Nonlinear Wrinkling Dynamics of a Multi-Component Vesicle (2D)

Abstract

This paper investigates wrinkling dynamics of two-dimensional multicomponent vesicles subjected to time-dependent extensional flow. By employing a non-stiff, pseudo-spectral boundary integral approach, we inspect the wrinkling patterns that arise due to negative surface tension and differential bending within a two-phase system. We focus on the formation and evolution of the wrinkling behaviors under diverse phase concentrations, extensional rates, and vesicle sphericity. Our findings demonstrate that for slightly perturbed circular vesicles, the numerical simulations align well with perturbation theory. For elongated vesicles, the wrinkling patterns vary significantly between phases, primarily influenced by their respective bending moduli. In weak flows, buckling behaviors are observed for elongated vesicles, where the membrane bends inward in regions with lower bending modulus.

1. Introduction

A vesicle is a small, viscous droplet encompassed by a highly flexible bilayer lipid membrane, commonly known as a liposome. Its structure mimics the composition of biological cells or cellular organelles and possesses unique properties with broad applications. Immersed in liquid, vesicles interact with surrounding fluids, adopting various shapes [1,2,3,4]. Extensive experimental, theoretical, and numerical investigations have been carried out to comprehend vesicle movement [5,6,7,8,9,10]. For example, a vesicle in shear flow shows different motion patterns depending on the viscosity ratio of internal and external fluids, excess surface area, and the strength of the applied flow. Recent studies have observed the non-stationary wrinkling dynamics of vesicles in time-dependent elongation flows [10]. High-frequency wrinkles arise due to negative surface tension on the membrane when the flow direction changes abruptly. Turitsyn et al. probed into the effects of negative surface tension on membranes and developed a linear theory for the dynamics of wrinkled membranes [9]. Later, a series of numerical studies have been conducted for both 2D and 3D wrinkling dynamics [11,12,13].

As fundamental components of living cells and organisms, membranes are composed of a mixture of materials such as lipids, proteins, cholesterol, and ions, playing crucial roles in communication between internal and external biological environments (e.g., solute transport) [14,15,16]. Vesicle membranes exhibit liquid-like properties, resist bending, and are inextensible. Experiments conducted on giant unilamellar vesicles have demonstrated the existence of a wide range of behaviors exhibited by multicomponent vesicles [14,17]. These behaviors include spinodal decomposition into distinct surface domains (such as liquid-ordered and liquid-disordered phases), commonly referred to as rafts or domains, raft coarsening, viscous fingering, vesicle budding, fission, and fusion, all of which are observed with concomitant changes in membrane morphology [18,19,20,21,22]. Recent experiments show that the competition between intradomain electrostatic repulsions and interdomain line tension leads to domain shape transitions in phase-separating lipid monolayers [23], and protein phase separation on membrane is potentially relevant to various cellular protrusions [24]. There has been considerable theoretical and numerical researches on the bending energy of lipid bilayer biomembranes [25,26,27]. Recently, a theoretical model was proposed to describe the deformations and micrometric curvature sensitivity [28], and a deformable continuum model predicting curvature sensing behavior was proposed to account for the physical properties of the membrane and the helix insertion [29]. As a result, modeling vesicle dynamics involves solving a complex moving boundary problem where fluids, phase domains, and vesicle morphology are intricately coupled. Nevertheless, studies on the effects of inhomogeneous bending still remain limited.

In this paper, we utilize a thermodynamically consistent model to investigate vesicle dynamics within an incompressible viscous fluid [26,27]. Our primary interest is in the morphological transformations induced by extensional flow, notably the emergence of wrinkles on the vesicle membrane. We specifically examine the morphological evolution of vesicles subjected to time-dependent extensional flows and assess the effects of non-uniform bending moduli. Although it incorporates certain simplifications, the model effectively captures the nonlinear interactions among Stokes flow, vesicle shape changes, and the evolution of surface phases. We hope that this investigation lays the groundwork for a more detailed analysis of multicomponent vesicle systems.

The structure of the paper is as follows: In Section 2, we derive and nondimensionalize the governing equations. Section 3 reviews perturbation theory and presents the key findings. Section 4 describes the numerical algorithms employed. Numerical results are presented in Section 5. The paper concludes in Section 6 with a discussion of potential future work.

2. Materials and Methods

Consider a closed, multicomponent membrane filled with fluids of viscosity $\eta$ both inside and outside. The vesicle is subjected to a time-dependent extensional flow with strength $S^{\prime }\left(t\right)$, whose direction is switched at $t=0$. We present the dimensionless system using the characteristic length of the membrane l (e.g., $l\approx {10}^{-5}$ m) as the length scale, and the bending relaxation time ${\tau }_B=\eta l^3/\overline{B}$ as the time scale, where $\eta$ is the viscosity of the fluid, and $\overline{B}$ represents the bending modulus of the more rigid phase. It is worth noting that there are three different time scales: the bending relaxation time ${\tau }_B$, the extensional time scale ${\tau }_S=1/S^{\prime }$ which is detemined by the strength of the applied flow field, and the diffusion time scale ${\tau }_D$, which is defined below. Also, the excess length, defined as $\Delta =L/R-2\pi$ where L is the total arc length and $A=\pi R^2$ is the total area, represents a measure of the vesicle’s circularity.

2.1. Flow Field

Let $\Sigma$ represent the interface, e.g., the membrane, between the fluids in and outside the vesicle. We employ the Stokes equations to model the ambient fluid within low Reynolds number regime:

$$\nabla ·\mathbf{T}=0\mathrm{and}\nabla ·\mathbf{u}=0\mathrm{in}{\mathbb{R}}^2\setminus \Sigma ,$$

(1)

where $\mathbf{T}=-p\mathbf{I}+\eta (\nabla \mathbf{u}+\nabla {\mathbf{u}}^T)$ is the viscous stress tensor, $\mathbf{u}$ is the velocity, and p is the pressure. The volume constraint, or area constraint in two dimensions, is inherently satisfied due to the incompressibility condition in Equation (1).

Velocity continuity across the interface $\Sigma$ is expressed as

$${\left[\left[\mathbf{u}\right]\right]}_{\Sigma }\equiv \mathbf{u}{{|}_{\Sigma ,\mathrm{in}}-\mathbf{u}|}_{\Sigma ,\mathrm{out}}=\mathbf{0},$$

(2)

while a discontinuity in stress across the interface follows the Laplace-Young condition:

$${\left[\left[\mathbf{T}\mathbf{n}\right]\right]}_{\Sigma }=\mathbf{f},$$

(3)

where $\mathbf{f}$ is the nondimensional total force derived later in Equation (15), and $\mathbf{n}$ is the outward normal vector of the interface $\Sigma \left(t\right)$.

In the far-field, we consider a plane time-dependent elongational flow with a strength S, given by:

$${\mathbf{u}}^{\infty }\left(\mathbf{x}\right)=\mathrm{sign}\left(\mathrm{t}\right)S(-x,y)\mathrm{for}\parallel \mathbf{x}\parallel \to \infty ,$$

(4)

where ${\displaystyle \mathrm{sign}\left(t\right)=\left\{\begin{array}{cc}-1,\hfill & \hfill t<0\\ 1,\hfill & \hfill t\ge 0\end{array}\right.}$, and $S={\tau }_B/{\tau }_S$ is the nondimensional constant extesional rate. Due to the linear nature of the Stokes equations, we have:

$$\mathbf{u}={\mathbf{u}}^{\mathrm{ind}}+{\mathbf{u}}^{\infty }\mathrm{in}{\mathbb{R}}^2,$$

(5)

where ${\mathbf{u}}^{\mathrm{ind}}$ is the velocity field induced by the presence of the vesicle. Lastly, the interface $\Sigma \left(t\right)={\mathbf{x}}_{\Sigma }\left(t\right)$ moves with the fluid velocity by a no-slip boundary condition:

$${\displaystyle \frac{d{\mathbf{x}}_{\Sigma }}{dt}}{=\mathbf{u}|}_{\Sigma },$$

(6)

and the vesicle memrabe is locally inextensible:

$${\nabla }_{\mathbf{s}}·\mathbf{u}=0,$$

(7)

where the vector $\mathbf{s}$ is the counter clockwise tangential directioin of the interface $\Sigma$, and the operator ${\nabla }_{\mathbf{s}}=(\mathbf{I}-\mathbf{nn})\nabla$.

2.2. Material Field

In this study, we focus on vesicles whose membrane is composed of only two surface phases (or lipid components). Let $\psi (s,t)$ represent the mass concentration of the softer phase (liquid phase), where s is the arc length parameterizing the interface $\Sigma$. Then the concentration of the harder phase (e.g., solid or gel phase) reads $1-\psi (s,t)$. Also, the harder phase can be induced by other chemicals that interact with the membrane components and alter their physical properties. In order to simplfy the model, we assumpe no chemical reactions or heat conduction, that the total mass of each phase is conserved and distributed only on the interface. By doing so, the mass conservation can be expressed as:

$$M_{\psi }\left(t\right)\equiv \underset{\Sigma \left(t\right)}{\int }\psi (s,t)ds=M_{\psi }\left(0\right).$$

(8)

Note that in Equation (8), we simply assume that the density of each phase equals to one. Consequently, the phase concentration $\psi (s,t)$ evolves according to the convection-reaction-diffusion equation. Its local form in Eulerian corrdinates reads:

$${\psi }_t+\mathbf{u}·\nabla \psi -\mathbf{n}·\nabla \mathbf{u}·\mathbf{n}\psi ={\nabla }_{\mathbf{s}}·\mathbf{J},$$

(9)

where $\mathbf{J}$ denotes the surface flux that is derived below in Equation (16). The expression $-\mathbf{n}·\nabla \mathbf{u}·\mathbf{n}={\nabla }_{\mathbf{s}}·{\mathbf{u}}_{\mathbf{s}}+H\mathbf{u}·\mathbf{n}$, where ${\mathbf{u}}_{\mathbf{s}}=(\mathbf{I}-\mathbf{nn})\mathbf{u}$ is the tangential velocity on the interface, and H is the local mean curvature.

For an incompressible Stokes flow, $\mathbf{n}·\nabla \mathbf{u}·\mathbf{n}$ represents the local changing rate of the interfacial area (or the arc length in 2D). Therefore, this term in Equation (9) describes how the value of $\psi$ changes with respect to interface stretching. Since the vesicle membrane is assumed to be locally inextensible, Equation (9) becomes a diffusion equation and in Lagrangian coordinates it reads

$${\psi }_t={\nabla }_{\mathbf{s}}·\mathbf{J}.$$

(10)

2.3. Constitutive Relations

Following [26,27], the free energy of the system is expressed as

$$\begin{array}{ccc}\hfill E& =& E_b+E_{\psi }+E_{\lambda },\mathrm{with}\hfill \end{array}$$

(11)

$$\begin{array}{ccc}\hfill E_b& =& {\displaystyle \frac{1}{2}{\int }_{\Sigma \left(t\right)}B\left(\psi \right)H^2\mathrm{d}s,}\hfill \end{array}$$

(12)

$$\begin{array}{ccc}\hfill E_{\psi }& =& {\displaystyle \frac{a_0}{ϵ}}{\int }_{\Sigma \left(t\right)}\left(g\left(\psi \right)+{\displaystyle \frac{{ϵ}^2}{2}}{\left|{\nabla }_{\mathbf{s}}\psi \right|}^2\right)\mathrm{d}s,\hfill \end{array}$$

(13)

$$\begin{array}{ccc}\hfill \mathrm{and}{E}_{\lambda }& =& {\int }_{\Sigma \left(t\right)}\Lambda \mathrm{d}s.\hfill \end{array}$$

(14)

Here, $E_{\mathrm{b}}$ represents the non-dimensional bending energy, which varies with the local phase concentration $\psi$ via a linear function $B\left(\psi \right)=(1-\psi )B_2+\psi B_1$ for simplicity, where $B_1$ and $B_2$ are the bending moduli for the softer phase and harder phase, resepectively.

$E_{\psi }$ is the line energy associated with the surface phases. The term $g\left(\psi \right)$ is a double-well potential defined as ${\displaystyle g\left(\psi \right)=\frac{1}{4}{\psi }^2{(1-\psi )}^2}$, with the two minima leading to stable phases at $\psi =0$ and $\psi =1$. The parameter $\epsilon ={\epsilon }^{\prime }/l$ is a small non-dimensional constant quantifying the excess energy due to surface gradients, where ${\epsilon }^{\prime }$ is the corresponding dimensional parameter. Such scaling ensures that $E_{\psi }$ approaches a finite constant as $\epsilon \to 0$. The non-dimensional size of the line energy is given by $a_0=a_0^{\prime }l/B_1$, where $a_0^{\prime }$ is a measure of the dimensional line energy.

Finally, $E_{\lambda }$ denotes the energy associated with membrane tension $\Lambda$, acting as a Lagrange multiplier to enforce local arc length inextensibility.

Taking the variation of the total membrane energy with respect to the position of the interface, we get the force density acting on the membrane $\mathbf{f}=-\delta E/\delta \Sigma$, given by:

$$\begin{array}{cc}\hfill {\displaystyle \frac{\delta E}{\delta \Sigma }}=& \left(-{\left(B\left(\psi \right)H\right)}_{ss}-{\displaystyle \frac{B\left(\psi \right)}{2}}{H}^3+{\displaystyle \frac{a_0}{\epsilon }}\left(g\left(\psi \right)-{\displaystyle \frac{{\epsilon }^2}{2}}{\psi }_s^2\right)H+\Lambda H\right)\mathbf{n}\hfill \\ \hfill & +\left(\left({\displaystyle \frac{a_0}{\epsilon }}\left({\displaystyle \frac{dg\left(\psi \right)}{d\psi }}-{\epsilon }^2{\psi }_{ss}\right)+{\displaystyle \frac{1}{2}}{\displaystyle \frac{dB\left(\psi \right)}{d\psi }}{H}^2\right){\psi }_s-{\Lambda }_s\right)\mathbf{s}\hfill \end{array}$$

(15)

where ${(·)}_s$ is the differentiation with respect to the arc length.

For mass transportation, the surface flux is defined as:

$$\mathbf{J}=\nu {\nabla }_{\mathbf{s}}\mu ,$$

(16)

where $\nu ={\tau }_B/\left({\tau }_D{a}_0/\epsilon \right)$ is a positive, non-dimensional mobility coefficient, with ${\tau }_D$ being the diffusion timescale ${\tau }_D={\nu }^{\prime }{a}_0^{\prime }\left(l^2/{\epsilon }^{\prime }\right)$ and ${\nu }^{\prime }$ the dimensional mobility coefficient. For simplicity we set $\nu =1$, and other choices yield results that are qualitatively similar to those presented in the main text [26]. The chemical potential $\mu$ is then expressed as:

$$\mu ={\displaystyle \frac{\delta E}{\delta \psi }}={\displaystyle \frac{1}{2}}{\displaystyle \frac{dB\left(\psi \right)}{d\psi }}{H}^2+{\displaystyle \frac{a_0}{\epsilon }}\left({\displaystyle \frac{dg\left(\psi \right)}{d\psi }}-{\epsilon }^2{\psi }_{ss}\right).$$

(17)

If there is no energy added into the system (i.e., ${\mathbf{u}}_{\infty }=0$), we can substitute constitutive relations Equations (15) and (17) into the following equation,

$${\displaystyle \frac{dE}{dt}}={\int }_{\Sigma \left(t\right)}{\displaystyle \frac{d\psi }{dt}}{\displaystyle \frac{\delta E}{\delta \psi }}d\Sigma +{\int }_{\Sigma \left(t\right)}\mathbf{u}·{\displaystyle \frac{\delta E}{\delta \Sigma }}d\Sigma ,$$

(18)

and perform integration by parts using the divergence theorem. Then we get the following energy dissipation formula,

$${\displaystyle \frac{dE}{dt}}=-{\int }_{\Sigma }\nu {\left|{\nabla }_{\mathbf{s}}\mu \right|}^{2}d\Sigma -{\displaystyle \frac{1}{2}}{\int }_{\Sigma }\eta (\nabla \mathbf{u}+\nabla {\mathbf{u}}^T):(\nabla \mathbf{u}+\nabla {\mathbf{u}}^T)d\Sigma <0.$$

(19)

Thus, the constitutive assumptions are consistent with the second law of thermodynamics.

3. Review of Linear Analysis

In previous works [11,12], we conducted a linear stability analysis of a slightly perturbed circular vesicle with only one phase. Here, we briefly summarize the main results. The perturbed shape of the vesicle is described by $P(t,\theta ):r=R(1+P)$, where R is the radius of an equivalent circle, and the perturbation P is expanded in Fourier harmonics as:

$$P(t,\theta )=\sum _{-\infty }^{+\infty }{C}_k{P}_{k}exp\left(ik\theta \right).$$

(20)

The term $P_k\left(t\right)$ represents the amplitude of the k-th mode, and the coefficient ${\displaystyle C_k=\sqrt{\frac{\Delta }{\pi (k^2-1)}}}$ is utilized purely to simplify the formula.

The external flow field in polar coordinate is given by:

$$V_r=\omega \mathrm{sign}\left(t\right)S{\displaystyle \frac{r}{R}}cos2\theta ,V_{\theta }=-\omega \mathrm{sign}\left(t\right)S{\displaystyle \frac{r}{R}}sin2\theta ,$$

(21)

where the coefficient ${\displaystyle \omega =\frac{B}{R^3\eta }\sqrt{\frac{\Delta }{6\pi }}}$, and B is the bending modulus. The term $\mathrm{sign}\left(t\right)$ denotes the switch of flow direction at $t=0$.

The conservation of the vesicle’s arc length requires ${\sum }_{k\ne 0,1}{|P_k|}^2=2$ [11]. Solving the Stokes equation with the specified external flow, we can get the growth rate of the k-th mode perturbation:

$${\tau }_B{\dot{P}}_k=\mathrm{sign}\left(t\right)S\left({\delta }_{k,2}+{\delta }_{k,-2}\right)-(A_k\lambda +{\Gamma }_k)P_k,$$

(22)

where ${\delta }_{i,j}$ is the Kronecker delta, ${\displaystyle A_k=\left|k\right|/4}$, ${\displaystyle {\Gamma }_k=\left|k\right|(k^2-3/2)/4}$, and $\lambda$ is the nondimensional surface tension. The characteristic timescale ${\displaystyle {\tau }_B=R^3\eta /B}$. In a stationary state, e.g., ${\dot{P}}_k=0$, the vesicle adopts an ellipsoidal shape, predominantly storing excess arc length in harmonics $k=\pm 2$, with $|P_k|\ll 1$ for $\left|k\right|>2$.

For small S, higher mode harmonics ($\left|k\right|>2$) can not be excited after the direction of the flow is switched at $t=0$, and the dynamics of the vesicle become purely rotational, transitioning from one equilibrium state with $P_{\pm 2}=1$ (horizontal) to another $P_{\pm 2}=-1$ (vertical).

If S is sufficiently large, higher mode harmonics are excited once the direction of the flow is swithced at $t=0$, leading to a three-stage wrinkling dynamics [9,11,12]. In the first stage, most excess arc length remains in the second-order Fourier harmonics, and higher modes grow exponentially while surface tension remains nearly constant as $\lambda =-S/A_2$. The second stage starts once the exponential growth saturates. The surface tension $\lambda$ declines slowly and the most unstable mode shifts towards smaller k. At the end of this second stage, when the amplitude of the wrinkles peaks, the distribution of higher mode harmonics centers around $k^{*}\sim S^{1/3}$. The characteristic wavelength and time are given by:

$${\displaystyle \frac{1}{k^{*}}}\sim S^{-1/3}\mathrm{and}T\sim S^{-1}.$$

(23)

Finally, the wrinkles smooth out in the third stage, and the vesicle approaches a new stable state with $P_{\pm 2}=-1$. As shown in Figure 1.

4. Numerical Method

The numerical scheme is based on the work of [26,27]. Here we just introduce a brief summary of the method. Due to the terms with high-order spatial derivatives, the governing equations for both the membrane phase evolution and position evolution are numerically stiff. Therefore, explicit time-marching schemes that require extremely small time-step sizes are very expensive. Following [26], we use a small-scale decomposition approach for the time-marching scheme.

We use the boundary integral method to solve the Stokes equations [30], that the velocity field at any point $\mathbf{x}$ is represented by a boundary integral:

$$\mathbf{u}\left(\mathbf{x}\right)={\mathbf{u}}_{\infty }\left(\mathbf{x}\right)+{\displaystyle \frac{1}{4\pi \eta }}{\int }_{\Sigma }\mathbf{G}(\mathbf{x}-\mathbf{y})\mathbf{f}\left(\mathbf{y}\right)\mathrm{d}\Sigma \left(\mathbf{y}\right),$$

(24)

where Green’s function for the Stokes equation in free-space reads

$$\mathbf{G}\left(\mathbf{r}\right)=-log\rho \mathbf{I}+{\displaystyle \frac{\mathbf{r}\otimes \mathbf{r}}{{\rho }^2}},\rho ={\parallel \mathbf{r}\parallel }_2.$$

(25)

By construction, the Equation (24) satisfies the Stokes equations and the boundary conditions Equations (2) and (3). Taking the limit as $\mathbf{x}\to \Sigma$ and then applying the kinematic boundary condition Equation (6), we get an integro-differential equation for the moving membrane. Finally, we use a second-order method to update the material points on the membrane, by evolving the tangent angles of each point. For more details, please refer to [26].

5. Numerical Results

In this section, we present numerical results illustrating the contribution of phase domain to the vesicle dynamics. We first consider a nearly circular vesicle with only limited excess arc length, and compare our results with previous perturbation analysis [11,12]. Then we study the case of elongated vesicles with large excess arc length. In the following parts, the vesicle membrane is composed of two lipid phases, which are always denoted respectively by the red color ($B_1=0.3$) and the blue color ($B_2=1$) in the figures. The initial phase distribution at each point along the membrane is set to be a fixed concentration, e.g., $\psi =0$, $0.3$, $0.5$, and $0.7$ with small perturbation, e.g., ${10}^{-3}$ [26,27]. Therefore, the two phases are almost evenly distributed along the vesicle before the simulation starts. We put the vesicle in an extensional flow, i.e., $\dot{x}=Sx$, $\dot{y}=-Sy$, and run the simulation until the phase decomposition stops, e.g., significant phase boundaries separating the two phases emerge and the vesicle shape reaches an equilibrium. Then we reverse the direction of the flow and observe the dynamics of the vesicle.

Otherwise noted, the time step is set to be $\Delta t=0.5\times {10}^{-4}$, and the spatial resolution is set to be $N=1024$. The error tolerance for GMRES is ${10}^{-12}$ and a 25th order Fourier smoothing with the filter level ${10}^{-13}$ is used.

5.1. Vesicles with Limited Excess Arc Length

In the numerical tests, we use an ellipse with aspect ratio a:b = 1:0.8, i.e., $\Delta =0.059$ to represent a quasi-circular vesicle. To quantitatively describe the morphological evolution of the vesicle, we define a variable $\delta d$ to measure the maximum shape deviation from the equivalent circle with radius $R=\sqrt{A/\pi }$ as

$$\delta d\left(t\right)=max|r_i\left(t\right)-R|,$$

(26)

where $r_i\left(t\right)$ measures the distance of a material point $(x_i\left(t\right),y_i\left(t\right))\in \Sigma \left(t\right)$ from the geometric center of the vesicle $(x_c\left(t\right),y_c\left(t\right))$ at time t.

Perturbation theory indicates that for nearly circular vesicles, extensional flow with a large rate S triggers high-frequency wrinkles, divisible into initiation, development, and decay stages. While small S leads to purely rotational evolution without wrinkles [11,12]. As phase field energy $E_{\psi }$ was introduced into the model, the wrinkling dynamics became much more complicated. For a given shape, line energy $E_{\psi }$ is lower when there are fewer boundaries separating the two phases, and the bending energy $E_B$ is lower when the soft red phase accumulates in the higher curvature regions. Because the vesicle is nearly circular, the curvature differences among all points are small, making $E_{\psi }$ the dominant factor.

In Figure 2, we plot the evolution of the shape parameter $\delta d$ for $\psi =0,0.3,0.5$, and $0.7$ in both strong flow ($S=40$) and weak flow ($S=2$). The numerical results are highly consistent with the perturbation analysis when $\psi =0$. And the dynamics for different $\psi$ are roughly the same under the same extensional rate.

In Figure 3, we show the evolution of the global surface tension $\lambda \left(t\right)$ for $S=40$. $\lambda \left(t\right)$ which is computed as the mean value of the local surface tension, can be interpreted as a global Lagrange multiplier. The three-stage wrinkling behavior can be observed for all $\psi$ ranging from 0.3 to 0.7, and the values of $\lambda$ at $t=0^{-}$ (before the flow reversal) and $t\to \infty$ are the same for all cases, too. This is because the surface tension is determined by the shape of the vesicle and the external flow field. Note that the duration of the first stage for $\psi =0$ is longer than for other cases, because the red softer region is more prone to wrinkling when compressed by the fluid.

In Figure 4, we plot the detailed evolution sequences of vesicles for $\psi =0$, $0.3$, $0.5$, and $0.7$ under extensional rates $S=40$ and $S=2$. Before the flow is reversed at $t=0$, the softer red phase accumulates in the regions featuring higher curvature, while the harder blue phase accumulates in the lower curvature regions. Such a phase decomposition occurs universally for $\psi$ values ranging from 0.25 to 0.75. After the flow is reversed at $t=0$, because the values of the bending modulus for all the points are no longer uniform throughout the vesicle, the wrinkling dynamics become highly asymmetric. Specifically, for the case $S=40$ and $\psi =0.7$, high frequency wrinkles arise at the bottom of the vesicle when $t=0.0076$.

5.2. Dynamics for Elongated Vesicles

Next we investigate into the nonlinear dynamics of elongated vesicles. Here, the vesicle is configured to be an ellipse with an aspect ratio a:b = 1:0.25, that is, $\Delta =2.30$. The initial phase distribution at each point is also set to be a fixed concentration with a small perturbation. Again, we compare the wrinkling dynamics for different average concentrations $\psi =0,0.3,0.5,$ and 0.7 both in strong and weak flow.

5.2.1. Wrinkling Dynamics When $S=40$

In Figure 5, we plot $\delta d$ as a function of time for all $\psi$. The dynamics can be divided into the two stages, and the equilibrium shapes at $t=0^{-}$ and $t\to \infty$ for different $\psi$ are almost the same, because the dynamics are dominated by the external flow field.

Figure 6 shows the detailed morphological evolution of the vesicle for $\psi =0$, 0.3, 0.5 and 0.7. Note that there are four phase boundaries for the initial equilibrium state of the vesicle when $\psi =0.3$ and $0.5$, and the red phase accumulates around both the right and left high curvature ends. On the other hand, there are only two phase boundaries for the $\psi =0.7$ case, which is because the total amount of the blue phase is small and it can accumulate at one of the lower curvature regions. The dynamics of the $\psi =0$, 0.3, and 0.5 cases are highly symmetric, while the $\psi =0.7$ case is not.

Figure 7a shows the evolution of global surface tension for $\psi =0$, 0.3, 0.5 and 0.7. We find that the global surface tension $\lambda$ at $t=0^{-}$ and $t\to \infty$ is the same for all $\psi$, since the equilibrium shapes for all cases are the same. In Figure 7b, we plot the evolution of local surface tension $\lambda (\alpha ,t)$ for $\psi =0.3$. Here, $\alpha$ is the variable that denotes the position on the vesicle membrane, where $\alpha =0$ and $\alpha =1$ refer to the same point $(1,0)$ on the very right end at $t<0$, as shown in Figure 1. We find that $\lambda (0,t<0)=\lambda (1/4,t\to \infty )$ and $\lambda (1/4,t<0)=\lambda (0,t\to \infty )$, which indicates that the two equilibrium shapes at $t<0$ and $t\to \infty$ are the same, despite a $\pi /2$ rotation.

In Figure 8a, we compare the wrinkling behavior for $\psi =0.7$ with the wrinkling behavior of vesicles composed of only one particular phase under the same extensional rate $S=40$. For $\psi =0.7$, the wrinkles at the upper blue part are similar to the ones at the top of the blue vesicle, while the wrinkles in the lower red part are similar to the ones at the bottom of the red vesicle. This clearly indicates that the characteristic wavelength is well preserved for each phase respectively, which again proves that the wrinkling behavior is dominated by the elastic property and the external flow field. Also note that the wrinkling dynamics of the $\psi =0.7$ case involve a pure rotation caused by non-symmetric phase distribution, where the blue phase moves from the top to the left in the end, which is consistent with the evolution of local surface tension, as shown in Figure 8b.

5.2.2. The Buckling Dynamics When $S=2$

When S is small, $E_{\psi }$ dominates the dynamics. Since the dynamics are prolonged nearly 10 times compared with vesicles with the nearly circular shapes, the non-symmetric phase distributions affect the dynamics dramatically, as shown in Figure 9 and Figure 10. Specifically, for $0.3\le \psi \le 0.7$, the vesicle can buckle inward in the red, softer region.

In Figure 9, we plot $\delta d$ as a function of time for the $\psi =0,0.3,0.5$, and $0.7$ cases under the conditions of $S=2$ and $\Delta =2.30$. The equilibrium shapes at $t=0^{-}$ and $t\to \infty$ for different cases are quite different. Particularly for the $\psi =0.5$ case, the initial and the final equilibrium states are not the same, since the two soft red regions merge together during the evolution.

Figure 10 shows the detailed dynamics for the $\psi =0$, 0.3, 0.5, and 0.7 cases when $S=2$. For $0.3\le \psi \le 0.7$, phase decomposition starts immediately as the dynamics begin. Consistent with previous observations, the soft red phase accumulates in the higher curvature regions, while the hard blue phase accumulates in the lower curvature regions. After the direction of the flow is switched at $t=0$, the vesicles buckle inward at the softer red parts, and the more is the red phase, the more significant is the buckling. For $\psi =0.3$, the external fluid at the right-hand side pushes the membrane to bend inward. While for $\psi =0.5,0.7$, the softer red phase clusters in the upper region, where the vesicles slowly buckle inward. Nevertheless, for all cases, the vesicles are totally flattened in the end by the external flow.

6. Conclusions and Future Work

We have used a boundary integral method to simulate the dynamics of two-dimensional inextensible multicomponent vesicles in time-dependent extensional flows at finite temperatures. The fluids were governed by Stokes flow with singular forces, due to bending, elastic tension, and complex phase field. Since the vesicles are inextensible, the elastic tension is not an independent parameter, but rather is a Lagrange multiplier required to prevent stretching or compression of the vesicle boundary. We focus on the transient wrinkling instability, which occurs when an applied extensional flow is impulsively started, or the flow direction is suddenly reversed. The negative elastic tension generated by the inextensibility of the vesicle boundary drives high-frequency perturbations to compensate for the compressive stress caused by external flow field changes. When the vesicle aligns with the new flow field, the compressive force is released, and the elastic tension turns non-negative and the interface perturbation disappears.

For the two-phase vesicles considered in this work, the phases separate into regions that consist of one phase only, where the hard phase congregates in the low curvature region, and the soft phase congregates in the high curvature region. This asymmetry in the elastic configuration breaks the symmetry inherited from the applied extensional flow, leading to a more complex and dynamic evolution compared to the pure phase cases.

The findings presented herein add to the growing body of evidence highlighting the crucial role of material composition on the dynamics of biomembranes at microscopic scales. Essentially, we have discovered that the presence of compound phase domains serves as a potent symmetry-breaking mechanism for the vesicle interface, profoundly influencing its dynamical behavior. When the excess length is minimal, the asymmetry in vesicle shape arises inherently from inhomogeneous bending and phase segregation, resulting in a predictable limited response under strong flow conditions. Conversely, as the excess length increases, inhomogeneous bending becomes more significant, allowing the vesicle to undergo highly asymmetric wrinkling under strong flow or even buckling under weak flow, especially when the phases are unevenly distributed.

Several assumptions were made in this work. Firstly, we assumed a linear combination of the individual bending moduli to represent the bending modulus for the composite phase, although our simulation framework is capable of accommodating more sophisticated models. Secondly, we assumed equal viscosities for the interior and exterior fluids, noting that inhomogeneous bending may interact intricately with the wrinkling dynamics when there are viscosity contrasts. Additionly, wrinkles have also been observed in other flow fields such as time-dependent shear flow. Finally, although the simulation is conducted in two dimensions, the results are expected to be relevant to three-dimensional flows. And the analysis is valid for vesicles of multiple size and multilamelarity.