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A perturbation approach to the density functional theory of adsorption
Volume 93A, number 9
PHYSICS LETTERS
14 February 1983
A PERTURBATION APPROACH TO THE DENSITY FUNCTIONAL THEORY OF ADSORPTION Malcolm J. GRIMSON
Physics Laboratory, University of Kent, Canterbury, Kent, UK Received 20 September 1982
A modified treatment of an exact density functional theory for non-uniform fluids due to Saam and Ebner is presented and a perturbation method of solution is proposed which avoids the need for a local density approximation. For hard-core fluids near a hard wall the equation for the equilibrium density reduces to the wall-particle version of the RHNC approximation.
Knowledge of the density profde of a fluid in equilibrium at a solid surface is of considerable importance in studies of adsorption and thin liquid films. An important class of theories for the structure of non-uniform fluids are those based on the density functional formalism [1 ] in which a variational princii~le for the free energy determines the equilibrium density. Saam and Ebner [2,3] constructed a density functional expression for the free energy of a non-uniform fluid by a functional integration and subsequent local density approximation to the nonuniform direct correlation function c(r, r'). However, the application of this theory requires careful handling in order to avoid situations in which the local density at which the uniform direct correlation function c(Ir- r'l) is evaluated does not correspond to a thermodynamically stable state. In this letter a modified treatment of the formally exact density functional theory of Saam and Ebner [2] is presented and a perturbation method of solution is proposed which avoids any local density approximation. Consider a classical system of particles with fixed chemical potential # and temperature Tin the presence of an external potential v(r) which couples to the particle number density n(r). Saam and Ebner [2] have shown that for such a system there exists a free energy function ~2[n] whose minimum with respect to local variations of n(r) is the equilibrium grand free energy of the system. Explicitly
~(~ [n]
-~20) =
fdr tao r) -
1] In(r) - no]
+
f dr n(r) In [n(r)/no]
- f as f ddfdrdr'[n(r)-no]c(r,r';d)[n(r')-no] , 0
(1)
0
where {3-1 = kBT, c(r, r'; ¢0 is the non-uniform direct correlation function at a density n(r, or) and the parameter a describes an integration path in the space of density functions from a reference, uniform state of density n 0, external potential v = 0 and free energy ~20 to a final state of density n(r) and free energy ~2[n]. A theorem due to Mermin [4] ensures that I2[n] is unique and independent of the path chosen for the functional integrals. The result follows from the defining relation for the direct correlation function,
5 2g218n(r)Sn(r') = 8o(r)/Sn(r') = -kBr[8(r - r')/n(r') - c(r, r')]
(2)
•if it is assumed that c(r, r') is given at any arbitrary density. In order to implement their theory, Saam and Ebner [3] used the closure approximation 0 031-9163/83/0000-0000[$ 03.00 © 1983 North-Holland
479
Volume 93A, number 9
2 f d . f d: c(r,r'; 0
PHYSICS LETTERS
14 February 1983
(3)
= e(lr - r'i, ~),
0
where c(Ir - r'l, ~) is the direct correlation function of the uniform fluid at a bulk density ~ = [n(r) + n(r')]/2. While this approximate form of the density functional theory is in good agreement with simulation studies of a one-dimension fluid [5], the density profiles for studies of gas adsorption qualitatively differ from the simulations of Lane et al. [6]. It is generally acknowledged that the failure of the theory is due to the local density approximation (3) which may require c(r, ~) to be evaluated for metastable states [ 6 - 8 ] . An alternative closure was suggested by Grimson and Rickayzen [9] who took the density $ in eq. (3) to be that of the bulk fluid well away from the external perturbation. Using this closure the equation for the density in the density functional theory reduces to that obtained using the Ornstein-Zemike (OZ) equation with a hypernetted chain (HNC) closure [ 10] and following a linearisation of the free energy the density functional theory gives the same equation for the density as the OZ equation with a Percus-Yevick (PY) closure [10]. The closure approximation of Grimson and Rickayzen represents a replacement of c(r, r') in eq. (2) with its form in the limit of a vanishing external potential, e(lr - r'l, no). In order to capitalise on the results of this closure, the non-uniform direct correlation function c(r, r') is split into its uniform and non-uniform components with c(r, r') = c(Ir - r'[, no) + e(r, r'),
(4)
where e(r, r') contains all of the inhomogeneous contributions to c(r, r'). If it is assumed that e(r, r') is twice functionally differentiable with respect to the density, then it is possible to introduce an associated free energy contribution fir where 6 2 fir/6n(r) 6n(r') = k B T e(r, r').
(5)
Then the formally exact free energy given in eq. (1) may be rewritten as
[~(a - fir) = #a o + f dr tOu(r) - 1] [n(r) - no] + f dr n ( r ) I n [n(r)/no] - f d r dr' [n(r) - no] ¢([r - r'[, no)[n(r' ) - no].
(6)
The free energy (I2 - fir) is equivalent to the form used by Grimson and Rickayzen [9]. The equilibrium density which minimises the free energy ~ of eq. (6) with respect to variations in n(r) is given by [3o(r) - E(r) = - l n [ n ( r ) / n o ]
+
f drc(Ir
- r'l, no)[n(r') - no],
(7)
where
8 ~r/~n(r) = E(r).
(8)
The function/3-1E(r) is seen to have the role of an additional external potential in eq. (7) which is equivalent to the equation for density given by the OZ equation with a HNC closure when the external potential has the form o(r) - / 3 - 1 E ( r ) . The precise status of E(r)may be seen by considering the extemal potential to arise from a source particle in which case the external potential o mimics the pair potential u. It is then straightforward to show [ 11 ] that eq. (7) reduces to ln[h(r) + 11 + flu(r) = h(r) - c(r) + E(r), 480
(9)
Volume 93A, number 9
PHYSICS LETTERS
14 February 1983
where h(r) is the total correlation function. However, eq. (9) is identical to a resummation of the diagrammatic expansion of h(r) in powers of the density and we may identify E(r) as the bridge function or sum of non-nodal elementary graphs in the diagrammatic expansion [12]. The solution of eq. (7) for the density n(r) requires the specification of the bridge function E(r). Rosenfeld and Ashcroft [12] have shown that for uniform fluids E(r) is a highly universal and short'ranged function that exists at all densities. This led Nieminen and Ashcroft [8] to suggest that a local density approximation to the bridge function in the non-uniform system would be appropriate. However, it is possible to avoid any local density approximation by assuming that the universality property of the bridge functions holds even in the non-uniform context and use a perturbative method of solution. Let there be a reference system of bulk density no, characterised by a uniform direct correlation function Cref(lrl, no), subject to an external potential Oref~r). The equilibrium density of the reference system according to eq. (7) is given by 3Oref(r) - E r e f ( r ) = -ln[nref(r)/no] +
fdr'
Cref(Ir - r'l,
no)[nref(r') - n0].
(10)
The assumption of universality for the non-uniform bridge functions allows E(r) to be equated with Eref(r ) and from eqs. (7) and (10), /3[0(/') - Oref(r)] = -ln[n(r)/nref(r)] - n o fdr' [c([r - r'[, no) - Cref(lr - r'[, no) ]
+
fdr'
[c(Ir - r'[, no)n(r' ) - Cref(Ir - r'l, no)nref(r')].
(11)
Eq. (11) is the basic equation for the equilibrium density obtained in this perturbation approach to the density functional theory and may be solved when a suitable reference-system is available. In studies of the structure of a fluid at a solid planar surface the external potential is usually assumed to contain a hard wall part to ensure that the solid is impenetrable. If the pair potential u(r) is assumed to contain a hardcore interaction of hard-sphere diameter d, then a suitable reference system is available in the solution of the generalised mean spherical approximation (GMSA) closure to the OZ equation for the hard-sphere-hard-wall system [ 13]. The choice of a pair potential with a hard-sphere interaction avoids the necessity of introducing a second perturbation scheme associated with a softening of the interatomic core. When the hard planar wall is located at x = --d/2, n(x) = nref(x ) = 0 for x < 0 and Oref(X ) = 0 for X > 0. The equilibrium density will satisfy
(3o(x) = -ln[n(x)/nref(X)] +
dx' [c'(Ix - x'l, no)n(x') - Cref(lx - x'[, no)nref(X )] 0
I -- n o f dx' [c'(IX - x'l, no) - Cref(lx - x II, no)],
x > 0,
(12)
where
c'(x) = f r y dz c(r).
(13)
The connection of this density functional theory to other theories for the fluid structure at a hard wall follows from the wall-particle version of the OZ equation [10]
h(~ ) = ~(.~) + .o f dx' ~'(Ix - ~'l,.o)h(x'),
(14) 481
Volume 93A, number 9
PHYSICS LETTERS
14 February 1983
where h(x) and c(x) are the total and direct wall-particle correlation functions respectively. The equilibrium density profile is related to the wall-particle correlation function h(x) by the equation
n(x) = no[h(x ) + 1]
(15)
and substitution o f e q . (14) into eq. (12) gives f o r x > 0,
[3v(x) = -ln[n(x)/nref(X )] + n(x) - nref(X ) - [c(x) - Cref(X)].
(16)
However, eq. (16) constitutes the wall-particle renormalised hypemetted chain (RHNC) equation obtained by Sullivan and Stell [14]. Thus all of the results and conclusions of Sullivan and Stell regarding the wall-particle RHNC approximation are equally valid for this density functional theory. The most important finding o f Sullivan and Stell with regard to this theory was that the pressure obtained from the density profile did not coincide with the simulation values. Although, the present theory was in better quantitative agreement with the exact results than the pressures calculated from the density profiles for the PY and HNC closures to the OZ equation, which correspond to cruder approximate density functional theories [9]. Clearly, the source of the errors in this theory must lie in the assumption that E(r) = Eref(r ) and an improved treatment will require a future study of the bridge function in order to classify the status of the universality hypothesis in non-uniform fluids. The financial support of the SERC is gratefully acknowledged.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [101 [11] [121 [13] [ 14]
482
R. Evans, Adv. Phys. 28 (1979) 143. W.F. Saam and C. Ebner, Phys. Rev. A15 (1977) 2566. W.F. Saam andC. Ebner, Phys. Rev. A17 (1978) 1768. N.D. Mermin, Phys. Rev. 137 (1965) A1441. C. Ebner, M.A. Lee and W.F. Saam, Phys. Rev. A21 (1980) 969. J.E. Lane, T.H. Spurling, B.C. Freasier, J.W. Perram and E.R. Smith, Phys. Rev. A20 (1979) 2147. J. Fischer and M. Methfessel, Phys. Rev. A22 (1980) 2836. R.M. Nieminen and N.W. Ashcroft, Phys. Rev. A24 (1981) 560. M.J. Gfimson and G. Rickayzen, Mol. Phys. 42 (1981) 767. D. Henderson, F.F. Abraham and J. Barker, Mol. Phys. 31 (1976) 1291. J.K. Percus, in: The equilibrium theory of classical fluids, eds. H.L. Frisch and J.L. Lebowitz (Benjamin, New York, 1964). Y. Rosenfeld and N.W. Ashcroft, Phys. Rev. A20 (1979) 1208. E. Waisman, D. Henderson and J.L. Lebowitz, Mol. Phys. 32 (1976) 1373. D.E. Sullivan and G. SteU, J. Chem. Phys. 69 (1978) 5450.
PHYSICS LETTERS
14 February 1983
A PERTURBATION APPROACH TO THE DENSITY FUNCTIONAL THEORY OF ADSORPTION Malcolm J. GRIMSON
Physics Laboratory, University of Kent, Canterbury, Kent, UK Received 20 September 1982
A modified treatment of an exact density functional theory for non-uniform fluids due to Saam and Ebner is presented and a perturbation method of solution is proposed which avoids the need for a local density approximation. For hard-core fluids near a hard wall the equation for the equilibrium density reduces to the wall-particle version of the RHNC approximation.
Knowledge of the density profde of a fluid in equilibrium at a solid surface is of considerable importance in studies of adsorption and thin liquid films. An important class of theories for the structure of non-uniform fluids are those based on the density functional formalism [1 ] in which a variational princii~le for the free energy determines the equilibrium density. Saam and Ebner [2,3] constructed a density functional expression for the free energy of a non-uniform fluid by a functional integration and subsequent local density approximation to the nonuniform direct correlation function c(r, r'). However, the application of this theory requires careful handling in order to avoid situations in which the local density at which the uniform direct correlation function c(Ir- r'l) is evaluated does not correspond to a thermodynamically stable state. In this letter a modified treatment of the formally exact density functional theory of Saam and Ebner [2] is presented and a perturbation method of solution is proposed which avoids any local density approximation. Consider a classical system of particles with fixed chemical potential # and temperature Tin the presence of an external potential v(r) which couples to the particle number density n(r). Saam and Ebner [2] have shown that for such a system there exists a free energy function ~2[n] whose minimum with respect to local variations of n(r) is the equilibrium grand free energy of the system. Explicitly
~(~ [n]
-~20) =
fdr tao r) -
1] In(r) - no]
+
f dr n(r) In [n(r)/no]
- f as f ddfdrdr'[n(r)-no]c(r,r';d)[n(r')-no] , 0
(1)
0
where {3-1 = kBT, c(r, r'; ¢0 is the non-uniform direct correlation function at a density n(r, or) and the parameter a describes an integration path in the space of density functions from a reference, uniform state of density n 0, external potential v = 0 and free energy ~20 to a final state of density n(r) and free energy ~2[n]. A theorem due to Mermin [4] ensures that I2[n] is unique and independent of the path chosen for the functional integrals. The result follows from the defining relation for the direct correlation function,
5 2g218n(r)Sn(r') = 8o(r)/Sn(r') = -kBr[8(r - r')/n(r') - c(r, r')]
(2)
•if it is assumed that c(r, r') is given at any arbitrary density. In order to implement their theory, Saam and Ebner [3] used the closure approximation 0 031-9163/83/0000-0000[$ 03.00 © 1983 North-Holland
479
Volume 93A, number 9
2 f d . f d: c(r,r'; 0
PHYSICS LETTERS
14 February 1983
(3)
= e(lr - r'i, ~),
0
where c(Ir - r'l, ~) is the direct correlation function of the uniform fluid at a bulk density ~ = [n(r) + n(r')]/2. While this approximate form of the density functional theory is in good agreement with simulation studies of a one-dimension fluid [5], the density profiles for studies of gas adsorption qualitatively differ from the simulations of Lane et al. [6]. It is generally acknowledged that the failure of the theory is due to the local density approximation (3) which may require c(r, ~) to be evaluated for metastable states [ 6 - 8 ] . An alternative closure was suggested by Grimson and Rickayzen [9] who took the density $ in eq. (3) to be that of the bulk fluid well away from the external perturbation. Using this closure the equation for the density in the density functional theory reduces to that obtained using the Ornstein-Zemike (OZ) equation with a hypernetted chain (HNC) closure [ 10] and following a linearisation of the free energy the density functional theory gives the same equation for the density as the OZ equation with a Percus-Yevick (PY) closure [10]. The closure approximation of Grimson and Rickayzen represents a replacement of c(r, r') in eq. (2) with its form in the limit of a vanishing external potential, e(lr - r'l, no). In order to capitalise on the results of this closure, the non-uniform direct correlation function c(r, r') is split into its uniform and non-uniform components with c(r, r') = c(Ir - r'[, no) + e(r, r'),
(4)
where e(r, r') contains all of the inhomogeneous contributions to c(r, r'). If it is assumed that e(r, r') is twice functionally differentiable with respect to the density, then it is possible to introduce an associated free energy contribution fir where 6 2 fir/6n(r) 6n(r') = k B T e(r, r').
(5)
Then the formally exact free energy given in eq. (1) may be rewritten as
[~(a - fir) = #a o + f dr tOu(r) - 1] [n(r) - no] + f dr n ( r ) I n [n(r)/no] - f d r dr' [n(r) - no] ¢([r - r'[, no)[n(r' ) - no].
(6)
The free energy (I2 - fir) is equivalent to the form used by Grimson and Rickayzen [9]. The equilibrium density which minimises the free energy ~ of eq. (6) with respect to variations in n(r) is given by [3o(r) - E(r) = - l n [ n ( r ) / n o ]
+
f drc(Ir
- r'l, no)[n(r') - no],
(7)
where
8 ~r/~n(r) = E(r).
(8)
The function/3-1E(r) is seen to have the role of an additional external potential in eq. (7) which is equivalent to the equation for density given by the OZ equation with a HNC closure when the external potential has the form o(r) - / 3 - 1 E ( r ) . The precise status of E(r)may be seen by considering the extemal potential to arise from a source particle in which case the external potential o mimics the pair potential u. It is then straightforward to show [ 11 ] that eq. (7) reduces to ln[h(r) + 11 + flu(r) = h(r) - c(r) + E(r), 480
(9)
Volume 93A, number 9
PHYSICS LETTERS
14 February 1983
where h(r) is the total correlation function. However, eq. (9) is identical to a resummation of the diagrammatic expansion of h(r) in powers of the density and we may identify E(r) as the bridge function or sum of non-nodal elementary graphs in the diagrammatic expansion [12]. The solution of eq. (7) for the density n(r) requires the specification of the bridge function E(r). Rosenfeld and Ashcroft [12] have shown that for uniform fluids E(r) is a highly universal and short'ranged function that exists at all densities. This led Nieminen and Ashcroft [8] to suggest that a local density approximation to the bridge function in the non-uniform system would be appropriate. However, it is possible to avoid any local density approximation by assuming that the universality property of the bridge functions holds even in the non-uniform context and use a perturbative method of solution. Let there be a reference system of bulk density no, characterised by a uniform direct correlation function Cref(lrl, no), subject to an external potential Oref~r). The equilibrium density of the reference system according to eq. (7) is given by 3Oref(r) - E r e f ( r ) = -ln[nref(r)/no] +
fdr'
Cref(Ir - r'l,
no)[nref(r') - n0].
(10)
The assumption of universality for the non-uniform bridge functions allows E(r) to be equated with Eref(r ) and from eqs. (7) and (10), /3[0(/') - Oref(r)] = -ln[n(r)/nref(r)] - n o fdr' [c([r - r'[, no) - Cref(lr - r'[, no) ]
+
fdr'
[c(Ir - r'[, no)n(r' ) - Cref(Ir - r'l, no)nref(r')].
(11)
Eq. (11) is the basic equation for the equilibrium density obtained in this perturbation approach to the density functional theory and may be solved when a suitable reference-system is available. In studies of the structure of a fluid at a solid planar surface the external potential is usually assumed to contain a hard wall part to ensure that the solid is impenetrable. If the pair potential u(r) is assumed to contain a hardcore interaction of hard-sphere diameter d, then a suitable reference system is available in the solution of the generalised mean spherical approximation (GMSA) closure to the OZ equation for the hard-sphere-hard-wall system [ 13]. The choice of a pair potential with a hard-sphere interaction avoids the necessity of introducing a second perturbation scheme associated with a softening of the interatomic core. When the hard planar wall is located at x = --d/2, n(x) = nref(x ) = 0 for x < 0 and Oref(X ) = 0 for X > 0. The equilibrium density will satisfy
(3o(x) = -ln[n(x)/nref(X)] +
dx' [c'(Ix - x'l, no)n(x') - Cref(lx - x'[, no)nref(X )] 0
I -- n o f dx' [c'(IX - x'l, no) - Cref(lx - x II, no)],
x > 0,
(12)
where
c'(x) = f r y dz c(r).
(13)
The connection of this density functional theory to other theories for the fluid structure at a hard wall follows from the wall-particle version of the OZ equation [10]
h(~ ) = ~(.~) + .o f dx' ~'(Ix - ~'l,.o)h(x'),
(14) 481
Volume 93A, number 9
PHYSICS LETTERS
14 February 1983
where h(x) and c(x) are the total and direct wall-particle correlation functions respectively. The equilibrium density profile is related to the wall-particle correlation function h(x) by the equation
n(x) = no[h(x ) + 1]
(15)
and substitution o f e q . (14) into eq. (12) gives f o r x > 0,
[3v(x) = -ln[n(x)/nref(X )] + n(x) - nref(X ) - [c(x) - Cref(X)].
(16)
However, eq. (16) constitutes the wall-particle renormalised hypemetted chain (RHNC) equation obtained by Sullivan and Stell [14]. Thus all of the results and conclusions of Sullivan and Stell regarding the wall-particle RHNC approximation are equally valid for this density functional theory. The most important finding o f Sullivan and Stell with regard to this theory was that the pressure obtained from the density profile did not coincide with the simulation values. Although, the present theory was in better quantitative agreement with the exact results than the pressures calculated from the density profiles for the PY and HNC closures to the OZ equation, which correspond to cruder approximate density functional theories [9]. Clearly, the source of the errors in this theory must lie in the assumption that E(r) = Eref(r ) and an improved treatment will require a future study of the bridge function in order to classify the status of the universality hypothesis in non-uniform fluids. The financial support of the SERC is gratefully acknowledged.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [101 [11] [121 [13] [ 14]
482
R. Evans, Adv. Phys. 28 (1979) 143. W.F. Saam and C. Ebner, Phys. Rev. A15 (1977) 2566. W.F. Saam andC. Ebner, Phys. Rev. A17 (1978) 1768. N.D. Mermin, Phys. Rev. 137 (1965) A1441. C. Ebner, M.A. Lee and W.F. Saam, Phys. Rev. A21 (1980) 969. J.E. Lane, T.H. Spurling, B.C. Freasier, J.W. Perram and E.R. Smith, Phys. Rev. A20 (1979) 2147. J. Fischer and M. Methfessel, Phys. Rev. A22 (1980) 2836. R.M. Nieminen and N.W. Ashcroft, Phys. Rev. A24 (1981) 560. M.J. Gfimson and G. Rickayzen, Mol. Phys. 42 (1981) 767. D. Henderson, F.F. Abraham and J. Barker, Mol. Phys. 31 (1976) 1291. J.K. Percus, in: The equilibrium theory of classical fluids, eds. H.L. Frisch and J.L. Lebowitz (Benjamin, New York, 1964). Y. Rosenfeld and N.W. Ashcroft, Phys. Rev. A20 (1979) 1208. E. Waisman, D. Henderson and J.L. Lebowitz, Mol. Phys. 32 (1976) 1373. D.E. Sullivan and G. SteU, J. Chem. Phys. 69 (1978) 5450.