With the aid of the critical size of colloidal cluster, the critical volume fraction of phase transition of colloidal system is determined by the principle of entropy maximum and Carnahan-Starling （CS） state equation in this paper. In our discussion, no parameter is introduced externally, and our results are in good agreement with the experimental results.
In a colloidal system confined by a small cylindric pipeline, the depletion interaction between two large spheres is different to the system confined by two plates, and the influence on depletion interaction from the pipeline is related to both the size and shape of it. In this paper, the depletion interactions in the systems confined by pipelines of different sizes or different shapes are studied by Monte Carlo simulations. The numerical results show that the influence on depletion force from the cylindric pipeline is stronger than that from two parallel plates, and the depletion force will be strengthened when the diameter of the cylinder is decreased. In addition, we also find that the depletion interaction is rather affected if the shape change of the pipeline is slightly changed, and the influence on depletion force from the shape change is stronger than that from the size change.
In a charged colloidal system, the influence on depletion interaction between two like-charged macro-ions is studied through Monte Carlo simulation in this paper. The numerical results show that this depletion force is affected by both the electrostatic interactions between charged spheres and charged plates and by the geometrical factor of the two charged plates, and they further indicate that the influence of geometrical confinement on the depletion interaction is larger than that of electrostatic potential.
In a three-sphere system, the middle sphere is acted upon by two opposite depletion forces from the other two spheres. It is found that, in this system, the two depletion forces are coupled with each other and result in a strengthened depletion force. So the difference of the depletion forces of the three-sphere system and its corresponding two two-sphere systems is introduced to describe the coupling effect of the depletion interactions. The numerical results obtained by Monte- Carlo simulations show that this coupling effect is affected by both the concentration of small spheres and the geometrical confinement. Meanwhile, it is also found that the mechanisms of the coupling effect and the effect on the depletion force from the ~eometry factor are the same.
