On the linear instability of the rayleigh spheroidization process

Scripta METALLURGICA et ~%TERIALIA

Vol. 25, pp. 799-804, 1991 Printed in the U.S.A.

Pergamon Press plc All rights reserved

On the Linear Instability of the Rayleigh Spheroidization

Process

S. A. Hackney Department of Metallurgical Engineering Michigan Technological University Houghton, MI 49931 (Received January 6, 1991) (Revised January 22, 1991)

Introduction The problem of diffusional stability of infinitely long cylinders with respect to spheroidization has been analyzed in several different works (1-6). It is well known that long solid cylinders will break down into individual particles of spherical geometry under the action of an appropriate shape perturbation. The driving force for this process is considered to be the chemical potential gradients associated with variations in curvature which develop on the surface, or interface, in response to the perturbation. The resulting diffusional flow of material can lead to a breakdown of the rod morphology. From a practical point of view, the cylindrical geometry is one which occurs in fiber reinforced composites and rod shaped precipitates in eutectic/eutectold structures. The shape stability of such rods is an important aspect of high temperature mechanical behavior. The general method of study for this problem involves analyzing the time dependent behavior of a perturbation; which is an imposed, mass conserving and periodic surface wave of small amplitude. These surface undulations will be referred to here as Rayleigh surface waves. In most cases, attention is focused on the surface wavelength which gives the maximum growth rate of the perturbation. This choice has not been Justified by experimental results [6] on rod eutectics which show significant scatter in the instability wavelength, and little or no physical basis, such as a stability argument, has been presented for the operation of the "maximum growth rate hypothesis" [7] in the growth of periodic Rayleigh surface waves. Despite the many detailed studies on the behavior of Rayleigh surface waves, the possibility that the surface waves themselves might actually be unstable has not been considered in the literature. In this note, the stability of Rayleigh surface waves with respect to a very simple distortion is considered and the result is examined with respect to the first in situ experimental observations of shape instabilities in free standing rods. Experimental High purity Cu foil (99.999) obtained from Alfa Products was electropolished to perforation, ion milled for 30 seconds (to remove the anodic oxide), and heated to 500"C in situ using the Phillips 301 TEM hot stage. A periodic morphological instability at the edge of the foil perforation (treated elsewhere [8]) led to the development of cylindrical rods (as determined by tilting about the rod axis) normal to the original foil edge. An example of this phenomenon is shown in Fig. i. The cylindrical rods were directly observed to undergo Rayleigh spheroidization and the wavelengths of the rod instability behind the initial tip blunting of the rods were measured. Rod diameters were determined at the midpoints between the Rayleigh surface wave peak and valley. Measurements were made only on the rods which were

799 0036-9748/91 $3.00 + .00 Copyright (¢) 1991 Pergamon Press plc

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SPHEROIDIZATION

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d e t e r m i n e d to be p e r p e n d i c u l a r to the electron beam (as d e t e r m i n e d by tilting p e r p e n d i c u l a r to the rod axis). The e x p e r i m e n t a l l y determined w a v e l e n g t h to rod d i a m e t e r ratios are presented in Table 1 and show significant scatter (as

in [6]). Analysis In this note, the works of Nichols [I], Nichols and Mulllns [2], and Sekerka and Marinis [6] are used as a point of departure. As in aspects of these works, it is assumed that surface d i f f u s i o n is the primary mechanism of mass t r a n s p o r t and the standard llnearizlng approximations of small slope and isotropic surface properties are made. To study the morphological stability of an infinite cylinder, the shape must be "perturbed". The standard p e r t u r b a t i o n has been of the form [1,2,6] r(z,t)

= R

+ 6(t)coswz

(i)

O

where R

is the initial,

unperturbed

radius,

~ is the magnitude of the

O

perturbation,

w is the w a v e n u m b e r

and z is the axial direction.

It has been shown [1,2] that the time dependent change in the radius of the p e r t u r b e d rod a s s o c i a t e d with a d i v e r g e n c e in the surface flux is given by (for small slope, isotropic surface properties, etc.) dr 82K HE = S 8z 2 where B is a constant defined in [1,2,6] K = rl

2d2r dz

or, in expanded

(2) and K is the curvature,

form

l_ - f(z)

K = Ro

R 2

d2r

- dz 2

(3)

O

where f(z) is the z-dependent part of the function r. The combination of equations (I), (2) and (3) has been studied numerous times [1,2,6] resulting in an e x p r e s s i o n for the growth rate of the perturbation

6 -- w 2 s [

~i 2 - w 2

1

(4)

O

where ~ is the time d e r i v a t i v e of 8. This indicates that a range of w a v e n u m b e r s exist, w < 1/Ro, w h i c h will cause the rod to spheroidlze . usual approach

is to specify the crltical wavelength,

The

~ = 2s/w c = 2~ Ro, C

above w h i c h s p h e r o i d l z a t l o n may occur and the w a v e l e n g t h growth rate of the instability t = V~ 2~R .

for the maximum

In order to study the stability of these surface waves, the perturbation of r in e q u a t i o n (i) must be further perturbed. One p a r t i c u l a r l y simple technique is to distort the w a v e l e n g t h so that it changes with position and then study the time dependent behavior of this w a v e l e n g t h distortion. An equation w h i c h accomplishes this is the periodic w a v e l e n g t h distortion

= ~O + ~cos(./L)z where ~ = 2L/n with n an integer,

(5) A << be, L >> R

O

wavelength.

and I O

For A << ~o' the w a v e l e n g t h

distortion

is the u n p e r t u r b e d

O

leads to a periodic

v a r i a t i o n in the w a v e n u m b e r which may be a p p r o x i m a t e d by the expression (Taylor series)

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SPHER01DIZATION

2~ w = Io + Acos(r~/L)z = Wo

801

2~Acos (w/L) z ~ 2

(6)

o

Thus,

the

initial

r(z,o)

condition

takes

the

form

= Ro + 6cos

[w ° - 2~Ac°s(~/L)z I ~

in

series

(~)

1 z

o

or,

expanding

a Taylor

about

w o

r(z,o)

= R o + ~cos w oz + z62~_____~A ~ 2 sin (WoZ) cos

(~/L)z

(8)

o

w h i c h is an even function. S u b s t i t u t i o n of e q u a t i o n gives the partial d i f f e r e n t i a l e q u a t i o n 0r 8t

(3) into e q u a t i o n

(2)

02r] Oz2 ~

B [04r + ~ [@z 4 Ro

(9)

to be solved using p e r i o d i c b o u n d a r y c o n d i t i o n s over the p e r i o d -L to L. E q u a t i o n (7) or (8) cannot simply be s u b s t i t u t e d into equation.(9) as this results in s e c u l a r (unbounded terms) in e x p r e s s i o n s for ~ and A. Instead, e q u a t i o n (8) m u s t be t r e a t e d as an initial c o n d i t i o n for the general s o l u t i o n to e q u a t i o n (9). Equation

(9) admits a s o l u t i o n of the form

r = #(z)

exp

(-~t)

S u b s t i t u t i o n of e q u a t i o n

taz 4

(i0) (i0) into e q u a t i o n

(9) gives the e q u a t i o n

R 2 az2J o

For the h a r m o n i c s o l u t i o n c o n s i s t e n t w i t h periodic b o u n d a r y c o n d i t i o n s and an even initial condition, ¢ is given as ¢ = A cos wmz, and the eigenvalue, ~, m

is d e t e r m i n e d as

~4

~=B

m

- ~ R

12]

(12)

Wm

o

The linear c o m b i n a t i o n of p o s s i b l e solutions w h i c h satisfy p e r i o d i c b o u n d a r y c o n d i t i o n s at -L and L and initial c o n d i t i o n (8) leads to the s o l u t i o n r = R o

+ Z A cos w z exp Bt m=1 m m

w m2

R

1 2 _ Wm o

where w

is m ~ / L and A m

is a Fourier c o e f f i c i e n t d e t e r m i n e d

from the initial

m

c o n d i t i o n in the usual m a n n e r (see Table II). It can be v e r i f i e d that this s o l u t i o n is a c o r r e c t s o l u t i o n by s u b s t i t u t i o n into e q u a t i o n (9). This s o l u t i o n can also be o b t a i n e d by the independent m e t h o d of p e r t u r b a t i o n expansion. It should be noted that e q u a t i o n (13) can also be used to study the c o n d i t i o n in w h i c h only a single harmonic mode is present as an initial c o n d i t i o n as in [1,2,6] and then only one Fourier c o e f f i c i e n t w o u l d be

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SPHER01DIZATION

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25, No.

non-zero. The r e s u l t i n g e q u a t i o n is the same result as o b t a i n e d by the m e t h o d of analysis used by p r e v i o u s authors [1,2,6] for this special initial condition. E q u a t i o n (13) can be s e p a r a t e d into a term w h i c h d e s c r i b e s the growth of the average h a r m o n i c w i t h wavelength, I = 2L/n (n an integer), and the terms O

due to the distortion; all o t h e r harmonics w i t h m~n. To d e t e r m i n e if a given w a v e n u m b e r w (= wo) = n ~ / L is stable, the time d e p e n d e n t b e h a v i o r of the harmonic modes in e q u a t i o n (13) w h i c h result from the w a v e l e n g t h d i s t o r t i o n is examined. This time d e p e n d e n c e of the w a v e l e n g t h d i s t o r t i o n is g i v e n by e q u a t i o n (13) e v a l u a t e d for al~ m except m = n. Thus, the d i s t o r t i o n terms will grow as long as I/Ro 2 • w , and As m 0. The values for A, are shown in Table If. It is e v i d e n t that the w a v e l e n g t h d i s t o r t i o n terms of initial m a g n i t u d e A will grow for I ~ m < ~ and d e c a y for all o t h e r p o s s i b l e values O

of m. This result is i n d e p e n d e n t of the value of n, w h i c h d e t e r m i n e s the average harmonic. This result implies that all w a v e l e n g t h s are u n s t a b l e w i t h respect to the type of initial c o n d i t i o n p e r t u r b a t i o n p r o p o s e d in e q u a t i o n (8), even I . The result that ~ is u n s t a b l e w i t h respect to p e r t u r b a t i o n ~X

max

has not been d e m o n s t r a t e d previously. D i s c u s s i o n and S u m m a r y The analysis leading to e q u a t i o n (13) allows the study of the effect of d i s t o r t i o n s on the p e r i o d i c w a v e f o r m of R a y l e i g h surface waves. Confidence in the method of s o l u t i o n can be d e v e l o p e d by n o t i n g that e q u a t i o n (13) gives the same result as p r e v i o u s authors w h e n the initial c o n d i t i o n is a simple harmonic. The fact that s i g n i f i c a n t scatter is o b s e r v e d in the e x p e r i m e n t a l l y m e a s u r e d i n s t a b i l i t y w a v e l e n g t h s (as in [6]) can be r a t i o n a l i z e d in terms of the analysis, w h i c h shows that no w a v e l e n g t h is stable w i t h r e s p e c t to perturbation, i.e., the w a v e l e n g t h d i s t o r t i o n s will grow w i t h time for all values of 2L/n. If the results of this linear analysis are e x t r a p o l a t e d into the n o n l i n e a r regime, the s u g g e s t i o n could be made that the w a v e l e n g t h at s p h e r o i d i z a t i o n is the result of r a n d o m "noise" in spatial v a r i a t i o n of the radius a s s o c i a t e d w i t h the formation of the rod. For example, t e m p e r a t u r e v a r i a t i o n s d u r i n g the g r o w t h of the rod e u t e c t i c s studied in [6] or initial variations in foil thickness p r i o r to the f o r m a t i o n of the free s t a n d i n g rods studied here will lead to spatial v a r i a t i o n s in rod radius. In this sense, the w a v e l e n g t h at s p h e r o i d i z a t i o n is a s t o c h a s t i c v a r i a b l e d e p e n d e n t on the random form of the initial c o n d i t i o n of the spatial v a r i a t i o n of rod radius. The i n s t a b i l i t y of the p e r i o d i c s o l u t i o n for the R a y l e i g h s p h e r o i d i z a t i o n process can be p l a c e d in a general context. The s t a b i l i t y of g e n e r i c p e r i o d i c solutions was e x a m i n e d in 1965 by Eckhaus [9] and the growth of a p e r i o d i c p e r t u r b a t i o n on p e r i o d i c initial conditions, as d e v e l o p e d here, are r e f e r r e d to as Eckhaus instabilities. Eckhaus i n s t a b i l i t i e s are known to occur in a v a r i e t y of p h y s i c a l p r o c e s s e s and have been treated in fluid flow [i0] and s o l i d i f i c a t i o n [ii] phenomena. It is evident from [I0] and m a n y other treatments that the finite size of the s y s t e m (the rod in this case) can effect the b e h a v i o r of the Eckhaus instability. E x p e r i m e n t a l and t h e o r e t i c a l efforts to c o r r e l a t e the s p h e r o i d i z a t i o n w a v e l e n g t h s c a t t e r to the rod length are c u r r e n t l y in progress. Acknowledgments The author w o u l d like to thank P r o f e s s o r s J. K. Lee, T. H. Courtney, and R. F. Sekerka for c r i t i c a l reviews of the m a n u s c r i p t and several e n l i g h t e n i n g discussions. This r e s e a r c h was s u p p o r t e d by the D i v i s i o n of M a t e r i a l s Sciences of the U. S. D e p a r t m e n t of Energy.

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References [i] F. A. Nichols, Doctoral Thesis, Carnegie-Mellon University, Pittsburgh, PA (1964). [2] F. A. Nichols and W. W. Mullins, Trans, AIME 233, 1840 (1965). [3] H. E. Cline, Acta. Met., 19, 481 (1971). [4] J. W. Cahn, Scripta Met., 13, 1069 (1979). [5] T. H. Courtney, New Developments and Applications in Composites, (editors D. Kuhlmann-Wilsdorf and W. C. Harrigan, Jr.) TMS-AIME Warrendale, PA, I (1979). [6] R. F. Sekerka and T. F. Marlnis, In Situ Composites IV, (editors F. D. Lemkey, H. E. Cline, and M, Mclean) Elsevier Science, NY, 315 (1982). [7] C. Zener, Trans. AIME, 167, 550 (1946). [8] S. A. Hackney, to be submitted to Materials Science and Eng. [9] W. Eckhaus, Studies in Non Linear Stability Theory, Springer-Verlag, NY ( 1 9 6 5 ) , p. 63-98. [i0] W. Zimmerman and L. Kramer, J. de Physique, 46, no. 3, 343 (1985). [11] A. J. Simon, J. Bechhoefer, A. Libchaber, Phys. Rev. Letters, 61 (22), 2574 (1988).

Table I*. 3.3 3.3 3.3

Experimentally Diameter. 3.8 3.8 3.8

3.9 4.0 4.3

Determined

4.5 5.0 5.0

Ratios of Instability Wavelength/Rod

6.3

Ratio predicted by maximum growth rate hypothesis = 4.44

*Measurements were made using concentric circles on the TEM screen. The error in this technique limits the accuracy of the ratios to two significant figures. Table

II.

Calculated

Forms of the Fourier Coefficients,

A m

4L6A 2

Icos(n+l+m), [ ~+i~

+

cos(n+l-m)n n+l-m

+ cos(n-l+m), n-l+m

+ cos(n-l-m)~] ~-[:-~ ) m = n

cos(n-l+m)~

+ cos(n-l-m)~

]

o

4L6A

[cos(n+l+m)~ n+l--~

+

n- l+m

n- l-m

m=n+l

o

4L6Af 2

[Ic°s(n+l+m)~-n¥1-~ + cos(n+l-m)Un+l_m + cos(n-l+m)~n_l+m ]

m = n -i

o

4L6A 2 o

[oos,n+1+m ~+-i7~

+

cos(n+l-m)~ n+l-m

+

cos(n+l-m)~ n-l+m

+

cos(n-l-m)~ m : n n-l-m ] m • n + 1 m n - 1

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Fig. Fig. i.

la

~

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Fig. Ib

A bright field TEM micrograph of cylindrical filaments produced during in situ heating by the capillary instability at the edge of a Cu (black area) foil. Many of the cylindrical filaments in (a) and (b) are undergoing Rayleigh spheroidization.