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Critical Point Corrections for Two- and Three-Dimensional Systems G. L. Aranovich and M. D. Donohue* Department of Chemical Engineering, The Johns Hopkins University, Baltimore, Maryland 21218 Received July 15, 2002. In Final Form: December 9, 2002 Adsorption, the equilibrium between two-dimensional and three-dimensional fluids, is considered in the framework of Ono-Kondo lattice theory. A critical point correction to Ono-Kondo theory is proposed, and new approximations for binodals and spinodals are derived. These approximations describe the phase behavior near the critical point more accurately including two-dimensional and three-dimensional binodals. They also improve the prediction of adsorption behavior.
Introduction 1-3
is Modeling phase behavior of fluids at interfaces necessary to understand molecular mechanisms of surface tension in liquids and adsorption behavior at surfaces and interfaces.4,5 Important features of interfaces include density gradients and oscillatory density profiles near the surfaces of liquids6-10 and the interrelationship between two- and three-dimensional behaviors.11-15 The role of dimensionality is not limited to geometry of the system but also includes correlation effects on heterogeneous surfaces16,17 and in nanopores.18-20 Dimensionality also affects critical phenomena including long-range critical fluctuations21 and universal scaling laws.22 Studying crossover between mean-field behavior and critical behavior23 provides important insights into the thermodynamics of fluids and liquid mixtures over a wide * Author to whom correspondence should be addressed. (1) Cheng, E.; Cole, M. W. Langmuir 1989, 5, 616. (2) Schwennicke, C.; Schimmelpfennig, J.; Pfnur, H. Phys. Rev. B 1993, 48, 8928. (3) Fainerman, V. B.; Miller, R. J. Colloid Interface Sci. 2000, 232, 254. (4) Adamson, A. W. Physical Chemistry of Surfaces; John Wiley & Sons: New York, 1976. (5) Jaycock, M. J.; Parfitt, G. D. Chemistry of Interfaces; John Wiley & Sons: New York, 1981. (6) Rice, S. A.; Gryko, J.; Mohanty, U. In Fluid Interfacial Phenomena, sect. 6; Croxton, C. A., Ed.; John Wiley: New York, 1986. (7) Osborn, T. R.; Croxton, C. A. Mol. Phys. 1980, 40, 1489. (8) Croxton, C. A. Liquid-State Physics-a Statistical Mechanical Introduction; Cambridge University Press: Cambridge, 1974. (9) Snook, I. K.; van Megen, W. J. J. Chem. Phys. 1980, 72, 2907. (10) Cheng, E.; Swift, M. R.; Cole, M. W. J. Chem. Phys. 1993, 99, 4064. (11) Gunster, J.; Stultz, J.; Krischok, S.; Goodman, D. W. Chem. Phys. Lett. 1999, 306, 335. (12) Mazeas, I.; Pelerin, P.; Sellami, H.; Hamraoui, A.; Olier, R.; Privat, M. Langmuir 1999, 15, 2879. (13) Durbin, M. K.; Malik, A.; Richter, A. G.; Yu, C.-J.; Eisenhower, R.; Dutta, P. Langmuir 1998, 14, 899. (14) Ruckenstein, E.; Bhakta, A. Langmuir 1994, 10, 2694. (15) Hill, T. L. J. Chem. Phys. 1947, 15, 767. (16) Aranovich, G. L.; Donohue, M. D. J. Chem. Phys. 1996, 104, 3851. (17) Schlangen, L. J. M.; Koopal, L. K. Langmuir 1996, 12, 1863. (18) Radhakrishnan, R.; Gubbins, K. E. Phys. Rev. Lett. 1997, 79, 2847. (19) Dunne, J. A.; Myers, A. L.; Kofke, D. A. Adsorption 1996, 2, 41. (20) Kaneko, K.; Cracknell, R. F.; Nicholson, D. Langmuir 1994, 10, 4606. (21) Evans, R. Mol.r Phy. 1981, 42, 1169. (22) Fisher, M. E.; de Gennes, P.-G. C. R. Acad. Sci. Paris 1978, 287, 207. (23) Anisimov, M. A.; Povodyrev, A. A.; Kulikov, V. D.; Sengers, J. V. Phys. Rev. Lett. 1995, 75, 3146.
range of conditions.24 However, rigorous approaches are limited and cannot be applied without further development to complex systems including self-assembled layers,25 colloid systems,26 and nanophases27,28 where objects of different scales (nano, meso, and macro) and different dimensionalities are in equilibrium. For complex systems, mean-field approaches turn out to be useful,29-31 but the description of the system behavior close to the critical point has to be corrected. In this paper, we propose a simple method of correcting critical points for mean-field theory of adsorption. For this purpose, we consider Ono-Kondo lattice density functional theory32,33 which describes density gradients near phase boundaries and in nanoscale pores.34-37 In the classical Ono-Kondo model,32 the vapor-liquid interface is represented by a lattice where each site can be either occupied by a molecule or empty. Also, Ono-Kondo theory has been applied to liquid-solid and gas-solid interfaces.34-39 Classical Ono-Kondo Theory Consider a one-component lattice gas in contact with a surface (hard wall). There are interactions between the nearest neighbors with being the energy of adsorbate(24) Anisimov, M. A.; Kiselev, S. B.; Sengers, J. V. Physica A 1992, 188, 487. (25) Chakraborty, A. K.; Golumbfskie, A. J. Annu. Rev. Phys. Chem. 2001, 52, 537. (26) Russel, W. B.; Saville, D. A.; Schowalter, W. R. Colloidal Dispersions; Cambridge University Press: New York, 1989. (27) Gale, P. A. Philos. Trans. R. Soc. London, Sect. A 2000, 358, 431. (28) Kiely, C. J.; Fink, J.; Brust, M.; Bethell, D.; Schiffrin, D. J. Nature 1998, 396, 444. (29) Aranovich, G. L.; Donohue, M. D. J. Chem. Phys. 2002, 116, 7255. (30) Aranovich, G. L.; Donohue, M. D. J. Chem. Phys. 2001, 115, 5331. (31) Aranovich, G. L.; Hocker, T.; Wu, D.-W.; Donohue, M. D. J. Chem. Phys. 1997, 106, 10282. (32) Ono, S.; Kondo, S. Molecular Theory of Surface Tension in Liquids. In Encyclopedia of Physics; Flu¨gge, S., Ed.; Springer: Berlin, 1960; Vol. 10, p 134. (33) Rowlinson, J. S.; Widom, B. Molecular Theory of Capillarity; Clarendon Press: Oxford, 1982. (34) Lane, J. E. In Adsorption from Solution at a Solid/Liquid Interface, sect. 3; Parfitt, G., Rochester, C., Eds.; Academic Press: London, 1983. (35) Lane, J. E.; Johnson, C. H. J. Aust. J. Chem. 1967, 20, 611. (36) Aranovich, G. L. Langmuir 1992, 8, 736. (37) Hocker, T.; Aranovich, G. L.; Donohue, M. D. J. Chem. Phys. 1999, 111, 1240. (38) Altenberger, A. R.; Stecki, J. Chem. Phys. Lett. 1970, 5, 29. (39) Yashonath, S.; Sarma, D. D. Chem. Phys. Lett. 1984, 110, 265.
10.1021/la020640a CCC: $25.00 © 2003 American Chemical Society Published on Web 02/07/2003
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Langmuir, Vol. 19, No. 6, 2003 2163
exchange of a molecule with a vacancy
Ma + Vb f Va + Mb
Figure 1. Cross section of monolayer between two walls.
(2)
where M is the adsorbate molecule and V is the vacancy (empty site) that it fills (and vice versa). If this exchange occurs at equilibrium, then
∆U - T∆S ) 0
(3)
where ∆U and ∆S are the enthalpy and entropy changes. The value of ∆S can be represented in the form
∆S ) kB ln W1 - kB ln W2
Figure 2. Typical solution for Ono-Kondo equation.
adsorbate interactions and S being the interaction energy for adsorbate molecules at the adsorbent surface. (Both and S are negative for attractive forces.) For one layer between two walls (Figure 1), the Ono-Kondo equation can be written in the following form:
ln
x1(1 - xb) xb(1 - x1)
+
z3 z1S z2 + x1 x )0 kT kT kT b
where W1 is the number of configurations where site in the adsorbed layer is occupied by an adsorbate molecule and the site in the bulk is empty and W2 is the number of configurations where the site in the bulk is occupied by an adsorbate molecule and site in the adsorbed layer is empty. If the overall number of configurations for the system is W0, then
Adsorption Equilibrium Consider taking an adsorbate molecule from the adsorbed layer between two walls (Figure 1) and moving it to an empty site in the bulk. This is equivalent to the (40) Aranovich, G. L.; Donohue, M. D. Phys. Rev. E 1999, 60, 5552. (41) Aranovich, G. L.; Donohue, M. D. Comput. Chem. 1998, 22, 429. (42) De Oliveira, M. J.; Griffiths, R. B. Surf. Sci. 1978, 71, 687. (43) Wagner, P.; Binder, K. Surf. Sci. 1986, 175, 421. (44) Aranovich, G. L.; Donohue, M. D. J. Colloid Interface Sci. 1997, 189, 101.
W1/W0 ) x1(1 - xb)
(5)
W2/W0 ) xb(1 - x1)
(6)
and
(1)
In this equation, x1 is the density or fraction of sites occupied by molecules in the adsorbed layer, xb is the fraction of sites occupied with fluid molecules in the bulk, z3 is the coordination number for three-dimensional lattice (z3 ) 6 for cubic lattice), z2 is the monolayer coordination number (z2 ) 4 for square lattice), z1 is the number of molecule-surface bonds (z1 ) 2 for one layer between two walls as shown in Figure 1), k is Boltzmann’s constant, and T is the absolute temperature. Equation 1 relates the density in the adsorbed layer with the density in the bulk. In earlier publications, we presented a numerical method of solving the Ono-Kondo equations when there are multiple solutions.40,41 A typical solution of eq 1 is illustrated in Figure 2 for z1 ) 2, z2 ) 4, z3 ) 6, /kT ) -1.3, and S/kT ) -0.5. In Figure 2, points A and B indicate spinodals for two-dimensional condensation (in the monolayer) and points C and D indicate spinodals for three-dimensional condensation (in the bulk). Hence, the Ono-Kondo equation not only gives the adsorption isotherm but it also relates phase behavior in the adsorbed layer to the phase behavior in the bulk.37 However, the Ono-Kondo model gives classical, meanfield phase diagrams and critical points.42-44 In this paper, we propose a correction to the Ono-Kondo equation to describe phase behavior near the critical point more accurately.
(4)
Substituting eqs 5 and 6 into eq 4, we obtain
∆S ) k ln[x1(1 - xb)/(1 - x1)xb]
(7)
The change in enthalpy, ∆U, can be written as
∆U ) Ea - Eb
(8)
where Ea is the configurational energy of a molecule in the adsorbed layer and Eb is the configurational energy of a molecule in the bulk. In the classical (mean-field) approximation
z1S z2x1 Ea )kT kT kT
(9)
z3xb Eb )kT kT
(10)
and
Substituting eqs 9 and 10 into eq 8 and then eqs 8 and 7 into eq 3 gives the classical Ono-Kondo equation (1) for a monolayer between two walls. Equilibrium between Phases in the Bulk Here, we assume molecules are in the two-phase region for the three-dimensional lattice fluid. Consider taking a molecule from phase 1 and moving it to an empty site in phase 2. This is equivalent to the exchange of a molecule with a vacancy
M1 + V2 f V1 + M2
(11)
If this exchange occurs at equilibrium, then eq 3 also is valid, and it results in the following equation instead of eq 1
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ln
xb′(1 - xb′′) xb′′(1 - xb′)
+
z3 z3 xb′ x ′′ ) 0 kT kT b
Aranovich and Donohue
where xb′ and xb′′ are densities of phase 1 and phase 2. Note that one-component lattice gas with nearest neighbors interactions is equivalent to classical Ising model45 with symmetric phase diagram. Assuming that xb′′ ) 1 - xb′, eq 12 can be written in the following form
2 ln
xb′ (1 - xb′)
+
z3 (2x ′ - 1) ) 0 kT b
(13)
which is the Bragg-Williams binodal.45
Differentiation of eq 1 with respect to xb gives
(14)
f3(xb,T) )
(15) f2(x1,T) )
(16)
Note that eqs 15 and 16 are the Bragg-Williams spinodals for the three-dimensional and two-dimensional lattices respectively.45 Equation 14 can be written in a more general way
dx1 f3(xb,T) ) dxb f2(x1,T)
(17)
(18)
(19)
give the spinodals for three-dimensional and twodimensional lattices, respectively. Improving the functions f3(xb,T) and f2(x1,T) (compared to eq 14) would allow more accurate predictions for equilibrium between adsorbed and bulk phases.
βb(T) ) -2Rb(T)/3
(25)
βa(T) ) -2Ra(T)/3
(26)
and
Therefore, the equations for spinodals are
z3 1 - 2Rb(T)xb(1 - xb) ) 0 (27) + xb(1 - xb) kT
and
and
f2(x1,T) ) 0
z2 1 - 2Ra(T)x1 - 3βa(T)x12 + x1(1 - x1) kT (24)
From statistical mechanics of the lattice gas,45 it is wellknown that both spinodals have to be symmetric. Hence
f3(xb,T) )
where equations
f3(xb,T) ) 0
z3 1 - 2Rb(T)xb - 3βb(T)xb2 + xb(1 - xb) kT (23)
and
and dx1/dxb ) ∞ if
z2 1 )0 + x1(1 - x1) kT
(22)
So
From eq 14, it follows that dx1/dxb ) 0 if
z3 1 )0 + xb(1 - xb) kT
(21)
represent more exact expressions for Ea and Eb. Here Ra, βa, Rb, and βb are temperature-dependent coefficients. Note that the quadratic and higher terms in eqs 20 and 21 reflect correlations that become important near the critical point but which are not taken into account in the classical mean-field Ono-Kondo model. Using eqs 20 and 21 instead of eqs 9 and 10 and leaving only linear, quadratic, and cubic terms, we obtain instead of eq 14
z3 1 - 2Rb(T)xb - 3βb(T)xb2 + dx1 xb(1 - xb) kT ) z2 dxb 1 + - 2Ra(T)x1 - 3βa(T)x12 x1(1 - x1) kT
Spinodals
z3 1 + dx1 xb(1 - xb) kT ) z2 dxb 1 + x1(1 - x1) kT
Eb z3xb )+ Rb(T)xb2 + βb(T)xb3 + ... kT kT
(12)
f2(x1,T) )
z2 1 - 2Ra(T)x1(1 - x1) ) 0 (28) + x1(1 - x1) kT
At the critical points, x1 ) 0.5 and xb ) 0.5. These requirements give from eqs 27 and 28
z2 Ra(T2cr) ) 8 + 2 kT2cr
(29)
z3 Rb(T3cr) ) 8 + 2 kT3cr
(30)
Improving Functions f3(xb,T) and f2(x1,T) Consider eqs 9 and 10 as linear approximations for an expansion of the configurational energy in powers of density. Then, the series
z1S z2x1 Ea )+ Ra(T)x12 + βa(T)x13 + ... (20) kT kT kT and (45) Hill, T. L. An Introduction to Statistical Thermodynamics; Addison-Wesley: London, 1960.
where T3cr and T2cr are critical temperatures for threeand two-dimensional lattices, respectively. In the vicinity of the critical point, there is a significant range of concentrations where the phase diagram is flat (and where T is almost constant). In this range
Ra(T) ≈ Ra(T2cr)
(31)
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Langmuir, Vol. 19, No. 6, 2003 2165
Rb(T) ≈ Rb(T3cr)
(32)
Away from the critical point, the corrections -2Rb(T)xb(1 - xb) and -2Ra(T)x1(1 - x1) to the functions f3(xb,T) and f2(x1,T) vanish because factors xb(1 - xb) and x1(1 x1) become small. Therefore, using approximations 31 and 32 in eqs 27 and 28 should be reasonable over a wide range of densities and temperatures. Plugging eqs 29 and 30 into eqs 27, 28, 31, and 32 gives the following equations for spinodals (for two- and three-dimensional cases, respectively):
-
kT )
-
kT )
z2
(
)
(33)
(
)
(34)
z2 1 - 4x1(1 - x1) 4 + x1(1 - x1) kT2cr z3
z3 1 - 4xb(1 - xb) 4 + xb(1 - xb) kT3cr
Similarly, using the linear, quadratic, and cubic terms in eq 21 results in corresponding corrections to eq 12 and the following equation instead of eq 13
(
)[ ]
Figure 3. Binodals (thick lines) for two-dimensional lattice with z2 ) 4: (A) predicted by eq 37, (B) quasi-chemical, (C) Bragg-Williams; solid circles show exact binodal from ref 45, thin lines show spinodals predicted by eqs 16 and 33.
z3 xb (1 + z3 (2xb - 1) + 8 + 2 2 ln kT (1 - xb) kT3cr xb)2 - xb2 +
2 3 2 x - (1 - xb)3 ) 0 (35) 3 b 3
which gives the binodal for three-dimensional lattice
-
kT )
z3(2xb - 1) xb z3 2 2 2 ln + 8+2 1 - 2xb + xb3 - (1 - xb)3 1 - xb 3 3 kT cr
(
3
)[
]
(36) For the two-dimensional case, eq 36 can be rewritten in the following form:
-
kT )
z2(2x1 - 1) x1 z2 2 2 2 ln + 8+2 1 - 2x1 + x13 - (1 - x1)3 cr 1 - x1 3 3 kT2
(
)[
]
Figure 4. Binodals (thick lines) for three-dimensional lattice with z3 ) 6: (A) predicted by eq 36, (B) quasi-chemical, (C) Bragg-Williams; solid circles show Monte Carlo simulation data for binodal from ref 46, thin lines show spinodals predicted by eqs 15 and 34.
(37) Scaling Behavior Near the Critical Point Figures 3 and 4 show binodals predicted by eqs 36 and 37 for two- and three-dimensional lattices, the exact 2D binodal45 and Monte Carlo simulation data for the 3D binodal.46 Also shown are binodals for quasi-chemical and Bragg-Williams approximations and spinodals predicted by eqs 15, 16, 33, and 34. As shown by Figures 3 and 4, this new model predicts binodals which are in a good agreement with exact results for 2D and with Monte Carlo simulation data for 3D. Though these results are quite good, there is a significant error at intermediate densities. (46) Lambert, S. M.; Soane, D. S.; Prausnitz, J. M. Fluid Phase Equilib. 1993, 83, 59.
Approximations 36 and 37 based on eqs 31 and 32 give binodals which are exact at the critical point and in the limits of small and large densities. However, there are significant errors at intermediate densities (Figures 3 and 4). To reduce these errors, we consider correction to eqs 31 and 32 taking into account the temperature dependence of the coefficients Ra and Rb. Consider expansion of Ra(T) and Rb(T) for subcritical temperatures in powers of ta and tb
ta )
T2cr - T T2cr
(38)
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Figure 5. Binodal for two-dimensional (square) lattice: (a) eq 51, (b) eq 37; solid circles show exact binodal from ref 45. Here Ba ) 3.5, and νa ) 0.25.
tb )
T3cr - T cr
T3
Rb(T) ) Re(tb) + Si(tb)
-
(40) (41)
∞
Aaitai ∑ i)0 Abitbi ∑ i)0
2 ln
(43)
Si(ta) ) Bataνa
(44)
Si(tb) ) Bbtbνb
(45)
with 1 > νa > 0 and 1 > νb > 0. Equations 40-45 give the following expansions for Ra(T) and Rb(T):
Ra(T) ) Ra(T2cr) + Bataνa + Aa1ta + Aa2ta2 + ... (46) Rb(T) ) Rb(T3cr) + Bbtbνb + Ab1tb + Ab2tb2 + ... (47)
xb 1 - xb
z3(2xb - 1)
(
[
z3
[
z2
) ][
T3cr - T
+ 8+2 + Bb kT3cr
cr
T3
νb
1 - 2xb +
]
2 3 2 x - (1 - xb)3 3 b 3
(50)
kT )
(42)
∞
Re(tb) )
kT )
2 ln
where Re and Si are regular and singular components defined as
Re(ta) )
(49)
With approximations 48 and 49 instead of 31 and 32, eqs 36 and 37 can be rewritten in the following form:
in the form
Ra(T) ) Re(ta) + Si(ta)
Rb(T) ≈ Rb(T3cr) + Bbtbνb
(39)
x1 1 - x1
+ 8+2
z2(2x1 - 1)
cr
kT2
(
+ Ba
) ][
T2cr - T T2
cr
νa
1 - 2x1 +
]
2 3 2 x - (1 - x1)3 3 1 3
(51)
In the limit of small ta and tb, eqs 50 and 51 give
FaL - FaG ) const taνa/2
(52)
FbL - FbG ) const tbνb/2
(53)
where FaL ) 0.5 + δx1 (liquid density for 2D phase), FbL ) 0.5 + δxb (liquid density for 3D phase), FaG ) 0.5 - δx1 (gas density for 2D phase), FbG ) 0.5 - δxb (gas density for 3D phase), δx1 and δxb are variations of x1 and xb near the critical value (0.5). Equations 52 and 53 coincide with well-known equation for densities of coexisting phases near the critical point if νa/2 and νb/2 coincide with critical exponent β for two and three dimensions respectively.33 Then
Equations 31 and 32 used only the first terms in the expansions 46 and 47. Now, we consider a second approximationsfirst two terms of the expansionsswhich gives
νa ) 2βa
(54)
νb ) 2βb
(55)
Ra(T) ≈ Ra(T2cr) + Bataνa
Note that βa ) 1/8 and βb ≈ 0.325 for 2D and 3D Ising models, respectively.
(48)
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Langmuir, Vol. 19, No. 6, 2003 2167
Figure 6. Binodal for three-dimensional (cubic) lattice: (a) eq 50, (b) eq 36; solid circles show Monte Carlo simulation data for binodal from ref 46. Here Bb ) 2, and νb ) 0.65.
Figure 6 gives the binodal for the three-dimensional (cubic) lattice predicted by eq 50 with Bb ) 2 and νb ) 0.65 compared to Monte Carlo simulation data. Also shown are predictions of eq 36. As seen from Figure 6, approximation 49 gives excellent agreement between theoretical binodal and Monte Carlo simulations. Critical Behavior Correction to Ono-Kondo Theory Taking into account the linear, quadratic, and cubic terms of eqs 20 and 21 and using eqs 3, 7, 8, 25, and 26 we obtain Figure 7. Solutions predicted by eqs 1 and 58 for z1 ) 2, z2 ) 4, z3 ) 6, /kT ) -1.2, S/kT ) -2.0, Ba ) 3.5, Bb ) 2, νa ) 0.25, and νb ) 0.65.
ln
x1(1 - xb) xb(1 - x1)
+
z1S z2 2 + x - Ra(T)x12 + Ra(T)x13 kT kT 1 3 z3 2 x + Rb(T)xb2 - Rb(T)xb3 ) 0 (56) kT b 3
where Ra(T) and Rb(T) are temperature-dependent coefficients vanishing at T f ∞. However, we will consider the range of temperatures where eqs 48 and 49 still are valid. For this range, using eqs 29, 30, 48, and 49 in eq 56 gives
ln
x1(1 - xb) xb(1 - x1)
(
Ba Figure 8. Solution predicted by eqs 1 and 58 for z1 ) 2, z2 ) 4, z3 ) 6, /kT ) -0.8, S/kT ) -2.0, Ba ) 3.5, Bb ) 2, νa ) 0.25, and νb ) 0.65.
Figure 5 shows binodal for two-dimensional (square) lattice predicted by eq 51 at Ba ) 3.5 and νa ) 1/4. Also shown is exact binodal45 and binodal predicted by eq 37. As seen from Figure 5, using the appoximation 48 instead of 31 makes the error in predictions very small.
cr
T2
[
z2 z1S z2 + x1 - 8 + 2 + kT kT kT cr
)] (
T2 - T cr
+ νa
x12 1 -
(
Bb
)
[
2
z3 z3 2 x x + 8+2 + 3 1 kT b kT cr
)] (
T3cr - T T3cr
νb
xb2
)
3
2 1 - xb ) 0 (57) 3
Note that eqs 44 and 45 are written in terms of subcritical temperatures. Therefore, eq 57 also is valid only for subcritical temperatures. In more general form, it can be rewritten as
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Figure 9. Adsorption isotherms predicted by eq 58 at S/kT ) -2.0 and various /kT: -1.7 (1), -1.75 (2), -1.76 (3), -1.8 (4), -2.0 (5).
ln
x1(1 - xb) xb(1 - x1)
|
Ba′
cr
+
|]
T2 - T T2
cr
νa
[
z2 z1S z2 + x - 8+2 + kT kT 1 kT cr
(
x12 1 -
|
Bb′
)
[
2
z3 z3 2 x x + 8+2 + 3 1 kT b kT cr
|]
T3cr - T T3cr
νb
(
xb2
)
3
Figure 9 shows solutions of eq 58 for z1 ) 2, z2 ) 4, z3 ) 6, Ba ) 3.5, Bb ) 2, νa ) 0.25, νb ) 0.65, S/kT ) -2.0 and various /kT below and above T2cr. As illustrated in Figure 9, eq 58 correctly predicts a two-phase region in adsorbed layer at T < T2cr and a one-phase region in adsorbed layer at T > T2cr.47
2 1 - xb ) 0 (58) 3
where Ba′ ) Ba and Bb′ ) Bb for subcritical temperatures, and Ba′ ) -Ba and Bb′ ) -Bb for supercritical temperatures (Ba and Bb are critical amplitudes). For a two-dimensional square lattice, z2 ) 4, /kT2cr ≈ -1.76, and νa ) 0.25 and for three-dimensional cubic lattice, z3 ) 6, /kT3cr ≈ -0.89, and νb ) 0.65. Figure 7 illustrates solutions predicted by eqs 1 and 58 for z1 ) 2, z2 ) 4, z3 ) 6, /kT ) -1.2, S/kT ) -2.0, Ba ) 3.5, Bb ) 2, νa ) 0.25, and νb ) 0.65. As illustrated in Figure 7, the classical Ono-Kondo equation incorrectly predicts a two-phase region in the adsorbed layer at a temperature that is supercritical for the two-dimensional phase. Equation 58 predicts a more correct behaviorsno phase transitions at this temperature. Figure 8 illustrates predictions by eqs 1 and 58 for z1 ) 2, z2 ) 4, z3 ) 6, /kT ) -0.8, S/kT ) -2.0, Ba ) 3.5, Bb ) 2, νa ) 0.25, and νb ) 0.65. As illustrated in Figure 8, the classical Ono-Kondo equation predicts a two-phase region in the bulk phase at a temperature that is supercritical for the three-dimensional phase; eq 58 does not have this problem.
Conclusions Equations 50 and 51 describe binodals for 2D and 3D lattices very accurately. They represent a general equation where coordination number, critical temperature, critical exponent, and critical amplitude depend on type of lattice and on dimensionality of the system. These approximations are exact in the limits of small and large densities and in the vicinity of the critical point. Equation 58 predicts more correct phase behavior both in the adsorbed layer and in the bulk with exact critical parameters. Acknowledgment. Support by the Division of Chemical Sciences of the Office of Basic Energy Sciences, U.S. Department of Energy, under Contract DE-FG0287ER13777, by the National Science Foundation under Grant BES-9910174, and by E.I. du Pont de Nemours and Co. is gratefully acknowledged. LA020640A (47) Masel, R. Principles of Adsorption and Reaction on Solid Surfaces, section 4.8.1; John Wiley: New York, 1996.
