# Self diffusion of reversibly aggregating spheres

###### Abstract

Reversible diffusion limited cluster aggregation of hard spheres with rigid bonds was simulated and the self diffusion coefficient was determined for equilibrated systems. The effect of increasing attraction strength was determined for systems at different volume fractions and different interaction ranges. It was found that the slowing down of the diffusion coefficient due to crowding is decoupled from that due to cluster formation. The diffusion coefficient could be calculated from the cluster size distribution and became zero only at infinite attraction strength when permanent gels are formed. It is concluded that so-called attractive glasses are not formed at finite interaction strength.

###### pacs:

05.10.Ln, 82.70.Dd, 82.70.Gg^{†}

^{†}preprint: APS/123-QED

## I Introduction

Reversible aggregation of small particles in solution is a common phenomenon. It leads to different equilibrium states depending on the volume fraction () of the particles and the strength of the interaction energy (). Weak attraction results in the formation of transient aggregates at low and a transient percolating network at high , while strong attraction may drive phase separation into a high and a low density liquid 433 (); 867 (); 823 (); 869 (); 652 (); 488 (); 913 (); 489 (); 288 (); 896 (); 791 (); 834 (); 878 (); 833 (); 790 (); 876 (); 898 (); 926 (); 907 (); 899 (); 938 (); 891 (); 927 (); 931 (); 933 (); 912 (); 883 (); 910 (); 955 (); 831 (); 832 (). The strength of the interaction and thus the equilibrium properties are determined by the ratio of the bond formation () and the bond breaking () probability 943 (); 942 (); 545 (); 413 (); 270 (); 269 (); 267 (); 411 (); 415 (); 478 (); 754 (); 725 (); 753 (); 901 (), while the kinetics of such systems depend on the absolute values of and . Two limiting cases may be distinguished: diffusion limited cluster aggregation (DLCA) for which a bond is formed at each collision () and reaction limited cluster aggregation (RLCA) for which the probability to form bonds goes to zero ().

The average long time self diffusion coefficient () of non-interacting hard spheres decreases with increasing volume fraction 877 (); 872 (); 1009 (); 1010 (); 993 () and the system forms a glass above a volume fraction of about 0.585. For a recent review of theories and experiments on the glass transition in colloids see Sciortino and Tartaglia952 (). Mode coupling theory predicts that goes to zero at a critical volume fraction following a power law:

(1) |

The pictorial view is that particles become trapped in cages formed by neighbouring particles. Experiments 994 () and computer simulations 907 (); 992 () confirmed this behaviour over a range of though with a larger value of (0.585). However, the diffusion coefficient is not truly zero at , and some mobility is still possible between and the volume fraction of random close packing (). The reason is that fluctuations of the cage size around the average value, allow particles to "hop" from one cage to another. Mode coupling theory uses the static structure factor as input and therefore cannot include the effect of fluctuations and heterogeneity.

Introducing reversible bond formation between the hard spheres has two consequences. Firstly, the structure factor is modified, because on average more particles will be within each others bond range. Secondly, the diffusion of bound particles becomes correlated for some duration that is related to the bond life-time. Obviously, the latter effect leads to a decrease of , because clusters of bound particles move more slowly than individual spheres and not at all in the case when they form a percolating network. However, modification of the structure may also lead to an increase of compared to the hard sphere case 907 (); 926 (); 923 (), because the accessible volume for a particle in which it can move increases. In other words, the average cage size increases. This effect can be clearly seen for the self diffusion of tracer particles in a medium of fixed spherical obstacles. For a given volume fraction of obstacles, is smaller when the obstacles are randomly distributed than when they form a percolating network tracer ().

An increase of with increasing attraction has actually been observed for concentrated suspensions of hard spheres with weak short range attraction where the particles can freely move within the attraction range. With further increase of the interaction energy the effect on the structure weakens, but the bond life-time increases so that decreases again. These effects have also been found in molecular dynamics simulations of such systems 907 (); 926 (); 923 (). Mode coupling theory describes this behaviour purely on the basis of the changes in the static structure factor and predicts complete arrest at finite interaction strength even at low volume fractions 1029 (). Systems arrested by attraction are called attractive glasses 1030 () to distinguish them from the ordinary repulsive glasses 994 ().

Obviously, irreversible aggregation, i.e. infinite attraction strength, leads to arrest at any volume fraction when all spheres become part of the percolating network 802 (). Generally, arrested systems at low volume fractions due to irreversible aggregation are called gels. In analogy, percolated systems with finite interaction strength may be called transient gels, though in practice one would only do that only if the bond life-time is very long so that the system flows very slowly. The issue addressed here is how varies with increasing attraction strength and specifically whether becomes zero in transient gels with a finite bond life-time. In other words, whether transient gels can be attractive glasses. Even if, as for repulsive glasses, does not truly become zero due to hopping processes, the question remains whether can be usefully described as going to zero at a finite critical interaction energy following a power law over a broad range of :

(2) |

As mentioned above, phase separation occurs when the attraction is strong, and needs to be avoided if one wishes to study the effect of strong attraction on at low volume fractions. This can be done to some extent by limiting the maximum number of bound neighbours below 6 936 (); 884 () or by introducing a correlation between the bond angles with different neighbours 921 (). One can also introduce a long range repulsive interaction to push phase separation to lower concentrations 923 (); 885 (); 826 (). Phase separation disappears completely if the bonds are rigid and the bond range is zero, because only binary collisions can occur so that the average coordination number cannot exceed two 901 (). For this model can be studied at any interaction strength and volume fraction without the interference of phase separation.

An extensive study of the structural properties of such systems was published earlier 901 (). It was shown that the equilibrium properties depend on the ratio and the elementary step size of the Brownian motion. As long as is much smaller than the average distance between the particles, the equilibrium properties are determined by a single parameter called the escape time () that is a combination of , and . is defined as the excess time it takes for two particles at contact to become decorrelated compared to the situation where no bonds are formed (). If the interaction range is finite and if is much smaller than the interaction range, then the escape time becomes independent of and the equilibrium properties are the same as for spheres interacting with a square-well potential.

Here we present results of the mean square displacement (MSD) of the particles as a function of the interaction strength for systems forming rigid bonds with zero and finite interaction strength. The main conclusion of this work is that introducing attraction leads to a decrease of without any sign of arrest at finite interaction strength. As mentioned above, there are two different mechanisms for slowing down: one is controlled by the bond life-time and the other is controlled by crowding. We show here that for rigid bond formation with the two mechanisms are independent and can be factorized. Since the first mechanism leads to complete arrest only for irreversible aggregation, we propose that the expression "attractive glass" is abandoned in favour of the expression "gel" that is commonly used to describe irreversibly bound percolating networks. The results presented here resemble to some extent those obtained recently by Zaccarelli et al.936 () for attractive spheres with limited valency. The main difference is that in those simulations the bonds were not rigid and localized motion was possible even at infinite bond strength. We will briefly discuss the effect of bond flexibility.

## Ii Simulation method

The simulation method for aggregation with zero interaction range has already been detailed elsewhere 901 (). Initially, non-overlapping spheres with unit diameter are positioned randomly in a box with size , using periodic boundary conditions, so that . Different sizes between and were used, and the results shown here are not influenced significantly by finite size effects. times a particle is chosen and moved a distance in a random direction. When the movement leads to overlap with another sphere it is truncated at contact. After this movement step all spheres in contact are bound with probability leading to the formation of clusters that are defined as sets of bound particles. In the next movement step, times a cluster is chosen and moved a distance in a random direction with a probability that is inversely proportional to the diameter of the cluster, thus simulating non-draining within the clusters. In the following cluster construction step, all bound particles are broken with probability and bonds are formed for new contacts with probability . Movement and cluster formation steps are repeated until all particles have formed a single cluster for irreversible aggregation or until equilibrium is reached for reversible aggregation.

The diffusion coefficient was calculated as and the time unit was defined as the time needed for a single particle to diffuse its own diameter so that the diffusion coefficient at infinite dilution is . In this article we will present diffusion coefficients normalized by . For the case of irreversible () diffusion limited () aggregation, the kinetics of cluster growth were the same as predicted from the Smolechowski equations for DLCA if was chosen sufficiently small802 (). As mentioned in the introduction, the equilibrium properties for reversible aggregation with zero interaction range are determined by the escape time 901 ():

(3) |

At equilibrium, a distribution of self-similar transient clusters is formed together with a transient percolating network if and are sufficiently large.

Reversible aggregation with finite interaction range was simulated by forming bonds with probability when the centre to centre distance between particles is less than 955 (). Bonds were again broken with probability . In this case the equilibrium properties are independent of the step size if . The system is equivalent to particles interacting through a square well attraction with interaction energy , where is the probability that particles within each others interaction range are bound: . is given in units of the thermal energy and is equivalent to the inverse temperature that is sometimes used to express the interaction strength. The equilibrium properties of systems with different interaction range are close if they are compared at the same second virial coefficient ()787 (). is the sum of the excluded volume repulsion and the square well attraction: . and is determined both by the interaction strength and the interaction range 892 ():

(4) |

in units of the particle volume

## Iii Results

### iii.1 Zero interaction range

Fig. 1 shows the mean square displacement, i.e. , of spheres at for different values of with , starting from a random distribution. Initially, the particles move freely until they begin colliding with other particles. This leads to slowing down of the MSD even in the absence of attraction (). At long times is again proportional to , but the long time diffusion coefficient () is smaller. Fig. 2 shows that the dependence of the normalized diffusion coefficient of hard spheres in the absence of attraction () on is in good agreement with literature results 877 (); 872 (); 1009 (); 1010 (); 993 ().

Introducing attraction slows down the MSD, because transient clusters are formed that move more slowly and above a critical value of a transient percolating network is formed. Particles that are part of the transient network are immobile and need to break bonds before they can diffuse. In the limit of , corresponding to irreversible DLCA, all particles are permanently stuck when they become part of the percolating network. Naturally, the value of where the particles get stuck decreases with increasing volume fraction, see Fig 3.

Once the system has reached equilibrium one can start again measuring the MSD. Fig. 4 shows the MSD for the same systems as in Fig 1, but starting from the equilibrium state. In this case the initial diffusion is slower than that of individual spheres, since the system contains clusters and for a percolating network. After some time the clusters collide causing a further slowing down of the MSD, but simultaneously, for , bonds are formed and broken. Of course, for the same long time diffusion coefficient is obtained independent of the starting configuration, compare Fig. 1. Remarkably, the effect of particle collisions on the slowing down of the MSD is the same at different . Therefore the MSD obtained at different can be superimposed within the statistical error by simple time shifts (see Fig. 4). The implication is that can be factorized in terms of the short time diffusion coefficient () and :

(5) |

At short times, i.e. before bond breaking becomes significant and before the particles collide, the MSD is determined by free diffusion of the clusters. Consequently, can be calculated from the size distribution of the clusters () which is a function of and :

(6) |

where is the average free diffusion coefficient of clusters with aggregation number . If a percolating network is present we need to consider only the sol fraction of mobile clusters ():

(7) |

where is the number average aggregation number of the sol fraction. In the absence of hydrodynamic interactions so that . For the more realistic case of non-draining clusters is inversely proportional to the radius of the clusters so that decreases more weakly with increasing .

Figure 5 shows as a function of for and . We have calculated from the cluster size distribution using Eq. (6) and was calculated using Eq. (5). Comparison with the simulation results, see Fig (5), demonstrates that the effect of attraction on is indeed fully determined by the cluster size distribution. Initially, decreases with increasing , because increases until at a critical value of () a percolating network is formed of immobile particles. is and for and , repectively. At the percolation threshold the weight average aggregation number diverges 901 (), but and thus remain finite. The maximum value of is obtained at the percolation threshold, beyond which it decreases with increasing volume fraction. Beyond the percolation threshold the sol fraction decreases much more strongly with increasing attraction than so that continues to decrease. However, it is obvious that will become zero only if , i.e. at infinite attraction strength. At large values of , decreases linearly with increasing . As expected, at lower volume fractions stronger attraction is needed to cause significant slowing down.

The MSD shown in Fig. 4 represent an average over all particles, and do not show how the displacement of the particles is distributed. In the absence of attraction the probability distribution that a particle has moved a distance at a given time () is given by:

(8) |

because each particle is equivalent. However, in the presence of attraction the displacement is highly heterogeneous, because the particles belong to clusters of different sizes. We have illustrated this for at . In this case almost all of the particles belong to the gel fraction and cannot move until their bonds are broken. Thus at short times the displacement is highly heterogeneous with a large fraction of particles that do not move at all and a small fraction of sol particles () that diffuse freely until they collide. With increasing time, more and more of the sol particles collide with the percolating network and stop moving, while more and more gel particles break lose and start diffusing. Fig 6 shows how the fraction of mobile particles that has moved after a time () increases with increasing .

Figure 7 compares at different times with the distribution that would have been found for same if all particles had been equivalent. Note that the area under the curves is equal to . The heterogeneity of the displacement decreases with increasing time until for all particles have formed and broken bonds many times so that and is given by Eq. 8 . Similar observations were made at .

Figure 7 resembles the results presented by Puertas et al. 925 (). However, they used molecular dynamics simulations for which bound particles move freely as long as the displacement does not involve breaking of bonds, as will be discussed below. For this reason they observed a peak situated at a small that represents the displacement of bound particles. A second peak was found at larger values of representing the sol fraction and moved to longer distances with increasing time. The amplitude of this peak increased with increasing time as more and more particles break their bonds.

### iii.2 Finite interaction range

The effect of finite interaction range was tested for and . We found also for this case that could be factorized into and , see Eq. 5, and that the effect of attraction on is fully determined by the free diffusion of the clusters. However, the dynamic heterogeneity that was important for the zero range interaction disappeared rapidly with increasing interaction range especially at higher concentrations, because monomers and clusters can move only a short distance before they interact. Individual particles exchange rapidly between different clusters including the gel fraction, at least for . Therefore each particle explores more rapidly the different dynamics of the system.

Recently, it was shown that the cluster distribution in equilibrium is similar for the two interaction ranges if compared at the same values of the second virial coefficient, at least for up to 955 (). We therefore expect that is the same at the same . Fig. 8 shows as a function of for 3 different volume fractions with and . As mentioned in the introduction, for lower volume fractions the slowing down of can be studied only over a limited range before phase separation occurs. As expected, decreases with increasing and the decrease is more important at higher concentrations. Again no sign of critical slowing down of the MSD was observed. For and the dependence of on was similar for and , but for it was stronger for than for . At high volume fraction is no longer the main parameter that determines the cluster size distribution because higher order interaction becomes important.

Recently, Foffi et al. 912 () reported a simulation study of the effect of the interaction range on for hard spheres with a square well interaction. Molecular dynamics simulations were used that gave the same equilibrium structures as with the method used here. However, molecular dynamics simulations gives different dynamics so that absolute values of cannot be compared with the results presented here. Nevertheless, they also found that the MSD of particles was independent of the interaction range if compared at the same value of and they argued that was the same because the number of bonds and the bond life-time was determined by . We expect that deviations will be found also with this method at higher concentrations and larger interaction ranges when higher order interaction becomes important.

## Iv Discussion

There are two causes for the decrease of in a system of attractive hard spheres. The first one is collisions with other spheres. This effect of crowding increases with increasing volume fraction and leads to strong decrease of near that can be described to some extent by mode coupling theory, as mentioned in the Introduction. The second cause is bond formation, which leads to the formation of transient clusters and gels. This effect increases with increasing attraction. The diffusion at short times is not influenced by collisions so that in the absence of attraction is equal to the free diffusion of the particles. The decrease of with increasing attraction is caused solely by cluster formation and can be calculated from the cluster size distribution. Since for any finite interaction there is a finite fraction of free particles and clusters, only becomes zero when the interaction is infinitely strong. The decrease of with increasing interaction strength is mainly determined by , see Fig. 9 and it is dominated by the decrease of the sol fraction for strong attraction when is close to unity.

The subsequent decrease of the diffusion coefficient from to at long times is caused by crowding. An important observation is that the slowing down caused by crowding is independent of the attraction strength for . The reason is that by definition for reversible DLCA the reversibility is expressed as soon as collisions occur and the memory of the connectivity is lost. Consequently, diffusion is slowed down by attraction, but does not become zero as long as the attraction is finite. The situation is different for reversible RLCA in which case the collisions occur between long-lived clusters and between the clusters and the percolating network. For given attraction strength is the same for reversible DLCA and RLCA, but for RLCA the ratio decreases with decreasing bond formation probability. This situation will be explored elsewhere.

The results presented here apparently contradict molecular dynamics simulations that showed critical slowing down at finite interaction strength 883 (); 923 (). The main difference between our simulation method and molecular dynamics is that rigid bonds are formed so that only cluster motion is possible. In the molecular dynamics simulations bound particles are still allowed to diffuse freely as long as no bonds are broken. The implication is that is equal to the free particle diffusion independent of the interaction strength. Consequently is faster and is even larger than for weak attraction. We have included bond flexibility in the simulations with finite interaction strength by allowing free diffusion for bound particles as long as it does not lead to bond breaking. Details of these simulations will be reported elsewhere. Here, we only mention that including bond flexibility increases . It is clear, that allowing more freedom for movement does not lead to a critical slowing down of the diffusion at a finite interaction range so that the apparent contradiction with molecular dynamics simulations persists. But, a close look at the molecular dynamics simulation results shows that they can also be interpreted in terms of a power law decrease. For instance, Puertas et al. 923 () simulated hard spheres with a short range depletion interaction caused by the addition of polymers. In Fig. 10 we have replotted as a function of the polymer volume fraction () for two different particle volume fractions. The authors interpreted the data in terms of Eq.2. At each concentration the smallest values of deviated from this expression, which was attributed to "hopping". It is clear, however, from Fig. 9 that the decrease of at large can also be described as a power law even for the smallest value of implying that only at infinite attraction.

Very recently, Zaccarelli et al. 936 () made a detailed study of the dynamics in attractive hard sphere systems with limited valence (3 and 4) using molecular dynamics simulations. They observed that for strong attraction the variation of could be described by a power law in terms of at least for implying that arrest only occurred at infinite attraction. On the other hand they could describe as a function of in terms of Eq. 1. was almost constant when increasing the attraction and only for strong attraction did they find a weak decrease of , but the< extrapolation was uncertain in this case. The authors used the expression reversible gel for systems that arrested at and glass (attractive or repulsive) for systems that arrested at . The expression attractive glass was introduced to describe the arrest that occurs for a given volume fraction at a finite attraction energy. It is clear from the present study that attractive glasses in this sense are only formed for irreversible aggregation.

## V Conclusion

Reversible cluster aggregation of hard spheres leads to equilibrium systems containing transient clusters and, above a critical interaction strength, a transient percolating network. If the aggregation is diffusion limited and the bonds are rigid, then the effect of attraction on is decoupled from the effect of crowding. The latter is equal to that of non-interacting hard spheres, while the former is fully determined by the cluster size distribution. The self diffusion coefficient of the spheres decreases with increasing attraction, but becomes zero only for irreversible aggregation, contrary to predictions from mode coupling theory. Therefore attractive glasses in the sense of systems that are dynamically arrested by a finite interaction energy do not exist.

###### Acknowledgements.

This work has been supported in part by a grant from the Marie Curie Program of the European Union numbered MRTN-CT-2003-504712.## References

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