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Wetting and Melting in a Methane/Graphite System? M. S. Pettersen,* M. J. Lysek, and D. L. Goodstein California Institute of Technology 114-36, Pasadena, California 91125 Received October 27, 1988 We report a new thermodynamic analysis of melting in thick adsorbed films and experimentalresults of an NMR survey of melting in methane f i i adsorbed on graphite. We fiid that solid-liquid stratification, related to surface melting in the bulk, may occur in a system that also exhibits triple point dewetting. Each phenomenon has distinctive asymptotic thermodynamic behavior in the thick film limit. We predict the existence of surface tension induced surface freezing, symmetric with surface melting, and new first-order phase transitions at coexistence between stratified and homogeneous films. The NMR data are used to show that solid methane wets graphite to at least 50 layers and that capillary condensation is not nucleated in ow system (methane on Grafoil). We c o n f i i that solid-liquid stratification does not occur in this system. Instead, melting is a first-order phase transition between homogeneous phases of solid and liquid, which we trace from a triple point dewetting transition at bulk coexistence down to 1.8 layers, where the latent heat of melting is known to have vanished. We conclude that melting in this system has a tricritical point at about 4 layers. Finally, we observe an increase in molecular mobility associated with the surface layers of the roughened solid phase. In this paper we report the results of a study of the system methane adsorbed on graphite in the multilayer regime near the triple point. Multilayer adsorbed films offer the intriguing possibility of studying matter between the two-dimensional monolayer regime and the three-dimensional bulk limit. In particular, we are interested in the melting transition. Because the dislocation unbinding model of melting given by Kosterlitz and Thousless' can only be solved theoretically in two dimensions, it would be particularly interesting to know whether the mechanism of melting in three dimensions can be related experimentally to that in two dimensions. The system methane on graphite seems uniquely suited to such a study. It has been argued2 that, in general, it should be impossible to grow a thick film of solid phase. The attractive potential of the substrate causes the first few layers to form under strain, and the energy cost of growing the strained solid inhibits the growth of arbitrarily thick films. The limit to growth, however, may be very high, as much as 20-30 layers, at which point conventional techniques cannot distinguish a thick wetted film from a bulk crystallite. The models of Phillips3 show that the methane/graphite system is exceptional in that promotion of molecules out of the first layer relieves the strain in the first layer and allows epitaxial growth of methane at its bulk lattice constant. We shall present novel evidence that methane wets graphite to at least 50 layers. There are a few systems that are believed to form wetted f i in the solid phase: In several of these, however, such as neon and argon on graphite5 and methane on Mg0: the melting transition is thought to be preempted by another transition, called surface melting. In surface melting, the solid-vapor surface is energetically unstable with respect to a system where the surface is covered with a thin layer of liquid. As the triple point Ttis approached in temperature from below, the liquid layer grows until it consumes the solid. When this phenomenon is present, the interfacial transition obscures any effects due to the intermediate dimensionality of the film. Of the presently known systems, methane on graphite alone appears both to wet in the solid phase and not to undergo surface *Current address: Dept. of Physics, Ohio State University, Columbus, OH 43210. 'Presented a t the symposium on "Adsorption on Solid Surfaces", 62nd Colloid and Surface Science Symposium, Pennsylvania State University, State College, PA, June 19-22, 1988; W. A. Steele, Chairman.
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melting. It may, therefore, be uniquely suitable for studies of melting in the multilayer regime. In order to understand the interrelationship between surface melting and melting in adsorbed films, we have constructed a continuum thermodynamic model that should be exact in the limit of thick films. Picturing a film consisting of a layer of liquid on top of a denser layer of solid on top of the substrate, we calculate the surface excess correction to the grand free energy as a sum of three sets of terms: the surface tensions of bulk media at each of the interfaces, a correction to the surface tensions to account for the fact that the media in contact at the interface are not infinitely thick,' and a term to account for the energy cost of forming a condensed phase, liquid or solid, compared to the free energy of the gas. The resulting free energy is then minimized with respect to the thickness of liquid and solid strata, resulting in the pair of equations
where the superscripts 1, s, and w indicate liquid, solid, and substrate. p is the density of the bulk phase, 1.1~is the chemical potential at coexistence between the bulk phase and the vapor, zi is the thickness of the stratum in the appropriate phase, and the ACik are constants depending on the van der Waals interaction between adsorbate molecules and various substrate media. These equations, relating z1and z, to 1.1 and T, represent the equation of state for a surface-melted film, analogous to the Frenkel-Halsey-Hill (FHH) equation: which describes the thickness of a homogeneous, unstratified film. Indeed, eq l a and (1)Kosterlitz, J. M.; Thousless, D. J. J.Phys. C 1973,6,1181. Nelson, D. R.; Halprin, B. I. Phys. Rev. B 1979,19,2457. Young,A. P. Phys. Reu. B -... 1979. 19. - ,1855. ---(2) Huse, D. A. Phys. Rev. B 1984,29, 6985. Gittes, F. T.; Schick, M. Phys. Rev. B 1984,30,209. (3) Phillips, J. M. Phys. Rev. B 1986, 34, 2823. (4)Bienfait, M.; Seguin, J. L.; Suzanne, J.; Lerner, E.; Krim, J.; Dash, J. G. Phys. Reu. B 1984,29,983. (5) Zhu,D.-M.; Dash, J. G. Phys. Rev. Lett. 1986,57,2959. Zhu,D.-M.; Dash, J. G. Phys. Reu. Lett. 1988, 60, 432. (6) Bienfait, M. Europhys. Lett. 1987, 4 , 79. (7) de Gennes, P.-G. J. Phys. (Les Ulis, Fr.) 1981, 42, 1377. (8) Frenkel, J. Kinetic Theory of Liquids; Oxford University Press: London, 1949. Halsey, G. D., Jr. J. Chem. Ph.ys. 1948.16.931. Hill, T. L. J. Chem. Phys. 1949, 17, 590. I
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Figure 1. Predicted heat capacity signal of surface melting for coverages of 10, 8, 6, 4 , and 2 layers (top to bottom).
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l b reduce to the FHH equation if we set p , = p1 and pOs = p,-,'. Alternatively, taking the limit z, m, we get the FHH equation for a liquid film on a solid substrate of the same substance. Equations l a and l b predict that when this kind of liquid-solid stratification occurs a layer of liquid may persist well below Tt (surface melting) and a layer of solid well above Tt (surface freezing). Typical heat capacity curves are shown in Figure 1 for several film thicknesses. The peaks seen there do not mark a phase boundary but are rather a consequence of the motion of the liquid-solid interface toward the substrate as the temperature is raised. Nevertheless, the difference between the peak temperature, T p ,and the bulk triple point temperature, T,, scales as ITp- Tt]a n-3,where n is the total amount adsorbed. In the thinnest films, there is little motion of the interface, and the peaks wash out because the temperature-dependent bulk-phase contributions to the free energy, which drive melting, become unimportant, and the behavior is dominated by the surface tensions, assumed temperature independent, which favor liquid-solid-substrate stratification. When the free energy of a surface-melted film is compared to that of a homogeneous, unstratified film, the phase diagrams of Figure 2 arise. Similar phase diagrams have been suggested by Pandit and F i ~ h e r .The ~ results depend on two combinations of surface tensions, 6 = usw + usg- ulw- ulg and 6' = - ad - uk, where u is the surface tension and the subscripts as before represent the phases solid, liquid, substrate, and gas (s, 1, w , g). The parameter 6 represents the difference in surface tension between solid and liquid films, and 6' represents the amount by which surface melting is energetically favored in the bulk solid. In Figure 2a, a stratified film is never stable, and the film melts directly from solid (SF) to liquid (LF) in a first-order phase transition, as does the bulk (BS BL). Figure 2b shows a case in which surface melting (SM) spans the bulk triple point. This state could more precisely be called surface melting below Tt and surface freezing above. The heat capacity curves of Figure 1are calculated for the surface-melted portion of this phase diagram. Figure 2c shows a case in which surface melting only occurs below T,. An analogous surface freezing phase above Tt could be generated by considering values of 6 and 6' < 0. All of the phase boundaries in Figure 2a,b,c are first-order phase transitions and should be observable in thermody-
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Figure 2. Hypothetical phase diagrams. BS = bulk solid, BL = bulk liquid, SF = solid film, LF = liquid film, SM = surfacemelted film; between the points marked W nonwettin occurs. (a) 6 = 0.21K/A2,6' = 0; (b) 6 = Q.1K/A2,6' = 0.!2K/Af; (c) 6 = 0.45K/A2, 6' = 0.35Kf A'. namic measurements. The exact locations of the phase boundaries depend on the values of 6 and 6'. All of the diagrams can be considered dependable only in the limit IP
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In Figure 2a and 2c, the liquid film (LF) makes direct contact with the bulk solid (BS) between the points marked W in both cases. These regions of coexistence between' liquid film and bulk solid are instances of nonwetting that occur even though the SF, LF, and SM states each wet the substrate. This phenomenon is known as triple point dewetting. Quite general argumentdo show that a transition between wetting and nonwetting phases must approach the bulk coexistence curve tangentially as ITw - TI a ( p o - p ) 2 / 3 0: n-2. The behavior seen in Figure 2a has been reported for the case of methane adsorbed on graphite." The data for neon on graphite and argon on graphite5 appear to be consistent with Figure 2b. We are not aware of any reported instance of Figure 2c, but this case does show that surface melting and triple point dewetting can occur in the same system, at least in principle. Having now established the general characteristics of phase diagrams near the bulk triple point, we turn to the results of an NMR survey of the system of methane on graphite. Figure 3 shows data for the spin-lattice relaxation time, T I ,as a function of temperature for a variety of film thicknesses. The shortening of T I in thin films compared to the bulk is a well-known phenomenon, but the dependence on film thickness has not previously been studied. We can estimate the dependence of the effect as follows. Let us consider a paramagnetic impurity or a localized, unpaired electron trapped in a defect state in ~~
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(10) Pettersen, M. S.; Lysek, M. J.; Goodstein, D. L.Surf. Sci. 1986,
175, 141.
(11) Lysek, M. J.; Pettersen, M. S.; Goodstein, D. L. Phys. Lett. A (9) Pandit, R.; Fisher, M.
E.Phys. Reu. Lett. 1983, 51, 1772.
1986, 115, 340.
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Langmuir, Vol. 5, No. 3, 1989 565
crystallites or capillary condensation). We measure Tl by the saturation recovery technique. All of our data can be fit by recovery curves with a single time constant, as shown in Figure 4 for a film of eight layers (which should be the \ O t \ worst case for capillary condensation-half the adsorbate should lie in filled pores and half in the film). Interdiffusion between bulk and film might produce an averaging between the two Tl values, but the condition for this not to occur, Nih(DTlfi1m)1/2
