Is Water at the Graphite Interface Vapor-like or Ice-like? - The Journal

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Is Water at the Graphite Interface Vapor-Like or Ice-Like? Yuqing Qiu, Laura Lupi, and Valeria Molinero J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.7b11476 • Publication Date (Web): 03 Jan 2018 Downloaded from http://pubs.acs.org on January 4, 2018

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The Journal of Physical Chemistry

Is Water at the Graphite Interface Vapor-like or Ice-like? Yuqing Qiu, Laura Lupi, and Valeria Molinero*

Department of Chemistry, The University of Utah, 315 South 1400 East, Salt Lake City, Utah 84112-0850, United States

Abstract Graphitic surfaces are the main component of soot, a major constituent of atmospheric aerosols. Experiments indicate that soots of different origins display a wide range of abilities to heterogeneously nucleate ice. The ability of pure graphite to nucleate ice in experiments, however, seems to be almost negligible. Nevertheless, molecular simulations with the monatomic water model mW predict that pure graphite nucleates ice. According to Classical Nucleation Theory, the ability of a surface to nucleate ice is controlled by the binding free energy between ice immersed in liquid water and the surface. To establish whether the discrepancy in freezing efficiencies of graphite in mW simulations and experiments arises from the coarse resolution of the model or can be fixed by reparameterization, it is important to elucidate the contributions of the water-graphite, water-ice and ice-water interfaces to the free energy, enthalpy and entropy of binding for both water and the model. Here we use thermodynamic analysis and free energy calculations to determine these interfacial properties. We demonstrate that liquid water at the graphite interface is not ice-like or vapor-like: it has similar free energy, entropy, and enthalpy as water in the bulk. The thermodynamics of the water-graphite interface is well reproduced by the mW model. We find that the entropy of binding between graphite and ice is positive and dominated, in both experiments and simulations, by the favorable entropy of reducing the ice-water interface. Our analysis indicates that the discrepancy in freezing efficiencies of graphite in experiments and the simulations with mW arises from the inability of the model to reproduce the free energy of the ice-graphite interface. This transferability issue is intrinsic to the resolution of the model, and arises from its lack of rotational degrees of freedom.

* corresponding author, e-mail: [email protected]

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1. Introduction. It has been argued that hydrophobic surfaces promote vapor-like fluctuations in the interfacial water.1 Graphite is considered a prototypical example of hydrophobic surface,2-3 although some studies argue that is more hydrophilic than previously expected.4 Molecular simulations with the monatomic water model mW indicate that the first layer of interfacial water on graphite has domains with order intermediate between ice and liquid,5 that these domains are the birthplace of ice nuclei,6 and that the degree of ice-like order at the surface increases with supercooling.7 This poses the question of whether water at the graphite interface is more or less ordered than in the bulk, and to which extent the simulations reproduce the experimental surface entropy of water at the graphite interface and its temperature dependence. Answering these questions is the first goal of this work. Classical Nucleation Theory (CNT) indicates that surfaces that nucleate ice, must bind to ice.8 The stronger the ice nucleus binds to the ice-liquid interface, the lower is the free energy barrier of nucleation, and the warmer the temperature at which the surface promotes the nucleation of ice.8-9 The best ice nucleating agents, such as ice nucleating proteins10-12 and monolayers of long-chain alcohols,8, 13 achieve a strong binding free energy to ice by providing ordered arrangements of hydroxyl groups that match the order of water molecules in specific faces of ice.8, 10-13 The regular arrangement of OH groups results in strong hydrogen bonds between ice and the surface that acts as a template for ice formation. Graphite, on the other hand, cannot template the ice structure.5 Nevertheless, graphitic surfaces nucleate ice in simulations with the mW model.5-7,

9, 14-18

The

simulations indicate that the ice nucleation rate as a function of the water-carbon attraction is nonmonotonous: first increases, and then decreases sharply before further increasing again.15 This implies that the binding free energy of ice to graphite changes non-monotonically as the graphitic surface becomes more hydrophilic. The second goal of this work is to compute the binding free energy of graphite to mW ice, determine its enthalpic and entropic contributions, and elucidate how do these contributions change with the strength of the water-carbon attraction. The ability of graphite to nucleate ice in experiments, however, is not well established. Most evidence suggests that graphite has little or no ice nucleation ability,19-22 although soot containing graphitic surfaces can nucleate ice at temperatures up to 16 K above the homogeneous limit.19, 23-25 These results suggest that the binding free energy of ice to graphite is closer to zero in experiments than in simulations. The third goal of this study is to elucidate which contribution to the free energy


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- entropy or enthalpy- is responsible for this difference between the experiments and the mW model, and to determine whether the discrepancy arises from the interaction potentials or from the low resolution of the coarse-grained model.

2. Simulation models and methods. Water is modeled with the monatomic water model mW,26 which has been extensively used for simulations of water and ice.5-8, 14, 26-44 Graphite is modeled as a single rigid layer of graphene following ref. 5, and its interaction with water is represented by the two-body term of the StillingerWeber (SW) potential45 with water-carbon interaction size σw-c = 0.32 nm, which reproduces the water-carbon distances of atomistic models.46 The strength of the water-carbon interaction εw-c = 0.13 kcal mol-1 was parameterized5 to reproduce the contact angle of water droplets on graphite, θ = 86°.3 The other parameters in the SW potential for water-carbon are the same as in ref. 45. We also consider graphitic surfaces with stronger water-carbon attractions, εw-c = 0.20 and 0.25 kcal mol-1, for which we compute the freezing temperatures following the same protocols as in ref. 7. We compute the surface tensions of graphite-ice and ice-water interfaces with the Mold Integration method.47-48 To compute the surface tension of the ice-water interface, γice-water, we build a layer of 288 potential energy wells (“molds”) in a 5 nm × 5 nm area, where each mold is in the position of the oxygen in a single bilayer of the basal plane of ice. This mold is immersed in a simulation cell with dimensions 5 nm × 5 nm × 5 nm containing 4608 water molecules in the liquid state. We implement the interaction of the potential well and water as a continuous function, following refs.

47-48

, using the tabulated pair-potential function of LAMMPS.49 To determine the

largest well radius 𝑟!! at which no induction time is required for the growth of ice, which we identify with the CHILL+ algorithm,34 we run simulations with various well radii, scanning from 0 to 1 Å, every 0.01 Å, with the depth of the well fixed at εm = 8 RT. For each radius we perform five independent molecular dynamics simulations starting with different random velocities. Each simulation is evolved for 1 ns in the NVT ensemble at 273 K, the temperature of ice-liquid coexistence for the mW model.35 The temperature is controlled with Nose-hoover thermostat50-51 with a relaxation time constant of 0.5 ps. The equations of motion are integrated with the velocity Verlet algorithm using periodic boundary conditions and a time step of 0.5 fs (this small step is required to integrate the steep interactions of mW with the mold). We find 𝑟!! = 0.53 ± 0.01 Å,


3


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smaller than the 𝑟!! = 1.00 Å reported for mold integration of the basal plane of mW using Monte Carlo simulations.47 We propagate the error bar in the surface tension from the error bar of 𝑟!! . The free energy of ordering water into ice through the molds is integrated for the potential wells width 𝑟! = 0.8, 0.9, 1.0, 1.1, 1.2 Å for ice-water interface, in each case turning on the potential energy wells from zero to its maximum depth, εm = 8 RT. The values of ε scanned for each system are listed in Supp. Table S1. For each value of εm of each 𝑟! , we count the number of wells filled with water every 1 ps and averaged this count over a 1 ns simulation. We verify that with the depth of εm = 8 RT, the averaged number of filled wells equals the number of wells in the system, 288. The free energy for each 𝑟! corresponds to twice the ice-water surface free energy, 𝛾!!! (𝑟! ). These values are then fitted to a straight line (Supp. Figure S1) to extrapolate to the thermodynamic value of 𝛾!!! = 35.0 ± 0.2 mJ m2 at 𝑟!! = 0.53 Å. This value agrees well with the γi-w = 34.5±0.8 mJ m-2 determined for mW exposing also the basal plane, by mold integration using a Monte Carlo code.47 This validates our implementation of Mold Integration through molecular dynamics in LAMMPS. To compute the binding of ice to graphite at 273 K, we build a mold surface of the basal plane of ice composed of 288 potential energy wells with area 5 nm × 5 nm in the first contact layer on graphite. The mold and the graphite interface are exposed to bulk water containing 4116 molecules. The integral computed is 𝛾!"#$ (𝑟!! ) = γi-w + γg-i – γg-w, where γi-w, γg-i and γg-w are the surface tension of the ice-water, graphite-ice and graphite-water interfaces, respectively. The maximum depth is set to εm = 8 RT; the averaged number of filled wells is computed with the same procedure as described for the ice-water interface. We find that 𝑟!! increases with increasing εw-c. We determine that for εw-c = 0.13 kcal mol-1, 𝑟!! = 0.48±0.01 Å, and the free energy is integrated for 𝑟! = 1.00, 1.05, 1.10, 1.15, 1.20 Å. For εw-c = 0.20 kcal mol-1 we determine 𝑟!! = 0.77 ± 0.01 Å and integrate the free energy for 𝑟! = 1.05, 1.10, 1.15, 1.20, 1.25 Å. For εw-c = 0.25 kcal mol-1, we determine 𝑟!! = 0.88 ± 0.01 Å, and integrate the free energy for 𝑟! = 1.13, 1.15, 1.20, 1.22, 1.25 Å. Further increase of εw-c would result in a large 𝑟!! that would lead to the capture of more than one water molecule per well in the thermodynamic integration, invalidating the method. Hence, the mold integration cannot be used to compute the surface tension of strongly attractive graphitic surfaces. Supp. Figure S1 shows the fit and values of the 𝛾!"#$ (𝑟!! ) for each εw-c. We compute the binding


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free energy ΔGbind of each surface from 𝛾!!"# (𝑟!! ) of that surface and the ice-liquid surface tension

γi-w, ΔGbind = γg-i – γg-w – γi-w = 𝛾!"#$ (𝑟!! ) - 2γi-w. We quantify the ice-like tetrahedral order of water and ice with the bond order parameter q3, as in ref. 7. The average values of q3 for interfacial and bulk water as a function of temperatures are taken from ref. 7. Here we compute the distribution of q3 of mW ice at the graphitic surfaces and compare it with that of bulk ice in order to estimate the surface entropy of graphite-ice interface. We first crystallize a box of 4116 water molecules in contact with a 5 nm × 5 nm graphite surface. We select the water molecules in two bilayer of ice on each side of the graphitic surface, and compute their q3 values every 10 ps for 10 ns at 270 K. As the q3 order parameter considers the four closest neighbors to a given molecule, and the closest layer of ice to graphite is undercoordinated, to make a fair comparison to the order parameter and its fluctuations at the graphite interface we select only two bilayers of ice from a trajectory of bulk ice, and compute the q3 value of them every 10 ps for 10 ns at 270 K, while only accounting for neighbors within the two bilayers. The resulting histograms of q3 for ice at the graphitic surfaces and in bulk are plotted in Supp. Figure S2. To measure the contact angle θ of mW water droplets on graphite and its dependence on temperature, we place a droplet containing 5241 water molecules on a 24 nm × 24 nm graphite surface. After equilibrating the droplet for 5 ns, we compute θ every 1 ps and averaged over 3 ns. The contact angles are computed as in ref. 7, setting the temperature of the droplet to 218, 220, 230, 240, 250, 260, 270, 280 and 298 K.

3. Results and discussion A. Water at the graphite surface has similar entropy and enthalpy than in bulk water. The surface entropy of the graphite-water interface, Sg-w = (∂S /∂Ag-w)p,T = - (∂γg-w/∂T)p,Ag-w, is a measure of the difference in entropy of water at the surface and in the bulk of the liquid. If water at the graphite interface were ice-like, the surface entropy would be negative; if it were vaporlike, it would be positive. In what follows we use thermodynamic relations to derive an expression of

γg-w in terms of the surface tension of the water-vapor interface γw-v and the contact angle θ of liquid water on graphite and compute the surface entropy of water at the graphite interface from the temperature dependence of these properties.


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Figure 1a sketches the process of adhesion of a slab of liquid water to the graphite surface. The free energy per area associated to that process is the adhesion free energy, ΔGadhesion, which we express in terms of the surface tensions of the graphite-water interface γg-w, the graphite-vapor interface γg-v, and the water-vapor interface γw-v: ΔGadhesion = γg-w – γg-v – γw-v

a)

water

ΔGadhesion

graphite

b)

graphite water

(1)

water graphite

ΔGbind

ice

water graphite ice

Figure 1. Processes of adhesion of water to graphite and binding of ice to graphite discussed in this study. a) Process of binding a slab of water to a graphitic surface; we call the free energy per area of this process the adhesion free energy, ΔGadhesion. b) Process of binding a graphite surface initially immersed in liquid water to ice; we call the free energy per area for this process the binding free energy, ΔGbind.

We write ΔGadhesion as a function of γw-v and the contact angle θ assuming, as in ref. 52, that Young’s equation is valid and that the reversible work of spreading vapor on graphite, γg-v, is zero: ΔGadhesion = γg-w – γw-v = -γw-v × (1+cosθ).

(2)

We then rearrange eq. 2 to express γg-w as a function of γw-v and θ:

γg-w = - γw-v × cosθ.

(3)

The hydrophobicity of the surface determines the sign of the surface free energy between graphite and water, γg-w. Hydrophobic surfaces (θ > 90°; cosθ < 0) destabilize water and have γsurface-w > 0, while hydrophilic surfaces (θ < 90°; cosθ > 0) stabilize water and have γsurface-w < 0. Graphite is


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borderline between hydrophobic and hydrophilic, with contact angle θ = 86°.3 The angle θ and water-vapor surface tension γw-v for the coarse-grained model at 273 K are in good agreement with the experimental values (Table 1). As a result, the water-graphite surface tension γg-w for the mW model (-4.7 mJ m-2) is essentially the same as for water (-5.3 mJ m-2). The negative sign of γg-w indicates that water at the graphite interface at 273 K is slightly more stable than in the bulk.

Table 1. Thermodynamics of the water-vapor and water-graphite interfaces at 273 K. γw-v

θ

(mJ m-2)

a)

γg-w

Sw-v

Sg-w

Hg-w

(mJ m-2)

(µJ m-2

(µJ m-2 K-

(mJ m-2

K-1)

1)

K-1)

water

76a

86°c

-5.3

141d

-9.8

-8.0

mW

68b

86°

-4.7

69e

-4.8

-6.0

ref. 53, b)ref. 54, c) ref. 55, d) ref. 53, e) ref. 54

We obtain the surface entropy of liquid water at the graphite interface, Sg-w, by deriving eq. 3 with respect to temperature: Sg-w = -Sw-v × cosθ + γw-v ( where Sw-v =−(

!!!!! !"

! !"# ! !"

)!,!!!! ,

(4)

)!,!!!! is the surface entropy of the water-vapor interface. The contact angle

of mW water on graphite is independent of temperature in the range from 218 to 298 K (Supp. Figure S2). That reduces eq. 4 to Sg-w = -Sw-v × cosθ. We are not aware of experimental measurements of the contact angle of water on graphite as a function of temperature. Here we assume θ is independent of temperature, at least in the immediacy of the melting temperature, also in experiments. The surface entropy of the liquid-vapor interface, Sw-v, is always positive for both water and mW. The magnitude of this quantity, however, is underestimated by the mW model because it does not have rotational degrees of freedom54 (Table 1 and Figure 2). We find that while


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the tetrahedral ordering on interfacial water increases on cooling; it does it in a way that mimics the ordering in bulk liquid water7 (Figure 3). This explains why the surface entropy of water at the graphite surface is quite insensitive to temperature (Figure 2). Comparing the surface entropies of water at the graphit and vapor interfaces (Table 1), it is clear that not only the signs are opposite but also that the magnitude of Sg-w is very small, because θ of water on graphite is close to 90o. We conclude that water at the graphite interface is neither ice-like nor vapor-like; its free energy, entropy and enthalpy (Table 1) are very similar to that of water in the bulk liquid.

a) water

b) mW

Figure 2: Temperature dependence of surface entropies and binding entropies of a) water and b) the mW model. Entropy of binding of graphite to ice (black), entropy of the water-vapor interface (blue), entropy of the graphite-water interface (orange), and entropy of the ice-water interface (red). Si-w decreases on supercooling, following the anomalous behavior of the heat capacity of supercooled liquid water. The lowering of the iceliquid surface entropy is reflected almost symmetrically in the entropy of binding ice to graphite, ΔSbind.

The analysis of this section indicates that water at the graphite interface is only slightly more stable than in the bulk. The stabilization of water at the surface is driven by a favorable –but also small– surface enthalpy (Table 1). The coarse-grained model represents well the stability of water at the graphite interface, γg-w, because mW reproduces well the water-vapor surface tension and –by construction– it matches the experimental contact angle of water on graphite. The entropic and enthalpic contributions to the surface free energy of water at the graphite interface are also well reproduced by the mW model, but only because a small value of cosθ (~0.07 for graphite) disguises the large differences in surface entropy of the water-vapor interface in mW and real water (Table 1). mW would not reproduce well the surface entropy and enthalpy of water at surfaces that are very


8

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hydrophilic or very hydrophobic if the parameterization is performed to reproduce the contact angle of liquid water with the surface. Alternative parameterizations based on the thermodynamics of the interfaces may not reproduce the contact angle of very hydrophobic or hydrophilic surfaces.56 Fully atomistic water models generally reproduce the surface tension of the water-vapor interface within 20% of the experimental value, and reproduce well the surface entropy of that interface.54 From that, we expect that fully atomistic models parameterized to reproduce the contact angle of water on graphite (or any other rigid surface), will properly capture the experimental thermodynamics of interfacial water.

Figure 3. Temperature dependence of the tetrahedral ice-like ordering (measured by the bond-order parameter q3) for mW water in the first layer in contact with graphite (black circles) is the same as for the bulk liquid (red circles). Adapted from ref. 7, with permission.

B. The binding of graphite to ice is favored by entropy. The entropy associated to the binding of graphite from water to ice (Figure 1b) is ΔSbind = Sg-i – Sg-w – Si-w, (5) where Sg-w is the graphite-water interface derived in section A and Sg-i and Si-w are the surface entropies of the graphite-water and ice-water interfaces, respectively. Replacing Sg-w with eq. 4 we obtain: ΔSbind = Sg-i + Sw-v cosθ – Si-w. (6) As water molecules at the ice-graphite interface have order and fluctuations similar to that in bulk ice (Supp. Figure S3), we approximate Sg-i ≈ 0. Sg-w is also small for water on graphite (section A), therefore the surface entropy of the ice-water interface, Si-w, controls the sign and magnitude of the entropy of binding graphite to ice, ΔSbind.


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We obtain the surface entropy of the water-ice interface, Si-w, from the temperature !

!

dependence of the ice-water surface tension γi-w, using Turnbull’s heuristic relation57, γi-w = λΔHm𝑣!! , where λ is a constant equal to 0.32 for water, ΔHm is the melting enthalpy of ice, and vm its molar volume. This relation has been shown to be exact for hard spheres58 and a very good approximation for water.59 The expression of the ice-liquid surface entropy becomes: Si-w = - (

!!!!! !" !

!!!!

)! = -λ (

!

!

!" !

!

!

!

!

!

!!

)! 𝑣!! + λ𝑣!! ( !"! )! Δ𝐻! !

!

= - λ𝑣!! Δ𝐶! + λ𝑣!! 𝛼Δ𝐻! , !

(7)

where Δ𝐶! is the difference in heat capacity between ice and liquid and 𝛼 is the thermal expansion coefficient of ice. The small value of 𝛼 makes the second term negligible (this would still be the case !

!

if vm referred to the volume of liquid water). Using λ𝑣!! = γi-w(Tm)/ΔHm(Tm), the surface entropy of the ice-liquid interface reduces to a product of the difference in heat capacity between ice and liquid and a constant given by the ratio of the surface tension of the ice-water interface and the enthalpy of melting, both evaluated at the melting point: Si-w = -

!!!! (!! ) !!! (!! )

×Δ𝐶! . (8)

The entropy of the ice-water interface, Si-w, is always negative in the experiments and the simulations (Figure 2) because the heat capacity of liquid water is always larger than the heat capacity of ice. The mW model reproduces well the Tm of ice,35 the enthalpy of melting at Tm,26 and the iceliquid surface tension (Table 2). However, the mW model does not reproduce well ΔCp, because the monatomic model cannot account for the rotational contributions to the heat capacity of the liquid phase,44, 60 resulting in a surface entropy Si-w that is almost a quarter of that in water (Table 2). We conclude that the lack of hydrogen atoms in the mW model limits its ability to reproduce the magnitude of the surface entropy of the ice-liquid interface.


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Table 2. Thermodynamics of the ice-water interface and ice melting, and of binding graphite to ice, at the corresponding melting temperatures: 273.15 K for water,61 273 K for mW,35 252 K for TIP4P/2005, 62and 232 K for TIP4P.62 ΔHm

γi-w

ΔCp

Si-w

ΔSbind

-TΔSbind

Hi-w

(kJ mol-1)

(mJ m-2)

(J K-1 mol-1)

(µJ m-2 K-1)

(µJ m-2 K-1)

(kJ mol-1 nm-2)

(mJ m-2 K-1)

water

6.0a

32d

40.6a

-215

225

-37.0

-26.7

mW

5.3b

35

7.2b

-44

49

-8.1

23.0

TIP4P

4.39c

27.2e

22.2c

-137

145f

-20.3g

-4.6

TIP4P/2005

4.85c

28.9e

29.7c

-177

185f

-28.1g

-15.7

a)

ref 62, b) ref 44, c) ref. 62, d) average value of refs. 63-64, e) ref. 47, f) Si-w = 112 µJ m-2 K-1 for TIP4P and Si-w = 113 µJ m-2 K-1 for TIP4P/2005 are estimated from γw-v(T) provided in ref.65 We assume the contact angle for water on a graphite surface is parameterized to reproduce the experimental contact angle, θ = 86° and computed their binding entropy ΔSbind.

The binding of graphite to ice is favored by entropy for both water and the mW model at the melting temperature, because the binding involves the reduction of the entropically unfavorable ice-water interface. As mW underestimates the magnitude of Si-w, it also severely underestimates ΔSbind (Figure 2 and Table 2). The mW model, however, reproduces44 the anomalous power-law increase of the heat capacity of supercooled liquid water,66 correctly predicting the sharp downturn of Si-w and concomitant increase in the entropic driving force for binding of graphite to ice at deep supercooling (Figure 2). We conclude that the binding of graphite to ice is favored by entropy at the melting temperature and under supercooled conditions because it decreases the unfavorable iceliquid interface. The surface entropy of the ice-liquid interface has been estimated for the fully atomistic water models TIP4P/ice, TIP4P/2005 and TIP4P from ice nucleation data obtained through the seeding method.67 In that procedure, γi-w at a given temperature is deduced from size of the critical ice nuclei and the thermodynamic driving force for crystallization, assuming the validity of Classical Nucleation Theory. Further assuming that the surface tension is a linear function of temperature,


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resulted in Si-w -130 µJ m-2 K-1 for TIP4P/ice, -250 µJ m-2 K-1 for TIP4P, and -380 µJ m-2 K-1 for TIP4P/2005.67 It was interpreted that the large variability of surface entropies arise from uncertainties in the determination of the ice-liquid surface tensions.67 Eq. 8 provides an alternative method to compute Si-w. We estimate Si-w to be -137 µJ m-2 K-1 for TIP4P and -177 µJ m-2 K-1 for TIP4P/2005, at their corresponding melting points (232 and 252 K,62 respectively), using the iceliquid surface tensions of the water models at their melting points from ref.

47

and enthalpies of

melting and heat capacities (assumed to be temperature independent) from ref.

62

. While more

accurate estimates could be obtained using the heat capacities evaluated at the corresponding melting points of the models, our analysis suggests that the differences in surface entropy of the iceliquid interface of these atomistic water models at their melting points is smaller than previously reported. C. The binding of graphite to ice is favored by enthalpy in the simulations but not in experiments. The binding free energy ΔGbind has two contributions, the binding enthalpy ΔHbind and the binding entropy ΔSbind. The enthalpic term cannot be computed directly from the simulations. Here, we first compute the binding free energy of ice to graphite for mW using thermodynamic integration and estimate it for water using experimental freezing temperatures, and then determine ΔHbind = ΔGbind + TΔSbind with the entropic contribution determined in section B. The binding free energy between graphite and ice per unit area is defined as: ΔGbind = γg-i – γg-w – γi-w, (9) where γg-i, and γi-w are the surface tension of the graphite-ice and ice-water surfaces (Figure 1b). The binding free energy of an ice nucleus to graphite can be computed from the difference in the heterogeneous and homogeneous freezing temperatures, ΔTf, assuming the validity of Classical Nucleation Theory.8 ΔTf = Thet – Thom = 12 ± 2 K for mW,5 resulting in ΔGbind-nucleus = -15.5 kJ mol-1 nm-2.8 However, ΔGbind-nucleus includes two contributions: the free energy of binding per area, ΔGbind, and a contribution from the graphite-ice-water contact line:8 ΔGbind-nucleus = ΔGbind + τ × L/A, where

τ is the line tension and L the contact line of the graphite-ice-water interface and A is the binding area. To determine ΔGbind from the freezing temperature, we need to know the value of the line


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tension τ of the three phase interface. Instead, we compute the binding free energy between extended mW ice and graphite surfaces at the ice-water melting point with the mold integration method.47-48 We find ΔGbind = -19.5 kJ mol-1 nm-2. The comparison between ΔGbind and ΔGbind-nucleus indicates that τ is positive – destabilizes the binding – and between 1 and 10 pN. We determine a favorable ΔHbind = -11.4 kJ mol-1 nm-2 for the binding of graphite to mW ice, from the ΔGbind computed through thermodynamic integration and the ΔSbind of section B. Table 3 shows that both the enthalpic and entropic contributions to binding mW ice to graphite are favorable and comparable in magnitude. Table 3. Thermodynamics of binding graphite and other graphite graphitic surfaces to ice at 273 K. System εw-c

θa (kcal water

mW

ΔTf (K)

mol-1)

ΔGbind

-TΔSbind

ΔHbind

γg-i

(kJ mol-1

(kJ mol-1

(kJ mol-1

(mJ m-2)

nm-2)

nm-2)

nm-2)

graphite

86°

~0a

> -4

-37.0

> 33

> 20.1

0.13

86°

12±2b

-19.5

-8.1

-11.4

-2.1

0.20

0

19±2

-27.0

< -18.6

> -8.4

< -75.8

0.25

0

21±2

-29.4

< -18.6 c

> -10.8

< -79.8

a)

estimated from ref 20, b) from ref 7, c) for surface that induce complete wetting, the binding entropy ΔSbind computed with eq. 6 with contact angle θ = 0° is an upper limit of the true value.

The binding free energy of graphite to ice has not been measured in experiments. The poor ability of graphite to nucleate ice in experiments20 suggests that ΔGbind-nucleus ≈ 0 or larger. Assuming that τ of the graphite-water-ice interface in experiments is similar to that for the mW model, we estimate that ΔGbind > - 4 kJ mol-1 nm-2. Using the free energy and the entropy of binding derived from experimental data (Table 3), we conclude that ΔHbind must be positive, i.e. unfavorable to binding. This is opposite to what we find for the mW model. We conclude that mW does not


13


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reproduce the free energy, enthalpy or entropy of binding of graphite to ice (Table 3): it overestimates the strength of the binding free energy, underestimates the favorable entropic contribution, and predicts that the binding is favored by enthalpy while the experimental data suggests that it is not. To understand the origin of these discrepancies we examine the individual contributions of the graphite-ice, graphite-water and water-ice interfaces. We have shown in section B that the discrepancy in ΔSbind originates from the ice-liquid interface, and is a consequence of the monatomic nature of the mW model. The discrepancy in ΔGbind arises from the free energy cost of the graphiteice interface, which is predicted to be slightly favorable to binding by mW and unfavorable by the experimental data (Table 3). These comparisons point to a lack of transferability to the graphite-ice interface of the coarse-grained interaction potentials developed to reproduce the contact angle of the graphite-water interface. This limitation is intrinsic to the resolution of the model; it cannot be fixed by re-parameterization of the mW-graphite interaction without impacting the agreement in the contact angle θ. Atomistic water models that reproduce θ should reproduce the ΔSbind. It is, however, an open question whether that suffices to reproduce the surface tension of graphite-ice interface γg-i and ΔHbind with non-polarizable rigid models. D. ΔG bind changes non-monotonously with the strength of water-carbon attraction. The nucleation rate of ice on graphitic surfaces in simulations with the mW model changes non-monotonically with increasing strength of the water-carbon interaction, εw-c.15 Starting with the

εw-c that reproduces the experimental contact angle for water on graphite, 0.13 kcal mol-1,5 the nucleation rate first increases with εw-c – reaching a maximum at εw-c = 0.52 kcal mol-1 – before plunging to a minimum at εw-c = 0.78 kcal mol-1, wherefrom the nucleation rate increases again with

εw-c.15 We compute ΔGbind for surfaces with εw-c = 0.20 and 0.25 kcal mol-1 with the mold integration method and find that ΔGbind becomes more negative with increasing εw-c (Table 3), consistent with the increase in nucleation rates upon this change in interactions in ref. 15 and the increase in freezing efficiencies in ref.

7

and Table 3. If the line tension τ of the ice-graphite-water interface were

insensitive to εw-c, then the binding free energy should present a local minimum at εw-c = 0.52 kcal mol-1, where the nucleation rate is maximum, followed by a local maximum at εw-c = 0.78 kcal mol-1, where the nucleation rate is minimum. In what follows we provide evidences that this non


14

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monotonous trend in the free energy arises from opposing trends in the change in enthalpy and entropy of binding with the strength in the water-carbon attraction. Increasing εw-c results in enhanced order of mW water in the first contact layer with the surface15 and increased layering of water at the interface.7 This implies that the surface entropy of the surface-water interface becomes more negative with increasing εw-c, consistent with the predictions based on the hydrophilicity of the surface (eq. 4). The decrease in the water-graphite surface entropy leads to larger entropy gains on binding (Table 3), as water is released from a structured state at the surface to the bulk of the liquid. Based on the structural evolution of the first layer of water in contact with graphite in the liquid and crystal states,7, 15 we expect the entropy of binding to increase monotonously with εw-c up to about 0.75 kcal mol-1. Beyond that point, the water molecules in the first contact layer are constrained so strongly that they are not released upon ice binding: they form an adsorbed monolayer that tiles the surface with pentagons in what appears to be a 32.4.3.4 pattern.15 This pattern, also called Cairo pentagonal tiling or σ phase, has been previously reported for a confined ice bilayer under lateral compression.68 As the first layer water pentagons passivates the graphitic surface,15, nucleation of ice on graphitic surfaces with εw-c ≥ 0.78 kcal mol-1 starts on the second layer of interfacial water. We propose that because that first ordered layer of interfacial water is not released when ice binds to graphitic surfaces with εw-c ≥ 0.78 kcal mol-1, the binding entropy ΔSbind should decrease sharply upon the formation of the passivating layer, and then slowly increase again with further increase in εw-c as the 2nd layer becomes more structured.15 This would result in a nonmonotonous, almost discontinuous, behavior of the entropy of binding as the effective surface where the ice binds changes from graphitic to a monolayer tiled with irregular water pentagons. We have shown that the free energy of binding changes non-monotonously with εw-c, first decreasing for εw-c < 0.52 kcal mol-1 and then increasing for 0.52 < εw-c < 0.78 kcal mol-1, while ΔSbind must increase monotonously in all this range. Then, ΔHbind must increase, albeit slower than ΔSbind, when εw-c increases in this range. This implies that the enthalpy of the liquid decreases faster with increasing εw-c than the enthalpy of the ice, destabilizing the binding of ice to the surface upon increase in the water-carbon attraction.


15


Page 16 of 22

Moreover, the ice nucleation rate at the surface tiled with the crystal of pentagons (εw-c = 0.78 kcal mol-1) is 106 m-3s-1 slower than at the graphite surface (εw-c = 0.13 kcal mol-1).15 As a surface tiled with water must have lower contact angle than graphite, eq. 4 and eq. 6 indicate it must have more favorable entropy of binding to ice. We conclude that the enthalpy of binding of graphite to the surface passivated with a monolayer of pentagons is less favorable than to the bare graphitic surface.

4. Conclusions In this work we use thermodynamics and data from experiments and simulations to compute the free energy, enthalpy and entropy for the water-graphite interface. We find that the surface free energy of water at the graphite interface is negative but very small, i.e. water at this surface has stability comparable to that in the bulk liquid. This is a result of the contact angle of water on graphite θ being very close to 90o, and is well reproduced by the mW model. In both the model and experiments, the small negative free energy of the graphite-water interface is favored by enthalpy and disfavored by entropy. Nevertheless, both these components are small, from which we conclude that liquid water at the graphite interface is not ice-like or vapor like: it has thermodynamics similar to that in the bulk liquid. The mW model parameterized to reproduce the contact angle of water on graphite, reproduces well the thermodynamics of the water-graphite interface, mostly because the small value of cosθ conceals a large underestimation in the entropy of the liquid-vapor interface Sw-v by the mW model. The underestimation of Sw-v by mW arises from the resolution of the model, due to its lack of rotational degrees of freedom, and cannot be improved through re-parameterization. This implies that monatomic or coarser water models should not reproduce the thermodynamics of liquid water in contact with strongly hydrophilic or strongly hydrophobic surfaces. On the other hand, our analysis suggest that fully atomistic models, which typically reproduce well the surface entropy and free energy of the water-vapor interface, should correctly capture the interfacial thermodynamics of liquid in contact with surfaces parameterized to reproduce the experimental contact angles. Ice can be heterogeneously nucleated by graphitic surfaces in simulations with the mW model.5-7, 9, 14-18 It has been shown that an increase in the mW water-carbon attraction results in a non-monotonous trend in the ice nucleation rates.15 This implies8 that the free energy of binding of


16

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mW ice to graphitic surfaces is a non-monotonous function of the water-surface attraction. Our analysis indicates that this non-monotonic behavior results from opposing trends of the entropic and enthalpic contributions: an increase in water-carbon attraction decreases more the enthalpy of liquid water than ice, disfavoring the binding, but it also orders more the first layer of water at the interface, leading to larger entropy gains upon ice binding. Contrary to the simulations with the mW model, experiments suggest that graphite is a poor ice-nucleating surface.19-22 This implies that while the binding free energy of graphite to ice is negative for mW, it is almost zero or higher for water. Our analysis suggests that the discrepancy arises from the inability of mW to reproduce the free energy of the ice-graphite interface, despite reproducing the free energy of the water-graphite interface. This lack of transferability of the model cannot be fixed by re-parameterization of the water-carbon interactions; is intrinsic to the coarse resolution of the model. As accurate predictions of the binding free energy to ice are key for the forecast of heterogeneous freezing temperatures of surfaces,8 it would be important to determine in future work whether reproducing the contact angle of water on a surface is a sufficient condition for a fully atomistic model to capture the thermodynamics of binding of the surface to ice.

Supporting Information. Radii of the mold wells (Fig. S1) and scanned values of water-well attraction (Table S1) in the Mold integration method, distribution of order parameter q3 for simulations of ice at the graphite surface and in bulk (Fig. S2), and the dependence of the contact angle of liquid water on graphite in the coarse-grained simulations (Fig. S3).

Acknowledgements. We thank Paul DeMott and Alexei Kiselev for valuable discussions and explanations on the experimental measurements of ice nucleation on graphitic surfaces. This work was supported by the National Science Foundation through award CHE-1305427 “Center for Aerosol Impacts on Climate and the Environment”. We acknowledge the Center for High Performance Computing at The University of Utah for technical support and a grant of computer time.


17


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(62) Vega, C.; Abascal, J. L. F. Simulating Water with Rigid Non-Polarizable Models: A General Perspective. Physical chemistry chemical physics : PCCP 2011, 13, 19663. (63) Ketcham, W.; Hobbs, P. An Experimental Determination of the Surface Energies of Ice. Philos. Mag. 1969, 19, 1161-1173. (64) Hardy, S. A Grain Boundary Groove Measurement of the Surface Tension between Ice and Water. Philos. Mag. 1977, 35, 471-484. (65) Sakamaki, R.; Sum, A. K.; Narumi, T.; Yasuoka, K. Molecular Dynamics Simulations of Vapor/Liquid Coexistence Using the Nonpolarizable Water Models. J. Chem. Phys. 2011, 134, 124708. (66) Angell, C.; Shuppert, J.; Tucker, J. Anomalous Properties of Supercooled Water. Heat Capacity, Expansivity, and Proton Magnetic Resonance Chemical Shift from 0 to-38%. J. Phys. Chem. 1973, 77, 3092-3099. (67) Espinosa, J. R.; Sanz, E.; Valeriani, C.; Vega, C. Homogeneous Ice Nucleation Evaluated for Several Water Models. J. Chem. Phys. 2014, 141, 18C529. (68) Johnston, J. C.; Kastelowitz, N.; Molinero, V. Liquid to Quasicrystal Transition in Bilayer Water. J. Chem. Phys. 2010, 133, 154516.

TOC Graphic bulk water graphite ice

entropy driven bulk water water molecules

graphite ice


22

Is Water at the Graphite Interface Vapor-like or Ice-like? - The Journal

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