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Langmuir 2009, 25, 2090-2100
Monte Carlo Simulation of Surfactant Adsorption on Hydrophilic Surfaces T. Zehl,* M. Wahab, P. Schiller, and H.-J. Mo¨gel Institute of Physical Chemistry, TU-Freiberg, Leipziger Strasse 29, 09596 Freiberg, Germany ReceiVed April 24, 2008. ReVised Manuscript ReceiVed December 3, 2008 Monte Carlo simulations have been carried out to study the adsorption behavior of small flexible amphiphilic molecules on solid surfaces from aqueous solutions. A simple coarse-grained solvent-free off-lattice model, with a square-well pair potential and hard core excluded volume effect, has been used. Adsorption isotherms for weakly and strongly hydrophilic homogeneous surfaces have been determined. The adsorbed layer displays a coexistence region with an upper critical point. Below the critical temperature a densely packed patch coexists with a two-dimensional gas-analogous phase. Above the critical temperature, a percolating network forms at higher surfactant concentrations. Depending on the ratio between the strength of the hydrophobic effect and the adsorption energy, a large variety of associates has been observed. Monolayers, bilayers, admicelles, small clusters, and percolating networks as typical associate structures have been found. In the four-region model, which is extended by the coexistence region, a characteristic adsorbed layer structure for each region can be detected. Intermediate structure types have been produced by variation of the adsorption energy.
1. Introduction The adsorption of amphiphilic molecules on solid surfaces from aqueous solutions is the crucial effect for a number of technical applications of surfactants covering a wide range of processes such as hydrophobization of surfaces, stabilization of colloidal dispersions, protection of surfaces against particular chemical reactions, ore flotation, paint technology, and enhanced oil recovery. For this reason, much work has been done to study and understand basic rules of the surfactant adsorption from aqueous solutions on inorganic solid surfaces. Typical adsorbents with well characterized hydrophilic surfaces are mineral oxides as alumina, silica, rutile, and mica or technical products as silicon wafers. In technical applications, the adsorption can be used to achieve an appropriate partition of surfactants between the solution phase and the adsorbate in favor of the adsorbed phase. A second group of applications utilizes particular properties of the adsorbed layer, which are caused and modified by the layer structure. While the first aspect can be well described by measuring the adsorption isotherms in depletion experiments, it is more difficult to obtain experimental data that are suitable to draw reliable conclusions on the structure of the adsorbate. Helpful experimental techniques are the atomic force microscopy,1 fluorescence decay spectroscopy,2,3 ellipsometry,4 Fourier transform infrared spectroscopy/attenuated total internal reflection spectroscopy,5 and neutron reflection technique.6 Some of these methods are restricted to the investigation of systems above the critical micelle concentration. Analyzing the experimental results can be supplemented by theoretical evaluations and modeling to elucidate the adsorbed layer structure and its properties. The huge number of measurements of adsorption isotherms is regularly evaluated in comprehensive review articles.7-12 * Corresponding author. E-mail:
[email protected]. (1) Manne, S.; Gaub, H. E. Science 1995, 270, 1480. (2) Levitz, P.; Van Damme, H.; Keravis, D. J. Phys. Chem. 1984, 88, 2228. (3) Levitz, P.; Van Damme, H J. Phys. Chem. 1986, 90, 1302. (4) Tiberg, F. J. Chem. Soc., Faraday Trans. 1996, 92, 531. (5) Singh, P. K.; Adler, J. J.; Rabinovich, Y. I.; Moudgil, B. M. Langmuir 2001, 17, 468. (6) Lee, E. M.; Thomas, R. K.; Cummins, P. G.; Staples, E. J.; Penfold, J.; Rennie, A. R. Chem. Phys. Lett. 1989, 162, 196. (7) Zhu, B.-Y.; Gu, T. AdV. Colloid Interface Sci. 1991, 37, 1.
Isotherms are frequently grouped with regard to their shape as S-type, L-type, or SL-type ones. The Langmuir isotherm (Ltype) arises from the S-type one by a particular parameter choice. S-type isotherms can be divided in 4 regions when plotted on a log-log scale. According to this four-region model,13 the surfactants adsorb as individual molecules on single surface sites at low surfactant concentrations. This first region is governed by Henry’s law. It is characterized by the linear relation between surfactant surface concentration and solution concentration. After a second concentration region with strong rise of adsorption follows a third region with essentially weaker increasing of adsorption up to reaching a plateau region with constant adsorbed amount. In some cases, the fourth region can contain a weak maximum before arriving at the plateau. The four-region model comprises the single molecule adsorption in the Henry region and the onset of the cooperative hydrophobic interaction that already starts at concentrations much smaller than the critical micelle concentration. The cooperativity causes the surfactants to associate into clusters, patches and adsorbed two-dimensional phases of various structures. Typical suggested structures are patches of monolayers, bilayers, and admicelles. Monolayer patches of molecules with head directed toward the solid surface and with tail protruding into the liquid are termed hemimicelles.14 Admicelles are micelles which are anchored by one or more surfactant heads on the surface. Particularly for the adsorption of ionic surfactants at oppositely charged solid surfaces, the S-shape of many experimental adsorption isotherms have been explained by supposing surface aggregates being two-dimensional analogues of bulk associates. It should be noted, however, that the assumed types of surface aggregates, namely monolayers, bilayers, admicelles, and hemimicelles, are idealized borderline cases. In dependence on the strength of the hydrophobic effect, (8) Zhang, R.; Somasundaran, P. AdV. Colloid Interface Sci. 2006, 123–126, 213. (9) Somasundaran, P.; Huang, L. AdV. Colloid Interface Sci. 2000, 88, 179. (10) Atkin, R.; Craig, V. S. J.; Wanless, E. J.; Biggs, S. AdV. Colloid Interface Sci. 2003, 103, 219. (11) Paria, S.; Khilar, K. C. AdV. Colloid Interface Sci. 2004, 110, 75. (12) Rosen, M. J. Surfactants and Interfacial Phenomena; Wiley-Interscience: New York, 2004. (13) Somasundaran, P.; Fuerstenau, D. W. J. Phys. Chem. 1966, 70, 90. (14) Gaudin, A. M.; Fuerstenau, D. W. Trans. AIME 1955, 202, 958.
10.1021/la8020595 CCC: $40.75 2009 American Chemical Society Published on Web 01/21/2009
Simulation of Surfactant Adsorption
which is influenced by the molecular properties, and the adsorption energy, a large diversity of intermediate structures is expected to occur. Temperature and surfactant concentration specify the actual type of the observed structure. Most of experimental methods are not capable to determine the layer structure, but they can give advices for discriminating between plausible structure models. Theoretical considerations and computer modeling can essentially support the interpretation of experimental results. There are three different types of theoretical approaches to tackle the problem. The first one bases on the assumption of some kind of chemical equilibrium between surfactants in the solution and an admicelle which is anchored by one molecule on the solid surface.15 The second one is the so-called twodimensional condensation theory,16,17 which is in fact an extension of the Fowler isotherm that includes lateral molecular interactions to the surfactant adsorption.18,19 Apart from the Frumkin-Fowler ansatz there are also published other analytical self-consistent field calculations.20-22 A third approach is based on the assumption of two-dimensional molecular clusters which interact as hard particles on the surface.23-25 Their phase behavior has been described by the scaled particle theory. The BET-theory has been adapted to the surfactant adsorption on solids.26 In addition to analytical evaluations, surfactant adsorption can be investigated by Monte Carlo27-30 or Molecular Dynamics simulations.31 Computer experiments provide immediate and comprehensive information on the structure of adsorption layers. In this paper, we present results of off-lattice Monte Carlo simulations for the surfactant adsorption on hydrophilic solid surfaces with a simple coarse-grained surfactant model. This model was already used for investigating the self-assembling of surfactants in aqueous solutions.32 It provided similar results as Molecular Dynamics simulations that use a more complicated potential.33 The model system is an off-lattice version of a previously introduced lattice model used for surfactant adsorption.29,30 The simulation results reproduce the typical isotherm shape and give some insight into the large variety of possible structures in adsorbed surfactant layers. Given a simple molecular structure, both the adsorption energy and the strength of the hydrophobic effect determine basically the possible types of the surfactant layer structure. The structures actually formed in surfactant layers are ruled by the temperature and the surfactant concentration. Our simulation results support two-dimensional (15) Zhu, B.-Y.; Gu, T. J. Chem. Soc., Faraday Trans I 1989, 85, 3813. (16) Cases, J. M.; Villieras, F. Langmuir 1992, 8, 1251. (17) Hankins, N. P.; O′Haver, J. H.; Harwell, J. H. Ind. Eng. Chem. Res. 1996, 35, 2844. (18) Scamehorn, J. F.; Schechter, R. S.; Wade, W. H. J. Colloid Interface Sci. 1982, 85, 463. (19) Harwell, J. H.; Hoskins, J. C.; Schechter, R. S.; Wade, W. H. Langmuir 1985, 1, 251. (20) Bo¨hmer, M. R.; Koopal, L. K. Langmuir 1992, 8, 1594. (21) Bo¨hmer, M. R.; Koopal, L. K. Langmuir 1992, 8, 2649. (22) Bo¨hmer, M. R.; Koopal, L. K.; Janssen, R.; Lee, E. M.; Thomas, R. K.; Rennie, A. R. Langmuir 1992, 8, 2228. (23) Lajtar, L.; Narkiewicz-Michalek, J.; Rudzinski, W.; Partyka, S. Langmuir 1993, 9, 3174. (24) Drach, M.; Lajtar, L.; Narkiewicz-Michalek, J.; Rudzinski, W.; Zajac, J. Colloids Surf. A 1998, 145, 243. (25) Drach, M.; Andrzejewska, A.; Narkiewicz-Michalek, J. Appl. Surf. Sci. 2005, 252, 730. (26) Gu, T.; Rupprecht, H.; Galera-Gomez, P. A Colloid Polym. Sci. 1993, 271, 799. (27) Wijmans, C. M.; Linse, P. J. Phys. Chem. 1996, 100, 12583. (28) Zheng, F.; Zhang, X.; Wang, W. Langmuir 2008, 24, 4661. (29) Reimer, U.; Wahab, M.; Schiller, P.; Mo¨gel, H.-J. Langmuir 2001, 17, 8444. (30) Reimer, U.; Wahab, M.; Schiller, P.; Mo¨gel, H.-J. Langmuir 2005, 21, 1640. (31) Shinto, H.; Tsuji, S.; Miyahara, M.; Higashitani, K. Langmuir 1999, 15, 578. (32) Zehl, T.; Wahab, M.; Mo¨gel, H.-J.; Schiller, P. Langmuir 2006, 22, 2523. (33) Cooke, I. R.; Kremer, K.; Deserno, M. Phys. ReV. E 2005, 72, 11506.
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Figure 1. (a) Sketch of the flexible molecule model HTT consisting of two hydrophobic segments T (yellow) and a hydrophilic head segment H (blue). The angle β can vary between 0° and 120°. (b) Contour plot of the adsorption energy for a head at a distance z of the surface which is represented by surface spheres (---).
condensation theories. A two-phase region with an upper critical point is found in the adsorbed layer. The corresponding equilibrium is simulated in the canonical ensemble. Layer structures are also studied in the grand canonical ensemble. The paper is organized as follows: in the second section, the model and the simulation conditions are explained. In the following part, the adsorption isotherms and adsorbed surfactant layer structures obtained from simulations upon different adsorption conditions are presented and discussed. The paper is closed by conclusions.
2. Model and Simulation Details The thermodynamic equilibrium of surfactant adsorption from aqueous solutions on solid surfaces is governed by complex molecular interactions responsible for the hydrophobic effect and the adsorption competition between surfactants and water. On the atomistic scale of resolution, the hydrophobic effect is well-known to result from different attractive interactions between polar and nonpolar molecular parts as well as from considerable restructuring of the water network in the environment of an amphiphilic molecule. Consequently, on the thermodynamic level, the hydrophobic effect has to be described as an entropic one.34 For this reason, computer simulations of the aggregation behavior in surfactant solutions require applying effective interactions between molecular segments of coarse-grained models. Effective interactions could be formally obtained by integration procedures suitable to average over degrees of freedom on the atomic scale.35 In our simulations, we use a water-free coarse-grained model. The complex balance of molecular interactions is transformed into a simple mutual attraction between hydrophobic molecular parts.36 The amphiphilic molecule model is a chain of hydrophobic and hydrophilic spheres. The hydrophilic sphere represents the headgroup including some solvation shell together with the counter-ions. Each hydrophobic sphere resembles several CH2groups. All spheres are impenetrable and equally sized. To study the basic features of surfactant adsorption, we use the simplest flexible molecule consisting of one hydrophilic head (H) and two hydrophobic tail (T) segments (see Figure 1a). In addition to the excluded volume interaction, the only intermolecular interaction is the attraction between the segments of different molecules. Defining central forces between them, the effective pair interaction is expressed as square-well potential (34) Tanford, C. The Hydrophobic Effect: Formation of Micelles and Biological Membranes; Krieger Publ. Comp.: Malabar, 1991. (35) Nielsen, S. O.; Klein M. L. A Coarse Grain Model for Lipid Monolayer and Bilayer Studies. In Bridging Time Scales: Molecular Simulations for the Next Decade; Nielaba, P., Mareschal, M. , Ciccotti G., Eds.; Springer: Berlin, 2002. (36) Reimer, U.; Zehl, T.; Wahab, M.; Schiller, P.; Mo¨gel, H.-J. Colloid Surf. A 2006, 290, 25.
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{
∞ for reσ u(r) ε for σ < r e σ + d 0 for r>σ+d
Zehl et al.
(1)
where σ describes the sphere diameter, characterizes the attraction strength, and d the range of particle attraction. The adsorbing solid surface is built up by a surface of spheres packed in a rectangular lattice arrangement. These spheres interact with hydrophilic head spheres via square well potential (1), where is replaced by a different parameter ad. The attractive interaction between one head segment and the surface requires a more sophisticated graphic representation (Figure 1b). The adsorption energy comprises all pair interactions between a surfactant head and surface spheres. Due to the packing of surfaces spheres, the adsorption energy varies locally. The adsorption energy depends on the head-surface distance z and on lateral coordinates x, y. The contour plot shown in Figure 1b refers to the case that the head is localized in a (z,x)-plane which coincides with the centers of next neighbor surface spheres. The effective Hamiltonian is defined as the sum over all interactions. Interaction energies as well as the temperature T are scaled by the same arbitrary units so that they are dimensionless. The simulations are carried out in a cubic box with an edge length L ) 48σ. Periodic boundary conditions in two directions x, y lateral to the adsorbing surface are applied. In the direction z normal to the surface, a layer of fixed water-like segments is added to close the upper side of the box. The simulations were performed in the canonical or grand canonical ensemble. The Monte Carlo procedure is based on the Metropolis algorithm.37-39 As starting point for the Monte Carlo simulations, the molecules were placed randomly in the simulation box. Using the Metropolis algorithm, the coordinates of the molecules were manipulated by at least 15*106 Monte Carlo steps (MCS) to reach thermodynamic equilibrium. A Monte Carlo step is a sequence of n attempts to move molecules, with n being the number of molecules. After this equilibration, further 10 × 106 MCS are applied to generate equilibrium configurations, which are used to determine aggregation structures and to measure geometric and thermodynamic properties. The average over a large number of equilibrium system configurations is denoted by 〈〉 in the formulas. The surfactant bulk concentration φ is determined in a subvolume of the box placed between fictive planes at the heights z ) 20σ and z ) 40σ. φ is expressed in molecule numbers N per volume according to
φ)
〈N〉 48220σ3
(2)
This definition ensures good averaging without strong boundary effects. The surface concentration R is evaluated from the number of molecules Nads in clusters which contact the solid surface per surface area
R)
〈Nads 〉 482σ2
(3)
A molecule is allocated to a definite cluster if one of its hydrophobic segments interacts with a hydrophobic segment of another molecule in the same cluster. The orientation order parameter S is evaluated from the angle θ between the axis perpendicular to the solid surface and the head-tail vector between (37) Metropolis, N. A.; Rosenbluth, A. W.; Rosenbluth, M. N.; Teller, A. H.; Teller, E. J. Chem. Phys. 1953, 21, 1087. (38) Frenkel, D.; Smit, B. Understanding molecular simulation; Academic Press: London, 2002. (39) Landau, D. P.; Binder, K. Monte Carlo Simulations in Statistical Physics; Cambridge Univ. Press: Cambridge, U.K., 2000.
Figure 2. Adsorption isotherms for different temperatures T. Closed symbols represent NVT and open symbols µVT simulation data. The enlarged section shows the Henry region. Key: red, coexistence curve; blue, isotherms above critical temperature; R, surfactant surface concentration; φ; surfactant bulk concentration.
the centers of the terminal spheres in the model molecule as the average of the second Legendre polynomial P2(cos θ)
1 S ) (3〈cos2 θ〉 - 1) 2
(4)
For evaluating the head density profile PHEAD(z) of adsorbed layers, the box is divided in fictive layers with width ∆z. PHEAD(z) is the number of head segments in the zth layer divided by the number of all head segments in the adsorbed layer. Finally, we define the hydrophobicity as the hydrophobic surface area fraction which is determined by the following procedure: The (x,y)-surface plane is divided into equal quadratic cells with the edge length σ/2. A probe sphere with radius σ approaches the adsorbate layer until a surfactant segment or a surface segment is contacted. The path of the approach is antiparallel to the cell normal which is positioned with its origin in the center of each cell. If a tail segment is hit by this move the cell is considered as hydrophobic, in the other case as hydrophilic. The hydrophobicity is defined as the fraction of hydrophobic cells.
3. Results and Discussion The adsorption of surfactants from aqueous solutions on hydrophilic solid surfaces has been studied by Monte Carlo simulations both in the canonical and in the grand canonical ensemble. The canonical ensemble with constant volume, constant temperature, and constant number of surfactant molecules in the simulation box represents the experimental conditions when adsorption isotherms are recorded by utilizing the depletion method. In this method, the adsorbens is added to the solution with a certain initial surfactant concentration. After finishing the adsorption process, the amount of adsorbed surfactant is calculated from the difference of surfactant concentration in the bulk solution before and after adsorption. The partition of the surfactant between the surface layer and the solution volume at constant temperature is quantitatively described by the adsorption isotherm with surfactant surface concentration R and bulk concentration φ as variables. Adsorption isotherms and structures of adsorbed surfactant layers have been determined for two different parameter sets of the square-well pair potential. The interaction parameters ) -1.0, d ) 0.73 correspond to molecules which are capable to form bilayer structures as well as micelles in the bulk phase, whereas the parameter set ) -3.1, d ) 0.1 belongs to molecules which prefer to associate into micelles, in accord with the phase diagram (see Figure 2 in ref 32). The adsorption of surfactant
Simulation of Surfactant Adsorption
molecules that form layer structures is studied for several values of the adsorption energy parameter ad. 3.1. Adsorption of Molecules Which Are Capable To Form Bilayer Structures as well as Micelles. 3.1.1. Adsorption Isotherms in the Context of the Four-Region Model. First we consider the adsorption behavior of the surfactants using the parameter set ) -1 and d ) 0.73 for the square well potential. For the adsorption energy parameter ad ) -0.5, the isotherms evaluated from simulations in the canonical and the grand canonical ensembles are depicted in Figure 2. Two different shapes of isotherms can be distinguished. At high temperatures, the curves are S-shaped. The positive slope near their inflection point continuously rises with decreasing temperature. For lower temperatures, there is a second kind of isotherms. Simulations within the grand canonical ensemble with fixed chemical potential of the surfactant provide isotherms with a jump in the region of medium surfactant bulk concentrations. Obviously, there are two adsorption isotherm branches indicating a first order phase transition in the adsorbed surfactant layer at low temperatures. Both branches of the isotherms have been obtained and used to construct the coexistence curve with an upper critical point (Figure 2). The critical data are estimated as Tc ) 1.19, Rc ) 0.33, and φc ) 0.005. With the exception of the coexistence region, the isotherms from simulations in the canonical and the grand canonical ensembles coincide. It should be mentioned that, in contrast to the interpretation of the adsorption isotherm obtained from lattice MC simulations in,28 both isotherm branches are absolutely stable. This statement also holds for adsorbed patches in the coexistence region. In this region, curves obtained from the canonical ensemble display a very steep incline or even a weakly backward course. Such isotherms with a part directed backward have also been obtained from Monte Carlo simulations of surfactant adsorption in lattice models.28,29 This behavior is analogous to the decreasing concentration of single surfactant molecules with increasing surfactant concentration in solutions just above the critical micelle concentration. Such behavior has been detected experimentally40,41 and in computer simulations, too.42-48 This effect can be ascribed to the cooperative nature of surfactant association and comes from using concentration instead of activity as thermodynamic variable. Negative slopes of isotherms are typical for the equilibration in multiphase systems with conserved particle number, where interacting particles are partitioned between different phases. The thermodynamic features of surfactant adsorption are confirmed by visualizing qualitatively different patterns of surface coverage at low and high temperatures in Figure 3. At low temperatures, a two-dimensional patch of a dense phase coexists with a dilute gas-analogous phase on the surface. In the high temperature region, small clusters are formed. With increasing concentration, this dilute phase transforms into a two-dimensional percolating surfactant network up to saturation at higher concentrations without passing any discontinuous phase transition. The percolation threshold coincides with the inflection point of the adsorption isotherm. We can distinguish three different cases (40) Kahlweit, M.; Teubner, M. AdV. Colloid Interface Sci. 1980, 13, 1. (41) Johnson, I.; Olofsson, G.; Jo¨nsson, B. J. Chem. Soc., Faraday Trans 1 1987, 83, 3331. (42) Wijmans, C. M.; Linse, P. J. Chem. Phys. 1997, 106, 328. (43) Brindle, D.; Care, C. M. J. Chem. Soc. Faraday Trans. 1992, 88, 2163. (44) Wang, Y.; Mattice, W.; L.Napper, D. H Langmuir 1993, 9, 66. (45) Rodriguez-Guadarrama, L. A.; Talsania, S. K.; Mohanty, K. K.; Rajagopalan, R. Langmuir 1999, 15, 437. (46) Von Gottberg, F. K.; Smith, K. A.; Hatton, T. A. J. Chem. Phys. 1997, 106, 9850. (47) Zaldivar, M.; Larson, R. G. Langmuir 2003, 19, 10434. (48) Mackie, A. D.; Panagiotopoulos, A. Z.; Szleifer, I. Langmuir 1997, 13, 5022.
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Figure 3. Snapshots of aggregates adsorbed on the surface above and below the critical temperature at different surfactant surface concentrations R (red, surface; blue, head sphere; yellow, tail sphere).
of multiphase equilibrium: (a) two-phase equilibrium between a low density surface phase and the bulk solution, (b) two-phase equilibrium between high density surface phase and surfactant solution, and (c) three-phase equilibrium between low density surface phase, high density surface phase, and the bulk solution. Our off-lattice simulations of the surfactant adsorption on hydrophilic surfaces confirm the main thermodynamic features resulting from earlier Monte Carlo simulations on a simple cubic lattice.29,49-51 It concerns the general shape of adsorption isotherms and the existence of a critical point. Coarse-grained flexible amphiphilic molecules consisting of two hydrophobic and a hydrophilic segment were used to simulate the adsorption equilibrium between the bulk solution and the surface layer in a simple cubic lattice model. The sub- and supercritical isotherms are very similarly shaped to those presented in Figure 2. Concerning the structure changes of adsorbed surfactant layers with the surface concentration below and above the critical temperature, we also find a qualitative agreement to the lattice simulation results for the distinct adsorption regimes. Recently, Zheng et al. 28 published results of Monte Carlo simulations for the adsorption of 6- and 7-segment surfactant molecules on hydrophilic surfaces. They used the Larson model52 with 26 nearest neighbors per lattice site in combination with a gauge cell method.53,54 Their isotherms and the general structure of the adsorbed surfactant layers qualitatively agree with the results of Reimer et al.29,49-51 However, because of misleading interpretations in28 incorrect conclusions have been drawn there. This point is discussed in more detail in the Supporting Information. The adsorption isotherms from our simulations presented in Figure 2 are also in accord with the thermodynamic adsorption model of Cases and Vielleras.8 This model is based on a modification of the Frumkin-Fowler theory55 which takes into account lateral interactions between adsorbate molecules in the Bragg-Williams approximation for lattice models. Various selfconsistent field theories20-22,56-58 provide similar thermodynamic results. S-shaped adsorption isotherms are typical for the four(49) Reimer, U. Ph.D. Thesis, TU Freiberg, 2002. (50) Mo¨gel, H.-J.; Schiller, P.; Reimer, U.; Wahab, M. 11th Int. Conf. Surf. Coll. Sci. 2003 Foz do Iguacu (Brazil). (51) Mo¨gel, H.-J.; Schiller, P.; Zehl, T.; Reimer, U.; Wahab Sitzungsberichte der Sa¨chs. Akad. Wiss. Math.-nat. Klasse 2005, 130 Heft 2. (52) Larson, R. G. J. Chem. Phys. 1992, 96, 7904. (53) Neimark, A. V.; Vishnyakov, A. J. Chem. Phys. 2005, 122, 054707. (54) Neimark, A. V.; Vishnyakov, A. J. Chem. Phys. 2005, 122, 174508. (55) Desjonqueres, M.-C.; Spanjaard, D. Concepts in Surface Physics; Springer: Berlin, 1996. (56) Evers, O. A.; Scheutjens, J. M. H. M.; Fleer, G. J. Macromolecules 1990, 23, 5221. (57) Leermakers, F. A. M.; Scheutjens, J. M. H. M. J. Chem. Phys. 1988, 89, 3264. (58) Leermakers, F. A. M.; Scheutjens, J. M. H. M. J. Collod Interface Sci. 1990, 136, 231.
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Table 1. Partition Coefficient K, the Upper Limits of the Henry Region r1 and O1 for Different Temperatures T ) 1.00 φ1 R1 K
-4
5.17 × 10 2.59 × 10-3 4.947
T ) 1.10 -4
7.07 × 10 2.82 × 10-3 3.864
T ) 1.15 -4
8.87 × 10 3.22 × 10-3 3.605
region model59,60 termed according to the four isotherm regions in log-log-scale plots. It is appropriate to discuss different structures of adsorbed phases together with the corresponding sections of the adsorption isotherm. The only modification needed is the inclusion of an additional phase coexistence in the region II at low temperatures. Henry Region (I). At low surfactant bulk concentrations, there is a Henry adsorption regime. Single molecules are adsorbed on the surface forming a two-dimensional gas-analogous phase. The surfactant partition between the surface phase and the bulk solution obeys Henry’s law with the partition coefficient K(T) ) R/φ, which remains constant up to a surface concentration R1(T) and a solution concentration φ1(T). Their values are listed in Table 1. Surfactant adsorption is a complex process which consists mainly in the displacement of ions or water molecules by the head groups of single molecules. It is accompanied by the rearrangement of the counter-ions, by the change of the dynamic structure of the water network, various dissociation reactions and so on. All these processes attribute to the displacement enthalpy, which can be determined by calorimetric measurements61-64 or from the temperature dependence of Henry’s partition coefficient. In our simulations, the energy change due to complex adsorption phenomena is represented by the effective interaction energy ad multiplied by the number of contacts between a head segment and surface segments. According to the relation,
d(ln K) ∆Hisost ) dT T2
(5)
we get from our simulations the molar isosteric displacement enthalpy ∆Hisost ) -2.36. This value is equivalent to 4.72 contacts of an adsorbed molecule with neighbored surface spheres. It corresponds to the exothermic effect obtained experimentally for the initial anchoring of surfactant molecules on the solid surface.55 The Henry region is limited by the perceptible onset of the hydrophobic effect, which results in cluster formation on the surface at surfactant surface concentrations larger than R1. The experimental differential enthalpy of displacement tends to zero or even to small positive values at concentrations beyond the Henry region. Region II. There is the common view that the dramatic increase of the adsorption in this region is caused by surfactant clustering on the surface due to the hydrophobic effect. However, the associates which give reason for increasing slope of the isotherms are supposed to possess very different structures.7,13,19,60,65,66 In our simulations, we observe small clusters which increase in number and size with increasing bulk concentration. At subcritical temperatures, this stage is limited by the coexistence curve. At supercritical temperatures, this cluster phase transforms continu(59) Fuerstenau, D. W.; Jang, H. M. Langmuir 1991, 7, 3138. (60) Fan, A.; Somasundaran, P.; Turro, N. J Langmuir 1997, 13, 506. (61) Denoyel, R.; Rouquerol, J. J. Colloid Interface Sci. 1991, 143, 555. (62) Stodghill, S. P.; Smith, A. E.; O’Haver, J. H. Langmuir 2004, 20, 11387. (63) Partyka, S.; Rudzinski, W.; Brun, B.; Clint, J. H Langmuir 1989, 5, 297. (64) Pettersson, A.; Rosenholm, J. B. Langmuir 2002, 18, 8436. (65) Gao, Y.; Du, J.; Gu, T. J. Chem. Soc., Faraday Trans. 1 1987, 83, 2671. (66) Yeskie, M. A.; Harwell, J. H. J. Phys. Chem. 1988, 92, 2346. (67) Koopal, L. K.; Lee, E. M.; Bo¨hmer, M. R. J. Colloid Interface Sci. 1995, 170, 85.
T ) 1.20 -4
7.13 × 10 2.36 × 10-3 3.259
T ) 1.25
T ) 1.30 -4
10.73 × 10 3.31 × 10-3 3.104
12.63 × 10-4 3.60 × 10-3 2.830
Figure 4. Contour plot of the patch thickness and cross section of the bilayer in the marked region for a snapshot at T ) 1.0, R ) 0.231.
ously with increasing concentration up to the percolation threshold. The percolation threshold data coincide with the inflection point of the adsorption isotherm. In contrast to the structure assumption by Zhu et al.,7 we find no clusters with typical micelle structure in this region. Furthermore, the cluster size distributions are strictly monotonic decreasing at each point of the isotherm. There is no sharp micelle size distribution on the surface. The structures of the adsorbed surfactant layers obtained from the lattice and the off-lattice model are essentially different. Due to orientation restrictions of the lattice model in,29 structures have been obtained which could be considered as admicelles below and as incomplete bilayer above the percolation threshold at supercritical temperatures. The off-lattice model shows more reliable that a percolating network of admicelles instead of a bilayer is formed above the critical temperature. The common features of our off-lattice simulation results and lattice simulation findings28,29,50,51 in region II are the onset of twodimensional cluster formation with weak cluster- cluster interactions. Below the critical temperature the cluster formation proceeds until attaining the coexistence curve, while above the critical temperature the layer is filled up until percolation is achieved. We do not agree with the interpretation in,28 where the percolation threshold is considered as starting point of a morphological transition (for details see Supporting Information). Coexistence Region IIa. Below the critical temperature, coexistence between two two-dimensional surfactant phases and the surfactant solution can be observed. The gas-analogous surface phase is in equilibrium with a compact patch, which increases up to the complete coverage of the surface with rising molecule number in the box. The occurrence of the condensed phase requires a minimal patch size to compensate the boundary energy, e.g., at T ) 1.0 about 50 molecules are necessary. The patch is an asymmetric bilayer, where the bottom layer possesses both a larger area density and a stronger orientation order than the upper layer. The upper layer displays surface undulations (Figure 4). In lateral direction, the boundary region of the patch differs from the interior in area density and energy, as it is shown in Figure 5. This observation can be quantitatively described by the
Simulation of Surfactant Adsorption
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Figure 5. (a) Local area density F(D) and (b) Excess energy Eexc(D) in patches as function of distance D of the patch boundary at T ) 1.0, R ) surfactant surface concentration.
Figure 6. Excess energy per surfactant ∆Eexc versus number of particles Np in the patch at T ) 1.0: shadowed, patch region; nonshadowed, region of layers with hole.
local area density F(D) and the excess energy per molecule Eexc(D) where D is the distance to the patch boundary. To determine these functions, first a quadratic grid is overlaid on the patch. The centers of the grid sectors at the patch boundary have the coordinate D ) 0. The internal sectors have coordinates corresponding to the minimal distance D of any boundary segment. For each D, the number of molecules N(D), the energy per molecule E(D), and the area of all sectors A(D) are evaluated. The averaged ratio 〈N(D)/A(D)〉 is the local area density F(D), and the average 〈E(D)/N(D)〉 is the mean energy per molecule at distance D to the patch boundary. The local excess energy per molecule Eexc(D) is defined as difference between 〈E(D)/N(D)〉 and the energy per molecule in a bilayer at the coexistence curve. A further excess energy per surfactant molecule ∆Eexc can be considered for the whole patch. ∆Eexc is the difference of energies per surfactant molecule between the values for the given patch and the layer at the coexistence curve. As expected, this excess energy decreases with increasing patch size Np (Figure 6). Due to the periodic boundary conditions applied to the simulation box, the patch can transform from an island into a percolating film with a hole. The transition between the two shapes is indicated by the inflection point of the adsorption isotherm. The structural properties of bilayer patches qualitatively resemble those of the corresponding patches in lattice simulations.29,50,51 The morphological transition from circular or spherical micelles to cylindrical micelles claimed in ref 28 is caused by a misleading
interpretation for lattice simulation results (for details see Supporting Information). Region III. Starting from the binodal concentration at low temperatures and from the inflection point of supercritical isotherms, further incorporation of surfactant molecules into the adsorbed layer diminishes with rising bulk concentration. This is accompanied by a moderate restructuring toward more compactness up to approaching adsorption saturation. However, the structures of the adsorbed layer are qualitatively different at low and high temperatures. First let us compare the layer structures at the beginning of region III at T ) 1.0 and T ) 1.3. The contour plots of the layer thickness and cross sections are shown for two snapshots in Figure 7. At subcritical temperatures, asymmetric bilayers are observed. By measuring the head density profile along the surface normal, one finds undulations which could be interpreted as worm-like hemimicelles adsorbed on the compact hydrophobic monolayer next to the solid. Above the critical temperature, a holey structure of loosely connected surfactant clusters which do not have any preferred size is observed. The head density profiles indicate how the structure varies with temperature. In contrast to the high temperature cluster structure, the bilayers have no head segments in their centers. The asymmetry of the layer structures is remarkable in both cases and becomes less pronounced with increasing temperature as long as the bilayer exists. At higher temperatures, when a cluster network occurs, the asymmetry increases with increasing temperature due to the drastic drop of the orientation order in the bottom layer. The ratio of the peak heights in the head density profile linearly decreases up to the critical temperature. As it can be seen in Figures 8, both halves differ in orientation order and surface concentration. Below the critical temperature, the bottom part of the layer is more densely packed, whereas above Tc the upper half-is denser. The addition of further surfactant molecules leads to more compact layer structures. The orientation order and the surface concentration increase in both layer halves. However, the qualitative differences between low temperature and high temperature structures are preserved in the whole region up to attaining the adsorption saturation. This behavior can be quantitatively verified by considering the head density profiles PHEAD(z) in normal direction. The head density profile is little changed with increasing surfactant concentration, but it depends strongly on the temperature. The asymmetry remains unchanged. Comparing the adsorbate structure resulting from our simulations in the present off-lattice model with that obtained from previous lattice simulations,29,49 one finds qualitative coincidence. The main difference is the lack of a phase transition in the bilayer
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Figure 7. Contour plots for the thickness and cross section of the layer in the marked region (a) at T ) 1.0 on the coexistence curve for surface concentration R ) 1.002. (b) at T ) 1.3 for the surface concentration R ) 0.290 at the inflection point of the isotherm.
Figure 8. (a) Orientation order parameter S for upper and bottom halves of the adsorbed layers at different temperatures T and surfactant bulk concentrations φ. (b) Surface concentration R for upper and bottom halves of the adsorbed layers at different temperatures
at low temperatures, which leads to a more dense structure in the lattice model. It is not clear whether this phase transition is an artifact. Probably, compared to lattice models, the bilayer in off-lattice models becomes gradually more compact without phase transition because of the lack of restrictions on density variation. The morphological transition stated in ref 28 for the corresponding supercritical temperature region has not been observed in our off-lattice simulations (for details see Supporting Information). Region IV. Region IV is a plateau region where the adsorbed layer is unable to uptake further surfactant molecules from the solution. Below the critical temperature, it starts slightly above the critical micelle concentration for the solution. Above the critical temperature, there is only an asymptotic approach to this regime at high solution concentrations. How do these results correspond to experimental experience and theoretical concepts on the relations between the shape of adsorption isotherm and adsorbate structure? The S-shaped isotherms, which have been measured in a large number of experiments to investigate the adsorption of ionic surfactants on oppositely charged surfaces and of uncharged surfactants on hydrophilic surfaces, could be interpreted as supercritical isotherms. There are also some experimental results for adsorption of nonionic surfactants3,11,22 which are consistent with the jump-like course of subcritical isotherms in accord with our simulation results. However, most experimental adsorption
data clearly indicate a strong but smooth increase of surfactant surface concentration with increasing bulk concentration. There can be different reasons for this disagreement between theoretical predictions of computer simulations and experimental findings. First, it can be argued that in the coarse-grained model a lot of molecular details and thermodynamic peculiarities are neglected. Some of them, e.g., the change of surface potential upon adsorption, specific interactions between counterions and surface, and screening effects caused by the ionic strength, can essentially affect the adsorption behavior as has been shown and discussed in ref 67. A second argument to explain why experimental adsorption isotherms do not resemble the shape of our subcritical isotherms is the surface heterogeneity. This will be discussed below in section , where simulation results of the adsorption behavior on heterogeneous surfaces are presented. Experimental methods which determine the layer structure of the adsorbed surfactants usually yield the mean layer thickness and, at the best, the mean thicknesses of hydrophilic and hydrophobic sublayers. Because of the averaging over relatively large areas, it is frequently difficult or impossible to determine the lateral molecular organization within the layer. Atomic force microscopy is the only method which up to now allows one a sufficiently detailed measurement of the surfactant adsorbate structure with lateral resolution in molecular scales. However, this method requires a solid surface with sufficiently small
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Figure 9. Simulated adsorption isotherm (points) and curve fitted with equation (6) R∞ ) 0.68, k1 ) 16.31, k2 ) 2.72 × 10,9 and Na ) 5.77.
roughness. For this reason, the question about the correspondence between the shape of the adsorption isotherm and the surfactant layer structure is open yet, although there are many attempts based on plausible explanations to relate the shape of the adsorption isotherm with suggested layer structures, e.g., admicelles, hemimicelles, monolayers, and bilayers. 3.1.2. Shape of Isotherms and Structure of Corresponding Adsorbed Layers. Let us discuss the question whether the characteristics of the adsorption isotherm allows us to draw reliable conclusions on the structure of the adsorbed surfactant layer. For this purpose, we compare our results to a model15 which presupposes a completely different structure of the adsorption layer. In this model, it is assumed that admicelles with a significant size cover the solid surface. The formation of admicelles is described as equilibrium between one adsorbed monomer and (Na-1) monomers from the solution characterized by the mass action constant k2. We have fitted our simulated supercritical isotherms to the formula15
(
R∞k1φ R)
)
1 + k2φNa-1 Na
1 + k1φ(1 + k2φNa-1)
(6)
where R and φ are the surface surfactant concentration and the surfactant concentration in the solution, respectively. The aggregation number in the supposed admicelles is Na, and k1 is the equilibrium constant for the adsorption of a monomer. Our simulation data at T ) 1.30 and the fit with equation (6) are shown in Figure 9. Obviously, the isotherm (6) fits our data pretty well, even in the Henry region. The fit provides the aggregation number Na ) 5.77. The layer structure assumed for deriving the isotherm with the mass action model is in complete contradiction with the structures observed in our simulations. For each point of the isotherm below the percolation threshold, the cluster size analysis resulting from simulations shows a monotonously decreasing size distribution. Above this surface concentration, there is a loose percolating network. However, extracting the averaged cluster sizes from simulations, we obtain roughly the same value Na ) 6. Thus two distinct models, namely the mass action model for adsorbed admicelles15 and our microscopic adsorption model based on hydrophobic interactions, produce nearly the same isotherm for differently structured adsorption layers. 3.1.3. Influence of Adsorption Energy on the Structure of Adsorbed Layers. The influence of the adsorption energy
Figure 10. Adsorption isotherms at T ) 1.0 for different adsorption energy parameters ad: red, coexistence curve.
Figure 11. Snapshots of adsorbed aggregates at T ) 1.0 for ad ) -1.5 and different surfactant surface concentrations R.
parameter ad on the shape of the adsorption isotherm and the structure of the adsorbed layer is studied for the subcritical isotherm at T ) 1.0. If the adsorption energy parameter ad is changed from -0.25 to -1.5, the adsorption isotherms are shifted to smaller surfactant bulk concentrations. Figure 10 illustrates that the difference between the surface concentrations in the coexisting two-dimensional phases decreases with increasing absolute value of ad. Furthermore, the adsorption isotherms are changed from S-shaped ones to isotherms with an additional inflection point. For especially high adsorption energy, the isotherm has a marked two step character with an S-shape part after the coexistence region. Isotherms with an additional inflection point have been observed experimentally in case of adsorption of cationic surfactants on silica,68-71 on cellulose72 and for nonionic surfactants on kaolin.61 This particular behavior becomes more pronounced with rising absolute value of the adsorption energy. The snapshots for ad ) -1.5 in Figure 11 show that a monolayer patch forms in equilibrium with the twodimensional gas-analogous phase. The patch is preserving its monolayer structure until the whole solid surface is covered. Only a few surfactant molecules are adsorbed on the hydrophobic patch surface. In a second step, at higher surfactant concentrations, (68) Trompette, J. L.; Zajac, J.; Keh, E.; Partyka, S. Langmuir 1994, 10, 812. (69) Wittrock, C.; Kohler, H.-H.; Seidel, J. Langmuir 1996, 12, 5550. (70) Seidel, J.; Wittrock, C.; Kohler, H.-H. Langmuir 1996, 12, 5557. (71) Fleming, B. D.; Biggs, S.; Wanless, E. J. J. Phys. Chem. B 2001, 105, 9537. (72) Alila, S.; Boufi, S.; Belgacem, M. N.; Beneventi, D. Langmuir 2005, 21, 8106.
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Figure 12. Orientation order parameter of adsorbed layers at T ) 1.0 for different adsorption energy parameters ad: (a) upper leaflet; (b) bottom leaflet.
Figure 13. Cross sections of intermediate structures of the adsorbed surfactant layers as function of adsorption energy at T ) 1.0.
the adsorption goes on by increasing the surfactant density in the second leaflet of the adsorbed layer until forming a percolating coverage. Finally, an asymmetric bilayer forms. The percolation threshold is indicated by the second inflection point of the adsorption isotherm. Simulated structures agree well with structures concluded from electrokinetic measurements which were carried out simultaneously with adsorption experiments.68 The electrophoretic mobility of silica particles tends to zero if the coverage of the negatively charged silica surface by cationic surfactants becomes complete. This can be assessed as monolayer formation with tails protruding into the solution phase. At higher surfactant concentrations, the particles possess an increasing positive charge indicating bilayer formation. In the high concentration region above the cmc, the adsorbed surfactant molecules self-assemble into bilayers independently of the adsorption strength. As depicted in Figure 12, the adsorption energy affects mainly the orientation order in the bottom part of the adsorbed surfactant layer, whereas the order in the upper half-is nearly uninfluenced. In the low concentration region, we obtain bilayer patches at low adsorption energy and monolayer patches at high adsorption energy. For adsorption energies in between, the whole spectrum of intermediate structures from monolayers up to bilayers can be obtained (Figure 13). This is mainly of interest for applications, e.g., for flotation and dispersion
Figure 14. Adsorption isotherm at T ) 1.0 for heterogeneous surface consisting of for quadratic equally sized domains with different adsorption energy parameters ad ) -0.5, -0.75, -1.0, -1.5 (filled circles) and superposition of equally weighted adsorption isotherms for corresponding single domains (open circles).
stabilization, which utilize the hydrophobicity and screening function of the adsorbed layer. Let us return to the question how surface heterogeneity influences the shape of subcritical isotherms. The superposition of simulated jump-like isotherms with 4 different adsorption energies already yields rather smooth curves as seen in Figure 14. This example can be ascribed to particles with large distinct crystal surfaces exposed to the bulk solution. Each surface domain has an adsorption energy for its own so that the concentration at which a dense phase patch is formed varies from patch to patch. Furthermore, we also have considered adsorption on a heterogeneous surface which is piecemeal composed of 4 small quadratic domains on a particle. A simulation that assumes the same adsorption energies as for the isotherm constructed by superposition provides a still smoother S-shaped isotherm. This comes from the boundary effects between different domains. As a consequence, in both cases the jump-like region of the adsorption isotherm is smeared out on heterogeneous surfaces due to the superposition of locally varying adsorption conditions. Actually, real solid surfaces are heterogeneous so that a set of domains with various adsorption energies is to taken into account. 3.1.4. Hydrophobicity of Adsorbed Layer Structures. An interesting feature of the adsorption behavior is the resulting hydrophobicity of the covered surface (Figure 15a). With rising temperature, the fraction of hydrophobic area in the adsorbate
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Figure 15. (a) Hydrophobicity as function of bulk concentration φ at different temperatures (ad ) -0.5). (b) Hydrophobicity of the surface covered with surfactants at T ) 1.0 for different adsorption energy parameters ad.
Figure 16. Adsorption isotherm for T ) 1.0, ad ) -0.5 for model molecules with potential parameters ) -3.1 and d ) 0.1.
grows. At each temperature, the maximum hydrophobicity is reached at the coexistence curve. This behavior can be explained by the change of the orientation order parameter and the surface concentration in the surfactant leaflet exposed to the bulk solution. The descend of the hydrophobicity with increasing surfactant concentration results from the rising compactness of the top layer.
For different adsorption energies ad and subcritical temperatures, the hydrophobicity goes through a maximum in the low concentration region. Experimental measurements of the flotation rate give evidence that the hydrophobicity has a maximum at a certain surfactant concentration.73 The maximum hydrophobicity rises remarkably with adsorption strength and is shifted toward smaller surfactant surface concentrations. At high surfactant surface concentrations, the hydrophobicity reaches a saturation limit, which is nearly the same for each adsorption energy parameter ad (Figure 15b). 3.2. Adsorption of Molecules Which Prefer Association into Micelles. Finally, we studied the adsorption of molecule models with hydrophobic interaction parameters ) -3.1 and d ) 0.1. This parameter set belongs to the region in the phase diagram where micelles are stable in the solution phase. The corresponding isotherm at T ) 1.0 and ad ) -0.5 in Figure 16 is similarly shaped as the corresponding subcritical isotherm in Figure 2. However, the adsorbed surface layer has a completely different structure. The surfactants are organized in worm-like admicelles both below and above the critical micelle concentration (see Figures 17). At high concentrations, there are long meandering worm-like admicelles comparable to those found in some structure detections by AFM measurement.74-79
Figure 17. Contour plots for height and cross sections for admicelles built from molecules in the marked region with potential parameters ) -3.1 and d ) 0.1. (a) R ) 0.335; (b) R ) 0.804.
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The adsorption of surfactants on hydrophilic surfaces from aqueous solutions has been studied by using off-lattice Monte Carlo simulations. The simplest flexible amphiphilic molecule model consisting of two tail and one head spherical segments furnished with excluded volume interaction and square-well potential is utilized to mimic the hydrophobic effect. Adsorption isotherms together with associated structures of the adsorption layer have been obtained by simulations in the canonical and grand canonical ensembles. The adsorption isotherms are in good agreement with the phase separation model for lattice systems18,19 and confirm the four-region description8 above the critical temperature. In dependence on the ratio between the strength of the hydrophobic effect and the adsorption energy, a large diversity of adsorbed layer structures has been observed, i.e., adsorbed single molecules, two-dimensional clustered phases, monolayers, cluster networks, bilayers with various internal modifications, and worm-like admicelles. Under convenient temperature and
concentration conditions, any structures intermediate between them can be obtained. It has been shown that it is impossible to draw reliable conclusions on the structure of adsorption layer on the basis of thermodynamic data alone. Different layer structures may fit very well to the same adsorption isotherm. Our simulations make it possible to give a unified description of the adsorption thermodynamics in the framework of the 4-region model and the phase separation model by superposing the results for domains with different adsorption energies. The adsorption layers are asymmetric with regard to their orientation order and their surface area density. Under some conditions, the bilayer can be regarded as hemimicelles adsorbed on a bottom monolayer. The hydrophobicity goes through a maximum at low concentrations. At saturation, it attains a constant value which is independent of the adsorption energy and the temperature. The off-lattice simulations qualitatively confirm essential results for the adsorption of surfactants in equivalent lattice models. It is shown that the diversity of structures of adsorbed surfactant layers suggested by experimental observations can be explained by a very simple model without sophisticated additional assumptions.
(73) Schwarz, R.; Heckmann, K.; Strnad, J. J. Colloid Interface Sci. 1988, 124, 50. (74) Lamont, R. E.; Ducker, W. A. J. Am. Chem. Soc. 1998, 120, 7602. (75) Schulz, J. C.; Warr, G. G. Langmuir 2002, 18, 3191. (76) Grant, L. M.; Tiberg, F.; Ducker, W. A. J. Phys. Chem. B 1998, 102, 4288. (77) Dong, J.; Mao, G. Langmuir 2000, 16, 6641. (78) Grant, L. M.; Ducker, W. A. J. Phys. Chem. B 1997, 101, 5337. (79) Grant, L. M.; Ederth, T.; Tiberg, F. Langmuir 2000, 16, 2285.
Supporting Information Available: Figures showing head density profiles PHEAD(z) of adsorbed surfactant layers for different temperatures T and text giving comments on the results of lattice Monte Carlo simulations in ref 28. This material is available free of charge via the Internet at http://pubs.acs.org.
4. Summary
LA8020595
