Study On The Phase Diagram Of Single-opening Bubble Under BC Model

In 1998,A.Saitoh et al.first observed cup-shaped vesicles by opening a small hole in a closed vesicle.In 2005,T.Umeda and others obtained outward convex cup shapes and other shapes by numerical calculation which is in agreement with the experiment.The phase diagram for these shapes is given in the ADE model.The reduced area they studied were all less than 1.So far there is no detailed study for free vesicles with one hole for larger reduced area difference in the BC model,and no phase diagram in the BC model for these vesicles.In order to obtain the phase diagram for free vesicles with one hole,one need to study every branch solution systematically.Huang cong et al.obtained the open dumbbell-shaped solution and open prolate solution under the SC model.According to the relationship between SC model and BC model,we can obtain the solution directly under the BC model through Legendre transformation.However,in the SC model,since for the same spontaneous curvature there may exist multiple solutions,one solution cannot evolve continuously when the solution is folded.Therefore,we developed a method in the ensemble with fixed opening radius,which is called ensemble ?.In the previous literature,the method of fixing linear force coefficient is called ensemble ?.According to the relationship between the two ensembles,when the solution in the ensemble ? is folded,it can be found in the ensemble ?,which greatly simplifies the calculation process.Our main conclusions are as follows:1.Under the SC model,Huang cong et al.obtained two solutions,and the relationship between the two solutions is confusing.By using the method in ensemble ?,we found that the concave cup solution obtained by liang et al.with near-free(adhesion radius approaching zero)also exists for the open-end vesicles,this perfect the AB solution.The relationship between the two solutions is studied systematically.with the increase of spontaneous curvature a recombination phenomenon appears between the two solutions.This is a new phenomenon that has not been found in previous studies on closed vesicles.2.Calculation on the free vesicles with one hole under the BC model shows that the concave cup solution and other branch solutions for near freed status by Liang et al.still exist for the free status.We also found some other branch solutions.Starting the spherical and prolate vesicles of nearly closed vesicles,we clarify the relationship of these solutions by analyzing the curvature and Gaussian angle.This lays a foundation to further determine the stability region(i.e.phase diagram)in the BC model.3.Based on systematic study of the solutions found above,we found that the concave cup solution(called A branch)and open dumbbell solution(called B branch)are solutions in the BC model.For small line tension coefficient,there is a unique shape corresponding to each reduced area difference(i.e.,no folding occurs)in AB branch solution,which is consistent with the previous calculation results for closed vesicles.However,for larger line tension coefficient,the folding behavior of AB branch begins to appear,and the shape transformation is discontinuous.With the increase of the line tension coefficient,the folding behavior of AB branch begins to change.This is quite different from previous understanding of the BC model.4.By systematically scanning of the phase space(which depends on the reduced area difference and the reduced line tension coefficient),we excluded the existence area of the self-intersecting dumbbell shape,and determined the stable region of A branch and B branch in the phase space.By analyzing the discontinuous phase transition of A branch and B branch under different line tension coefficients,the phase diagram vesicles with one hole in the BC model was finally drawn.