Vortices on curved surfaces

Topological defects in thin films coating a deformed substrate interact with the underlying curvature. This coupling mechanism influences the shape of biological structures and provides a new strategy for the design of interfaces with prescribed functionality. In this article, a mathematical formalism based on the method of conformal mapping that is presented permits the calculation of the energetics of disclinations, dislocations, and vortices on rigid substrates of spatially varying Gaussian curvature. Special emphasis is placed on determining the geometric force exerted on vortices in curved superfluid films. This force, which attracts (repels) vortices towards regions of negative (positive) Gaussian curvature, is an illustration of how material shape can influence quantum mechanical degrees of freedom.

Figure 1

Cross sections of two regions in which one can study vortex energetics. (a) A region with nonparallel boundaries. The vortex is pushed to the right. This tendency can be interpreted in terms of either a drive toward a shorter length or as the local induction force due to the curvature in the vortex enforced by the boundaries. (b) A cross section of a constant thickness layer of helium bounded above by air and below by the substrate. The vortices keep a constant length and remain straight while moving around. Hence there is no local induction force or thickness-variation force to overwhelm the geometrical forces that we focus on.

Figure 2

A comparison between the forces on vortices due to curvature and due to boundaries. Left: the flow around a vortex situated on the side of a Gaussian bump, calculated with the methods described in this paper. The low density of flow lines above the vortex indicates a lower velocity and thus a higher pressure, leading to the repulsion represented in Eq. (12), $-\nabla (-\pi K{n}^2{U}_G)$. Right: the analogous flow around a vortex in a disk with a solid boundary. The attraction to the boundary is also seen to result from high speeds, since the flow lines are compressed in between the vortex and the boundary.

Figure 3

Vortices on a Gaussian bump. (a) A bumpy surface shaped as a Gaussian. (b) Top view of (a) showing a schematic representation of the positive and negative intrinsic curvature as a nonuniform background “charge” distribution that switches sign at $r=r_0$. The varying density of + and − signs tries to mimic the changing curvature of the bump.

Figure 4

Plot of the interaction energy $E\left(r\right)=-\pi K{U}_G\left(r\right)$ between a singly-quantized vortex and a Gaussian bump with $\alpha =1$. The energy is measured in units of $K$ and the radius is measured in units of $r_0$. Note that the force points away from the bump and has its maximum strength near $r_0$.

Figure 5

An azimuthally symmetric substrate and its downward projection on a flat plane. The shaded strip surrounding $P$ is more stretched than the one surrounding $Q$ despite their projections onto the plane having the same area. As discussed in the text, it follows that the energy stored in the field will be lower if the center of the vortex is located at $Q$ rather than $P$. From 127.

Figure 6

Plot of vortex trapping surface.

Figure 7

Plot of the curvature of the $\lambda =1$ vortex trapping surface, measured in units of ${(\alpha ∕r_0)}^2$.

Figure 8

Plot of the geometric potential for the $\lambda =1$ vortex trapping surface.

Figure 9

A saddle surface with $\lambda =17$; this parameter value is just large enough to destabilize a vortex at the center.

Figure 10

Plots of the sign of the curvature, with white for positive curvature. (a) The Gaussian bump, and (b) and (c) the saddle surfaces with $\lambda =1$ and 17, respecitvely. Because of the lack of symmetry in (c), the center point becomes a saddle point of the energy; the vortex is pushed away by the strong positive curvature in the ellipsoidal regions at positive and negative $y$.

Figure 11

The geometry-defect interaction energy of a vortex on the saddle surface with $\lambda =17$. One notices a slight instability in the $x$ direction.

Figure 12

Plot of minus the geometric potential $-U_G\left(r\right)$ for $\alpha =0.5,1,1.5,2$. The arrow indicates increasing $\alpha$. The radial coordinate $r$ is measured in units of $\lambda$ and $r_0=\lambda$.

Figure 13

Plot of the area of a cup of radius $r$ for $\alpha =0.5,1,1.5,2$. The arrow indicates increasing $\alpha$. The radial coordinate $r$ is measured in units of $\lambda$ and $r_0=\lambda$.

Figure 14

Plot of $E\left(r\right)$ measured in units of $K={\hslash }^2{\rho }_s∕m^2$ as $\alpha$ is varied. In these units, the thermal energy $k_{B}T$ is less than 0.02 below the Kosterlitz-Thouless transition, for $150\text{−}\mathrm{\AA }$ films. The radial coordinate $r$ is measured in units of $\lambda$ and $r_0=\lambda ∕2$. Note that this plot is a 2D slice of a 3D potential. For $\alpha <{\alpha }_c$, $E\left(r\right)$ is approximately a paraboloid while, for $\alpha >{\alpha }_c$, we have a Mexican hat potential.

Figure 15

Plot of $E\left(r\right)$ in units of ${\hslash }^2{\rho }_s∕m^2$ vs $r$ as $r_0$ is varied. The aspect ratio is kept fixed at $\alpha =2$ while the range of the geometric potential (corresponding to the width of the bump) is varied so that $r_0=0.2,0.4,0.6,0.8,1$ in units of $\lambda$. As $r_0$ decreases, the geometric force becomes stronger, so the system goes through a transition analogous to the one displayed in Fig. 14.

Figure 16

Graphical method for determining equilibrium positions of one vortex. The equilibrium position is at the intersection of $\sin \theta \left(r\right)∕2$ and $r∕\lambda$. If we fix $r_0$ and set $\alpha =1$, the rotational frequency will control the position of the vortex. The four lines correspond to $\Omega =\hslash ∕m{r}_0^2,\hslash ∕4m{r}_0^2$ (which is the critical frequency ${\Omega }_c$), $\hslash ∕25m{r}_0^2,\hslash ∕100m{r}_0^2$.

Figure 17

Arrangements of six vortices that can occur on a curved surface, comparing tightly confined vortices (the top row) to more dispersed vortices (the bottom row). A circle of radius $r_0$ is drawn to give a sense of the size of the bump. The upper row shows the patterns which occur at large angular frequencies $(\Omega =9{\hslash }^2∕m{r}_0^2)$. The transition from the pentagon to the rectangle with two interior points is discontinuous, and there is a range of aspect ratios $2.1<\alpha <2.7$, where both configurations are metastable. The third configuration is nearly degenerate with the second configuration. The lower row shows the configurations which occur for $\Omega ={\hslash }^2∕m{r}_0^2$ as $\alpha$ increases. The first transition is continuous and caused by the central vortex’s being repelled from the top by the geometric interaction. The third configuration is similar to the second large $\Omega$ configuration but the effect of the geometric repulsion is seen in its asymmetry.

Figure 18

The flow in a wiggly annular region (the target substrate $T$), obtained by conformally mapping the flow from a circular annulus (the reference substrate $R$). Since every pair of radial spokes in the first picture comprises the same energy, this is true in the conformal image as well. The small region in the constriction on the right manages this by compensating for its small area by having a high flow speed.

Figure 19

Using conformal mapping to find the geometric energy of vortices on a complicated target surface $T$ by comparison to a simple surface $R$. The target surface, left, has the topology of an infinite plane but is distorted by a three-dimensional lump. The reference plane $R$ (shown in the plane of the page) is on the right. $T$ is split up into portions $I$ and $O$ and different maps are used to map each to $R$. $I$ consists of the interior of radius $l$ disks (the circles in the figure with thick boundaries) in $T$, and is mapped rigidly to the heavily demarcated disks in $R$. $O$ consists of the exterior of the disks in $T$ (in gray), and is mapped conformally to the gray portion of $R$. $I$ and $O$ contain the same kinetic energy as their images in $R$, but the images do not fit together perfectly. Thus the difference in energy between the flows in $T$ and in $R$ is determined by calculating how much energy is contained in the annuli in $R$, which are either left bare or covered twice.

Figure 20

Vortex and its streamlines on an “Enneper disk.”

Figure 21

An illustration of the period lattice of the target surface, projected into the $x\text{−}y$ plane. The coordinates ${\lambda }_1,{\mu }_1$ and ${\lambda }_2,{\mu }_2$ describe the basis vectors, which we do not assume to be orthogonal.

Figure 22

A periodic surface with a square lattice but not square symmetry. This surface is illustrated by its contour plot. It has the two reflectional symmetries but not the 90° symmetry of a square lattice; hence, generically, the conformal image will have a rectangular lattice.

Figure 23

A capped cylinder; a cylinder of length $2H$ is closed off by hemispheres at the north and south poles of radius $R$. The circulation around every lattitude is the same.

Figure 24

The flow on an azimuthally symmetric surface, described by the coordinates $(z,r)$ and an azimuthal angle $\varphi$ (not shown). (a) The surface is defined as the surface of revolution of the curve $r\left(z\right)$ in the $r\text{−}z$ plane. The other two images show the flow on an ellipsoid and compare the flow pattern predicted by the band model (b) and the exact solution determined by conformal mapping (c). The flow lines in the band between the two vortices become close to horizontal and are approximately azimuthally symmetric. Beyond the vortices, they are spaced far apart, indicating a vanishingly small speed for a greatly elongated surface.

Figure 25

The rotational and fluid energy (units of $K$) as a function of ${\sigma }_1=2{\sigma }_{\mathrm{eq}}-{\sigma }_2=\sigma$ (units of $R$) for $H=3.5R$ and $m\omega ∕\hslash =0.49R^{-2},0.61R^{-2},0.74R^{-2}$. The middle curve, roughly at $\omega ={\omega }_b$, shows the last position where the vortex is stable as $\omega$ is decreased.

Figure 26

Applying the method of images to a pointed sphere. (a) The process of folding a hemisphere into a pointed sphere, bounded at the north and south poles by two 180° disclinations. The north and south poles move outward along the axis, while the latitudes stay horizontal. The Gaussian curvature is invariant because the decreasing curvature of the lines of longitude is compensated by the tighter curvature around the lines of lattitude. (b) A top view of vortices during the furling up of one hemisphere. The first stage shows both the $(0<\alpha <\pi )$ hemisphere that will be retained and the other hemisphere together with two defects and their images. The hemisphere is sealed up into the pointed sphere so that the left and right halves of the cut (which appears as a horizontal diameter in this top view) are brought together to form the seam on the pointed sphere; simultaneously, the defects $Q_1,Q_2$ move to positions ${\mathbf{u}}_1$ and ${\mathbf{u}}_2$ on the pointed sphere. The furling process leads to a continuous flow pattern on the pointed sphere. For example, the two points marked with an $˟$ on the cut in the original sphere are sealed together. Both points feel a strong flow, the one on the right because it is close to the vortex at $Q_1$ and the one on the left because it is close to this vortex’s image, and these flows are equal.

Figure 27

How to form cones of negative curvature. One complete sheet of paper is slit and an angle is cut out of an additional sheet of paper. The edges labeled 1 are taped together, and then the edges labeled 2 are taped. The circular arcs join together to form an extra-large circle. The cone angle is $\theta =2\pi +\beta$, and cones with even larger cone angles can be formed using additional sheets.

Figure 28

Definition plot for a laminating film. $h\left(\mathbf{x}\right)$ is the height of the substrate above the horizontal surface at a point $\mathbf{x}=(x,y)$, and $h_L\left(\mathbf{x}\right)$ is the height of the upper surface of the film. $D\left(\mathbf{x}\right)$ is the thickness of the film which (if the film has a slowly varying thickness) is given by $[h_L\left(\mathbf{x}\right)-h\left(\mathbf{x}\right)]\cos \theta \left(\mathbf{x}\right)$, where $\theta \left(\mathbf{x}\right)$ is the local inclination angle of the substrate.

Figure 29

The disjoining pressure is the interaction between $\mathbf{r}$ and the substrate, which is below the surface $S$ in the figure. This can be calculated using the result for the interaction with an imaginary flat substrate (below the tangent plane $S_{\tan }$) and subtracting the very narrow shaded region.

Figure 30

The resting force (dashed line) and minus the rotational force (solid lines) between two vortices on an ellipsoid of aspect ratio 3.5. The rotational forces are shown for ${\Omega }_a$ (top curve), ${\Omega }_b$ (bottom curve), and an intermediate value of $\Omega$. The vortex equilibrium positions are the intersection points. For smaller values of $\Omega$, the attraction between the vortices always overcomes the rotational force, causing them to annihilate. For larger values of $\Omega$, the rotational force overcomes the attraction, causing the vortices to move to opposite poles.

Figure 31

Illustration of the combined rotational and resting energies for different frequencies $\Omega$ and $H=3.5R$. The second curve corresponds to a frequency greater than ${\Omega }_b$, the frequency where the rotation keeps vortices from annihilating. Although it looks practically flat on this scale, the energy has a curvature of about $10K∕R^2$ at the off-center equilibrium position indicated. The third curve corresponds to $\Omega ={\Omega }_a$, where the vortices are first pushed to the ends of the ellipsoids. The angular frequencies are given in units of $\hslash ∕m{R}^2$.

Figure 32

The combined rotational and kinetic energy for vortices on a sphere, where there are no stable off-axis positions. The energy is graphed as a function of ${\sigma }_1={\sigma }_{\mathrm{eq}}-s∕2$. From the top, the curves correspond to $\Omega =0,0.4\hslash ∕m{R}^2,0.52\hslash ∕m{R}^2,1.5\hslash ∕m{R}^2$. The third of these corresponds to $\Omega ={\Omega }_a={\Omega }_b$. In the last curve, the vortices are trapped near ${\sigma }_1=0$ (i.e., at the poles) by the rotational energy.

Figure 33

Construction of the largest circle centered at a point on a cone with $\theta <\pi$. The cone is cut open and flattened so that the center is on the bisector of the angle.

Figure 34

A surface with the maximum geometric force. (a) The surface which contains an isometric disk of radius $R$ and has the maximum geometrical force. A semicircle (with films on both sides) is connected by a neck $A$ to a plane. If a single vortex is placed at $B$, the force on the vortex approaches $4\pi K∕R$ as the edges of the surface become sharper. (b) An unfolded image of the flow pattern.