Multiplicity of Inertial Self-Similar Conical Shapes in an Electrified Liquid Metal

Liquid-metal ion sources (LMIS) are widely used for ion implantation in semiconductors, focused ion-beam systems for lithographic patterning and now even small in-space thruster units for precision pointing. Above a critical field strength, a liquid metal just prior to ion emission forms a conical protrusion, which undergoes accelerated sharpening of the tip due to field self-enhancement. Despite decades of interest in this phenomenon, the influence of inertial effects on the shape and flow field prior to emission remain poorly understood. Zubarev [Formation of conic cusps at the surface of liquid metal in electric field, JETP Lett. 73, 544 (2001)] showed that when local Maxwell and capillary forces prevail at an electrified tip of a perfectly conducting liquid in inviscid flow, the asymptotic behavior of the potential fields and interface shape far from the tip can be described by a one-parameter family of self-similar series solutions describing a dynamic Taylor cone with radially convergent flow toward the tip. The Maxwell and capillary pressure undergo divergent growth in finite time, characteristic of blowup phenomena in self-focusing singularity flows. Suvorov and Zubarev [Formation of the Taylor cone on the surface of liquid metal in the presence of an electric field, J. Phys. D: Appl. Phys. 37, 289 (2004)] later found solutions that retained inertial effects and showed that the self-focusing, divergent dynamics leading to power-law growth are preserved. In this work, we focus especially on the influence of inertial effects and extend the analysis to include time-reversal symmetry. We provide an interweaved procedure for calculating the coefficients of the series expansions for the potential fields and interface shape and use these to deduce a compact relation for bounds on the minimum and maximum liquid apex height achievable. We then develop a boundary integral patching technique, which yields the complete self-similar solution, valid throughout the near- and far-field domain. The resulting two-parameter family of numerical solutions reveals a multiplicity of fluid configurations, which we coin subconical, superconical, and mixed conical, that exhibit not only tip sharpening but tip bulging, tip separation flow, interface stagnation points, and receding shapes reminiscent of recoil after capillary pinchoff. The existence of such multiple configurations may ultimately help explain experimental observations during LMIS operation involving tip pulsation, droplet emission, liquid recoil and collapse.

Figure 1

(a) High-voltage (1 MeV) transmission electron micrograph of a conical protrusion in molten $\mathrm{Au}\mathrm{Ge}$ undergoing ion emission (45 $\mu \mathrm{A}$). Reproduced with permission from Fig. 6(b) of Ref. [27]. (b) Scanning electron micrograph imaged after solidification showing numerous protrusions in an electrically stressed sample of molten titanium. The formations resulted from exposure of molten titanium to large electric field gradients in the hybrid damped rf structures within the 30-GHz Compact Linear Collider at CERN. The surface rf fields, estimated to be roughly 95–135 MV/m, are applied for a duration of 70 ns every 20 ms. Reproduced with permission from Fig. 9 of Ref. [37].

Figure 2

Top image: laboratory frame description. An axisymmetric protrusion, emanating from a pool of perfectly conducting liquid (yellow) held at constant potential and of volume ${\mathrm{\Omega }}_{\mathrm{liq}}$ with surface boundary $\mathrm{\Gamma }(X,T)$ (cyan), accelerating toward a flat circular counter electrode (gray). The volume of the interstitial vacuum layer is denoted (${\mathrm{\Omega }}_{\mathrm{vac}}$). Nondimensional Cartesian coordinates are denoted $X=(X,Y,Z)$. The surface unit normal is $N$. The point $X_C$ (red dot) denotes the intersection of the central vertical axis with the tangent to the conical envelope defined by the classic Taylor cone with interior half-angle ${\theta }_T\cong {49.2923}^{\circ }$. At the point $X_C$, the Maxwell and capillary pressure diverge to infinity. Bottom image: self-similar frame of reference depicted in spherical coordinates where $\mathit{\chi }=(X_C-X)/{\tau }^{2/3}$ and $\tau =T_C-T$. The boundary $\gamma (X)$ (cyan) denotes the interface separating the liquid volume ${\omega }_{\mathrm{liq}}$ from the vacuum volume ${\omega }_{\mathrm{vac}}$. The surface unit normal is $n$. The exterior polar angle ${\theta }_0=\pi -{\theta }_T$.

Figure 3

Illustrations of asymptotic self-similar solutions for three flow regimes showing the electric potential $\psi$ (solid black lines), electric field $|\mathrm{\nabla }\varphi |$ (dashed black lines), fluid pressure $p$ (solid red lines—equal increments) and fluid velocity $\mathrm{\nabla }\psi$ (black arrows). The liquid (yellow) and vacuum domains are separated by an interface (cyan), which as $\mathsf{r}\to \mathrm{\infty }$ describes a conical interface $z=h(\mathsf{r}\mathsf{)}\mathsf{=}\mathsf{r}\cos {\theta }_{\mathsf{0}}$, where ${\theta }_0=\pi -{\theta }_T\cong 130.7077$ and ${\theta }_T$ is the class Taylor angle. The apex (red point) of the conical envelope defines the blowup point. (a) Classic static Taylor solution in self-similar frame. (b) Inertialess self-similar solutions from Eqs. (32)–(34). (c) Example of inertial self-similar solution from Eqs. (65)–(67).

Figure 4

Flow chart depicting the interweaved procedure for computing the coefficients $a_k$, $b_k$, $c_k$. Arrows represent the equations required for computing the coefficient shown to the next higher order as required by simultaneous solution of the equipotential condition in Eq. (59a) (blue arrows), the kinematic boundary condition in Eq. (59g) (red arrows), and the surface Bernoulli equation given by Eq. (53) (green arrows).

Figure 5

Flow configurations resulting from inertia effects showing relative motion between fluid in the tip and nearby bulk region. Dashed line (red) represents the asymptotic conical envelope with interior half-angle ${\theta }_T$. Curved lines (cyan) denote the interface of the electrified liquid. (a) Subconical configuration in which both the tip and bulk fluid together advance or together recede from the conical envelope, which is always above the moving liquid. (b) Superconical configurations in which both the tip and bulk fluid together advance or together recede from the conical envelope, which is always below the moving liquid. (c) Example of a mixed-conical configuration in which the tip and bulk fluid move in opposite directions, with some portions of the liquid interface below and some above the conical envelope.

Figure 6

Illustrations of the geometry for estimating the distance between the maximum liquid height and the apex of the asymptotic conical envelope (red dot) with ${\theta }_0=\pi -{\theta }_T$. Volumes indicated by ${\omega }_T$, ${\omega }_D$, and ${\hat{\omega }}_D$ can be positive or negative depending on whether the liquid is advancing toward the conical apex point from below (subconical configuration) or advancing toward the intersection point from above (superconical configuration)—see Fig. 5. (a) The volume ${\omega }_T$ (hatched gray lines) (interface unit normal $n_T$) denotes the interstitial volume above the liquid boundary $\gamma$ (cyan curve) and below the conical envelope boundary ${\gamma }_T$ (dashed black line) defined by the exterior angle ${\theta }_0$. The coordinate ${\mathit{\chi }}_{\ast }=({\mathsf{r}}_{\mathsf{\ast }}\mathsf{,}{\theta }_{\mathsf{0}}\mathsf{)}$ denotes the point of intersection between ${\gamma }_{\mathrm{liq}}$ (dashed black curve) and $\gamma$. The boundary element ${\gamma }_{\ast }$ (magenta line) with unit normal $n_{\ast }$ is a small vertical line bridging $\gamma$ and ${\gamma }_T$ at ${\mathit{\chi }}_{\ast }$. (b) The volume ${\omega }_D$ (red hatched lines) represents the interstitial volume between $\gamma$ and ${\gamma }_D$ (dashed red line). The branch of ${\gamma }_D$ to the right is defined by the curve $z=c_{0}r+c_1{r}^{-1/2}$. (c) The domain ${\hat{\omega }}_D$ (hatched lines) represents the small volume exterior to $\gamma$, which is enclosed within the gray rectangular region with horizontal length $r=\hat{a}$ and vertical length $z=z_{max}$. The variable $\hat{a}$ denotes the radial coordinate corresponding to the turning radius of ${\gamma }_D$. The variable $z_{\max }$ is related to the minimum elevation of $\gamma$ at $r=0$ as described in the text.

Figure 7

Sketches showing two computational domains, ${\omega }_{\mathrm{liq}}$ and ${\omega }_{\mathrm{vac}}$, and corresponding far-field boundary conditions used for numerical solution (in the self-similar frame) of the interface shape, velocity field, and electric potential. (a) Liquid domain of volume ${\omega }_{\mathrm{liq}}$ bounded above by the boundary $\gamma$ (teal solid line) and below by the boundary ${\gamma }_{\mathrm{liq}}$ (black dashed line). Vacuum domain of volume ${\omega }_{\mathrm{vac}}$ bounded above by the boundary ${\gamma }_{\mathrm{vac}}$ (magenta solid line) and below by the boundary $\gamma$. Domain truncation point denoted by ${\mathit{\chi }}_{\ast }$. Spherical coordinates indicated by $(\mathsf{r}\mathsf{,}\theta \mathsf{)}$. (b) Boundary conditions for velocity potential $\psi$ satisfying kinematic condition from Eq. (59g) on $\gamma$ and ${\psi }_{\mathrm{\infty }}$ from Eq. (69) on ${\gamma }_{\mathrm{liq}}$. (c) Boundary conditions for electric field potential $\varphi$ subject to $\varphi =0$ on $\gamma$ and $\mathrm{\partial }{\varphi }_{\mathrm{\infty }}/\mathrm{\partial }n$ from Eq. (70) on ${\gamma }_{\mathrm{vac}}$. (d) Axisymmetric cylindrical geometry for boundary integral patching technique described in Sec. 5; $\beta =\pi$ along smooth boundary segments and $\beta =\pi /2$ at truncation point ${\mathit{\chi }}_{\ast }$.

Figure 8

Numerical solutions (in cylindrical coordinates) obtained from the boundary integral patching technique showing the liquid interface shape (cyan curve), velocity field (black arrows), and pressure contours (dark gray curves, constant increments, red $>$ blue) for the free parameter values $(a_0,b_0)$ indicated. (a) Left image: subconical configuration for $a_0=-1.36$ and $b_0=1.15$. Right image: superconical configuration for $a_0=0$ or $a_1=0.23$ and $b_0=1.15$. (b) Left image: mixed-conical configuration for $a_0=-1.36$ and $b_0=0$. Right image: superconical configuration for $a_0=-1.36$ and $b_0=1.15$.

Figure 9

(a) Numerical solutions (in cylindrical coordinates) obtained from the boundary integral patching technique showing the liquid interface shape (cyan curve), velocity field (black arrows), and pressure contours (dark gray curves, constant increments, red $>$ blue) for $a_0=-2.370$ and $b_0=1.757$. Superposed dots (black) on the liquid interface represent data points extracted from the inset image in Fig. 1 of Ref. [36]. (b) Boundary data along $\gamma$ from the numerical solutions in (a) showing the interface shape $h(\gamma )$ (blue line), interface velocity potential $\psi (\gamma )$ (red line) and interface electric field strength $(\mathrm{\partial }\varphi /\mathrm{\partial }\mathbf{n})(\gamma )$ (green line) with increasing distance $r$. Thicker line segments at large radii represent the leading-order solutions ($k=0$) evaluated from Eqs. (69)–(71) and Eqs. (72)–(74) given by ${\psi }_{\mathrm{\infty }}=0.989703a_0{\mathsf{r}}^{\mathsf{1}\mathsf{/}\mathsf{2}}$, $\mathrm{\partial }{\varphi }_{\mathrm{\infty }}/\mathrm{\partial }\mathsf{r}\mathsf{=}\mathsf{0.848}\mathsf{582}{\mathsf{b}}_{\mathsf{0}}{\mathsf{r}}^{\mathsf{-}\mathsf{1}\mathsf{/}\mathsf{2}}$, and $h_{\mathrm{\infty }}=c_{0}r$ for $a_0=-2.370$, $b_0=1.757$, and $c_0=0.860437$.