- Email: [email protected]
The effect of temperature gradient on the stability of a sinusoidal surface profile
Scripta METALLURGICA
Vol. 17, pp. 1289-1292, Printed in the U.S.A.
1983
Pergamon Press Ltd.
THE EFFECT OF TEMPERATUREGRADIENT ON THE STABILITY OF A SINUSOIDAL SURFACE PROFILE J. M. Yu*, M. A. Eshelman and R. Trivedi Ames Laboratory, USDOE, and Department of Materials Science and Engineering, lowa State University, Ames, IA 50011
(Received May 31, 1983) (Revised August 22, 1983) Introduction For an i s o t r o p i c surface energy, a rough surface when annealed w i l l become smooth so as to minimize i t s surface area. Mullins (I) has shown that a sinusoidal p r o f i l e w i l l decay with time according to the equation A(t) = a(o) exp[-K(m)t] (I) where K(~ is the decay constant which is a function of m = 2~/~, X being the wavelength of the periodic p r o f i l e , A(t) is the amplitude of the sine wave at time t , and A(O) is the amplitude of the i n i t i a l p r o f i l e described by y = A(O)sin~. Thus, when a given sinusoidal surface is annealed at a constant temperature, the wave decays in a manner which decreases i t s amplitude but does not a l t e r i t s wavelength. Such a behavior of the sine p r o f i l e has been v e r i f i e d experimentally by a number of investigators (2-5). On the other hand, the e f f e c t of temperature gradient on the r e l a x a t i o n of a surface p r o f i l e has not been studied before. We have carried out experimental studies to characterize the changes in sinusoidal p r o f i l e as a function of time when a constant temperature gradient was imposed across the sample. Under such conditions, a sinusoidal p r o f i l e was found not to decay exponentially with constant wavelength. Instead, the p r o f i l e became unstable and broke up into small c y l i n d r i c a l caps. These caps i n i t i a l l y tended to align themselves in the d i r e c t i o n of the temperature gradient. Consequently, the temperature gradient caused a one-dimensional periodic p r o f i l e to breakup i n t o a two dimensional array before the e n t i r e p r o f i l e decayed to flatness. The main purpose of t h i s paper is to report this new experimental observation on the s t a b i l i t y of a periodic p r o f i l e in the presence of a temperature gradient. Experimental Procedure Experimental work was carried out on single crystals of vanadium with (111) and ( i i ~ ) surface o r i e n t a t i o n s . The surface of the sample was f i r s t polished mechanically through Linde A and then electropolished at -70°C in a methyl alcohol-6% perchloric acid s o l u t i o n . The specimen was then cleaned in methanol in an ultrasonic bath to remove any e l e c t r o l y t e adhering to the surface. Sinusoidal p r o f i l e s were then obtained on the electropolished surface of the sample by the photo-etching technique described by Bonzel and Gjostein (3). A series of sinusoidal p r o f i l e s with wavelengths varying from 12-32 ~m were produced on a given surface (5). These sinusoidal p r o f i l e s were photo-etched on the ( I i 0 ) surface in three d i f f e r e n t d i r e c t i o n s , i . e . , [001], [ l l h ] and [111], The sample was annealed in an u l t r a h i g h vacuum and was heated by electron bombardment. I n i t i a l l y , the surface of the sample was cleaned by heating to 1500°C for 30 seconds to evaporate a few surface layers. Next, the sample was heated to 1150°C with the annealing temperature maintained constant. During this annealing period, a regular sinusoidal p r o f i l e was developed and the amplitude decay, measured by the laser d i f f r a c t i o n technique, was found to follow equation ( i ) . After the i n i t i a l anneal, a f i n i t e temperature gradient of about 50°C/cm was imposed across the length of the specimen and the change in p r o f i l e was studied in situ by the laser d i f f r a c t i o n technique. Observations using optical microscope were made at the end of the annealing period. Results During the annealing at a constant temperature, the laser d i f f r a c t i o n pattern showed a one-dimensional array of spots and the amplitude decay was calculated by measuring the order of *Present address:
Poly-Solar Inc., C~rland, TX.
1289 0036-9748/83 $3.00 + .00
1290
TENPERATURE GRADIENT AND S T A B I L I T Y
OF SURFACE PROFILE
Vo'l.
17,
No.
11
the most i n t e n s e d i f f r a c t i o n beam as a f u n c t i o n of t i m e [ 6 ] . When a t e m p e r a t u r e g r a d i e n t was imposed on the sample, the d i f f r a c t i o n spots i n i t i a l l y became somewhat d i f f u s e d and in some cases a r r a y s of spots appeared in the d i r e c t i o n normal t o the i n i t i a l diffraction array. Such a d i f f r a c t i o n p a t t e r n i n d i c a t e d t h a t the t e m p e r a t u r e g r a d i e n t a l t e r e d the one dimensional s i n e wave p r o f i l e t o some p e r i o d i c s t r u c t u r e in two d i r e c t i o n s . The specimen was then taken out of the u l t r a h i g h vacuum system and observed under o p t i cal microscope. The appearance of the s u r f a c e p r o f i l e is shown in Fig. I . The p i c t u r e s r e p r e sent the top view o f the s u r f a c e so t h a t the l i n e s are along the c r e s t of the sine wave. Note t h a t the i n s t a b i l i t y in the p r o f i l e occurs f o r a l l w a v e l e n g t h s . The l a r g e r wavelength p r o f i l e s are seen t o break up more s l o w l y than the s m a l l e r wavelength p r o f i l e s .
= 26 ~n
FIG. 1.
X = 20 ~n
~ = 16 ~
X = 12 ~m
The effect of temperature gradient on the break-up of sinusoidal profiles of different wavelengths. The top pictures are for time = 200 hours. The bottom pictures are for time = 300 hours.
The break-up of the surface p r o f i l e into cylindrical segments is shown in Fig. 2. Here, also note that the small wavelength profiles broke up much faster than the higher wavelength profiles. Furthermore the i n i t i a l l y cylindrical segments became spherical in shape with time. It can also be seen that these spherical caps tended to line up in the direction which appears to be close to the direction of the imposed temperature gradient. The temperature gradient was imposed along the [110] direction and the p r o f i l e breakup was high for lines along [110] direction compared to the lines along the [111] direction. No breakup of the p r o f i l e was observed for lines in the [001] direction. These observations are shown in Fig. 3. Thus, the i n s t a b i l i t y of the p r o f i l e was observed only when some f i n i t e temperature gradient existed along the crest of the sine wave. The above results were obtained for a sinusoidal p r o f i l e on the (110) surface. Since the surface diffusion coefficient is s l i g h t l y anisotropic on this surface orientation, the anisotropy may also contribute to the break up of the p r o f i l e . Therefore, experiments were carried out on the (111) surface of vanadium. These experiments showed the same behavior of p r o f i l e break up on the (111) surface as on the (110) surface. Consequentlywe conclude that the i n s t a b i l i t y of the p r o f i l e occurs due to the presence of a temperature gradient.
Vol.
17, No.
Ii
TEMPERATURE
GRADIENT AND STABILITY OF SURFACE PROFILE
1291
[
= 32 ~ FIG. 2.
~ = 26 ~m Break up of s i n u s o i d a l
profiles
under the e f f e c t
a
FIG, 3.
~ = 20 um of a 50°C/cm t e m p e r a t u r e g r a d i e n t .
b
The e f f e c t of the d i r e c t i o n of t e m p e r a t u r e g r a d i e n t on the break up of a s i n u s o i d a l profile. Temperature g r a d i e n t is (a) along the l i n e s (b) p e r p e n d i c u l a r t o the lines. Discussion
We s h a l l now r a t i o n a l i z e e x p e r i m e n t a l o b s e r v a t i o n s in terms of a simple model. Consider a segment o f the s i n e p r o f i l e along the c r e s t as a h a l f c y l i n d e r which is placed on a f l a t s u r face, Such a c y l i n d e r d u r i n g a n n e a l i n g w i l l tend t o become f l a t due t o the d e s i r e d decrease in the surface a r e a . Nichols and M u l l i n s (7) showed t h a t a c y l i n d r i c a l sample of c o n s t a n t volume w i l l break up i n t o small segments i f the l e n g t h o f t h e c y l i n d e r was g r e a t e r than 2~R, where R is
1292
TEMPERATURE
GRADIENT AND STABILITY
OF SURFACE
PROFILE Vol.
17, No.
ii
the radius of'a cylinder. This result is identical to the one obtained by Lord Rayleigh (8) for the i n s t a b i l i t y of a long c y l i n d r i c a l l y shaped column ofliquid. Thus itcanbereasonedthata sinusoidal p r o f i l e can minimize i t s surface energy by reducing i t s amplitude and/or by breaking up into c y l i n d r i c a l p a r t i c l e s . Experimental conditions w i l l determine which of these two mechanisms w i l l dominate. Under isothermal conditions, the decay rate of the sinusoidal p r o f i l e w i l l be s u f f i c i e n t l y large so that the reduction in surface energy occurs p r i m a r i l y by wave decay. When a temperature gradient is imposed, the thermal f l u x along the lines (or along the crest of the wave) can make i t possible for the p r o f i l e to break up into c y l i n d r i c a l segments. When large temperature gradients are present along the crest of the wave, the thermal f l u x can be s i g n i f i cantly greater than the c a p i l l a r i t y f l u x so that the volume of the c y l i n d r i c a l segment w i l l remain constant and the c y l i n d r i c a l element w i l l undergo Rayleigh i n s t a b i l i t y . For the i n i t i a l p r o f i l e of 20 ~n wavelength, and 1.2 ~m amplitude, theoretical calculations for vanadium show that the thermal f l u x (9) and the c a p i l l a r i t y f l u x ( i ) w i l l be equal when the temperature gradient is about 19°C/cm. Since a temperature gradient of 50°C/cm was used in our experiments, the thermal f l u x which causes the p r o f i l e to break up would be s i g n i f i cantly larger than the i n i t i a l c a p i l l a r i t y f l u x for the decay of the p r o f i l e . Consequently, the volume change of the c y l i n d r i c a l segment w i l l be n e g l i g i b l e . In f a c t , our experimental results showed that there was no decay of the p r o f i l e u n t i l the break up had already taken place. In summary, experimental results have been presented which show that an i n i t i a l sinusoidal p r o f i l e decays by a simple reduction in i t s amplitude i f the surface is annealed isothermally. However, when a temperature gradient is imposed across the surface, the sinusoidal prof i l e breaks up into c y l i n d r i c a l segments which then become semi-spherical in shape before f i n a l decay to flatness. The rate of break up is found to be faster for smaller wavelength profiles. Acknowledgements This work was supported by Des Laboratory, which is operated for USDOE by lowa State University, under contract No. W-7405-Eng-82, supported by the Director of Energy Research, Office of Basic Energy Sciences, WPAS-KE-02-OI. References I. 2. 3. 4. 5. 6. 7. 8. 9.
W. W. Mullins, J. Appl. Phys., 30, 77 (1959). P. S. Maiya and J. M. Blakely, T . Appl. Phys., 38, 698 (1967). H. P. Bonzel and N. A. Gjostein, Phys. Stat. SOTS. 25, 209 (1968). H. P. Bonzel, A. M. Franken and W. Schwarting, SurTT. Sci, 8__77,13 (1979). J. M. Yu and R. T r i v e d i , Surf. Sci, 125, 396 (1983). H. P. Bonzel and N. A. Gjostein, J. Appl. phys., 39, 3480 (1968). F . A . Nichols and W. W. Mullins, Trans. Met. Soc~'-AIME, 233, 1840 (1965). Lord Rayleigh, Proc. London Math. Soc., i0, 4 (1878). P. G. Shewmon, Diffusion in Solids, p. 18~, McGraw-Hill, New York (1963).
Vol. 17, pp. 1289-1292, Printed in the U.S.A.
1983
Pergamon Press Ltd.
THE EFFECT OF TEMPERATUREGRADIENT ON THE STABILITY OF A SINUSOIDAL SURFACE PROFILE J. M. Yu*, M. A. Eshelman and R. Trivedi Ames Laboratory, USDOE, and Department of Materials Science and Engineering, lowa State University, Ames, IA 50011
(Received May 31, 1983) (Revised August 22, 1983) Introduction For an i s o t r o p i c surface energy, a rough surface when annealed w i l l become smooth so as to minimize i t s surface area. Mullins (I) has shown that a sinusoidal p r o f i l e w i l l decay with time according to the equation A(t) = a(o) exp[-K(m)t] (I) where K(~ is the decay constant which is a function of m = 2~/~, X being the wavelength of the periodic p r o f i l e , A(t) is the amplitude of the sine wave at time t , and A(O) is the amplitude of the i n i t i a l p r o f i l e described by y = A(O)sin~. Thus, when a given sinusoidal surface is annealed at a constant temperature, the wave decays in a manner which decreases i t s amplitude but does not a l t e r i t s wavelength. Such a behavior of the sine p r o f i l e has been v e r i f i e d experimentally by a number of investigators (2-5). On the other hand, the e f f e c t of temperature gradient on the r e l a x a t i o n of a surface p r o f i l e has not been studied before. We have carried out experimental studies to characterize the changes in sinusoidal p r o f i l e as a function of time when a constant temperature gradient was imposed across the sample. Under such conditions, a sinusoidal p r o f i l e was found not to decay exponentially with constant wavelength. Instead, the p r o f i l e became unstable and broke up into small c y l i n d r i c a l caps. These caps i n i t i a l l y tended to align themselves in the d i r e c t i o n of the temperature gradient. Consequently, the temperature gradient caused a one-dimensional periodic p r o f i l e to breakup i n t o a two dimensional array before the e n t i r e p r o f i l e decayed to flatness. The main purpose of t h i s paper is to report this new experimental observation on the s t a b i l i t y of a periodic p r o f i l e in the presence of a temperature gradient. Experimental Procedure Experimental work was carried out on single crystals of vanadium with (111) and ( i i ~ ) surface o r i e n t a t i o n s . The surface of the sample was f i r s t polished mechanically through Linde A and then electropolished at -70°C in a methyl alcohol-6% perchloric acid s o l u t i o n . The specimen was then cleaned in methanol in an ultrasonic bath to remove any e l e c t r o l y t e adhering to the surface. Sinusoidal p r o f i l e s were then obtained on the electropolished surface of the sample by the photo-etching technique described by Bonzel and Gjostein (3). A series of sinusoidal p r o f i l e s with wavelengths varying from 12-32 ~m were produced on a given surface (5). These sinusoidal p r o f i l e s were photo-etched on the ( I i 0 ) surface in three d i f f e r e n t d i r e c t i o n s , i . e . , [001], [ l l h ] and [111], The sample was annealed in an u l t r a h i g h vacuum and was heated by electron bombardment. I n i t i a l l y , the surface of the sample was cleaned by heating to 1500°C for 30 seconds to evaporate a few surface layers. Next, the sample was heated to 1150°C with the annealing temperature maintained constant. During this annealing period, a regular sinusoidal p r o f i l e was developed and the amplitude decay, measured by the laser d i f f r a c t i o n technique, was found to follow equation ( i ) . After the i n i t i a l anneal, a f i n i t e temperature gradient of about 50°C/cm was imposed across the length of the specimen and the change in p r o f i l e was studied in situ by the laser d i f f r a c t i o n technique. Observations using optical microscope were made at the end of the annealing period. Results During the annealing at a constant temperature, the laser d i f f r a c t i o n pattern showed a one-dimensional array of spots and the amplitude decay was calculated by measuring the order of *Present address:
Poly-Solar Inc., C~rland, TX.
1289 0036-9748/83 $3.00 + .00
1290
TENPERATURE GRADIENT AND S T A B I L I T Y
OF SURFACE PROFILE
Vo'l.
17,
No.
11
the most i n t e n s e d i f f r a c t i o n beam as a f u n c t i o n of t i m e [ 6 ] . When a t e m p e r a t u r e g r a d i e n t was imposed on the sample, the d i f f r a c t i o n spots i n i t i a l l y became somewhat d i f f u s e d and in some cases a r r a y s of spots appeared in the d i r e c t i o n normal t o the i n i t i a l diffraction array. Such a d i f f r a c t i o n p a t t e r n i n d i c a t e d t h a t the t e m p e r a t u r e g r a d i e n t a l t e r e d the one dimensional s i n e wave p r o f i l e t o some p e r i o d i c s t r u c t u r e in two d i r e c t i o n s . The specimen was then taken out of the u l t r a h i g h vacuum system and observed under o p t i cal microscope. The appearance of the s u r f a c e p r o f i l e is shown in Fig. I . The p i c t u r e s r e p r e sent the top view o f the s u r f a c e so t h a t the l i n e s are along the c r e s t of the sine wave. Note t h a t the i n s t a b i l i t y in the p r o f i l e occurs f o r a l l w a v e l e n g t h s . The l a r g e r wavelength p r o f i l e s are seen t o break up more s l o w l y than the s m a l l e r wavelength p r o f i l e s .
= 26 ~n
FIG. 1.
X = 20 ~n
~ = 16 ~
X = 12 ~m
The effect of temperature gradient on the break-up of sinusoidal profiles of different wavelengths. The top pictures are for time = 200 hours. The bottom pictures are for time = 300 hours.
The break-up of the surface p r o f i l e into cylindrical segments is shown in Fig. 2. Here, also note that the small wavelength profiles broke up much faster than the higher wavelength profiles. Furthermore the i n i t i a l l y cylindrical segments became spherical in shape with time. It can also be seen that these spherical caps tended to line up in the direction which appears to be close to the direction of the imposed temperature gradient. The temperature gradient was imposed along the [110] direction and the p r o f i l e breakup was high for lines along [110] direction compared to the lines along the [111] direction. No breakup of the p r o f i l e was observed for lines in the [001] direction. These observations are shown in Fig. 3. Thus, the i n s t a b i l i t y of the p r o f i l e was observed only when some f i n i t e temperature gradient existed along the crest of the sine wave. The above results were obtained for a sinusoidal p r o f i l e on the (110) surface. Since the surface diffusion coefficient is s l i g h t l y anisotropic on this surface orientation, the anisotropy may also contribute to the break up of the p r o f i l e . Therefore, experiments were carried out on the (111) surface of vanadium. These experiments showed the same behavior of p r o f i l e break up on the (111) surface as on the (110) surface. Consequentlywe conclude that the i n s t a b i l i t y of the p r o f i l e occurs due to the presence of a temperature gradient.
Vol.
17, No.
Ii
TEMPERATURE
GRADIENT AND STABILITY OF SURFACE PROFILE
1291
[
= 32 ~ FIG. 2.
~ = 26 ~m Break up of s i n u s o i d a l
profiles
under the e f f e c t
a
FIG, 3.
~ = 20 um of a 50°C/cm t e m p e r a t u r e g r a d i e n t .
b
The e f f e c t of the d i r e c t i o n of t e m p e r a t u r e g r a d i e n t on the break up of a s i n u s o i d a l profile. Temperature g r a d i e n t is (a) along the l i n e s (b) p e r p e n d i c u l a r t o the lines. Discussion
We s h a l l now r a t i o n a l i z e e x p e r i m e n t a l o b s e r v a t i o n s in terms of a simple model. Consider a segment o f the s i n e p r o f i l e along the c r e s t as a h a l f c y l i n d e r which is placed on a f l a t s u r face, Such a c y l i n d e r d u r i n g a n n e a l i n g w i l l tend t o become f l a t due t o the d e s i r e d decrease in the surface a r e a . Nichols and M u l l i n s (7) showed t h a t a c y l i n d r i c a l sample of c o n s t a n t volume w i l l break up i n t o small segments i f the l e n g t h o f t h e c y l i n d e r was g r e a t e r than 2~R, where R is
1292
TEMPERATURE
GRADIENT AND STABILITY
OF SURFACE
PROFILE Vol.
17, No.
ii
the radius of'a cylinder. This result is identical to the one obtained by Lord Rayleigh (8) for the i n s t a b i l i t y of a long c y l i n d r i c a l l y shaped column ofliquid. Thus itcanbereasonedthata sinusoidal p r o f i l e can minimize i t s surface energy by reducing i t s amplitude and/or by breaking up into c y l i n d r i c a l p a r t i c l e s . Experimental conditions w i l l determine which of these two mechanisms w i l l dominate. Under isothermal conditions, the decay rate of the sinusoidal p r o f i l e w i l l be s u f f i c i e n t l y large so that the reduction in surface energy occurs p r i m a r i l y by wave decay. When a temperature gradient is imposed, the thermal f l u x along the lines (or along the crest of the wave) can make i t possible for the p r o f i l e to break up into c y l i n d r i c a l segments. When large temperature gradients are present along the crest of the wave, the thermal f l u x can be s i g n i f i cantly greater than the c a p i l l a r i t y f l u x so that the volume of the c y l i n d r i c a l segment w i l l remain constant and the c y l i n d r i c a l element w i l l undergo Rayleigh i n s t a b i l i t y . For the i n i t i a l p r o f i l e of 20 ~n wavelength, and 1.2 ~m amplitude, theoretical calculations for vanadium show that the thermal f l u x (9) and the c a p i l l a r i t y f l u x ( i ) w i l l be equal when the temperature gradient is about 19°C/cm. Since a temperature gradient of 50°C/cm was used in our experiments, the thermal f l u x which causes the p r o f i l e to break up would be s i g n i f i cantly larger than the i n i t i a l c a p i l l a r i t y f l u x for the decay of the p r o f i l e . Consequently, the volume change of the c y l i n d r i c a l segment w i l l be n e g l i g i b l e . In f a c t , our experimental results showed that there was no decay of the p r o f i l e u n t i l the break up had already taken place. In summary, experimental results have been presented which show that an i n i t i a l sinusoidal p r o f i l e decays by a simple reduction in i t s amplitude i f the surface is annealed isothermally. However, when a temperature gradient is imposed across the surface, the sinusoidal prof i l e breaks up into c y l i n d r i c a l segments which then become semi-spherical in shape before f i n a l decay to flatness. The rate of break up is found to be faster for smaller wavelength profiles. Acknowledgements This work was supported by Des Laboratory, which is operated for USDOE by lowa State University, under contract No. W-7405-Eng-82, supported by the Director of Energy Research, Office of Basic Energy Sciences, WPAS-KE-02-OI. References I. 2. 3. 4. 5. 6. 7. 8. 9.
W. W. Mullins, J. Appl. Phys., 30, 77 (1959). P. S. Maiya and J. M. Blakely, T . Appl. Phys., 38, 698 (1967). H. P. Bonzel and N. A. Gjostein, Phys. Stat. SOTS. 25, 209 (1968). H. P. Bonzel, A. M. Franken and W. Schwarting, SurTT. Sci, 8__77,13 (1979). J. M. Yu and R. T r i v e d i , Surf. Sci, 125, 396 (1983). H. P. Bonzel and N. A. Gjostein, J. Appl. phys., 39, 3480 (1968). F . A . Nichols and W. W. Mullins, Trans. Met. Soc~'-AIME, 233, 1840 (1965). Lord Rayleigh, Proc. London Math. Soc., i0, 4 (1878). P. G. Shewmon, Diffusion in Solids, p. 18~, McGraw-Hill, New York (1963).