A. F. M. Barton Victoria University of Wellington Wellington, New Zeoland
The Description of Dynamic Liquid Structure
Structure in a crystalline solid is an ordering of atoms or molecules which ~ e r s i s t sfor lona- neriods of time and ex. tends for distances which are long on a molecular scale. The uarticles oscillate about lattice sites, but only occasionally undergo displacements to different positions: Structure in a liquid means something very different. The environment of-any molecule is cont&ually changing, and after a certain time has elapsed the arrangement hears no simple relation to the positions a t the time the observation started. Liquid structure is a dynamic structure, and refers to the relative positions of the molecules a t a given instant rather than those positions which persist for long periods of time. Another way of qualitatively describing liquid structure is to say that it has no long range order as in a solid, but like the solid structure it possesses short range order. This short range order is ~ h o r t both in position because of packing requirements, and in time because of the dynamic nature of liquid structure. At the one extreme of times which are long and distances which are verv- large on a molecular scale. the equations of hydrodynamics are valid. Just as important for-an understanding of the liquid state is the behavior a t the other extreme of very short times and distances. A crystalline solid acts as a diffraction grating for elecsuch as tromagnetic waves such as X-rays or neutrons with a wavelength comparable with interatomic spacing. Because of the short range correlation that exists between the positions of the molecules in a liquid, it too can cause diffraction to a certain extent. The scattering of radiation is one of our main sources of information about the dynamic liquid state (1, 2). Another important technique is computer simulation (3, 4). This account has the aim neither of describing these practical methods nor of reviewing the experimental results, hut of outlining the formalism currently in use and describing in general terms the nature of the various correlation or distribution functions which describe the correlation between molecules. First the correlation between molecular positions will be discussed, followed by time correlation functions, and finally the more general distribution functions involving both position and time will he introduced.
X Figure 1. Diagrammatic representation of volume element dr and vector r.
-
Correlation in Posilion: Molecular Distribution Functions
Molecular aistribution functions describe the prohahility of occurrence of a particular arrangement of N molecules in thermal equilibrium in the liquid (2, 5). There is a series of distribution functions, of which the simplest is the one molecule distribution function n"l(r~)dr, (1) This is proportional to the probahility of finding one molecule in a volume element dxldyldzl = drl specified by vector rl from an arbitrary origin (see Fig. 1). Since probability is a pure number (between 0 and I ) , n"' ( r l ) has the dimensions of reciprocal volume or number density and in an isotropic fluid a t equilibrium it must equal the mean number density p defined as the total number of molecules Ndivided by the total volume V
Figure 2. Diagrammatic representation of vectors and volume elements associated with the two molecuie distribution functions (after Rowlinson (51).
,,'"(.,)
= p = N -
v
(2)
The function n"' (TI) is therefore independent of the position of rl. The two-molecule distribution function
is proportional to the probability of finding a molecule at rz in a volume element drz if at the same time there is a molecule in volume element dr1 a t rl (Fig. 2). If the fluid is isotropic, n!Z' (rl,r2) is a function only of the scalar separation r = lrl - rzl and not of the actual positions of rl and rz, i.e., it is independent of the direction of r . It is frequently useful to have a function which is normalized to unity at large values of r. If r is large, the probability of simultaneous occupancy is the product of independent occupancies (there is no long range order or correlation between the two molecules) lim n@'(r)dr,dr?= (ni')dr,)(nl'ldr,) , .,, (4) = p~dr,dr* i.e.
Volume 50. Number 2, February 1973 / 119
tion vector r,, the instantaneous number density p(r) of the system is
,In this equation 6(r - ri) is the Dirac delta function, i.e., if the ith atom lies in an arbitrarily small volume element dr, then 6(r - r , ) d r = 1, but if it lies outside the element, 6(r - r J d r = 0. (The instantaneous number density p(r) should not he confused with the molecular distribution functions n'l', n'z', etc., which also have dimensions of number density.) The fluctuation in this instantaneous number density is defined as
r/o Figure 3. The pair distribution function far liquid argon at 143 K and 43 cm3 mol-I (after Rawiinson ( 5 ) ) .
and we call the dimensionless distribution function defined in this way the pair distribution function g ( r ) = p-%W(r) = (!!)-'n(2)(r) (6) N Another way of describing g(r) is as follows: given a molecule with center a t the origin, pg(r)dr is the average number of other molecules whose centers lie in an element d r about the position r. An equivalent definition of g(r) will be introduced and explained below
The function g(r) is sometimes called the radial distribution function, but the description "radial" should strictly be reserved for the quantity 4arzpg(r). Figure 3 shows a typical pair distribution function: it can be seen that the correlation in position a t short distances fades away over several molecular diameters to a limiting long distance value of g(r) = 1. The pair distribution function may be obtained experimentally from X-ray and neutron scattering experiments (Z), or by calculation in computer simulation experiments on systems with simplified intermolecular potentials (3). Sometimes the structural information is expressed in terms of the "total correlation function" defmed as
M r ) = p(r) - p (11) i.e., in any volume element dr, Gp(r)dr is the amount by which the density p(r) differs from the mean density p. Using the symbol (A) to represent an ensemble average of the function A, we can express the fact that on average the number density has the value p by (6dr))= 0 or ( ~ ( r )=) p Time-Dependent Correlation Functions
(12)
Up to now in this account only the relative positions of molecules in a liquid have been discussed. Consider now a molecule i which a t time t = 0 is located at a position rd0) with various properties such as velocity vi(0), and which at time t has position ri(t), velocity vi(t), etc. The rate a t which the velocities and other characteristic properties of the particles change as a result of interactions with other particles is described by time autocorrelation functions. The velocity autocorrelation function z(t) for example is defined as
where the angular brackets denote averages over an equilibrium ensemble of initial conditions. The velocity autocorrelation function for liquid argon (6) is shown in Figure
h(r)=g(r)-1 i.e., the value of g(r) in excess of its random value of unity. The function n'2' (r) or g(r) is not sufficiently detailed for some purposes: although i t describes the probability of finding molecules a t a distance r from a chosen molecule, it says nothing about the angular distribution of the neighbors around the central molecule. The three-molecule distribution function drldr2dr3 (8) n'3'(r~,r~,r3) is proportional to the probability of a third molecule being in dra at r3 when the first two molecules are at rl and rz. This distribution function contains information about angular distribution but is more difficult to study experimentally. An approximation for nl3' which is expected to be valid at low densities but to become poorer a t increasing densities is the "superposition" approximation
,,q r,,r2,r8
= n"~(r~.~~)n~~'(r,,r~)n'~~(r~,r~)
P3
(9)
The d;stribution of molecules in a liquid may also be described in terms of density fluctuations. This is done as follows for a liquid consisting of a collection of N molecules in a volume V. If the center of the ith molecule has a posi1 2 0 /Journal
of Chemical Education
Figure 4. The velocity autocorrelation function for liquid argon at 94.4 K and 29 cm3 m o i ' from a molecular dynamics calculation (6).
4. The function is equal to unity a t zero time, and decays to zero as time advances and "memory" of the initial conditions is lost. The negative portion of the function ("anticorrelation") is due to the trapping of a particle within a cage formed by its neighbors, resulting in frequent reversals of velocity. The macroscopic transport coefficients such as the diffusion coefficient D, shear viscosity coefficient q s , thermal conductivity r , which describe the linear response of a system to a n external force have been shown to he related to the auto-correlation function for fluctuation of an appropriate quantity in that system in thermal equilihrium in the ahsence of the force. These relations, generally under the title of the fluctuation-dissipation theorem, have been discussed in detail by several authors (7). This means that the same information is accessible from a study of macroscopic response to an applied force as is ohtained by observing the microscopic fluctuations by a technique such as radiation scattering. Transport phenomena, which involve transport of matter, momentum or energy, are described as in the following examples
Figure 5 . The mean square displacement of atoms in liquid argon at 94.4 K and 29 cm3 m o l ' from a molecular dynamics calculation (6)
where u,( t) is the x component of the velocity a t time t
where 01, is the x,y component of the stress tensor and M is the molecular mass; and K
=M m
P
1-
dt
(16)
where J, is the heat current or flux in the x direction. Reference to Figure 4 and eqn. (14) shows that D may he evaluated if the velocity autocorrelation function is known, and also that the effect of negative regions in z(t) is to reduce the value of D as would he expected intuitively from the "cage" effect. The value of D for liquid argon is established from the autocorrelation function within a time of the order of 10-12 sec. Another way of describing the motion of particles in a system is by the mean square displacement
A simpler function is useful for the description of many transport properties G(r, -
r,,fr
- t , ) = G ( r , t ) = l[f(r,,p,,t,;rr,pr.ti)dp,,dp2 (19)
This time dependent pair distribution function, known as the Van Hove distribution function, depends only on space differences, r = r, - r2, and time differences, t = t2 - tl, because the system is assumed to he in thermal equilihrium. The formal definition of G(r,t) is (2)
This may he compared with eqn. (10). Corresponding to the instantaneous numher density
there is the number density function The behavior of (r)2as a function of time for liquid argon is shown in Figure 5. Again the macroscopic properties may he obtained from the limiting long time (-10-l2 sec) hehavior: the slope o f t h e linear portion is related to the diffusion coefficient by
The i a n d j terms in eqn. (20) may be separated
and so an alternative form for G(r,t) is where C is a constant. Although asymptotic behavior is achieved a t t -10-'2 sec, the value of the root mean square displacement a t 2.5 x 10-12 sec (-2A) is only ahout one half of the nearest neighbor distance (3.7A for liquid argon a t this temperature) and so the memory of the identlty of the nearest neighbor is not lost. Information of this type may he obtained more accurately from time dependent pair correlation functions (see below). General Pair Distribution Function A general description of the position and momentum of each molecule a t each instant of time is ~ r o v i d e dbv the general pair distribution function: f ( r l , p l i t l ; r ~ , ~ 2 . t 2is) the probability of finding a molecule a t r 2 with momentum p2 a t time tz if there was a molecule at rl with momentum p l a t time t,
At sufficiently large values of t, for a homogeneous liquid the two molecules are statistically independent, so G(r,t) (equ. (23))approaches p
It is now possible to divide G(r,t) into asymptotic ( G ( r , t ) )and correlation (G'(r,t)) terms G7r.t)
-
=
G(r,tl
=
G(r,tl -
G,(r,t) p
(25)
The correlation term indicates the deviation in the density of a system from the average density, and a t long times tends to zero. Volume 50. Number 2,
February 1973 / 121
At large t or large r the self term tends to zero (very low probability) and the distinct term tends to the average density of the system. Figure 6 shows a qualitative description of self and distinct components of G(r,t) a t times which are short, similar and long compared with the relaxation time T of the system. At very short times (Fig. 6a) the sharp peak a t the origin is the molecule marking the origin, and the fluctuating curve is g(r). The relation between g(r) and G(r,t) may be shown in the following way. If G(r,t) a t time zero (G(r,O)) is divided into self and distinct parts there is ohtained the alternative definition for the pair distribution function, g(r).
t
0
Gs
r
-
Figure 6. D i a g r a m m a t i c representation of self (G,) a n d distinct ( G o ) l e r m s 01 t h e s p a c e - t i m e density correlation function at times 1 which are (a) short. i b ) similar, a n d (c) long c o m p a r e d with t h e relaxation t i m e r of t h e system.
At time zero the sharp G,(r,O) peak a t the origin in Figure 6 becomes a delta function, and the fluctuating G&,O) curve is g(r). The normalization procedures cause g(r) to a while G(r.0) tends to the approach unity in the limit r value p . After a time long (Fig. 6c) G, approaches zero and Gd approaches the mean number density for all values of
-
..
It is h o ~ e dthat this brief account will serve as an adequate introduction for those interested in studying more deeply the rapidly growing chemical physics and physical chemistry literature on the dynamic behavior of liquids. Literature Cited
Van Hove (8) described the division of G(r,t) into a "self' term (subscript s) with i = j, and a "dcstinct" term (subscript d ) with i Z j. The self term reflects the prohahility of finding a particular molecule a t the position r a t time t if it was located a t the origin at time zero, and it therefore describes the diffusion of the particular molecule away from the origin. The distinct term describes the probability of finding a molecule a t position r a t time t if a different molecule was located a t the origin a t time zero
122 /Journal ot Chemical Education
ill ICeelstafl. P. A . and Schofield. 1'. Contemp Ph?hvr. 6 . 274. 15:l fl%Sl: Mountain. R. U.. C r i l Km.. SiiIidStolr Sri.. I.i(LSiil1. (21 E~d~,.fl: P. A.. "A" lnfroduciion to the liquid State." Academic Press. London. 1967
is1 Mflonald.1. H.. andsheer. K..O u o n RPi...21.28RIL9701. i4i Temperley, H. K. Y.. Howlinson. .I. S.. and Rushhmske. C. 5 . IEditorrI. "Physics of Simple Liquids". Nonh~Holland. Amsterdam. 116" iai R O W I ~ .I. ~ ~s.. . in B C ~ ~ J. I ~N.. Y .i ~ d i t ~'.lissayr ~i. in rhernirtry". VOI. I. Asmi" I "n.ir." ........Plsrr ...., . .,. , .197" .. .., n I . 161 Rahman. A l'h>r. Kri,.. 13% A415 i 1961). ~ rA..~ f f , i7l For erample. Zwanrig. H.. Ann. Rrc. Phvr Chem.. 16. 67 119651; ~ ~ ~ l P. Rep ROEP h b . 21.333t19Mii: Kubo. H..Rep R u g . Fhvl.. 29.255i19FCIl (81 VanH0W.l l'hys. Hei, ,95.249119541.
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