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Universal pinch off of rods by capillarity-driven surface diffusion
Scripta Materialia, Vol. 39, No. 1, pp. 55– 60, 1998 Elsevier Science Ltd Copyright © 1998 Acta Metallurgica Inc. Printed in the USA. All rights reserved. 1359-6462/98 $19.00 1 .00
Pergamon PII S1359-6462(98)00127-4
UNIVERSAL PINCH OFF OF RODS BY CAPILLARITYDRIVEN SURFACE DIFFUSION Harris Wong1,*, Michael J. Miksis1, P.W. Voorhees2 and Stephen H. Davis1 1
Department of Engineering Sciences and Applied Mathematics and 2Department of Materials Science and Engineering, Northwestern University, Evanston, IL 60208, USA (Received January 29, 1998) (Accepted in revised form March 22, 1998) Introduction
Interfacial energy is a central factor in setting the morphology of phases and in determining the stability of equilibrium morphologies. For example, a cylinder of radius a with surface energy is unstable when subject to infinitesimal axisymmetric perturbations of wavelengths greater than 2pa. These initially small perturbations grow into the nonlinear regime and lead eventually to the cylinder breaking up into a series of isolated spherical particles, an equilibrium configuration that minimizes the surface energy. The cylinder can be liquid [1, 2] or solid [3], and the critical wavelength is independent of the mass-transport mechanism. This so-called Rayleigh or pinching instability has been predicted and observed to occur in other contexts. For example, thin strips on substrates breakup as well, whether the strips are liquid rivulets [4, 5] or solid lines [6]. Since the morphological evolution is driven simply by the minimization of surface energy, it is not surprising that the Rayleigh instability and the subsequent pinch off is observed in an enormous variety of two-phase mixtures, such as composites composed of rods embedded in a matrix [7, 8], and cylindrical pore channels in ceramics [9]. Despite the breath of systems which undergo this type of capillary-driven morphological evolution, a clear picture of the evolution process away from the linear regime is not well understood. This is because the presence of interfacial energy introduces strong nonlinearities into the equations, which typically require full numerical solutions. The morphological evolution of rods to near pinch off has been simulated numerically when the mass moves by surface diffusion [8, 10, 11]. While these studies describe the evolution of rods in the nonlinear regime, the dependence of the evolution pathway on the initial perturbations imposed on the rod is unclear. In addition, it has not been possible to address the last stages of the break-up process, due to the small length scales and large interfacial velocities that are present. Thus, the evolution of the morphology near the topological singularity where the rod finally fissions or pinches off remains unexplored. Here we examine the morphological evolution of a rod via capillary-driven surface diffusion as it both approaches and departs the topological singularity of pinch off. During the final stages of pinching the neck radius approaches zero, and self-similar solutions are sought. Since cones are the only axisymmetric self-similar geometry, we propose that, as seen far from the neck, the morphology
* Permanent address: Mechanical Engineering Dept., Louisiana State University, Baton Rouge, LA 70803,USA.
55
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Figure 1. Cross sections of a film rod (a) before breakup, (b) at the moment of breakup, and (c) after breakup. Near the pinch-off point, the rod can be approximated as two opposing cones with half-cone angle a. The origin of the cylindrical coordinates (r, z) coincides with the tip of the cones. The surface unit normal vector n forms an angle f with the r-axis.
asymptotes to two opposing cones. Numerical evidence [8, 11] supports this assertion. This self-similar local solution implies that the surface morphology near pinch off can assume a time-independent form under a proper scaling, despite the large surface deflections and the strongly nonlinear nature of the evolution near the pinching singularity. Moreover, only one cone angle is found from solving the scaled equations. Therefore, for any system and for a wide array of initial perturbations of the rod there is a single similarity solution both entering and departing the pinching singularity. The surface morphology in the last stages of pinching is in this sense universal. Formulation Assume that the mass of the rod moves mainly by surface diffusion, and that the surface energy is isotropic. The local normal interfacial velocity, un, is given by [12]: un 5 B ¹2s k,
(1)
where ¹s is the surface gradient operator, and k is the surface curvature. B 5 DsgnV /kT, where Ds is the surface diffusivity, g is the surface energy, n is the number of diffusing atoms per unit area, V is the atomic volume, k is Boltzmann’s constant and T is temperature. Arc-length coordinates are used to accommodate the complicated surface morphologies before and after breakup. The film surface is located by the cylindrical coordinates (r, z), with origin at the tip of the cone as shown in Fig. 1. The surface normal makes an angle f with the r-axis. The surface coordinates (r, z) and angle f are functions of time t and arc length s, and are related by geometry: 2
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UNIVERSAL PINCH OFF OF RODS
r 5 sin f, s
57
z 5 cos f. s
(2a,b)
If p 5 rer 1 zez is the position vector of a surface point, then p/t is the surface velocity at constant arc length and is usually not in the normal direction. Given the normal vector n 5 (cosf)er 2 (sinf)ez, we can calculate un 5 n ● p/t, and k 5 ¹s ● n, where ¹s 5 es/s 1 (eu/r)/u, with u being the azimuthal coordinate around the z-axis [13]. Equation (1) then becomes ~cos f!
S
2 r z sin f 2 ~sin f! 5 2B 2 1 t t s r s
DS
D
f cos f 2 . s r
(2c)
At the moment of breakup, t 5 T, and the film consists of two opposing cones touching at the tip (Fig. 1(b)):
f(T, s) 5 a,
r(T, s)5s sina,
z(T, s) 5 s cosa.
(3a,b,c)
Before breakup (t , T), the boundary conditions specify that the film profile is symmetric about the plane z 5 0 (Fig. 1(a)): At s 5 0,
f 5 0,
2f 5 0. s2
z 5 0,
(4a)
The last condition requires zero mass flux at the symmetric plane. After breakup (t . T), the boundary conditions specify that the film profile has rotational symmetry about the z-axis: At s 5 0,
f 5 p/2,
2f 5 0. s2
r 5 0,
(5a,b,c)
Far from the point of breakup, the film is not disturbed:
f 3 a as s 3 `.
(6)
Self-Similar Profile before Breakup (t < T) We determine the surface morphology prior to breaking (t , T) by following the analysis of Wong et al. [14], and defining a set of self-similar variables: S5
s , B (T 2 t)1/4 1/4
Z(S) 5
R(S) 5
z(t, s) , B (T 2 t)1/4
r(t, s) B (T 2 t)1/4 1/4
F(S) 5 f(s, t).
1/4
(7a,b) (7c,d)
Equations (2) then become dR 5 sinF, dS 4
S
DS
d2 sin F d 2 1 dS R dS
dZ 5cosF dS
D
dF cos F 2 - RcosF 1 ZsinF 5 0. dS R
The initial and boundary conditions in (3), (4), and (6) combine to give that at S 5 0
(8a,b)
(8c)
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Figure 2. Self-similar film profiles before and after breakup. Only one self-similar solution is found for pinching. This solution yields a half-cone angle a 5 46.04°. The post-breakup self-similar profile corresponding to this angle is also shown. The self-similar variables R and Z before and after breakup are defined differently in (7) and (9).
F 5 0,
Z 5 0,
d2F 5 0, dS2
(8d,e,f)
and that as S 3 `, F 3 a.
(8g)
Equations (8) are solved using a fourth-order Runge-Kutta scheme with step size DS 5 0.001. At S 5 0, two additional conditions are needed to start the integration. We vary R and dF/dS at the symmetry plane S 5 0 until the boundary condition at infinity, applied at S 5 35, is satisfied. Only one solution is found with R(0) 5 0.7016287178 and dF/dS(0) 5 0.9003221268. This gives a 5 46.04°. The self-similar profile is plotted in Fig. 2. The presence of a unique cone angle during pinching implies that the topological singularity introduces a universality in the morphology. Given an unbroken solid with axial symmetry, the pinching process increases the local slope of the interface until it reaches the critical value. This is consistent with the gradual increase observed in the time-dependent numerical simulations of Sekerka & Marinis [8] and Coleman et al. [11]. From their published figures, we measure the half-cone angles near the onset of pinch off and got values of 44°-45° from Sekerka & Marinis (Fig. 3) and 42°-45° from Coleman et al. (Figs. 2 and 5). Detailed comparison of film profiles is not possible due to a lack of resolution of their figures. Nevertheless, the presence of half-cone angles less than 46.04° is consistent with our predictions that the cone angles must approach the critical value during the evolution process. Thus, the influence of the initial conditions becomes progressively weaker as the topological singularity is approached. The existence of a self-similar morphology close to pinch off also allows us to address the dynamics of pinching. All characteristic lengths in the system scale as (T 2 t)1/4. Therefore, for any system, the curvature at the neck diverges to infinity as (T 2 t)21/4, whereas the thickness of the neck of the rod goes to zero as (T 2 t)1/4. Although the scaling of the morphology with time in the neighborhood of the singularity is unique, the actual values of these quantities are system dependent through the presence of B in the self-similar variables.
Vol. 39, No. 1
UNIVERSAL PINCH OFF OF RODS
59
Self-Similar Profile after Breakup (t > T) To examine the morphological evolution following pinch-off (t . T), a different set of self-similar variables is needed: S5
s , B (t 2 T)1/4
R(S) 5
1/4
Z(S) 5
z(t, s) , B1/4(t 2 T)1/4
r(t, s) B (t 2 T)1/4 1/4
F(S) 5 f(s, t).
(9a,b)
(9c,d)
The governing equations (2) reduce to dR 5 sinF, dS 4
S
DS
sin F d d2 1 dS2 R dS
dZ 5cosF dS
D
dF cos F 2 - RcosF 2 ZsinF 5 0. dS R
(10a,b)
(10c)
The boundary conditions (3), (5), and (6) become that at S 5 0, F 5 p/2,
R 5 0,
d2F 5 0, dS2
(10d,e,f)
and that as S 3 `, F 3 a.
(10g)
Equations (10) are solved using the same shooting method as discussed above. At S 5 0, however, R 5 0 and (10c) is singular. To circumvent this singularity, the integration is started at S 5 So . 0 using the following local expansion around S 5 0: F5
p Zo 1 CoS 1 S3 1 . . . 2 96
(11)
where Co is the scaled tip curvature (5dF/dSuS50) and Zo 5 Z(0) is the scaled tip location. Given a value of Co and Zo, (11) is used to start the integration at S 5 So 5 0.02. (Reducing So to 0.01 has no effect on the results.) Values of Co and Zo are varied in order to satisfy the boundary condition at infinity and to give a specific cone angle. The resulting self-similar profile for t . T with a half-cone angle of 46.04° is given in Fig. 2. This is the profile that should be observed immediately following pinch off. While it is possible to find similarity solutions for a wide range of cone angles when t . T [10, 15], since there is only one cone angle possible for t , T, the surface morphology immediately after pinch off must be consistent with the t , T solution and assume the form given in Fig. 2. Thus the evolution of the surface both prior and post pinch off are system independent and universal. Conclusions We have derived local similarity solutions for the axisymmetric pinch off of rods when the morphological evolution is by capillarity-driven surface diffusion. These local solutions describe the approach to and departure from the topological singularity where a rod pinches into two separate bodies. During
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pinching, the self-similar surface profile far away from the neck approaches two opposing cones with a unique half-cone angle of 46.04°. It is thus likely that all rods must pinch off with this cone angle. This assertion is supported by several numerical simulations. After pinch off, the smoothening of the cone tip is again self-similar. The results obtained here for rods also apply to the pinch off of cylindrical pore channels. Acknowledgments This research was supported by Department of Energy grant DE-FG02-95ER25241. References 1. 2. 3. 4. 5. 6. 7. 8.
9. 10. 11. 12. 13. 14. 15.
L. Rayleigh, Proc. London Math. Soc. 10, 4 (1879). S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, Clarendon Press, Oxford (1961). F. A. Nichols and W. W. Mullins, Trans. Metall. Soc. AIME. 233, 1840 (1965). S. H. Davis, J. Fluid Mech. 98, 225 (1980). J. B. Culkin and S. H. Davis, AIChE J. 30, 263 (1984). M. S. McCallum, P. W. Voorhees, M. J. Miksis, S. H. Davis, and H. Wong, J. Appl. Phys. 79, 7604 (1996). H. E. Cline, Acta Metall. 19, 481 (1971). R. F. Sekerka and T. F. Marinis, in Proceedings of International Conference on Solid-Solid Phase Transformations, ed. H. I. Aaronson, D. E. Laughlin, R. F. Sekerka, and C. M. Wayman, pp. 67– 84, The Metallurgical Society of AIME, Pittsburgh (1981). L. Kulinsky, J. D. Powers, and A. M. Glaeser, Acta Mater. 44, 4115 (1996). F. A. Nichols and W. W. Mullins, J. Appl. Phys. 6, 1826 (1965). B. D. Coleman, R. S. Falk, and M. Moakher, SIAM J. Sci. Comput. 17, 1434 (1996). W. W. Mullins, J. Appl. Phys. 28, 333 (1957). H. Wong, D. Rumschitzki, and C. Maldarelli, Phys. Fluids. 8, 3203 (1996). H. Wong, M. J. Miksis, P. W. Voorhees, and S. H. Davis, Acta Mater. 45, 2477 (1997). H. Wong, M. J. Miksis, P. W. Voorhees, and S. H. Davis, Engineering Science and Applied Mathematics Department, Northwestern University, Technical Report 9606 (1997).
Pergamon PII S1359-6462(98)00127-4
UNIVERSAL PINCH OFF OF RODS BY CAPILLARITYDRIVEN SURFACE DIFFUSION Harris Wong1,*, Michael J. Miksis1, P.W. Voorhees2 and Stephen H. Davis1 1
Department of Engineering Sciences and Applied Mathematics and 2Department of Materials Science and Engineering, Northwestern University, Evanston, IL 60208, USA (Received January 29, 1998) (Accepted in revised form March 22, 1998) Introduction
Interfacial energy is a central factor in setting the morphology of phases and in determining the stability of equilibrium morphologies. For example, a cylinder of radius a with surface energy is unstable when subject to infinitesimal axisymmetric perturbations of wavelengths greater than 2pa. These initially small perturbations grow into the nonlinear regime and lead eventually to the cylinder breaking up into a series of isolated spherical particles, an equilibrium configuration that minimizes the surface energy. The cylinder can be liquid [1, 2] or solid [3], and the critical wavelength is independent of the mass-transport mechanism. This so-called Rayleigh or pinching instability has been predicted and observed to occur in other contexts. For example, thin strips on substrates breakup as well, whether the strips are liquid rivulets [4, 5] or solid lines [6]. Since the morphological evolution is driven simply by the minimization of surface energy, it is not surprising that the Rayleigh instability and the subsequent pinch off is observed in an enormous variety of two-phase mixtures, such as composites composed of rods embedded in a matrix [7, 8], and cylindrical pore channels in ceramics [9]. Despite the breath of systems which undergo this type of capillary-driven morphological evolution, a clear picture of the evolution process away from the linear regime is not well understood. This is because the presence of interfacial energy introduces strong nonlinearities into the equations, which typically require full numerical solutions. The morphological evolution of rods to near pinch off has been simulated numerically when the mass moves by surface diffusion [8, 10, 11]. While these studies describe the evolution of rods in the nonlinear regime, the dependence of the evolution pathway on the initial perturbations imposed on the rod is unclear. In addition, it has not been possible to address the last stages of the break-up process, due to the small length scales and large interfacial velocities that are present. Thus, the evolution of the morphology near the topological singularity where the rod finally fissions or pinches off remains unexplored. Here we examine the morphological evolution of a rod via capillary-driven surface diffusion as it both approaches and departs the topological singularity of pinch off. During the final stages of pinching the neck radius approaches zero, and self-similar solutions are sought. Since cones are the only axisymmetric self-similar geometry, we propose that, as seen far from the neck, the morphology
* Permanent address: Mechanical Engineering Dept., Louisiana State University, Baton Rouge, LA 70803,USA.
55
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Vol. 39, No. 1
Figure 1. Cross sections of a film rod (a) before breakup, (b) at the moment of breakup, and (c) after breakup. Near the pinch-off point, the rod can be approximated as two opposing cones with half-cone angle a. The origin of the cylindrical coordinates (r, z) coincides with the tip of the cones. The surface unit normal vector n forms an angle f with the r-axis.
asymptotes to two opposing cones. Numerical evidence [8, 11] supports this assertion. This self-similar local solution implies that the surface morphology near pinch off can assume a time-independent form under a proper scaling, despite the large surface deflections and the strongly nonlinear nature of the evolution near the pinching singularity. Moreover, only one cone angle is found from solving the scaled equations. Therefore, for any system and for a wide array of initial perturbations of the rod there is a single similarity solution both entering and departing the pinching singularity. The surface morphology in the last stages of pinching is in this sense universal. Formulation Assume that the mass of the rod moves mainly by surface diffusion, and that the surface energy is isotropic. The local normal interfacial velocity, un, is given by [12]: un 5 B ¹2s k,
(1)
where ¹s is the surface gradient operator, and k is the surface curvature. B 5 DsgnV /kT, where Ds is the surface diffusivity, g is the surface energy, n is the number of diffusing atoms per unit area, V is the atomic volume, k is Boltzmann’s constant and T is temperature. Arc-length coordinates are used to accommodate the complicated surface morphologies before and after breakup. The film surface is located by the cylindrical coordinates (r, z), with origin at the tip of the cone as shown in Fig. 1. The surface normal makes an angle f with the r-axis. The surface coordinates (r, z) and angle f are functions of time t and arc length s, and are related by geometry: 2
Vol. 39, No. 1
UNIVERSAL PINCH OFF OF RODS
r 5 sin f, s
57
z 5 cos f. s
(2a,b)
If p 5 rer 1 zez is the position vector of a surface point, then p/t is the surface velocity at constant arc length and is usually not in the normal direction. Given the normal vector n 5 (cosf)er 2 (sinf)ez, we can calculate un 5 n ● p/t, and k 5 ¹s ● n, where ¹s 5 es/s 1 (eu/r)/u, with u being the azimuthal coordinate around the z-axis [13]. Equation (1) then becomes ~cos f!
S
2 r z sin f 2 ~sin f! 5 2B 2 1 t t s r s
DS
D
f cos f 2 . s r
(2c)
At the moment of breakup, t 5 T, and the film consists of two opposing cones touching at the tip (Fig. 1(b)):
f(T, s) 5 a,
r(T, s)5s sina,
z(T, s) 5 s cosa.
(3a,b,c)
Before breakup (t , T), the boundary conditions specify that the film profile is symmetric about the plane z 5 0 (Fig. 1(a)): At s 5 0,
f 5 0,
2f 5 0. s2
z 5 0,
(4a)
The last condition requires zero mass flux at the symmetric plane. After breakup (t . T), the boundary conditions specify that the film profile has rotational symmetry about the z-axis: At s 5 0,
f 5 p/2,
2f 5 0. s2
r 5 0,
(5a,b,c)
Far from the point of breakup, the film is not disturbed:
f 3 a as s 3 `.
(6)
Self-Similar Profile before Breakup (t < T) We determine the surface morphology prior to breaking (t , T) by following the analysis of Wong et al. [14], and defining a set of self-similar variables: S5
s , B (T 2 t)1/4 1/4
Z(S) 5
R(S) 5
z(t, s) , B (T 2 t)1/4
r(t, s) B (T 2 t)1/4 1/4
F(S) 5 f(s, t).
1/4
(7a,b) (7c,d)
Equations (2) then become dR 5 sinF, dS 4
S
DS
d2 sin F d 2 1 dS R dS
dZ 5cosF dS
D
dF cos F 2 - RcosF 1 ZsinF 5 0. dS R
The initial and boundary conditions in (3), (4), and (6) combine to give that at S 5 0
(8a,b)
(8c)
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Figure 2. Self-similar film profiles before and after breakup. Only one self-similar solution is found for pinching. This solution yields a half-cone angle a 5 46.04°. The post-breakup self-similar profile corresponding to this angle is also shown. The self-similar variables R and Z before and after breakup are defined differently in (7) and (9).
F 5 0,
Z 5 0,
d2F 5 0, dS2
(8d,e,f)
and that as S 3 `, F 3 a.
(8g)
Equations (8) are solved using a fourth-order Runge-Kutta scheme with step size DS 5 0.001. At S 5 0, two additional conditions are needed to start the integration. We vary R and dF/dS at the symmetry plane S 5 0 until the boundary condition at infinity, applied at S 5 35, is satisfied. Only one solution is found with R(0) 5 0.7016287178 and dF/dS(0) 5 0.9003221268. This gives a 5 46.04°. The self-similar profile is plotted in Fig. 2. The presence of a unique cone angle during pinching implies that the topological singularity introduces a universality in the morphology. Given an unbroken solid with axial symmetry, the pinching process increases the local slope of the interface until it reaches the critical value. This is consistent with the gradual increase observed in the time-dependent numerical simulations of Sekerka & Marinis [8] and Coleman et al. [11]. From their published figures, we measure the half-cone angles near the onset of pinch off and got values of 44°-45° from Sekerka & Marinis (Fig. 3) and 42°-45° from Coleman et al. (Figs. 2 and 5). Detailed comparison of film profiles is not possible due to a lack of resolution of their figures. Nevertheless, the presence of half-cone angles less than 46.04° is consistent with our predictions that the cone angles must approach the critical value during the evolution process. Thus, the influence of the initial conditions becomes progressively weaker as the topological singularity is approached. The existence of a self-similar morphology close to pinch off also allows us to address the dynamics of pinching. All characteristic lengths in the system scale as (T 2 t)1/4. Therefore, for any system, the curvature at the neck diverges to infinity as (T 2 t)21/4, whereas the thickness of the neck of the rod goes to zero as (T 2 t)1/4. Although the scaling of the morphology with time in the neighborhood of the singularity is unique, the actual values of these quantities are system dependent through the presence of B in the self-similar variables.
Vol. 39, No. 1
UNIVERSAL PINCH OFF OF RODS
59
Self-Similar Profile after Breakup (t > T) To examine the morphological evolution following pinch-off (t . T), a different set of self-similar variables is needed: S5
s , B (t 2 T)1/4
R(S) 5
1/4
Z(S) 5
z(t, s) , B1/4(t 2 T)1/4
r(t, s) B (t 2 T)1/4 1/4
F(S) 5 f(s, t).
(9a,b)
(9c,d)
The governing equations (2) reduce to dR 5 sinF, dS 4
S
DS
sin F d d2 1 dS2 R dS
dZ 5cosF dS
D
dF cos F 2 - RcosF 2 ZsinF 5 0. dS R
(10a,b)
(10c)
The boundary conditions (3), (5), and (6) become that at S 5 0, F 5 p/2,
R 5 0,
d2F 5 0, dS2
(10d,e,f)
and that as S 3 `, F 3 a.
(10g)
Equations (10) are solved using the same shooting method as discussed above. At S 5 0, however, R 5 0 and (10c) is singular. To circumvent this singularity, the integration is started at S 5 So . 0 using the following local expansion around S 5 0: F5
p Zo 1 CoS 1 S3 1 . . . 2 96
(11)
where Co is the scaled tip curvature (5dF/dSuS50) and Zo 5 Z(0) is the scaled tip location. Given a value of Co and Zo, (11) is used to start the integration at S 5 So 5 0.02. (Reducing So to 0.01 has no effect on the results.) Values of Co and Zo are varied in order to satisfy the boundary condition at infinity and to give a specific cone angle. The resulting self-similar profile for t . T with a half-cone angle of 46.04° is given in Fig. 2. This is the profile that should be observed immediately following pinch off. While it is possible to find similarity solutions for a wide range of cone angles when t . T [10, 15], since there is only one cone angle possible for t , T, the surface morphology immediately after pinch off must be consistent with the t , T solution and assume the form given in Fig. 2. Thus the evolution of the surface both prior and post pinch off are system independent and universal. Conclusions We have derived local similarity solutions for the axisymmetric pinch off of rods when the morphological evolution is by capillarity-driven surface diffusion. These local solutions describe the approach to and departure from the topological singularity where a rod pinches into two separate bodies. During
60
UNIVERSAL PINCH OFF OF RODS
Vol. 39, No. 1
pinching, the self-similar surface profile far away from the neck approaches two opposing cones with a unique half-cone angle of 46.04°. It is thus likely that all rods must pinch off with this cone angle. This assertion is supported by several numerical simulations. After pinch off, the smoothening of the cone tip is again self-similar. The results obtained here for rods also apply to the pinch off of cylindrical pore channels. Acknowledgments This research was supported by Department of Energy grant DE-FG02-95ER25241. References 1. 2. 3. 4. 5. 6. 7. 8.
9. 10. 11. 12. 13. 14. 15.
L. Rayleigh, Proc. London Math. Soc. 10, 4 (1879). S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, Clarendon Press, Oxford (1961). F. A. Nichols and W. W. Mullins, Trans. Metall. Soc. AIME. 233, 1840 (1965). S. H. Davis, J. Fluid Mech. 98, 225 (1980). J. B. Culkin and S. H. Davis, AIChE J. 30, 263 (1984). M. S. McCallum, P. W. Voorhees, M. J. Miksis, S. H. Davis, and H. Wong, J. Appl. Phys. 79, 7604 (1996). H. E. Cline, Acta Metall. 19, 481 (1971). R. F. Sekerka and T. F. Marinis, in Proceedings of International Conference on Solid-Solid Phase Transformations, ed. H. I. Aaronson, D. E. Laughlin, R. F. Sekerka, and C. M. Wayman, pp. 67– 84, The Metallurgical Society of AIME, Pittsburgh (1981). L. Kulinsky, J. D. Powers, and A. M. Glaeser, Acta Mater. 44, 4115 (1996). F. A. Nichols and W. W. Mullins, J. Appl. Phys. 6, 1826 (1965). B. D. Coleman, R. S. Falk, and M. Moakher, SIAM J. Sci. Comput. 17, 1434 (1996). W. W. Mullins, J. Appl. Phys. 28, 333 (1957). H. Wong, D. Rumschitzki, and C. Maldarelli, Phys. Fluids. 8, 3203 (1996). H. Wong, M. J. Miksis, P. W. Voorhees, and S. H. Davis, Acta Mater. 45, 2477 (1997). H. Wong, M. J. Miksis, P. W. Voorhees, and S. H. Davis, Engineering Science and Applied Mathematics Department, Northwestern University, Technical Report 9606 (1997).