6 b5 3 b e /R 3. A1 Inserting the expresssion for the local diffusion coefficient in one cell equation A5 in the equation of the effective global diffusion coefficient and using the result for f (b) we have D D R 3 2 b 5 R 6 3 b /R 3 1 e 1R/b db. 3 A11

13 P. A. Netz and T. Dorfmüller: Diffusion in gels 9233 Solving this integral by using the variable Kb 3 R 3 and changing the corresponding limits in the integral we obtain an expression for D/D D e 2 e E D 1 2, where E 1 x x e u A12 u du is the exponential integral function. This is the same equation as deduced by Johansson et al. 4 but with a different definition of. is related to the geometry of the cell. It is not surprising that the form of the analytical expression is the same. In the case of the cylindrical cell model, with cell radius b applied to describe diffusion of a tracer with radius R through randomly oriented fibres with volume fraction and radius a, it is possible to define a function x,y x2 y 2 and it is easy to show that and g cc R,b dra,a dra 1 b,a, A13 A14 Db 1 D 1 Ra,a, A15 b,a gr dera,a. A16 dra For the case of spherical symmetry we have neglected the dimension of the obstacles they reduce to points we have x,y 4 x3 3 y 3 and similar relations for the equivalent quantities g sc R,b dr,r* dr 1 b,r*, A17 A18 and Db D 1 1 R,R* b,r*, A19 gr der,r*. A2 dr For the other model structures it was impossible to obtain an analytical expression for g(r) and therefore equation A6 had to be solved numerically to obtain f (b), which has been inserted into equation A7, also numerically solved. 1 R. Ghez, A Primer of Diffusion Problems Wiley, New York, A. H. Muhr and J. M. V. Blanshard, Polymer 23, L. Johansson, U. Skantze, and J.-E. Löfroth, Macromolecules 24, L. Johansson, C. Elvingson, and J.-E. Löfroth, Macromolecules 24, R. I. Cukier, Macromolecules 17, A. G. Ogston, Trans. Faraday Soc. 54, A. G. Ogston, B. N. Preston, J. D. Wells, and J. McK. Snowden, Proc. R. Soc. London, Ser. A 333, L. Johansson and J.-E. Löfroth, J. Chem. Phys. 98, M. J. Saxton, Biophys. J. 66, Y. Suzuki and I. Nishio, Phys. Rev. B 45, L. Pavesi and A. Rigamonti, Phys. Rev. E 51, M. Köpf, C. Corinth, O. Haferkamp, and T. F. Nonnenmacher, Biophys. J. 7, S. Havlin and D. Ben-Avraham, Adv. Phys. 36, P. A. Netz and Th. Dorfmüller, J. Chem. Phys. 13, M. J. Saxton, Biophys. J. 7, A. R. Bizzarri and S. Cannistraro, Phys. Rev. E 53, R B. Jönsson, H. Wennerström, P. G. Nilsson, and P. Linse, Colloid Polym. Sci. 264, S. Hanna, W. Hess, and R. Klein, Physica A 111, B. Cichocki and K. Hinsen, Physica A 166, P. Pfeifer and M. Obert, The Fractal Approach to Heterogeneous Chemistry Wiley, New York, 1989, p I. C. Kim and S. Torquato, J. Chem. Phys. 96, P. A. Netz, Ph.D. thesis, Universitaet Bielefeld, Bielefeld, S. Torquato and S. B. Lee, Physica A 167, S. Torquato, B. Lu, and J. Rubinstein, Phys. Rev. A 41, L. G. Nilsson, L. Nordenskioeld, P. Stilbs, and W. H. Braunlin, J. Phys. Chem. 89, M. J. Saxton, Biophys. J. 64, M. J. Saxton, Biophys. J. 52, H. Fricke, Phys. Rev. 24, M. J. Saxton, Biophys. J. 61, L. T. Fan, D. Neogi, and M. Yashima, Elementary Introduction to Spatial and Temporal Fractals Springer, Berlin, 1991.