Water in Nanopores: III. Surface Phase Transitions of Water on

Raluca Musat , Jean Philippe Renault , Marco Candelaresi , D. Jason Palmer , Sophie Le Caër , Roberto Righini , Stanislas Pommeret. Angewandte Chemie...
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J. Phys. Chem. C 2007, 111, 15716-15725

Water in Nanopores: III. Surface Phase Transitions of Water on Hydrophilic Surfaces† Ivan Brovchenko* and Alla Oleinikova Physical Chemistry, Dortmund UniVersity, Otto-Hahn-Strasse 6, Dortmund D-44227, Germany ReceiVed: May 15, 2007; In Final Form: August 19, 2007

Water in hydrophilic pores shows a rich phase diagram due to the appearance of the surface phase transitions. By simulation studies of water in various hydrophilic pores, we have analyzed the evolution of the water phase diagram with the strengthening of a water-wall interaction potential U0. A second-order wetting transition is observed rather close to the liquid-vapor critical temperature when U0 ≈ -2 kcal/mol. The wetting temperature rapidly decreases with the strengthening of a water-wall interaction. A first-order wetting transition, accompanied by a prewetting transition, is observed when the hydrophilicity of the walls is within the range of -4 e U0 e -3 kcal/mol. At U0 e -4 kcal/mol, the temperature of the wetting transition is below the bulk freezing temperature, and one or two layering transitions appear instead of the prewetting transition. Coexistence curves of the first layering transition are well described by the two-dimensional Ising model with critical exponent β ) 0.125. The critical temperature of a layering transition is not sensitive to the pore size and shape. It is always above the bulk freezing temperature and varies from about 400 K at U0 ≈ -3 kcal/mol to about 330 K for quasi-two-dimensional water.

1. Introduction Adsorption of water from the air on hydrophilic surfaces occurs in various natural processes on the earth. Obviously, the presence of water clusters, water layer(s), or a macroscopic water film on the surface essentially modifies the system properties. For example, hydration water at the biosurfaces is crucial for biofunctions, which appear above some critical hydration level, when the surface is covered homogeneously by the hydrogenbonded water network.1,2 Studies of water adsorption at hydrophilic surfaces are important in various fields of science and technology. The character of the adsorption of fluids on strongly attractive surfaces depends on the occurrence of the surface transitions, such as layering, prewetting, and wetting. To predict the behavior of water near various surfaces, it is necessary to know the surface phase diagram of water, which shows how the surface transitions depend on the fluid-wall interaction. In particular, it shows the evolution of the temperature of the wetting transition and the critical temperatures of the layering and prewetting transitions with strengthening of the fluid-wall interaction. The surface phase diagram was studied for Ising magnets3,4 and for some model fluids,5 but it is not known for water. The available theory of the surface transitions3,6-8 allows one to expect the following scenarios for the adsorption of a fluid at subcritical temperatures. A macroscopically thick liquid film appears at a solid surface in a saturated vapor phase with increasing temperature due to the wetting transition at T ) Tw. The thickness of this film may develop continuously (secondorder wetting transition) or may jump discontinuously (firstorder wetting). In the latter case, Tw is the temperature of a triple point where a bulk liquid-vapor phase transition meets a prewetting transition, which marks the coexistence of a vapor phase with a liquid film. The coexistence curve of a prewetting †

Part of the “Keith E. Gubbins Festschrift”. * To whom correspondence should be addressed. E-mail: brov@heineken. chemie.uni-dortmund.de.

transition ends at the critical temperature, which should be lower than the bulk critical temperature. The temperature Tw of the wetting transition decreases with strengthening of the fluidwall interaction and may be lower than the bulk freezing temperature. At some particular strength of the fluid-wall attraction, the prewetting transition is replaced by a sequence of layering transitions. The first layering transition is a 2D condensation of about one monolayer of fluid molecules at the solid surface. The second and subsequent layering transitions correspond to the condensation of a fluid layer at the surface of mono- or multilayer film. Both prewetting and layering transitions are the quasi-2D first-order phase transitions, which occur out of the bulk liquid-vapor coexistence at undersaturated vapor pressures. The quasi-2D character of the layering and prewetting transitions is due to the finite width of fluid films. Their critical points and asymptotic critical behavior belong to the universality class of the 2D Ising model. The layering transitions were studied experimentally for fluids adsorbed at highly homogeneous and planar crystalline surfaces of graphite, lamellar halides, metal oxides, and so forth. In the adsorption isotherm, a layering transition appeared as a sharp vertical step, providing about monolayer coverage of the surface. Such kinds of behavior were reported for water adsorption on the surfaces of salt crystals and on the hydroxylated surfaces of some metal oxides. However, the critical temperature Tc,1 of the first layering transition of water was estimated only for water on the hydroxylated surface of Cr2O3 (Tc,1 ≈ 305 K9) and on the surface of NaF (Tc,1 > 308 K10). The critical temperatures of the layering transition were observed below the bulk freezing temperature for noble gases, molecular hydrogen, molecular nitrogen, methane, and methyl chloride and above the bulk freezing temperature for water, ethylene, ethane, propane, molecular oxygen, and alcohols (see ref 11 for a review of experimental studies). The critical temperature Tc,1 of the first layering transition of fluids is typically about 0.30-0.55 of the bulk liquid-vapor critical temperature T3D. In particular, Tc,1 depends strongly on the dimensional incompatibility between

10.1021/jp073751x CCC: $37.00 © 2007 American Chemical Society Published on Web 10/06/2007

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the adsorbate molecules and substrate.12 For comparison, the critical temperature of a 2D Lennard-Jones fluid is about 0.40T3D.13 With increasing of the layer number, the critical temperature of the layering transition may increase or decrease when approaching the roughening temperature. Some of the coexistence curves of the layering transitions obtained experimentally14-17 were fitted to the simple scaling law

(

)

Tc,1 - T p∼ Tc,1

β

(1)

where p is the order parameter, which measures a dissimilarity of the coexisting phases, and β is the critical exponent. The values of β obtained from the fits varied from about 0.10 to 0.20, in reasonable agreement with β ) 0.125, expected for the critical behavior of 2D systems. The are several experimental observations of the prewetting phase transition in one-component fluids (see ref 18 for more details). The temperature of the first-order wetting transition Tw, where vapor, adsorbed film, and liquid coexist, varies from about 0.38T3D for helium on cesium19 to 0.89T3D for mercury on sapphire.20 The critical temperature Tc,p of the corresponding prewetting transitions varies from 0.48T3D to 0.99T3D and even exceeds the bulk critical temperature (Tc,p ) 1.05T3D) in the case of mercury on molybdenum.21 Note that the prewetting transition as well as the wetting transition of water was never observed experimentally. The sequence of layering transitions was obtained for the lattice-gas model by various theoretical and simulation methods (see, for example, refs 4, 11, and 22-29 for more details). For strong surface potentials, the critical temperatures of the layering transitions are close to the critical temperature of the 2D system, and they slightly increase with layer number, approaching the roughening temperature.24 The surface heterogeneity causes a decrease of Tc,1 and may result in the disappearance of the firstorder layering transition.29 With the weakening of a substrate potential, Tc,1 increases, and condensation of two or more subsequent layers could occur simultaneously.23,26,29 For yet weaker substrate potentials, the prewetting transition, that is, a condensation of a film of a several molecular layers width, appears instead of the sequence of layering transitions.24,26-28,30,31 The discrete nature of the lattice models yields an infinite sequence of layering transitions. In continuum models, the layering transitions are promoted by the density oscillations near the wall. As these oscillations decay rather quickly in a liquid coexisting with a vapor, only a finite sequence of the layering transitions can be expected for fluids. Layering and prewetting transitions of various model fluids were studied using density functional theories7,28,32-46 and by computer simulations.47-63 In computer simulations, surface phase transitions can be studied in a slit-like or cylindrical geometry, with periodic boundary conditions in two and one directions, respectively. In most simulation studies, cited above, a slit pore with two asymmetric walls (an adsorbing wall and a hard wall) was used, and the chemical potential of a fluid was varied in order to get the surface transition. The disadvantage of this approach is that the phase diagram of a fluid in such a pore remains unknown, and a surface transition may be mixed up with a capillary condensation. Alternatively, the surface transitions can be studied by the simulations of the phase diagram of confined fluid. In this case, the surface phase transitions and the capillary condensation can be clearly distinguished. By using this approach, we have found the layering and prewetting transitions of water near hydrophilic surfaces in simulations.11,58-60 In the

present paper, we study how these transitions depend on the strength of the water-wall interaction, pore size, and geometry. Additionally, the liquid-vapor phase transition of 2D water is analyzed. These results form the basis for the construction of the surface phase diagram of water. 2. Methods The layering and prewetting transitions of water were studied by simulations of the TIP4P64 and ST265 water models confined in cylindrical pores with radii Rp from 12 to 25 Å and slit pores of a width of Hp ) 24 Å. A spherical cutoff of 12 Å for both the Coulombic and LJ parts of the water-water interaction potential was used. In accordance with the original parametrization of the TIP4P and ST2 models, no long-range corrections were included. The interaction between the water molecules and the surface was described by a (9-3) LJ potential

Uw(r) ) [(σ/r)9 - (σ/r)3]

(2)

where r is the distance from the water oxygen to the pore wall and parameter σ is fixed at 2.5 Å. As the appearance of the surface phase transitions is expected when the well depth U0 of the water-wall potential is comparable with the energy of the pair water-water interaction, the parameter  was varied to change U0 from -1.93 to -7.70 kcal/mol. Besides, we have simulated a liquid-vapor transition of 2D water on a substrate with infinite attraction of oxygen atoms (in the limit U0 f -∞). In this case, all oxygen atoms were located in one plane, while molecular rotations were not restricted. The (9-3) LJ potential with σ ) 2.5 Å resulted from the integration over a semi-infinite solid, formed by uniformly distributed LJ particles with σ ) 3.5 Å. Numerical integrations over (12-6) LJ particles forming a cylindrical pore showed that the value of σ remained practically the same as that for the semi-infinite solid, whereas the well depth U0 increased by a factor of about 1.4 and 1.2 in cylindrical pores with radii of Rp ) 12 and 25 Å, respectively. The average water density in the pore was calculated, assuming that a pore volume until the distance σ/2 ) 1.25 Å from the pore wall was occupied by water. Coexistence curves of water in pores in a wide temperature range were obtained by Monte Carlo (MC) simulations in the Gibbs ensemble,66,67 which provides direct equilibration between the coexisting phases. Due to the occurrence of the surface phase transitions, up to three two-phase regions were obtained for some systems. In these cases, simulations in the Gibbs ensemble were performed separately for each of these regions by appropriate choice of the average water density of the simulation system. The number of water molecules in the two simulation cells varied from about 400 in simulations of the layering transition in the smallest cylindrical pores with Rp ) 12 Å up to about 5000 in the simulations of the liquid-vapor coexistence in the largest cylindrical pore with Rp ) 25 Å. Typically, several dozens of the successful molecular transfers between the coexisting phases per molecule were provided in the course of the simulations in the Gibbs ensemble. This number was minimal (about 2) for the low-temperature simulations of the coexistence between the water film and liquid in the large pores and maximal (several hundreds) for the simulations of the layering transitions. An essential number of the molecular transfers (several transfers per molecule) was achieved even in the case of the coexistence between a 2D water vapor and a 2D ice at T ) 255 K. Insertion of water molecules into 2D ice was possible due to the presence of several vacancies. The details of the efficient techniques for molecular transfer may be found

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Figure 1. Coexistence curves of TIP4P water in cylindrical pores of radius Rp ) 25 Å with different strengths U0 of the water-wall interaction. Liquid-vapor coexistence is shown by open circles. Coexistence between a water film and a liquid water is shown by black closed circles. The coexistence curve corresponding to the prewetting transition is shown by blue circles. Horizontal lines indicate the estimated temperature of the wetting transition.

in ref 60. The ratio between the lateral size of the slit pore and its width Hp always exceeded 1 and was essentially higher for the high-temperature phases with one or two adsorbed water layers and for the low-temperature vapor phase. Similar values were provided for the ratio between the length of the cylindrical pore and its diameter. The density profiles of the coexisting phases were obtained by MC simulations in the constant-volume ensemble, using the average densities of these phases obtained in the Gibbs ensemble simulations. 3. Results 3.1. Phase Diagrams of Water in Pores. Coexistence curves of water in cylindrical pores with radius Rp ) 25 Å for two strengths of the water-wall interaction, U0 ) -1.93 and -3.08 kcal/mol, are shown in Figure 1. In the pore with a relatively weak water-wall potential (U0 ) -1.93 kcal/mol), liquid water coexists with a vapor in a wide temperature range from 250 to 510 K. When the temperature increases to T ) 515 K, the density of the vapor phase sharply increases from 0.040 to 0.149 g/cm3, indicating formation of a water film adsorbed on the pore walls in a vapor phase. This is a signature of the wetting transition, which occurs upon heating along the pore coexistence curve at the wetting temperature Tw ≈ 512.5 ( 2.5 K. In the case of a first-order wetting transition, a prewetting transition should split from the liquid-vapor transition at Tw. However, with used temperature step in 5°, we did not get coexistence between a vapor and a water film, corresponding to the possible prewetting transition. This indicates on the second-order character of the wetting transition. Besides, near the wall, the water density in the adsorbed film and in the coexisting liquid phase differs noticeably. A qualitatively similar picture was observed for the lattice-gas model, which exhibits a second-order wetting transition.68 In an infinite system, a macroscopically thick wetting film grows continuously when

Brovchenko and Oleinikova approaching the temperature of the second-order wetting transition, whereas in the pore, it is suppressed due to the difference in the chemical potentials of the liquid-vapor coexistence in the pore and in the bulk. Therefore, a second-order wetting transition appears as a shoulder on the vapor branch of the liquid-vapor coexistence curve of confined water (see upper panel of Figure 1). A similar effect of the critical wetting transition on the shape of the pore coexistence curve was seen for the lattice-gas model.68 Note that there are no indications of the wetting transition for water in the cylindrical pore of the same size with an essentially weaker water-wall interaction U0 ) -0.39 kcal/mol.69 With the strengthening of a water-wall potential from U0 ) -1.93 to -3.08 kcal/mol, the phase diagram of water changes drastically (Figure 1, lower panel). The direct coexistence of a liquid with a vapor can be obtained in the temperature interval from 200 to 300 K (open circles in Figure 1). Starting from T ) 250 K and up to the pore critical temperature, there is a coexistence between a liquid phase and a water film adsorbed on the pore wall (closed black circles in Figure 1) In the temperature range 250 < T < 435 K, there is a first-order surface phase transition between a vapor phase and a water film (blue circles in Figure 1). In the temperature interval from T ) 250 to 300 K, three kinds of coexistence (vapor-liquid, vaporfilm, and film-liquid) can be obtained in simulations. Obviously, some of these thermodynamic states are metastable, and a triple point at temperature Tt, where vapor coexists with the water film and with liquid, is expected. There is one transition between stable phases below Tt, and there are two transitions between stable phases at T g Tt. The temperature Tt of the triple point may be estimated as Tt ) 290 ( 15 K. Analysis of the density profiles of water in the coexisting liquid phase and water film evidences that two water layers near the wall are almost identical in both phases. This means that the surface phase transition between a vapor and a water film, which occurs out of the bulk coexistence (blue circles in Figure 1), is a prewetting transition. Accordingly, the respective triple point temperature Tt should be considered as the temperature of the first-order wetting transition. In an infinite system, one may expect the appearance of a macroscopically thick wetting layer at Tt upon heating along the liquid-vapor coexistence curve. In the pore, the thickness of a wetting layer at Tt is limited to about two water layers. A similar wetting layer was observed in the simulations of water in a smaller pore of radius Rp ) 12 Å with U0 ) -3.85 kcal/mol.60 When the water-wall interaction becomes stronger (U0 ) -3.85 kcal/mol), the prewetting transition in the pore of a radius Rp ) 25 Å is still present (see blue circles in the upper panel of Figure 2). This transition meets the liquid-vapor transition (black circles) in the triple point at Tt,1 ≈ 225 K. At another triple point at Tt,2 ≈ 260 K, the prewetting transition splits into two layering transitions. The first layering transition of water (red circles), which is a 2D condensation of water at the pore wall, is seen in the temperature range from 275 to 400 K. Besides, in the narrow temperature range above Tt,2, there is a second layering transition, which is 2D condensation of a second layer of water on the first layer of water adjusted to the pore wall (green circles). Both the prewetting transition and the second layering transition disappear in a strongly hydrophilic cylindrical pore of the same radius Rp ) 25 Å but with U0 ) -4.62 kcal/mol (Figure 2, lower panel). Condensation of the first water layer at the pore walls occurs via the first layering transition, whereas

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Figure 3. Layering transition of TIP4P water in cylindrical pores of various radii Rp with the water-wall interaction strength U0 ) -4.62 kcal/mol.

Figure 2. Coexistence curves of TIP4P water in cylindrical pores of radius Rp ) 25 Å with different strengths U0 of the water-wall interaction. Liquid-vapor coexistence is shown by open circles. Coexistence between a water film and a liquid water is shown by black close circles. The first and second layering transitions and the prewetting transition are shown by red, green, and blue circles, respectively.

the second water layer grows continuously with pressure. There are no triple points in this system, at least at T g 300 K. 3.2. First Layering Transition. First, we examine the effect of the pore size and geometry on the coexistence curve of the first layering transition. For this purpose, we use the layering transitions in the cylindrical pores of a radius Rp ) 25 Å, shown in Figure 2, and the layering transitions in various cylindrical and slit pores reported in our previous studies.58-60 As the density of a water film in terms of the average pore density is strongly effected by the pore size and shape, the analysis is performed using the surface density F*, which shows the number of water molecules per 1 Å2 of the surface. Taking into account that water oxygens in the first layer are localized at about 3 Å from the pore wall, F* for the first layering transition in cylindrical pores was calculated using a cylinder with radius (Rp - 3 Å) as the adsorbing surface. In Figure 3, we show the first layering transition of water in four cylindrical pores of various radii with the same water-wall interaction U0 ) -4.62 kcal/mol. The diameter Fd ) (F/2 + F/1)/2 of the coexistence curve, which is the average of the surface density F/1 of a vapor and the surface density F/2 of a coexisting water layer, is shown by the open circles in Figure 3. Typical distributions of water molecules in the coexisting phases may be seen from the density profiles shown for some temperatures in the left panel of Figure 4. The maximum of a density profile at the distance of about 3 Å from the pore wall, which reflects the strong localization of water molecules in the minimum of the water-wall interaction potential, is present in both phases. Upon heating, the density of a monolayer phase slowly decreases (see Figure 3), and the localization of water near a pore wall slightly weakens (see Figure 4). In a vapor phase, the height of the density maximum is very small at low temperatures and becomes comparable with one in a monolayer phase only in the close proximity of the critical temperature Tc,1 of the layering transition.

Figure 4. Density profiles of water in the two coexisting phases of the first (left panel) and second (right panel) layering transitions in a cylindrical pore with a radius of Rp ) 12 Å with U0 ) -4.62 kcal/ mol. Lower- and higher-density phases are shown by red and blue lines, respectively. The profiles for lower temperatures are shifted vertically by 3 and 6 g cm-3.

The shape of the coexistence curve is determined by the temperature dependences of the order parameter ∆F and of the diameter Fd. The order parameter of the layering transition ∆F ) (F/2 - F/1)/2 measures the dissimilarity between the coexisting phases and is equal to zero at T g Tc,1. Below Tc,1, ∆F should follow the universal scaling law

∆F ) (F/2 - F/1)/2 ) B

(

)

Tc,1 - T Tc,1

β

) Bτβ

(3)

where β is the critical exponent of the order parameter, τ is a reduced temperature deviation from Tc,1, and B is a coefficient. As the layering transition should belong to the universality class of the 2D Ising model, we impose the value of the critical exponent β ) 0.125 in the analysis. We have found that the temperature dependence of the order parameter ∆F of the layering transition agrees with the theoretical expectations within

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Brovchenko and Oleinikova to the higher density in small pores. The temperature dependence of Fd was fitted to the regular equation

Fd ) (F/2 + F/1)/2 ) F/c (1 + A1τ + A2τ2)

Figure 5. Order parameter ∆F of the first layering transition of water as a function of the reduced temperature τ of TIP4P water in cylindrical and slit pores with water-wall interaction strengths of U0 ) -4.62 (1-5), -3.85 (6), and -7.70 kcal/mol (7). The pore sizes are Rp ) 12 (1,7), 15 (2), 20 (3), and 25 Å (4,6), and H ) 15 Å (5). The solid lines represent eq 3 with the slope β2D ) 0.125 expected in 2D systems and amplitudes B and critical temperatures Tc,1 given in Table 1.

the accuracy of simulations. This conclusion is supported by Figure 5, where ∆F(τ) is shown as a function of the reduced temperature τ in a double logarithmic scale. The temperature range used in the fits and the values of fitting parameters are shown in Table 1. Evidently, the critical temperature of the layering transition is practically independent of the pore size and is equal to Tc,1 ) 400.8 ( 0.8 K in all studied cylindrical pores with U0 ) -4.62 kcal/mol. In a slit pore of the same hydrophilicity, the critical temperature was found to be equal to 395.0 ( 0.2 K, that, is 5° lower. The diameters Fd(τ) of the layering transitions in cylindrical pores of various radii decrease with increasing temperature (Figure 3) and, upon approaching Tc,1, show an anomalous trend

(4)

where F/c is a critical surface density and A1 and A2 are coefficients. The fitting parameters as well as the estimated critical surface densities are shown in Table 2. In all pores with U0 ) -4.62 kcal/mol, the estimated values of F/c are within the interval 0.044 e F/c e 0.049 Å-2. To test the effect of the strength of the water-wall interaction on the layering transition, we performed simulations of water in the pores of radius Rp ) 12 Å with a much stronger interaction potential U0 ) -7.70 kcal/mol. Two coexistence curves of the layering transition for U0 ) -4.62 and -7.70 kcal/mol are compared in the left panel of Figure 6. An increase of the water-wall interaction strength suppresses the critical temperature from Tc,1 ) 401.6 ( 0.9 to 360.3 ( 0.9 K (see Table 1). The temperature dependence of the order parameter ∆F in the pore with U0 ) -7.70 kcal/mol closely follows the asymptotic 2D behavior from the critical temperature down to about 200 K (Figure 5). The diameter Fd of the layering transition in this pore shows a strong anomalous bend toward higher density upon approaching Tc,1 and cannot be satisfactory fitted by the regular eq 4. With the strengthening of the fluid-wall potential, in the limit U0 f -∞, the layering transition should approach the liquid-vapor transition of a 2D fluid. The phase diagram of a 2D water, with water oxygen located in a plane but without restrictions on molecular rotations, is shown in Figure 7. A liquid phase of a 2D water exists in a temperature interval from the critical temperature, which we have estimated as 330 ( 5 K, to the freezing temperature. Strong hysteresis was observed in the temperature range from 260 to 300 K, indicating first-order liquid-solid phase transition. 2D ice represents a square-like lattice with long-range order, whereas chain-like hydrogen-bonded patterns are the dominant structure of 2D liquid.11 The freezing temperature may be estimated as a center of the hysteresis temperature interval, that is, at about 280 K. Therefore, 2D water exists in a liquid state within a rather narrow temperature interval of about 50°. Note that we did not observe a freezing of the quasi-2D water monolayers adsorbed at hydrophilic surfaces with U0 g -7.70 kcal/ mol.

TABLE 1: The Values of Parameters Obtained from Fits of the Order Parameter ∆G for the Prewetting and First and Second Layering Transitions to the Eqs 3 and 5; the Parameters Fixed in the Fits are Shown by Italics pore size

model

U0 ((kcal)/(mol))

range of fit (K)

Tc (K)

β

B (Å-2)

0.179(9) 0.275(8) 0.125

0.1257(29) 0.1344(12) 0.112(2)

Rp ) 12 Å Rp ) 25 Å Hp ) 24 Å

TIP4P TIP4P ST2

-3.85 -3.08 -3.08

Prewetting Transition 250 e T e 385 390.1(4) 250 e T e 435 435.9(4) 335 e T e 350 353.8(10)

Rp ) 12 Å Rp ) 15 Å Rp ) 20 Å Rp ) 25 Å Hp ) 24 Å Rp ) 12 Å Rp ) 25 Å

TIP4P TIP4P TIP4P TIP4P TIP4P TIP4P TIP4P

-4.62 -4.62 -4.62 -4.62 -4.62 -7.70 -3.85

First Layering Transition 200 e T e 400 401.6(9) 100 e T e 400 401.1(5) 150 e T e 400 400.0(2) 225 e T e 400 400.2(4) 175 e T e 395 395.0(2) 225 e T e 355 360.3(9) 275 e T e 395 403.5(18)

0.125 0.125 0.125 0.125 0.125 0.125 0.125

0.0619(6) 0.0626(3) 0.0623(3) 0.0619(2) 0.0627(6) 0.0627(8) 0.0704(6)

Rp ) 12 Å Rp ) 15 Å Hp ) 24 Å Rp ) 12 Å Rp ) 25 Å

TIP4P TIP4P TIP4P TIP4P TIP4P

-4.62 -4.62 -4.62 -7.70 -3.85

Second Layering Transition 225 e T e 330 340(2) 225 e T e 300 340(2) 250 e T e 315 315.2(8) 250 e T e 325 340(2) 265 e T e 275 276(3)

0.125 0.125 0.125 0.125 0.125

0.059(13) 0.027(5) 0.0272(10) 0.030(1) 0.0320(41)

b1

13.8 0.676 0.709 0.676

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TABLE 2: The Values of the Critical Density G/c Estimated from the Temperature Dependence of the Diameter Gd or Obtained from the Fit of Gd to Eq 4 pore size

U0 ((kcal)/(mol))

F/c (Å-2)

A1

A2

Rp ) 12 Å Rp ) 25 Å Hp ) 24 Å

Prewetting Transition -3.85 0.0765(39) 1.46(51) -3.08 0.0565(19) 3.60(57) -3.08 0.0800(20) 0.52(30)

-1.34(85) -3.81(65)

Rp ) 12 Å Rp ) 15 Å Rp ) 20 Å Rp ) 25 Å Hp ) 24 Å Rp ) 12 Å Rp ) 25 Å

First Layering Transition -4.62 0.0494(4) 0.281(28) -4.62 0.0478(3) 0.376(18) -4.62 0.0485(11) -4.62 0.0439(10) 1.11(27) -4.62 0.0440(10) -7.70 0.0508(12) -3.85 0.0515(5) 0.623(27)

-1.01(54)

Rp ) 12 Å Rp ) 15 Å Hp ) 24 Å Rp ) 12 Å Rp ) 25 Å

Second Layering Transition -4.62 0.171(4) 0.056(8) -4.62 0.192(4) 0.020(6) -4.62 0.190(2) 0.075(15) -7.70 0.172(3) 0.049(12) -3.85 0.167(5) 0.195(30)

-0.18(6)

In the large pore of a radius Rp ) 25 Å, the layering transition occurred already when U0 ) -3.85 kcal/mol (Figure 2, upper panel). The coexistence curves of a layering transition in terms of a surface density for this pore and for the pore of the same size but with U0 ) -4.62 kcal/mol are compared in the right panel of Figure 6. Weaker localization of the adsorbed water layer near the wall for the less attractive surface caused an increase of its average density. Accordingly, the coefficient B (Table 1) and the critical density F/c (Table 2) were higher for the weaker water-wall interaction. Note that the critical temperature of the layering transition decreased by only 3° when U0 changed from -3.85 to -4.62 kcal/mol. 3.3. Second Layering Transition. The second layering transition was found for a water-wall interaction strength of U0 ) -4.62 kcal/mol in cylindrical pores of radii Rp ) 12 and 15 Å, and it was not detected in larger cylindrical pores of radii Rp ) 20 and 25 Å (see Figure 2 (lower panel) in this paper and Figure 11 in ref 60). It was also found for water confined in the slit pore with a width of Hp ) 24 Å with the same waterwall interaction (Figure 11 in ref 60) and in a cylindrical pore with Rp ) 25 Å and U0 ) -3.85 kcal/mol (Figure 2). Density profiles in the coexisting phases of the second layering transition in a cylindrical pore of radius Rp ) 12 Å are shown in the right panel of Figure 4. Obviously, the water density in the first layer is almost the same in the two coexisting

Figure 6. Effect of the strength of the water-wall interaction on the layering transition of TIP4P water in cylindrical pores with radii Rp ) 12 and 25 Å.

Figure 7. Phase diagram of quasi-two-dimensional TIP4P water with all oxygens located in one plane (U0 f -∞). The data points obtained with increasing and decreasing temperature are shown by red and blue circles, respectively.

Figure 8. Coexistence curve of the second layering transition of water in a slit pore of width Hp ) 24 Å and U0 ) -4.62 kcal/mol (open circles). Fits of the coexistence curve with parameters shown in Tables 1 and 2 are shown by the solid line. The diameter Fd and its fit to the eq 4 are shown by solid squares and the dashed line, respectively.

phases, and this observation remains valid at all temperatures studied. The second density maximum at about 5.8 Å from the surface indicates localization of water molecules in the second layer. There is a noticeable amount of water in the second layer in the lower-density phase even at low temperatures. The second maximum of the density profiles is not as pronounced as the first maximum near a pore wall. The second layering transition in the slit pore of a width Hp ) 24 Å is shown in Figure 8 in terms of a surface density. It occupies the density interval from 0.16 to 0.22 Å-2, indicating that the high-density phase corresponds to the surface density of about two layers of water molecules. Although the second layering transition is expected to be two-dimensional, the order parameter shows significant deviation from the expected asymptotic power law (eq 3). Therefore, the temperature dependence of the order parameter in the temperature range 250 e T e 315 K was fitted to the extended scaling equation, which includes one correction to scaling

∆F ) Bτ0.125 (1 + b1τ)

(5)

15722 J. Phys. Chem. C, Vol. 111, No. 43, 2007

Figure 9. Coexistence curves of the second layering transition of water in cylindrical pores of radius Rp ) 12 Å for water-wall interaction strengths of U0 ) -4.62 (blue squares) and -7.70 kcal/mol (red circles). The diameters Fd are shown by respective solid symbols. Fits of the coexistence curves F* ) Fd ( ∆F and fits of the diameters with parameters shown in Tables 1 and 2 are shown by solid and dashed lines, respectively.

where b1 is the coefficient of the correction term. The obtained values of the critical temperature Tc,2 and of the fitting coefficients B and b1 are shown in Table 1. The diameter of the second layering transition decreases slightly with temperature and may be described by eq 4. The fitting equations for two coexisting phases of the second layering transition F* ) Fd ( ∆F with parameters given in Table 2 are shown in Figure 8. For the analysis of the surface density at the second layering transition in cylindrical pores, F* was calculated using a cylinder with radius Rp ) 4.5 Å as the adsorbing surface, taking into account that the center of a water bilayer is located at about 4.5 Å from the pore wall. In cylindrical pores with U0 ) -4.62 kcal/mol, the shape of the coexistence curve of the second layering transition is far from the shape expected for 2D systems (see Figure 9). Imposing the critical exponent β ) 0.125, the order parameter ∆F was fitted to eq 5 to estimate the values of the critical temperature Tc,2. It was found to be almost the same (Tc,2 ≈ 340 K) in cylindrical pores of radii Rp ) 12 and 15 Å and ∼25° lower in slit pores of width Hp ) 24 Å at the same water-wall interaction strength U0 ) -4.62 kcal/mol. The effect of the water-wall interaction on the second layering transition may be studied by comparison of this phase transition in two pores of the same geometry. Two coexistence curves, corresponding to the second layering transitions in a cylindrical pore of radius Rp ) 12 Å with a water-wall strengths of U0 ) -4.62 and -7.70 kcal/mol, are compared in Figure 9. The strengthening of a water-wall interaction makes this transition essentially wider in terms of a surface density (coefficient B in eq 5 increases) and closer to the phase transition in 2D systems (correction term b1 in eq 5 decreases). Fits of ∆F and Fd to eqs 5 and 4 indicate the closeness of their critical temperatures within the accuracy of simulations. The critical density of the second layering transition is only slightly higher in the pore with a stronger water-wall attraction. In the largest studied pore of radius Rp ) 25 Å, the second layering transition is observed at weaker interaction of U0 ) -3.85 kcal/mol (see Figure 2). The estimated critical temperature of this transition ∼276 K is noticeably lower than that in all other pores. 3.4. Prewetting Transition. A prewetting transition of water as well as a temperature of a wetting transition along the liquid-

Brovchenko and Oleinikova

Figure 10. Coexistence curves of ST2 water in a slit pore of width Hp ) 24 Å with a water-wall interaction strength of U0 ) -3.08 kcal/ mol. Liquid-vapor coexistence is shown by black circles. The coexistence between the water film and liquid water is shown by red circles. The coexistence curve of a prewetting transition is shown by blue circles.

Figure 11. Density profiles of water in the two coexisting phases of water in cylindrical pore with radius Rp ) 12 Å and with U0 ) -4.62 kcal/mol. Profiles for lower- and higher-density phases are shown by red and blue lines, respectively. Profiles for lower temperatures are shifted vertically on 3 and 6 g cm-3.

vapor coexistence curve is extremely sensitive to the waterwall interaction. In the particular case of a large cylindrical pore with Rp ) 25 Å, the prewetting transition is not seen when U0 g -1.93 kcal/mol, and it splits into layering transitions already at U0 ≈ -3.5 kcal/mol. We have succeeded in finding a prewetting transition of ST2 water confined in a slit pore of Hp ) 24 Å with U0 ) -3.08 kcal/mol (see Figure 10). We estimate the temperature of the triple point of the prewetting transition and the liquid-vapor phase transition as Tt ) 335 ( 5 K. This value is comparable with the temperature Tt ) 295 ( 15 K of the triple point of TIP4P water in a cylindrical pore of a radius Rp ) 25 Å with the same water-wall interaction. The density profiles of two water phases coexisting at the prewetting transition are shown in Figure 11. Two water layers adsorbed on the pore wall are seen at low temperatures when they coexist with a very low density vapor. Upon heating, the density of both layers decreases, whereas more water molecules appear

Water in Nanopores

Figure 12. Prewetting transition of TIP4P water in cylindrical pores of radii Rp ) 12 and 25 Å and of ST2 water in a slit pore of width Hp ) 24 Å. Diameters of the coexistence curve of the prewetting transition are shown by squares, and their fits to eq 4 with parameters given in Table 2 are shown by solid lines. The fit of the coexistence curve in a slit pore with a 2D value of the critical exponent β ) 0.125 is shown by the dashed line.

in a vapor phase. Note that the water density in the second layer is noticeably lower than that in the first layer when the temperature approaches the critical temperature of the prewetting transition. As far as the prewetting transition is a surface phase transition, it was analyzed in terms of a surface density, which, for cylindrical pores, was defined similarly to that of the case of the second layering transition (see above). The prewetting transitions in a large pore of radius Rp ) 25 Å with U0 ) -3.08 kcal/mol and in a narrow pore of radius Rp ) 12 Å with U0 ) -3.85 kcal/mol are shown in Figure 12 together with a prewetting transition for ST2 water. The temperature dependence of the order parameter ∆F of the prewetting transition was fitted to eq 3 with β as an adjustable parameter. We have found that β ) 0.125 is appropriate to describe the temperature dependence of ∆F in a slit pore only, whereas a larger value of β is obtained in both cylindrical pores (see Figure 13 for the dependence of ∆F on τ on a double logarithmic scale). The trend toward 3D behavior with β ≈ 0.326 is stronger in a larger pore due to a thicker prewetting layer. The critical temperature of the prewetting transition varies from Tc,p ≈ 354 K in the slit pore to Tc,p ≈ 436 K in the large cylindrical pore (Table 1). The value F/c of the critical surface density of a prewetting transition decreases with the increasing of its critical temperature (Figure 12). Similar to the layering transition, the coexistence curve diameter Fd of the prewetting transition shows an anomalous bend toward higher density in the narrow cylindrical pore when approaching Tc,p (see Figure 12). 4. Discussion Our simulations evidence that water exhibits surface phase transitions such as layering and prewetting near a smooth hydrophilic surface. The first layering transition, which is a firstorder phase transition from vapor to about one monolayer of water adsorbed at a hydrophilic surface, is observed in a wide temperature range when the water-wall interaction U0 is stronger than about -4 kcal/mol. The order parameter of the first layering transitions in various pores follows the critical behavior of the 2D Ising model with the critical exponent β ) 0.125, in accordance with theoretical expectations. In pores with

J. Phys. Chem. C, Vol. 111, No. 43, 2007 15723

Figure 13. Order parameter ∆F of the prewetting transition as a function of the reduced temperature τ for TIP4P water in cylindrical pores (two upper panles) and for ST2 water in a slit pore (lower panel). The values of the critical temperature of the prewetting transitions Tc,p and the slopes β of the solid straight lines are given in Table 1. The dashed lines show the slope β2D ) 0.125 expected in 2D systems.

Figure 14. Master plot of the layering transition of water in pores of various size and shape with a water-wall interaction strength of U0 ) -4.62 kcal/mol. The densities of the coexisting phases and the coexistence curve diameter are rescaled by the critical density F/c ) 0.044 Å-2 and shown by solid and open symbols, respectively. The solid lines represent the densities of the coexisting phases F/1,2 ) Fd ( ∆F, where ∆F is described by eq 3 with β2D ) 0.125 and B ) 1.55F/c and the diameter Fd (dashed line) is described by eq 4 with an added singular term of ∼τ0.25.

the same water-wall interaction, the density and structure of the coexisting phases and the critical temperature of the first layering transition are found to not be sensitive to the pore size and geometry. Such universality of the first layering transition is supported by the master plot where the fluid density is normalized by the critical density F/c , and the temperature is normalized by the critical temperature Tc,1 (Figure 14). The coexistence curves and diameters of the layering transitions in five different pores with U0 ) -4.62 kcal/mol coincide well when the critical density F/c ) 0.044 Å-2 and the amplitude B ) 1.55F/c . The diameter of the master coexistence curve shows a negative anomaly with approaching T1c , which can be satisfactorily described by the singular term ∼τ2β ) τ0.25.70

15724 J. Phys. Chem. C, Vol. 111, No. 43, 2007 The critical temperature of the first layering transition Tc,1 decreases with the strengthening of a water-wall interaction. A similar trend was observed for the lattice-gas model71,72 and for the LJ fluid.47 It reflects an improving two dimensionality of the system due to the stronger localization of molecules in a plane parallel to the pore wall. The highest value of Tc,1 ≈ 404 K was observed in the pore with a relatively weak waterwall interaction U0 ) -3.85 kcal/mol. With the strengthening of U0 to -4.62 kcal/mol, Tc,1 shifted only slightly to about 400 K. Near a more hydrophilic surface with U0 ) -7.70 kcal/ mol, Tc,1 ≈ 360 K. Finally, the lowest limit of the critical temperature of the layering transition at a smooth surface is a critical temperature T2D of the corresponding 2D fluid. This temperature is about 330 K, that is, 0.57T3D. Therefore, the critical temperature of the first layering transition of water may be expected to be in the interval 0.57T3D < Tc < 0.70T3D. The bulk freezing temperature of the TIP4P water model is about 0.41T3D,73 whereas the layering transition of the same water model may occur up to 0.69T3D, and there is no freezing of the adsorbed water layer for U0 g -7.70 kcal/mol. Therefore, there is a rather wide temperature interval where the layering transition of water is a phase transition between two fluid phases. Available experimental estimations of the critical temperature of the layering transition for water give the values Tc,1 g 0.48T3D,9,10 which are also above the bulk freezing temperature. Lower experimental values of Tc,1 could be explained by the restrictions on the rotation of water molecules and by the roughness of the surfaces in experiments. Both factors are absent in our simulation studies. Rotational motion of molecules adsorbed at the surface effectively decreases the two-dimensional character of the system and should cause an increase of Tc,1. This could explain why the critical temperatures of the layering transition of spherically symmetrical molecules (such as noble gases, for example) are below the freezing temperature, whereas for strongly asymmetrical molecules (ethylene, ethane, propane, water, and alcohols), the situation is opposite (see ref 11 for more details). We did not observe a triple point of the first layering transition and the liquid-vapor phase transition, where a monolayer of water coexists with liquid water and vapor. A second layering transition, that is, a quasi-2D condensation of about water monolayer on a substrate, which is already covered by one liquid layer of water, was seen in some slit and cylindrical pores when the water-wall interaction varied from U0 ) -3.85 to -7.70 kcal/mol. It may possess a triple point both with the first layering transition and with the liquid-vapor transition. The critical temperature of the second layering transition is always lower than the critical temperature of the of the first one, and it was found between 0.48T3D and 0.59T3D. This reflects the expected approach of the critical temperature of the layering transition to the roughening temperature with the increase of layer number. Similar behavior was found by density functional calculations for a strongly associative LJ fluid in pores.39-42 The existence of the second layering transition, its critical temperature, and the properties of the coexisting phases were found to be strongly sensitive to the strength of a water-wall potential, pore size, and shape. Even near strongly hydrophilic surfaces, there were only two noticeable density oscillations of liquid water when it was in equilibrium with a saturated vapor or with a saturated bulk water. Water properties (e.g., orientational ordering) in the third and subsequent layers are close to the bulk ones.59,60,74-76 This could explain why water does not show the third and subsequent layering transitions.

Brovchenko and Oleinikova When the water-wall interaction is comparable with the water-water pair interaction (i.e., U0 ≈ -3 kcal/mol), a prewetting transition, which is a condensation of water film of about two layers thick, splits from the liquid-vapor phase transition. The prewetting transition seems to be rather sensitive to a water model. Near the same wall, its critical temperature Tc,p is by about 23% higher for TIP4P than that for the ST2 model, whereas their bulk critical temperatures differ by 5% only. In the presence of a prewetting transition, there is a a triple point, where vapor, water film, and the liquid phase coexist. A similar triple point was observed in computer simulations of the lattice-gas model.30,31 The temperature Tt of the triple point in the pore is a temperature where we can expect a first-order wetting transition in a semi-infinite system. This temperature drastically depends on the water-wall potential U0. When U0 ≈ -4 kcal/mol, Tt ≈ 225 K, and the prewetting transition starts to disappear by splitting into two layering transitions. With a weakening of a water-wall interaction, the first-order wetting transition becomes of a second order, and the prewetting transition disappears completely within the interval -3 < U0 < -2 kcal/mol. Such evolution agrees with the theoretical expectations for the evolution of the temperature and order of the wetting transition with the weakening of the surface field.3 In our simulations, the thickness of a wetting layer above the wetting temperature does not exceed two molecular layers, that is, it remains microscopic. There are two possibilities to explain this behavior. First, this may reflect suppression of a wetting layer by confinement, which causes shift of the chemical potential relative to the bulk liquid-vapor coexistence.8 If so, a macroscopic wetting layer may be expected in a semi-infinite system, and a thicker wetting layer should be detected in simulations of water in very large pores, where the liquidvapor coexistence is closer to the bulk one. Alternatively, the wetting transition at Tt may be a transition between two microscopic films of different thickness even in a semi-infinite system. Such a scenario assumes that this wetting transition is caused by the short-range fluid-wall interaction, and it may be followed by continuous growth of the wetting film when approaching the temperature of the second wetting transition, that is, the temperature of the critical wetting transition governed by the long-range fluid-wall forces.77,78 Taking into account that the wetting transition of water with the formation of a thick liquid film was never observed experimentally, the second scenario seems to be more realistic. Acknowledgment. We acknowledge financial support from DFG (SPP 1155) and from BMBF (Grant 01SF0303). References and Notes (1) Rupley, J. A.; Careri, G. AdV. Protein Chem. 1991, 41, 37. (2) Oleinikova, A.; Brovchenko, I. Mol. Phys. 2006, 104, 3841. (3) Nakanishi, H.; Fisher, M. E. Phys. ReV. Lett. 1982, 49, 1565. (4) Binder, K.; Landau, D. P. Phys. ReV. B 1988, 37, 1745. (5) van Swol, F.; Henderson, J. R. Phys. ReV. A 1989, 40, 2567. (6) Cahn, J. W. J. Chem. Phys. 1977, 66, 3667. (7) Ebner, C.; Saam, W. F. Phys. ReV. Lett. 1977, 38, 1486. (8) Dietrich, S. Phase Transitions and Critical Phenomena; 1988; Vol. 12, pp 1-218. (9) Morishige, K.; Kittaka, S.; Morimoto, T. Surf. Sci. 1981, 109, 291. (10) Morishige, K.; Kittaka, S.; Morimoto, T. Surf. Sci. 1982, 120, 223. (11) Brovchenko, I.; Oleinikova, A. In Handbook of Theoretical and Computational Nanotechnology; Rieth, M., Schommers, W., Eds.; American Scientific Publishers: Stevenson Ranch, CA, 2006; Vol. 9, Chapter 3, pp 109-206. (12) Millot, F.; Larher, Y.; Tessier, C. J. Chem. Phys. 1982, 76, 3327. (13) Panagiotopoulos, A. Z. Int. J. Thermophys. 1994, 15, 1057.

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