4.2.1 Case of $\unicode[STIX]{x1D70E}_{r}=1$

For $\unicode[STIX]{x1D70E}_{r}=1$ , the deformation of a spherical capsule with either a Skalak or a neo-Hookean membrane attains an intermediate maximum prolate deformation at $t=132$ and slowly reaches a steady-state deformation at $Ca=0.25$ considering $\unicode[STIX]{x1D716}_{r}=1$ , ${\hat{C}}_{m}=50$ and ${\hat{G}}_{m}=0$ (figure 3). Both Skalak and neo-Hookean capsules show nearly identical dynamics of deformation. Although there is no dielectric constant or conductivity contrast between the internal and external fluids, a steady-state prolate deformation is observed. The normal electric fields are identical in the two fluids at all times; the tangential field though is discontinuous, such that the external fluid has a higher magnitude than the inner fluid owing to the transmembrane potential. At $t=0$ , the normal (outward) and tangential fields (anticlockwise) are equal for both internal and external fluids on account of the same conductivities and dielectric constants, and hence no net Maxwell stress is generated, and deformation is absent. As time progresses, the normal electric field falls to zero, over the membrane charging time, and at still longer times a fully charged non-conducting membrane leads to zero current. The tangential field, on the other hand, continues to be negative (anticlockwise), with the inner field always lower than the outer, and the difference between the two is proportional to the transmembrane potential. At long times, the inner tangential field is zero, and the outer is $-(3/2)\sin \unicode[STIX]{x1D703}$ , corresponding to the transmembrane potential (figure 23 a in § A.2.2). Thus, the normal Maxwell stresses arise purely due to the tangential field (normal stress is proportional to $E_{n}^{2}-E_{t}^{2}$ , and $E_{n}$ is the same in the inner and outer surfaces). Therefore the normal stresses are always compressive, since the tangential electric field on the outer surface is always higher (figure 23 b in § A.2.2). The net normal compressive stress can be written as an isotropic compressive part and a tensile part corresponding to the second Legendre polynomial of $\cos \unicode[STIX]{x1D703}$ . This tensile part leads to prolate deformation. The tangential stresses are zero at $t=0$ and $t=\infty$ , because of the absence of electric contrast and normal electric fields being zero, respectively. The tangential stresses are from the equator to the pole at intermediate times (figure 23 c in § A.2.2).

4.2.2 Case of $\unicode[STIX]{x1D70E}_{r}>1$

Figure 3 shows that for $\unicode[STIX]{x1D70E}_{r}=10$ the capsule initially undergoes a high prolate deformation and then it slowly relaxes back to a steady-state prolate shape. At very short times $t<t_{MW}$ , the two fluids act as dielectrics, and since there is no dielectric contrast, the deformation is zero. The two fluids act as leaky dielectrics at time $t\sim O(t_{MW})$ , and their conductivity ratio determines their deformation and increases as $\unicode[STIX]{x1D70E}_{r}$ increases. The membrane impedance is small since $t_{MW}<t_{cap}$ , and the membrane is not charged. Therefore, the electric field distribution, as well as the Maxwell stresses are similar to that observed in a liquid drop suspended in an ambient fluid and subjected to an electric field. The current continuity condition means that, when the inner fluid is more conducting ( $\unicode[STIX]{x1D70E}_{r}=10$ ), the polarization vector inside the drop is aligned in the direction of the electric field, whereby there are net positive charges at the north pole and net negative charges at the south pole. Thus, unlike the $\unicode[STIX]{x1D70E}_{r}=1$ case, even the normal field contributes to normal stresses and the normal stresses are predominantly tensile (figure 24 b in § A.2.2), hence a strong prolate deformation is observed. At $t>t_{cap}$ , the normal electric stress at the poles and the tangential stress over the interface disappear as the membrane becomes fully charged. Therefore, the prolate steady-state deformation is mainly due to the compressive electric stress, which is maximum at the equator (figure 24 b in § A.2.2). Note that prolate deformation is observed even in the absence of conductivity contrast, as discussed earlier for $\unicode[STIX]{x1D70E}_{r}=1$ and later elaborated for the case of $\unicode[STIX]{x1D70E}_{r}<1$ .

Figure 4.

Shape evolution of a capsule with Skalak membrane at $\unicode[STIX]{x1D70E}_{r}=10$ for $Ca=0.25$ .

Intermediate shapes in the dynamics of the deformation of an elastic capsule from an initial spherical shape to the final steady-state shape at $Ca=0.25$ are shown in figure 4 for $\unicode[STIX]{x1D70E}_{r}=10$ . A spherical capsule undergoes an intermediate maximum prolate deformation (figure 4 b) before reaching the steady-state deformation (figure 4 d).

4.2.3 Case of $\unicode[STIX]{x1D70E}_{r}<1$

When the external fluid is more conducting ( $\unicode[STIX]{x1D70E}_{r}=0.1$ ), at short times, the polarization vector is from north pole to south pole, opposite in direction to the applied field, that is, the north pole has a net negative charge, and the south pole has a net positive charge. Thus for $t\sim O(t_{MW})$ , the net Maxwell stress at the membrane interface is compressive and leads to an oblate shape. Note that the normal electric field leads to compressive normal stress, and the isotropic electric pressure is also compressive (see figure 25 a–c in § A.2.2 for variation of transmembrane potential, normal and tangential electric stresses with time on a static sphere).

Interestingly, irrespective of the conductivity ratio, the final steady-state deformations are the same and are prolate (figure 3). This independence of the final steady-state shape and deformation with respect to $\unicode[STIX]{x1D70E}_{r}$ can be easily understood: At long times the non-conducting membrane acts like a dielectric, thereby rendering the outer normal electric field zero as well as the corresponding tangential stress also zero (figures 23 c, 24 c and 25 c in § A.2.2). The Maxwell stress is, therefore, only normal (figures 23 b, 24 b and 25 b in § A.2.2). Moreover, since the normal electric field is zero, the normal electric stress is compressive (figures 23 b, 24 b and 25 b in § A.2.2) and is maximum at the equator, the deformation is always prolate.

Thus an oblate–prolate transition is observed at long times for $\unicode[STIX]{x1D70E}_{r}<1$ . A closed-form expression for the transition time is difficult to obtain on account of the equation resulting from $DD=0$ being implicit in time. The physics seems to be governed by the normal Maxwell stress going to zero, which in the case of SEM leads to

(4.1) $$\begin{eqnarray}t\propto t_{cap}\log \frac{\unicode[STIX]{x1D70E}_{r}+2\sqrt{2-\unicode[STIX]{x1D70E}_{r}^{2}}}{(2+\unicode[STIX]{x1D70E}_{r})}.\end{eqnarray}$$

This leads to a value of around $t=173$ for $\unicode[STIX]{x1D70E}_{r}=0.1$ $(t_{cap}=525)$ , which is of the same order as $t=243$ seen in figures 3 and 5(a). Thus the transition time can be expected to be of this order, although not exactly, since it also depends on the capillary number as well as on the ratio of hydrodynamic and electric time scales.

Figure 5.

(a) Time required for oblate to prolate shape transition as a function of capillary number at $\unicode[STIX]{x1D70E}_{r}=0.1$ . (b) Time required to attain intermediate maximum deformation at $\unicode[STIX]{x1D70E}_{r}=10$ (– – –) and $\unicode[STIX]{x1D70E}_{r}=0.1$ (——).

The Skalak and the neo-Hookean models do not show much difference for $\unicode[STIX]{x1D70E}_{r}>1$ but show significant differences in the maximum deformation for $\unicode[STIX]{x1D70E}_{r}<1$ , and expectedly the strain-softening neo-Hookean model shows higher deformation than the Skalak model. The time required to reach the maximum deformation is a complex interplay of the membrane charging and the fluids charging time scales, that is, $t_{MW}$ and $t_{cap}$ , and is found to obey $t_{MW}<t<t_{cap}$ . Thus it is higher for $\unicode[STIX]{x1D70E}_{r}=0.1$ as compared to that for $\unicode[STIX]{x1D70E}_{r}=10$ . The time required to reach the maximum deformation decreases with an increase in the capillary number (figure 5 b) since typically the deforming forces are larger than the restoring forces.

Figure 6.

Shape evolution of a capsule with Skalak membrane at $\unicode[STIX]{x1D70E}_{r}=0.1$ for $Ca=0.25$ .

Intermediate shapes for the dynamics of the deformation of an elastic capsule from a spherical shape to the steady-state shape at $Ca=0.25$ are shown in figure 6 for $\unicode[STIX]{x1D70E}_{r}=0.1$ . While for $\unicode[STIX]{x1D70E}_{r}>1$ prolate shapes (figure 4 a–d) are observed, oblate shapes are seen (figure 6 b–d) for $\unicode[STIX]{x1D70E}_{r}<1$ until the membrane is fully charged. To understand this better, deformation studies at a still higher capillary number ( $Ca=0.45$ ) are conducted with an emphasis on $\unicode[STIX]{x1D70E}_{r}<1$ .

4.3.1 Case of $\unicode[STIX]{x1D70E}_{r}<1$

Figure 7.

Shape evolution of a capsule with Skalak membrane at $\unicode[STIX]{x1D70E}_{r}=0.1$ for $Ca=0.45$ .

Figure 8.

Streamline plot (curves with arrow, shown only at right side of axis of symmetry), velocity profile (short arrows, magnitude represents relative extent of flow, shown only at left side of axis of symmetry) and electric stress (arrows from the interface) for $Ca=0.45$ and $\unicode[STIX]{x1D70E}_{r}=0.1$ at different times. Inset shows the variation of meridional $T_{s}$ (— ⋅ —), azimuthal $T_{\unicode[STIX]{x1D713}}$ (– – –) and mean $T_{m}$ (——) membrane tensions along the arclength.

The temporal evolution of a spherical capsule with $\unicode[STIX]{x1D70E}_{r}=0.1$ at a high capillary number $(Ca=0.45)$ is shown in figure 7. In figure 8(a–g) electric tractions, streamlines and velocity profiles are shown for figure 7(b–h), respectively, and meridional elastic tension $T_{s}$ , azimuthal elastic tension $T_{\unicode[STIX]{x1D713}}$ and mean elastic tension $T_{m}=(T_{s}+T_{\unicode[STIX]{x1D713}})/2$ are shown in the insets. The same data for $Ca=0.25$ and $\unicode[STIX]{x1D70E}_{r}=0.1$ corresponding to figure 6(b–f) are shown in figure 9(a–e).

Figure 8(a) shows that the oblate shape at $\unicode[STIX]{x1D70E}_{r}=0.1$ is associated with compressive normal Maxwell stress such that their distribution is of the second Legendre kind; the flow is from poles to equator due to tangential stresses, thereby assisting the formation of oblate shapes. The elastic tensions (meridional and azimuthal) are tensile for oblate shapes at short times.

Figure 8(b) shows further increase in the oblate deformation of the capsule because the normal Maxwell stress is compressive such that it is maximum at the poles and minimum at the equator. However, importantly, an onset of flow reversal (flow from the equator to poles) is observed. A complex interplay of flow induced by both Maxwell stress as well as the elastic stress due to relaxation of the stretched capsule (indicated by the meridional tension also becoming tensile at the equator) leads to an initiation of biconcave shape formation of the capsule as shown in figure 8(c).

At $t=75$ , for $Ca=0.45$ , although the flow is from the equator to the poles, the large compressive normal Maxwell stress at the poles continues to bring together the poles, resulting in a biconcave shape (figure 8 d). The biconcave shape is necessary to generate enough curvature so that the elastic stresses can balance the electric stress. On the other hand, for $Ca=0.25$ , the compressive Maxwell stresses at the poles are not relatively high enough to cause a biconcave shape (figure 9).

For $Ca=0.45$ , at $t=120$ the charging of the membrane leads to substantial compressive electric stress at the equator, with a corresponding lowering of stress at the poles, which results in further squaring of the capsule (figure 8 e). This is indicated by a greater region of high azimuthal tension around the equator. A peak in the azimuthal tension is now seen at the shoulders. The depression of the biconcave shape relaxes since the normal Maxwell stress at the poles reduces. For biconcave, squaring and hexagonal shapes observed at intermediate times, the meridional tension is compressive at the equator while the azimuthal tension remains tensile. The relaxation of the poles continues at $t=150$ , leading to hexagonal shapes (figure 8 f). The shoulders seem to get arrested, probably due to the absence of electric stress, such that at $t=\infty$ the electric stresses are compressive, which, as discussed earlier, can be decomposed into an isotropic part and a tensile part $P_{2}(\cos \unicode[STIX]{x1D703})$ , which disappears at $54^{\circ }$ , resulting in a prolate shape (figure 8 g). In the case of $Ca=0.25$ though, the insufficient compressive electrical stresses at the poles lead to an absence of biconcave and hexagonal shapes (figure 9). This results in oblate shapes directly transforming into square shapes, wherein higher compressive electrical stress at the equator as compared to the poles leads to relaxation to a prolate steady-state shape. At the steady prolate shape, the azimuthal tension is compressive at the equator while meridional tension is tensile across the entire arclength (insets of figures 8 g and 9 e).

A similar study for a neo-Hookean capsule has been conducted and reported in the supplementary material (§ B.3). The salient features of the dependence of the deformation on the constitutive law of the membrane are (i) all the shapes observed for a Skalak membrane are also observed in the case of a neo-Hookean membrane, albeit at a lower $Ca$ , e.g. the deformation seen in Skalak membrane at $Ca=0.45$ is observed at $Ca=0.29$ for a neo-Hookean membrane when $\unicode[STIX]{x1D70E}_{r}=0.1$ , and (ii) a neo-Hookean membrane can undergo breakup (numerical) at a much lower $Ca$ number as discussed in § 4.5.

Figure 9.

Streamline plot (curves with arrow, shown only at right side of axis of symmetry), velocity profile (short arrows, magnitude represents relative extent of flow, shown only at left side of axis of symmetry) and electric stress (arrows from the interface) for $Ca=0.25$ and $\unicode[STIX]{x1D70E}_{r}=0.1$ at different times. Inset shows the variation of meridional $T_{s}$ (— ⋅ —), azimuthal $T_{\unicode[STIX]{x1D713}}$ (– – –) and mean $T_{m}$ (——) membrane tensions along the arclength.

Figure 10.

Variation of transmembrane potential on the interface ( $\unicode[STIX]{x1D703}=0$ at north pole and $\unicode[STIX]{x1D703}=\unicode[STIX]{x03C0}$ at south pole) for the shapes at $t=0$ (⋯⋯),  $t=50$ (– – –),  $t=120$ (— ⋅ —), $t=230$ (— ⋅ ⋅ —) and $t=\infty$ (——) obtained during the deformation of a capsule at $Ca=0.45$ and $\unicode[STIX]{x1D70E}_{r}=0.1$ .

The variation of the transmembrane potential of the interface is shown in figure 10 for the intermediate shapes observed during the deformation of a capsule at $Ca=0.45$ and $\unicode[STIX]{x1D70E}_{r}=0.1$ . From figure 10 it can be observed that initially the capsule remains uncharged ( $\unicode[STIX]{x1D719}_{m}=0$ at $t=0$ ), then the transmembrane potential gradually increases, suggesting an increase in the charging of the membrane, and eventually at $t=\infty$ the membrane becomes fully charged. In all the cases, it can be observed that the poles always attain the maximum transmembrane potential, suggesting the possibility of electroporation at the poles. Moreover, the transmembrane potential at the poles is higher than $3/2$ at long times due to the deviation of the shape from sphericity.

Figure 11.

Variation of normal electric stress (– – ⋅ – –), normal elastic stress (— ⋅ —), normal hydrodynamic stress (– – –), stress due to bending rigidity (——), tangential electric stress (⋯⋯), tangential elastic stress (— ⋅ ⋅ —) and tangential hydrodynamic stress (— —) at the interface ( $\unicode[STIX]{x1D703}=0$ at north pole and $\unicode[STIX]{x1D703}=\unicode[STIX]{x03C0}$ at south pole) for the (a,d) biconcave shape at $t=75$ , (b,e) square shape at $t=120$ and (c,f) steady prolate shape at $t=\infty$ in the deformation of a capsule at $Ca=0.45$ and $\unicode[STIX]{x1D70E}_{r}=0.1$ .

To understand the relative contribution of the membrane, hydrodynamic and electric stresses, figure 11 show these values in the case of a Skalak membrane where biconcave, square and steady-state prolate shapes are observed. In figure 11, the mean electric and elastic stresses are subtracted from their respective values. While in the biconcave case the hydrodynamic stress is minimal due to the onset of flow reversal, it is significant in the square shapes, indicating the role of hydrodynamic stress in shape transition from biconcave to spheroid. The hydrodynamic stress is absent in the steady-state prolate shape. Thus the normal as well as the tangential force balance is essentially a cancellation of the electric and membrane stresses. Among the membrane stresses, the bending contribution is seen in the biconcave and square shapes, whereas it is minimal in the steady prolate shape. The bending contribution is observed at regions of high curvature.

4.3.2 Case of $\unicode[STIX]{x1D70E}_{r}>1$

Figure 12.

Shape evolution of a capsule with Skalak membrane at $\unicode[STIX]{x1D70E}_{r}=10$ for $Ca=2$ .

In figure 12 the dynamics of a spherical capsule with a Skalak membrane at a high capillary number ( $Ca=2$ ) for $\unicode[STIX]{x1D70E}_{r}=10$ is shown. Unlike the case for $\unicode[STIX]{x1D70E}_{r}=0.1$ , and as discussed earlier, a capsule does not attain oblate intermediate shapes for $\unicode[STIX]{x1D70E}_{r}=10$ . Initially, because of high tensile electric traction, a capsule exhibits large prolate deformation (figure 13 a,b). Eventually, because of the compressive electric tractions (maximum at the equator and minimum at the poles) at long times, it relaxes back to a steady-state prolate shape (figure 13 e) through intermediate shapes shown in figure 13(c,d). At short times, the meridional and azimuthal tensions are tensile. While the meridional tension is maximum at the equator and smallest at the poles (figure 13 a,b), it is vice versa for the azimuthal tension. At intermediate times, azimuthal tension is negligible around the equatorial region. At steady state, the elastic tensions are similar to that at $\unicode[STIX]{x1D70E}_{r}=0.1$ . Simultaneously, the flow changes from equator to poles at short time (figure 13 a), from poles to equator at intermediate times (figure 13 b–d), and eventually disappears at long times (figure 13 e).

Figure 13.

Streamline plot (curves with arrow, shown only at right side of axis of symmetry), velocity profile (short arrows, magnitude represents relative extent of flow, shown only at left side of axis of symmetry) and electric stress (arrows from the interface) for $Ca=2$ and $\unicode[STIX]{x1D70E}_{r}=10$ at different times. Inset shows the variation of meridional $T_{s}$ (— ⋅ —), azimuthal $T_{\unicode[STIX]{x1D713}}$ (– – –) and mean $T_{m}$ (——) membrane tensions along the arclength.