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A molecular dynamics investigation of the surface correlations in liquid argon
Volume 35A, number 5
PHYSICS LETTERS
o v e r which the s a m p l e was a b s o r b i n g latent heat. Allowance was made for the slight r o u n d i n g effect at the s t a r t and end of the f i r s t - o r d e r change in d e r i v i n g the total t i m e i n t e r v a l At of about 19 m i n u t e s for the l a t t e r . In e v a l u a t i n g the latent heat a b s o r b e d , L = At (dQ/dt)mean , the m e a n value for the r a t e of l o s s of heat over r e g i o n s 1 and 2 was used. T h i s value, (dQ/dT)mean , i s the m e a n of Cp( T 1) (d T/dt), an--d-Cp( T 2) (dT/dt) 2 where Cp(T1) and ~.P(T2) a r e the heat c a p a c i t i e s above and below T ~ and ( d T / d t ) l , and (dT/dt) 2 are the t e m p e r a t u r e d r i f t r a t e s i n these r e s p e c t i v e r e gions. The r e s u l t i s a latent heat of 0 . 4 7 ± 0 . 1 cal/mole. We should also m e n t i o n that this s a m e c h r o m i u m s a m p l e displayed f i r s t - o r d e r effects in other p h y s i c a l p r o p e r t i e s , i n c l u d i n g i t s e l e c t r i c a l r e s i s t i v i t y at T N. The s a m e method a s d e s c r i b e d h e r e i n has f u r t h e r m o r e been s u c c e s s f u l l y applied to obtaining the latent heat of the a ~ r t r a n s i t i o n in c e r i u m [11].
28 June 1971
T h i s r e s e a r c h was supported by the National R e s e a r c h Council of Canada.
References [1] A. Arrott, S.A. Werner and H. Kendrick, Phys. Eev. Letters 14 (1965) 1022. [2] E. W. Lee and M. A. Asgar, Phys. Rev. Letters 22 (1969) 1436. [3] T. Matsmnoto and T. Mitsui, J. Phys. SOc. Japan 27 (1969) 786. [4] J. S. Imai and Y. Sawada, Phys. Letters 34A (1971) 333; D. I. Bo[ef and J. de Klerk, Phys. Rev. 129 (1963) 1063. [5] M. Shimizu, J.AppLPhys. 39 (1968) 1101. [6] J.C.Kimball, Phys.Rev. 183 (1969)533. [7] R. H. Beaumont, H. Chihara and J. A. Morrison, Phil.Meg. 5 (1960) 188. [8] M. B. Salamou, D.S. Simons and P. R. Gamier, Solid State Commun. 7 (1969) 1035. [9] G.T. Meaden and N. H. Sze, J. Low Temp. Phys. 1 (1969). [10] N. H. Sze, G.T. Meaden and C. H. Chin, Cryogenics 9 (1969) 369. [11] N.H. Sze and G.T. Meaden, to be published.
A MOLECULAR DYNAMICS INVESTIGATION OF THE SURFACE CORRELATIONS IN L I Q U I D A R G O N C. A. CRQXTON and R. P. FERRIER
Cave~Ush Labwratory, University of Cambridge, England Received 19 May 1971
A 2--dimensional molecular dynamics simulation of the llquid-vapour interface of argon at the triple point yields a strongly oscillatory density profile. It is found that the dynamical properties in the transition ~one also show quasi-crystalline characteristics normal to the interface.
A m o l e c u l a r d y n a m i c s s i m u l a t i o n of the liquidvapour i n t e r f a c e for a t w o - d i m e n s i o n a l L e n n a r d J o n e s s y s t e m of a r g o n a t o m s at 94.4°K i s r e p o r t . ed. Spatial and d y n a m i c a l coupling of the bulk and s u r f a c e s t a t e s i s achieved by setting up a t h e r m a l d i s r u p t i o n m a t r i x (TDM) which s u b j e c t s the u p p e r m o s t 30 A of the liquid to c o o p e r a t i v e bulk motions. The TDM p o s s e s s e s c h a r a c t e r i s tic i d e n t i f i a b l e as a c o r r e l a t i o n length and a r e l a x a t i o n t i m e r e p r e s e n t a t i v e of the c o o p e r a t i v e b e h a v i o u r of c l u s t e r s of a t o m s n e a r the liquid s u r f a c e . The e q u i l i b r i u m s y s t e m will b e i s o c h o r i c and adiabatic and hence l s o e n e r g e t l c , and so the TDM is s u b j e c t to conditions of both p a r t i cle and e n e r g y c o n s e r v a t i o n . Atoms in the vapour 330
phase a r e e l a s t i c a l l y r e f l e c t e d f r o m the top of the a r r a y , whilst the u s u a l periodic b o u n d a r y conditions a r e applied to the s i d e s of the a r r a y . The configuration is topologically i d e n t i c a l to the s u r f a c e of a c y l i n d e r , the top p e r i m e t e r of which i s p e r f e c t l y r e f l e c t i n g , and the lower subject to the TDM. T h e atoms are released from pseudo-random s i t e s , when they s p o n t a n e o u s l y develop into a s t r o n g l y l a y e r e d i n t e r p h a s e , the p e r i o d of the d e n s i t y o s c i l l a t i o n s being about 3.8 ]L, the f i r s t c o o r d i n a t i o n d i s t a n c e in liquid argon. S e v e r a l atomic l a y e r s develop, and the s t r u c t u r e t a i l s away into the vapour region. The evolution of the d y n a m i c a l q u a n t i t i e s -
Volume 35A, number 5
PHYSICS LETTERS
the v e l o c i t y a u t o c o r r e l a t i o n , its F o u r i e r t r a n s f o r m , the s p e c t r a l density, and the m e a n s q u a r e d i s p l a c e m e n t - all show a n i s o t r o p i c behaviour. The velocity a u t o c o r r e l a t i o n shows s t r o n g o s c i l l a t i o n s for t i m e s up to n e a r l y 5 x 10 -12 sec, which is a p p r o x i m a t e l y five t i m e s g r e a t e r than that found by R a h m a n [1] in bulk liquid argon. C e r t a i n l y d e c o r r e l a t i o n is c o n s t r a i n e d to a twod i m e n s i o n a l plane and this might be c o n s i d e r e d to p r o t r a c t the c o r r e l a t i o n , but the a u t o c o r r e l a tion p a r a l l e l to the liquid s u r f a c e shows much s h o r t e r phonon l i f e t i m e s and is quite l i q u i d - l i k e . The F o u r i e r t r a n s f o r m of the o s c i l l a t o r y autoc o r r e l a t i o n shows d i s c r e t e peaks in the v i b r a t o r y r e g i o n i n d i c a t i v e of a s i m p l e l a t t i c e s t r u c t u r e . On the other hand, the e x i s t e n c e of l o w - f r e q u e n c y diffusive m o d e s is evidenced by the n o n - z e r o value of the function in the l i m i t o~ -. 0. The m e a n s q u a r e d i s p l a c e m e n t s n o r m a l and p a r a l l e l to the liquid s u r f a c e yield asymptotic diffusion c o n s t a n t s of 1.05 x 10 -4 c m 2 / s e c and 5.22 x 10 -5 c m 2 / s e c r e s p e c t i v e l y . That the n o r real diffusion c o n s t a n t i s g r e a t e r than the l a m e l l a r v a l u e is at f i r s t sight s u r p r i s i n g . However, on the b a s i s of a jump diffusion model for motion n o r m a l to the s u r f a c e a m e a n step length I of 3.8 JL with a phonon life t i m e 7o ~ "z.a~ x 1 "u- 12 sec y i e l d s a diffusion c o n s t a n t in good a g r e e m e n t with the ' e x p e r i m e n t a l ' value. L a m e l l a r diffusion, however, i s e s s e n t i a l l y l i q u i d - l i k e with l - ~ 1 /~ and ~'o ~ 10-12 sec. O b s e r v a t i o n of the individual t r a j e c t o r i e s shows that the a t o m s make r a p i d i n t e r p l a n a r e x c u r s i o n s n o r m a l to the s u r f a c e , and
28 June 1971
as such s u b s t a n t i a t e s the jump diffusion model. The s i g n i f i c a n c e of o r d e r e d s t a t e s in the v i c i n ity of the i n t e r f a c e is c o n s i d e r a b l e for the thermod y n a m i c functions of the liquid surface. It may be shown that u n d e r c e r t a i n c i r c u m s t a n c e s the effect of a highly s t r u c t u r e s i n t e r p h a s e is to g e n e r a t e a m a x i m u m in the s u r f a c e t e n s i o n - t e m p e r a t u r e c h a r a c t e r i s t i c . Whether such an i n v e r s i o n e x i s t s , and how far beyond the t r i p l e point depends s e n s i t i v e l y upon the d e t a i l s of the density profile. White [2, 3] in his s e r i e s of p r e c i s i o n equilibrium m e a s u r e m e n t s of T(T) shows that c e r t a i n liquid m e t a l s do indeed show this i n v e r s i o n , and conc l u d e s that this is d i r e c t l y r e l a t e d to the s u r f a c e e x c e s s entropy in the i n t e r p h a s a l region. It m u s t be e m p h a s i s e d , however, that any g e n e r a l c o n c l u s i o n s about t h r e e - d i m e n s i o n a l s y s t e m s b a s e d on these t w o - d i m e n s i o n a l s i m u l a t i o n s m u s t be subject to c e r t a i n r e s e r v a t i o n s . In p a r t i c u l a r a given c o r r e l a t i v e effect is likely to be enhanced in a t w o - d i m e n s i o n a l s y s t e m ; n e v e r t h e l e s s , t h e s e c o m p u t a t i o n s give an i n d i c a t i o n of the tendency for density o s c i l l a t i o n s to develop at he l i q u i d - v a p o u r i n t e r f a c e .
References [1] A. Rahman, Phys. Rev. 136 (1964) A405. [2] D. W. G. White, Metals, materials and metallurgical reviews (July, 1968). [3] D.W.G.White, Trans. Metall. Soc. AIME 236 (1966) 796.
331
PHYSICS LETTERS
o v e r which the s a m p l e was a b s o r b i n g latent heat. Allowance was made for the slight r o u n d i n g effect at the s t a r t and end of the f i r s t - o r d e r change in d e r i v i n g the total t i m e i n t e r v a l At of about 19 m i n u t e s for the l a t t e r . In e v a l u a t i n g the latent heat a b s o r b e d , L = At (dQ/dt)mean , the m e a n value for the r a t e of l o s s of heat over r e g i o n s 1 and 2 was used. T h i s value, (dQ/dT)mean , i s the m e a n of Cp( T 1) (d T/dt), an--d-Cp( T 2) (dT/dt) 2 where Cp(T1) and ~.P(T2) a r e the heat c a p a c i t i e s above and below T ~ and ( d T / d t ) l , and (dT/dt) 2 are the t e m p e r a t u r e d r i f t r a t e s i n these r e s p e c t i v e r e gions. The r e s u l t i s a latent heat of 0 . 4 7 ± 0 . 1 cal/mole. We should also m e n t i o n that this s a m e c h r o m i u m s a m p l e displayed f i r s t - o r d e r effects in other p h y s i c a l p r o p e r t i e s , i n c l u d i n g i t s e l e c t r i c a l r e s i s t i v i t y at T N. The s a m e method a s d e s c r i b e d h e r e i n has f u r t h e r m o r e been s u c c e s s f u l l y applied to obtaining the latent heat of the a ~ r t r a n s i t i o n in c e r i u m [11].
28 June 1971
T h i s r e s e a r c h was supported by the National R e s e a r c h Council of Canada.
References [1] A. Arrott, S.A. Werner and H. Kendrick, Phys. Eev. Letters 14 (1965) 1022. [2] E. W. Lee and M. A. Asgar, Phys. Rev. Letters 22 (1969) 1436. [3] T. Matsmnoto and T. Mitsui, J. Phys. SOc. Japan 27 (1969) 786. [4] J. S. Imai and Y. Sawada, Phys. Letters 34A (1971) 333; D. I. Bo[ef and J. de Klerk, Phys. Rev. 129 (1963) 1063. [5] M. Shimizu, J.AppLPhys. 39 (1968) 1101. [6] J.C.Kimball, Phys.Rev. 183 (1969)533. [7] R. H. Beaumont, H. Chihara and J. A. Morrison, Phil.Meg. 5 (1960) 188. [8] M. B. Salamou, D.S. Simons and P. R. Gamier, Solid State Commun. 7 (1969) 1035. [9] G.T. Meaden and N. H. Sze, J. Low Temp. Phys. 1 (1969). [10] N. H. Sze, G.T. Meaden and C. H. Chin, Cryogenics 9 (1969) 369. [11] N.H. Sze and G.T. Meaden, to be published.
A MOLECULAR DYNAMICS INVESTIGATION OF THE SURFACE CORRELATIONS IN L I Q U I D A R G O N C. A. CRQXTON and R. P. FERRIER
Cave~Ush Labwratory, University of Cambridge, England Received 19 May 1971
A 2--dimensional molecular dynamics simulation of the llquid-vapour interface of argon at the triple point yields a strongly oscillatory density profile. It is found that the dynamical properties in the transition ~one also show quasi-crystalline characteristics normal to the interface.
A m o l e c u l a r d y n a m i c s s i m u l a t i o n of the liquidvapour i n t e r f a c e for a t w o - d i m e n s i o n a l L e n n a r d J o n e s s y s t e m of a r g o n a t o m s at 94.4°K i s r e p o r t . ed. Spatial and d y n a m i c a l coupling of the bulk and s u r f a c e s t a t e s i s achieved by setting up a t h e r m a l d i s r u p t i o n m a t r i x (TDM) which s u b j e c t s the u p p e r m o s t 30 A of the liquid to c o o p e r a t i v e bulk motions. The TDM p o s s e s s e s c h a r a c t e r i s tic i d e n t i f i a b l e as a c o r r e l a t i o n length and a r e l a x a t i o n t i m e r e p r e s e n t a t i v e of the c o o p e r a t i v e b e h a v i o u r of c l u s t e r s of a t o m s n e a r the liquid s u r f a c e . The e q u i l i b r i u m s y s t e m will b e i s o c h o r i c and adiabatic and hence l s o e n e r g e t l c , and so the TDM is s u b j e c t to conditions of both p a r t i cle and e n e r g y c o n s e r v a t i o n . Atoms in the vapour 330
phase a r e e l a s t i c a l l y r e f l e c t e d f r o m the top of the a r r a y , whilst the u s u a l periodic b o u n d a r y conditions a r e applied to the s i d e s of the a r r a y . The configuration is topologically i d e n t i c a l to the s u r f a c e of a c y l i n d e r , the top p e r i m e t e r of which i s p e r f e c t l y r e f l e c t i n g , and the lower subject to the TDM. T h e atoms are released from pseudo-random s i t e s , when they s p o n t a n e o u s l y develop into a s t r o n g l y l a y e r e d i n t e r p h a s e , the p e r i o d of the d e n s i t y o s c i l l a t i o n s being about 3.8 ]L, the f i r s t c o o r d i n a t i o n d i s t a n c e in liquid argon. S e v e r a l atomic l a y e r s develop, and the s t r u c t u r e t a i l s away into the vapour region. The evolution of the d y n a m i c a l q u a n t i t i e s -
Volume 35A, number 5
PHYSICS LETTERS
the v e l o c i t y a u t o c o r r e l a t i o n , its F o u r i e r t r a n s f o r m , the s p e c t r a l density, and the m e a n s q u a r e d i s p l a c e m e n t - all show a n i s o t r o p i c behaviour. The velocity a u t o c o r r e l a t i o n shows s t r o n g o s c i l l a t i o n s for t i m e s up to n e a r l y 5 x 10 -12 sec, which is a p p r o x i m a t e l y five t i m e s g r e a t e r than that found by R a h m a n [1] in bulk liquid argon. C e r t a i n l y d e c o r r e l a t i o n is c o n s t r a i n e d to a twod i m e n s i o n a l plane and this might be c o n s i d e r e d to p r o t r a c t the c o r r e l a t i o n , but the a u t o c o r r e l a tion p a r a l l e l to the liquid s u r f a c e shows much s h o r t e r phonon l i f e t i m e s and is quite l i q u i d - l i k e . The F o u r i e r t r a n s f o r m of the o s c i l l a t o r y autoc o r r e l a t i o n shows d i s c r e t e peaks in the v i b r a t o r y r e g i o n i n d i c a t i v e of a s i m p l e l a t t i c e s t r u c t u r e . On the other hand, the e x i s t e n c e of l o w - f r e q u e n c y diffusive m o d e s is evidenced by the n o n - z e r o value of the function in the l i m i t o~ -. 0. The m e a n s q u a r e d i s p l a c e m e n t s n o r m a l and p a r a l l e l to the liquid s u r f a c e yield asymptotic diffusion c o n s t a n t s of 1.05 x 10 -4 c m 2 / s e c and 5.22 x 10 -5 c m 2 / s e c r e s p e c t i v e l y . That the n o r real diffusion c o n s t a n t i s g r e a t e r than the l a m e l l a r v a l u e is at f i r s t sight s u r p r i s i n g . However, on the b a s i s of a jump diffusion model for motion n o r m a l to the s u r f a c e a m e a n step length I of 3.8 JL with a phonon life t i m e 7o ~ "z.a~ x 1 "u- 12 sec y i e l d s a diffusion c o n s t a n t in good a g r e e m e n t with the ' e x p e r i m e n t a l ' value. L a m e l l a r diffusion, however, i s e s s e n t i a l l y l i q u i d - l i k e with l - ~ 1 /~ and ~'o ~ 10-12 sec. O b s e r v a t i o n of the individual t r a j e c t o r i e s shows that the a t o m s make r a p i d i n t e r p l a n a r e x c u r s i o n s n o r m a l to the s u r f a c e , and
28 June 1971
as such s u b s t a n t i a t e s the jump diffusion model. The s i g n i f i c a n c e of o r d e r e d s t a t e s in the v i c i n ity of the i n t e r f a c e is c o n s i d e r a b l e for the thermod y n a m i c functions of the liquid surface. It may be shown that u n d e r c e r t a i n c i r c u m s t a n c e s the effect of a highly s t r u c t u r e s i n t e r p h a s e is to g e n e r a t e a m a x i m u m in the s u r f a c e t e n s i o n - t e m p e r a t u r e c h a r a c t e r i s t i c . Whether such an i n v e r s i o n e x i s t s , and how far beyond the t r i p l e point depends s e n s i t i v e l y upon the d e t a i l s of the density profile. White [2, 3] in his s e r i e s of p r e c i s i o n equilibrium m e a s u r e m e n t s of T(T) shows that c e r t a i n liquid m e t a l s do indeed show this i n v e r s i o n , and conc l u d e s that this is d i r e c t l y r e l a t e d to the s u r f a c e e x c e s s entropy in the i n t e r p h a s a l region. It m u s t be e m p h a s i s e d , however, that any g e n e r a l c o n c l u s i o n s about t h r e e - d i m e n s i o n a l s y s t e m s b a s e d on these t w o - d i m e n s i o n a l s i m u l a t i o n s m u s t be subject to c e r t a i n r e s e r v a t i o n s . In p a r t i c u l a r a given c o r r e l a t i v e effect is likely to be enhanced in a t w o - d i m e n s i o n a l s y s t e m ; n e v e r t h e l e s s , t h e s e c o m p u t a t i o n s give an i n d i c a t i o n of the tendency for density o s c i l l a t i o n s to develop at he l i q u i d - v a p o u r i n t e r f a c e .
References [1] A. Rahman, Phys. Rev. 136 (1964) A405. [2] D. W. G. White, Metals, materials and metallurgical reviews (July, 1968). [3] D.W.G.White, Trans. Metall. Soc. AIME 236 (1966) 796.
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