Research On The Shape And Phase Transition Of Multi-degree-of-freedom Bubble

The study of adhering open vesicles is of great significance in biology and medicine.In the past,the adhesion of closed vesicles were studied in depth,while Dong Ni et al studied the shape equation of adhesive open vesicles,and the convex cup-shaped solution were founded.For the first time,Liang Yuefeng has studied the solution and phase transition behavior of the adhesive open membrane bubble under the BC model.At present,there is no systematic studyed on the solution set,the properties and the phase transition of the adhesive open membrane bubble.Because the ADE model is generally accepted as a model closer to the real membrane bubble,it is undoubtedly more important to study the phase transition of the membrane bubble.Compared with free open vesicles,the adhesive open vesicles has one more freedom,that is adhesion radius Rcon and the opening radius Ropen of adhering open vesicles can be changed at the same time.It is a multi degree of freedom problem,and the solution is complicated.The ADE model has one more parameter than the BC model,that is the nonlocal area difference elastic modulus(the BC model only corresponds to the limit case when it tends to infinity)so the calculation is greatly increased.The cup shaped membrane bubble is founded and the solution is relatively clear for the reduction area difference Aa<1.However the solution is relatively complex for ?a>1.So this paper only studies for ?a>1.In this paper,the properties of the every branch can be clearly distinguished by studying the free open vesicles under BC model by the relaxation method.The results show that there are four branching solutions in the opening membrane vesicles,represented by the letters A,B,C,D.A bifurcation represents the concave cup and the continuous evolution obtained the neck self-intersection dumbbell solution,B bifurcation represents the dumbbell shaped solution,C bifurcation represent the concave oblate ellipsoid and continuous evolution obtained oblate ellipsoid of self-cross at the opening,D bifurcation solution represent convex oblate ellipsoid.Then the mapping relationship between open and closed vesicles from BC model to ADE model is compared.It is found that the mapping has greatly changed due to Rcon degree of freedom,and the " Gibbs wing " behavior appeared in the closed vesicles,accompanied by discontinuous phase change,while the open vesicles was folded under two models,the folding point was different,and the " Gibbs wing " was not appeared,and there is no additional discontinuous phase transition for the same branch solution.Based on the research of the BC model,the study on the fixed adhesion coefficient ensemble(Ensemble I)is carried out.The focus is on the study of the shapes and phase transition of the vesicles under the ADE model.Firstly,the relationship between the reduced energy and the reduced relaxation area difference?a0 is studied.In order to explain the behavior of the solution,the behavior of the B-branching under the BC model is studied.Finally,the phase transition caused by adhesion coefficient is studied in two models,found that with the increase of ? the free opening dumbbell bubble continuous transition to the adhesion of the open dumbbell vesicles,continued to increase ? discontinuous transition into the concave cup shape for BC model and ADE model.It is proved that the adhesion membrane bubble is more stable than the free membrane bubble.