Dynamics of fluid vesicles in shear flow: Effect of membrane viscosity and thermal fluctuations

The dynamical behavior of vesicles is investigated in simple shear flow. A simulation technique is presented that combines a three-dimensional particle-based mesoscopic model (multiparticle collision dynamics) for the solvent with a dynamically triangulated surface model for the membrane. In this model, thermal fluctuations of the solvent and of the membrane are consistently taken into account. The membrane viscosity can be varied by changing the bond-flip rate of the dynamically triangulated surface. Vesicles are found to transit from steady tank-treading to unsteady tumbling motion with increasing membrane viscosity. At small reduced volumes, the shear induces a transformation from a discocyte to a prolate shape at low membrane viscosity. On the other hand, at high membrane viscosity, the shear induces a transformation from prolate to discocyte, or tumbling motion accompanied by shape oscillations between these two states. Thermal fluctuations induce intermittent tumbling and smooth out the transitions. This effect can be understood from a simplified stochastic model.

Figure 1

Dependence of the membrane viscosity ${\eta }_{\mathrm{mb}}$ on the probability $\psi$ for the selection of a bond for a bond-flip attempt.

Figure 2

(Color online) Free energy $F\left(\alpha \right)$ as a function of the asphericity $\alpha$ for the reduced volumes $V^{*}=0.59$ (solid line), 0.66 (dashed line), and 0.78 (dash-dotted line) in the absence of shear flow. Error bars are shown for a few data points.

Figure 3

(Color online) Snapshot of a discocyte vesicle under simple shear flow at the reduced shear rate ${\dot{\gamma }}^{*}=0.92$ and ${\eta }_{\mathrm{mb}}^{*}=0$. The arrows represent the velocity field in the $xz$ plane.

Figure 4

(Color online) Dependence of the average inclination angle $⟨\theta ⟩(-\pi ∕2⩽\theta <\pi ∕2)$ and average angular velocity $⟨\omega ⟩$ on the reduced volume $V^{*}$ at the reduced shear rate ${\dot{\gamma }}^{*}=0.92$ and ${\eta }_{\mathrm{mb}}^{*}=0$. Circles and squares represent prolate and discocyte vesicles, respectively. The solid and dashed lines are calculated by KS theory for prolate and oblate ellipsoids, respectively.

Figure 5

(Color online) Dependence of the average inclination angle $⟨\theta ⟩$ on the membrane viscosity ${\eta }_{\mathrm{mb}}^{*}$ for ${\dot{\gamma }}^{*}=0.92$ and various volumes $V^{*}$. Circles and squares represent prolate and discocyte vesicles at $V^{*}=0.59$, respectively. Triangles, diamonds, crosses, and pluses represent vesicles at $V^{*}=0.66$, 0.78, 0.91, and 0.96. The solid and dashed lines are calculated by KS theory for prolate ($V^{*}=0.59$, 0.66, 0.78, 0.91, and 0.96) and oblate ellipsoids $(V^{*}=0.59)$, respectively. The dash-dotted lines are calculated by KS theory and Eq. (19) with rotational Péclet number $\chi =600$, for $V^{*}=0.66$ and 0.78.

Figure 6

(Color online) Time dependence of asphericity $\alpha$ and inclination angle $\theta$, for $V^{*}=0.78$, ${\eta }_{\mathrm{mb}}^{*}=1.62$, and ${\dot{\gamma }}^{*}=0.92$. The dashed lines are obtained from Eqs. (18, 19) with ${\zeta }_{\alpha }^{*}=100$, ${\zeta }_{\theta }^{*}=30$, $A^{*}=18$, and $B\left(\alpha \right)=1.0$.

Figure 7

Relative membrane viscosity ${\eta }_{\mathrm{mb}}^{\mathrm{c}}$, at which tumbling motion is first observed for ${\dot{\gamma }}^{*}=0.92$. The simulation data (엯) are compared with the results of KS theory (solid line) for prolate shapes.

Figure 8

(Color online) Dependence of the average angular velocity $⟨\omega ⟩$ on the membrane viscosity ${\eta }_{\mathrm{mb}}^{*}$ for shear rate ${\dot{\gamma }}^{*}=0.92$ and various volumes $V^{*}$. Circles and squares represent prolate and discocyte vesicles at $V^{*}=0.59$, respectively. Triangles, diamonds, crosses, and pluses represent vesicles at $V^{*}=0.66$, 0.78, 0.91, and 0.96. The solid and dashed lines are calculated by KS theory for prolate ($V^{*}=0.59$, 0.66, 0.78, 0.91, and 0.96) and oblate ellipsoids $(V^{*}=0.59)$, respectively.

Figure 9

(Color online) Dynamical phase diagram of a vesicle in shear flow, for reduced volume $V^{*}=0.59$. Symbols show simulated parameter values, and indicate tank-treading discocyte and tank-treading prolate (엯), tank-treading prolate and unstable discocyte (▵), tank-treading discocyte and tumbling (transient) prolate (◻), tumbling with shape oscillation (◇), unstable stomatocyte $(+)$, stable stomatocyte $(\times )$, and near transition (∎). The dashed lines are guides to the eye.

Figure 10

(Color online) Time dependence of asphericity $\alpha$ and inclination angle $\theta$, for $V^{*}=0.59$, ${\eta }_{\mathrm{mb}}^{*}=0.87$. The dashed lines are obtained from Eqs. (18, 19) with $B\left(\alpha \right)=1.12-0.14\alpha$. The other parameters are set to the same values in Fig. 6. Snapshots of the vesicle cross section in the $xz$ plane are also shown.

Figure 11

(Color online) Time dependence of asphericity $\alpha$ and inclination angle $\theta$, for $V^{*}=0.59$ and ${\eta }_{\mathrm{mb}}^{*}=0.34$. The dashed and solid lines represent the vesicle at ${\dot{\gamma }}^{*}=1.84$ and 3.68, respectively. Snapshots of vesicles (a tank-treading stomatocyte at $t∕\tau =45$, and a vesicle during the stomatocyte-to-prolate transformation at $t∕\tau =23$) are also shown.

Figure 12

(Color online) Dependence of (a) the mean angle $⟨\theta ⟩(-\pi ∕2⩽\theta <\pi ∕2)$ and (b) the tumbling frequency $f_{\mathrm{tmb}}$ on $B$ obtained from Eq. (19) with various relative shear rates $\chi$. The errors are less than (a) $0.0003\pi$ and (b) 0.2.

Figure 13

(Color online) Dependence of the dimensionless tumbling frequency $⟨f_{\mathrm{tmb}}⟩∕D_{\theta }$ on the rotational Péclet number $\chi$, for different values of $B$ close to the transition point, as obtained from Eq. (19). The errors are less than 0.1.

Figure 14

(Color online) Dependence of the inclination angle $\theta$ on the viscosity ${\eta }_{\mathrm{in}}$ of an internal fluid at $V^{*}=0.59$ and ${\eta }_{\mathrm{mb}}^{*}=0$ or 0.5 in KS theory. The solid and dashed lines represent prolate and oblate ellipsoids, respectively.

Figure 15

(Color online) Asphericity $\alpha$ of free-energy minima with each radius of gyration $R_{\mathrm{g}}∕R_0$ at $V^{*}=0.59$. The solid and dashed lines are the deepest and second minima, respectively. The inset shows the free energy of the asphericity $\alpha$ at $R_{\mathrm{g}}∕R_0=1.1825$ (along the thick vertical line).

Figure 16

Finite-size effects of the inclination angle $⟨\theta ⟩$ for $V^{*}=0.59$, ${\eta }_{\mathrm{mb}}^{*}=0$, and $\dot{\gamma }=0.92$ in the prolate phase. Circles, diamonds, squares, and triangles represent simulation results for $R_0∕L_y=R_0∕L_z=0.24$, 0.28, 0.19, and 0.16, respectively.