Hydrodynamics and rheology of a vesicle doublet suspension

The dynamics of an adhesive two-dimensional vesicle doublet under various flow conditions is investigated numerically using a high-order, adaptive-in-time boundary integral method. In a quiescent flow, two nearby vesicles move slowly toward each other under the adhesive potential, pushing out fluid between them to form a vesicle doublet at equilibrium. A lubrication analysis on such draining of a thin film gives the dependencies of draining time on adhesion strength and separation distance, which are in good agreement with numerical results. In a planar extensional flow, we find that a stable vesicle doublet forms only when two vesicles collide head-on around the stagnation point. In a microfluid trap where the stagnation of an extensional flow is dynamically placed in the middle of a vesicle doublet through an active control loop, novel dynamics of a vesicle doublet are observed. Numerical simulations show that there exists a critical extensional flow rate above which adhesive interaction is overcome by the diverging stream, thus providing a simple method to measure the adhesion strength between two vesicle membranes. In a planar shear flow, numerical simulations reveal that a vesicle doublet may form provided that the adhesion strength is sufficiently large at a given vesicle reduced area. Once a doublet is formed, its oscillatory dynamics is found to depend on the adhesion strength and their reduced area. Furthermore the effective shear viscosity of a dilute suspension of vesicle doublets is found to be a function of the reduced area. Results from these numerical studies and analysis shed light on the hydrodynamic and rheological consequences of adhesive interactions between vesicles in a viscous fluid.

Figure 1

(a) The adhesion potential $\varphi \left(z\right)$ in Eq. (2). Distances less than $\delta$ are repulsive and distances greater than $\delta$ are attractive. (b) Each point on a vesicle is repelled or attracted by all points on the other vesicles. (c) The resulting adhesive force is found by integrating over all points on all other vesicles [Eq. (3)]. The color is the dot product of the adhesion force with the vector joining the vesicles. Therefore, the green and yellow regions are repelled by the other vesicle while the blue regions are attracted to the other vesicle.

Figure 2

Scaling of the time it takes for two vesicles to reach equilibrium. (a) Scaling with respect to $\delta$ with $\mathcal{H}=0.1$. (b) Scaling with respect to $\mathcal{H}$ with $\delta =0.4$. The arrows indicate the time when the equilibrium is reached within $1\%$.

Figure 3

The equilibrium configurations of a doublet of identical vesicles under adhesion in a quiescent flow at three values of $\mathrm{\Delta }A$. The vesicle reduced areas of $\mathrm{\Delta }A=0.95$, 0.75, and 0.6 are labeled. The vesicle length is fixed at $2\pi$, the dimensionless bending modulus is ${\kappa }_b=1$, the Hamaker constant is $\mathcal{H}=5$, and the separation distance is $\delta =0.2$. The color coding is the tension along the vesicle. The overall equilibrium shapes of two vesicles under adhesive interactions vary with the reduced area. The right figure shows the membrane shapes in the contact region. The color coding is the adhesive force projected along the center-of-mass vector.

Figure 4

(a) The total bending energy (solid curve) and adhesion energy (dash-dotted curve) plotted against the reduced area $\mathrm{\Delta }A$. (b) The sum of the two energies plotted against the reduced area $\mathrm{\Delta }A$. The two vesicles in the doublet are of the same length and reduced area. The Hamaker constant is $\mathcal{H}=5$ and the separation distance is $\delta =0.2$.

Figure 5

The total bending energy (a) and adhesion energy (b) of a vesicle doublet at equilibrium vs the adhesion strength $\mathcal{H}$. The two identical vesicles in the doublet have a length of $2\pi$ and a reduced area of $\mathrm{\Delta }A=0.95$, with separation distance $\delta =0.2$ (triangles), 0.3 (crosses), and 0.4 (circles).

Figure 6

The dynamics of a vesicle pair in a planar extensional flow. The Hamaker constant is $\mathcal{H}=0.7$ for the first three rows, and $\mathcal{H}=0.6$ for the bottom row. From top to bottom, the initial vertical displacement is $\mathrm{\Delta }y=0$, 0.02, 0.1, and 0.1. The reduced area is $\mathrm{\Delta }A=0.75$, the extension rate $\dot{\epsilon }=0.09$, and the adhesion distance $\delta =0.4$. The color coding is the tension.

Figure 7

The definition of the inclination angle of a vesicle doublet at the center of an extensional flow. An inclination angle of $\pi$ indicates a doublet whose long axis is orthogonal to the diverging direction, while an inclination angle of $\pi /2$ indicates a doublet whose short axis is orthogonal to the diverging direction. The square mark is a stagnation point and the round point is located at the center of the vesicle. The initial configurations in this section all have an inclination angle of $\theta =\pi$.

Figure 8

A vesicle doublet in an extensional flow with an initial inclination angle $\theta =\pi$. The reduced area is $\mathrm{\Delta }A=0.7$, the Hamaker constant is $\mathcal{H}=0.7$, the separation distance is $\delta =0.4$, and the extension rate is $\dot{\epsilon }=0.02$ (top), $\dot{\epsilon }=0.07$ (middle), and $\dot{\epsilon }=0.1$ (bottom). The center of the vesicle, denoted with a black square, is the stagnation point. The color coding is the tension.

Figure 9

A vesicle doublet in an extensional flow with an initial inclination angle $\theta =\pi$. The reduced area is $\mathrm{\Delta }A=0.8$, the Hamaker constant is $\mathcal{H}=0.7$, the separation distance is $\delta =0.4$, and the extension rate is $\dot{\epsilon }=0.1$. The center of the vesicle, denoted with a black square, is a stagnation point. The color coding is the tension.

Figure 10

The long-time behavior of two vesicles in a fluid trap with Hamaker constant $\mathcal{H}=0.7$ and separation distance $\delta =0.4$. (a) The inclination angle of a doublet formed by vesicles of reduced area $\mathrm{\Delta }A=0.7$ in an extensional flow with an extension rate $\dot{\epsilon }$. For all extension rates, the vesicle's orientation angle tends to a value in $(\frac{\pi }{2},\pi )$. (b) The final inclination angle of the doublets for a variety of reduced areas. At larger extension rates, the doublet is broken by the flow. (c) A phase diagram of the long-time behavior of a fluid trap formed by two adhering vesicles in an extensional flow. Reduced areas and extension rates that do not form a doublet are marked in red, and those that do form a doublet are marked in blue. For the blue marks, the size is proportional to the final inclination angle. Squares indicate that the doublet had not reached an equilibrium inclination angle in the given time horizon.

Figure 11

The formation of a weakly adhered doublet in a shear flow. The reduced area is $\mathrm{\Delta }A=0.9$, the Hamaker constant is $\mathcal{H}=0.7$, the separation distance is $\delta =0.4$, and the shear rate is $\dot{\gamma }=0.5$. The interaction between the two vesicles is a sliding motion, and the dynamics has a period of about 42. The color coding is the tension.

Figure 12

The formation of a strongly adhered doublet in a shear flow. The reduced area is $\mathrm{\Delta }A=0.9$, the Hamaker constant is $\mathcal{H}=2.1$, the separation distance is $\delta =0.4$, and the shear rate is $\dot{\gamma }=0.5$. Since the contact region is unchanged throughout the simulation, the dynamics of the doublet resembles a rigid body motion with a period of about 42. The color coding is the tension.

Figure 13

(a) The distance between a pair of vesicles in a planar shear flow with shear rate $\dot{\gamma }=0.5$ and separation distance $\delta =0.4$. The different lines correspond to a linear spacing of Hamaker constants ranging from $\mathcal{H}=0.1$ and 1.0. (b) The time required for a doublet to complete a single rotation. The reduced area is $\mathrm{\Delta }A=0.90$ and the separation distance is $\delta =0.4$. The different lines correspond to different shear rates and separation distances.

Figure 14

The phase diagram of vesicle hydrodynamics in a shear flow with $\delta =0.1$ (a) and $\delta =0.4$ (b) with reduced area $\mathrm{\Delta }A=0.9$. The critical Hamaker constant $\mathcal{H}$ for binding of two vesicles in a shear flow depends on the shear rate $\dot{\gamma }$. For a doublet to form, lower separation distances require a stronger Hamaker constant at a given shear rate. The stability limit appears to be linear, indicating that there is potentially a stability criterion that depends on a dimensionless number involving the ratio of the Hamaker constant and shear rate.

Figure 15

(a) The intrinsic viscosity of a tank-treading vesicle (blue) and a doublet (red). The shear rate is $\dot{\gamma }=0.5$, the Hamaker constant is $\mathcal{H}=0.7$, and the separation distance is $\delta =0.4$. The black marks denote intrinsic viscosity values computed by Ghigliotti et al. [19] (cf. Fig. 5). (b) The decomposition of the intrinsic viscosity into the contributions for the bending and tension (black) of the vesicles, and the contribution from the vesicle adhesion (green). Also included is the intrinsic viscosity of a single tank-treading vesicle (blue).