Decay of Correlations in Bulk Fluids and at Interfaces - American

Chapter 12

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Decay of Correlations in Bulk Fluids and at Interfaces: A Density-Functional Perspective R. Evans and R. J . F. Leote de Carvalho HH Wills Physics Laboratory, University of Bristol, Bristol BS8 1TL, United Kingdom

Density functional methods can be used to show that for classical fluids with short-ranged interatomic potentials the length scales which describe the asymptotic decay of the one-body density profile at a fluid interface are the same as those which determine the decay of the two-body correlation function g(r) of the bulk fluid. This general result has striking implications for a variety of interfacial properties - the possible occurrence of oscillations in the liquid-vapour density profile, the occurrence of wetting and layering phase transitions at substrate-fluid interfaces and the nature of the decay of the solvation force for a confined fluid. For fluids whose potentials have power-law decay the density profiles and g(r) exhibit power-law decay at longest range. However, for liquid densities the intermediate range damped oscillatory decay is governed by the same leading-order pole structure that describes short-ranged forces. Recent work on the decay of correlations i n binary liquid mixtures and in ionic fluids is also described. Consider a simple (atomic) fluid adsorbed at a solid substrate or wall. The average (one-body) density profile p(r) w i l l usually exhibit pronounced structure reflecting the ordering or layering that arises from packing effects, ie. from short-range correlations between the atoms. The precise form of the density profile must depend on the details of the wall-fluid potential V(r), the fluid-fluid interatomic potential and the thermodynamic state point. If the bulk fluid is a high density liquid a large number of oscillations develop in the profile, whereas i f the bulk is a dilute gas, and there is incomplete wetting of the wall-gas interface by l i q u i d , only a few oscillations, close to the wall, should occur. One might expect that the asymptotic decay of the profile into the bulk fluid far from the wall should be less dependent on the precise form of V(r) ; rather it should be determined primarily by the properties of the bulk fluid. Recent work has sought to elucidate different classes of asymptotic behaviour and the physical factors which determine these. A key result, obtained from density functional techniques, is that for short-ranged interatomic potentials the length scales which determine the longest range decay of a planar wallfluid interfacial profile are identical to those which characterise the longest range decay of the radial distribution function g(r) of the bulk liquid. Thus, determining

0097-6156/96/0629-0166$15.00/0 © 1996 American Chemical Society In Chemical Applications of Density-Functional Theory; Laird, B., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1996.

12.

EVANS & LEOTE DE CARVALHO

Decay of Correlations in Bulk Fluids

167


the different classes of asymptotic decay for g(r) is sufficient to determine the decay of planar interfacial profiles. This observation has important consequences for a variety of fluid interfacial phenomena. It leads to predictions for the existence (or non-existence) of oscillations in the density profiles at liquid-vapour and liquidliquid interfaces and in the solvation force for fluids confined between two parallel walls. It also has implications for wetting and layering transitions at wall-fluid interfaces. In this lecture we review several aspects of the general theoretical framework for describing the decay of inhomogeneous fluid structure in terms of the decay of g(r). The presentation is necessarily brief; further details of the theory and results of calculations can be found in (1 - 4). Asymptotics of the Bulk Pair Correlation Function g(r) for a Pure Fluid. The asymptotic decay of g(r) is most easily determined from the (bulk) OrnsteinZernike (OZ) equation, which relates the total pair correlation function h(r) =g(r) - 1 to the direct correlation function c(r). In Fourier space this is: h( )=-3sL=if—k

il

q

W

\-pc{q) p{\-pc{q)

(i)

J

(

)

where p is the number density and denotes the Fourier transform. It follows that A

We must distinguish between fluids for which c(r) is short ranged (finite ranged or exponentially decaying) and those for which c(r) decays as a power law. Recall that for fluids away from their critical points, diagrammatic analysis shows that c(r) —» - /J0(r) as r —> °°, where β = (k T)~ and 0(r) is the interatomic pairwise potential. B

l

I f c(r) decays faster than a power law it follows

Short-Ranged Potentials.

(1-4) from equations 1 and 2 that the asymptotics of rh(r) are determined completely by the poles of h(q) at complex q=a= α, + ι α

0

satisfying

1 - pe(a)=0

(3)

There can be no poles lying on the real axis apart from the liquid-vapour spinodals ( a = 0 ) and an infinite ranged oscillatory solution (4) often found at very high density [a

Q

= 0 , α , Φ θ ) . A pole can lie on the imaginary axis where it givesriseto

pure exponential decay of rh(r), or poles can lie off the imaginary axis where they yield exponentially damped oscillatory decay. In the latter case the poles occur as conjugate pairs: a = ± a , + i a . Once the poles have been determined (from knowledge of c(q) at a given p, Τ ) contour integration can be used (2,3) to obtain 0

In Chemical Applications of Density-Functional Theory; Laird, B., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1996.

CHEMICAL APPLICATIONS OF DENSITY-FUNCTIONAL THEORY

168

η where q is the η th pole and R is the residue of qc(q) n

n

/ ( l - pc(q)) at q . n

F o r the

model potentials and the closure approximations that have been studied there is an infinite number of poles but the longest range part of h(r) is determined by the pole or poles with the smallest value of cc , the inverse decay length. T w o scenarios are 0

found: (a) a pole lying on the imaginary axis q =ia Downloaded by UNIV MASSACHUSETTS AMHERST on October 9, 2012 | http://pubs.acs.org Publication Date: May 5, 1996 | doi: 10.1021/bk-1996-0629.ch012

n

0

has the lowest value and the

ultimate decay is purely exponential rh(r) ~ Ae " ° , r - » oo e

(5)

r

or (b) a conjugate pair of poles has a smaller imaginary part ά imaginary axis and the ultimate decay is

0

rh(r) ~ Âe~ ° c o s (a,r - θ), a

r

r

than that on the

oo

(6)

where Θ is a phase angle. The amplitudes A and λ and the phase are given in terms of the residues. The line i n the p, Τ plane where ά =a marks the crossover, at longest range, from damped oscillatory to pure exponential decay of rh(r). This is referred to as the Fisher-Widom (FW) line after the authors who first surmised such crossover should occur (5). Crossover arises as a result of competition between repulsive and attractive components of the interatomic potential. A t high Τ and p repulsion dominates (the negative portion of a typical c(r)) whereas at low Γ and p attraction dominates (the positive portion) (1). The F W line has been calculated (7) for a ( σ , 3 σ / 2 ) square-well fluid of well-depth ε, using a random phase approximation for the direct correlation function: 0

0

(7)

( # ) . 2

The second term i n equation 10 is the

contribution from the (complex) pole with the smallest value of a . 0

expected to have α ~2πΙσ. χ

This pole is

Other poles w i l l make further damped oscillatory

contributions but, provided they are well-separated, the leading-order pole should give the dominant oscillatory contribution. Those readers with interest in the history of liquid state science might note that it is thirty years since Enderby et.al. (8) properly identified the ultimate power-law decay of h(r), following an earlier observation by W i d o m (70) .Verlet, i n his famous 1968 paper (77) on the molecular dynamics of the Lennard-Jones fluid, was probably the first to enquire just how damped oscillations, which must occur at short and intermediate range, can be separated from the ultimate power-law decay of h(r). W e believe equation 10 is the appropriate prescription. H o w accurate is this approximation? This question was examined at some length i n reference (3). Explicit calculations, within the random phase approximation, for a model potential with hard-sphere repulsion and an attractive -a /r 6

6

tail show that equation 10

provides a very accurate fit to the 'exact' h(r), obtained from numerical Fourier transform of h(q) given by equation 1, for r>2a.

A t high densities a

0

is not

particularly large and 5(0) is small, with the result that the second term i n equation 10 dominates until τ ~ 2 5 σ , after which the power-law decay takes over. A t low densities a

Q

is much larger and 5(0) is increased, so that the oscillations in h(r) are

eroded much faster than at high densities (3).


171


172


Comparing equations 8 and 10 it is clear that the most significant difference between short and long-ranged potentials is the replacement of the pure exponentially decaying term by a power-law contribution. A s a consequence no sharp F W line can be defined in the long-ranged case, since there can be no crossover from pure exponential to damped oscillatory decay at any range; recall that there is no pole on the imaginary axis. However, this does not mean that intermediate-range structure w i l l not reflect the thermodynamic state. A s indicated above, at high densities the second term of equation 10 provides a very accurate description of intermediate range oscillatory structure of h(r) and the power-law contribution does not manifest itself until very large separations. A t lower densities the damped oscillatory term decays much faster and the power-law contribution has a larger amplitude (controlled by 5(0)) thereby reducing the number o f discernible oscillations. W h i l e there is no sharp F W line, there should be a crossover region in the p,T plane which marks the erosion of oscillations at intermediate range. Support for this view can be gleaned from two sources: (a) results for h(r) extracted from neutron-diffraction data taken for N e and X e along their saturated liquid curves (72) shows that intermediate range damped oscillatory decay persists up to T~0.95T , whereas for higher temperatures the oscillatory decay seems to C

disappear, (b) early calculations of h(r), based on the optimised cluster theory, for the full Lennard-Jones potential indicated crossover from damped oscillatory to monotonie decay at intermediate range as the density was reduced (13). For a given p , Γ one can estimate the separation r a t which the oscillations w i l l become indiscernible by equating the magnitudes of the two terms in equation 10. In computer simulation the pairwise potential is necessarily truncated. For the Lennard-Jones fluid the cut-off separation is often R =2.5a or 3.0σ. Such a short-ranged model w i l l only have the same intermediate range oscillatory structure as the full Lennard-Jones potential i f the dominant complex poles (near α =2π/σ) of both models lie very close together and the corresponding residues are very close. Results, based on the random phase approximation, suggest that this is indeed the case (3). c

χ

Asymptotics of hy(r) for a Binary Fluid Mixture. The analysis described in the previous section can be extended to mixtures. The generalisation of the O Z equation can be expressed as (11)

h^N^IDiq) where i, j runs over the species labels. Although the numerator

(q) is different

for different correlation functions, the denominator D(q) is common.

It follows

that all the hy(q) exhibit the same pole structure, determined by the zeros of D(q). Thus, for short-ranged potentials,

where the (common) w pole is given by D[q )-0. Only the residues R^ and hence, the amplitudes, differ for different combinations of species. Since the th

n


1

12.

EVANS & LEOTE DE CARVALHO

Decay ofCorrelations in Bulk Fluids

longest range part o f h^r) is determined by the q with the smallest imaginary part, n

all h (r) i}

w i l l ultimately decay with the s a m e exponential decay length and

oscillatory wavelength. This fact appears to have been appreciated first by Martynov (14) but was first explained in (2). The pole structure in a mixture should be similar to that in a pure fluid so that, for short-ranged potentials, there should be a unique F W surface i n thermodynamic phase space separating regions of longest range pure exponential decay from those of exponentially damped oscillatory decay, applicable for a l l the h (r). If the poles i}

are simple, the formulae for the residues Rj allow one to derive simple relations linking the amplitudes associated with the decay. For the particular case of binary mixtures, where i,j run over the species a and b, one finds for pure exponential decay


j

r h ^ ^ A ^

(13)

4*^=4,*.

(14)

with

Whereas for damped oscillatory decay, where a conjugate pair contribute,

r^W^^.^cos^r-e..)

(15)

Â Â =À l

(16)

0.+β»=2β,*

(17)

with m

bb

a

and

Explicit formulae for the amplitudes A and phases 0 can be derived (2). The results expressed in equations 13 to 17 are very general. They should apply for the longest range decay o f correlations i n any binary fluid mixture where the interatomic forces are short-ranged. A t first sight, the existence of a common asymptotic form is counterintuitive - especially when one contemplates mixtures with widely differing atomic sizes. It is not obvious that such mixtures should have hq(r) with a common wavelength and decay length. The accuracy of equation 15 has been examined for hard-sphere mixtures in the Perçus-Yevick approximation. For a variety of extreme concentrations and ratios of hard sphere diameters, equation 15, corresponding to a single conjugate pair of complex poles, yields h (r) that are very close to the results obtained from numerical Fourier transform o f the O Z equations - at least for separations beyond the second maximum. Even the positions of the first and second maxima are given reliably (2). In other words the decay and the amplitude and phase relations are obeyed for separations down to second nearest neighbours. Note that the amplitude relations in equations 14 and 16, which have the form of a geometric mixing rule, hold irrespective of the m i x i n g rule for the strength of the ab attractive potential and that for the effective range of the ab repulsion. The details of the chemistry do not matter - provided the relevant poles are simple. D o these results have relevance for real (atomic) mixtures where dispersion forces are present? If there are well-separated pole structures for binary mixtures (which we expect) the contribution from the pole with the smallest value of a w i l l dominate tj

t>

i}

0


173

174


the intermediate range decay of h^r) and the remarkable relations expressed i n equations 13 to 17 should hold over a similar (intermediate) range to that found for the pure fluid. Thus, it is feasible that these results could provide some new insight into the interpretation of structural data (from neutron and X - r a y diffraction) on binary mixtures.


Ionic Liquids. The analysis described above is not immediately applicable to an ionic liquid. In such systems the existence of the long-ranged Coulomb forces between the ions means that the direct correlation functions c^r) decay very s l o w l y , as r~\ for large r. Thus, a priori, it is not obvious that the pairwise correlation functions hg(r) i n a binary ionic liquid, such as a molten salt or a (primitive) electrolyte, should exhibit the same type of decay, with the same amplitude and phase relations, as those (equations 13 to 17) which characterise a binary neutral mixture. That this is the case, is a consequence of electrostatic screening. The O Z equations for an ionic mixture are the same as those for any mixture but now the Fourier transforms of the c (r) must be separated into f>

Coulombic and non-Coulombic terms:

where eZ+ is the charge on the positive ion, eZ_ is that on the negative ion and ε = 4 π ε ε , with e the relative permittivity. Such a division follows naturally i n a density functional approach where the intrinsic free energy functional is written as the total electrostatic energy plus a remainder. c^(q), which is analytic i n q , denotes the short-ranged, non-Coulombic contribution; dispersion forces are not considered here. It is straightforward to show that the O Z equations can be written in the form of equation 11, with suitable definitions of N (q) and D(q) (75). 0

r

Γ

2

tj

The pole structure of an ionic liquid is different from that of its neutral counterpart. This is most clearly illustrated for the special case of a symmetric binary fluid where the ++ interionic potential is equal to the - potential, so that h (r)=h__(r). ++

there are only two independent pair correlation functions: h =[h D

++

-Λ _)/2. +

h =(h +h _)/2 s

s

Then and D

s

D

+

h measures correlations i n the total number density and h

measures correlations in the charge density. The poles of h (q) h (q)

++

(density) and of

(charge) are given by independent equations and are determined by different

physical considerations (75). Density poles exhibit similar structure to that found in a single-component neutral fluid and cross-over from pure exponential to damped oscillatory decay of rh (r), as the density of ions is increased at fixed Τ, occurs by the same mechanism as in the neutral fluid giving rise to a F W line. The charge poles behave differently. These depend primarily on the inverse Debye screening s

ι length κ =(4πρβ(Ζβ) 1 0

2

ε) . 2

Crossover from pure exponential to damped

oscillatory decay of rh (r) occurs at a particular value of K v i a a coalescence of a pair of pure imaginary poles followed by fission into a conjugate complex pair D

D



12. EVANS & L E O T E D E CARVALHO

(75). Such a mechanism was first described by Kirkwood (76) in a discussion of charge oscillations in strong electrolytes. The locus of points in the p,T plane where crossover occurs to charge oscillations at longest range is termed the Kirkwood (K) line. Figure 2 shows the results (75) of calculations of the FW and the Κ lines for the restricted primitive model (RPM), ie. charged hard-spheres of equal diameter R, based on the generalised mean spherical approximation (GMS A) which is the simplest thermodynamically self-consistent theory for an ionic liquid. Within the GMSA, crossover to charge oscillations is determined by the condition K R = 1.228 so the Κ line is a straight line in the p,T plane - see figure 2. It intersects the vapour branch of the coexistence curve close to the critical temperature. The FW line intersects the liquid branch but at a lower value of TIT than that in figure 1 for the (truncated) Lennard-Jones fluid. By combining results for the density and charge poles one can ascertain which of these determines the ultimate decay of h^(r) and h+_(r). A variety of crossover lines emerges (see figure 7 of (75)). These separate regions of the phase diagram where the decay of ion-ion correlation functions are dominated by monotonie charge, monotonie density, oscillatory charge or oscillatory density poles. The results suggest that 1:1 primitive model electrolytes should exhibit a different sequence of crossover, as the density of ions is increased at fixed 7\ from 2:2 electrolytes. Since leading order asymptotics provide an accurate description of both h (r) and h (r) for r>\.5R, the results also yield some fresh insight into the nature of density and charge oscillations for molten salts.


D

C

D

s

Asymptotics of Wall-Fluid Interfacial Profiles. We now discuss the repercussions of the results obtained for the behaviour of bulk correlation functions for the density profiles of fluids at interfaces. That there should be direct repercussions follows from the fact that the radial distribution of the bulk fluid g(r) = p(r)fp, where p(r) is the (inhomogeneous) density profile obtained by fixing an atom at the origin, thereby creating a spherically symmetric 'external' potential V(r) for the other atoms. (In the special case of pairwise potentials V(r) = 0(r) ). In other words g(r) can be regarded as a one-body density profile associated with a particular type of inhomogeneity. One might suppose that other types of external potential might give rise to the same type of asymptotic decay of the density profile - provided the external potential is sufficiently shortranged. As an example consider a pure liquid in a bulk state which is on the oscillatory side of the FW line. Then h(r) decays as in equation 6. If the wallfluid potential V(z), where ζ is the distance normal to the wall, is of finite range the density profile for this interface should decay as p{z)-p

~

A^e- ° cos(a z-d ), a

z

1

iyf

z-»~

(19)

where p is the bulk density. a and a are the same inverse length scales which determine the decay of rh(r). The amplitude and phase will be different from the corresponding bulk quantities in equation 6; they will depend on the form 0

x

oiV(z).

Equation 19 was derived starting from the exact integral equation for the density profile in planar geometry (4). A somewhat more revealing derivation can be


175


176


0.12

\FW

Κ

0.1

0.08

0.06

0.04 -

0.02 I 0

0.05

0.1

0.15

0.2

0.3

0.25

0.4

0.35

Ρ* Figure 2. Cross-over lines for the R P M calculated using the G M S A . T h e short-dashed curve is the F W line where cross-over from monotonie to damped oscillatory decay occurs for the total number density correlation function h (r). s

Onset of charge oscillations, in h (r), occurs on the dotted line K . The solid line is the liquid-vapour coexistence curve and the long-dashed line marks the D

accompanying spinodals. T* = k TeR/(eZ) temperature and total density. B

2

and p* = pR are the reduced 3


12.


EVANS & L E O T E D E CARVALHO

obtained from density functional theory. The density profile of an inhomogeneous fluid subject to an external potential V(r) satisfies the exact Euler-Lagrange equation (see eg. (77) or (18)) M=V(rO + ^ ( p ( r ) ) - i 8 - c « ( ( p ] ; r ) a

1

1

where μ is the chemical potential, μ (p)

= β~ 1η(Λ ρ) is the chemical potential of


( 1 )

3

ι

ίά

the ideal classical gas, and c

(20)

1

, the one-body direct correlation function, is the first

derivative of the excess (over ideal) Helmholtz free energy functional 7^ [p] : x

c

Decay of Correlations in Bulk Fluids and at Interfaces - American

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