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The depletion interactions of the three-sphere system in which the three spheres are on one line are studied by Monte Carlo simulations. The depletion interactions are determined by ARM, and the coupling effect was proved by the numerical result that the depletion interactions in the three-sphere system are larger than that of the corresponding two-sphere system. Furthermore, we find that the mechanisms of the coupling effect and the effect on depletion force from the geometry factor are the same. In addition, the numerical results also show that this coupling effect will be affected by both the volume fraction and separation of three-sphere system.

It’s well known that the depletion interaction arises when two large spheres are immersed in the sea of small spheres. The mechanism about the depletion force was first described by Asakura and Oosawa (AO) with the concept of excluded volume [

It is well know that a 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’ of radius r, and is usually determined through the acceptance ratio method (ARM).

For 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 [3,8,14],

where is the number of samples drawn out from N simulated samples, which generated with the potential where is not infinite; is the number of samples drawn out from N simulated samples, which generated with potential where is not infinite, and is the Fermi function, and C is a constant which is usually set to a value of 0 for a hard sphere system. Since the change of potential is only relates to the number of samples drawn out from the N simulated samples, therefore the depletion interaction can be got through the ARM conveniently.

In this paper, we consider the system composed of three large spheres schematically described by

For the sake of simplicity, the positions of spheres A and C are fixed when B moves from the contact of A to the middle point of A and C. h and H are the separations of A and B, A and C, respectively. In order to expose the coupling effects, a comparison of the depletion interactions f to that of the two-sphere system described by

In the simulations, we consider the depletion forces of the hard spheres systems with a cell box of size , in which the three large sphere are placed along direction. Obviously, the small spheres are randomly distributed around the macro spheres to form a fluid. The size ratio of the macroto micro-sphere is 5, and the number of the micro-ions N is determined by the

given volume fraction, defined as or for the system of threeor twosphere respectively, where is the total volume of the cell box, is for the volume of the micro-sphere, denotes the volume of the macro-sphere. As the systems under consideration are not confined by geometry factors, the period boundary condition is applied to all the three directions of X, Y and Z in the Monte Carlo simulations. In addition, the configurations of the micro-spheres are sampled according to the Metropolis algorithm with the two macro-spheres A and C fixed while B moves from the contact of A to the middle point of A and C. Each micro-sphere is orderly chosen involving a trial displacement. Except for an overlapping with the macro-spheres and the other micro-spheres, the new position of the micro sphere is randomly accepted. In our simulations, 1.0 × 10^{5} Monte Carlo steps (MCS) are used for the equilibrium of the system and other 3.0 × 10^{5 }MCS to collect data. In addition, the depletion potential is set as 0 while the two macro-spheres A and B are at contact, i.e., h = 0. In this way, the depletion potentials, then the depletion forces in both the twoand three-sphere systems, , , are determined by ARM, and the results of the two system corresponding to the volume fractions η = 0.116, 0.229 and 0.341 are shown in Figures 2(a) and (b), Figures 3(a) and (b), Figures 4(a) and (b), respectively. The separation of sphere A and C of above systems is H = 18r. In addition, the depletion potential F in unit of is plotted as a function of h, which is measured in unit of 2r, and the unit of depletion force is, where is

the number density of the small sphere. In Figures 2-4, the solid lines describe the depletion potentials or depletion forces of the systems composed of two-sphere, the dash-lines for that of the three-sphere systems’. From Figures 2-4, it’s evident that, no matter the volume fraction is large or small, the depletion forces of the threesphere systems are larger than that of the corresponding two-sphere system. This result is not in accordance with the common sense of physics related to forces, therefore it is very interesting. As is known that, in the view of physics, when one object is suffered two forces in opposite directions at the same time, the resultant force will be smaller than the larger component one. However, the case considered here is that the sphere B is acted by the two opposite depletion forces from A and C at the same time, but the total force described by the dashed lines in Figures 2(b), 3(b) and 4(b) is larger than the larger component force from sphere A. So the additivity of depletion interactions is proved to be not true, at least for the case considered in this paper. Furthermore, it is reasonable to think that the two depletion forces impressed on the sphere B may couple with each other, and finally result in a strengthened total depletion force. This is the

coupling effect of the depletion interactions. Unfortunately the mechanism for the coupling effect is still unknown. However, we believe that there are some relations between the coupling effect and the effect on depletion force from the geometry factor. In fact, as is known that the depletion force between two spheres will be strengthened if they are confined by geometry factors, such as by two plates [

On the other hand, from Figures 2(b) and 4(b) we also find that, in

−1.45, so the difference between them is 0.1; in

In addition, we also study the depletion interactions in the three-sphere system when the separation of spheres A and C is changed. So the cases of H = 22r, 18r, 16r, are studied. For simplicity, only the depletion forces are shown in Figures 5 and 6, which are corresponding to the volume fraction, 0.229. In both Figures 5 and 6, the dashed, dotted and solid lines are for H = 22r, 18r, 16r, respectively. From Figures 5 and 6, it is found that, with decrease of separation, the depletion force of the three-sphere system is increased; compared Figures 5 and 6, it is found that the depletion force will increase with the increase of volume fraction. So the coupling effect of the three-sphere system is related to the volume fraction and separation of the system.

In conclusion, we have investigated the depletion interactions among the three large spheres through Monte Carlo simulations. It is found that the depletion interactions can couple each other and result in a strengthened total depletion forces, and the couple effect will increase when the volume fraction increases from 0.116 to 0.229; the couple effect will also increase with decrease of the separation H of the system. In addition, it is also find that, if the third sphere of the three-sphere system is taken as a geometry factor to the other two spheres, the intrinsic or the mechanisms of the effect on depletion interactions from geometry factor and of the coupling effect are the same.