Self-diffusion of non-interacting hard spheres in particle gels

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Self-diffusion of non-interacting hard spheres in particle gels

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2011 J. Phys.: Condens. Matter 23 234115

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IOP PUBLISHING JOURNAL OFPHYSICS:CONDENSEDMATTER

J. Phys.: Condens. Matter 23 (2011) 234115 (10pp) doi:10.1088/0953-8984/23/23/234115

Self-diffusion of non-interacting hard spheres in particle gels

Jean-Christophe Gimel and Taco Nicolai

L’UNAM Université, Laboratoire Polymères, Colloides et Interfaces, UMR CNRS 6120 - Université du Maine, av. O. Messioen, 72085 Le Mans cedex 9, France E-mail:Jean-Christophe.Gimel@univ-lemans.frandTaco.Nicolai@univ-lemans.fr Received 15 October 2010, in final form 25 January 2011

Published 25 May 2011

Online atstacks.iop.org/JPhysCM/23/234115 Abstract

Different kinds of particle gels were simulated using a process of random aggregation of hard spheres. The mean square displacement of Brownian spherical tracer particles through these rigid gels was monitored and the average diffusion coefficient, normalized with the free diffusion coefficient (D), was obtained. For each gel structure the effect of the gel volume fraction (φ) and size ratio of the tracer (d) on the relative diffusion coefficient was investigated systematically. The volume fraction that is accessible to the tracers (φa) was determined in each case. D was found to be approximately the same ifφawas the same, independent ofφ, d and the gel structure. However a different behaviour is found if the tracers can penetrate the strands of the gel. A state diagram of d versusφis given that shows the critical values (dc,φc) at which all tracers become trapped. Different values are found for different gel structures. The

dependence of D onφ/φcis independent of d, while the dependence of D on d/dcis independent ofφ.

(Some figures in this article are in colour only in the electronic version)

1. Introduction

Self-diffusion of particles in gels is part of the wider research area of transport in complex media [1]. One category of gels are formed by random aggregation of particles such as latex spheres [2], clay [3] and proteins [4]. The structure of these particle gels has been studied in some detail and may be described in terms of self-similarity characterized by fractal dimensions [5].

Two limiting cases may be distinguished: diffusion limited cluster aggregation (DLCA) where each collision leads to bond formation; and reaction limited cluster aggregation (RLCA) where the probability to form a bond during a collision is very small [6,7]. Furthermore, one may distinguish systems that form rigid bonds from systems that form so-called slippery bonds. In the latter case bound particles can slide along the surface until they are locked in place by other bound particles, leading to locally denser structures [8].

The diffusion of tracers in particle gels has, as far as we know, not yet been studied in much detail. Elsewhere, we reported on numerical simulations of the diffusion of trace amounts of hard spheres with different diameter in gels formed

by DLCA and RLCA of hard spheres with rigid bonds [9].

The results may be compared with those obtained in systems consisting of randomly distributed hard spheres or overlapping spheres that have been studied before [10–20]. In each case the spherical obstacles were immobile and the only interaction between the obstacles and the tracer spheres was excluded volume. A remarkable finding of our previous study [9]

was that the self-diffusion coefficient normalized by the free diffusion coefficient (D) was to a large extent determined by the accessible volume fraction to the tracers (φa) for all systems studied. The idea that tracer diffusion through fixed obstacles is determined byφa was earlier suggested by others [16,21]

for different structures.

The focus of much of the earlier work was on the behaviour close to the critical volume fraction of the matrix where all tracers are trapped (φc). φc depends on the configuration of the obstacles and the size of the tracers, but the accessible volume fraction at φc was approximately 3%

in all cases. Very close to the critical point the diffusion of the tracers becomes anomalous and can be well described by using the percolation model for the structure of the accessible volume [9,12,18–20].

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Figure 1. Images of gels formed atφ=0.05 by (a) DLCA, (b) RLCA, (c) SDLCA and (d) FHS.

Anomalous tracer diffusion through immobile obstacles is interesting from a fundamental point of view and can be compared with anomalous diffusion that is observed close to the liquid–glass transition. The glass transition of dense suspensions of spheres has been widely studied in the past [22].

A more recent trend is to investigate the effect of immobilizing a fraction of the spheres. Tracer diffusion in particle gels can be considered as an extreme limit in which the fraction of freely moving spheres goes to zero [23–26]. However, the critical behaviour can only be observed over a very small range of volume fractions or tracer sizes. Here we report on a detailed investigation of Brownian tracer diffusion in particle gels over a wide range of volume fractions and tracer sizes. Contrary, to the model systems discussed by others, these particle gels correspond to real materials, see for examples [27,28].

The dependence of φa on the gel volume fraction (φ) is close for gels formed by DLCA and RLCA with rigid bonds and consequently the diffusion coefficients are similar.

For both gels φa at a given φ is only slightly larger than for randomly placed frozen hard spheres (FHS), which may be considered as the reference system [9]. The objective of the present study was to extend our investigation to a gel with a different dependence of φa on φ. Therefore we also simulated particle gels formed by DLCA with slippery bonds (SDLCA) [8] that are locally more dense and therefore have a significantly largerφa at a givenφ. We also made a more extensive and systematic investigation of the probe size dependence. This has allowed us to identify a peculiar effect of the probe size on the diffusion in SDLCA gels that can be explained in terms of diffusing through and between the matrix strands.

2. Simulation method

Gels were simulated by irreversible cluster–cluster aggregation (DLCA, RLCA and SDLCA) of hard spheres with diameter dg. This length scale is considered as the unit length and subsequently all distances are expressed relative to dg. Starting from a random distribution of hard spheres, simulations were performed in cubic boxes of size 50 with periodic boundary conditions until all particles were connected; see [29, 30]

and [8] for details. Images of the three types of gel atφ=0.05 are shown in figure1. The structure of the DLCA and RLCA gels is quite similar, but SDLCA gels have thicker strands and larger pores. For each system,φa was calculated as the probability that a tracer of size d could be randomly inserted in the box without overlapping the gel particles. It was calculated using 10⁷independent insertion trials. As a reference system we also simulated randomly distributed frozen hard spheres (FHS).

Once the gels were created, tracer diffusion was simulated by randomly inserting a hard sphere with diameter d in the accessible volume and moving it with small displacements s in random directions. If the displacement led to overlap, the movement was truncated at contact. The simulation time n was incremented at each trial of motion. The characteristic time, t0, is defined as the time needed for a free tracer (without the presence of the obstacles) to diffuse a squared distance equal to d². Using the above definitions, the mean square displacement (MSD),R², of a free tracer is given for n sufficiently large and s sufficiently small by:

R² =d² t

t0

=ns² (1) where t is the physical time. The MSD was calculated using

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J. Phys.: Condens. Matter 23 (2011) 234115 J-C Gimel and T Nicolai

Figure 2. (a) MSD of tracers with d=0.95 in SDLCA gels withφ=0.35 for various step sizes s increasing from top to bottom. The dotted straight line represents the MSD of a free tracer with same size and is given by equation (1). (b) The same data as (a) but using a

representation that shows both asymptotic behaviours (at very small and very large MSD).

periodic boundary conditions and several thousands (10³–10⁴) of independent paths of randomly inserted tracers. We checked for finite size effects by varying the box size, and all results shown here are not influenced by finite size effects.

In the presence of the obstacles, equation (1) remains valid at very short times before any collision occurs with the gel. When the tracers start to feel the obstacles the MSD is slowed down, but as long as the accessible volume percolates a diffusive process is recovered at long times:

R² =Dd² t

t0

=Dns²(t → ∞). (2) A direct consequence of equations (1) and (2) is that the simulation time needed to reach a given physical time is inversely proportional to the square of s. This means that in practice the duration over which the MSD can be probed in the simulations is limited by the value of s.

Figure 2(a) shows the MSD of 10³ independent tracers in a SDLCA gel withφ = 0.35, d = 0.95 and values of s between 0.005 and 0.3. The volume accessible to the tracers is around 5.6% in this case. For the smallest step size, the long time diffusive regime was not reached even after running the simulations for more than a month of CPU time. This can be seen more clearly in figure 2(b) where R²/(ns²)is plotted as a function of R². The second plateau, which gives D directly, is obtained in a reasonable simulation time only for larger values of s. Using 10³tracers leads to a relative error less than 10% in the determination of D (see figure2(b)). Figure3 shows that the asymptotic behaviour of the MSD at s → 0 can be obtained by a linear extrapolation, but this procedure is highly time consuming. In figure2we clearly see that for R²larger than 20, varying s only increases the physical time needed to reach a given R² without modifying its relative time dependence. Figure 4 shows that results obtained at s = 0.3 for R² larger than 20 can be superimposed onto those obtained at s = 0.005 over a more limited range of R² by simple vertical shifts. The superposition leads to a master curve that is very close to the result obtained with the

Figure 3. MSD for tracers with size d=0.95 in SDLCA gels with φ=0.35 as a function of s for various ns²as indicated in the figure.

Straight lines represent linear fits to the data.

extrapolation method. In this way the correct value of D can be obtained within 10% without the need for prohibitively long simulations.

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Figure 4. Evolution ofR²/(ns²)as a function ofR²for tracers with d=0.95 in SDLCA gels withφ=0.35. The blue dotted line represents results obtained at s=0.3 but shifted up by a factor 2.05 in order to give a good superimposition with data obtain at s=0.005 forR²larger than 20.

Thus in practice, for small MSD, the step size was chosen sufficiently small so that reducing it further had no effect on the results. For large MSD, larger step sizes were chosen and corrected by superposition with the results at smaller MSD obtained with a small step size.

3. Results

Figure5(a) shows D of tracers with d =1 as a function ofφ for DLCA, RLCA and SDLCA gels, while figure5(b) shows the dependence on d forφ=0.35. The results on SDLCA gels complement those shown in Babu et al [9] for DLCA, RLCA and randomly placed frozen hard spheres (FHS). D decreases with increasingφor d and becomes zero at the critical values

φcor dc. The dependence of D onφand d is close for DLCA and RLCA, but it is significantly different for SDLCA.

For DLCA gels, the dependence of D on φ was determined for three probe sizes, see figure 6(a), and the dependence on the probe size was determined at three volume fractions, see figure6(b). As might be expected, the decrease of D with increasing φis stronger for larger tracers and the decrease with increasing d is stronger at largerφ.

In figure7the results shown in figures5and6are plotted as a function ofφa. It is clear that the dependence of D on the structure, the volume fraction and the tracer size is to a large extent determined byφa. This was already reported for a more limited set of results in [9], where it was shown that tracer diffusion in FHS and frozen fully penetrable spheres also had the same dependence onφa. However, the spread of the data is larger than the uncertainty of the simulations, indicating that the correlation between D and φa is far from perfect.

A phenomenological equation was proposed to describe the relationship between D andφawithφ^ca the critical accessible volume fraction:

D= φa

φa−φa^c

φa(1−φa^c) _μ

. (3)

We have found that the critical value ofφais independent of d andφ: φa^c = 0.035 for SDLCA and φ^ca = 0.027 for DLCA. If the data are plotted as a function ofε=(φa−φa^c)/φ^ca, we observe the same behaviour close to the critical point for the two types of gel, see figure8.

φa can be determined far more quickly and more accurately than D and therefore the dependence of φa on d andφcould be studied systematically. Figure9(a) showsφa

as a function ofφfor DLCA gels for a range of tracer sizes.

In the limit of d → 0, φa = (1 −φ), while for larger probes φa decreases more quickly with increasing φ, due to excluded volume interactions. The data can be superimposed using scaling factors (ad), see figure9(b), which implies that increasing d is equivalent to increasing φ over the whole range of volume fractions and tracer sizes that were simulated.

Figure 5. (a) Dependence of D onφfor tracers with d=1 in gels formed by SDLCA, RLCA and DLCA as indicated in the figure.

(b) Dependence of D on the tracer size forφ=0.35 in gels formed by RLCA, SDLCA, DLCA as indicated in the figure. The dashed lines are guides to the eyes.

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Figure 6. Dependence of D in gels formed by DLCA: (a) onφfor tracers with different size d indicated in the figure, (b) on the tracer size for differentφindicated in the figure. The dashed lines are guides to the eyes.

Figure 7. (a) Dependence of D on the accessible volume fraction for different tracers, volume fractions and gel structures as indicated in the figure. Open symbols indicate results obtained from theφ-dependence at fixed values of d and filled symbols indicate results obtained from the d-dependence at fixed values ofφ. The solid lines represent equation (3) withφ^c_a=0.035 for SDLCA (red or grey in the printed version) andφa^c=0.027 for DLCA (black). The dashed lines indicate D=Dmφa²^/³for DLCA (black) and SDLCA (red or grey in the printed version) with d=1.0, see text. (b) The same data as in (a) in a double-logarithmic representation.

One may define an effective volume fraction as φe = adφ. φa decreases linearly with increasing φe up to φe ≈ 0.4, but the decrease is weaker at higher volume fractions when the excluded volume of different obstacles starts to overlap significantly.

The scaling factors ad are plotted as a function of d in figure 10. The increase of φe with the tracer size can be understood by considering the volume excluded by the obstacles for the centre of mass of the tracers. The volume excluded by a single spherical obstacle is proportional to (1+d)³. Thus for well separated spherical obstacles: ad = (1 + d)³. In the case that the obstacles are long strands of spheres: ad = (1 +3d +1.5d²). The latter situation applies approximately to DLCA gels for which most of the spheres have two neighbours [29]. The dashed line in figure10 represents ad =(1+3d+1.5d²)and describes the dependence

of the scaling factors on d very well for DLCA gels at least up to d=0.5.

RLCA gels gave results that are very close to DLCA gels and were not studied in more detail. For SDLCA gels the data could also be scaled. The master curve is almost the same as that of DLCA gels, see figure9(b), but the dependence of ad

onφis weaker, see figure 10. This can be explained by the formation of thicker strands by SDLCA with more overlapping exclude volume than the thin strands of DLCA gels. For FHS the dependence ofφaonφobtained with different probe sizes could not be scaled, see figure11. We cannot offer an explanation for the good scaling in the case of gels and the absence of scaling in the case of FHS.

The dependence ofφa on d is shown in figure12(a) for DLCA gels at differentφ. Initially,φadecreases linearly with increasing d. The results can be scaled using scaling factors 5

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Figure 8. Same data as in figure7(b) but plotted as a function of ε=(φa−φa^c)/φ^ca. The solid lines represent equation (3) with φa^c=0.035 for SDLCA (red) andφa^c=0.027 for DLCA (black).

a_φifφais normalized by(1−φ)so that it is unity at d =0, see figure12(b). One may define an effective tracer diameter:

de=a_φd. A master curve could also be obtained for SDLCA gels, but its shape was different, see figure12(b). The scaling factors (a_φ) are plotted in figure13. Again the results obtained with FHS could not be scaled, see figure14. In addition, the range over which φa decreases linearly with increasing d is very small for FHS.

φ^cais the value ofφaat which D becomes zero, which is about 0.03 depending weakly on the gel structure. The critical volume fractions (φc) and probe sizes (d_c) are interpolated from data shown in the graphs and are plotted in figure 15. For a few systems the critical values were obtained directly from the dependence of D on φor d and are indicated as stars in

Figure 10. Dependence on the tracer size of the scaling factors used to obtain the master curves shown in figure9(b) for DLCA (circles) and SDLCA (triangles) gels. The dashed line represents

a_d =(1+3d+1.5d²)and the solid line represents ad =(1+d)³.

figure15. The values obtained directly from measurements of D are in excellent agreement with those deduced fromφa. dc

decreases with increasing volume fraction and reaches about 0.3 at random close packing (φ = 0.64). A higher volume fraction is needed to trap particles in SDLCA gels than in DLCA gels, but the difference reduces for smaller tracers.

In figure 16(a) D is plotted as a function of φ/φc for different tracer sizes. A universal behaviour is observed, which is a consequence of the fact thatφa is a universal function ofφ/φc, see figure9. D is not a universal function of d/dc, becauseφadepends on(1−φ)and D decreases weakly with increasing φ at d = 0, see figure 6(b). However, if D is

Figure 9. (a) Dependence of the accessible volume fraction on the volume fraction for DLCA gels at different values of d between 0.1 and 1.0 in steps of 0.1. The dashed line representsφa=1−φ. (b) The same data as in (a) plotted as a function of adφ(circles). The master curve obtained for SDLCA is also shown (triangles). The solid line representsφa=1−a_dφ. The inset of (b) shows the same data in a

semi-logarithmic representation. The two horizontal dashed lines in the inset representφa=0.027 and 0.035, which are the critical accessible volume fractions for, respectively, DLCA and SDLCA.

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Figure 11. (a) Dependence of the accessible volume fraction on the volume fraction for FHS matrices at different values of d between 0.1 and 1.0 in steps of 0.1 (φa=1−φat d=0). Solid lines represent the analytical expression derived by Torquato [1]. (b) The same data as in (a) plotted as a function of adφ.

Figure 12. (a) Dependence of the accessible volume fraction on the tracer size for DLCA gels at different values ofφbetween 0.05 and 0.45 in steps of 0.05 (φa=1−φat d=0). (b) The same data as in (a) normalized by 1−φand plotted as a function of a_φd (spheres). The master curve obtained for SDLCA is also shown (triangles). The solid line representsφa=1−a_φd. The inset shows the same data in a

semi-logarithmic representation.

normalized by its value at d = 0 it is an universal function of d/dcindependent of φ. This is shown in figure 16(b) for DLCA gels at different volume fractions. The dependence of D/Dd=0 on d/dc for SDLCA gels was tested at only one volume fraction, but the same behaviour is expected at other volume fractions. A different dependence of D/Dd=0on d/dc

is found for SDLCA and DLCA gels, which is expected since the dependence ofφa/(1−φ)on d/dcwas also different, see figure12. The origin of this difference will be discussed below.

4. Discussion

We have simulated the diffusion of spherical tracers in rigid gels that were formed by random aggregation of hard spheres.

No interactions other than excluded volume were taken into

account, contrary to what was done recently by Jardat et al [31]

who studied self-diffusion coefficients of ions in the presence of charged obstacles. The present results extend those reported by Babu et al [9] and show that D depends on the volume fraction of the obstacles, the size of the tracers and the structure of the gel, but that it is in all cases determined to a large extent by the volume that is accessible to the probes. This is evident for randomly placed point-like obstacles [16] and fully penetrable spheres [15] because in both cases the system at a given φa is equivalent to diffusion of point tracers in a unique structure, determined only by the number density of the obstacles. Mittal et al [21] proposed for the special case that the tracers and the obstacles have the same size (d = 1) that D was approximately equal to D_m. φa²^/³, where D_m is the self-diffusion coefficient when all the matrix particles are mobile hard spheres. Sung and Yethiraj [20] made simulations 7

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Figure 13. Dependence on the volume fraction of the scaling factors used to obtain the master curves shown in figure12(b) for DLCA (circles) and SDLCA (triangles) gels.

of tracer diffusion through frozen spherical obstacles with different arrangements, and found that the simple relationship proposed by Mittal et al worked well at low volume fractions, but not close toφc. In agreement with Sung and Yethiraj [20], we found that this relationship does not work well, see figure7.

Here we have shown that the correlation betweenφaand D is not restricted to tracers with the same size as the obstacles.

The problem of relating the dynamic property D toφ, d and the gel structure therefore reduces to that of relating the structural propertyφatoφ, d and the gel structure. For point tracersφa =1−φand does not depend on the structure. We have found that the effect of increasing the probe size is to increase the effective volume fraction of the obstacles. For well separated spherical obstacles with unit size: φe=(1+d)³φ, so thatφa = 1−(1+d)³φ, but this relationship fails when

Figure 15. State diagram indicating the critical volume fractions and tracer diameters where all the probes become trapped, i.e. where φa=φa^c, for DLCA gels (circles) and SDLCA gels (squares). Open symbols represent critical values ofφand d obtained from the dependence ofφaonφ, while the crossed symbols represent values obtained from the dependence ofφaon d. The black stars represent values where D became 0 obtained from simulations of the MSD as a function ofφand d.

the excluded volume of the obstacles starts to interpenetrate.

For randomly distributed frozen hard spheres. Torquato has proposed an analytical solution forφa, see [1] chapter 5, which is in excellent agreement with our simulation results; see solid lines in figures11(a) and14(a).

Unfortunately, for gels formed by random aggregation of spherical particles no analytical solution forφa exists. Here we investigated the dependence ofφaon the volume fraction and the tracer size in detail for gels formed by DLCA and SDLCA that are representative for colloidal gels. A third type

Figure 14. (a) Dependence of the accessible volume fraction on the tracer size for FHS matrices at different values ofφbetween 0.05 and 0.45 in steps of 0.05 (φa=1−φat d=0). Solid lines represent the analytical expression derived by Torquato [1]. (b) The same data as in (a) normalized by 1−φand plotted as a function of a_φd.

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Figure 16. (a) Dependence of D onφ/φcfor DLCA (open symbols) and SDLCA (filled symbols) gels for tracer sizes indicated in the figure.

(b) Dependence on d/dcof D normalized by the value at d=0 for DLCA (open symbols) and SDLCA (filled symbols) gels at volume fractions indicated in the figure. The dashed lines are guides to the eye.

of gel formed by RLCA gave results that were close to those for DLCA gels. The dependence ofφaonφfor different tracer sizes could be scaled to form a master curve. For DLCA gels φawas close to that of linear strings of particles forφa >0.3 and the scaling factors were consistent with this structure for d < 0.5. SDLCA gels have a denser local structure and therefore the scaling factors had a weaker dependence on d.

Master curves could also be obtained for the dependence ofφaon d ifφawas normalized by(1−φ), which is the value ofφaat d=0. The scaling factors depended more strongly on φfor SDLCA gels than for DLCA gels. The superposition for the dependence ofφaboth onφand on d was very good down to the critical value ofφa, which is remarkable considering that the results obtained for FHS could not be scaled.

A consequence of the close correlation between D and φa is that not only is the dependence of φa onφ universal, but so is D. The correlation between D and φa is not perfect and therefore the universality is only approximate, but it nevertheless yields a good estimate of the dependence of D on φ and on d. This means that for the types of gels investigated here the diffusion coefficient of spherical particles can be estimated at all concentrations after measurements at only one or two concentrations. If the present simulation results are relevant for real systems one should be able to construct master curves of D versusφfor different tracer sizes, which is relatively easy to verify.

Contrary to the dependence of D onφ, the dependence of D/Dd=0on d is not the same for DLCA and SDLCA gels.

For SDLCA gels D/Dd=0 decreased initially steeply until a value of d ≈ 0.16. This can be explained by the fact that the thicker strands of the SDCA gels contain pores with a smallest diameter of around 0.16, see figure17. Only tracers with d < 0.16 can penetrate the strands of the SDLCA gels.

Clearly this initial decrease depends on the absolute value of d and cannot be a universal function of d/dc. However, the behaviour at larger d/dcis still expected to be universal. In the

Figure 17. Example of a 5% SDLCA gel with red tracers of size d=0.2.

case of SDLCA gels the effect of diffusion through the strands is relatively weak. A much stronger effect can be expected for heterogeneous gels formed by systems for which phase separation is arrested by gelation [32] or when the obstacles are themselves porous. We will explore this issue in future work.

So far we have only considered rigid gels. An increase of the mobility of the tracer particles can be expected if the matrix is flexible. The difference is especially large if one compares the diffusion of tracers with d = 1 in a system of FHS with that in a system of freely diffusing spheres at the same volume fraction. In the latter case the tracers become trapped only atφ ≈ 0.6. Preliminary results showed that also for DLCA there is a considerable increase of the D if the gel is rendered flexible. We will investigate this effect in detail elsewhere.

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5. Conclusion

Self-diffusion of spherical particles in gels formed by random aggregation of hard spheres is to a large extent determined by the volume fraction that is accessible to the particles.

The accessible volume has a universal dependence on the volume fraction independent of the probe size and a universal dependence on the probe size independent of the volume fraction. The concentration dependence was almost the same for particle gels with very different local structures.

However, the probe size dependence was different, reflecting the difference of the local structure. For small volume fractions or probe sizes the accessible volume of DLCA gels is the same as that of strands of spheres.

The relative variation of the diffusion coefficient with the volume fraction was approximately independent of the probe size if plotted as a function ofφ/φcand was almost the same for DLCA and SDLCA gels. The relative variation of the diffusion coefficient with the probe size was approximately independent of the volume fraction if plotted as a function of d/dc, but it was different for DLCA and SDLCA gels.

However, the universal dependence of D on d/dcis no longer valid if the gel is formed by strands that are themselves permeable to small tracers. In that case universality is obtained only for tracers that cannot penetrate the strands.

Probes with a diameter less than a third of the gel particles are not fully trapped in the gel even at concentrations approaching random close packing. The critical probe size increases with decreasing volume fraction and reaches that of the gel particle atφ =0.27 and 0.36 for DLCA and SDLCA gels, respectively. In order to trap particles in the gel higher volume fractions are needed for SDLCA gels than for DLCA gels.

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