1574
J. Phys. Chem. B 2007, 111, 1574-1581
Surface Melting of Octamethylcyclotetrasiloxane Confined in Controlled Pore Glasses: Curvature Effects Observed by 1H NMR Oleg V. Petrov, Dulce Vargas-Florencia, and Istva´ n Furo´ * DiVision of Physical Chemistry and Industrial NMR Centre, Department of Chemistry, Royal Institute of Technology, SE-10044 Stockholm, Sweden ReceiVed: August 24, 2006; In Final Form: December 1, 2006
We have measured the thickness of the pre-molten surface layer that appears at the interface of octamethylcyclotetrasiloxane (OMCTS) to the matrix in controlled pore glasses with pore diameters ranging 7.5-73 nm. Except for the glass with the largest pores, the layer thickness data for different pore diameters fall on a single master curve when plotted versus Tm - T, where Tm is the size-dependent volume melting point of the pore-confined OMCTS. Hence, at a single temperature, the surface layer thickness depends strongly on the curvature of the pore wall and therefore that of the solid-liquid interface. For temperatures where it exceeds two monolayers, the layer thickness depends logarithmically on Tm - T; for the glass with the largest pores, this turns into a power law with the exponent -1/2. The results are interpreted in terms of a continuous model of the solid-liquid interface with an arbitrary curvature. Because OMCTS is a weakly polar molecule with close to spherical shape, our data also lend themselves to Lennard-Jones type simulations.
Introduction The term “surface melting”, as well as “interfacial” or “premelting”, is related to the existence of a thermodynamically stable layer (hereafter called surface layer) on solid surfaces that has a liquid-like mobility at temperatures below the bulk melting point. The concept of surface melting has a long history since Faraday’s experiments on ice regelation;1-4 it took on its present form by the 1950s when a model of the surface layer on ice had been developed based on the difference between the molecular arrangements in the bulk and on the surface.5,6 Surface melting is, though, not limited to icesit has been demonstrated for metals, semiconductors, molecular solids, and rare gases.7-14 Pre-melting was recently imaged in 3D as occurring at dislocations within colloidal crystals.15 The surface layer is also known to have a bearing on gas adsorption,16 chemical reactions on surfaces,17 and friction of solids,4 and hence, it is important for metallurgy, coating technology, adhesion, wetting, and lubrication.18 Physically, the surface layer results from a binding of a molecule (or atom) at the crystal surface that, in response to the reduced number of molecules (atoms) that it may interact with, is weaker than binding in the bulk. The layer exhibits a considerable departure from the crystalline order and is, hence, often referred to as a quasi-liquid. Surface layers are likely to explain the absence of overheating in solids, for they can naturally nucleate the melting of the bulk.7,10,19 Curiously, the layer may diminish or be lacking at specific interfacial energies and at particular crystallographic faces.20-25 Unlike the apparently abrupt bulk melting, the surface one is a continuous process26 in which the surface layer thickens continuously with increasing temperature until it diverges at the bulk melting point. Indeed, in this picture, bulk melting is the divergent τ f ∞ limit of surface melting. At a particular temperature, the thickness τ of the layer takes a value that minimizes * Corresponding author:
[email protected]. Telephone: +46 8 790 85 92. Fax: +46 8 790 82 07.
the interfacial free energy. Conversely, by measuring τ, one can obtain information about the interfacial free energy. The temperature dependence of τ, measured for ice22,27-33 and lead20,21,34 for both single crystals and small particles is not universal but is described by either logarithmic20,21,31-33 or power13,27-30,35,36 laws with the exponent in the latter varying from -0.33 to -0.60. This diversity has been attributed to different types of interfacial interactions predominant in the different cases:37 short-range interactions lead to logarithmic and long-range ones to power-law growth. Another factor influencing the temperature dependence of thickness is the surface curvature.13,28,38,39 Thus, Kofman et al.38 have shown that, in the case of lead particles on a SiO2 substrate, the layer thickness is, at the same temperature, larger for nonspherical inclusions than for spherical ones of comparable sizes. Engemann13 has demonstrated that the logarithmic growth law fits extremely well for ice on a smooth silica substrate, although it turns into the power law on a rough substrate. Cahn et al.39 explored the effect of curvature to analyze the surface melting of ice confined between micron-sized spheres;28 their long-range interfacial potential leads to the a power-law growth. Porous materials are where both the temperature and curvature effects on the surface layer can be observed naturally. Water is still the most, if not the only, compound having been studied that way.36,40,41 NMR investigations of ice in porous Vycor glass indicated a surface layer of 0.5 nm thickness down to -80 °C for 4 and 10 nm pore samples but detected no liquid-like constituent at this temperature in 30 and 50 nm pore samples.36 In an NMR study of water frozen at -18 °C in controlled pore glasses,41 the fraction of surface layer in 24 nm pores was found to be 3 times greater than that in 73 nm pores versus the ratio 2.5 as would be expected from the specific surface area of these glassessa possible manifestation of some curvature effect. Although there is a large amount of other evidence for surface melting of water and organic compounds in pores42-49 and at least some rough estimates of the surface layer thickness have
10.1021/jp0654765 CCC: $37.00 © 2007 American Chemical Society Published on Web 01/26/2007
The Surface Melting of Octamethylcyclotetrasiloxane
J. Phys. Chem. B, Vol. 111, No. 7, 2007 1575
TABLE 1: Properties of the Controlled Pore Glasses Used in the Study label
d,a nm
CPG75 CPG115 CPG156 CPG237 CPG313 CPG729
7.5 11.5 15.6 23.7 31.3 72.9
∆d,b (% 6.0 7.3 5.8 4.3 5.6 6.4
S/V, 10-3/nm 326 224 126 76.5 58.0 31.6
κ,c 10-3/nm 93 64 36 22 17 9.0
a Nominal pore diameter as supported by the manufacturer. b Pore size distribution (80% of pores are within the interval). c Integral pore curvature calculated using the relationship 2κ/(S/V) ) 0.57.55
been presented in all of those cases, neither temperature nor curvature effects on the thickness have been studied thoroughly. In the present paper, we report a study of the surface melting of octamethylcyclotetrasiloxane (OMCTS) in controlled pore glasses. OMCTS is a low-polarity liquid consisting of globularly shaped molecules50-52 and is thus suitable for comparison with model “Lennard-Jones” liquids. A special motivation for this study was that OMCTS had been found to be a good probe material for NMR cryoporometry,53 where surface melting plays a role. Experimental OMCTS as obtained from Aldrich (98%) was imbibed into six uncoated controlled pore glasses (CPGs) from Millipore Corporation. The CPGs made in grains of 100-150 µm and with pores of 7.5, 11.5, 15.6, 23.7, 31.3, and 72.9 nm in nominal diameter were used as obtained (Table 1). Single-point BET surface area measurements were done by nitrogen with a FlowSorb III apparatus using the flowing gas method.54 OMCTS was poured directly over a CPG that was placed in a 5 mm NMR tube with up to 100% excess over an estimated pore volume to ensure complete pore filling and to provide a reference temperature in the form of the bulk melting during the experiments.55 To facilitate liquid penetration into the pores and to suppress possible air inclusions within the porous network, the tube was centrifuged for up to 12 h at room temperature. 1H NMR measurements were conducted in a Bruker DMX 500 spectrometer with 500 MHz frequency and equipped with a standard 5 mm probe. The temperature of the samples in the NMR probe was controlled by an accuracy of (0.1 K with a Bruker BVT2000 temperature controller. Absolute temperature calibration was accomplished by means of the temperaturedependent chemical shift difference between the methyl and hydroxyl resonances of methanol.56 The measurements were performed by slowly warming the deep-frozen samples from 230 K up to the pore melting temperature Tm. The latter was pore-size dependent and varied from 268 K for CPG75 with the 7.5 nm pores to 288 K for CPG729 with 72.9 nm pores, as it had been obtained previously.53 After any temperature change, a sufficiently long waiting time (g10 min) made certain that the sample reached thermal equilibrium. There, two separate experiments were performed: the acquisition of a free induction decay (FID) and the measurement of the transverse relaxation time T2. FID was acquired after a composite pulse57 consisting of three successive 90° pulses, each with τ ) 4.3 µs duration and a phase cycling that eliminated the tank-circuit ringing otherwise corrupting the initial part of FID; in this way, an uncorrupted FID could be recorded from 20 µs onward after the end of the last pulse. The applied 90° pulse length was tested and found to be sufficient for a full excitation of solid OMCTS. The T2 measurements were carried out to monitor the onset of
melting in pores (see below) and employed the Carr-PurcellMeiboom-Gill (CPMG) pulse sequence with an interpulse delay of 50 µs. The obtained relaxation curves were deconvoluted into relaxation-time distributions with the program UPEN,58 which in turn yielded the mean transverse relaxation time . Results Figure 1a presents typical FIDs M(t) measured below Tm. They consist of two components: a quickly decaying one arising from solid OMCTS and a slowly decaying one arising from the quasi-liquid OMCTS in the surface layer. Figure 1b shows the corresponding two-component spectra. Because the pore surface is much larger (>106) than the outer surface of the grains, we can safely consider the quasi-liquid signal as arising from the surface layer within the pores. We model M(t) by a sum41 of one Gaussian and one exponential function corresponding to the solid and liquid signals, respectively,
M(t) ) (1 - pL) exp{-(t/T*2S)2} + pL exp{-t/T2L} (1) where pL, T*2S, and T2L were fitting parameters. The fit had also included an amplitude factor by which all experimental and fitted data were normalized, in effect producing M(0) ) 1. The leastsquares fits of eq 1 to the two presented data sets are shown in Figure 1a by solid lines. Because both pore-filling and interstitial OMCTS contribute to the solid signal, pL was additionally normalized with the fraction of the pore-filling material. The latter was defined as the ratio r of the liquid signal intensities (i) on the plateau between Tm and the bulk melting point T0 where only the pore-filling liquid is molten and (ii) immediately above T0 where all liquid is molten. Figure 2 adopted from ref 53 illustrates the points at which those intensities were measured. As a first step toward the width of the surface layer, the derived pl ) pL/r provides us the volume fraction of the surface layer within pore-filling OMCTS (Figure 3a). We note that the present procedure of acquiring both solid and liquid signals followed by their separation based on eq 1 differs from the T2-filtering (in effect, cancellation of the solid signal) that has been used in ref 36. Despite being perhaps more complex, our approach has the important advantage that it is not sensitive to the temperature dependence of the nuclear magnetic susceptibility or, to a certain extent, spectrometer performance and, therefore, requires no temperature correction of the obtained data. Besides, solid OMCTS has a relatively long T*2S (∼20 µs) and a sufficiently effective suppression of the solid signal would also reduce the liquid signal (T2L g 300 µs, see Figure 4) by a few percent. Before analyzing pl(T), one has to ensure that it is defined over the temperature region where it is not contributed by OMCTS already molten within the volume of small pores. Because the pore size in CPGs shows a distribution (see Table 1), pl(T) in itself has no sufficiently distinctive crossover from surface melting to volume melting. To detect the onset of volume melting, we exploited instead the temperature dependence of T2L, which has a pronounced increase at this point (see Figure 4). The increase is due to the large difference in molecular mobility in the quasi-liquid surface layer and within the true liquid. Given T2L(T), we define, as illustrated in Figure 4, the onset of volume melting as the intercept of asymptotes for the low- and high-temperature parts of T2L(T). Throughout the paper, pl(T) is considered up to the volume melting temperatures thus defined. Figure 3a shows that, at a given temperature, pl is inversely correlated with the pore size or, in other words, directly
1576 J. Phys. Chem. B, Vol. 111, No. 7, 2007
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Figure 1. (A) The normalized free induction decay (FID) signals M(t) of OMCTS confined in porous glasses CPG75 (7.5 nm pores) and CPG729 (72.9 nm pores), acquired at 262 and 284 K, respectively. The long-time tails of the FIDs arise from the pre-molten surface layer. Those temperatures are ∼5 K below the respective volume melting points Tm within the pores. The solid lines are the least-square fits by eq 1. (B) Normalized spectra corresponding to the FIDs in (A). Note that both pore-filling and interstitial among the glass grains OMCTS contribute to the broad line arising from frozen OMCTS.
Figure 2. The melting curve of OMCTS in CPG115 recorded during an NMR cryoporometry experiment.53 The region where only porefilling OMCTS is molten is denoted by (i) and (ii) indicates the temperatures where all OMCTS is molten. The arrow is at the inflection point of the melting curve, which defines the volume melting temperature Tm in the pores. The insert magnifies the temperature interval where an abrupt transition from surface to volume melting occurs.
correlated with the surface-to-volume ratio S/V of the pore. To calculate τ from pl, we exploit well-known formulas for equidistant surfaces. For a convex one-hole pore
Ss ) S (1 - 2κt)
(2a)
Vs ) V - S(t - κt2)
(2b)
where S and V represent the surface area and volume of a pore, respectively, Ss ) Ss(t) and Vs )Vs(t), the surface and volume of the solid core filling the pore equidistantly by t from the pore wall,55 and κ is the integral mean curvature of the wall. At t ) τ
pl ) 1 - Vs/V ) (S/V)(τ - κτ2)
(3)
which, in turn, yields
τ ) (1 - x1 - 4plκV/S)/(2κ)
(4)
This formula enables us to estimate the curvature effect on the layer thickness, provided that κ is known. Assuming that 2κ/ (S/V) = 0.57, as previously established for CPG237 and CPG729,55 holds for all glasses under investigation, we obtain
Figure 3. (a) The volume fraction pl of the surface layer for OMCTS confined in different porous glasses as a function of temperature. The vertical dotted lines indicate average volume melting temperatures Tm within the pores of the different glasses, obtained from NMR cryoporometry experiments53 as illustrated in Figure 2. (b) Surface layer thickness τ calculated from (a) using eq 4. The horizontal dotted line indicates a thickness corresponding to the van der Waals diameter of OMCTS (0.74 nm24).
the κ values listed in Table 1. We note that the ratio 0.57 is close to that for cylinders (0.5), and the local tube-like geometry of pore CPGs was, indeed, either assumed or argued for in previous works.41,59 However, knowing κ, the cylindrical model is too simplified. With κ and S/V given in Table 1 and pl presented in Figure 3a, eq 4 provides τ shown in Figure 3b. We see that, at a particular temperature, the layer in smaller pores is thicker than
The Surface Melting of Octamethylcyclotetrasiloxane
Figure 4. Representative temperature dependences of the mean transverse relaxation time for the liquid fraction of OMCTS confined in the glasses CPG115 and CPG729. The sharp increase of T2 is attributed to the onset of pore melting (note the logarithmic scale on the axis). The dashed lines are asymptotes of the T2 curves at lower and higher temperatures. The arrows indicate the average volume melting temperatures Tm within the pores of the different glasses as obtained from the NMR cryoporometry experiments.53
J. Phys. Chem. B, Vol. 111, No. 7, 2007 1577 apparent linearity, respectively. In both cases, the temperature is relative to the volume melting point within the pores, Tm. With this shift of the temperature scale, the data for different pore sizes converge to a common master curve. Clearly, neither logarithmic nor power-law dependences fit the data over the entire temperature region. Nevertheless, logarithmic growth seems to be a good limiting behavior at temperatures close to Tm where τ is greater than two monolayers. The exception is OMCTS in CPG729, where the growth is better described by a power law with the exponent -1/2 (see Figure 7 below). To derive a model for τ(T), one has to minimize the free energy F ) F(T,t), written for a system with an arbitrary layer thickness t, with respect to t.19,38 The difficulty is to establish an F(T,t) that correctly accounts for molecular interactions in the interface and their dependence on t. For a semi-infinite system with a flat interface (κ ) 0), this problem has been solved within the density-functional approach.25,60-62 The extended solution, outlined in the Appendix, yields the free energy per unit area as
F ) Fb + γlw + γsl + ∆γe-2t/ξ
(5)
Here, Fb is a sum of bulk free energies of the solid and liquid phases, and the next two terms are the free energies associated with the liquid-wall (lw) and solid-liquid (sl) interfaces. The last term accounts for short-range molecular interactions in the vicinity of the wall, with ξ being the correlation length of the crystalline order in the layer and ∆γ ≡ γsw - γlw - γsl the free energy excess of a single solid-wall (sw) interface over a solidliquid-wall “sandwich”. Altogether, the first three terms in the sum represent the free energy that the system would have if it consisted of bulk-like phases divided by discontinuous interfaces; the last term introduces a blurred interface where the surface layer retains some crystalline order over the length scale of ξ. It is convenient to think of this term as an effective potential between sl and lw interfaces φ(t) ) ∆γe-2t/ξ, repulsive under the surface melting condition (∆γ > 0). At this point, we assume that the exponential form of φ(t) is valid not only for flat surfaces, but also for curved equidistant surfaces. Whether or not this assumption is sufficiently good for the systems investigated here will be verified below upon fitting the model to our data. With the assumption above, the free energy (now in J instead of J/m2 as in eq 5) becomes
F ) Vsµs/νs + (V - Vs) µl/νl + Sγlw + Ssγsl +S∆γe-2t/ξ (6) where µs,l are the bulk chemical potentials for the solid and the liquid, respectively, and Vs ≈ Vl ) V are their corresponding molar volumes. The temperature dependence of F is introduced through that of µ: Figure 5. Surface layer thickness τ as a function of Tm - T plotted on semilogarithmic (a) and logarithmic (b) scales, where Tm is given in Figure 3a.
in bigger pores. On the other hand, τ can attain a greater limiting value at pore melting for larger pores; for example, τ in CPG729 reaches approximately 4 nm and only 2 nm in CPG75. Accepting that the van der Waals diameter of OMCTS is ∼0.74 nm,24 the maximum layer thickness varies, therefore, for the glasses under investigation from approximately 2 to 5 monolayers. Discussion In Figure 5, τ is re-plotted on two different scales, appropriate for visualizing either logarithmic or power growth laws as
µs - µl ) ∆H(T/T0 - 1)
(7)
where ∆H is the molar latent heat of melting. Retaining only the t-dependent terms in eq 6, we have
F(T,t) ) (∆H/ν)(T/T0 - 1)Vs(t) + Ss(t)γsl + S∆γe-2t/ξ
(8)
where the dependences Ss ) Ss(t) and Vs ) Vs(t) are given by eq 2. Minimizing eq 8 with respect to t yields
∂F/∂t|t)τ ) 0 ) (1 - 2κτ)[(T/T0 - 1)∆H/ν] + 2κγsl + (2∆γ/ξ)e-2τ/ξ (9) For κτ , 1 and temperatures not in the close vicinity of the volume melting point within the pore, Tm, the term 2κτ in eq 9 can be ignored, which gives
1578 J. Phys. Chem. B, Vol. 111, No. 7, 2007
[
Petrov et al.
)]
(
γslT0ν ξ ξ∆H 0 τ ) - ln T T 2κ 2 2∆γT0ν ∆H
(10)
Taking into account that Tm in the κτ , 1 regime obeys the Gibbs-Thomson equation, T0 - Tm ) 2κγslT0V/∆H,55 eq 10 takes its final form
[
ξ∆H ξ (T -T) τ ) - ln 2 2∆γT0ν m
]
(11)
Thus, for κτ , 1, the model predicts a logarithmic growth of τ. Not surprisingly, at κ f 0, eq 11 coincides with the expression for flat surfaces,19 recalling that Tm ) T0 for κ ) 0. Equation 11 contains two parameters, ξ and ∆γ, which can be obtained through fitting eq 11 to the data. We note that eq 11 implies ∆γ > 0 as a surface melting condition. Out of the κτ , 1 regime, there is no simple analytical solution for τ as in eq 11. One analytical result that can be obtained, though, arises by exploiting the condition ∂2F/∂t2 ) 0 that, together with eq 9, defines the temperature for volume melting within the pore.55 Hence, by evaluating these two relations, one obtains that the surface layer thickness assumes, at T ) Tm, a finite value
τmax ≡ τ(Tm) ) 1/(2κ) - ξ/2 - K/(T0 - Tm)
(12)
With our parameters, this relation yields a τmax value that varies from 1.7 nm for CPG75 to 12.7 nm for CPG313. One should note that pore size distribution and the resulting distribution of Tm prevent us from performing an experimental comparison. We stress that the result in eq 12 rests on the validity of our initial assumptions such as the exponential form of the surface term in eq 5 and the independence of the enthalpy of fusion on the pore size. In particular, this latter assumption may not be valid.63 Equation 12 also yields that τmax diverges for flat surfaces with κ f 0, which is consistent with the notion of volume melting as divergent surface melting at T f Tm. In order to obtain the temperature dependence of τ at temperatures away from Tm and out of the κτ , regime, the data can be instead analyzed using the following numerical procedure. Let us rewrite eq 8 in the form
Figure 6. Illustrative fit of the data for CPG156 using the model function in eq 13; see details in text.
TABLE 2: Best-Fit Parameters ξ and ∆γ/γsl of the Model under Discussiona glass
ξ, nm
∆γ/γsl
R2 b
CPG75 CPG115 CPG156 CPG237 CPG313 CPG729c
1.30 1.36 1.52 1.48 1.52 2.22
1.06 1.04 1.25 1.41 1.12 0.83
0.984 0.999 0.997 0.997 0.996 0.967
a See eq 13. b The goodness-of-fit statistics defined as R2 ) 1 2 2 [(N - 1)/(N - 2)]∑(τi - τexp j - τexp i ) /∑(τ i ) , where N is the number of exp experimental points τi , τj is their mean value, and τi is the fit curve’s points. c The fit by the model is not credible.
F*(T,t) ) (t - κt2) [(T0 - T)/K] - 2κt + (∆γ/γsl)e-2t/ξ (13)
Figure 7. Surface layer thickness τ for CPG729 presented on a logarithmic scale.
where we have divided eq 8 by the constant γslS (which does not change the minimum condition for F) and denoted K ≡ γslT0V/∆H. Using the value K ) 70 K nm,53 we compute F*i ) F*i (Ti,t;ξ,∆γ/γsl) on a pre-defined linear grid of t with experimental temperatures Ti and arbitrary but reasonable values for ξ and ∆γ/γsl. Seeking a minimum of F*i gives us the temperature dependence of thickness τi ) τi(Ti;ξ,∆γ/γsl), which is fitted to the experimental one upon varying the parameters ξ and ∆γ/ γsl by the Simplex method.64 Results of these numerical fits are presented in Figure 6 and Table 2. As demonstrated by Figure 6, the nonlinearity of τ(T) on the semilogarithmic scale may be well explained as the effect of finite curvature of the surface. Except for CPG729, for which case the model is obviously inadequate, the fit provides 〈ξ〉 ) 1.45 nm for the average correlation length of the crystalline order that corresponds to two OMCTS layers. An increase of ξ by ca. 20% is observed when going from CPG75 to CPG313. This observation may be rationalized as an additional effect of
curvature: departing from the solid phase toward the wall, the molecules interact effectively more with a curved convex wall compared to a flat one. The fits also yield 〈∆γ/γsl〉 ) 1.14. Together with γsl ≈ 14 mJ/m2 for OMCTS,53 it results in ∆γ ≈ 16 mJ/m2. It is interesting to note that no surface melting of OMCTS was observed at its solid-vapor interface,24 implying that ∆γ may be negative in that case. This emphasizes the importance of the substrate to provide the necessary condition ∆γ > 0 for this phenomenon. As illustrated in Figure 7, τ(T) in CPG729 exhibits, as the only exception in our study, no logarithmic growth but instead a power-like one, at least in the vicinity of Tm. Such behavior is expected for dominating long-range interactions, e.g., dispersion forces.10 Indeed, by replacing the exponential potential in eq 8 with φ(t) ) ∆γσ2/(t2 + σ2), which corresponds to a repulsive dispersion force between two microscopic bodies with a limiting separation σ,10 yields
The Surface Melting of Octamethylcyclotetrasiloxane
τ)
[
∆H (Tm - T) 2∆γσ2T0ν
]
J. Phys. Chem. B, Vol. 111, No. 7, 2007 1579
-1/3
(13)
on the condition 2κτ , 1. In real systems, an interfacial potential should contain both a repulsive short-range term and an either attractive or repulsive long-range term,23,65 with the latter dominating at large enough separation of lw and sl interfaces. This may very well be the case for OMCTS in CPG729 where, at temperatures close to Tm, the layer thickness reaches 4 nm (see Figure 5). The crossover thickness depends on the Hamaker constant, determining the strength of the dispersion forces, and on the correlation length ξ. Unfortunately, the magnitude and sign of the Hamaker constant is not known for our system, and therefore, we cannot properly estimate the effect of long-range interactions. On the other hand, introducing the Hamaker constant in the model as a third fitting parameter (that is, in addition to ξ and ∆γ) is not without complications because the concept of a single Hamaker constant breaks down for an arbitrary thin surface layer.66 Conclusion Systematic investigations of the role of curvature in surface melting have been rare. The data we present in Figure 3 are therefore unique and represent a major advance. On the basis of those data and an extended density-functional model, we could gain one important insight: at moderate thicknesses of the quasi-liquid layer, that is, 2-4 monolayers of OMCTS, surface melting is dominated by repulsive short-range interfacial interactions. For flat surfaces, such interactions are known to yield a logarithmic dependence of the layer thickness upon temperature deviation from the bulk melting point T0. For curved surfaces, the model above predicts, in the first order, the same logarithmic dependence but upon temperature deviation from the curvature-dependent melting point Tm < T0 within the pore volume. The experimentally observed departure from such logarithmic behavior was well described by retaining higher order terms in the free-energy expression, which become significant at high wall curvatures. Modeling the repulsive shortrange interactions by an exponential potential, we also obtained two quantitative characteristics of the interface at which the surface melting occurs: (i) the free energy gain ∆γ ≡ γsw γlw - γsl that arises by having a quasi-liquid layer separating the solid from the wall and (ii) the correlation length ξ that shows how far into the surface-molten layer the molecular arrangement retains the memory of the adjacent crystalline order. In our largest investigated pores, we had a clear indication that long-range interactions may start to have an influence on surface melting if the quasi-liquid layer is sufficiently (>5-6 monolayers) thick. Lacking information about the Hamaker constant in the investigated system, we postpone a detailed analysis of the interplay between short- and long-range interactions for a future work. In this context, we note that our experimental protocol can, with trivial modifications, be applied for other liquids and, perhaps, also in particle sediments instead of porous glasses. Finally, we would like to point out that, as for any other method applied to measure the extension of pre-molten surface layers, our definition of layer width is dependent on the detection technique. Here, we distinguish between two pools of molecules on the basis of their molecular mobilities. The first, “rigid” pool contains those molecules that, on average, exhibit slower molecular motions. This results in a quick Gaussian-like decay of the transverse magnetization. The other, “quasi-liquid” pool contains more mobile molecules, which therefore exhibit a
slower exponential-like transverse decay. That our model functions in Figure 1a describe the signal well ascertains that this distinct two-pool model is an adequate description. However, other methods such as infrared spectroscopy or scattering techniques31-33,67-69 may set the limit between “solid” and “quasi-liquid” (i) elsewhere and (ii) based on criteria other than dynamical such as density. This should be kept in mind when making comparisons to relevant other data. Conversely, comparison of our NMR-based method with other ones is a pending task. Acknowledgment. We thank Professor E. Tosatti for useful comments and Dr. Susanna Wold for her help with the BET measurements. This work has been supported by the Swedish Science Council (VR) and the Knut and Alice Wallenberg Foundation. D.V.F. also appreciates Conacyt, Mexico for a scholarship. Appendix Here, we outline and extend the arguments25,60-62 that lead to a simple continuous model of the solid-liquid interface of variable thickness. Consider a semi-finite system adjacent to a flat wall. The free energy per unit area of such a system is expressed as a functional of the order parameter F(z)
F{F} )
∫f(F(z)) dz
(A.1)
where f(F(z)) is the free energy density at distance z from the surface. The order parameter is chosen to be zero for a pure liquid (Fl ) 0) and unity for a pure crystalline solid (Fs ) 1). If F(z) varies smoothly between these two values, one may approximate70 f(F(z)) ) fu(F) + J(dF/dz)2, where fu is the free energy density for a uniform system and J is a positive constant. For sake of simplicity, interactions between the wall and fluid are assumed to be sufficiently short-range that their contribution to F is simply φ(Fw), where Fw is F(z)0). Hence, one can write
F{F} ) φ(Fw) +
∫∞0 [fu(F) + J(dF/dz)2] dz
(A.2)
For F(z) that provides a minimum F{F} with a fixed Fw, the integrand in eq A.2 has to satisfy the Euler-Lagrange equation. In the absence of an explicit z-dependence of the integrand, this yields
J(dF/dz)2 ) fu(F) - fu(Fb) ≡ ∆fu(F)
(A.3)
where Fb denotes the bulk order parameter in either the liquid or solid phase where (dF/dz) f 0. At a temperature T ≈ T0, with T0 being the bulk melting point, one can take for fu(F) the simple parabolic form:
∆fu ) RF2 ∆fu ) R(Fcr - F)2
F e F˜ F g F˜
(A.4a) (A.4b)
where F˜ corresponds to a barrier between the liquid and solid states (Figure 8). Integrating the differential equation (A.3) with eq A.4 and the boundary conditions at z ) 0 and z f ∞ yield
F(z) ) F˜ e(z - t)/ξ
zet
F(z) ) Fcr - (Fcr - F˜ )e-(z - t)/ξ z g t
(A.5a) (A.5b)
where ξ ≡ xJ/R and t is the coordinate at which F(z) ) F˜ . The boundary conditions at z ) 0 imply the relationship
1580 J. Phys. Chem. B, Vol. 111, No. 7, 2007
Petrov et al.
Figure 8. Schematic representation of the free energy density ∆fu as a function of the order parameter F at the phase equilibrium temperature T ) T0 (adopted from ref 61).
Figure 9. The order parameter profile F(z) corresponding to eq A.5.
t ) ξ ln(F˜ /Fw), which clarifies that parameters Fw and t are interchangeable, and one can alter F by varying either Fw or t. The F(z) profile provided by eq A.5 is shown in Figure 9. Backward substitution of eq A.3 into eq A.2 enables one to write F as a sum of bulk and surface terms:
F ) Fb + Fs ) Fb + φ(Fw) + 2J
∫∞0 (dF/dz)2 dz
(A.6)
If the quasi-liquid layer adjacent to the wall is much thicker than the F(z) profile width, t . ξ (see Figure 9), there are two distinct interfaces, namely the liquid-wall and solid-liquid ones. In this case, eq A.6 can be expressed as
F ≈ Fb + γlw + γsl
(A.7)
where γsl and γlw are the experimentally well-defined free energies of the solid-liquid and liquid-wall interfaces, respectively. At the other limit t ) 0, one can write
F ≈ Fb + γsw
(A.8)
where γsw is the free energy of the solid-wall interface. The way that F approaches its limiting values is the question. If one assumes a parabolic dependence
φ(Fw) ) R1Fw2 + C ) R1F˜ 2e-2t/ξ + C
(A.9)
where R1 > 0 and C is a constant for φ(Fw), and integrates eq A.6, then
F ) Fb + C + 2xJR F˜ 2 + (R1 - xJR)F˜ 2e-2t/ξ
(A.10)
or taking into account eqs A.7-A.9,
F ) Fb + γlw + γsl + ∆γe-2t/ξ
(A.11)
where ∆γ ≡ γsw - γlw - γsl. Equation A.11 is the sought-after free energy per unit surface area of the system at T ≈ T0, with the quasi-liquid layer thickness t being a parameter. References and Notes (1) (2) (3) (4) (5) (6)
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