^{1}

^{2}

^{2}

^{*}

The coupling effects of depletion interactions in three-sphere systems with different size ratio of large- to small-sphere are studied by Monte Carlosimulations in this paper. The numerical results show that this coupling effect is affected by the size ratio of large- to small-sphere: the larger the size ratio is, the larger the coupling effect will be.

Recently, coupling effects of depletion interaction were reported in the three-sphere system or in the two-sphere system confined by two plates [1-3]. It was pointed out that when a large-sphere is suffered by more than one depletion force, these depletion forces will couple with each other and results in a strengthened depletion force. As a new finding, the coupling effect is a very interesting and significant topic for us to study, since the dynamic behavior of particles and the structure of the colloidal system will be affected by the strengthened depletion interactions. With the aid of the concept of excluded volume, Asakura and Oosawa (AO) suggested that, the mechanism about the depletion force is just related to geometrical factor, therefore it is very simple [

In this paper, the model and theory are presented in Section 2, and the numerical results and a simple discussion will be given in Section 3, finally a summary of our study will be given in Section 4.

It is well known that the hard spheres mixture is characterized by the pair potential of

where d is the distance between the two spheres in diameters and, respectively. The force exerted on the big sphere of radius R by a small sphere of radius r

can then be written as. Consequently, the depletion force is the total force acted on the large sphere from the small spheres, and can be usually determined through the acceptance ratio method (ARM): if the potential and partition function of two systems are and respectively, where and are the external potentials corresponding to the two large spheres located at different positions, the free energy difference between these two systems is given by the following expression [6,11,15],

where is the number of samples drawn out from simulated samples, which generated with the potential where is not infinite; is the number of samples drawn out from simulated samples, which generated with potential where is not infiniteand is the Fermi function, and C is a constant which is usually set to a value of 0 for a hard sphere system.

We note that the depletion force can be determined by the differential of the free energy of the system obtained through Equation (2). Besides, it is important to emphasize that if there are many depletion forces acted on a sphere at the same time, only the resultant force corresponding to the free energy will be determined by ARM.

For the three-sphere system, a parameter is introduced to describe the coupling effect:

where f is the resultant force of the three-sphere system described by

Obviously, if there is no coupling effect between and, , otherwise, so can be used to describe the coupling effect. For the sake of simplicity, we suppose that, in all the systems mentioned above, the positions of spheres A and C are fixed, only B moves from the position of the contact of A to the middle point of A and C, then the depletion forces f, , are determined by ARM through the systems described by Figures 1(a)-(c) respectively. The parameter h is the separation of A and B, and H is the separation of A and C.

In this section, only the unconfined three-sphere systems and the corresponding two two-sphere systems with size ratio of largeto small-sphere R/r = 3, 5, will be studied, respectively. In the simulations, the two or three large hard spheres are placed along direction of the cell box of size, but the small spheres are randomly distributed around the macro spheres to form a fluid; the number of micro-ions is determined by the given volume fraction, defined as or for the threeor two-sphere system respectively, where is the total volume of the cell box, is for the volume of the microsphere, denotes the volume of the macrosphere. As unconfined systems, the period boundary condition is applied to all the three directions of X, Y and Z in the Monte Carlo simulations. Then the configurations of the micro-spheres will be sampled according to the Metropolis algorithm [

tion interactions acted on the sphere B are coupled with each other. In addition, from Figures 6 and 7 we also find that, 1) is enlarged when the volume fraction of the system increases from 0.12 to 0.23; 2) is also enlarged when the size ratio of largeto smallsphere of the system of R/r is increased from 3 to 5. In other words, the larger the volume fraction is, the larger the coupling effect of the depletion interactions will be; the larger the size ratio is, the larger the coupling effect of the depletion interactions will be. This is very important for us to get the physical viewpoint of nucleation. Supposing in a binary colloidal system, free large-spheres are continually pushed together by small-spheres, a cluster consisting of the packed spheres is therefore built up. Furthermore, with the increasing size of the cluster, it will suffer more and more strengthened depletion interactions from the small-spheres, because both the depletion interactionsand the coupling effect of these depletion interactions will increase with the increase of the size ratio of the largeor small sphere to cluster. As a result, the cluster will grow larger and larger, even a nucleation will turn

out. In other words, the coupling effect of depletion interactions is helpful for the nucleation packing.

In conclusion, we have studied the depletion interactions in the unconfined three-sphere systems with different size ration through Monte Carlo simulations. It is found that this coupling effect is clearly and completely described by the parameter of the depletion force difference of the three-sphere system and the two corresponding twosphere systems. It is also found that the coupling effect of depletion force is affected by both the size ratio and the volume fraction of the system.

Project supported by the Scientific Research Fund of Hunan Provincial Education Department, China (Grant No. 10A075).