QUENCH ECHOES IN LENNARD-JONES AND RUBIDIUM FROZEN FLUIDS--TEMPERATURE DEPENDENCE OF THE ANHARMONIC BEHAVIOR

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QUENCH ECHOES IN LENNARD-JONES AND

RUBIDIUM FROZEN FLUIDS–TEMPERATURE

DEPENDENCE OF THE ANHARMONIC BEHAVIOR

G. Grest, S. Nagel, A. Rahman

To cite this version:

G. Grest, S. Nagel, A. Rahman. QUENCH ECHOES IN LENNARD-JONES AND RUBIDIUM

FROZEN FLUIDS–TEMPERATURE DEPENDENCE OF THE ANHARMONIC BEHAVIOR.

Jour-nal de Physique Colloques, 1980, 41 (C8), pp.C8-293-C8-296.

�10.1051/jphyscol:1980874�.

JOURNAL DE PHYSIQUE CoZZoque C8, suppZ6ment au n08, Tome 41, aoCt 1980, page

~ 8 - 2 9 3

QUENCH ECHOES I N LENNARD-JONES AND RUBIDIUM FROZEN FLUIDS--TEMPERATURE DEPENDENCE OF THE ANHARMONIC BEHAVIOR

G.S. G r e s t , S.R. ~ a ~ e l * and A.

ahm man**

Department o f Physics, Purdue Univ., West L a f a y e t t e , Indiana 47907.

*

James Franck I n s t i t u t e and Department o f Physics, Univ. o f Chicago, Chicago, IZZinois 60637.

x* Argonne National Laboratomj, Argonne, I 2 Zinois 60439.

Abstract.

-

We have found a new echo phenomenon t h a t appears i n m o l e c u l a r dynamics s i m u l a t i o n s o f a s o l i d . The echo appears i n t h e behavior o f t h e temperature a f t e r t h e system recovers from two a b r u p t quenches. This e f f e c t can be e x p l a i n e d i n terms o f c l a s s i c a l normal mode a n a l y s i s o f t h e system. We have a p p l i e d these echo techniques t o t h e study o f close packed glasses w i t h Lennard-Jones and r u b i - dium i n t e r a t o m i c p o t e n t i a l s . We have been a b l e t o study t h e degree o f anharmonic behavior i n t h e system as a f u n c t i o n of temperature as t h e sample i s heated from t h e glass t o t h e l i q u i d s t a t e . We have a l s o s t u d i e d t h i s behavior when t h e a t t r a c t i v e and r e p u l s i v e t e r n s i n t h e Lennard-Jones poten- t i a l a r e varied.

I n a r e c e n t paper,' a new echo phenomenon was r e p o r t e d t h a t can be observed i n molecular dynamics s i m u l a t i o n s o f amorphous and c r y s t a l l i n e s o l i d s . The echo occurs when t h e temperature (which i s s i m p l y p r o p o r t i o n a l t o t h e t o t a l k i n e t i c energy o f t h e system), i s q u i c k l y lowered two times i n succession w i t h an i n t e r v a l , t

,

between quenches. A t a t i m e ti a f t e r t h e second quench, i f t h e system i s allowed t o r u n f r e e l y , t h e tem- ~ e r a t u r e suPdenly drops and then r i s e s again. I n Fig. 1, we show t h e echo behavior o f t h e tem- o e r a t u r e a f t e r t h e second quench when an i n t e r v a l tl = 45At separates t h e two quenches. Here A t i s t h e t i m e s t e p used i n t h e i n t e g r a t i o n o f ?!ewtonls equation i n t h e molecular dynamics s i n u l a t i o n . 2 Note t h e l a r g e d i p i n temperature a t a t i m e tl a f t e r t h e second quench.

his

quench echo.can be e x p l a i n e d i n terms o f c l a s s i c a l normal mode a n a l y s i s . I n a completely harmonic s o l i d each mode w i l l v i b r a t e independently. The decay o f a mode

w i l l

be caused b y anharmonic i n t e r a c - t i o n s between t h e modes. The modes which have j u s t t h e r i g h t frequency, w = where n i s an

- 1

i n t e g e r , w i l l escape t h e second quench w i t h o u t l o s i n g any o f t h e i r k i n e t i c energy. A1T t h e o t h e r modes (those w i t h t h e "wrong" frequency) w i l l l o s e some energy d u r i n g t h e second quench. Assuming a completely harmonic system i t i s p o s s i b l e t o w r i t e down e x a c t l y t h e behavior o f t h e temperature as a f u n c t i o n o f time. A f t e r t h e f i r s t quench each mode c o n t r i b u t e s an amount t o t $ e k i n e t i c energy and t h e r e f o r e t o t h e tempera- t u r e depending on i t s amplitude a t t h e time o f t h e quench. A l l modes s t a r t o f f a f t e r ' t h e f i r s t quench w i t h t h e same phase. Hence

where Ai i s p r o p o r t i o n a l t o t h e amplitude o f mode

i

a t t = 0. A f t e r a second quench t h e new ampli- t u d e ' w i l l be Aicosw.tl so t h a t , measuring t i m e from t h e moment o f $he l a s t quench:

I f t h e r e a r e many modes i n t h e system, d i s t r i - b u t e d o v e r a smooth d e n s i t y o f s t a t e s , t h i s

expression leads t o a w e l l - d e f i n e d echo a t t = tl. Each a d d i t i o n a l quench w i l l i n t r o d u c e another

f a c t o r cos2wit2 where t2 is t h e i n t e r v a l between t h e l a s t two quenches

2 2 2 2

T ( t ) = GAi cos w.t cos w.t s i n w . t

i 1 1 1 2 1

where t i m e i s again measured from t h e l a s t quench. T h i s w i l l c r e a t e echoes a t tl (which we s h a l l r e f e r t o as t h e " s t i m u l a t e d echo"), t 2 and s m a l l e r ones a t

I

t2

+

tl

1

.

The degree t o which a p a r t i c - u l a r s o l i d i s anharmonic can be measured by observing whethe.r t h e echo behavior obeys these equations e x a c t l y .

Fig. 1.--Temperature versus t i m e a f t e r two- quenches w i t h an i n t e r v a l tl = 45At between them. Note t h e echo a t 45At.

I n t h e p r e s e n t oaoer we w i l l show t h e tem- p e r a t u r e dependence o f t h e quench echoes i n t h e g l a s s and l i q u i d phases o f two 500 p a r t i c l e systems i n t e r a c t i n g v i a d i f f e r e n t i n t e r a t o m i c p o t e n t i a l : a Lennard-Jones 6-12 p o t e n t i a l and an i n t e r a t o m i c o o t e n t i a l which corresponds t o t h a t found i n l i q u i d rubidium.3 The reduced temperature, g i v e n b y T* = k B T / ~ where E i s t h e

depth o f t h e i n t e r a t o m i c p o t e n t i a l w e l l , was v a r i e d . These two systems have been s t u d i e d previously4, 5,

6

and were found t o have v e r y

JOURNAL DE PHYSIQUE

cg-294

d i f f e r e n t p r o p e r t i e s i n t h e l i q u i d s t a t e . The l i q u i d r u b i d i u m sample showed w e l l - d e f i n e d modes o u t t o l a r g e wavevectors (wavelengths as small as 1.25 times t h e i n t e r p a r t i c l e soacing5) whereas t h e Lennard-Jones l i q u i d showed t h a t t h i s ohonon behavior d i s a ~ p e a r s much more r a o i d l y w i t h wave- v e c t o r (wavelengths o n l y as small as 5 times t h e i n t e r p a r t i c l e distance5).

By

s t u d y i n g t h e behavior o f t h e quench echoes i n these two systems we have found evidence o f a s i m i l a r d i f - ference between t h e b e h a v i o r o f these two systems i n t h e g l a s s phase a t temperatures much below t h e m e l t i n g temperature. The decay i n time o f t h e normal modes i n t h e Lennard-Jones system i s much f a s t e r than t h a t o f t h e modes i n t h e rubidium sample a t t h e same value o f t h e reduced tempera- t u r e . The echo behavior c o n t a i n s no i n f o r m a t i o n about t h e s p a t i a l d i s t r i b u t i o n of a mode b u t o n l y about i t s t i m e dependence.

There are s e v e r a l ways t o demonstrate t h e p r e s e n c e ' o f anharmonic b e h a v i o r i n a system using quench echoes. As can be seen from Eq. 3, i t does n o t m a t t e r f o r t h e subsequent behavior o f a harmonic system whether t h e i n t e r v a l tl comes b e f o r e o r a f t e r t h e i n t e r v a l t2 in the quench sequence t h a t c r e a t e s t h e echoes a t t T , t and

I t

+ tl

1 .

By quenching t h e same system ? i r s t wizh-the i n t e r v a l s i n one o r d e r and then again w i t h t h e i n t e r v a l s reversed, one can m o n i t o r t h e amount o f anharmonic behavior i n a system b y seeinc how d i f f e r e n t t h e response o f t h e

system i s t o t h e two d i f f e r e n t quench sequences.

F i g . 2.--The temperature versus time f o r t h e Lennard-Jones system quenched t h r e e times w i t h tl = 45At and t2 = 30At (dashed l i n e ) and w i t h tl = 30At and t 2

=

45At ( d o t t e d l i n e ) . The t ~ o curves a r e d i s p l a c e d v e r t i c a l l y . The i n i t i a l temperatures a r e (a) T* = 0.11, (b) T* = 0.46 and ( c ) T* = 1.2.

I n Fig. 2 we show t h e r e s u l t s o f t h i s behavior f o r i n t e r v a l s o f 30At and 45At f o r t h e Lennard- Jones system a t 3 temperatures. I n each case the curves a r e displaced v e r t i c a l l y from each o t h e r f o r comparison and t h e lower curve i n each s e t i s f o r tl = 45At and t2 = 30At and t h e upper curve i s f o r tl = 30At and t 2 = 45At. A t very low s t a r t i n g temperatures, we can observe no d i f f e r e n c e between t h e two curves. However a t a s t a r t i n g temperature o f T* = 0.11 (Fig. 2a)

,

we a l r e a d y see t h a t t h e depth o f t h e two echoes i s d i f f e r e n t i n t h e two curves i n d i c a t i n g a s i z a b l e m o u n t o f anharmonicity. Even a t t h i s low tem- o e r a t u r e (% 1/8 o f t h e m e l t i n g temperature o f t h e

P.c.c. c r y s t a l ) t h e b e h a v i o r o f t h e s o l i d shows t h a t approximately 20% of t h e energy o f a mode i s rlissi,pated per period. A t h i g h e r temperatures

we see t h a t t h e anharmonic behavior becomes pro- g r e s s i v e l y more pronounced u n t i l , i n t h e l i q u i d a t T* = 1.2 we have d i f f i c u l t y seeing any echo behavior a t a1 1.

I n Fig. 3, we show t h e behavior o f t h e r u b i d i u m sample a f t e r a s i m i l a r s e r i e s o f quenches, The lowest temperature s t u d i e d was T* = 0.15. Although t h i s temperature i s above t h a t f o r Fig. ?a-where T* = 0.11, i t i s c l e a r t h a t t h e r e i s considerably l e s s anharmonic behavior than i n t h e Lennard-Jones system s i n c e t h e two curves show much s m a l l e r d e v i a t i o n s from each o t h e r . As t h e temoerature i s r a i s e d we c o n s i s t e n t l y see t h a t

t h e rubidium system i s much more harmonic than t h e Lennard-Jones one a t a comparable temperature.

F i g . 3.--The r e s u l t s o f a s i m i l a r quench sequence as i n F i g . 2 when a p p l i e d t o t h e rubidium sample. The i n i t i a l temperatures a r e (a) T* = 0.15, (b) T* = 0.5 and ( c ) T* = 1.1.

Lennard-Jones sample a t T* = 0.11 and t h e r u b i d i u m sample a t T* = 0.15. We can determine from t h e slope o f t h e decay o f t h e echo t h e e x t e n t o f anharmonic mode coupling. t4e see t h a t t h e echo i n rubidium decays much more s l o w l y than i n t h e Lennard-Jones sample. This i s c o n s i s t e n t w i t h what was found from t h e data i n F i g s . 2 and 3.

Fig. 4.--The decay o f t h e s t i m u l a t e d echo as a f u n c t i o n o f t f o r t h e rubidium sample a t T* = 0.15 ( t r i a n g l e s f and f o r t h e Lennard-Jones sample a t T = 0.11 ( c i r c l e s ) . I n each case t l = 30At. The conclusion from t h i s a n a l y s i s o f the quench echoes i s t h a t t h e Lennard-Jones glass i s much more anharmonic than t h e rubidium. T h i s anhar- m o n i c i t y becomes more and more pronounced as t h e temperature i s r a i s e d towards t h e l i q u i d phase. These r e s u l t s suggest t h a t i n the l i q u i d the sound modes o f t h e Lennard-Jones f l u i d would be auch more q u i c k l y damped than i n r u b i d i c m i n p a r t due t o t h e g r e a t e r anharmonicity i n t h e former system. T h i s i s what i s p a r t i a l l y responsible f o r t h e f a c t t h a t sound modes are n o t observed a t l a r g e wavevectors i n Lennard-Jones f l u i d s .

Fig. 5.--The decay o f t h e simulated echo i n d i f f e r e n t p o t e n t i a l s as a f u n c t i o n o f t 2 a t T* = 1 1 . I n a l l cases tl = 30At. The r e s u l t s are

f o r t l i e 10-6

Lennard Jones (open c i r c l e s ) ; 12-6 Lennard-Jones (closed c i r c l e s ) and 12-8 Lennard-Jones (closed t r i a n g l e s ) .

I n order t o understand i n more d e t a i l t h e r o l e t h a t t h e p o t e n t i a l p l a y s i n t h e damping o f nodes we have s t u d i e d the behavior of t h e stimu- l a t e d echo f o r s e v e r a l d i f f e r e n t p o t e n t i a l s . I n each case we s t a r t w i t h i d e n t i c a l c o n f i g u r a - t i o n s o f t h e atoms so t h a t t h e e f f e c t o f t h e p o t e n t i a l on forming t h e s t r u c t u r e i s n o t included. The i n i t i a l s t a r t i n g temperature o f t h e system i s a l s o t h e same f o r a l l t h e systems studied. The p o t e n t i a l s we have used a r e s i m i l a r t o t h e Lennard-Jones b u t w i t h d i f f e r e n t values -For t h e exponents i n o r d e r t o vary t h e r e l a t i v e s t r e n g t h s o f t h e r e p u l s i v e and a t t r a c t i v e f o r c e s : V ( r ) a ($)" - We have s t u d i e d n = 12,

m

= 6 ;

n

= 10, m = 6; and

n = 12, m = 8. The r e s u l t s a r e shown i n Fig. 5. I t i s c l e a r t h a t t h e l a r g e r the value o f (m

+

n) ( i .e., t h e steeper t h e p o t e n t i a l ) t h e f a s t e r i s t h e damping o f t h e echo. The deaay o f t h e rubidium echo i n Fig. 4 i s c l o s e s t t o t h a t o f t h e system w i t h t h e l o w e s t value o f ( n

+

m). The steepness o f t h e p o t e n t i a l i s a p p a r e n t l y what i s ~ r i m a r i l y r e s p o n s i b l e f o r t h e anharmonic c o u p l i n g of nodes. This conclusion i s s i m i l a r t o t h a t o f !laan

u.

,4 who showed t h a t t h e sound modes i n t h e l i q u i d were r e l a t i v e l y i n s e n s i t i v e t o t h e a t t r a c t i v e p a r t o f t h e p o t e n t i a l .7 They con- cluded t h a t i t was t h e r e l a t i v e steepness o f t h e r e p u l s i v e core o f t h e Lennard-Jones p o t e n t i a l comnared t o t h a t o f rubidium which caused t h e dramatic e f f e c t o f t h e sound modes being h i g h l y damned i n t h e former w h i l e n o t i n t h e l a t t e r . I n t h e present paper we have extended t h e i r r e s u l t s and have shown how t h e anharmonicity and mode c o u p l i n g i n a g l a s s y system e x i s t s even i n the s o l i d s t a t e and how they grow w i t h temperature The steepness of t h e p o t e n t i a l i s i m ~ o r t a n t f o r the damping o f the sound modes i n t h e glass because i t causes each atom t o f e e l a s t r o n g departure from a p u r e l y harmonic p o t e n t i a l and thus couples t h e v a r i o u s normal modes. !Je presume t h a t t h e f a s t decay of t h e sound modes i n t h e l i q u i d s t a t e i s i n p a r t due t o s i m i l a r e f f e c t s as observed i n t h e glass.

S.R.N. acknowledges support from an A l f r e d P. Sloan Foundation f e l l o w s h i p . T h i s work was s u ~ p o r t e d in p a r t by NSF Grant DMR77-09931 NSF- MRL DMR77-23798 and by t h e U.S. Department o f Energy.

1. G. S . Grest, S . R. Nagel and A. Rahman, t o be published.

2. For a Lennard-Jones system t h e i n t e r a t o m i c 0 1 2 - 0 6

p o t e n t i a l i s given by V = 4&[(--)

_(F)

_1.

The t i m e step of i n t e g r a t i o n used was A t = 0.01 (!)1/2 where M i s t h e mass of t h e p a r t i c l e . S i m i l a r l y f o r t h e rubidium p o t e n t i a l cr and E a r e t h e u n i t s of l e n g t h and

energy and t h e A t used was dt = 0 . 0 0 7 5 ( E ) ~ / ~ . E

3.

D.

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358 (1971).

D.

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2983 (1970).

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A .

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1667 (1974).

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Moeller-l$Ienghoffer,

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7.

In t h i s context t h e a t t r a c t i v e Dart of t h e

p o t e n t i a l implies t h e region where