Langmuir 2005, 21, 3475-3479
3475
On Numerical Classification of Solution Adsorption Isotherms† Marcellus T. Coltharp* Division of Mathematics and Science, Kentucky State University, Frankfort, Kentucky 40601 Received October 5, 2004. In Final Form: January 19, 2005 To numerically classify solution adsorption isotherms, a difference or deviation measure, DSmc (the relative difference between two sums of the adsorption maximum’s characteristics of selectivity isotherms × 100), is derived. The measure is applicable to completely miscible binary solutions on solids. This quantity evaluates the difference between an adsorption system and the ideal adsorption system (ideal adsorbed and bulk phases, homogeneous surface, and equal molar area solution components) at the point of maximum adsorption. For model systems, DSmcs are calculated at several levels of surface heterogeneity (Gaussian distribution of surface energy) and for different signs of phase nonideality (regular solution phases) on a homogeneous surface and on a simple two-site-type heterogeneous surface. All heterogeneous surfaces have negative DSmc values, but nonideal phases have DSmcs with signs opposite to the sign of deviation from Raoult’s law. DSmcs from both U- and S-shape isotherms are reported for 16 experimental systems consisting of hydrocarbon mixtures and both alcohol + hydrocarbon and alcohol + water solutions or acetone + carbon tetrachloride on several silica gels and a variety of carbons.
Introduction Commonly, classification of isotherms, most often surface excess isotherms, is an informative step in analysis of adsorption from completely miscible binary solutions onto solids. Such classification is useful in evaluation of surface energy distributions,1 in development or extension of theoretical models,2-4 in practical applications,5-7 and for other purposes.8 The available classification methods are qualitative. The simplest and oldest method is by general shape, i.e., either a U or an S shape is found. Slightly more complex is categorizing by one of the SchayNagy shape subtypes9 which involves a check of isotherm linearity. Most laborious is deciding whether an ideal adsorption system (IAS) is fit (for U shapes only) because this involves trial and error calculations. (The IAS consists of a homogeneous surface with ideal bulk and ideal adsorbed phases and equal molar area mixture components.) The first and last classification methods, at least, also apply to selectivity isotherms, surface phase mole fraction vs bulk phase mole fraction. (The surfaces phase mole fraction is found from the surface excess.) It would be helpful to have a quantitative means of classifying or ranking isotherms according to sign and extent of deviation from the IAS. A simple numerical ranking would aid the often difficult comparisons between
members of a set of theoretical10 or experimental isotherms.11 Further, such classifying would provide useful background information for determination of distributions of surface energy when desired or calculation of adsorbed phase activity coefficients when appropriate. In this paper, a measure or indicator of deviation from the IAS, DSmc, is proposed. The measure is derived and is tested with selectivity isotherms from both model systems and real systems. The central part of the derivation is the reference condition, which is found from the derivative of the IAS expression for the adsorbed phase composition at the adsorption maximum. From the model data, the measure’s dependence on two well-known factors that affect adsorption, surface heterogeneity and phase nonideality, is established. The first factor is represented by Gaussian distributions of surface energies and a twosite-type surface and the second by regular solutions with both positive and negative deviation from Raoult’s law. Next, the effects of combination of the factors, which is more physically realistic, are found. Here, the model consists of a regular solution with either type of deviation from Raoult’s law on the two-site-type surface. Finally, the measure is applied to literature experimental systems consisting of hydrocarbon solutions or polar molecules + hydrocarbons or + carbon tetrachloride on silica gels or carbons. Relations
* Present address: c/o Mayhew, 1400 Kenesaw Ave., Apt. 12-G, Knoxville, TN 37919. E-mail:
[email protected]. † Parts of this paper were presented at the 77th Colloid and Surface Science Symposium, Atlanta, GA, June 2003; Paper 385. (1) Heuchel, M.; Brau¨er, P.; Szombathely; M. v., Messow, U.; Einicke, W.-D.; Jaroniec, M. Langmuir 1993, 9, 2547. (2) Podkoscielny, P.; Dabrowski, A.; Leboda, R. Colloids Surf., A 2001, 182, 219; Chem. Abstr. 2001, 135, 25156z. (3) Heuchel, M. Langmuir 1997, 13, 1150. (4) Michalek. J.; Zajac, J.; Rudzinski W. Langmuir 1990, 6, 1505. (5) Bo´ta, A.; La´szlo´, K.; Nagy, L. G.; Copitzky, T. Langmuir 1997, 13, 6502. (6) Everett, D. H. Pure Appl. Chem. 1981, 53, 2181. (7) Kiraly, Z.; Dekany, I.; Mastalir, A.; Bartok, M. J. Catal. 1996, 161, 401. (8) Kalies, G.; Brau¨er, P.; Messow, U. J. Colloid Interface Sci. 1999, 214, 344. (9) Schay, G.; Nagy, L. G. J. Chim. Phys. 1961, 58, 149.
Derivation of the Deviation Indicator DSmc. Establishing the Reference Condition. At the interface between a solution of components 1 and 2 and a homogeneous surface, adsorption at temperature T can be described by the phase exchange reaction.12 Here, the reactants (1)σ and (2)l are in equilibrium with the products (10) Dabrowski, A.; Jaroniec, M.; Os´cik, J. Multilayer and Monolayer Adsorption from Liquid Mixtures of Nonelectrolytes on Solid Surfaces. In Surface and Colloid Science; Matijevic, E., Ed.; Plenum Publishing: New York, 1987; Vol. 14, Chapter 2; pp 124, 140. (11) Coltharp, M. T.; Hackerman, N. J. Colloid Interface Sci. 1973, 43, 176. Isotherm characteristics were plotted vs surface oxygen or surface carbon atom type to avoid confusion in comparison between or among isotherms. (12) Everett, D. H. Trans. Faraday Soc. 1964, 60, 1803.
10.1021/la047539l CCC: $30.25 © 2005 American Chemical Society Published on Web 03/02/2005
3476
Langmuir, Vol. 21, No. 8, 2005
Coltharp
Samx,id ) 1
(5)
This expression gives the reference condition for evaluation of deviations of real systems from ideal behavior. DSmc Defined. The deviation measure DSmc, the difference between the sum of the adsorption maximum’s characteristics (the mole fraction of the same component in both phases at the adsorption maximum) for a system and the sum of the same characteristics for the IAS relative to the latter, is defined as a percent by
DSmc )
Figure 1. Model system selectivity isotherms with adsorption maxima designated: (dash-dot curve) two-site-type surface with ideal phases, -∆σ°alo 0.8368 kJ/mol, -∆σ°ahi 7.531 kJ/ mol, Θlo ) Θlhi ) 0.50; (dash curve) IAS, -∆σ°alo, 0.8368 kJ/ mol. Temperature for both systems was 298.2 K. The location of an adsorption maximum is indicated by a v. Terms here are defined after eq 8.
(1)l
σ
and (2) . The σ and l represent the adsorbed phase and the bulk phase, respectively. The equilibrium constant for the IAS version of this reaction is
K)
X1lX2σ X1σX2l
(1)
where XiR is a mole fraction, i specifies component 1 or 2, and R specifies phase σ or l. From this relation, the Everett equation for adsorbed phase composition
Xσ )
KX 1 + X(K - 1)
(2)
can be found. For brevity from here on, component 2 is understood and X represents Xl, i.e., only superscripts and subscripts needed for clarity are retained. If Xσ from eq 2 is plotted vs X, as in Figure 1, the U-shaped selectivity isotherm formed has a point at which adsorption Xσ - X is at a maximum (designated by v in Figure 1). The slope of the curve at this point, designated by amx, is parallel to the adsorption azeotrope line (Xσ ) X) whose slope is 1. Thus, at the adsorption maximum
|
dXσ dX
amx
)1
K1/2 - 1 K-1
DSmc ) (Samx - 1) × 100
(7)
Bounds of DSmc. At the lower bound of a selectivity isotherm Xσ vs X, as Xamx f 0, Xamxσ f 0; hence Samx f 0 and DSmc f -100%. Also, at the upper bound, as Xamx f 1, Xamxσ f 1, thus Samx f 2 and DSmc f 100%. Model System Relations to Test DSmc. Heterogeneous Surfaces. On a homogeneous surface, a surface with only one type of adsorption site, the equilibrium constant for ideal phases is related to the surface energy difference, -∆σ°a, by
K ) exp
-∆σ°a RT
(8)
where -∆σ° is -(σ2° - σ1°), σi° is the surface (interfacial) energy per area for pure component i at the liquid/solid interface, a is the partial molar area of a component (here, equal areas), R is the gas constant, and T the absolute temperature. For the simplest type of heterogeneous surface, there are two types of sites, one with low surface energy difference -∆σ°alo at a fractional surface coverage Θlo, and the other with high surface energy difference -∆σ°ahi and a coverage Θhi. The overall composition of the adsorbed phase Xσ in this case is given by
(9)
(3)
(4)
has physical meaning. The id specifies the IAS. If eq 4 is substituted into eq 2 and the sum Samx,id of the adsorption maximum’s characteristics Xamx,idσ and Xamx,id is taken, subsequent substitutions and rearrangement yield (13) (K - 1)Xamx2 + 2Xamx - 1 ) 0.
(6)
Here, Samx is defined as the sum of the adsorption maximum’s characteristics for any specific completely miscible system, and Samx,id is as defined above. Substitution of eq 5 into eq 6 gives the evaluation form
Xσ ) ΘloXloσ + ΘhiXhiσ
When this condition is used in the derivative of eq 2 with respect to X and rearrangements are made, a quadratic equation in Xamx is found.13 Only the positive root of the quadratic
Xamx,id )
Samx - Samx,id × 100 Samx,id
for selected -∆σ°a pairs. Only the 50% coverage is evaluated in this study. For a more general description of adsorption, the adsorption integral14 is used. One version of the discrete form of this integral is n
Xσ )
Xkσfk∆(-∆σ°a) ∑ k)1
(10)
where Xkσ is the local adsorption isotherm on site type k, fk is the distribution function over the n total -∆σ°ak (14) Szombathely, M. v.; Brau¨er, P.; Jaroniec, M. J. Comput. Chem. 1992, 13, 17.
Classification of Solution Adsorption Isotherms
Langmuir, Vol. 21, No. 8, 2005 3477
values, and ∆(-∆σ°a) is the surface energy increment (quadrature weight). A Gaussian distribution was assumed.15 Nonideal Phases. For nonideal phases, activities aiR in each phase replace concentrations in eq 1 with each determined by an activity coefficient γiR according to aiR ) γiRXiR. The regular solution approximation was used for the γiR values, since its simple form yields key real solution behavior. This approximation is a lattice model based on the difference in interaction energy between a reference molecule in the bulk solution with its nearest neighbors, also here the difference in interaction energy between a reference molecule in the surface layer with its nearest neighbors in the same layer and with the nearest neighbors in the adjacent bulk layer, and the interaction energy between a similar reference molecule with its nearest neighbors in a pure liquid phase of each solution component. In this approximation, for the bulk phase
γil ) exp[wTl(1 - Xil)2]
surface
solution
DSmc/%
degree of heterogeneity changed Gaussianb s ) (0.8368c s ) (1.255c s ) (1.674c s ) (2.092 c two site typed homog.e homog.e
ideal ideal ideal ideal ideal
-0.8 -2.4 -5.8 -11.3 -23.0
phase nonideality changed reg., wT ) -1 reg., wT ) 1
6.3 -2.6
heterogeneity fixed, nonideality changed reg., wT ) -1 two site typed -14.5 reg., wT ) 1 two site typed -25.5 a T, 298.2 K. Isotherms are U shape. b Gaussian centered at -∆σ°a, 4.184 kJ/mol. c kJ/mol. d kJ/mol: -∆σ°alo, 0.8368, -∆σ°ahi, 7.531; Θlo ) Θhi ) 0.50. e -∆σ°a, 4.184 kJ/mol.
(11)
in which wTl is the temperature-dependent interaction energy parameter (difference in interaction energy divided by RT) and the subscript i indicates component i of the i, j pair. For the adsorbed phase
γiσ ) exp(lwTσ(1 - Xiσ)2 + mwTl(1 - Xil)2)
(12)
where wTσ is the temperature-dependent interaction energy parameter (interaction energy difference divided by RT) and the multiples l and m are the fractions of nearest neighbors to a molecule in the adsorbed layer and in the adjacent bulk layer,16 respectively. After the relevant forms of eq 11 and eq 12 are substituted into eq 1 (homogeneous surface) and rearranged
exp wT(l(2Xσ - 1) + (m - 1)(2X - 1)) )
Table 1. DSmcs of Model Systems: Homogeneous or Heterogeneous Surfaces and Ideal or Nonideal Phasesa
Xσ(X - 1) K(Xσ - 1)X (13)
is found; here wTσ ) wTl is assumed. Cubic close packed values, l of 1/2 and m of 1/4, were used. Xσs here were found by recursion. Heterogeneous Surfaces and Nonideal Phases. For the most complex model examined, both phases nonideal and a heterogeneous surface, Xσ values from eq 13 (regular phases) were used in eq 9 (two site type surface) for each site type. Evaluation of DSmc. The slope condition, eq 3, was used to find the adsorption maximum’s characteristics of each selectivity isotherm. For model systems, Xσ vs X table values in the vicinity of a maximum were fit with a quadratic, and its derivative was taken. For experimental (literature) systems,17 the condition was used graphically with the aid of a 45°-90°-45° triangle, a ruler, and
dividers. By means of the property that a straight line crossing two parallel lines forms equal opposite angles, the parallel to the no adsorption line extended was constructed. A 45° line from the no adsorption line was first drawn, then the desired parallel (tangent to the selectivity isotherm) was found by moving the triangle up and down the ruler edge on the constructed line. Here, the triangle had a 45° corner (the opposite angle) on the ruler edge. The tangent point provides the axes coordinates of the adsorption maximum’s characteristics. If a selectivity isotherm is S shape, it has the features of a U shape plus two more, a second point with a slope of 1 and an adsorption azeotrope point, or no adsorption point. To resolve which of the slope condition points is taken as the adsorption maximum, a convention is needed. At one point, adsorption of component 2 is positive and at the other it is negative (the first point occurs at the maximum of an S-shape surface excess isotherm and the second at its minimum). It was assumed that the larger absolute adsorption |Xσ - X| indicates the adsorption maximum, thus component 2 was assigned to be the component which gives the positive difference such that (Xσ - X)amx > |Xσ - X|amn where amn designates the smaller difference (surface excess isotherm minimum). For a sense of the values involved, the DSmc of the heterogeneous system, the upper curve in Figure 1, is -23.0% because Xamxσ is 0.560 and Xamx 0.210. In comparison, DSmc for the IAS system, the lower curve with Xamxσ of 0.542 and Xamx of 0.458, is 0.0%, as expected. The uncertainty of DSmc from model evaluations is (0.2% and that from literature isotherm characteristics ranges from about (0.5 to (1.0%. The latter uncertainties are larger due to imprecisions in geometrical evaluation of characteristics in small literature figures. Results and Discussion
(15) The Gaussian form used was f ) s-1(2π)-1/2 exp {-(2s2)-1 [(-∆σ°a) - (-∆σ°ac)]2} where s is the standard deviation and -∆σ°ac the central value. The normalization condition, the sum of the n quantities fk∆(-∆σ°a) is 1, holds here. Each isotherm was found by multiplying a 19 × 20 matrix times a 20-element distribution vector. Each element in the matrix is a value from eq 2 for a given Xl (l is an index) and Kk (eq 8) times the surface energy increment. (16) Reference 10, pp 125-126. (17) In most publications, the isotherms displayed are surface excess isotherms, surface excess (mmol/g) vs X, not selectivity isotherms. (To convert from surface excess to Xσ, the area of the solid, the partial molar area of each solution component and the adsorbed layer thickness or equivalently the filled surface layer amounts for each component (or simply the number of moles in the adsorbed phase) are needed for the most often used evaluations.) All the literature isotherms used here, however, were selectivity isotherms.
Model Systems. Table 1 displays DSmc values for model systems with different degrees of surface heterogeneity and/or type of phase nonideality. For ideal phases, both varieties of heterogeneous surfaces, those with a Gaussian distribution of energies and the two-site type, show negative deviation from the IAS. As heterogeneity, the range of surface site energies indicated by the standard deviation s in the Gaussian series, increases, DSmc gets more negative. On the most heterogeneous Gaussian surface, DSmc reaches around a tenth of its total negative range and the DSmc for the two-site-type surface is about twice that size.
3478
Langmuir, Vol. 21, No. 8, 2005
Coltharp
Table 2. DSmcs Calculated from Literature Isotherms: Effect of Surface Change or of Solution Change surface
solution
Table 3. DSmcs Calculated from Literature Isotherms: Various Solutions/Silica Surfacesa
DSmc/%
surface changed, solution fixeda carbon black benzene + cyclohexane active carbon benzene + cyclohexane
-1.5 -22.0
surface fixed, solution changedb silica gel, Si 40 2-propanol + n-heptane silica gel, Si 40 methanol + benzene
-2.2 -7.0
a U-shape isotherms, the first component listed is preferentially adsorbed, 298 K, surface area not given.18 b U-shape isotherms, preference convention as in footnote a, 293 ( 0.2 K; surface area, 814 m2/g.19
With nonideal (regular) phases on a homogeneous surface, there is also deviation from the IAS, but now deviation is in both directions. The sign of DSmc is opposite to the sign of deviation from Raoult’s law. That is, the system with negative deviation from Raoult’s law has a positive DSmc whereas the system with positive deviation has a negative DSmc. The absolute DSmc values here are small compared to the most heterogeneous Gaussian surface and the two-site-type surface. There is asymmetry here, however, since negative deviation from Raoult’s law gives a DSmc about twice the absolute DSmc for positive deviation. At the bottom of the table, each of the most involved systems, regular solution phases on a two-site-type surface, has a DSmc that consists of the DSmc of the twosite-type system with the DSmc of the respective nonidealty contribution added almost exactly. Thus, the contributions from surface heterogeneity and phase nonideality are additive. Literature Experimental Systems. Surface Changed or Solution Changed. In Table 2, a simple comparison check of the model systems’ results is made. Here, the effects of surface heterogeneity and nonideality in the phases are demonstrated using DSmcs of two pairs of literature isotherms. The DSmcs are those of a solution of hydrocarbons on each of a carbon heterogeneity pair and two alcohol + hydrocarbon solutions separately on a silica gel. For both sets, DSmcs are negative, as expected for heterogeneous surfaces and phases with positive deviation from Raoult’s law.20-22 In going from the carbon black to the active carbon, heterogeneity increases and DSmc gets more negative, as anticipated. The alcohol systems on the silica gel differ in DSmc, as expected because change in the alcohol alters the interactions between solution components as does change in the hydrocarbon. Various Solutions on Various Silica Gels. Table 3 gives DSmcs for several polar molecule + hydrocarbon (or + carbon tetrachloride) solutions and a hydrocarbon pair on silica gels. All but one of the DSmcs are negative. The DSmc range is from about positive 2% to almost 30% of the total negative range. Although most of the systems have absolute values smaller than 12%, one system has about twice this value and another two and a half times it. For the approximately homologous alcohol series (S2, S4-6), as carbon atom number increases DSmc gets more negative. (The “approximately” applies because there is no twocarbon alcohol and the propanol is not primary.) (18) Nagy, L. G.; Schay, G. Acta Chim. Hung. 1963, 39, 365. (19) Goworek, J.; Derylo-Marczewska, A.; Borowka, A. Langmuir 1999, 15, 6103. (20) Hildebrand, J. H.; Scott, R. L. The Solubility of Nonelectrolytes, 3rd ed.; Dover: New York, 1964 (reprint of 1950 ed.), p 221. (21) Allen, B. B.; Lingo, S. P.; Felsing, W. A. J. Phys. Chem. 1939, 43, 425. (22) Lange’s Handbook of Chemistry, 13th ed.; Dean, J. A., Ed.; McGraw-Hill: New York, 1985; pp 10-63, 10-66 to 68.
system
surface
S1 S2 S3 S4 S5 S6
silica gelb silica gelc silica gel, SG 2d silica gel, Si 100e silica gelc silica gelc
solution
DSmc/ %
acetone + carbon tetrachloride 2.4 methanol + toluene -4.7 benzene + cyclohexane -6.7 2-propanol + n-heptane -10.7 1-butanol + toluene -23.2 1-amyl alcohol + toluene -29.1
a All isotherms are U shape. The first component listed is preferentially adsorbed. In the remaining footnotes, the first number is the temperature and the second surface area. b 298 K, 150.1 m2/g.23 c 293.2 ( 0.2 K, 345 m2/g.24 d 303 ( 0.2 K, 380-420 m2/g.25 e 293.2 ( 0.2 K, 348 m2/g.19
At best, each small negative DSmc indicates a surface with a low degree of heterogeneity because, besides this factor’s contribution, there is also a negative contribution due to the positive deviation from Raoult’s law.20,22,26,27 For the carbon tetrachloride solution, the positive DSmc cannot be explained by the limited model system results because for this system also the deviation from an ideal solution26 is positive. Various Solutions on Various Carbons. In Table 4, DSmcs are shown for hydrocarbon solution pairs and both an alcohol-hydrocarbon and an alcohol-water solution on a variety of carbon surfaces. The range of DSmcs is from near -2 to -56%, almost double the negative range on the silica gels. Here, only one value is less than -12% and the rest are spread fairly evenly through the range. The wider negative DSmc range indicates that the carbons have a wider heterogeneity range than the silica systems have. Here too, the solutions deviate positively from Raoult’s law;20,22,26,28 thus each has a negative contribution from nonideality. The effects of heterogeneity are clear for these carbon systems because the descending DSmc sequence is roughly in order of increasing heterogeneity.31 The sequence starts with graphite and Graphon, carbons with nearly homogeneous surfaces of mostly interlinked hexagons in graphitic sheets. Then comes a carbon with some surface oxygen groups followed by a more porous carbon, active carbon, that likely has more surface oxygen functionalities. Next there is a surface in which the hexagon sheet edges have lost their oxygen functionalities during high-temperature heat treatment. Finally, the sequence ends with charcoal, a highly porous carbon with many surface oxygen groups that is adsorbing from the most complex solution on the list, a mixture of two hydrogen bonding compounds. Summary and Conclusion In addition to the qualitative methods of classifying adsorption systems currently available, a quantitative (23) Nasuto, R. Rocz. Chem. 1977, 51, 525. (24) Oscik, J.; Goworek, J. Pol. J. Chem. 52 1978, 52, 525. (25) Kagiya, T.; Sumida, Y.; Tachi, T. Bull. Chem. Soc. Jpn. 1972, 45, 1643. (26) Azeotropic Data. III; Horsley, L. H. (Compiler); American Chemical Society: Washington, DC, 1973; pp 51, 289, 373. (27) For only 1-amyl alcohol + toluene, a nonazeotrope,26 the indication of deviation from Raoult’s law is not direct. Since lower alcohol + aromatic hydrocarbon mixtures have positive heats of mixing,28 it is likely that the amyl alcohol solution also has a positive heat of mixing, thus also a positive deviation. (28) Rowlinson, J. S. Liquids and Liquid Mixtures; Buttersworth: London, 1959; pp 148, 178, 180. (29) Ash, S. G.; Brown, R.; Everett, D. H. J. Chem. Soc., Faraday Trans. 1 1975, 71, 123. (30) Kiselev, A. V.; Khopina, V. V. Trans. Faraday Soc. 1969, 65, 1936. (31) Coltharp, M. T.; Hackerman, N. J. Phys. Chem. 1968, 72, 1171.
Classification of Solution Adsorption Isotherms
Langmuir, Vol. 21, No. 8, 2005 3479
Table 4. DSmcs Calculated from Literature Isotherms: Various Solutions/Carbon Surfacesa system C1 C2 C3 C4 C5 C6
surface
solution
DSmc/%
Graphonc carbon black, heat treated: 800 °Cd active carbon* b carbon black, heat treated: 1700 °C* d charcoalb
benzene + cyclohexane benzene + cyclohexane toluene + n-heptane ethanol + benzene toluene + n-heptane ethanol + water
-1.8 -18.3 -25.0 -33.0 -37.7 -55.5
graphiteb
a Isotherms with an asterisk (*) are S shape; undesignated isotherms are U shape. The first component listed is preferentially adsorbed, or for S shapes is the major preferred component (component with the highest absolute adsorption). In the remaining footnotes, the first number is the temperature and the second surface area. b 298 K, area not given.18 c 313 K, 86 m2/g.29 d 293 K, area not given.30
measure DSmc that indicates a system’s deviation from ideal adsorption has been introduced, tested against model adsorption systems, and found applicable to actual systems. A given DSmc may include contributions from both surface heterogeneity and solution nonideality. Undoubtedly, because it is a single point measure, DSmc will not be sufficient for some purposes or some systems. However, even for such cases DSmcs should still be of help. DSmc values provide perspective for full isotherm comparisons as well as for recovery of surface energy
distributions and for evaluation of surface activity coefficients. As DSmcs accumulate, useful generalizations will be found. Further investigation should clarify DSmc dependence not only on surface heterogeneity and phase nonideality but also on molecular size differences of solution components. To test the tentative conclusions made here, DSmcs should be evaluated for a larger number of systems. LA047539L
