Journal of Colloid and Interface Science 449 (2015) 226–235
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Journal of Colloid and Interface Science www.elsevier.com/locate/jcis
Temperature dependence of the evaporation lengthscale for water confined between two hydrophobic plates Yuri S. Djikaev ⇑, Eli Ruckenstein ⇑ Department of Chemical and Biological Engineering, SUNY at Buffalo, Buffalo, NY 14260, United States
g r a p h i c a l a b s t r a c t Confined Water : Liquid or Vapor ? SHL
hydrophobic plate 2
2
hydrophobic plate 1
WATER SHL 1
a r t i c l e
i n f o
Article history: Received 27 October 2014 Accepted 20 January 2015 Available online 7 February 2015 Keywords: Water hydrogen bonding Confined liquid water Hydrophobic plates Density functional theory Density profiles Evaporation lengthscale Temperature effect Hydrophobic hydration Hydrophobic interactions
a b s t r a c t Liquid water in a hydrophobic confinement is the object of high interest in physicochemical sciences. Confined between two macroscopic hydrophobic surfaces, liquid water transforms into vapor if the distance between surfaces is smaller than a critical separation, referred to as the evaporation lengthscale. To investigate the temperature dependence of the evaporation lengthscale of water confined between two hydrophobic parallel plates, we use the combination of the density functional theory (DFT) with the probabilistic hydrogen bond (PHB) model for water–water hydrogen bonding. The PHB model provides an analytic expression for the average number of hydrogen bonds per water molecule as a function of its distance to a hydrophobic surface and its curvature. Knowing this expression, one can implement the effect of hydrogen bonding between water molecules on their interaction with the hydrophobe into DFT, which is then employed to determine the distribution of water molecules between two macroscopic hydrophobic plates at various interplate distances and various temperatures. For water confined between hydrophobic plates, our results suggest the evaporation lengthscale to be of the order of several nanometers and a linearly increasing function of temperature from T = 293 K to T = 333 K, qualitatively consistent with previous results. Ó 2015 Elsevier Inc. All rights reserved.
1. Introduction Liquid water in a hydrophobic confinement has long been the object of high scientific interest in physics, chemistry, and biology ⇑ Corresponding authors. E-mail addresses: [email protected] (Y.S. Djikaev), [email protected] (E. Ruckenstein). http://dx.doi.org/10.1016/j.jcis.2015.01.052 0021-9797/Ó 2015 Elsevier Inc. All rights reserved.
[1–8]. Such systems (either nano- or micro-scale) play an important role in many natural phenomena and technological processes, such as those in zeolites and clays [9], carbon nanotubes and nanofluidics [10–12], ion channels [13], living cells interior [14], water transport in plants [15], etc. Confined fluids (water being most interesting among them) exhibit quite unusual properties that are not observed in the bulk [16–24]. On the other hand, the peculiar properties of water near hydrophobic surfaces give rise
Y.S. Djikaev, E. Ruckenstein / Journal of Colloid and Interface Science 449 (2015) 226–235
to such amazing phenomena as hydrophobic hydration and hydrophobic interaction [25–32]. These are believed to constitute an important (if not crucial) element of a wide variety of physical, chemical, and biological phenomena [33–39], such as self-assembly of amphiphiles into micelles and membranes [40], gating of ion channels [41], etc. Hydrophobic interactions are also important in many technological and industrially significant processes (such as wetting, froth flotation, adhesion, etc.) [42]. Hydrophobic hydration is the thermodynamically unfavorable dissolution of a hydrophobic particle (microscopic or macroscopic), whereof the accommodation in water is accompanied by an increase in the associated free energy due to structural (and possibly energetic) changes in water around the hydrophobe. Since the total volume of water affected by two hydrophobes is smaller when they are close together than when they far away from each other, there appears an effective, solvent-mediated attraction between them which is referred to as hydrophobic attraction. Most properties of hydrophobic interactions may be unambiguously determined from the analogous properties of hydrophobic hydration; the former can be regarded as a partial reversal of the latter. Understanding these phenomena remains limited and the development of predictive models that are capable of estimating their dependence on temperature, pressure, and other thermodynamic variables of the system is important. Further progress in this direction requires more experimental and theoretical studies clarifying how such forces depend on external parameters, hydrophobe size and shape, and their surface properties. The simplest model of solvent-mediated interaction between large hydrophobes can be represented by liquid water confined between two infinitely parallel hydrophobic plates. This model has been the object of numerous theoretical and simulational studies [1–8]. It was observed that, even under conditions where the bulk liquid is stable, if the separation between two plates is sufficiently small, the liquid confined between the plates evaporates therefrom so that the system transforms into ‘‘vapor confined between two hydrophobic plates’’. The crossover separation, below which the liquid between two hydrophobic plates becomes metastable (despite being stable in the bulk) and transforms into vapor, is referred to as the evaporation length scale [6]. A similar, although not equivalent phenomenon has been observed in nanofluidics. For example, studying the filling of carbon nanotubes (CNTs, of diameter of about 1 nm) with water, it was found that the nature of water inside a CNT changes dramatically with the CNT diameter; water is stabilized in a vapor-like phase in smaller CNTs (of diameter 0.8–1.0 nm), in an ice-like phase for medium-sized CNTs (of diameter 1.1–1.2 nm), and in a liquid-like liquid phase for larger CNTs (of diameter greater than 1.4 nm) [7]. Although the evaporation lengthscale is an important quantity for the wide variety of aforementioned phenomena and processes involving hydrophobic and confinement effects, its dependence on various thermodynamic parameters has been hardly studied even for simplest systems. Only in Ref. [6] the comparison of water evaporation lengthscale with that of other common liquids was carried out in the framework of the classical thermodynamics by using actual thermophysical property data in a wide range of temperatures. In this paper, we attempt to shed more light on the temperature dependence of the evaporation lengthscale of water confined between two hydrophobic parallel plates. This can be achieved by examining the distribution of water molecules between the plates at different interplate distances in a range of temperatures of interest. Along with computer simulations, the formalism of the density functional theory (DFT) [43–46] is widely used for studying the fluid density profiles near rigid surfaces [47,48]. It usually treats the
227
interaction of fluid molecules with a foreign surface in the meanfield approximation whereby every fluid molecule is considered to be subjected to an external potential, due to its pairwise interactions with the molecules of an impenetrable substrate. This external potential gives rise to a specific contribution to the free energy functional. The minimization of the latter with respect to the number density of fluid molecules (as a function of the spatial coordinate r) provides their equilibrium spatial distribution. The application of the DFT formalism to water-like (non-spherical) molecules, whereof the pairwise intermolecular potential depends not only on the distance between molecules but also on their mutual orientations, is practically extremely difficult due to the multifold configurational integrals involved. Furthermore, the effect of a hydrophobic surface on the ability of fluid (water) molecules to form hydrogen bonds had been also ignored in the conventional DFT. These problems can be now partly alleviated by combining the DFT formalism with the recently developed probabilistic hydrogen bond (PHB) model [49–55] for hydrogen bonding between water molecules in the vicinity of a foreign surface. This model provides an analytic expression for the average number of hydrogen bonds that a water molecule can form as a function of its distance to the surface. Knowing this expression, one can implement the effect of the hydrogen bonding of water molecules on their interaction with the hydrophobic surface into DFT, which is then employed to determine the distribution of water molecules between two macroscopic hydrophobic infinitely large plates at various interplate distances and various temperatures. 2. The combination of the DFT formalism with the PHB model Before presenting our results for the temperature dependence of the evaporation lengthscale of water confined between two hydrophobic plates, let us outline the combination of the DFT formalism [43–48] and PHB model [49–55] which allows one to find the water density profiles between two plates while explicitly taking into account the effect of water–water hydrogen bonding on the hydrophobe-fluid interactions. We will first briefly describe the PHB model then present its implementation into the DFT formalism. 3. The outline of a probabilistic approach to water–water hydrogen bonding near a hydrophobic surface The PHB model [49,50] considers a water molecule, whereof the location is determined by its center, to have four hydrogen-bonding (hb) arms (each capable of forming a single hydrogen bond) of rigid and symmetric (tetrahedral) configuration with the interarm angles a ¼ 109:47 . Each hb-arm can adopt a continuum of orientations. For a water molecule to form a hydrogen bond with another molecule, it is necessary that the tip of any of its hb-arms coincide with the second molecule. The length of an hb-arm thus equals the length g of a hydrogen bond. The hydrogen bond length g is assumed to be independent of whether the molecules are in the bulk or near a hydrophobic sure of pairwise interactions between face. The characteristic length g water and molecules constituting the substrate (flat and large enough to neglect edge effects, with its location determined by the loci of the centers of its outermost, surface molecules) plays a simple role in the hydrogen bond contribution to hydration or e ) only determines the referhydrophobic interaction [50–52]. It ( g ence point for measuring the distance between water molecule and substrate, so it will be set equal to g. Consider two infinitely large hydrophobic plates, 1 and 2, immersed in liquid water (Fig. 1). Even if the intrinsic hydrogen
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the hb-arms of a vicinal water molecule cannot lead to the formation of hydrogen bonds. This constraint may depend on the variables r 1 ; r2 , whence the corresponding dependence of k1 ; k2 ; k3 , and k4 . The coefficients k1 ; k2 ; k3 , and k4 as functions of r 1 ; r 2 can be evaluated by using geometric considerations [55]. They all become equal to 1 if both r1 P 2g and r 2 P 2g, when Eq. (1) reduces to its 2
Fig. 1. A water molecule (shown as a disk S) in the vicinity of two hydrophobic plates 1 and 2, at a distance r 1 from plate 1 and a distance r 2 from hydrophobe 2; the distance between plates is d. A layer of thickness g from x ¼ g to x ¼ 2g; near plate 1 is referred to as the solute hydration layer of that plate (SHL1). The SHL2 is defined in a similar way.
bonding ability of a water molecule is not affected by a hydrophobe, in its vicinity a ‘‘vicinal’’ water molecule forms a smaller number of bonds than in bulk because the hydrophobic surface restricts the configurational space available to other water molecules necessary for a vicinal water molecule to form hydrogen bonds (‘‘missing neighbor effect’’). The actual number of hydrogen bonds, that a particular BWM can form, depends on both its location and its orientation. The probabilistic model allows one to obtain an analytic expression for the average (‘‘average’’ with respect to all possible orientations of the water molecule) number of bonds ns that a vicinal water molecule (in the vicinity of a hydrophobe) can form as a function of its distance to the hydrophobe [50–52] and hydrophobe radius [53,54] (in case of a spherical hydrophobe). When the water molecule is close enough to two plates, ns will depend on the distance between water molecule and each hydrophobe [55]. The location of each plate will be determined by the location of the plane formed by centers of its outermost (facing the interplate space) molecules. The distance between the plates will be denoted by d, whereas the distance from the center of a selected water molecule S to the plate i (i ¼ 1; 2) will be denoted by r i . Clearly, only two out three variables d; r1 , and r 2 are independent; we will choose them to be r1 and r2 . Thus, ns ¼ ns ðr 1 ; r 2 Þ. Using the PHB approach, let us represent the function ns ¼ ns ðr 1 ; r 2 Þ as 2
3
4
ns ¼ k1 b1 þ k2 b1 þ k3 b1 þ k4 b1 ;
ð1Þ
where b1 is the probability that one of the hb-arms (of a bulk water molecule) can form a hydrogen bond and the coefficients k1 ; k2 ; k3 , and k4 depend on the variables r 1 ; r 2 , and so does ns . Eq. (1) assumes that the intrinsic hydrogen-bonding ability of a water molecule (i.e., the tetrahedral configuration of its hb-arms and their lengths and energies) is unaffected by its proximity to the hydrophobes so that the latter only restrict the configurational space available to other water molecules necessary for the selected water molecule to form hydrogen bonds. Thus, near the hydrophobes some orientations of
3
4
bulk analog, nb ¼ b1 þ b1 þ b1 þ b1 , with nb being the number of hydrogen bonds per bulk water molecule. Since experimental data on nb (and even its temperature dependence) are readily available, one can find b1 as a positive solution (satisfying the condition 0 < b1 < 1) of the latter equation. Thus, Eq. (1) provides an efficient pathway to ns . Consider a layer of thickness g from r i ¼ g (minimal distance between water molecule and plate i) to r i ¼ 2g near the plate i (i ¼ 1; 2) and denote it by SHLi (standing for ‘‘Solute Hydration Layer i’’). The quantities ns ; k1 ; k2 ; k3 ; k4 are functions of both variables r1 and r 2 only for water molecules located in the overlap region of SHL1 and SHL2. If a water molecule belongs only to SHL1 but not to SHL2, then ns ; k1 ; k2 ; k3 ; k4 are functions of only one variable, r1 (see Refs. [13,14]); likewise, if a water molecule belongs only to SHL2 but not to SHL1, then the quantities ns ; k1 ; k2 ; k3 ; k4 are functions of only one variable, r 2 . Denote the energy of a bulk (water–water) hydrogen bond by eb < 0 and the energy of a vicinal hydrogen bond (involving at least one molecule from SHL1 or SHL2) by es < 0. The vicinal hydrogen bond may be slightly altered energetically compared to the bulk one, but the direction (either enhancement or weakening) of such alteration is still an object of contention and hence will be neglected, as previously [51–55]. In the PHB approach, there is no restriction on the energies of bulk or vicinal (water–water) hydrogen bonds, so that the approach is valid independent of whether b < s or b ¼ s , or b > s . 3.1. Hydrogen bond contribution to the interaction of a water molecule with a hydrophobic plate The deviation of ns from nb and the deviation of es from eb give rise to a hydrogen bond contribution to the external potential field whereto a water molecule is subjected in the vicinity of two hydrophobes. This contribution, U hb ext , can be determined as hb U hb ext U ext ðr 1 ; r 2 Þ ¼
1 ðes ns eb nb Þ 2
ð2Þ
The first term on the RHS of this equation represents the total energy of hydrogen bonds of the water molecule near the hydrophobes, whereas the second term is the energy of its hydrogen bonds in bulk (at both r 1 ! 1 and r 2 ! 1); the factor 1=2 is needed to prevent double counting the energy because every hydrogen bond and its energy, either es or eb , are shared between two molecules. The dependence of U hb ext on r 1 and r 2 may be due not only to the function ns , but also to the r-dependence of es . In the PHB model ns ¼ nb when both r1 P 2g and r 2 P 2g, hence it is reasonable to assume that
es ¼ eb as well when both r1 P 2g and r2 P 2g. Thus, U hb ext is a very hb short-ranged function of r’s, such that U ext ¼ 0 if both r1 P 2g and r2 P 2g. As the energetic alteration of a vicinal hydrogen bond (compared to the bulk one) is still uncertain, it will be neglected in numerical calculations and es will be assumed to be equal to eb . 3.2. Formalism of density functional theory The fluid density distribution near a rigid surface can be efficiently studied by using computer simulations or DFT [46–48]. Although the application of DFT methods to water-like fluids is
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freight with significant difficulties (mentioned above), they can be circumvented by using the PHB model for water molecules [51– 55]. The DFT formalism usually treats the interaction of fluid molecules with a foreign surface in the mean-field approximation whereby every fluid molecule is considered to be subjected to an external potential, due to its pairwise interactions with the molecules of an impenetrable substrate [46–48]. This external potential gives rise to a specific contribution to the free energy functional. The minimization of the latter with respect to the number density of fluid molecules (as a function of the spatial coordinate r) provides their equilibrium spatial distribution. The effect of a hydrophobic surface on the ability of fluid (water) molecules to form hydrogen bonds had been until recently ignored in the conventional DFT.
F h ½qðrÞ ¼
Z
dr f i ðqðrÞÞ þ
Z
e ðrÞÞ; dr qðrÞ Dwh ð q
229
ð7Þ
where
f i ðqÞ ¼ kB T q ðlnðK3 qÞ 1Þ 1=2
2
(with K ¼ ðh =2pmkB TÞ being the thermal de Broglie wavelength of a model molecule of mass m, and h and kB being Planck’s and Boltzmann’s constants, respectively) is the free energy density of an ideal gas of density q and
Dwh ðqÞ ¼
1
q
½f h ðqÞ f i ðqÞ
Now, however, taking into account U hb ext , one can represent the overall external potential U ext U ext ðr 1 ; r 2 Þ, whereto a water molecule is subjected between two hydrophobes, as [51–55]
is the configurational part of the free energy of hard sphere fluid per molecule. e ðrÞ is determined in terms of qðrÞ via an The weighted density q implicit equation
hb U ext ¼ U pw ext þ U ext ;
e ðrÞ ¼ q
U pw ext
ð3Þ
U pw ext ðr 1 ; r 2 Þ
where represents pairwise interactions of a fluid molecule with the molecules of the substrate as well as the effect of the latter on the pairwise interactions between fluid molecules themselves. The grand thermodynamic potential X of a nonuniform single component fluid of volume V at a temperature T and (externally imposed) chemical potential l (an open thermodynamic system), subjected to an external potential field U ext , can be represented as a functional of the number density qðrÞ of fluid molecules
X½qðrÞ ¼ F ½qðrÞ þ
Z
dr U ext ðrÞqðrÞ l
Z
dr qðrÞ;
ð4Þ
where F ½qðrÞ, the intrinsic Helmholtz free energy of the system as a functional of qðrÞ, includes the contributions from fluid–fluid interactions as well as the ideal gas term. Conventionally, F ½qðrÞ is divided into two parts,
F ½qðrÞ ¼ F ri ½qðrÞ þ F a ½qðrÞ;
ð5Þ
where F a ½qðrÞ arises from attractive forces between fluid molecules, whereas F ri ½qðrÞ represents the repulsive forces (and includes the ideal gas contribution as well). In the van der Waals approximation the attractive forces are treated in a mean-field fashion, while the repulsive interactions are often modeled as those of hard spheres, so that Eq. (5) is approximated by
F ½qðrÞ ¼ F h ½qðrÞ þ
1 2
ZZ
drdr0 qðrÞqðr0 Þ/a ðjr r0 jÞ;
Z
e ðrÞÞ; dr0 qðr0 Þwðjr0 rj; q
e ðrÞÞ is the weight function. Although in more where wðjr0 rj; q e ðrÞÞ depends on sophisticated versions [44,45] of WDA wðjr0 rj; q e ðrÞ, we will hereafter adopt its simpler version wherein the weight q e ðrÞ. function is independent [45,46] of q In a grand canonical ensemble (corresponding to an open thermodynamic system under constant l, V, and T) the equilibrium density profile is obtained by minimizing the functional X½qðrÞ with respect to qðrÞ, i.e., by solving the Euler–Lagrange equation dX=dq ¼ 0 with respect to qðrÞ. The Euler–Lagrange equation can be written as
l ¼ kB T lnðK3 qðrÞÞ þ Wðr; qðrÞÞ;
ð9Þ
where
Z e ðrÞÞ Wðr; qðrÞÞ ¼ UðrÞ þ dr0 qðr0 Þ/a ðjr r0 jÞ þ Dwh ð q Z e ðr0 ÞÞwðjr0 rjÞ þ dr0 qðr0 Þ Dw0h ð q and Dw0h ðqÞ dDwh ðqÞ=dq. If the constraint of constant l (external chemical potential) is replaced by a given value of the fluid density far away from the impenetrable wall, q1 ¼ qðrÞjjrj¼1 , the Euler–Lagrange equation can be written in the form
qðrÞ ¼ q1 exp ð6Þ
where F h ½qðrÞ is the intrinsic Helmholtz free energy functional of a hard sphere fluid and /a ðjr r0 jÞ is the attractive part of the interaction potential between two fluid molecules located at r and r0 . As the free energy functional of a three-dimensional hardsphere fluid is not known exactly, further approximations are needed. The simplest ansatz for F h ½qðrÞ that incorporates shortranged correlations recurs [44–46] to a ‘‘smoothed’’ or ‘‘weighted’’ e ðrÞ which is a non-local functional of the actual density density q e ðrÞ at point r can be regarded qðrÞ. Namely, the weighted density q as a mean density obtained by averaging the actual density qðrÞ with an appropriate ‘‘weight’’ function over an appropriate volume in the vicinity of r. This non-local approximation for F h ½qðrÞ is often referred to as a weighted density approximation (WDA) [44–46]. A key requirement of WDA is that the weight function e ðrÞ be sufficiently and averaging volume should ensure that q smooth so that F h ½qðrÞ can be calculated in a form formally reminiscent of local density approximation. In the framework of WDA, the intrinsic Helmholtz free energy functional of hard sphere fluid is represented in the form
ð8Þ
1 ðWðr; qðrÞÞ Wðr; q1 ÞÞ : kB T
ð10Þ
In the canonical ensemble (corresponding to a closed thermodynamic system with constant number of molecules N, volume V, and temperature T), the chemical potential l, that appears in Eqs. (4) and (9), is not known in advance. Instead, it plays the role of a Lagrange multiplier corresponding to the constraint of fixed number of molecules in the system:
N¼
Z
dr qðrÞ:
V
This equation can be used to determine the Lagrange multiplier l, i.e., the chemical potential in the system (see, e.g., Refs. [45,52]). The Euler–Lagrange equation for the density profile in the canonical ensemble can be written as
qðrÞ ¼ N R
exp½Wðr; qðrÞÞ=kB T : dr exp½Wðr; qðrÞÞ=kB T
ð11Þ
This expression has the familiar form of a ‘‘generic’’ one-particle distribution function in classical statistical mechanics [56]. In the particular case of (fluid) water between two hydrophobic plates, one can use the planar symmetry of the system and choose
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the Cartesian coordinates so that the plates are parallel to the y–z plane, with the plate 1 at x ¼ 0, plate 2 at x ¼ d, and the molecules of the fluid occupying the ‘‘interplate space’’ 0 < x < d. The equilibrium density profile obtained from Eq. (9) or Eq. (10) or Eq. (11) is then a function of a single variable x, i.e., qðrÞ ¼ qðxÞ. 4. Numerical evaluations For a numerical illustration, we considered the model fluid water confined between two infinitely large hydrophobic plates, at five temperatures: T ¼ 293 K, 303 K, 313 K, 323 K, and 333 K. The calculations were carried out at the vapor–liquid equilibrium conditions. At a given temperature, the densities qv and ql of coexisting vapor and liquid, respectively, are determined by solving the equations lðq; TÞjq¼qv ¼ lðq; TÞjq¼ql ; pðq; TÞjq¼qv ¼ pðq; TÞjq¼ql , requiring the chemical potential l lðq; TÞ and pressure p pðq; TÞ to be the same throughout both coexisting phases. The number of hydrogen bonds per bulk water molecule as a function of temperature was approximated using a linear leastmean squares fit of the data provided in [57] (nb ¼ 3:69 at 0 °C, nb ¼ 3:59 at 25 °C, and nb ¼ 3:24 at 100 °C), resulting in nb nb ðTÞ ¼ anT bnT ðT 273:15Þ, with anT ¼ 3:69577 and bnT ¼ 0:00453846. The temperature dependence of ns is uniquely determined by the function nb ðTÞ, because b1 in Eq. (1) is unambiguously determined by the thermodynamic state of the bulk water (temperature, pressure, etc.) as the solution of the equation 2
3
4
nb ¼ b1 þ b1 þ b1 þ b1 satisfying the constraint 0 < b1 < 1 (for details see Refs. [49,50]). The dependence of b on T was approximated by the linear function
b b ðTÞ ¼ a b ðT 273:15Þ; with the constants a ¼ 3:82 1013 erg and b ¼ 4:98 1016 erg/ K determined from the data provided in Refs. [58,59], respectively. The ratio es =eb was assumed to be temperature-independent. The pairwise interactions between two water molecules were modeled with the Lennard-Jones (LJ) potential. The energy parameter ww of the LJ potential was adjusted to be 5:4 1014 , which differs from its values used in the computer simulations (Monte Carlo or molecular dynamics) of various water models, in which
ww
ranges 14
[57f]
from
5:31 1015
erg
(ST2
model)
to
1:47 10 erg (SWM4-NDPmodel) to 2:54 1013 erg (SSD model). Such a modification was needed to ensure that the phase diagram of model water more or less resembles that of real water. In the DFT formalism this can be achieved only by adjusting the single intermolecular potential describing water–water interactions, whereas in computer simulations water–water interactions are usually described by the combination of LJ and electrostatic potentials; hence the difference in the energy parameters of the respective LJ potentials. The length parameter r of the LJ potential also has different values in different water models 3b in the range from 3.02 Å (SSD model) to 3.18 Å (SWM4-NDP model). On the other hand, the length g of the hydrogen bond (i.e., the distance between the oxygen atoms of two hydrogen-bonded water molecules) is reported [57f] to be about 2.98 Å. Since r’s (of various water models) and g are so close to each other, we assumed for our model r ’ g ’ 3:18 Å. The attractive part /at of pairwise water–water interactions was modeled via the Weeks–Chandler–Anderson perturbation scheme [60]. The interaction potential between water molecule and molecule of a hydrophobe was assumed to be of LJ type with an energy parameter ewp and a length parameter g. Integrating this interaction with respect to the position of the molecule of a plate over the semi-infinite plate volume for both plates 1 and 2, one can obtain the pairwise contribution U pext into U ext . We assumed the
dimensionless number density of molecules in the hydrophobic plates to be qp g3 1 and considered ewp =eww ¼ 0:75 to mimic their hydrophobicity. The chemical potential of a uniform hard sphere fluid lh and the configurational part Dwh Dwh ðq; TÞ of the free energy of a hard sphere fluid were modeled in the Carnahan–Starling approximation [44,61], whereas for the weight function e ðrÞÞ in Eq. (8) we adopted a q e -independent version wðjr0 rj; q [44,46]
Dwh ¼ kB T
n ð4 3nÞ ð1 nÞ2
;
wðr12 Þ ¼
3
pg4
ðg r12 ÞHðg r 12 Þ;
with HðuÞ being the Heaviside (unit-step) function. In order to find the equilibrium density profile of (fluid) water confined between two hydrophobic plates, we have solved Eq. th
(9) by iterations. Namely, the density profile qi ðxÞat the i iteration is found from the density profile qi1 ðxÞ at the previous, ði 1Þth iteration:
kB T lnðK3 qi ðrÞÞ ¼ l Wðr; qi1 ðrÞÞ
ð12Þ
At any temperature T and interplate distance d, the initial profile was taken to be of the shape shown in Fig. 2, which corresponds to some nonequilibrium liquid ‘‘forcefully’’ confined between the plates. Since the calculations are carried out in an open system (grand canonical ensemble) under conditions of phase coexistence, such an initial profile is necessary to ensure that, if the liquid between the plates can be stable, the corresponding solution of Eq. (12) will not be missed during the iteration procedure. As the iterations proceed, the initial profile gradually transforms into the equilibrium one. The Euler–Lagrange equation (Eq. (9)) for a grand canonical ensemble under conditions of phase coexistence may have two equilibrium solutions: one for the vapor phase, the other for the liquid phase. At large interplate distances, the first solution attained during the iterations is that of the stable liquid water confined between the plates. However, at small interplate distances, the first solution attained in the course of iterations corresponds to the metastable (or even unstable) liquid between the plates; such a solution gradually decays, ceding place to the second solution which corresponds to a vapor-like phase confined between the plates. The results of numerical calculations are shown in Figs. 3–7. Fig. 3 presents the density profiles of water molecules between two hydrophobic plates at temperature T ¼ 293 K for various interplate distances d (hereinafter, all densities and distances/ lengths are shown in dimensionless units, qg3 ; x=g, and d=g,
0.8
0.6
d
0.4
7.0
0.2
0.0 0
1
2
3
4
5
6
7
Fig. 2. The typical initial density profile for the approximation procedure in Eq. (12), shown as the dimensionless density qg3 vs dimensionless distance x=g to plate 1 for the case where the interplate distance is d=g ¼ 7:0.
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(a)
(b) 0.8
0.0025 0.6
0.0020 0.0015
d
0.4
d
0.0010
10
5 0.2
0.0005 0.0000
0.0 0
1
2
3
4
5
0
2
4
6
8
10
0.8
0.0025 0.0020
0.6
0.0015
d
0.4
d
0.0010
15
8 0.2
0.0005 0.0
0.0000 0
2
4
6
0
8
0.0025
2
4
6
8
10
12
14
0.8
0.0020 0.6 0.0015
d
0.4
d
0.0010
20
9 0.2
0.0005 0.0000
0.0 0
2
4
6
8
0
5
10
15
20
Fig. 3. The density profiles of water molecules between two hydrophobic plates at temperature T ¼ 293 K for various interplate distances d (shown as qg3 vs x=g, with x being a distance from plate 1). Every figure panel represents a profile for its own d as indicated therein. Fig. 3a corresponds to d < de , whereas Fig. 3b is for d > de .
respectively); note the different scales for the ordinate coordinates in Fig. 3a and b. Every figure panel represents a profile for its own d as indicated therein. Fig. 3a corresponds to d < de , where de is the characteristic lengthscale of water confined between two infinitely large hydrophobic plates (the quantitative definition of de is discussed below). As clear, at d < de , water confined between the plates is in the vapor-like phase. On the other hand, the profiles in Fig. 3b correspond to d > de ; they represent a liquid-like phase confined between the plates. Although Fig. 3 shows the typical behavior of density profiles with changing distance between the plates for any temperature, the actual value of the characteristic ‘‘evaporation’’ lengthscale de (however defined) will depend on the temperature of the system. In Fig. 4a–e, the density profiles of water confined between two plates at the interplate distance d ¼ 9g are plotted for various temperatures (indicated in the figure panels). The dashed lines indicate the average density, whereas the filled circle emphasizes the data
point corresponding to the middle between the plates. As clear, at this interplate distance the space between the plates is filled with water in a vapor-like phase, for all temperatures. This suggests that de > 9g in the considered temperature range. As expect v of water molecules ed for the vapor phase, the average density q increases with increasing temperature; so does the mid-point den^v . sity q In Fig. 5a-e, the density profiles of water confined between two plates at the interplate distance d ¼ 11g are plotted for various temperatures (indicated in the figure panels). Again, the dashed lines indicate the average density, whereas the filled circle emphasizes the data point corresponding to the middle between the plates. At this interplate distance water confined between the plates is in a liquid-like phase, for all temperatures. This suggests that de < 11g for all the temperatures considered. As expected l decreases with increasfor the liquid phase, the average density q ^v . ing temperature, and so does the mid-point density q
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0.0025 0.004 0.0020 0.003 0.0015 0.002
0.0010
a T 293 K 0.001
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c T 313 K
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4
0.007 0.006 0.005 0.004 0.003 0.002 0.001
e T 333 K
0.000 0
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Fig. 4. The density profiles of water confined between two plates at the interplate distance d ¼ 9g for various temperatures (indicated in the figure panels).
Figs. 4 and 5 indicate that, in the temperature range from 293 K to 333 K. the evaporation lengthscale de (however defined) is in the range 9 < de =g < 11. Provided that there is a more or less rigorous definition of de , it is thus necessary to determine the density profiles qðzÞ on an adjustable, progressively finer grid in the interval of interplate distances initially from d ¼ 9g to d ¼ 11g. The data, presented in Figs. 3–5, suggest two alternative methods to define and determine the evaporation lengthscale de of liquid (water) confined between two hydrophobic plates. One , method is based on the d-dependence of the average density q ^. the other on the d-dependence of the mid-point density q vs d and q ^ vs d are plotted (as solid In Fig. 6, the functions q squares and disks, respectively) for the temperature T ¼ 293 K. Fig. 6a shows the data for interplate distances d < de , so that the fluid confined between the plates is in a vapor like phase, whereas Fig. 6b presents data for interplated distances d > de with the fluid and between the plates in a liquid-like phase. Similar behavior of q q^ as functions of d will be also observed for other temperatures. (solid squares) At a given temperature, the average density q monotonically increases with increasing d, but exhibits a sharp increase from values, characteristic for a vapor phase, to values,
characteristic of a liquid phase. This sharp increase occurs in a very narrow interval of d’s whereof the middle can be defined as the evaporation lengthscale de . This definition lacks mathematical rigor, but can be useful in characterizing the phenomenon of interest. On the other hand, at a given temperature, as d increases, the ^ (disks) first slowly decreases (its values being mid-point density q characteristic of a vapor phase), but after attaining a minimum, it sharply increases to values, characteristic of a liquid phase then continues its slow monotonic increase. This sharp increase occurs in a very narrow interval of d’s. Again the middle can be defined as the evaporation lengthscale de . However, unlike the function ¼q ðdÞ, the function q ^ ¼q ^ ðdÞ has a unique point, namely, the q location of the minimum, which can be defined to be the evaporation lengthscale de of liquid from between the hydrophobic plates. Although the latter definition is mathematically more rigorous and hence formally preferable, its practical use is rather doubtful, because it involves the DFT calculations of the density profiles between the plates on a rather fine grid in some interval of interplate distances d (the interval which itself has to be identified via the DFT calculations of the profiles as well). Therefore, from the ¼q ðdÞ and q ^ ¼q ^ ðdÞ practical point of view, the both functions q
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0.8 0.8 0.6
0.6
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0.4
c T 313 K
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a T 293 K
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0.0
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0.7 0.6 0.5 0.4 0.3
e T 333 K
0.2 0.1 0.0 0
2
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6
8
10
Fig. 5. The density profiles of water confined between two plates at the interplate distance d ¼ 11g for various temperatures (indicated in the figure panels).
can be used to loosely identify the evaporation lengthscale as the middle of the interval of d wherein the functions sharply rise from their vapor-like-density values to the liquid-like-density values. Using such a definition, we have determined the evaporation lengthscale de of model water confined between two hydrophobic plates for five different temperatures (293 K, 303 K, 313 K, 323 K, and 333 K). The results are presented in Fig. 7. For the model water and model hydrophobes that we studied, the combined DFT/PHB model predicts de to be of the order of several nanometers (de 3 nm). In the temperature range considered de is a relatively weak function of T. As the temperature increases by about 14 % from 293 K, the evaporation length increases by about 7 %. Note that the error bars of every data point in Fig. 7 are about 0.1% (i.e., de was determined with the relatively accuracy of 0.1%). The predictions of our model for de are in a partial agreement with the previously reported results. In Refs. [1,26] the evaporation lengthscale was estimated to be of the order of 10 nm. On the other
hand, in Ref. [6] the evaporation lengthscale and its temperature dependence for water and several other organic liquids were studied by using the formalism of classical thermodynamics with the actual vapor–liquid surface tension data (at atmospheric conditions) for real substances. The magnitude of de for water was found to be significantly larger, of the order of 1 lm, more in line with the number quoted in Ref. [31]. Furthermore, as the water temperature increases from about 273 K to about 298 K, de was also reported to mildly decrease [6]. As the water temperature increased from about 298 K up to its boiling point 373 K (at atmospheric pressure 1 bar), the evaporation lengthscale monotonically, but progressively steeper increased, naturally diverging at the boiling point (similar behavior was observed also for the organic liquids as well) [6]. Such a T-dependence of de for a real water is qualitatively consistent with the temperature dependence of de predicted by the DFT/PHB approach for the model water confined between model hydrophobic plates in the temperature range that we studied.
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(a) 0.0025 0.0020 T 293 K; d de vapor midpoint density
0.0015
average density
0.0010
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3
4
5
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7
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9
10
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T 293 K; d de liquid midpoint density
0.4 0.3
average density
10
12
14
16
18
20
22
and midpoint density q ^ of the fluid confined between two Fig. 6. Average density q plates as a function of the interplate distance d (solid squares and disks, respectively) for the temperature T ¼ 293 K.
10.6 10.4
We have found that there may be two alternative pathways to define and determine the evaporation lengthscale of water between hydrophobic plates, one based on the average density of water molecules between the plates the other based on the local density in the middle between the plates. From the practical point of view, both definitions are virtually equivalent to loosely identify the evaporation lengthscale as the middle of the interval of interplate distances wherein these quantities charply rise from their vapor-like-density values to the liquid-like-density values. Using such a definition, we have determined the evaporation lengthscale of model water confined between two model hydrophobic plates in the temperature range from 293 K to 333 K. Our combined DFT/PHB model predicts the evaporation lengthscale to be of the order of several nanometers and to be a relatively weakly increasing function of temperature. These predictions are in a partial qualitative agreement with expectations. Although there are large quantitative variations in the value of the evaporation lengthscale reported in different sources, it should be emphasized, that these may be due to the inadequacy of the comparison of results (either simulational or theoretical or experimental) obtained for different systems. It is not quite straightforward to compare the results for solvophobic and confinement phenomena when two different sets of data are of different origins. Extreme care must hence be taken if the two sets are for different solvents (or solvent models) or/and for different (in shape/size) solutes/hydrophobes or/and for different solute–solvent interactions. References [1] [2] [3] [4] [5] [6] [7] [8]
10.2 10.0
[9] [10]
9.8
[11] [12] [13]
9.6 9.4 290
300
310
320
330
340
[14] [15] [16]
TK [17] Fig. 7. The temperature dependence of the evaporation lengthscale de of model water confined between two hydrophobic plates in the range from T = 293 K to T = 333 K.
5. Concluding remarks Although the evaporation lengthscale is an important quantity for the wide variety of phenomena and processes involving liquids in solvophobic confinement, its dependence on various thermodynamic parameters has been hardly studied even for simplest systems. We have attempted to shed more light on the temperature dependence of the evaporation lengthscale of water confined between two hydrophobic parallel plates, by using the combination of the DFT formalism and PHB model. Such a combination allows one to find the water density profiles between two plates while explicitly taking into account the effect of water–water hydrogen bonding on the hydrophobe-fluid interactions.
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Journal of Colloid and Interface Science www.elsevier.com/locate/jcis
Temperature dependence of the evaporation lengthscale for water confined between two hydrophobic plates Yuri S. Djikaev ⇑, Eli Ruckenstein ⇑ Department of Chemical and Biological Engineering, SUNY at Buffalo, Buffalo, NY 14260, United States
g r a p h i c a l a b s t r a c t Confined Water : Liquid or Vapor ? SHL
hydrophobic plate 2
2
hydrophobic plate 1
WATER SHL 1
a r t i c l e
i n f o
Article history: Received 27 October 2014 Accepted 20 January 2015 Available online 7 February 2015 Keywords: Water hydrogen bonding Confined liquid water Hydrophobic plates Density functional theory Density profiles Evaporation lengthscale Temperature effect Hydrophobic hydration Hydrophobic interactions
a b s t r a c t Liquid water in a hydrophobic confinement is the object of high interest in physicochemical sciences. Confined between two macroscopic hydrophobic surfaces, liquid water transforms into vapor if the distance between surfaces is smaller than a critical separation, referred to as the evaporation lengthscale. To investigate the temperature dependence of the evaporation lengthscale of water confined between two hydrophobic parallel plates, we use the combination of the density functional theory (DFT) with the probabilistic hydrogen bond (PHB) model for water–water hydrogen bonding. The PHB model provides an analytic expression for the average number of hydrogen bonds per water molecule as a function of its distance to a hydrophobic surface and its curvature. Knowing this expression, one can implement the effect of hydrogen bonding between water molecules on their interaction with the hydrophobe into DFT, which is then employed to determine the distribution of water molecules between two macroscopic hydrophobic plates at various interplate distances and various temperatures. For water confined between hydrophobic plates, our results suggest the evaporation lengthscale to be of the order of several nanometers and a linearly increasing function of temperature from T = 293 K to T = 333 K, qualitatively consistent with previous results. Ó 2015 Elsevier Inc. All rights reserved.
1. Introduction Liquid water in a hydrophobic confinement has long been the object of high scientific interest in physics, chemistry, and biology ⇑ Corresponding authors. E-mail addresses: [email protected] (Y.S. Djikaev), [email protected] (E. Ruckenstein). http://dx.doi.org/10.1016/j.jcis.2015.01.052 0021-9797/Ó 2015 Elsevier Inc. All rights reserved.
[1–8]. Such systems (either nano- or micro-scale) play an important role in many natural phenomena and technological processes, such as those in zeolites and clays [9], carbon nanotubes and nanofluidics [10–12], ion channels [13], living cells interior [14], water transport in plants [15], etc. Confined fluids (water being most interesting among them) exhibit quite unusual properties that are not observed in the bulk [16–24]. On the other hand, the peculiar properties of water near hydrophobic surfaces give rise
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to such amazing phenomena as hydrophobic hydration and hydrophobic interaction [25–32]. These are believed to constitute an important (if not crucial) element of a wide variety of physical, chemical, and biological phenomena [33–39], such as self-assembly of amphiphiles into micelles and membranes [40], gating of ion channels [41], etc. Hydrophobic interactions are also important in many technological and industrially significant processes (such as wetting, froth flotation, adhesion, etc.) [42]. Hydrophobic hydration is the thermodynamically unfavorable dissolution of a hydrophobic particle (microscopic or macroscopic), whereof the accommodation in water is accompanied by an increase in the associated free energy due to structural (and possibly energetic) changes in water around the hydrophobe. Since the total volume of water affected by two hydrophobes is smaller when they are close together than when they far away from each other, there appears an effective, solvent-mediated attraction between them which is referred to as hydrophobic attraction. Most properties of hydrophobic interactions may be unambiguously determined from the analogous properties of hydrophobic hydration; the former can be regarded as a partial reversal of the latter. Understanding these phenomena remains limited and the development of predictive models that are capable of estimating their dependence on temperature, pressure, and other thermodynamic variables of the system is important. Further progress in this direction requires more experimental and theoretical studies clarifying how such forces depend on external parameters, hydrophobe size and shape, and their surface properties. The simplest model of solvent-mediated interaction between large hydrophobes can be represented by liquid water confined between two infinitely parallel hydrophobic plates. This model has been the object of numerous theoretical and simulational studies [1–8]. It was observed that, even under conditions where the bulk liquid is stable, if the separation between two plates is sufficiently small, the liquid confined between the plates evaporates therefrom so that the system transforms into ‘‘vapor confined between two hydrophobic plates’’. The crossover separation, below which the liquid between two hydrophobic plates becomes metastable (despite being stable in the bulk) and transforms into vapor, is referred to as the evaporation length scale [6]. A similar, although not equivalent phenomenon has been observed in nanofluidics. For example, studying the filling of carbon nanotubes (CNTs, of diameter of about 1 nm) with water, it was found that the nature of water inside a CNT changes dramatically with the CNT diameter; water is stabilized in a vapor-like phase in smaller CNTs (of diameter 0.8–1.0 nm), in an ice-like phase for medium-sized CNTs (of diameter 1.1–1.2 nm), and in a liquid-like liquid phase for larger CNTs (of diameter greater than 1.4 nm) [7]. Although the evaporation lengthscale is an important quantity for the wide variety of aforementioned phenomena and processes involving hydrophobic and confinement effects, its dependence on various thermodynamic parameters has been hardly studied even for simplest systems. Only in Ref. [6] the comparison of water evaporation lengthscale with that of other common liquids was carried out in the framework of the classical thermodynamics by using actual thermophysical property data in a wide range of temperatures. In this paper, we attempt to shed more light on the temperature dependence of the evaporation lengthscale of water confined between two hydrophobic parallel plates. This can be achieved by examining the distribution of water molecules between the plates at different interplate distances in a range of temperatures of interest. Along with computer simulations, the formalism of the density functional theory (DFT) [43–46] is widely used for studying the fluid density profiles near rigid surfaces [47,48]. It usually treats the
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interaction of fluid molecules with a foreign surface in the meanfield approximation whereby every fluid molecule is considered to be subjected to an external potential, due to its pairwise interactions with the molecules of an impenetrable substrate. This external potential gives rise to a specific contribution to the free energy functional. The minimization of the latter with respect to the number density of fluid molecules (as a function of the spatial coordinate r) provides their equilibrium spatial distribution. The application of the DFT formalism to water-like (non-spherical) molecules, whereof the pairwise intermolecular potential depends not only on the distance between molecules but also on their mutual orientations, is practically extremely difficult due to the multifold configurational integrals involved. Furthermore, the effect of a hydrophobic surface on the ability of fluid (water) molecules to form hydrogen bonds had been also ignored in the conventional DFT. These problems can be now partly alleviated by combining the DFT formalism with the recently developed probabilistic hydrogen bond (PHB) model [49–55] for hydrogen bonding between water molecules in the vicinity of a foreign surface. This model provides an analytic expression for the average number of hydrogen bonds that a water molecule can form as a function of its distance to the surface. Knowing this expression, one can implement the effect of the hydrogen bonding of water molecules on their interaction with the hydrophobic surface into DFT, which is then employed to determine the distribution of water molecules between two macroscopic hydrophobic infinitely large plates at various interplate distances and various temperatures. 2. The combination of the DFT formalism with the PHB model Before presenting our results for the temperature dependence of the evaporation lengthscale of water confined between two hydrophobic plates, let us outline the combination of the DFT formalism [43–48] and PHB model [49–55] which allows one to find the water density profiles between two plates while explicitly taking into account the effect of water–water hydrogen bonding on the hydrophobe-fluid interactions. We will first briefly describe the PHB model then present its implementation into the DFT formalism. 3. The outline of a probabilistic approach to water–water hydrogen bonding near a hydrophobic surface The PHB model [49,50] considers a water molecule, whereof the location is determined by its center, to have four hydrogen-bonding (hb) arms (each capable of forming a single hydrogen bond) of rigid and symmetric (tetrahedral) configuration with the interarm angles a ¼ 109:47 . Each hb-arm can adopt a continuum of orientations. For a water molecule to form a hydrogen bond with another molecule, it is necessary that the tip of any of its hb-arms coincide with the second molecule. The length of an hb-arm thus equals the length g of a hydrogen bond. The hydrogen bond length g is assumed to be independent of whether the molecules are in the bulk or near a hydrophobic sure of pairwise interactions between face. The characteristic length g water and molecules constituting the substrate (flat and large enough to neglect edge effects, with its location determined by the loci of the centers of its outermost, surface molecules) plays a simple role in the hydrogen bond contribution to hydration or e ) only determines the referhydrophobic interaction [50–52]. It ( g ence point for measuring the distance between water molecule and substrate, so it will be set equal to g. Consider two infinitely large hydrophobic plates, 1 and 2, immersed in liquid water (Fig. 1). Even if the intrinsic hydrogen
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the hb-arms of a vicinal water molecule cannot lead to the formation of hydrogen bonds. This constraint may depend on the variables r 1 ; r2 , whence the corresponding dependence of k1 ; k2 ; k3 , and k4 . The coefficients k1 ; k2 ; k3 , and k4 as functions of r 1 ; r 2 can be evaluated by using geometric considerations [55]. They all become equal to 1 if both r1 P 2g and r 2 P 2g, when Eq. (1) reduces to its 2
Fig. 1. A water molecule (shown as a disk S) in the vicinity of two hydrophobic plates 1 and 2, at a distance r 1 from plate 1 and a distance r 2 from hydrophobe 2; the distance between plates is d. A layer of thickness g from x ¼ g to x ¼ 2g; near plate 1 is referred to as the solute hydration layer of that plate (SHL1). The SHL2 is defined in a similar way.
bonding ability of a water molecule is not affected by a hydrophobe, in its vicinity a ‘‘vicinal’’ water molecule forms a smaller number of bonds than in bulk because the hydrophobic surface restricts the configurational space available to other water molecules necessary for a vicinal water molecule to form hydrogen bonds (‘‘missing neighbor effect’’). The actual number of hydrogen bonds, that a particular BWM can form, depends on both its location and its orientation. The probabilistic model allows one to obtain an analytic expression for the average (‘‘average’’ with respect to all possible orientations of the water molecule) number of bonds ns that a vicinal water molecule (in the vicinity of a hydrophobe) can form as a function of its distance to the hydrophobe [50–52] and hydrophobe radius [53,54] (in case of a spherical hydrophobe). When the water molecule is close enough to two plates, ns will depend on the distance between water molecule and each hydrophobe [55]. The location of each plate will be determined by the location of the plane formed by centers of its outermost (facing the interplate space) molecules. The distance between the plates will be denoted by d, whereas the distance from the center of a selected water molecule S to the plate i (i ¼ 1; 2) will be denoted by r i . Clearly, only two out three variables d; r1 , and r 2 are independent; we will choose them to be r1 and r2 . Thus, ns ¼ ns ðr 1 ; r 2 Þ. Using the PHB approach, let us represent the function ns ¼ ns ðr 1 ; r 2 Þ as 2
3
4
ns ¼ k1 b1 þ k2 b1 þ k3 b1 þ k4 b1 ;
ð1Þ
where b1 is the probability that one of the hb-arms (of a bulk water molecule) can form a hydrogen bond and the coefficients k1 ; k2 ; k3 , and k4 depend on the variables r 1 ; r 2 , and so does ns . Eq. (1) assumes that the intrinsic hydrogen-bonding ability of a water molecule (i.e., the tetrahedral configuration of its hb-arms and their lengths and energies) is unaffected by its proximity to the hydrophobes so that the latter only restrict the configurational space available to other water molecules necessary for the selected water molecule to form hydrogen bonds. Thus, near the hydrophobes some orientations of
3
4
bulk analog, nb ¼ b1 þ b1 þ b1 þ b1 , with nb being the number of hydrogen bonds per bulk water molecule. Since experimental data on nb (and even its temperature dependence) are readily available, one can find b1 as a positive solution (satisfying the condition 0 < b1 < 1) of the latter equation. Thus, Eq. (1) provides an efficient pathway to ns . Consider a layer of thickness g from r i ¼ g (minimal distance between water molecule and plate i) to r i ¼ 2g near the plate i (i ¼ 1; 2) and denote it by SHLi (standing for ‘‘Solute Hydration Layer i’’). The quantities ns ; k1 ; k2 ; k3 ; k4 are functions of both variables r1 and r 2 only for water molecules located in the overlap region of SHL1 and SHL2. If a water molecule belongs only to SHL1 but not to SHL2, then ns ; k1 ; k2 ; k3 ; k4 are functions of only one variable, r1 (see Refs. [13,14]); likewise, if a water molecule belongs only to SHL2 but not to SHL1, then the quantities ns ; k1 ; k2 ; k3 ; k4 are functions of only one variable, r 2 . Denote the energy of a bulk (water–water) hydrogen bond by eb < 0 and the energy of a vicinal hydrogen bond (involving at least one molecule from SHL1 or SHL2) by es < 0. The vicinal hydrogen bond may be slightly altered energetically compared to the bulk one, but the direction (either enhancement or weakening) of such alteration is still an object of contention and hence will be neglected, as previously [51–55]. In the PHB approach, there is no restriction on the energies of bulk or vicinal (water–water) hydrogen bonds, so that the approach is valid independent of whether b < s or b ¼ s , or b > s . 3.1. Hydrogen bond contribution to the interaction of a water molecule with a hydrophobic plate The deviation of ns from nb and the deviation of es from eb give rise to a hydrogen bond contribution to the external potential field whereto a water molecule is subjected in the vicinity of two hydrophobes. This contribution, U hb ext , can be determined as hb U hb ext U ext ðr 1 ; r 2 Þ ¼
1 ðes ns eb nb Þ 2
ð2Þ
The first term on the RHS of this equation represents the total energy of hydrogen bonds of the water molecule near the hydrophobes, whereas the second term is the energy of its hydrogen bonds in bulk (at both r 1 ! 1 and r 2 ! 1); the factor 1=2 is needed to prevent double counting the energy because every hydrogen bond and its energy, either es or eb , are shared between two molecules. The dependence of U hb ext on r 1 and r 2 may be due not only to the function ns , but also to the r-dependence of es . In the PHB model ns ¼ nb when both r1 P 2g and r 2 P 2g, hence it is reasonable to assume that
es ¼ eb as well when both r1 P 2g and r2 P 2g. Thus, U hb ext is a very hb short-ranged function of r’s, such that U ext ¼ 0 if both r1 P 2g and r2 P 2g. As the energetic alteration of a vicinal hydrogen bond (compared to the bulk one) is still uncertain, it will be neglected in numerical calculations and es will be assumed to be equal to eb . 3.2. Formalism of density functional theory The fluid density distribution near a rigid surface can be efficiently studied by using computer simulations or DFT [46–48]. Although the application of DFT methods to water-like fluids is
Y.S. Djikaev, E. Ruckenstein / Journal of Colloid and Interface Science 449 (2015) 226–235
freight with significant difficulties (mentioned above), they can be circumvented by using the PHB model for water molecules [51– 55]. The DFT formalism usually treats the interaction of fluid molecules with a foreign surface in the mean-field approximation whereby every fluid molecule is considered to be subjected to an external potential, due to its pairwise interactions with the molecules of an impenetrable substrate [46–48]. This external potential gives rise to a specific contribution to the free energy functional. The minimization of the latter with respect to the number density of fluid molecules (as a function of the spatial coordinate r) provides their equilibrium spatial distribution. The effect of a hydrophobic surface on the ability of fluid (water) molecules to form hydrogen bonds had been until recently ignored in the conventional DFT.
F h ½qðrÞ ¼
Z
dr f i ðqðrÞÞ þ
Z
e ðrÞÞ; dr qðrÞ Dwh ð q
229
ð7Þ
where
f i ðqÞ ¼ kB T q ðlnðK3 qÞ 1Þ 1=2
2
(with K ¼ ðh =2pmkB TÞ being the thermal de Broglie wavelength of a model molecule of mass m, and h and kB being Planck’s and Boltzmann’s constants, respectively) is the free energy density of an ideal gas of density q and
Dwh ðqÞ ¼
1
q
½f h ðqÞ f i ðqÞ
Now, however, taking into account U hb ext , one can represent the overall external potential U ext U ext ðr 1 ; r 2 Þ, whereto a water molecule is subjected between two hydrophobes, as [51–55]
is the configurational part of the free energy of hard sphere fluid per molecule. e ðrÞ is determined in terms of qðrÞ via an The weighted density q implicit equation
hb U ext ¼ U pw ext þ U ext ;
e ðrÞ ¼ q
U pw ext
ð3Þ
U pw ext ðr 1 ; r 2 Þ
where represents pairwise interactions of a fluid molecule with the molecules of the substrate as well as the effect of the latter on the pairwise interactions between fluid molecules themselves. The grand thermodynamic potential X of a nonuniform single component fluid of volume V at a temperature T and (externally imposed) chemical potential l (an open thermodynamic system), subjected to an external potential field U ext , can be represented as a functional of the number density qðrÞ of fluid molecules
X½qðrÞ ¼ F ½qðrÞ þ
Z
dr U ext ðrÞqðrÞ l
Z
dr qðrÞ;
ð4Þ
where F ½qðrÞ, the intrinsic Helmholtz free energy of the system as a functional of qðrÞ, includes the contributions from fluid–fluid interactions as well as the ideal gas term. Conventionally, F ½qðrÞ is divided into two parts,
F ½qðrÞ ¼ F ri ½qðrÞ þ F a ½qðrÞ;
ð5Þ
where F a ½qðrÞ arises from attractive forces between fluid molecules, whereas F ri ½qðrÞ represents the repulsive forces (and includes the ideal gas contribution as well). In the van der Waals approximation the attractive forces are treated in a mean-field fashion, while the repulsive interactions are often modeled as those of hard spheres, so that Eq. (5) is approximated by
F ½qðrÞ ¼ F h ½qðrÞ þ
1 2
ZZ
drdr0 qðrÞqðr0 Þ/a ðjr r0 jÞ;
Z
e ðrÞÞ; dr0 qðr0 Þwðjr0 rj; q
e ðrÞÞ is the weight function. Although in more where wðjr0 rj; q e ðrÞÞ depends on sophisticated versions [44,45] of WDA wðjr0 rj; q e ðrÞ, we will hereafter adopt its simpler version wherein the weight q e ðrÞ. function is independent [45,46] of q In a grand canonical ensemble (corresponding to an open thermodynamic system under constant l, V, and T) the equilibrium density profile is obtained by minimizing the functional X½qðrÞ with respect to qðrÞ, i.e., by solving the Euler–Lagrange equation dX=dq ¼ 0 with respect to qðrÞ. The Euler–Lagrange equation can be written as
l ¼ kB T lnðK3 qðrÞÞ þ Wðr; qðrÞÞ;
ð9Þ
where
Z e ðrÞÞ Wðr; qðrÞÞ ¼ UðrÞ þ dr0 qðr0 Þ/a ðjr r0 jÞ þ Dwh ð q Z e ðr0 ÞÞwðjr0 rjÞ þ dr0 qðr0 Þ Dw0h ð q and Dw0h ðqÞ dDwh ðqÞ=dq. If the constraint of constant l (external chemical potential) is replaced by a given value of the fluid density far away from the impenetrable wall, q1 ¼ qðrÞjjrj¼1 , the Euler–Lagrange equation can be written in the form
qðrÞ ¼ q1 exp ð6Þ
where F h ½qðrÞ is the intrinsic Helmholtz free energy functional of a hard sphere fluid and /a ðjr r0 jÞ is the attractive part of the interaction potential between two fluid molecules located at r and r0 . As the free energy functional of a three-dimensional hardsphere fluid is not known exactly, further approximations are needed. The simplest ansatz for F h ½qðrÞ that incorporates shortranged correlations recurs [44–46] to a ‘‘smoothed’’ or ‘‘weighted’’ e ðrÞ which is a non-local functional of the actual density density q e ðrÞ at point r can be regarded qðrÞ. Namely, the weighted density q as a mean density obtained by averaging the actual density qðrÞ with an appropriate ‘‘weight’’ function over an appropriate volume in the vicinity of r. This non-local approximation for F h ½qðrÞ is often referred to as a weighted density approximation (WDA) [44–46]. A key requirement of WDA is that the weight function e ðrÞ be sufficiently and averaging volume should ensure that q smooth so that F h ½qðrÞ can be calculated in a form formally reminiscent of local density approximation. In the framework of WDA, the intrinsic Helmholtz free energy functional of hard sphere fluid is represented in the form
ð8Þ
1 ðWðr; qðrÞÞ Wðr; q1 ÞÞ : kB T
ð10Þ
In the canonical ensemble (corresponding to a closed thermodynamic system with constant number of molecules N, volume V, and temperature T), the chemical potential l, that appears in Eqs. (4) and (9), is not known in advance. Instead, it plays the role of a Lagrange multiplier corresponding to the constraint of fixed number of molecules in the system:
N¼
Z
dr qðrÞ:
V
This equation can be used to determine the Lagrange multiplier l, i.e., the chemical potential in the system (see, e.g., Refs. [45,52]). The Euler–Lagrange equation for the density profile in the canonical ensemble can be written as
qðrÞ ¼ N R
exp½Wðr; qðrÞÞ=kB T : dr exp½Wðr; qðrÞÞ=kB T
ð11Þ
This expression has the familiar form of a ‘‘generic’’ one-particle distribution function in classical statistical mechanics [56]. In the particular case of (fluid) water between two hydrophobic plates, one can use the planar symmetry of the system and choose
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the Cartesian coordinates so that the plates are parallel to the y–z plane, with the plate 1 at x ¼ 0, plate 2 at x ¼ d, and the molecules of the fluid occupying the ‘‘interplate space’’ 0 < x < d. The equilibrium density profile obtained from Eq. (9) or Eq. (10) or Eq. (11) is then a function of a single variable x, i.e., qðrÞ ¼ qðxÞ. 4. Numerical evaluations For a numerical illustration, we considered the model fluid water confined between two infinitely large hydrophobic plates, at five temperatures: T ¼ 293 K, 303 K, 313 K, 323 K, and 333 K. The calculations were carried out at the vapor–liquid equilibrium conditions. At a given temperature, the densities qv and ql of coexisting vapor and liquid, respectively, are determined by solving the equations lðq; TÞjq¼qv ¼ lðq; TÞjq¼ql ; pðq; TÞjq¼qv ¼ pðq; TÞjq¼ql , requiring the chemical potential l lðq; TÞ and pressure p pðq; TÞ to be the same throughout both coexisting phases. The number of hydrogen bonds per bulk water molecule as a function of temperature was approximated using a linear leastmean squares fit of the data provided in [57] (nb ¼ 3:69 at 0 °C, nb ¼ 3:59 at 25 °C, and nb ¼ 3:24 at 100 °C), resulting in nb nb ðTÞ ¼ anT bnT ðT 273:15Þ, with anT ¼ 3:69577 and bnT ¼ 0:00453846. The temperature dependence of ns is uniquely determined by the function nb ðTÞ, because b1 in Eq. (1) is unambiguously determined by the thermodynamic state of the bulk water (temperature, pressure, etc.) as the solution of the equation 2
3
4
nb ¼ b1 þ b1 þ b1 þ b1 satisfying the constraint 0 < b1 < 1 (for details see Refs. [49,50]). The dependence of b on T was approximated by the linear function
b b ðTÞ ¼ a b ðT 273:15Þ; with the constants a ¼ 3:82 1013 erg and b ¼ 4:98 1016 erg/ K determined from the data provided in Refs. [58,59], respectively. The ratio es =eb was assumed to be temperature-independent. The pairwise interactions between two water molecules were modeled with the Lennard-Jones (LJ) potential. The energy parameter ww of the LJ potential was adjusted to be 5:4 1014 , which differs from its values used in the computer simulations (Monte Carlo or molecular dynamics) of various water models, in which
ww
ranges 14
[57f]
from
5:31 1015
erg
(ST2
model)
to
1:47 10 erg (SWM4-NDPmodel) to 2:54 1013 erg (SSD model). Such a modification was needed to ensure that the phase diagram of model water more or less resembles that of real water. In the DFT formalism this can be achieved only by adjusting the single intermolecular potential describing water–water interactions, whereas in computer simulations water–water interactions are usually described by the combination of LJ and electrostatic potentials; hence the difference in the energy parameters of the respective LJ potentials. The length parameter r of the LJ potential also has different values in different water models 3b in the range from 3.02 Å (SSD model) to 3.18 Å (SWM4-NDP model). On the other hand, the length g of the hydrogen bond (i.e., the distance between the oxygen atoms of two hydrogen-bonded water molecules) is reported [57f] to be about 2.98 Å. Since r’s (of various water models) and g are so close to each other, we assumed for our model r ’ g ’ 3:18 Å. The attractive part /at of pairwise water–water interactions was modeled via the Weeks–Chandler–Anderson perturbation scheme [60]. The interaction potential between water molecule and molecule of a hydrophobe was assumed to be of LJ type with an energy parameter ewp and a length parameter g. Integrating this interaction with respect to the position of the molecule of a plate over the semi-infinite plate volume for both plates 1 and 2, one can obtain the pairwise contribution U pext into U ext . We assumed the
dimensionless number density of molecules in the hydrophobic plates to be qp g3 1 and considered ewp =eww ¼ 0:75 to mimic their hydrophobicity. The chemical potential of a uniform hard sphere fluid lh and the configurational part Dwh Dwh ðq; TÞ of the free energy of a hard sphere fluid were modeled in the Carnahan–Starling approximation [44,61], whereas for the weight function e ðrÞÞ in Eq. (8) we adopted a q e -independent version wðjr0 rj; q [44,46]
Dwh ¼ kB T
n ð4 3nÞ ð1 nÞ2
;
wðr12 Þ ¼
3
pg4
ðg r12 ÞHðg r 12 Þ;
with HðuÞ being the Heaviside (unit-step) function. In order to find the equilibrium density profile of (fluid) water confined between two hydrophobic plates, we have solved Eq. th
(9) by iterations. Namely, the density profile qi ðxÞat the i iteration is found from the density profile qi1 ðxÞ at the previous, ði 1Þth iteration:
kB T lnðK3 qi ðrÞÞ ¼ l Wðr; qi1 ðrÞÞ
ð12Þ
At any temperature T and interplate distance d, the initial profile was taken to be of the shape shown in Fig. 2, which corresponds to some nonequilibrium liquid ‘‘forcefully’’ confined between the plates. Since the calculations are carried out in an open system (grand canonical ensemble) under conditions of phase coexistence, such an initial profile is necessary to ensure that, if the liquid between the plates can be stable, the corresponding solution of Eq. (12) will not be missed during the iteration procedure. As the iterations proceed, the initial profile gradually transforms into the equilibrium one. The Euler–Lagrange equation (Eq. (9)) for a grand canonical ensemble under conditions of phase coexistence may have two equilibrium solutions: one for the vapor phase, the other for the liquid phase. At large interplate distances, the first solution attained during the iterations is that of the stable liquid water confined between the plates. However, at small interplate distances, the first solution attained in the course of iterations corresponds to the metastable (or even unstable) liquid between the plates; such a solution gradually decays, ceding place to the second solution which corresponds to a vapor-like phase confined between the plates. The results of numerical calculations are shown in Figs. 3–7. Fig. 3 presents the density profiles of water molecules between two hydrophobic plates at temperature T ¼ 293 K for various interplate distances d (hereinafter, all densities and distances/ lengths are shown in dimensionless units, qg3 ; x=g, and d=g,
0.8
0.6
d
0.4
7.0
0.2
0.0 0
1
2
3
4
5
6
7
Fig. 2. The typical initial density profile for the approximation procedure in Eq. (12), shown as the dimensionless density qg3 vs dimensionless distance x=g to plate 1 for the case where the interplate distance is d=g ¼ 7:0.
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(a)
(b) 0.8
0.0025 0.6
0.0020 0.0015
d
0.4
d
0.0010
10
5 0.2
0.0005 0.0000
0.0 0
1
2
3
4
5
0
2
4
6
8
10
0.8
0.0025 0.0020
0.6
0.0015
d
0.4
d
0.0010
15
8 0.2
0.0005 0.0
0.0000 0
2
4
6
0
8
0.0025
2
4
6
8
10
12
14
0.8
0.0020 0.6 0.0015
d
0.4
d
0.0010
20
9 0.2
0.0005 0.0000
0.0 0
2
4
6
8
0
5
10
15
20
Fig. 3. The density profiles of water molecules between two hydrophobic plates at temperature T ¼ 293 K for various interplate distances d (shown as qg3 vs x=g, with x being a distance from plate 1). Every figure panel represents a profile for its own d as indicated therein. Fig. 3a corresponds to d < de , whereas Fig. 3b is for d > de .
respectively); note the different scales for the ordinate coordinates in Fig. 3a and b. Every figure panel represents a profile for its own d as indicated therein. Fig. 3a corresponds to d < de , where de is the characteristic lengthscale of water confined between two infinitely large hydrophobic plates (the quantitative definition of de is discussed below). As clear, at d < de , water confined between the plates is in the vapor-like phase. On the other hand, the profiles in Fig. 3b correspond to d > de ; they represent a liquid-like phase confined between the plates. Although Fig. 3 shows the typical behavior of density profiles with changing distance between the plates for any temperature, the actual value of the characteristic ‘‘evaporation’’ lengthscale de (however defined) will depend on the temperature of the system. In Fig. 4a–e, the density profiles of water confined between two plates at the interplate distance d ¼ 9g are plotted for various temperatures (indicated in the figure panels). The dashed lines indicate the average density, whereas the filled circle emphasizes the data
point corresponding to the middle between the plates. As clear, at this interplate distance the space between the plates is filled with water in a vapor-like phase, for all temperatures. This suggests that de > 9g in the considered temperature range. As expect v of water molecules ed for the vapor phase, the average density q increases with increasing temperature; so does the mid-point den^v . sity q In Fig. 5a-e, the density profiles of water confined between two plates at the interplate distance d ¼ 11g are plotted for various temperatures (indicated in the figure panels). Again, the dashed lines indicate the average density, whereas the filled circle emphasizes the data point corresponding to the middle between the plates. At this interplate distance water confined between the plates is in a liquid-like phase, for all temperatures. This suggests that de < 11g for all the temperatures considered. As expected l decreases with increasfor the liquid phase, the average density q ^v . ing temperature, and so does the mid-point density q
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0.0025 0.004 0.0020 0.003 0.0015 0.002
0.0010
a T 293 K 0.001
0.0005 0.0000
c T 313 K
0.000 0
2
4
6
8
0
0.0030
2
4
6
8
6
8
0.005
0.0025
0.004
0.0020 0.003 0.0015 0.0010
0.002
b T 303 K
0.001
0.0005 0.0000
d T 323 K
0.000 0
2
4
6
8
0
2
6
8
4
0.007 0.006 0.005 0.004 0.003 0.002 0.001
e T 333 K
0.000 0
2
4
Fig. 4. The density profiles of water confined between two plates at the interplate distance d ¼ 9g for various temperatures (indicated in the figure panels).
Figs. 4 and 5 indicate that, in the temperature range from 293 K to 333 K. the evaporation lengthscale de (however defined) is in the range 9 < de =g < 11. Provided that there is a more or less rigorous definition of de , it is thus necessary to determine the density profiles qðzÞ on an adjustable, progressively finer grid in the interval of interplate distances initially from d ¼ 9g to d ¼ 11g. The data, presented in Figs. 3–5, suggest two alternative methods to define and determine the evaporation lengthscale de of liquid (water) confined between two hydrophobic plates. One , method is based on the d-dependence of the average density q ^. the other on the d-dependence of the mid-point density q vs d and q ^ vs d are plotted (as solid In Fig. 6, the functions q squares and disks, respectively) for the temperature T ¼ 293 K. Fig. 6a shows the data for interplate distances d < de , so that the fluid confined between the plates is in a vapor like phase, whereas Fig. 6b presents data for interplated distances d > de with the fluid and between the plates in a liquid-like phase. Similar behavior of q q^ as functions of d will be also observed for other temperatures. (solid squares) At a given temperature, the average density q monotonically increases with increasing d, but exhibits a sharp increase from values, characteristic for a vapor phase, to values,
characteristic of a liquid phase. This sharp increase occurs in a very narrow interval of d’s whereof the middle can be defined as the evaporation lengthscale de . This definition lacks mathematical rigor, but can be useful in characterizing the phenomenon of interest. On the other hand, at a given temperature, as d increases, the ^ (disks) first slowly decreases (its values being mid-point density q characteristic of a vapor phase), but after attaining a minimum, it sharply increases to values, characteristic of a liquid phase then continues its slow monotonic increase. This sharp increase occurs in a very narrow interval of d’s. Again the middle can be defined as the evaporation lengthscale de . However, unlike the function ¼q ðdÞ, the function q ^ ¼q ^ ðdÞ has a unique point, namely, the q location of the minimum, which can be defined to be the evaporation lengthscale de of liquid from between the hydrophobic plates. Although the latter definition is mathematically more rigorous and hence formally preferable, its practical use is rather doubtful, because it involves the DFT calculations of the density profiles between the plates on a rather fine grid in some interval of interplate distances d (the interval which itself has to be identified via the DFT calculations of the profiles as well). Therefore, from the ¼q ðdÞ and q ^ ¼q ^ ðdÞ practical point of view, the both functions q
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0.8 0.8 0.6
0.6
0.4
0.4
c T 313 K
0.2
a T 293 K
0.2
0.0
0.0 0
2
4
6
8
0
10
2
4
6
8
10
8
10
0.8 0.6 0.6 0.4
0.4
d T 323 K
0.2
b T 303 K
0.2
0.0
0.0 0
2
4
6
8
0
10
2
4
6
0.7 0.6 0.5 0.4 0.3
e T 333 K
0.2 0.1 0.0 0
2
4
6
8
10
Fig. 5. The density profiles of water confined between two plates at the interplate distance d ¼ 11g for various temperatures (indicated in the figure panels).
can be used to loosely identify the evaporation lengthscale as the middle of the interval of d wherein the functions sharply rise from their vapor-like-density values to the liquid-like-density values. Using such a definition, we have determined the evaporation lengthscale de of model water confined between two hydrophobic plates for five different temperatures (293 K, 303 K, 313 K, 323 K, and 333 K). The results are presented in Fig. 7. For the model water and model hydrophobes that we studied, the combined DFT/PHB model predicts de to be of the order of several nanometers (de 3 nm). In the temperature range considered de is a relatively weak function of T. As the temperature increases by about 14 % from 293 K, the evaporation length increases by about 7 %. Note that the error bars of every data point in Fig. 7 are about 0.1% (i.e., de was determined with the relatively accuracy of 0.1%). The predictions of our model for de are in a partial agreement with the previously reported results. In Refs. [1,26] the evaporation lengthscale was estimated to be of the order of 10 nm. On the other
hand, in Ref. [6] the evaporation lengthscale and its temperature dependence for water and several other organic liquids were studied by using the formalism of classical thermodynamics with the actual vapor–liquid surface tension data (at atmospheric conditions) for real substances. The magnitude of de for water was found to be significantly larger, of the order of 1 lm, more in line with the number quoted in Ref. [31]. Furthermore, as the water temperature increases from about 273 K to about 298 K, de was also reported to mildly decrease [6]. As the water temperature increased from about 298 K up to its boiling point 373 K (at atmospheric pressure 1 bar), the evaporation lengthscale monotonically, but progressively steeper increased, naturally diverging at the boiling point (similar behavior was observed also for the organic liquids as well) [6]. Such a T-dependence of de for a real water is qualitatively consistent with the temperature dependence of de predicted by the DFT/PHB approach for the model water confined between model hydrophobic plates in the temperature range that we studied.
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Y.S. Djikaev, E. Ruckenstein / Journal of Colloid and Interface Science 449 (2015) 226–235
(a) 0.0025 0.0020 T 293 K; d de vapor midpoint density
0.0015
average density
0.0010
0.0005
3
4
5
6
7
8
9
10
(b) 0.8 0.7 0.6 0.5
T 293 K; d de liquid midpoint density
0.4 0.3
average density
10
12
14
16
18
20
22
and midpoint density q ^ of the fluid confined between two Fig. 6. Average density q plates as a function of the interplate distance d (solid squares and disks, respectively) for the temperature T ¼ 293 K.
10.6 10.4
We have found that there may be two alternative pathways to define and determine the evaporation lengthscale of water between hydrophobic plates, one based on the average density of water molecules between the plates the other based on the local density in the middle between the plates. From the practical point of view, both definitions are virtually equivalent to loosely identify the evaporation lengthscale as the middle of the interval of interplate distances wherein these quantities charply rise from their vapor-like-density values to the liquid-like-density values. Using such a definition, we have determined the evaporation lengthscale of model water confined between two model hydrophobic plates in the temperature range from 293 K to 333 K. Our combined DFT/PHB model predicts the evaporation lengthscale to be of the order of several nanometers and to be a relatively weakly increasing function of temperature. These predictions are in a partial qualitative agreement with expectations. Although there are large quantitative variations in the value of the evaporation lengthscale reported in different sources, it should be emphasized, that these may be due to the inadequacy of the comparison of results (either simulational or theoretical or experimental) obtained for different systems. It is not quite straightforward to compare the results for solvophobic and confinement phenomena when two different sets of data are of different origins. Extreme care must hence be taken if the two sets are for different solvents (or solvent models) or/and for different (in shape/size) solutes/hydrophobes or/and for different solute–solvent interactions. References [1] [2] [3] [4] [5] [6] [7] [8]
10.2 10.0
[9] [10]
9.8
[11] [12] [13]
9.6 9.4 290
300
310
320
330
340
[14] [15] [16]
TK [17] Fig. 7. The temperature dependence of the evaporation lengthscale de of model water confined between two hydrophobic plates in the range from T = 293 K to T = 333 K.
5. Concluding remarks Although the evaporation lengthscale is an important quantity for the wide variety of phenomena and processes involving liquids in solvophobic confinement, its dependence on various thermodynamic parameters has been hardly studied even for simplest systems. We have attempted to shed more light on the temperature dependence of the evaporation lengthscale of water confined between two hydrophobic parallel plates, by using the combination of the DFT formalism and PHB model. Such a combination allows one to find the water density profiles between two plates while explicitly taking into account the effect of water–water hydrogen bonding on the hydrophobe-fluid interactions.
[18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36]
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