Spheroidal and conical shapes of ferrofluid-filled capsules in magnetic fields

We investigate the deformation of soft spherical elastic capsules filled with a ferrofluid in external uniform magnetic fields at fixed volume by a combination of numerical and analytical approaches. We develop a numerical iterative solution strategy based on nonlinear elastic shape equations to calculate the stretched capsule shape numerically and a coupled finite element and boundary element method to solve the corresponding magnetostatic problem and employ analytical linear response theory, approximative energy minimization, and slender-body theory. The observed deformation behavior is qualitatively similar to the deformation of ferrofluid droplets in uniform magnetic fields. Homogeneous magnetic fields elongate the capsule and a discontinuous shape transition from a spheroidal shape to a conical shape takes place at a critical field strength. We investigate how capsule elasticity modifies this hysteretic shape transition. We show that conical capsule shapes are possible but involve diverging stretch factors at the tips, which gives rise to rupture for real capsule materials. In a slender-body approximation we find that the critical susceptibility above which conical shapes occur for ferrofluid capsules is the same as for droplets. At small fields capsules remain spheroidal and we characterize the deformation of spheroidal capsules both analytically and numerically. Finally, we determine whether wrinkling of a spheroidal capsule occurs during elongation in a magnetic field and how it modifies the stretching behavior. We find the nontrivial dependence between the extent of the wrinkled region and capsule elongation. Our results can be helpful in quantitatively determining capsule or ferrofluid material properties from magnetic deformation experiments. All results also apply to elastic capsules filled with a dielectric liquid in an external uniform electric field.

Figure 1

Illustration of the parametrization in cylindrical coordinates $\left(r,z,\phi \right)$ and the contour line with arc length $s$. The complete capsule is obtained by revolution of the red contour line, while the angle $\psi$ describes its slope. This contour line is calculated numerically. The polar radius is called $a$, while $b$ denotes the equatorial radius.

Figure 2

Numerical results for the magnetic field distribution and capsule shape (two-dimensional projection) for a capsule filled with a ferrofluid with a susceptibility of $\chi =21$. The ratio of Young's modulus and surface tension is $Y_{2\text{D}}/\gamma =100$. The external magnetic field $H_0$ is uniform and points in the upward direction. Arrows indicate the local direction of $H$; the color codes for the absolute value of $H$ in units of $H_0$. The (a) spherical capsule and (b) spheroidal capsule have uniform fields inside, while the field in the (c) conical-shaped capsule increases strongly in the tips. The elongations $a/b$ (ratio of the polar radius to the equatorial radius) are (a) $a/b=1$, (b) $a/b=2.26$, and (c) $a/b=5.38$. The magnetic Bond numbers $B_m$ [see the definition in Eq. (23)] are (a) $B_m=0$, (b) $B_m=262.4$, and (c) $B_m=702.2$.

Figure 3

Three-dimensional illustration of a wrinkled capsule. The length $L_{\text{w}}$ of the wrinkles is measured as the length of the region, where ${\tau }_{\phi }+\gamma <0$. The wrinkling wavelength is not determined explicitly here.

Figure 4

Elongation $a/b$ of a capsule filled with a ferrofluid with $\chi =21$ as a function of $B_m/[Y_{2\text{D}}/\gamma +(5+\nu )]$ for different values of $Y_{2\text{D}}/\gamma$ in the region of small deformations. The solid line describes the linear approximation from Eq. (25). The best agreement between the numerical data and the linear approximation is given for a purely elastic system without wrinkling effects (closed purple circles). Wrinkling effects lead to considerable deviations (squares).

Figure 5

Comparison of numerically calculated $r\left(z\right)$ contour of a capsule with $Y_{2\text{D}}/\gamma =100$ and $\chi =21$, for a value $B_m$ chosen such that the elongation is $a/b=2$ (the inset shows the location of the pictured shapes in the $B_m\text{−}a/b$ plane) with a spheroid. The shape calculated without wrinkling (blue solid line) shows very good agreement with a spheroid of the same volume and elongation (red dashed line). Taking wrinkling into account leads to visible deviations (green dotted line).

Figure 6

(a) Stretch factors in the meridional direction ${\lambda }_s\left(s_0\right)$ following the whole contour line from the south pole $\left(s_0=0\right)$ to the equator $\left(s_0/R_0=\pi /2\right)$ for $Y_{2\text{D}}/\gamma =100$ and $\chi =21$. The left scale (red dashed line) gives almost constant stretch factors for a spheroidal shape with $a/b=2$. The right scale (blue solid line) gives diverging stretch factors for a conical shape with $a/b=5.34$. (b) Logarithmic plot of ${\lambda }_s\left(s_0\right)$ near the tip for $s_0/R_0<{10}^{-1}$. The function ${\lambda }_s\left(s_0\right)=\mathrm{const}{s}_0^{-\beta }$ [see Eq. (29)] was fitted to the data of the conical shape, which gave $\beta =0.562$, corresponding to an angle $\alpha =25.{98}^{\circ }$ in Eq. (30). (c) Zoom in to the tip of the contour line $z\left(r\right)$ for the conical shape; the half opening angle is $\alpha \approx {25}^{\circ }$.

Figure 7

Elongation $a/b$ of a capsule filled with a ferrofluid with $\chi =21$ as a function of magnetic Bond number $B_m$ for different values of the dimensionless elastic parameter $Y_{2\text{D}}/\gamma$. The magnetic Bond number is rescaled by $Y_{2\text{D}}/\gamma +(5+\nu )$, which is motivated by the small field behavior [see Eq. (25)]. The solid lines describe the theoretical results from approximative energy minimization (see Sec. 3b). Open (closed) symbols denote numerical data for increasing (decreasing) $B_m$. The agreement is good for small elongations; the approximation fails for higher elongations, especially at the shape transition (close-up in the right diagram), where $a/b$ jumps for small changes of $B_m$. Hysteresis effects are clearly visible in that area. There are two sets of numerical data for $Y_{2\text{D}}/\gamma =\infty$: Square data points are based on the modified shape equations that take wrinkling into account, while diamonds are calculated without wrinkling. There are also two sets of data without elasticity: The upper data points (black) describe a droplet with a real conical tip with a cone angle of $\psi \left(0\right)=70.2^{\circ }$, as it was given in Ref. [32]; for the lower points (blue) we used the boundary condition $\psi \left(0\right)=0$. Dashed lines indicate the position of shape transitions. Above these lines, shapes are conical, while they are spheroidal below. The markers (b) and (c) correspond to the shapes in Fig. 2.

Figure 8

Meridional stretch factor ${\lambda }_s$ at the capsule pole $s_0=0$ as a function of Bond number $B_m$ for $Y_{2\text{D}}/\gamma =100$ and $\chi =21$. The stretch factor clearly exhibits a jump at the location of the discontinuous shape transition and hysteretic behavior.

Figure 9

(a) Critical Bond numbers $B_{m,c1}$ (lower data points) and $B_{m,c2}$ (upper data points) for varying $Y_{2\text{D}}/\gamma$ with $\chi =21$. The solid lines describe the prediction by the approximative energy minimization for spheroidal shapes. Both critical Bond numbers increase for increasing $Y_{2\text{D}}/\gamma$. In the region $B_{m,c1}<B_m<B_{m,c2}$ there are hysteresis effects in the spheroidal-conical shape transition. (b) Relative size $\mathrm{\Delta }{B}_{m,c}$ of the hysteresis area for a wider range of $Y_{2\text{D}}/\gamma$.

Figure 10

Extent of the wrinkled region represented by the polar angle ${\theta }_{\text{w}}$ as a function of $B_m/[1+(5+\nu )\gamma /Y_{2\text{D}}]$. The lines are the linear response result (38), crosses and stars are numerical data points for different values of $Y_{2\text{D}}\gamma$, which all collapse to the linear response result. The red (dashed) line gives the asymptotic result ${cos}^2{\theta }_{\text{w}}=5/9$ for large values of $B_m/[1+(5+\nu )\gamma /Y_{2\text{D}}]$ and for purely elastic capsules $\left(\gamma /Y_{2\text{D}}=0\right)$.

Figure 11

Relative wrinkle length $L_{\text{w}}/L$ as a function of elongation $a/b$ for spheroidal capsules with fixed $\chi =21$ and different values of $Y_{2\text{D}}/\gamma$. There are no wrinkles $\left(L_{\text{w}}/L=0\right)$ for $Y_{2\text{D}}/\gamma \lesssim 8.93$. The range of $L_{\text{w}}/L>0$ and the extent of wrinkles increase with $Y_{2\text{D}}/\gamma$ until they converge to an asymptotic curve for thick shells.

Figure 12

Illustration of the geometry at the capsule's south pole (not true to scale) for (a) a conical tip and (b) the spheroidal reference shape.

Figure 13

Comparison of data from Fig. 9 for $l_0=0.1$ (blue) with data for $l_0=0.2$ (red). There is an increasing deviation for higher values of $Y_{2\text{D}}/\gamma$.