J. Phys. Chem. C 2007, 111, 15505-15512
15505
Structure of Hard Spheres near a Hard Wall and in a Pore from the Residual Chemical Potential† Toma´ sˇ Boublı´k* Department of Chemistry, J. E. Purkinje UniVersity, Usti nad Labem, Czech Republic ReceiVed: March 24, 2007; In Final Form: May 12, 2007
The structure of inhomogeneous systemsshard sphere near a planar hard wall and hard sphere in a pores was studied employing the formula for the background correlation function in terms of the residual chemical potentials of hard spheres and that of a hard dumbbell (or a combined body). The enlarged hard dumbbell is considered for determination of the basic geometric quantities: volume, surface area, and mean radius (the mean curvature integral divided by 4π for convex bodies). A novel method to determine the mean radius is proposed. Detailed discussion of the way of determining the single geometric quantities in special cases of systems (hard sphere + hard wall) and (hard sphere + spherical pore) is given. The predicted density profiles or distribution functions compare well with pseudoexperimental data in the range of the reduced distances x e 2.
1. Introduction Study of the structure of inhomogeneous systems, that is, behavior of hard sphere (HS) fluids near a hard wall, HS in slots, cylindrical, or spherical pores has attracted considerable interest of scientists for several decades. This is due to the importance of such systems in both theory and practice. Knowledge of the structure of these systems, expressed mainly trough the distribution function or the HS density profile, plays an important role in understanding processes such as solvatation, adsorption, and the behavior of fluids in zeolites and membranes or biological cells. From the point of view of theory, the structure of inhomogeneous systems and the related thermodynamic functions are more complex than those known for bulk pure fluids or mixtures. Pseudoexperimental data obtained by Monte Carlo simulations in the grand canonical ensemble are available for fluids of hard spheres1-6 (both pure and mixtures), Lennard-Jones fluids,7,8 and more complex systems.9,10 Theoretical methods used to describe the structure of hard spheres near a hard wall include the use of the virial expansion at low densities and solution of the integrodifferential BBGKY or integral OrnsteinZernike (OZ) equations (with the Percus-Yevick (PY), HNC, or general mean spherical (GMSA) closures).11,12 At present, the most frequently used methods follow from the density functional theory (DFT).13-16 DFT has developed from the early formulation for homogeneous fluids17 to a versatile approach applicable to variety of inhomogeneous systems with excellent results. Solutions of both the OZ integral equation or DFT expressions for HS density are obtained by numerically employing an itterative approach and thermodynamic and structural expressions for the correspondent homogeneous systems. Recently,18 we proposed a different, much simpler method of determining the distribution function/density profile on the basis of expressions for the residual chemical potential and geometric quantities of HSs and hard dumbbell (or other combined hard body). The method stems from the study of †
Part of the special issue “Keith E. Gubbins Festschrift”. * E-mail:
[email protected].
Meeron and Siegert,19 who proposed a relation between the background correlation function (cavity function), Y, and residual chemical potentials, ∆µ, of hard spheres and hard dumbbell. Ballance and Speedy20 and Labı´k et al.21 employed for the determination of ∆µ the virial expansion, whereas the present author22 used a closed formula in terms of the hard body volumes, surface areas, and mean radii. Formerly,22 we approximated the geometric quantities of the combined body by those of the corresponding prolate spherocylinders. In the recent paper,18 we employed instead of spherocylinder the enlarged fused hard sphere model. The use of this model avoids the disadvantage of the previous approach, the ill-behavior of the predicted background correlation function at the reduced distance x ) 2. The presently considered model enables also an easy formulation of the expressions for the distribution functions of inhomogeneous systems. 2. Theory In the present method of determining the hard sphere distribution functions, originated from the work of Meeron and Siegert19 and Labı´k et al.,21 we start from the expression for the background correlation function (cavity function)
Y(r) ) exp[u(r)/kT] g(r) in terms of the residual chemical potentials of two hard spheres, ∆βµhs, and hard dumbbell, ∆βµd, in the form d hs ln Yij ) ∆βµhs i + ∆βµi - ∆βµ
(1)
For hard spheres, the background correlation function, Y, in the range of the reduced distances x ) l/σ g 1 is equal to the radial distribution function, g, whereas for x < 1, Y determines the direct correlation function. In the previous contribution,18 we studied two “selfconsistent” expressions for the chemical potential, βµ, of hard body systems, one of them possessing a form
10.1021/jp0723362 CCC: $37.00 © 2007 American Chemical Society Published on Web 07/19/2007
15506 J. Phys. Chem. C, Vol. 111, No. 43, 2007
Boublı´k
β∆µ ) -ln(1 - y) + 7y/(1 - y) + y2(15/2 - y)/(1 - y)2 + y3(8/3 - y/3)/(1 - y)3 (the corresponding HS compressibility factor at y ) 0.5 equals 13.167). In order to get better agreement of the predicted higher virial coefficients with the computer data, we slightly modified the last term of the above equation; the used hard body expression reads as
[
y (1 - y)
][ 2
[
]
y [3(R* + S*) + V *] + (1 - y) 3 y (Q* + 2S*) - + 3V* + 2 3 3 y y 2 - V* (2) 3 (1 - y)
β∆µ ) -ln(1 - y) +
(
[
)
]
]( )
In the above equation, y stands for the packing fraction, and R*, S*, V* are the reduced geometric quantities of hard bodies: the mean radius (the mean curvature integral divided by 4π in the case of convex bodies), surface area, and volume. The geometric functional Q* is connected with the overlap of three hard bodies, and approximations for it will be discussed in the next part. For hard spheres of diameter σ, we have R* ) 2R/σ ) 1, S* ) S/πσ2 ) 1, V* ) V/(πσ3/6) ) 1; the quantity Q (for which a relation S/(4π) e Q e R2 holds true) in the case of hard spheres possesses value Q* ) 1, too. Equation 2 holds for a hard dumbbell, corresponding to a pair of overlapping hard spheres, too. Instead of the hard dumbbell, however, we employed in ref 18 and here the enlarged hard dumbbell (EHD), a body “perceived” by the other hard spheres of the studied system. EHD originates when a probe hard sphere is rolled over a hard dumbbell. Its surface area and volume are evaluated by the method of Connolly.23,24 The mean radius can be determined on the basis of expressions for the surface areas, as will be discussed in the following part. If we denote properties of the EHD by superscript d and define ∆X/ij ) X/i + X/j - Xd*, we can write
y [3(∆R* + ∆S*) + ∆V*] + ln Y ) -ln(1 - y) + (1 - y) 3 y y2 (∆Q* + 2∆S*) - + 3∆V* + 2 2 3 (1 - y)
[
(
]
)
y3 y 2 - ∆V* (3) 3 3 (1 - y)
(
)
To this end, the knowledge of differences in the geometric functionals is sufficient for the determination of the background correlation function, Y, as the function of the center-to-center distance, l, or the reduced distance, x ) l/σ. We will discuss the evaluation of the geometric quantities of the EHD (and pure hard bodies) in detail in the next section. 3. Geometric Functionals of Hard Bodies 3.1. Pure Hard Spheres. In the case of pure hard sphere systems, all the reduced geometric quantities of the couple of hard spheres are equal to 2. The surface area and volume of EHD can be easily determined by the method proposed by Connolly: accordingly, the EHD model is divided into two convex spherical parts, and one saddle part between (see Figure 1). (In the case of pure fluid r1 ) r2 ) rprobe ) r and θ1 ) θ2 ) θ). The convex and saddle parts are separated by two cones (with axes lying in the line connecting the centers of hard spheres, l). The radius of torus (perpendicular to l) and the side
Figure 1. Enlarged hard dumbbell composed of two spheres with radii r1, r2. rp, probe radius; X, tore radius.
of the cone determine angle θ. Its value follows from a relation θ ) arcsin(x/2). If θ e π/3, then the knowledge of this angle is sufficient to evaluate Sd and Vd. The surface area of a convex part is
S1 ) 2πr2(1 + sin θ)
(4)
and that of one-half of the saddle part is
S2 ) 2πrprobe
∫0θ(X - rprobe cos γ) dγ )
2πr(2rθ cos θ - r sin θ) (5)
In the case of the one-component HS system rprobe ) r and X ) 2r cos θ. The total surface area is S ) 2(S1 + S2). Similarly, the total volume, V ) 2(V1 + V2), follows from expressions
V1 )
2π 3 1 r 1 + sin θ + cos2 θ sin θ 3 2
[
(
)
V2 ) πr X2 sin θ - Xr(cos θ sin θ + θ) +
(6)
]
r2 (2 sin θ + cos2 θ sin θ) (7) 3 Determination of the mean radius of a (nonconvex) enlarged hard dumbbell is not unique. In our recent paper,18 we considered three simple approximations proposed previously within our study of the hard body virial coefficients. Here, we present, however, a more fundamental approach based on the fact that the mean radius, R, of a smooth convex body (spheres, spherocylinders, etc.) is related to the surface area through a relation
R ) (1/8π)(∂S/∂r)
(8)
Because in our EHDs the convex parts are actually spherical caps and the saddle parts are obtained by revolving a part of a circle around axis l, we can evaluate R as a sum of contributions of convex and saddle parts. It is important to realize that we differentiate S under condition of constant angle θ. Thus, for a convex part (part of HS), we have
1 R1 ) (1/8π)(∂S1/∂r) ) r(1 + sin θ) 2
(9)
In the case of a saddle part, we have to realize that with an increase in the radius of the convex part, the probe radius decreases. Therefore,
R2 ) -(1/8π)(∂S2/∂rprobe) ) -(Xθ/4 - r sin θ/2) ) 1 r(-θ cos θ + sin θ) (10) 2
Structure of Hard Spheres near Hard Wall, in Pore
J. Phys. Chem. C, Vol. 111, No. 43, 2007 15507
The basic expression for Rd is obtained as a sum of contributions for two convex and two saddle parts,
Rd ) r(1 + 2 sin θ - θ cos θ)
Rs )
(11)
The above formulas hold true for all the reduced distances x e x3. For x > x3 (or θ > π/3), the enlarged dumbbell decomposes into two hard spheres with cusps, and the lower bound of the integral in eq 5 changes from 0 to φ (the same is true for similar integral in an expression for Vd) (see Figure 1). Taking into account that cos φ ) 2 cos θ, the final relationships for Rd and Sd read as
2Rd/σ ) 1 + 2 sin θ - sin φ - (θ - φ) cos θ S /(πσ ) ) 1 + sin φ + 2(θ - φ) cos θ d
where Rs and βs are nonsphericity parameters of the HS solution
2
r/s s/s 3V/s
βs )
q/s s/2 s
with geometric functionals r* ) 2rs/Fσ1 ) ∑xiRi/r1 ) ∑xipi, s/s ) q/s ) ∑xip2i , and V/s ) ∑ixip3i . The reduced differences of the geometric quantities are defined as
∆R/ij ) ∆Rij/r/s
(12)
∆S/ij ) ∆Sij/s/s
(13)
From the modified integrals in eq 7, the expression for Vd valid for the whole interval of x e 2 is
∆V/ij ) ∆Vij/V/s
Vd/(πσ3/6) ) 1 + 2 sin θ - sin φ - 3(θ - φ) cos θ + 2 cos2 θ(2 sin θ - sin φ) (14)
∆Q/ij ) ∆Qij/s/s
By subtracting expressions 12-14 from 2, we obtain differences ∆R, ∆S, and ∆V. It is interesting to note that all three differences are equal to zero for x g 2. The effect of other HSs on Y (of the considered pair) is included in the geometric quantity Q, which varies in the range (S/4π) e Q e R2. Wertheim25 proposed a prescription for Q of hard convex bodies in the form of an expansion; simpler approximations in the closed form were proposed, either as the arithmetic26 or geometric mean27 of the limiting values. Here, we use an approximation,
Q ) R(S/4π)1/2ξ
Values of ∆X (X ) R*, S*, V*, Q*) are obtained as differences of pmi + pmj (where m ) 1, 2, 3) and the corresponding geometric functionals of the enlarged dumbbell of pair ij; namely,
2Rdii/σ11 ) pi + (pi + p3) sin θi - p3 sin φi 1 (p + p3)(θi - φi) cos θi (18) 2 i 2Rdij/σ11 )
1
2
[
∑ pi + (pi + p3) sin θ3i - p3 sin φ3i -
2 i)1
(15)
1
where ξ ) (S/4πR2)n and n ) 1/4 (or n ) 1/6). This approximation works well for x e 2; in the range x > 2, where ∆Q rules the course of the background correlation function, no suitable approximation is available at present. [In our earlier work,28 we proposed an expression for extension of the g function beyond the distance of the first minimum, xm, in the form g ) 1 + A exp[-B(x - xm)] cos[π(x - xm)/c] with A ) g(xm) - 1, B ) 1.2(1 - y), and c related to xc for which g(xc) ) 1]. 3.1.1. Hard Sphere Mixtures. Next, we consider the binary mixture of hard spheres with different diameters, σ1 and σ2, with ratios p2 ) σ2/σ1 (and p1 ) σ1/σ1 ) 1) and mole fraction of component (2) x2. We approximate the probe sphere radius, r3, as
2
r3 ) (1 - x2)r1 + x2r2 and
p3 ) (1 - x2)p1 + x2p2
[
(
]
)
βsy3
Sdij/(πσ112) )
1
y 2 - ∆V* (16) 3 (1 - y)3
(
)
]
(19)
2
∑[pi2 + (pi2 - p32) sin θ3i +
2 i)1
p32 sin φ3i + p3(pi + p3)(θ3i - φ3i) cos θ3i] (21) and
Vdii/(πσ113/6) ) pi3 + (pi3 + p33) sin θi - p33 sin φi 3 2 p (p + p3)(θi - φi )cos θi + 2 3 i 1/2(pi + p3)2 cos2 θi[(pi + p3) sin θi - p3 sin φi] (22) 1
2
[
∑ pi3 + (pi3 + p33) sin θ3i - p32 sin φ3i -
2 i)1
3
y ln Yij ) -ln(1 - y) + [3R (∆R* + ∆S*) + ∆V*] + (1 - y) s 3 y y2 βs(∆Q* + 2∆S*) - + 3Rs∆V* + 2 3 (1 - y)2
(pi + p3)(θ3i - φ3i) cos θ3i
Sdii/(πσ112) ) pi2 + (pi2 - p32) sin θi + p32 sin φi + p3(pi + p3)(θi - φi) cos θi (20)
Vdij/(πσ113/6) )
The extension of eq 3 to mixtures reads as
(17)
9V/2 s
p32(pi + p3)(θ3i - φ3i) cos θ3i +
2
]
1/2(pi + p3)2 cos2 θ3i{(pi + p3) sin θ3i - p3 sin φ3i} (23) For angles φi (and φ3i) in the case of mixtures holds true
p3 cos φi ) (pi + p3) cos θi
15508 J. Phys. Chem. C, Vol. 111, No. 43, 2007
Boublı´k
Figure 2. Hard sphere near a planar hard wall. S′, contact circle; z, hard sphere distance from a wall.
3.2. Hard Sphere near a Hard Wall. Behavior of a hard sphere near a smooth planar hard wall can be, in principle, described as a limiting behavior of the hard sphere mixture where the diameter of component 2 tends to infinity. In this case, angle θ2 ) π/2, and the volume of the single large sphere and the corresponding part of the dumbbell compensate themselves in the expression for ∆V. The value of angle θ ) θ1 follows from
θ ) arctan[w/(1 - w2)1/2]
(24)
where w ) z - 0.5 and z denotes the reduced distance of HS from the wall (see Figure 2). For the reduced volume of the combined body, we can write
3π + θ - 2φ cos θ + 22 3 (25) 2 cos2 θ sin θ - sin φ + 2
( (
Vd/(πσ3/6) ) 1 + sin θ - sin φ -
)
Figure 3. Hard sphere in a spherical pore. R, pore radius; γ and γ′, angles.
(In the case of a hard sphere near a hard wall, similarly as for a pure hard sphere system, we need not divide the single differences by r/s , s/s , and V/s ). 3.2.1. Hard Sphere Mixture near a Hard Wall. Similarly as for hard sphere mixtures, we assume that p1 ) 1, p2 ) σ2/σ1, and p3 ) (1 - x2)p1 + xp2; y denotes the packing fraction, y ) F∑xiVi. For differences of the single geometric functionals one obtains
1 2Rdi /σ11 ) (pi + p3)(1 + sin θi) - p3 sin φi 2 π 1 π (p + p3) + θi - 2φi cos θi + (pi + p3) cos θi (31) 4 i 2 4 2∆Ri/σ11
)
and
( ) 1 ) [(p - p ) - (p + p ) sin θ + 2p sin φ + 2 π 1 (p + p )( + θ - 2φ ) cos θ ] (32) 2 2 i
3
i
i
3
3
i
3
i
i
i
i
Similarly,
3π + θ - 2φ cos θ 22 3 2 cos2 θ sin θ - sin φ + (26) 2
∆V/(πσ3/6) ) -sin θ + sin φ +
(
)
(
)
In the case of surface area, its value for the large sphere (with r2 f ∞) differs only by a relatively small circle, S′ (with a radius 2r cos θ), from the corresponding part of the given enlarged hard dumbbell. We can thus consider only S′ in determination of ∆S. Within this approach, we write
π S /(πσ ) ) sin φ + + θ - 2φ cos θ + cos2 θ (27) 2 d
2
(
∆S/(πσ2) ) 1 - sin φ -
)
θ (π2 + θ - 2φ) cos θ + cos(28) 2
For the mean radius, we have
2Rd/σ ) 1 + sin θ - sin φ -
1π + θ - 2φ cos θ + 22 π cos θ (29) 2
(
)
1 2 (p - p32)(1 + sin θi) + 2p32 sin φi + 2 i 1 π p3(pi + p3) + θi - 2φi cos θi + (pi + p3)2 cos 2θi (33) 2 2 Sdi /(πσ112) )
[
(
]
)
1 2 p + p32 - (pi2 - p32) sin θi 2 i π 1 2p32 sin φi - p3(pi + p3) + θi - 2φi cos θi - (pi + 2 2
∆Si/(πσ112) )
[
(
)
]
p3)2 cos2 θi (34) and
{
1 3 (p - p33) - (pi3 + p33) sin θi + 2 i 3 3 π p (p + p3) + θi - 2φi cos θi 2 3 i 2 3 1 2 2 (pi + p3) cos θi (pi + p3) sin θi - p3 sin φi + p3 + 2 2
∆Vi/(πσ113/6) )
(
)
[
2p33
] sin φ } (35) i
If (pi + p3) cosθi > p3, then φi ) 0; for larger θi,
cos φi ) (pi + p3) cos θi/p3
and
2∆R/σ ) -sin θ + sin φ +
1π + θ - 2φ cos θ (30) 22
(
)
holds true. Similarly, as in the case of the hard sphere mixtures, the single differences must be divided by r/s , s/s and V/s
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3.3. Hard Sphere in a Spherical Pore. A description of the behavior of HS in a spherical pore differs from that of a system of hard spheres near a planar hard wall in two aspects: (i) the surface S′ is not a planar circle, but a concave cup of a large sphere, and (ii) thevolume of the enlarged combined hard body includes a spherical segment of radius R and angle γ′ (see Figure 3). If we denote the reduced radius R* ) R/σ, then the inner surface of the mentioned part is equal to S′ ) 2πσ2R*2(1 cos γ). Taking this contribution into account, the geometric functionals Rd and Sd are given by expressions
1 π 1 2Rd/σ ) (1 + cos γ) + sin θ - sin φ - θ + + γ 2 2 2 π 2φ cos θ + R* sin γ - R* (1 - cos γ) (36) 2
(
)
1 Sd/(πσ2) ) (1 - cos γ) + sin φ + 2 π θ + + γ - 2φ cos θ + 2R*2(1 - cos γ) (37) 2
(
)
When we consider the volume of the enlarged combined hard body (ECB), we approximate the part of the volume above the tangent plane to the probe spheres, Vp (by which the ECD of this system differs from that of the system (HS + wall); see Figure 3), by a contribution
)
(
)
In the case of θ < π/3 φ ) 0, for θ > π/3, ECB decomposes into two parts: HS with the cusp and the corresponding opposite part. ∫ f(x) dx and ∫ f2(x) dx now decompose into couples with bounds (φ, θ) and (φ, (π/2 + γ)), similar to the case of the system (HS + planar wall). Angle γ′ in eq 38, which determines the volume of a spherical segment, is related to γ by the expression R* cos γ′ ) [(R* (1/2)) cos γ + (1/2)]. 3.3.1 Mixture of Hard Spheres in a Spherical Pore. The relationships for the geometric functionals of the title system are just simple extensions of those for the HS mixture near a hard wall:
1 (p + p3 cos γi) + (pi + p3) sin θi 2 i π 1 2p3 sin φi - (pi + p3) + θi + γi - 2φi cos θi + 2 2
2Rdi /σ11 )
[
(
)
]
πR* sin γi - 2R* (1 - cos γi) (42) Similarly,
[
(
)
]
4R* (1 - cos γi) (43) 2
3 π V /(πσ /6) ) 1 + sin θ - sin φ - θ + - 2φ cos θ + 2 2 3 2 2 cos θ sin θ - sin φ + + 2 1 2 3 4R* 1 - cos γ′ - sin γ′cos γ′ (38) 2
(
3
(
)
)
(
)
In eq 38, angle γ′ (which determines the volume of a spherical segment) is related to γ by
[(
R* cos γ′ ) R* -
1 1 cos γ + 2 2
)
]
Volume Vp is slightly smaller than the full one; however, differences are negligible, with the exception of the smallest radii, R*, where we employ an approximation cos γapp ) (2 cos γ′ + cos γ)/3. Then
1 2∆R/σ ) (1 - cos γ) - sin θ + sin φ + 2 1 π θ + + γ - 2φ cos θ - R* (1 - cos γ) (39) 2 2
(
)
1 ∆S/(πσ2) ) (1 + cos γ) - sin φ 2 π θ + + γ - 2φ cos θ + 2R*2(1 - cos γ) (40) 2
(
and
)
1 2 (p - p32 cos γi) + (pi2 - p32) sin θi + 2 i π 2p32 sin φi + p3(pi + p3) + θi + γi - 2φi cos θi + 2
2
Then d
(
(
Sdi /(πσ112) )
Vp ) (2π/3)R (1 - cos γ′ - sin γ′ cos γ′/2) 3
3 π θ + - 2φ cos θ 2 2 3 2 2 cos θ sin θ - sin φ + 2 1 4R*3 1 - cos γ′ - sin2 γ′ cos γ′ (41) 2
∆V/(πσ3/6) ) -sin θ + sin φ +
)
Then
1 (p - p3 cos γi) - (pi + p3) sin θi + 2 i π 1 2p3 sin φi + (pi + p3) + θi + γi - 2φi cos θi 2 2 R* (1 - cos γi) (44)
2∆Ri/σ11 )
[
(
]
)
1 2 (p + p32 cos γi) - (pi2 - p32) sin θi 2 i π 2p32 sin γi - p3(pi + p3) + θi + γi - 2φi cos θi 2
∆Si/(πσ112) )
[
(
)
]
4R*(1 - cos γi) (45) Finally,
{
1 3 (p - p33) - (pi3 + p32) sin θi + 2 i 3 2 π p (p + p3) + θi - 2φi cos θi 2 3 i 2 3 1 2 2 (pi + p3) cos θi (pi + p3) sin θi - p3 sin φi + p3 + 2 2 1 2 3 3 2p3 sin φi - 4R* 1 - cos γ′i - sin γ′i cos γ′i (46) 2
∆Vi/(πσ113/6) )
(
}
)
[
(
] )
4. Results 4.1. Distribution Function of Hard Spheres. Radial distribution functions (rdf) g in both the one-component and twocomponent hard sphere systems were studied in the previous
15510 J. Phys. Chem. C, Vol. 111, No. 43, 2007
Figure 4. Radial distribution function g(x) of pure hard spheres at Fσ3 ) 0.8; x ) l/σ.
paper.18 To verify the effect of the new formula for the mean radius, we determine rdf of pure hard spheres at Fσ3 ) 0.8. Comparison of the calculated g(x) with simulation data29 for x ∈ (1, 2) is shown in Figure 4. It is obvious that the method yields fair results at contact and at x ≈ 2. Slightly worse results are obtained at x ≈ 1.5. As discussed above, we ascribed the larger deviations at x ≈ 1.5 to inaccuracy of the approximation for Q at these distances. The considered approximation for Q also determines the range of applicability of the proposed method from a theoretical point of view. (In practice, an extension of g to distances x > 2 is possible by applying an expression proposed previously; cf. ref 28.) Application of this approach with parameters found from the original prescriptions [A ) -0.40, B ) 0.69, c ) 0.60] and from other sets of parameters is shown as dashed and dotted lines in Figure 4, too.
Boublı´k
Figure 5. Density profile F(z) of a hard sphere near a planar hard wall. Bulk density Fb ) 0.755. Simulation data and IE results (b and solid line) given in Figure 7 of Snook and Henderson12 are compared with curve from eq 3 (dashed-dotted line).
5. Hard Sphere near a Hard Wall Determination of the distribution function of a hard sphere near a planar hard wall and consequently the density profile F(z) ) Fb g(z) (where Fb denotes the bulk fluid density) by applying eqs 26-30 is straightforward. In Figures 5 and 6, we compare theoretical predictions of density profiles with simulation data of Snook and Henderson12 for two Fb values, Fb ) 0.755 and 0.91. In addition to simulation data, we present in Figure 5 the theoretical results of Snook and Henderson calculated from the OZ integral equation (IE) employing the GMSA or PY approximation (solid and broken curves on the whole interval x ∈ (0, 1.5)). (The dotted line depicts application of the extension formula with A ) -0.70, B ) -0.73, c ) 0.49). One can conclude that our theoretical results compare well with the simulation data and IE curves, even at the highest densities. On the other hand, the approximation proposed in ref 28 yields only a qualitative prediction. We also reached simialr conclusions for the newly studied system at Fb ) 0.8873. In Figure 7, we present simulation data and theoretical DFT curve (FMT-CS variant) taken from Figure 1 of ref 16, together with results from the present approach. One can see that larger deviations are found for x ∈ (1.2, 1.5) for which even DFT disagrees with the simulation results. Next, the binary mixture of hard spheres near a hard wall is considered. The above author16 considered the binary HS mixture (with the aspect ratio σ2/σ1 ) 2, mole fraction x2 )
Figure 6. Density profile F(z) of a hard sphere near a planar hard wall. Bulk density Fb ) 0.91.
0.2, and bulk density Fb ) 0.3209); he presented Monte Carlo data for F1(z) and F2(z) together with theoretical predictions from DFT. We apply our method to determine the density profiles of both components in the most important range of the reduced distances x ∈ (0.5,1.5). Our results are depicted in Figures 8 and 9. With the exception of the part for x ∈ (1.3, 1.5), they are practically indistinguishable from those found in ref 16. On the other hand, the extension formula yields rather poor values (see Figure 8). 5.1. Hard Sphere in Spherical Pore. The system (HS + spherical pore) can be considered as a mere modification of the treatment of the (HS + planar hard wall) system. Here, the wall is curved, and instead of the contact circle, we have to consider a contact spherical cup. (In the limit of very large radius, R, the relationships for all the geometric functionals converge to those for a planar wall.) We apply our method to the system studied by Zhou and Stell;11 namely, HS in a spherical pore of R* ) 6.5 (characterized in ref 11 by λ ) 1/13) at the bulk density Fb ) 0.86. The most important part of the density profile close to the wall, x ∈ (0.5, 1.5), is depicted
Structure of Hard Spheres near Hard Wall, in Pore
Figure 7. Density profile F(z) of a hard sphere near a planar hard wall. Bulk density Fb ) 0.8873. Simulation data and DFT (FMT-CS) results (b and solid line) given in Figure 1 of ref 16 are compared with dependence from eq 3 (dashed-dotted line).
Figure 8. Density profile F1(z) of small hard sphere in the HS mixture near a planar hard wall. Bulk density Fb ) 0.3209, σ2/σ1 ) 2, and mole fraction x2 ) 0.2. The notations are the same as in Figure 7.
J. Phys. Chem. C, Vol. 111, No. 43, 2007 15511
Figure 9. The same as in Figure 8 for F2(z) of big hard sphere.
Figure 10. Density profile F(z) of a hard sphere in a spherical pore of radius R/σ ) 6.5 at bulk density Fb ) 0.86.
in Figure 10. Fair agreement of the theory with pseudoexperimental data11 is obvious. 5.2. Hard Sphere in Cylindrical Pore. Because spherical and cylindrical pores in the framework of our approach do not differ substantially, we apply the variant proposed for HS in the spherical pore to a cylindrical one. We consider a pore of the reduced radius R* ) 4 and Fb ) 0.28731. Comparison of the theoretical dependence with MC data30 is shown in Figure 11. It appears that prediction for distances close to the contact point is slightly above the pseudoexperimental data. This trait becomes more pronounced with the increasing density. (If we consider R* ) 8, better agreement is found.) 6. Conclusion In this paper, we present some applications of the “geometric method” to characterize the structure of inhomogeneous hard sphere systems and comparison of its results with simulation data and predictions of the more complex methods: solution of the integral OZ equation and density functional theory. The latter theories (IE and DFT in the recent versions) are quite
Figure 11. Density profile F(z) of a hard sphere in a cylindrical pore of radius R/σ ) 4 at bulk density Fb ) 0.28731.
accurate on the large interval of distances (with exception of extremely dense systems). However, application of the latter methods to determine distribution functions or density profiles
15512 J. Phys. Chem. C, Vol. 111, No. 43, 2007 is often quite demanding (as the computer time and input data for the corresponding homogeneous systems is concerned). At present the “geometric method” is limited to the range of distances x ∈ (0, 2), yielding, thus, the direct correlation function, c, in the whole range x e 1 and the most important part of the distribution function. Determination of both of these functions (c and g) is straightforward and extremely simple both for pure fluids and mixtures. In this study, we focused our effort on the detailed derivation of the expressions for the geometric functionals of the enlarged hard dumbbell and other enlarged combined hard bodies. The expressions for geometric functionals enable application of the method proposed to determine the background correlation function and, consequently, g or F(z) from the residual chemical potentials of HS and the corresponding enlarged hard dumbbell (generally ECB). We propose a novel method to evaluate the mean radius from expressions for the surface areas of different partssconvex and saddlesof the enlarged combined hard bodies (including EHD). The method makes it possible to determine R* of variety of ECBs. With the new prescription, the considered “geometric method” yields a fair description of g(x) or F(z) in the most important range x ∈ (1, 2) or z ∈ (0.5, 1.5) both for pure HS fluid and mixtures. It is worth noting that characterization of the multicomponent mixtures does not bring additional complications, because the composition of mixtures (of reasonably different HSs) affects mainly the diameter of the probe hard sphere. Insufficient knowledge of the geometric quantity Q (for which we employed in all the studied cases the approximation Q* ) S* (R*2/S*)1/4) limits so far the use of the method to distances of x e 2. In practical applications, to determine g for x > 2, an empirical extension (cf. ref 28) can be considered, however, improved prescriptions (in comparison with those for homogeneous fluids) for its parameters must be formulated. Acknowledgment. This work has been financially supported by the Grant No. IAA 400 720 710 of the Grant Agency of the Czech Academy of Sciences.
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