Langmuir 2009, 25, 3577-3583
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Resolving the Inconsistency between Classical Diffusion and Adsorption G. L. Aranovich* and M. D. Donohue* Department of Chemical & Biomolecular Engineering, The Johns Hopkins UniVersity, Baltimore, Maryland 21218 ReceiVed NoVember 7, 2008. ReVised Manuscript ReceiVed December 16, 2008 The inconsistency between density profiles of fluids near surfaces and predictions of classical diffusion model is analyzed. A new diffusion equation and its solutions are proposed to reconcile adsorption behavior with predictions of the diffusion equation at the equilibrium limit. The classical phenomenological model of diffusion in fluids is based j , where V j is the characteristic velocity. on the concepts of the mean-free-path, λ, and diffusion coefficient, D ) (1/3)λV Using the limit of λ f 0 in the flux term gives classical diffusion equation, that is, Fick’s law. However, imposing j and λ, to one parameter, D ) (1/3)λV j . This is equivalent the limit of λ f 0 reduces two independent parameters, V j t, to only one length scale, (Dt)1/2, where t is time. Since the λ to reducing two independent length scales, λ and V length scale determines density profiles near surfaces, the classical diffusion model “loses” adsorption phenomena after applying the limit of λ f 0 and classical solutions are in conflict with adsorption at surfaces. Here, we show that relaxing the requirement of λ f 0 by using an exact (finite-difference) functional for the flux term fixes the problem. Solution of the finite-difference diffusion equation is analyzed. This solution allows boundary conditions consistent with density profiles in fluids near surfaces.
Inconsistency between Density Profiles of Fluid near Surfaces and Classical Diffusion Profiles at Equilibrium Limit Molecular-thermodynamic theories have been used extensively to model adsorption behavior of fluids1-3 and to analyze density profiles near solid surfaces.4-8 There are a number of important issues related to these profiles, including analysis of the Gibbs adsorption isotherms.9 Though there are fundamental aspects of adsorption behavior which are understood well,1-5 there are others that still remain unresolved. One of these problems is an inconsistency between the density distribution of fluids near surfaces and the solution of the phenomenological diffusion equation in the equilibrium limit.10,11 The density distribution of adsorbate near a surface can be found from rigorous theories (such as density functional theory, DFT3) or from simulations, and one might expect similar results from the solution of diffusion equations in the equilibrium limit. However, the classical diffusion theory fails to predict adsorption profiles. This paper discusses the reasons for this failure and ways to fix it. To illustrate the inconsistency between classical diffusion and adsorption, consider a semi-infinite fluid at equilibrium with a solid surface as shown in Figure 1. Fluid molecules interact with each other and with the surface. Assume that adsorbate-adsorbent (1) Hill, T. L. Statistical Mechanics, Principles and Selected Applications; McGraw-Hill: New York, 1956. (2) Steele, W. A. The Interaction of Gases with Solid Surfaces; Pergamon Press: Oxford, 1974. (3) Nicholson, D.; Parsonage, N. G. Computer Simulation and the Statistical Mechanics of Adsorption; Academic Press: London, 1982. (4) Ono, S.; Kondo, S. Molecular Theory of Surface Tension in Liquids; Springer: Gottingen, 1960. (5) Rowlinson, J. S.; Widom, B. Molecular Theory of Capillarity; Clarendon Press: Oxford, 1982. (6) Aranovich, G. L.; Donohue, M. D. Langmuir 2003, 19, 3822. (7) Aranovich, G. L.; Donohue, M. D. Phys. ReV. E 1999, 60, 5552. (8) Aranovich, G. L. J. Colloid Interface Sci. 1991, 141, 30. Aranovich, G. L. Langmuir 1992, 8, 736. (9) Donohue, M. D.; Aranovich, G. L. AdV. Colloid Interface Sci. 1998, 76-77, 137. (10) Chen, Y.; Aranovich, G. L.; Donohue, M. D. J. Colloid Interface Sci. 2007, 307, 34. (11) Aranovich, G. L.; Donohue, M. D. J. Phys. Chem. B 2007, 111, 9530.
Figure 1. Adsorption from a semi-infinite fluid on a solid surface. At equilibrium, the fluxes to and from the surface compensate each other.
interactions are short-range and the field of the adsorbent does not go beyond the first layer of adsorbed molecules. The density profile is then a result of correlations due to adsorbate-adsorbate interactions (repulsive, attractive, or both). Depending on initial conditions, molecules can diffuse to and from the surface. At equilibrium, the fluxes to and from the surface compensate each other, and this results in a certain density distribution, F(x), near a surface. Figure 2 illustrates the typical density profile for hard spheresnearahardwall.3 AsseenfromFigure2,molecule-molecule and molecule-wall correlations result in a nonconstant density distribution near the surface. The classical diffusion equation for a fluid is written as12,13
∂F ∂ ∂F D ) ∂t ∂x ∂x
( )
(1)
where F(x,t) is the density profile and D is the diffusion coefficient. At steady-state, ∂F/∂t ) 0 and eq 1 becomes
10.1021/la803703h CCC: $40.75 2009 American Chemical Society Published on Web 02/17/2009
3578 Langmuir, Vol. 25, No. 6, 2009
∂ ∂F D )0 ∂x ∂x
( )
AranoVich and Donohue
(2)
For constant D, the general solution of this equation is F ) C1x + C2, where C1 and C2 are arbitrary constants, and the equilibrium limit is a particular case of steady state (with zero net flux). The only way to satisfy condition F(∞) ) F∞ is to put C1 ) 0 and C2 ) F∞; this does not allow for a density profile like that in Figure 2. Allowing D to depend on position can change the solution. For example, for fluids, D usually is expressed as13,14
1j D) V λ 3
(3)
j is the characteristic velocity of molecules and λ is the where V mean free path (which depends on density). The classical approximation for the density dependence of λ for hard spheres is that15-17
λ)
σ
√2πF
(4)
where σ is the size (diameter) of a molecule. Note that the reduced density, F, is defined using σ and the number of molecules per unit of volume, n:
F ) nσ3
(5)
Plugging eqs 3 and 4 in eq 2 results in
∂ K ∂F )0 ∂x F ∂x
( )
(6)
where
K)
jσ V 3π√2
(7)
For constant K, solution of eq 6 can be written as
F ) C2 exp(C1x)
(8)
with two arbitrary constants, C1 and C2. However, the condition F(∞) ) F∞ can be met only if C1 ) 0 and C2 ) F∞; this does not allow adsorption either. For a fluid in a symmetric, slitlike pore, the general solution of eq 2 allows only F ) const. This is because both F ) C1x + C2 (for D ) const) and F ) C2 exp(C1x) (for D defined by eqs 3 and 4) can be symmetric only if C1 ) 0 which results in F ) const. These elementary examples indicate that the classical diffusion model is not consistent with density profiles near
Figure 3. Classical model for diffusion in fluid.
surfaces. In other words, solutions of classical diffusion equations are in conflict with adsorption at the surfaces. Even in the case of hard spheres near a hard wall, there is a nonconstant density distribution near the surface3 and eq 2 cannot predict this. Various approaches have been developed to model density profiles near surfaces. These include (a) more complex forms of evolution equations written in terms of thermodynamic functions, such as the chemical potential, µˆ, and entropy, Sˆ (ref 14, chapter 7), and (b) rigorous theories such as those based on integral equations and density functionals.1-5,13 Evolution equations in terms of µˆ or Sˆ for a molecule-vacancy “binary” system can be written in a general way that takes into account adsorption at surfaces (boundary conditions); however, deriving density profiles would require presenting µˆ or Sˆ as functionals of the density distribution and would become an integral equation theory. These rigorous theories have well-known advantages, but they also have difficulties. In particular, the application of rigorous theories is limited due to mathematical complexities and lack of analytical solutions even for the simplest cases.13 Therefore, phenomenological approaches, such as the classical diffusion model, still are useful. In this paper, we discuss the reason of why the classical diffusion equation is inconsistent with rigorous theories and, generally, with adsorption behavior. To our knowledge, this issue has never been analyzed adequately. Hence, one must question whether the inability of classical diffusion theory to capture the essential physics near surfaces can be fixed in the framework of the same physical model.
Why Diffusion Theory Does Not Predict Density Profiles near Surfaces at Equilibrium Figure 3 illustrates the classical model of mass transport in a fluid [ref 15, p 88]. In this model, the number of molecules j n(x - λ,t), crossing some surface, S, from left to right is (1/6)V and the number of molecules crossing this surface from right to j n(x + λ,t). Therefore, the diffusion flux, F(x,t), is left is (1/6)V
Figure 2. Density profile for hard spheres of diameter σ near a hard wall from Monte Carlo simulations at F(∞) ≈ 0.57 [ref 3, p 272]. Here, x ) 0 corresponds to the centers of hard spheres sitting on the wall.
(12) Seizo Ito Diffusion Equations; American Mathematical Society: Providence, RI, 1992. Bass, R. F. Diffusions and Elliptic Operators; Springer: New York, 1998. (13) Present, R. D. Kinetic Theory of Gases; McGraw-Hill: New York, 1958; section 3.3. (14) Cussler, E. L. Diffusion: Mass Transfer in Fluid Systems; Cambridge University Press: Cambridge, 1997. (15) Boltzmann, L. Basic Equation for the Transport of Any Quantity by the Molecular Motion. Lectures on Gas Theory, Part I; University of California Press: Berkeley, Los Angeles, 1964; paragraph 11. (16) Cutchis, P.; van Beijeren, H.; Dorfman, J. R. Am. J. Phys. 1977, 45, 970. (17) Shutler, P. M. E.; Springham, S. V.; Martinez, J. C. Eur. J. Phys. 2006, 27, 923.
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1 j[ F(x, t) ) V n(x - λ, t) - n(x + λ, t)] 6
(9)
Note that the nature of the flux is nonlocal,18-20 that is, it depends on the densities at x ( λ, not on the density (or density gradient) at x. However, in the literature on the classical diffusion model (including Boltzmann’s derivation15), the finite difference n(x λ,t) - n(x + λ,t) is replaced by the derivative:
n(x - λ, t) - n(x + λ, t) ≈ -2λ
∂n ∂x
(10)
{
}
∂ 1 j[ V F(x + λ, t) - F(x - λ, t)] ) 0 ∂x 6
(13)
and the steady-state solution has to satisfy the following equation:
F(x + λ) - F(x - λ) ) const ) C
(14)
Comparison with eq 9 indicates that at equilibrium C ) 0 and there is no net flux. Then, in the equilibrium limit
and eq 9 is changed to
1 j ∂n(x, t) F(x, t) ) - V λ 3 ∂x
(9a)
Plugging this in the mass balance equation
∂n ∂F )∂t ∂x
(11)
results in eq 1 with F and D defined by eqs 5 and 3, respectively. Approximation 10 is exact only in the limit of λ f 0. Therefore, j finite with λ f 0 results in the unphysical keeping D ) (1/3)λV j requirement that V f ∞ (which was discussed in earlier publications18,19). Also, imposing λ f 0 reduces two independent j and λ, to one parameter, D ) (1/3)λV j . This is parameters, V j t, equivalent to reducing two independent length scales, λ and V 1/2 to only one length scale, (Dt) . However, the density profile near a surface is determined by the length scale of molecular size, σ. This is known from rigorous theories5 and from simulations of density profiles, for example, for hard spheres near a hard wall.3 From eq 4, σ ) λ2πF. Therefore, the σ-scale is equivalent to the λ-scale at fixed F, and the diffusing fluid “knows” the sizes of molecules from its rate of collision. This mechanism is embedded in the original physical model;13-15 however, after applying the limit of λ f 0 in the flux term, the model loses information about molecular sizes and about the density distribution near a surface. Note that using Fokker-Plank terms in the diffusion equation12 (adding interactions with adsorbent as an external field) does not fix this problem because these terms do not incorporate adsorbate-adsorbate correlations and do not have the “σ-scale”. As will be shown, the direct way to obtain nonconstant density profiles near surfaces is by using the finite-difference operator (9) instead of its differential approximation (10). Combining eqs 5, 9, and 11 results in
{
equation, but eq 1 fails to do this. Here, we show that eq 12 is able to predict nonconstant profiles near surfaces. At ∂F/∂t ) 0, eq 12 can be written in the following (steadystate) form:
}
∂F(x, t) ∂ 1 j[ ) V F(x + λ, t) - F(x - λ, t)] ∂t ∂x 6
(12)
The mathematical aspects of solving eq 12 include difficult problems such as getting analytical solutions and rigorous analysis for various boundary and initial conditions. There is no general mathematical theory for this kind of differential finite-difference equation. Here, we have a more modest task and focus on relaxing the unphysical requirement of λ f 0 which allows solutions of the diffusion equation consistent with density profiles near surfaces.
Finite-Difference Diffusion Equation Consistent with Density Profiles near Surfaces As seen from eqs 9-11, the diffusion equation represents a mass balance in a general form. Density profiles near surfaces should be particular cases of the solution to the mass balance (18) Aranovich, G. L.; Donohue, M. D. Phys. A 2007, 373, 119. (19) Aranovich, G. L.; Donohue, M. D. Mol. Phys. 2007, 105, 1085. (20) Cattaneo, C. Atti. Semin. Mat. Fis. UniV. Modena 1948, 3, 83.
F(x + λ) - F(x - λ) ) 0
(15)
The mass balance represented by eq 15 is less restrictive than eq 2, and this allows density profiles at surfaces other than F ) F∞. To analyze eq 15, consider Fourier techniques representing eigenfunctions in the form:
Fk ) Ak cos(ωkx)
(16)
Plugging eq 16 in eq 15 gives
cos ωk(x + λ) - cos ωk(x - λ) ) 0
(17)
Since cos R - cos β ) 2sin[(R + β)/2] sin[(β - R)/2], eq 17 can be transformed to
sin(ωkλ) sin(ωkx) ) 0
(18)
Therefore, sin(ωkλ) ) 0 and eigenvalues are
ωk )
kπ λ
(19)
Then, the density profile satisfying eq 15 is ∞
F(x) )
∑ Ak cos kπx λ
(20)
k)0
Note that values of ωk ) kπ/λ can be eigenvalues only if λ(x) ) const. This is an analogue of D ) const in the classical diffusion equation, though there are physical situations where this can be a meaningful approximation.14 However, our goal is not getting a general solution for eq 15; it is rather limited to eliminating the inconsistency of the classical diffusion model, so there can be nonconstant density profiles near surfaces. Here, we will show that solution 20 can be consistent with any profile imposed by surfaces. Consider a fluid in a symmetric, slitlike pore as shown in Figure 4. In this figure, walls are at planes A and E, plane C is in the middle where x ) 0, and planes B and D are at the distance of λ from the walls, so the width of this pore is greater than or equal to 2λ. Except for some special cases (such as proximity to a critical point), the influence of the walls is limited to a distance less than λ, so this influence is between planes A and B and between planes D and E. Mathematically, eq 20 is a solution of eq 15 for x between planes B and D; otherwise, either x + λ or x - λ is beyond the walls of the pore. Since eq 20 defines a continuous function with an arbitrary coefficient, Ak, consider continuation of eq 20 beyond the planes B and D, up to the walls, A and E. This continuation has to coincide with the profiles near the surfaces. In Figure 4, the distance between planes B and D, dBD, can vary. When dBD ) 0, planes B and D coincide and the width of (21) Aranovich, G. L.; Donohue, M. D. J. Colloid Interface Sci. 1997, 189, 101.
3580 Langmuir, Vol. 25, No. 6, 2009
AranoVich and Donohue
Figure 4. Fluid in a symmetric, slitlike pore.
Figure 5. Fluid near a reflective surface at 0 < x < λ.
the pore is 2λ. For this case, F(x) can be designated as F0(x) and function 20 can be viewed as an expansion of F0(πx/λ) in Fourier (cosine) series over the interval from x ) -λ to x ) +λ. Therefore, at k g 1, coefficients, Ak, are
Ak )
1 π
∫-ππ F0(ξ) cos kξ dξ
(21)
∫-ππ F0(ξ) dξ
(22)
and
A0 )
1 2π
Therefore, at dBD ) 0, function 20 coincides with an arbitrary profile, F0(x), if coefficients, Ak, are determined by eqs 21 and 22. Note that A0 is the average density in the pore.
Plugging eq 16 in eq 23 gives cos ωk(x + λ) - cos ωk(λ x) ) 0 which coincides with eq 17, because cos ωk(x - λ) ) cos ωk(λ - x). Therefore, eq 20 is a solution for both eq 15 and eq 23, and continuation of function 20 over the range of 0 < x < λ corresponds to the mass balance near the surface. The reflecting boundary condition given by eq 23 describes a simplified model, and real surface conditions can be significantly more complex. However, our goal here is not to derive a model for adsorption or density profile near a surface. Rather, the goal is to bridge adsorption behavior to the phenomenological diffusion model in the equilibrium limit; this is possible by using the finite-difference functional for the flux term instead of the classical approximation (eq 10).
Mass Balance near Surfaces Mathematically, eq 20 is a general solution to diffusion eq 15 in the range of x where the distance to a wall is not less than λ. Continuation of this function (20) up to the walls allows imposing any boundary conditions by using eqs 21 and 22. To illustrate that such a continuation is not only fitting F0(x) but also represents a mass balance near the surface, consider a fluid (such as hard spheres3) near a reflective wall22 as shown in Figure 5. In Figure 5, plane A with x ) 0 is the surface of reflective wall, a variable coordinate x, with x < λ, defines plane C, and plane D is at the distance of λ from plane C. Plane B is at the distance of λ - x from the wall. When molecules move to the left from plane B and then move to the right after reflection from the wall (from plane A), the traveling distance is also λ. Therefore, planes B and D are at the same traveling distance from the plane C (distance λ). Plane D of Figure 5 plays the role of plane S′′ of Figure 3, and plane B of Figure 5 is an analogue of plane S′ of Figure 3. Therefore, to modify the classical calculation illustrated in Figure 3 in proximity to the surface as shown in Figure 5, mass balance at point x should be determined by F(x + λ) - F(λ - x) instead of F(x + λ) - F(x - λ). This gives instead of eq 15
F(x + λ) - F(λ - x) ) 0
Approximation for Small dBD As shown previously, eqs 20-22 give a solution to both eq 15 and eq 23 at dBD ) 0. Now, consider nonzero dBD, but suppose that it is small compared to λ. Assuming that the influence of the surface is limited to a distance less than λ, conditions at the boundaries can be written as
(
F(x) ) F0 x -
where (dBD/2) < x < (dBD/2) + λ, and
(
F(x) ) F0 x +
(22) Boltaks, B. I. Diffusion in Semiconductors; Academic Press: New York, 1963; Chapter IV.
dBD 2
)
(24)
)
(25)
where -λ - (dBD/2) < x < (-dBD/2). Then, for x > 0, eqs 20 and 24 give ∞
F(x) )
∑ Ak cos
k)0
kπ(x - dBD ⁄ 2) λ
(26)
and, for x < 0, eqs 20 and 25 result in
(23)
where 0 < x < λ and where continuation of function 20 is considered.
dBD 2
∞
F(x) )
∑ Ak cos
k)0
kπ(x + dBD ⁄ 2) λ
(27)
where the coefficients Ak are defined by eqs 21 and 22. Equations 26 and 27 also can be written in the following (unified) form:
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Langmuir, Vol. 25, No. 6, 2009 3581
Figure 6. Density profile between walls for F0(ξ) ) [exp(ξ) + exp(-ξ)]/ 2 at λ ) 10 and various dBD: 0 (1), 2 (2), and 4 (3). ∞
F(x) )
∑ Ak cos
k)0
kπ(|x| - dBD ⁄ 2) λ
(28)
While we considered the cases of x > 0 and x < 0 separately, the validity of this solution around the point x ) 0 needs to be verified by plugging solution 28 into eq 15. This gives: ∞
[
∑ Ak cos
k)0
kπ(|x + λ| - dBD ⁄ 2) λ kπ(|x - λ| - dBD ⁄ 2) ) 0 (29) cos λ
]
which can be rewritten as ∞
[
kπdBD kπdBD kπ|x + λ| + sin sin 2λ λ 2λ k)0 kπdBD kπdBD kπ|x - λ| kπ|x - λ| )0 cos - sin sin cos λ 2λ λ 2λ (30)
]
Recall that dBD is small compared to λ. Therefore, in eq 30, one can neglect terms containing sin(kπdBD/2λ) which is reasonable if terms with small k dominate. This gives instead of eq 30
∑ A [cos kπ|xλ+ λ| - cos kπ|xλ- λ| ]cos k
k)0
seen from Figure 6, increasing dBD results in growing error at x ) 0. This is because solution 28 is approximate requiring small dBD. Profiles in Figure 6 illustrate that this finite-difference diffusion model, unlike the classical model, can exhibit nonconstant density profiles near walls. Generally, eq 20 can predict more sophisticated profiles, including oscillations and multiple solutions due to phase transitions in adsorbed layers. However, in this paper, the goal is more modest, that is, to show that the finite-difference diffusion equation does not have conflicts with boundary conditions near surfaces.
Hard Molecules between Hard Walls
∑ Ak cos kπ|xλ+ λ| cos
∞
Figure 7. Density profile for hard discs between hard walls at a bulk density of 0.75. Shown are predictions of eq 20 (first 50 terms) for λ ) 4.5 and coefficients Ak determined by numerical integration from eqs 21 and 22; solid circles are Monte Carlo simulations.23
kπdBD )0 2λ
(31)
Since cos R - cos β ) 2sin[(R + β)/2] sin[(β - R)/2], eq 31 can be transformed to ∞
∑ Ak[2sin kπ(|x - λ|2λ+ |x + λ|) sin kπ(|x - λ|2λ- |x + λ|) ]
k)0
cos
kπdBD ) 0 (32) 2λ
Note that either |(|x - λ| + |x + λ|)/2λ| ) 1 or |(|x - λ| - |x + λ|)/2λ| ) 1. Therefore, either sin[kπ(|x - λ| + |x + λ|)]/2λ ) 0 or sin[kπ(|x - λ| - |x + λ|)]/2λ ) 0, which proves equality 32 and proves that eq 28 is solution of eq 15. Since function 28 also satisfies boundary conditions 24 and 25, there is no inconsistency between the diffusion model and density profiles near surfaces. Figure 6 illustrates density profiles predicted by eq 28 at λ ) 10, and various dBD: 0 (1), 2 (2), and 4 (3). For this illustration, we chose F0(ξ) ) (eξ + e- ξ)/2 which gives, from eqs 21 and 22, A0 ) (eπ - e-π)/2π and Ak ) (- 1)k(eπ - e-π)/π for k g 1. As
Hard molecules, such as hard discs and hard spheres, between hard walls are ideal examples of (adsorbate) fluid with no external field from the adsorbent. Despite the fact that there is no external field, there are nonconstant density distributions due to molecule-molecule and molecule-surface correlations. Results from Monte Carlo simulations for these systems indicate significant oscillations in the density distributions between walls.3 In Figure 7, the points represent Monte Carlo simulations obtained by Snook and Henderson23 for hard discs between hard walls at a bulk density of 0.75. As seen from Figure 7, the density near the walls (at x/σ ) (4.5) is about five times larger than that at the middle of the pore, and, obviously, the density profile is nothing like the constant density profile predicted by the classical model (eq 2). Figure 7 also shows predictions of eq 20 (with the first 50 terms) for λ ) 4.5σ and coefficients Ak determined by numerical integration from eqs 21 and 22. To calculate coefficients Ak of eq 20, numerical integration of eqs 21 and 22 was performed, using points from the Snook and Henderson simulations.23 Ideally, using data for F0(ξ) in eqs 21 and 22 should give accurate F(x) because eq 20 is a (cosine) Fourier transform of function F(x). However, in Figure 7, there is significant scatter due to numerical errors coming from (a) using discrete points from the Monte Carlo simulations instead of continuous F0(x); (b) errors in numerical integration in eqs 21 and 22; and (c) truncation of serious in eq 20 to 50 terms. Note that the errors in Figure 7 are not limited to numerical scatter from the calculations. As mentioned earlier, eq 16 represents eigenfunctions only if λ(x) ) const. Moreover, we assumed that λ ) 4.5σ, which makes the width of the pore equal (23) Snook, I. K.; Henderson, D. J. Chem. Phys. 1978, 68, 2134.
3582 Langmuir, Vol. 25, No. 6, 2009
AranoVich and Donohue ∞
F(x + x0) ) A0 +
∑ Ak cos
k)1
kπ(x + x0) λ
(38)
which can be transformed to ∞
F(x + x0) ) A0 +
kπx + B*k sin ∑ [A*k cos kπx λ λ ]
(39)
k)1
where A*k ) Ak cos(kπx0/λ) and B*k ) -Ak sin(kπx0/λ).
Empirical Use of eq 36 at λ ) λ(x)
Figure 8. Dependence of F on x/σ. The solid line represents predictions of eq 39 with A0 ) 0.05 and A1 ) 0.8. The solid circles are predictions of the Ono-Kondo model21 for a density of 0.05, assuming hexagonal structure and energies of molecule-molecule and molecule-surface interactions being -0.2kBT and -2.8kBT, respectively.
2λ; this simplifies calculations because eqs 20-22 become a (cosine) Fourier transform of F(x). In general, Figure 7 illustrates that the oscillating density profile, observed in the Monte Carlo simulations,23 can be consistent with eq 15 in the equilibrium limit. Obviously, this (oscillating) density distribution does not satisfy classical diffusion eq 2. So, diffusion eq 12, unlike the classical diffusion eq 1, is consistent with adsorption behavior of hard molecules between hard walls.
For Bk ) 0, eq 36 gives eq 20, coinciding with the oscillating part of the DFT density profiles (see, e.g., equation 7.55 in ref 3 regarded by Nicholson and Parsonage as the best analytical approximation for density profiles in fluids near a surface). Trivial solution F ) F∞ (which is the only solution of the classical eq 2) is a particular case of series 20 at A0 ) F∞ and Ak ) Bk ) 0 and for all k g 1. The coefficients, Ak and Bk, of the series in eq 36 allow a significant degree of freedom compared to the solution of eq 2. In fact, eq 36 is consistent with various types of adsorption behavior, including oscillations of profiles near surfaces, multilayer adsorption, and phase transitions in layers. To illustrate the character of predictions from eq 36, plug approximation 4 in eq 36 for λ ) λ(x), which gives ∞
F(x) )
∑
k)0
Generalization of eq 20 Following the logic of Fourier techniques, eq 20 can be generalized to include sine terms because eigenfunctions
Fk ) Bk sin(ωkx)
(33)
[
Ak cos
k√2π2Fx k√2π2Fx + Bk sin σ σ
sin(ωkλ) cos(ωkx) ) 0
(35)
which is satisfied if eigenvalues, ωk, are defined by eq 19. Therefore, a more general form for the solution of eq 15 can be written as ∞
F(x) ) A0 +
kπx + Bk sin ∑ (Ak cos kπx λ λ )
(36)
k)1
Equation (36) is a more general solution of diffusion eq 15. Though it goes beyond the scope of this paper, eq 15 can be used for analysis of adsorption in asymmetric slitlike pores. For dBD ) 0, the coefficients, Bk, that determine the density profile F0(x) in an asymmetric pore are
Bk )
1 π
∫-ππ F0(ξ) sin kξ dξ
F(x) ≈ A0 + A1 cos
(34)
Since sin R - sin β ) 2sin[(R - β)/2] cos[(R + β)/2], eq 34 can be transformed to
√2π2Fx σ
(41)
Figure 8 shows dependence of F on x/σ. In this figure, the solid line represents predictions of eq 39 with A0 ) 0.05 and A1 ) 0.8. The solid circles are predictions of the Ono-Kondo lattice model21 for a density of 0.05, assuming hexagonal structure and energies of molecule-molecule and molecule-surface interactions being -0.2kBT and -2.8kBT, respectively. As seen from Figure 8, eq 41 provides a qualitatively reasonable density profile near a surface. To analyze possible phase transitions in adsorbed layers (with stepped isotherms),7,9 assume that coefficients Ak are functions of the bulk density, F∞, and temperature, T. Then, taking the derivative in eq 40 with respect to F∞ gives
[
∞ ∂Ak ∂F k√2π2Fx ) cos ∂F∞ k)0 ∂F∞ σ
∑
Ak
(37)
and eqs 21 and 22 define coefficients Ak. Note that eq 20 is equivalent to eq 36 if the coordinate origin is shifted by a constant value. For example, using x + x0 instead of x in eq 20 gives
(40)
Consider a fluid in equilibrium with a surface which is at x ) 0. Assume that the most significant terms are at k ) 0 and k ) 1. However, term B1 sin(2π2Fx/σ) is less significant compared to A1 cos(2π2Fx/σ) near x ) 0. Then
also are solutions of eq 15. Plugging eq 33 in eq 15 gives
sin ωk(x + λ) - sin ωk(x - λ) ) 0
]
k√2π2x ∂F k√2π2Fx sin + σ ∂F∞ σ
]
∂Bk k√2π2x ∂F k√2π2Fx k√2π2Fx sin cos (42) + Bk ∂F∞ σ σ ∂F∞ σ which gives
Classical Diffusion and Adsorption Inconsistency ∞
∂F ) ∂F∞
∂A
√2π2Fx ∂Bk k√2π2Fx sin + σ ∂F∞ σ
∑ ∂F∞k cos k
k)0
∞
(
)
√2π2x k√2π2Fx k√2π2Fx 1+ k Ak sin - Bk cos ∑ σ k)1 σ σ (43)
Equation 43 indicates that ∂F/∂F∞ has singularities as ∞
1+
Langmuir, Vol. 25, No. 6, 2009 3583
(
)
√2π2x k√2π2Fx k√2π2Fx k Ak sin )0 - Bk cos ∑ σ k)1 σ σ (44)
Leaving only terms with k ) 1 in eq 44 gives
√2π2x √2π2Fx √2π2x f(x, F) ) 1 + A1 sin B1 σ σ σ √2π2Fx cos ) 0 (45) σ Equation 45 illustrates that phase transitions are possible at various distances to the surface, x. Figure 9 gives f(x,F) from eq 45 as a function of x/σ for A1 ) B1 ) 0.8 and F ) 0.5. As shown by Figure 9, there are (quasi)periodic singularities which can be interpreted as two-dimensional phase transitions in a multilayered adsorbate with steps in the isotherms.6-9
Conclusion Solutions of the classical phenomenological diffusion model (i.e., eq 1) are not consistent with density profiles near surfaces. In particular, at steady state, one-dimensional Fick’s equation is a second order differential equation whose general solution has two arbitrary constants to be found from boundary conditions. However, any choice of the arbitrary constants is in conflict with the density oscillations observed near surfaces.
Figure 9. Dependence of f(x,F) from eq 45 on x/σ for A1 ) 1, B1 ) 0.3, and F ) 0.5.
This study indicates that this fundamental problem arises from the classical approximation 10 which implies the limit of λ f 0 in the flux term. To fix the problem, we propose using the more correct finite difference n(x - λ,t) - n(x + λ,t) in the flux term. This results in the finite-difference diffusion eq 12 and finitedifference eq 15 in the equilibrium limit. Such a correction does not change the molecular mechanism of diffusion but gives a more accurate equation for the mass balance by eliminating the unnecessary approximation 10. This eliminates the inconsistency with density profiles near surfaces. The nature of this finite-difference diffusion equation allows less restrictive solutions 20 and 36 which do not conflict with boundary conditions imposed by surfaces. These solutions are consistent with various types of adsorption behavior, including multilayer adsorption and phase transitions in adsorbates with stepped isotherms. LA803703H
