Comparison of the intensity correlation function and ... - Science Direct

Physica A 202 (1994) 145-164 North-Holland

B

~Z~

SDI: 0378-4371(93)E0334-B

Comparison of the intensity correlation function and the intermediate scattering function of fluids: a molecular dynamics study of the Siegert relation H a r a l d Voigt and Siegfried Hess Institut fiir Theoretische Physik, Technische Universitiit Berlin, PN 7-1, Hardenbergstr. 36, D-10623 Berlin, Germany Received 22 July 1993

The Siegert relation is a link between the intensity correlation function I2(k, t) and the intermediate scattering function F(k, t). This relation is commonly used in the analysis of dynamic light scattering data. Molecular dynamics computer simulation renders possible a test of this relation since F(k, t) and 12(k, t) are directly accessible. In addition we study the influence of the size of the "scattering volume" for which F(k, t) and 12(k, t) are extracted. The validity of the Siegert relation is confirmed for "large" scattering volumes while for small systems deviations are found. These deviations are caused on one hand by number fluctuations and on the other hand by the dynamics of the local order in the fluid.

1. Introduction

Dynamic light scattering (DLS) experiments are a standard method to probe the dynamics of colloidal dispersions and of a great variety of complex fluids [1-3]. The results inferred from the measured intensity correlation function I2(k, t) are frequently interpreted in terms of the intermediate scattering function F(k, t) by using the Siegert relation. This is remarkable since in essence 3- and 4-particle correlations are expressed in terms of 2-particle correlations. It is the purpose of the present article to test this relation by molecular dynamics (MD) simulations of simple fluids. The functions I2(k, t) and F(k, t) are extracted from the simulation and compared for "scattering volumes" of different size. The validity of the Siegert relation is confirmed for large scattering volumes where the standard argument of independently fluctuating quantities obeying Gaussian statistics is fulfilled. On the one hand this is quite pleasing since the standard procedures of the interpretation of DLS data are valid, on the other hand it is disappointing that information on 3- and 0378-4371/94/$07.00 © 1994- Elsevier Science Publishers B.V. All rights reserved

H. Voigt, S. Hess / A MD study of the Siegert relation

146

4-particle correlations cannot be obtained. The situation is different for small scattering volumes. Here deviations from the Siegert relation are expected and have been looked for in experiments [4-7]. In the MD simulation a deviation from the Siegert relation is found for small scattering volumes which is mainly due to the fluctuation of a number of particles in the scattering volume. Recent fiber-optical setups [1,2] can in fact probe very small scattering volumes where these deviations might be of importance. This paper is organized as follows. In the next section a proof of the Siegert relation is given and some of its implications are discussed. We also present modified scattering functions which facilitate the analysis of deviations from the Siegert relation by correcting for the influence of number fluctuations. In section 3 we describe our MD simulations whose results are presented in section 4. A summary will be given in section 5.

2. The Siegert relation

2.1. Scattering functions The intermediate scattering function F(k, t) is defined as 1 ~/N,ot

F(k, t):= ~- \j,~ bj(t) G(O) exp{-ik. [rj(t) - rAO)}\/, b](t):=

{A'

, j>JV(t),

(1)

where rj(t) and rk(O ) are the positions of the jth and kth particles at times t and 0, respectively. By N(t) we denote the instantaneous number of particles in the fixed (but open) scattering volume V and by Ntot the total number of particles in the system. The average of Ac is given by N = ( X ( t ) ) = p V with the equilibrium density p. We use a counting function bi(t ) so that all sums are running over the same range. The scattering vector k is given by the difference of the wavevectors of the incident and the scattering light: k = ksc - k i n c. The wavevector k is related to the scattering angle 0 by Ikl = 21ki.¢1 sin(½0) (elastic scattering). In light scattering experiments it is easier to measure the time correlation of scattered intensity, which is proportional to the intensity correlation function I2(k, t), defined by

H. Voigt, S. Hess / A MD study of the Siegert relation

147

/ Nt°t Nt°t I2(k, t) := ~ - '/, -a l,,,=1 bj(O) bk(O ) b,(t) b,n(t ) × exp(ik • [rj(O) - rk(O)] } exp ik. [r,(t)

-

rm(t)]}) .

(2)

The Siegert relation links the intensity correlation Iz(k, t) to the intermediate scattering function F(k, t) and the static structure factor S(k) = F(k, 0) according to

I2(k, t) = F2(k, t) + S2(k).

(3)

The relation is assumed to be valid for a scattering volume which is large in the sense that it consists of many independently scattering particles. It is commonly used to infer F(k, t) from the measured I2(k, t) [2,3]. Since interacting particles in liquids are radially and angularly correlated at least over the range of some particle diameters, the linear dimension of the scattering volume should be large compared with this correlation length. It can be roughly estimated by the decay of the pair distribution function to g(r) ~ 1. The name "Siegert relation" can be traced back to an article by A.J.F. Siegert from 1943 entitled: " O n the fluctuations in signals returned by many independently moving scatterers", where the conditional probability for the scattered field E 2 at time t 2 given the field E 1 at t~ is derived [8]. Frequently eq. (3) is referred to as a consequence of the "Gaussian statistics" of the scattered radiation. The Siegert relation expresses a 4-particle-function by a 2-particle-function and sets thereby a limit to information about 3- and 4-particle correlations, which at least in principle could be gained from the intensity correlation function [6,7]. In our molecular dynamics simulations we extract the scattering functions F(k, t) and I2(k , t) for different sizes of the scattering volume to detect possible deviations from the Siegert relation, when higher correlations cannot be neglected. Before results of our MD simulations are presented a short derivation of the Siegert relation following standard arguments and some of its implications seems to be appropriate.

2.2. Proof and implications of the Siegert relation The scattering functions F(k, t), S(k) and I2(k, t) can be written as even moments of the microscopic density in k-space pk(t), which is defined as

H. Voigt, S. Hess / A MD study of the Siegert relation

148 Nt°t

ok(t) := ~ b,(t) e x p [ - i k - r,(t)], 1=1

bl(t) :=

{10 , '

l> k -1, the bi(t ) and exp[ik, ri(t)] are uncorrelated, (ii) for large k-values most of the terms in

exp{ik • [rj(O) - rk(O)] } exp{ik- [re(t) - rm(t)] } j, - 1 l,

-1

are averaged out, only number fluctuation (j = k, l - - m ) and self-diffusion terms (j = m ~ k = l) survive, (iii) Fs(k, t) decays rapidly compared to particle number fluctuations, so that for the amplitude factor containing the bj(t) one may take the value at t = 0. In the limit of large scattering volume where ( N ( t ) N ( O ) ) ~ N 2 and (N(?¢"1 ) ) / N 2 = 1 relation (9) reduces to g(Z)(k, t) = 1 + IFs(k, 012 .

(10)

From the Siegert relation, on the other hand, one would have obtained this result (10) with the approximation F(k, t)/S(k)~-Fs(k, t) irrespective of the number of particles in the scattering volume [2]. Thus deviations from the Siegert relation are expected when fluctuations of the number of particles in the scattering volume are important. In practice when the time scales for (N(0) N(t)) and Fs(k, t) are well separated and the number of particles in the scattering volume is nearly constant over the time of a scattering experiment, the simplified expression (10) can still be used for the analysis of the dynamics and the extraction of a diffusion coefficient [2,3]. The intensity correlation can also be written in terms of fluctuations of the instantaneous values of the basic scattering functions Ntot

O°(k; s) := Nj,~],k= bi(s) bk(S) exp(ik • [r](s)

--

rk(S)] }

and

J;(k, t; s) := -S-d,2

N j,k=l

with

bj(t + s) bk(S ) e x p { - i k • [ry(t + s) - rk(S)] }

H. Voigt, S. Hess / A MD study of the Siegert relation

150

T

F(k,t)=lim~

if ~ ( k , t ; s ) d s

T---~oo I

T

and

,f

S(k)=lrim- 7

0

5¢(k;s) ds:

0

I2(k , t) = S2(k) + (~9°(k; t + s) ~5¢(k; s)) , (11)

~e(k; s) := ~e(k; s) - s ( k ) ,

Ie(k, t) = F2(k, t) + ($~(k, t; s) $~(k, t; s)) , (12)

~ ( k , t; s) := ~(k, t; s) - F(k, t) .

Using the Siegert relation one infers from eqs. (11), (12) a connection between the scattering functions and their fluctuations:

F2(k, t) = (85¢(k; t) 85¢(k; 0)5,

S2(k) = (8o%(k, t; s) 8~(k, t; s ) ) .

(13)

Incidentally the given proof of the Siegert relation can be generalized to higher moments of the microscopic density pk(t). It is possible to calculate, e.g., the correlation ( ~ ( k , t; s) ~(k, t; 0)) from the averaged F(k, t). This is useful for an estimate of the error of MD results for scattering functions. More specifically, the variance 0-2 of the mean of an extracted function has to be corrected according to 0-2 = (r e/Ae)0- 2 with a correlation time %, the extraction timestep A e and the preliminary MD-given variance 0-2. A detailed analysis of MD data for F(k, t) confirmed the following result for %: c~

f F(k, t + s) F(k, t - s) + F(k, s) F(k, s) ds. %= [S(k)] 2+ [F(k, t)] 2

(14)

The correlation time for F(k, t) depends on k and (less strongly) on t. An equivalent calculation for the correlation time % for the intensity correlation function I2(k, t), where an 8-moment in pk(t) had to be resolved, leads to a rather complicated expression in F(k, t) and S(k), which also compared well to MD results.

2.3. Small scattering volume The particles in the first few coordination shells around a randomly picked central particle fills what we call, from now on, a small scattering volume. While relations (11) and (12) are still valid, for small scattering volumes the condition for the central limit theorem is not fulfilled in general, so the pk(t) need not be Gaussian distributed. Furthermore, the Siegert relation is not

H. Voigt, S. Hess / A MD study of the Siegert relation

151

obeyed at t = 0 for large k-values due to number fluctuations: in [9] the normalized intensity correlation function g(2)(t = O)= (5¢(k, 0)9~(k, t))/S:(k) for large k (where S(k)~-1) for a finite volume V, is given as

g(2)(O) = 2 + 2

Sv(O)

sV(o) N

:= 1 + -~

d3rl v

1

--N 1. As will be explained later the expected long time limit of the ratio for small k-values is below 1. The relaxation time seems to be independent of the k-value. Also the 148.1/4000 system shows a k-independent signal decaying exponentially with an amplitude (I2/I2)1,=01 ~ 5 × 10 -4, whereas in the case 500/4000 the signal fluctuates around 1 without any detectable initial decay since the fluctuation in the number of particles is smaller than in the previous cases.

1.~

1.~' k =8.435

L

k =22.493

1.002~

1.002L

]°°°1

~/,-

VV',"

"

" '54~V

0.9980[,0',,,,~,.~,,' ' '~'.'6''" '~'.'4""'~'.'2'""~i'0''" '~'.~'t 0.99801~,~,,,,,i 76,,,,,~74,,,,,~7,z',,,,,~,,0'""~'.~'t 1.0041.004/

k =36.550

1.002

1®2

• 'y 0.998

k =67.478

rv"

.. , ............ , .......................... 0.0. . . . . . . .0.8 1.6 2.4 3.2 4.0 4.8 t

0.998 .................................................. 0.0 0.8 1.6 2.4 3.2 4.0 4.8 t

Fig. 4. Ratio of the canonical over the modified intensity correlation function 12/l2 for the 62.5/4000 subsystem shown as a function of time for a k-value in a local minimum of S(k) (k = 8.4) and 3 values for which S(k) = 1 (cf. fig. 2). For the higher k-values the time evolution of the ratio is apparently independent of k.

H. Voigt, S. Hess / A MD study of the Siegert relation

157

On the other hand, for smaller subvolumes the number fluctuation contribution to the canonical intensity correlation function is clearly visible in the ratio I2/I2. In a logarithmic plot we recognize an exponential decay with time. In fig. 5 the ratio I2(k, t)/IE(k , t) is shown as function of time for the 7.8/500 case for two k-values around the main peak in S(k) and for two larger wavenumbers. Again an asymptotic approach to values below 1 ( k - - 5 . 6 , k = 8.4) and to 1 (k = 16.8, k = 61.8) is found. The deviation of the asymptotic ratio from 1 indicates that number fluctuations and density correlations are correlated for small k-values and short times. A corresponding analysis was made for the ratio of the intermediate scattering function F(k, t) over its modified version /3(k, t), cf. (16). For the cases of 18.2/500 and larger subvolumes we obtained remarkably well F//3 = 1 until the denominator goes through 0 due to noise. For the cases 7.8/500 and 4/500 the ratio increases slightly in time for k > 2"rr indicating a slower decay of /3 while for k < 2"rr the ratio F//3 decays very slowly. An example for F/F in the case 4/500 is depicted in fig. 6 as function of time and for two k-values. The k-dependence of the dynamics of the ratio F//3 again indicates the correlation between number fluctuations and density correlations for small scattering volumes. 1.035-

1.035k=8.435

k =5.623 1.020. ~

1.020.

1.005.

1.005.

0.9900.0.................................................. 0.8 1.6 2.4 3.2 4.0 4.8 t 0"9900.0'""0'.8'""i'.6'""21~(""3'.2'""4'.0'""4't.8' 1.035

1.035

k =61.855

k=16.869 1.020 ~

1.020 ~

1.005

1.005

°'99°o.6'""d'.i'""ilg" "i'.~'""~'.i'""~i~'""~'.i' t

0.990

ll,,,,,,,,,,IHlllllllllI,HIH'llJ'J'l''llll t 0.0H I p ,0.8 1.6 2.4 3.2 4.0 4.8

Fig. 5. Ratio of the canonical over the modified intensity correlation function 12/12 for different wavenumbers for the 7.8/500 subsystem plotted as a function of t for 2 wavevectors around the main peak in S(k) (k = 5.6, k = 8.4), an intermediate (k = 16.9) and a high k-value (k = 61.9). For the low wavenumbers the long time limit of 12/lz is different from 1 as explained in the text. The time evolution depends on k.

H. Voigt, S. Hess / A MD study of the Siegert relation

158

1.50-

1.020' k =7.029

A

1,25

k =10.543

1.005

1~

j,

0.990 -

1®

t 0.975 ,i ,, ,,, , , i , , , , , , , i , i l l , , , , , , i , i i , , , , , , , , 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

0.75 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , t 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Fig. 6. Ratio of the canonical over the modified intermediate scattering function F/F for the 4 / 5 0 0 subsystem as a function of time for two k-values right beyond the main peak in S(k). For short times the time evolution of the ratio depends on k, indicating a correlation between number fluctuations and density correlation in this small scattering volume.

4.3. The Siegert relation at t = 0

To test the Siegert relation at t = 0 means to compare the normalized intensity correlation g(2)=I2/S 2 to its Gaussian value g(2)(0)=2. For the larger systems (e.g. in the cases 4000/4000-148.1/4000) a deviation is nearly undetectable. In the 500 particle system with small subvolumes deviations from the Gaussian value become evident. Fig. 7 shows the ratios 12(k, O)/S2(k) and I2(k, 0)/S2(k) for the subvolumes 4/500-500/500. For k-values well beyond the main maximum in S(k) these ratios become independent of k. The deviations of these asymptotic values from the Gaussian value g(2)(0)= 2 increase with decreasing size of the scattering volume. In fig. 8 these deviations are plotted as functions of N, the average number of particles in the scattered volume, for the canonical (©) and modified ( ~ ) scattering functions (double logarithmic plot). The expected N-l-dependence [9] is found within statistical error only for the modified scattering functions whereas for the canonical functions it is more like N -3/4. 2.1"

2.10"

canonical scattering functions 2.0 ~ 1

|

1.95.

1.80'

1.9"

1.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10 20 30 40 50 60 70 80

k

~i Artlodified scattering functions

F-

1.65 i , , , , , , , , , U l l l l , , , , , , , , , , , , , , , , , , i , , , , , , , k 10 20 30 40 50 60 70 80

Fig. 7. Normalized intensity correlation function at t = 0 for subsystems of different size. For the subvolumes of the Nto t = 500 particle system the normalized intensity correlation function g(:)(k, O) is shown as function of k for the canonical and modified scattering functions. The large-k limit decreases with size of the scattering volume. The straight line indicates the Gaussian limit g(2)(O) = 2.

H. Voigt, S. Hess / A MD study of the Siegert relation

159

, 0 o -:

0

'0 j '

0 O O O O

,0200

,2

. . . . . . . .

IN

Fig. 8. High wavenumber limit of 2 - 12(k, O)/S2(k) for modified (~) and canonical (©) scattering functions in dependence on the size of the scattering volume. The deviation of the large-k limit of the normalized intensity correlation function gtZ)(O)from the Gaussian limit gt2)(0) = 2 is displayed as function of the average particle number N in the scattering volume. Notice the doublelogarithmic scale.

4.4. The Siegert relation for t >-0, large scattering volume In fig. 9 the intensity correlation function 12 and the squared intermediate scattering function F 2 are shown as functions of time for the 500/4000 system and for various k-values. In each graph the upper curve, which relaxes to a finite value, is Iz(k, t), the lower one, which approaches 0, is F2(k, t). The finite value which is shown as a horizontal line corresponds to S2(k). The smallest k-value presented (k = 0.7) is still in the hydrodynamic regime where damped oscillations are observed. The slow decay of the scattering functions is seen for k = 6.3 very close to the main peak in S(k). The decay is faster for k = 3.5 and the higher k-values. A qualitative inspection shows that the Siegert relation is well obeyed within statistical error in this case. For the 62.5/500 and 62.5/4000 cases also no deviation from the Siegert relation can be detected by looking at the difference 12 - F 2 - S 2 or the ratio Iz/(F 2 + $2). On the other hand we know from I2/I 2 that there is an offset for t = 0 and a number fluctuation part in 12 which cannot be present in F 2. But since 12/I2 - 1 is of the order of 10 -3 it is practically not detectable here due to the noise of the data. The situation is different for small scattering volumes.

4.5. The Siegert relation for t >I 0, small scattering volume For 18.5/500 and smaller subvolumes deviations from the Siegert relation in 2 (as well as in I 2 / ( F 2 + $ 2 ) ) can be detected. In fig. 10 the difference 12 - F 2 - S 2 as a function of time is plotted for the 4/500 case and for various k-values. There is a pronounced negative deviation for short times and a positive one at intermediate times slowly decaying to 0; the long time

12-Fz-S

H. Voigt, S. Hess / A M D study o f the Siegert relation

160

0.015"

1.2" O

k =3.514

k =0.703 0.8'

\

0.4' ~

0.010'

t 0.005.

~

0.0 m Mlll.,,,,,,,,,,,lltllllllm,,,,,,,,,lllllll 00 08 16 2 4 32 4 0 48 t

°'°°°o.dl'"d'.~'""~'.'g'"'~'.'~""~i'~""~'.'d'"t'~i'~ 1.2-

15--

k =8.435

k =6.326 0.8-

0.4-\

HH,HHH,I,,,mHHHHHI,H,,,HHHHHHH 00 08 16 24 32 4 0 48 t

3-

--

v

v~-

\ 0.0

..............................................

00

08

16

24

32

40

'~'~'t

3k =12.652

k =17.572

2- ~ . _ ~

2-

1-

1"

00 0,~,.............................................. t 08 16 24 32 4 0 4 8

00.0,~.......................................... '~'~'t 0.8 1.6 2.4 3.2 4.0 .

Fig. 9. Intensity correlation function I2(k, t) and squared intermediate scattering function F2(k, t) displayed as functions of time for different k-values (500/4000 subvolume). The upper curve corresponds to 12. It decays to S2(k), which is shown as a straight line. Notice the vertical scale in the upper left graph.

decay seems to be independent of k. Except for the smallest k-value (k = 3.5, notice the different vertical scale) the positive deviation is practically absent in the corresponding curves involving the modified scattering functions as displaced in fig. 11. From the fact that the slowly decaying positive component of the deviation is missing in fig. 11, one can conclude that it is due to number fluctuations whose influence is corrected for in the modified scattering functions. The approach to 0 of the difference I2- ~2_ ~2 for the modified scattering functions occurs on the time scale of the free flight. The deviations from the Siegert relation as e.g. seen in fig. 10 can he interpreted in terms of the fluctuations of the instantaneous static structure factor as stated in eq. (11), yielding

H. Voigt, S. Hess / A MD study of the Siegert relation 6-

161

0.15k =3.514

k =7.029 0.0C

-0.15-

-3 ................................................. 0.0 0.8 1.6 2.4 3.2 4.0 4.8

t

0.05 "1

-0.30 I.................................................. t 0.0 0.8 1.6 2.4 3.2 4.0 4.8 0.075 -

~

k = 10.543

k =21.087

0.00

0.000

-0.05'

-0.075

-0.10 !.................................................. t -0.150 ................................................. t 0.0 0.8 1.6 2.4 3.2 4.0 4.8 0.0 0.8 1.6 2.4 3.2 4.0 4.8 0.075 k =56.231

0.075 k

k =84.347

0.000

-0.075.

-0.15~ .................................................. 0.0 0.8 1.6 2.4 3.2 4.0 4.8

-0.075 ]

t

-0.1501 .................................................. 0.0 0,8 1,6 2.4 3.2 4.0 4.8 t

Fig. 10. Deviation from the Siegert relation 12(k, t) - [FZ(k, t) + S2(k)] for small scattering volume (4/500), canonical scattering functions. The difference 12 - F 2 - S 2 is plotted as function of time from small (k = 3.5) to high (k = 84.3) k-values. The deviation from 0 shows that the Siegert relation is not fulfilled. A negative component in the deviation is seen for short times on a time scale decreasing with increasing wavenumber, while the positive part of the deviation is for longer times apparently becoming independent of k. It stems from number fluctuations.

I2(k, t) - F2(k, t) - S2(k) = ($9°(k; t) 5SV(k; 0 ) ) - F2(k, t) . Thus fluctuations of the static structure factor are smaller in a small scattering v o l u m e (as inferred f r o m the negative values at t = 0) than in a large o n e w h e r e it is given by the Siegert relation as (~Se(k; 0) 5St(k; 0 ) ) = S2(k). T h e d e c a y of the correlation (~5e(k; t) ~5e(k; 0 ) ) is s l o w e r in the small scattering v o l u m e than in the large s y s t e m for which o n e obtains f r o m eq. (11) (~9°(k; t) ~9°(k; 0 ) ) = F2(k, t). This is plausible since structure and d y n a m i c s in a small s u b v o l u m e are d e t e r m i n e d by the fact that m o s t o f the particles are interacting directly

162

H. Voigt, S. Hess / A MD study of the Siegert relation '~ 9. o

0.25" k =7.029 0.00

6"

-0.25" /

0

,,.i.,,,,,,,i,,,,,,,m,,,,,i....,.,,,,ll, 0.0 0.8 1.6 2.4 3.2 4.0 4.8 t

0.15-

-0.50

0.15k =10.543

0.00

/

k =21.087 0.00"

-0.15.

-0,15"

-0.30 , , , I . , i . l , , . , , , m . . . , , i m l , , . i . , , . , i , i 0.0 0.8 1.6 2.4 3.2 4.0 4.8 t

-0,30

0.15-

o.o""Bi~'""ii~'"~i~'"3T'iiS'"~i~t

0.15. k =56+231

0.00-

k =84.347 0 . 0 0

•

-"

"

-

.

.

.

.

.

.

-0.15-

-0.15"

-0.30 , , , , , , , , , , , i , , . . m , , , , , , , , , . , , , m , , , , , , , , . , 0.0 0.8 1.6 2.4 3.2 4.0 4.8 t

-0.30 .................................................. t 0.0 0.8 1,6 2.4 3.2 4.0 4.8

$2(2k)]

Fig. 11. Deviation from the Siegert relation l"2(k, t) - [F2(k, 02+ for small scattering volume (4/500), modified scattering functions. The difference [2 - / ~ - g is plotted as function of t for the same k-values as in fig. 10. The positive part of the deviation from the Siegert relation seen in fig. 10 has vanished for k-values beyond the main peak in S(k), leaving only the negative k-dependent component. with each other and are thereby strongly correlated. If the scattering v o l u m e comprises m o r e than a few coordination shells the number of all pairs in the scattering v o l u m e by far exceeds the number of correlated pairs. T h e n the Siegert relation is fulfilled and the scattering v o l u m e can be said to be larger than a correlation v o l u m e .

5. S u m m a r y The Siegert relation holds for systems which fulfill two conditions: negligible number fluctuations either in the sense that N ~ const. (which often is valid for

H. Voigt, S. Hess / A MD study of the Siegert relation

163

sufficiently large scattering volumes) or such that the time scale for number fluctuations is long compared to that of the considered scattering functions. The latter condition depends on the chosen wavevector. The second requirement is that there exist many "correlation areas". Now it is possible to say this more precisely: the contribution of uncorrelated pairs should be large compared to that of pairs or at least of pairs of directly correlated particles belonging to well-structured coordination shells around a common central particle. Otherwise for t = 0 and for short times (free flight time) the Siegert relation is not obeyed. This means that for short times even the use of the normalized intensity correlation functions would not lead to the correct intermediate scattering function. Size and lifetime of a correlation area should depend on density and the interaction potential. An attractive part in the interaction might stabilize and enhance the regions of correlated particles. It is intended to study these topics in further simulations of Lennard-Jones systems. The comparison of canonical and modified scattering functions as presented in section 4.2 confirmed that the assumption of uncorrelated amplitude (particle number) and phase (structure) in the intensity correlation function is justified if the scattering volume and the chosen wavevector obey the condition V1/3k >>1.

Acknowledgements Financial support by the Deutsche Forschungsgemeinschaft via the Sonderforschungsbereich SFB 335 "Anisotrope Fluide" and the "Graduiertenkolleg Polymerwerkstoffe" is gratefully acknowledged. We thank D. Horn and H. Wiese (BASF) for stimulating discussions.

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