Letter pubs.acs.org/JPCL
Statistical Mechanics of Multilayer Sorption: Surface Tension Anthony S. Wexler* and Cari S. Dutcher† Air Quality Research Center, University of California, Davis, California 95616, United States ABSTRACT: Mathematical models of surface tension as a function of solute concentration are needed for predicting the behavior of surface processes relevant to the environment, biology, and industry. Current aqueous surface tension−activity models capture either solutions of electrolytes or those of nonelectrolytes, but a single equation has not yet been found that represents both over the full range of compositions. In prior work, we developed an accurate model of the activity−concentration relationship in solutions over the full range of compositions by extending the BET (Brunauer, Emmett, Teller) and GAB (Guggenheim, Anderson, de Boer) isotherms to multiple monolayers of solvent molecules sorbed to solutes. Here, we employ similar statistical mechanical tools to develop a simple equation for the surface tension−activity relationship that differs remarkably from prior formulations in that it (1) works equally well for nonelectrolyte and electrolyte solutes and (2) is accurate over the full range of concentrations from pure solvent to pure solute. SECTION: Surfaces, Interfaces, Porous Materials, and Catalysis
S
urface processes are important to the environment,1 nanotechnology,2 microfluidics,3 and biology,4 any circumstance where the kinetics or thermodynamics of liquid surfaces differs substantially from that in the bulk and where the surface area per volume is large, such as systems of nanoparticles. For gas−liquid interfaces, surface tension can drive flows via the Marangoni effect, alter vapor pressures via the Kelvin effect, and indicate the surface−bulk partitioning of surface-active soluble compounds. Recent models have advanced our understanding of the surface tension−composition relationship. Dutcher and co-workers5 developed such a model for electrolytes in water over the full concentration range from dilute to the melt. However, this model does not represent the surface tension− composition relationship for nonelectrolytes, such as alcohols, in water where small activities result in a dramatic drop in surface tension. Nearly simultaneously, Wang and co-workers6 merged separate models of electrolytes in water, nonelectrolytes in water, and mixed solvents into a framework that is remarkably successful at representing the surface tension−activity relationship for a wide range of concentrations and compositions. However, their model necessitates solute− solute interaction terms and use of disparate expressions to represent the influences of nonelectrolyte and electrolyte solutes, factors that can lead to significant predictive errors when extrapolated to highly complex multicomponent systems. In parallel to these surface tension model advances, Dutcher and co-workers7−9 extended the BET,10 GAB,11−13 and Ally and Braunstein14 single-monolayer isotherms to multiple monolayers, thereby representing solute hydration in solution. This model, referred to here as DGWC, assumes that water molecules sorb to solute molecules in multiple monolayers in equilibrium with each other and bulk water. The distinction between the multilayer of GAB and the multiple monolayers of DGWC is that with GAB, all layers in the multilayer have the same energy, whereas in DGWC, each monolayer has its own energy and configurational entropy. DGWC accurately represents activity−composition relationships in aqueous © 2013 American Chemical Society
solutions for multiple solutes over the full concentration range from infinite dilution to the pure liquid solute. Because DGWC successfully represents the bulk thermodynamics of complex aqueous solutions based on a sorption isotherm formalism, the question naturally arises as to whether such a model framework can also be employed to represent the surface tension−composition relationship. Such is the goal of this Letter. Following Dutcher and co-workers,7−9 we first derive the configurational entropy but here for the vapor−liquid interfacial surface. Consider a surface populated by water molecules where the maximum number of water molecules at the surface is defined as NWS. If each solute molecule displaces r water molecules from the surface, the surface partition function, ΩS, is then ΩS =
NWS!(r) (NWS − rNSS) !(r) (rNSS)!(r)
(1)
where NSS is the number of the solute molecules displacing water molecules from the surface and the !(r) denotes a multifactorial with skip of r. Why use a multifactorial in eq 1? Consider a solute molecule displacing a water molecule from the surface. The solute molecule may displace multiple water molecules and on average may displace a noninteger number of water molecules. The r skip represents this average number of waters displaced from the surface by the solute. The bulk partition function, ΩB, represents the interactions of the solute in the surface layer and solute in the bulk ΩB =
(NSS + NSB)! NS! = NSS! NSB! NSS! (NS − NSS)!
(2)
Received: April 4, 2013 Accepted: May 2, 2013 Published: May 2, 2013 1723
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Because the surface tension of pure water, σW, and the area occupied by a water molecule, SW, are known, eq 9 has three unknowns for each solute, r, K, and C. If the pure solute surface tension, σS, is known, eq 9 can be solved for C and evaluated at the limit of pure solute, with C = 1 − [1 − (1 − K) exp(rSW(σW − σS)/kT)]/K, removing C as a parameter. Likewise, if the pure solute surface tension is not known, this equation can be used to cast it as a parameter instead of C. As expected, the change in surface tension due to each solute is inversely proportional to the effective area occupied at the surface by each solute molecule, rSW. As a reminder, the parameter r is the number of solvent molecules displaced from the surface by each solute molecule, and C and K are parameters related to the energy of the multilayer and the monolayer, respectively7−9,11−13 Equation 9 can be written in a form σ = σW − kT/(SWr) ln[1 + KCaS/(1 − KaS)] which is similar to eq 13 of Wang and coworkers,6 which, for a single solute, can be written as σ = σW + kTΓσ,0 ln[1 − KaS/(1 + KaS)]. Here, r and Γσ,0 play a similar mathematical role in both equations as does K. The Wang form lacks the C parameter, which, recall from the derivation of eq 7, is related to the equilibrium between the solute in the bulk and the surface. Figures 1−4 compare model predictions to measurements for representative electrolyte and nonelectrolyte systems. Table 1 lists the parameter values for the binary electrolyte and
where NSB is the number of solute molecules in the bulk and NS = NSS + NSB is the total number of solute molecules. Combining the two partition functions in eqs 1 and 2, and using Stirling’s approximation, gives ⎛1 ⎞ ⎛ NS ⎞ NWS ln ΩSΩ B ≈ NWS⎜ ln ⎟ ⎟ + NS ln⎜ NWS − rNSS ⎠ ⎝r ⎝ NS − NSS ⎠ ⎛ (N − rN )(N − N ) ⎞ SS S SS ⎟ + NSS ln⎜ WS 2 rN ⎠ ⎝ SS
(3)
Note that Stirling’s approximation for multifactorials of skip r and large N is ln(N!(r)) ≈ (N/r)ln(N). Again, following Dutcher and co-workers,7−9 the energy of the system is E = −NW SεWS − NWBεWB − NSSεSS − NSBεSB = −NWSΔεWS − NW εWB − NSSΔεSS − NSεSB
(4)
where εWS, εWB, εSS, and εSB are the energies per molecule associated with water at the surface, water in the bulk, solute at the surface, and solute in the bulk, respectively, NW = NWS + NWB is the total number of water molecules, ΔεWS = εWS − εWB, and ΔεSS = εSS − εSB. Combining eqs 3 and 4, the Gibbs free energy of the system is approximately G E ≈ − ln ΩSΩ B kT kT
Table 1. Parameter Values for Equation 9
where kT is the Boltzmann constant times temperature. To obtain surface tension, we take the derivative of G with respect to surface area. Taking SW to be the area occupied by each water molecule, the surface area, A, is the sum of the area occupied by solute at the surface, rSWNSS, and water at the surface, SW(NWS − rNSS), so that A = SWNWS σ = σW
rN ⎞ kT ⎛ ln⎜1 − SS ⎟ + rS W ⎝ NWS ⎠
solute (solvent)a glycerol (water) water (glycerol) ammonium nitrate (water) ethanol (water) potassium nitrate (water)
(5)
where σW is the surface tension of pure water and the equality σW = −ΔεWS/kT was obtained by evaluating the derivative of G with respect to A at the limit NS = 0. We obtain the solute activity, aS, by taking the derivative of G with respect to NS
KaS = 1 −
NSS NS
2 ΔεSS rNSS = kT (NS − NSS)(NWS − rNSS)
1 − KaS rNSS = 1 − KaS(1 − C) NWS
(6)
⎞ 1 − KaS kT ⎛ ln⎜ ⎟ rS W ⎝ 1 − KaS(1 − C) ⎠
C = −0.84, K = 0.28
2
0.25
71.98
2
0.23
3
0.18
1
4.6
2
0.06
116.6 (predicted) 22.2 119.9 (predicted)
alcohol systems employed in these figures. For all examples, the activity−composition relationship in the bulk solution was obtained using the DGWC model.7 For examples where water is the solvent, kT/SW = 41 mN/m, based on T = 298 K and the commonly assumed area of a water molecule, 0.1 nm2. Glycerol is miscible in water; therefore, it can be used to explore not only glycerol dissolved in water but also water dissolved in glycerol to test the model for nonaqueous solvents. Figure 1, a two-parameter fit of eq 9, shows water as the solvent on one curve and glycerol as the solvent on the other, from data summarized by Strey and co-workers.15 Both capture the inflection in the curve of the surface tension as a function of activity. Ammonium nitrate is a simple 1:1 electrolyte with a very high solubility and significant atmospheric importance. Figure 2 shows the three-parameter fit (r, K, and σ) of the model (eq 9) to International Critical Tables data16 for the range of concentrations from 0.5 to 14 M. For solutes that increase
(7)
(8)
which, when employed in eq 5, gives σ = σW +
r = 3.0
62
For water as the solvent, kT/Sw = 41 mN/m, and for water as a solute in glycerol, kT/Sw = 1.22 mN/m. b% RMSE is the root-mean-square meas 2 1/2 p (σfit )] / percent relative error, equal to 100% × [∑ni=1 i − σi np meas σi , where np is the number of data points in the fit. ∑i=1
By combining eqs 6 and 7 and rearranging, we obtain 1−
r = 33.5, K = 0.914 r = −1, K = 0.994 r = −6.8, K = 0.99
% RMSEb
a
where εSB/kT ln K. To find the equilibrium partitioning of solute between the surface and bulk layers, we minimize the Gibbs free energy with respect to the amount of solute at the surface by setting to zero the derivative of G/kT with respect to the amount of solute at the surface, NSS, resulting in C ≡ exp
parameters
number of fit parameters
pure solute surface tension (mN/m)
(9) 1724
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Figure 1. Surface tension of aqueous glycerol illustrating that eq 9 can be used with water or glycerol as the solute. The decreasing curve shows glycerol as the solute; the increasing curve shows water as the solute. The lines are eq 9 using the parameters in Table 1 with data summarized by Strey and co-workers15 (squares).
Figure 3. Surface tension of the water−ethanol solution as a function of ethanol activity, comparing eq 10 using the parameters in Table 1 (line) with data from Ernst and co-workers17 (squares).
Figure 4. Surface tension of the water−KNO3 solution as a function of KNO3(aq) activity, comparing eq 11 using the parameters in Table 1 (line) with data from Abramzon and Gaukhberg18 (squares).
Figure 2. Surface tension of aqueous ammonium nitrate as a function of solute activity, comparing eq 9 using the parameters in Table 1 (line) with data from ref 16 (squares).
system over the concentration range of 1−2.63 M. The water− KNO3 system brings up another limit to eq 9, analogous to that of ethanol, where in this case, CK/(1 − K) ≪ 1. Solutes that satisfy this condition provide another simplified form of eq 9
the surface tension, r values are negative, indicating that the surface is depleted in solute with respect to the bulk and that the surface tension of the solution increases with increasing solute concentration, a common occurrence among nonacidic electrolytes in water. The next examples illustrate two limiting cases for eq 9. Figure 3 shows the fit for ethanol. Straight chain alcohols are known to partition to the surface of water;17 these have extremely small values of K and correspondingly large values of C. Why are the values of K and C linked? For finite values of C, as K → 0, the right-hand side of eq 9 reduces to σ = σW, indicating that the solute has no substantial influence on surface tension. However, alcohols lower surface tension dramatically; therefore, C must be correspondingly large to counterbalance the very small value of K. In this limit of extremely small K and correspondingly large C values, eq 9 reduces to a simpler form σ = σW −
kT ln(1 + KaS) rS W
σ = σW −
kT CaS S W 1 − KaS
(11)
In this limit, if the pure solute surface tension is known, C = (1 − K)SW(σW − σS)/kT, resulting in a single fit parameter, K. The fits for the ammonium nitrate−water system and the potassium nitrate−water system can be used to predict the effective pure solute surface tension of these salts by extrapolating subsaturated aqueous solution surface tensions to that of the pure solute. Predicted surface tensions of the pure salt melt are 116.6 and 119.9 mN/m, respectively. Dutcher and colleagues5 also predicted these values by extrapolating data obtained for the pure salt melt to lower temperatures, obtaining 117.6 and 133.7 mN/m, respectively, at 293 K. The remarkable agreement for ammonium nitrate is partially due to its high solubility and relatively low melting point. The agreement for potassium nitrate is not as good, but considering that both extrapolations were quite far apart in temperature (for Dutcher’s model) and concentration (the current model), the agreement is remarkably good. The derivation presented here results in an equation for surface tension as a function of solute activity that, unlike any
(10)
In this limit, if the pure solute surface tension is known, then K = exp[rSW(σW − σS)/kT] − 1, resulting in a single fit parameter, r. Figure 4 compares the model to a fit of measurements reported by Abramzon and Gaukhberg18 for the water−KNO3 1725
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(9) Dutcher, C. S.; Ge, X.; Wexler, A. S.; Clegg, S. L. Statistical Mechanics of Multilayer Sorption: Extension of the Brunauer− Emmett−Teller (BET) and Guggenheim−Anderson−deBoer (GAB) Adsorption Isotherms. J. Phys. Chem. C 2011, 115, 16474−16487. (10) Brunauer, S.; Emmett, P. H.; Teller, E. J. Adsorption of Gases in Multimolecular Layers. J. Am. Chem. Soc. 1938, 60, 309−319. (11) Guggenheim, E. A. Applications of Statistical Mechanics; Clarendon Press: Oxford, U.K., 1966. (12) Anderson, R. B. Modifications of the Brunauer, Emmett and Teller Equation. J. Am. Chem. Soc. 1946, 68, 686−691. (13) de Boer, J. H. The Dynamical Character of Adsorption; Clarendon Press: Oxford, U.K., 1968. (14) Ally, M. R.; Braunstein, J. Statistical Mechanics of Multilayer Adsorption: Electrolyte and Water Activities in Concentrated Solutions. J. Chem. Thermodyn. 1998, 30, 49−58. (15) Strey, R.; Viisanen, Y.; Aratono, M.; Kratohvil, J. P.; Yin, Q.; Friberg, S. E. On the Necessity of Using Activities in the Gibbs Equation. J. Phys. Chem. B 1999, 103, 9112−2116. (16) Washburn, E. International Critical Tables of Numerical Data, Physics, Chemistry and Technology; McGraw-Hill: New York, 1926− 1930; Vol. 4. (17) Ernst, R. C.; Watkins, C. H.; Ruwe, H. H. The Physical Properties of the Ternary System Ethyl Alcohol−Glycerin−Water. J. Phys. Chem. 1936, 40, 627−635. (18) Abramzon, A. A.; Gaukhberg, R. D. Surface Tension of Salt Solutions. Russ. J. Appl. Chem. 1993, 66, 1315−1320.
prior model, works equally well for electrolytes in water, nonelectrolytes in water, and nonaqueous solvents. One monolayer and one multilayer, same as the GAB isotherm in solution, are sufficient in the statistical mechanics to accurately represent the surface tension−activity relationship. The differences between the model derived here and GAB arise from incorporating the effective relative size of solute and solvent molecules, a unique approach in this field. Although the examples focus on aqueous solutions, the water−glycerol example demonstrates that the choice of solute or solvent is arbitrary and suggests that eq 9 represents the surface tension− activity relationship for arbitrary solvents. In summary, eq 9 is an entirely new formulation that, with only 1−3 parameters captures the surface tension−activity relationship over the full concentration range from dilute to pure solute for single− solute electrolyte or nonelectrolyte solutions.
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Author Contributions
The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript. Funding
This work was supported by the Department of Energy Atmospheric Systems Research program and the Electric Power Research Institute. C.S.D. was supported through a National Science Foundation Atmospheric and Geospace Sciences Postdoctoral Research Fellowship. Notes
The authors declare no competing financial interest. † E-mail:
[email protected] (C.S.D).
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ACKNOWLEDGMENTS We would like to thank Simon Clegg for his helpful critiques of the manuscript.
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