Angle-dependent integral equation theory improves results of thermodynamics and structure of rose water model

Abstract

Orientation-dependent integral equation theory (ODIET) was applied to the rose water model. Structural and thermodynamic properties of water modeled with the rose model were calculated using ODIET and compared to results from orientation-averaged integral equation theory (IET) and Monte Carlo simulations. Rose water model is a simple two-dimensional water model where molecules of water are represented as Lennard–Jones disks with explicit hydrogen bonding potential in form of rose functions. Orientational dependency significantly improves IET, as the thermodynamic results obtained using ODIET are significantly more in agreement with results calculated using MC than in the case of the orientationally averaged version. At high temperatures, the agreement between the simulation and theory is quantitative; however, when temperatures lower, a slight deviation between results obtained with different methods appear. ODIET correctly predicts the radial distribution function; moreover, ODIet also enables the calculation of angular distributions. While the angular distributions obtained with ODIET are in qualitative agreement with distributions from MC simulations, the height of the peaks in angular distributions differs between methods. Using results from ODIET, the spatial distribution of water molecules was constructed, which aids in the interpretation of other structural properties. ODIET was also used to calculate fractions of molecules with different number of hydrogen bonds, which is in the agreement with the simulations. Overall, use of ODIET significantly improves the obtained results in comparison to standard IET.

I. INTRODUCTION

It is hard to find a substance that is more important in various systems than water. It has major role in living organisms. In industry, water is commonly used as a solvent due to its outstanding solvation properties and abundance in nature, which makes it the cheapest solvent. Moreover, water is also crucial in biological systems, both as the solvent and reactant in different biochemical reactions. Consequently, water has important effect on many of the systems in which it is present. Even though the molecule of water is one of the simplest molecules, its properties are not as simple. Water exhibits many anomalous properties, such as density maximum at 4 °C, negative thermal expansion coefficient, high heat capacity in liquid state, and high surface tension; these anomalous properties are also the one that makes water’s role so important in many systems.¹ The reason for water’s anomalous properties is in its ability to from strong orientation-dependent hydrogen bonds. The shape of the molecule and its tendency to form hydrogen bonds enable water to form large networks of molecules connected by hydrogen bonds; such networks are highly ordered and play important role in behavior of water.

When theoretical investigations of water and aqueous systems are conducted, different models are used to model water. There are many different models of water, which differ in their complexity and accuracy; moreover, different models can also bi-parametrized for use in different systems. In computer simulations of more complex systems where water appears as solvent two groups of models are often used, these are SPC² and TIP^3,4 models. These are relatively simple models where molecules are modeled using Lennard–Jones potential ad differently placed partial charges. Some of the models are also more complicated and consider things, such as flexible bonds and angles, polarization of molecule, etc.^5–9 Another group of water models are very simple models that are not atomistically realistic, such as Bol’s model,¹⁰ model by Smith and Nezbeda,¹¹ Mercedes–Benz model,^12,13 and rose model.¹⁴ In this work, the rose model of water is used. Rose water model is a simple water model that was made as mimic of the Mercedes–Benz model; however, the rose model was made in a such way that it is computationally less demanding and more flexible than the MB model. Both the MB and rose model represent a molecule of water as a two-dimensional Lennard–Jones disk that has added explicit hydrogen bonding potential, and the functions used in description of hydrogen bonding potential differ between models. The advantage of the models, such as MB and rose, is that they are computationally less demanding; thus, they enable us to explore wider ranges of condition in shorter time. Moreover, their two-dimensionality makes visualization of processes in the systems more simple. Another advantage of these models is also that they can be used in analytical theories, such as integral equation theory (IET)^15–19 and thermodynamic perturbation theory (TPT)^15,20–21 where they can also serve as a test ground for the development of such theories. Using Monte Carlo simulations, IET and TPT comparison between MB and rose model was made.²² Using TPT and MC, liquid vapor line and percolation line the rose model were determined,²³ and the hierarchy of anomalies was investigated.³¹ Moreover, using IET and MC properties of rose, particles in random porous media were studied.²⁴

In previous studies of rose water model where integral equation theory was used to calculate structural and thermodynamic properties, orientationally averaged version of IET was used. Orientationally averaged version of IET was successfully at predicting pair distribution functions between rose particles as the agreement with results from MC was good. However, IET was less successful with prediction of thermodynamic quantities as the agreement between quantities calculated with IET and MC was only semiquantitative and also less good than the agreement between TPT and MC. While at high temperature, orientationally averaged version of IET provides almost quantitative agreement of thermodynamics with MC results, and the results become worse as the temperature lowers.²² This happens due to orientational averaging used in IET, as in orientationally averaged version the arms that form hydrogen bonds are not fixed at angles 120° between them are.

Thus, in this study, our aim is to modify orientation-dependent version of IET (previously used for MB model) in order to be suitable for the rose model of water and then calculate structural and thermodynamic properties of water modeled by the rose model. The results from orientation-dependent IET and MC simulation were taken from previous paper.²² This paper is structured in the following way. In Sec. II, the we present the model used in this work. In Sec. III, the orientation-dependent integral equation theory is described. The results are presented and discussed in Sec. IV and later summarized in Sec. V.

II. THE MODEL

The rose water model is a simple two-dimensional model of water. The interesting name of the model comes from the use of rose functions in hydrogen bonding potential.¹⁴ According to the model, each molecule is described as a two-dimensional Lennard–Jones disk with added hydrogen bonding potential that enables the formation of hydrogen bonds,

$$U{(\overrightarrow{X_i},\overrightarrow{X_j})}_{\mathit{rose}}=U_{LJ}(r_{ij})+U_{HB}(\overrightarrow{X_i},\overrightarrow{X_j}),$$ (1)

where r_ij is the distance between centers of molecules i and j, and ${\vec{X}}_i$ and ${\vec{X}}_j$ are vectors of positions and orientations of molecules i and j. LJ part of the potential has a standard form

$$U_{LJ}(r_{ij})=4{ϵ}_{LJ}\left({\left(\frac{{\sigma }_{LJ}}{r_{ij}}\right)}^{12}-{\left(\frac{{\sigma }_{LJ}}{r_{ij}}\right)}^6\right).$$ (2)

Each molecule in interacting pair has its own independent contribution to hydrogen bonding potential, and the contribution of second molecule has no effect on contribution of the first,

$$U_{HB}(\overrightarrow{X_i},\overrightarrow{X_j})=U_{HB}(\overrightarrow{r_{ij}})+U_{HB}(\overrightarrow{r_{ji}}).$$ (3)

As a result of this independence, “half” hydrogen bonds can appear where one molecule is oriented favorable to from hydrogen bond and other molecule has unfavorable orientation for hydrogen bond formation. However, “half” bonds like this are not so common because the probability of formation of such bond is much smaller than probability to from the full hydrogen bond.

The contribution of one molecule to hydrogen bonding potential is a product of two terms, one is orientation dependent (U(θ_ij)) and the other depends on the distance between centers of interaction molecules (s(r_ij)),

$$U_{HB}(\overrightarrow{r_{ij}})=\frac{{ϵ}_{HB}}{2}*s(r_{ij})*U({\theta }_{ij}),$$ (4)

where $\overrightarrow{r_{ij}}$ is a vector between molecules i and j oriented in the body frame of molecule i, r_ij is the length of this vector, and θ_ij is the angle of orientation of the vector in body frame of molecule i. ϵ_HB is the HB energy parameter.

Orientational term of hydrogen bonding potential of molecule i depends on the position of molecule j with respect to the body frame of molecule i, or from other perspective, it depends on the orientation of molecule i with respect to the position of molecule j. As mentioned, the three-petal rose function is used in the orientational term of hydrogen bonding potential,

$$U({\theta }_{ij})=a_2{\sin }^2(3{\theta }_{ij})+a_1\sin (3{\theta }_{ij}),$$ (5)

where a₁ and a₂ are coefficients determining the angular-dependent shape of the potential by changing the contribution of two sinusoidal terms. In Cartesian coordinates, the function is written as

$$\begin{array}{ll}\hfill U_{HB}(\overrightarrow{r_{ij}})& =\frac{{ϵ}_{HB}}{2}*s(r_{ij})*\left(a_2*\frac{{(3x_{ij}^2{y}_{ij}-y_{ij}^3)}^2}{r_{ij}^6}+a_1*\frac{3x_{ij}^2{y}_{ij}-y_{ij}^3}{r_{ij}^3}\right),\hfill \end{array}$$ (6)

where $r_{ij}=\sqrt{x_{ij}^2+y_{ij}^2}$, x_ij and y_ij are cartesian coordinates of molecule j in body frame of molecule i.

Maximal value of each orientational dependent term is 1, so the maximal value of the sum of the contributions of both molecules is 2; therefore, normalization is made by adding $\frac{1}{2}$ into Eq. (4).

In distance dependent term, a double-sided cubic switching function is used

$$s(r_{ij})=\left\{\begin{array}{cc}0,\hfill & r_{ij}<r_l,\hfill \\ \frac{(r_l+2r_{ij}-3r_{HB}){(r_l-r_{ij})}^2}{{(r_l-r_{HB})}^3},\hfill & r_l\le r_{ij}<r_{HB},\hfill \\ \frac{(r_u+2r_{ij}-3r_{HB}){(r_u-r_{ij})}^2}{{(r_u-r_{HB})}^3},\hfill & r_{HB}\le r_{ij}<r_u,\hfill \\ 0,\hfill & r_u\le r_{ij},\hfill \end{array}\right.$$

where r_HB is the HB distance, and r_l and r_u are the lower and upper bounds of HB. Switching function is symmetrical, since |r_HB − r_l| = |r_HB − r_u| = r_FWHM, as r_FWHM is the “full width at half maximum” of peak formed by the function.

We chose parameters of rose potential in such way that the properties of the model resemble properties of Mercedes–Benz water model.^12,13 In further text, this parametrization of the rose model is referred to as MB parametrization of the model. The parameters of this parametrization are following: ϵ_LJ = 0.1, σ_LJ = 0.7, ϵ_HB = 1, r_HB = 1, r_FWHM = 0.2, a₁ = 0.6, and a₂ = −0.4. The Lennard–Jones part of the potential is the same as in MB model. We also use the second parametrization that we refer to as real parametrization. The difference is only in the Lennard–Jones part of the potential, the parameters in real parametrizations are ϵ_LJ = 0.2 and σ_LJ = 0.890 899. The result of this parametrization is that the minimum of LJ potential is at the length of hydrogen bond; moreover, the LJ contribution to total potential is larger than in MB parametrization.

III. THEORY

A. Monte Carlo simulations

We used Monte Carlo simulations with the Metropolis algorithm to obtain structural and thermodynamic properties of the rose water model using both parametrizations described above. Results from these simulations were used as reference points for comparisons to integral equation theories. Simulations were performed in NpT ensemble. Initially, 100 rose water particles were randomly placed in square basic cell, while minimum separation between the particles was no less than σ_LJ. Periodic boundary condition with the closest image convention were used to mimic macroscopic systems. Each simulation step consisted of attempts of one random rotation and one random translation of randomly selected molecule; therefore, in each simulation cycle, there were N translational and N rotational move attempts. To keep the constant pressure once per cycle, an attempt to change volume of the system was also made. First, 100 000 cycles were performed in order to equilibrate the system. Following was the sampling phase where ten series of 100 000 cycles were used to calculate structural and thermodynamic properties. We have run also simulations with 200 particles to make sure there was no size effect in results.

B. Orientation-dependent integral equation theory

Here, we summarize the orientation-dependent version of Wertheim’s multidensity Ornstein–Zernike (OZ) equation that we used to describe a pure Rose water model fluid. We used the same version of ODIET that was used for the description of the MB model,¹⁶ with some adjustments in order to be suitable for the rose model. The orientation-dependent multidensity OZ equation for system of rose model particles is the following:^16,25

$$\begin{array}{ll}\hfill h_{\alpha \beta }^{(W)}(r_{12},{\theta }_1,{\theta }_2)& =c_{\alpha \beta }^{(W)}(r_{12},{\theta }_1,{\theta }_2)\hfill \\ \hfill & +\frac{1}{2\pi }\int \sum _{\mu \nu }{c}_{\alpha \mu }^{(W)}(r_{13},{\theta }_1,{\theta }_3){\sigma }_{\mu \nu }\hfill \\ \hfill & \times h_{\alpha \beta }^{(W)}(r_{32},{\theta }_3,{\theta }_2)d{r}_{3}d{\theta }_3,\hfill \end{array}$$ (7)

where W indicates Wertheim’s formalism, α, β, μ, and ν indicate bonding sites of rose molecules, and σ_μν are Wertheim’s density parameters.^26,27 This equation is difficult to solve numerically; therefore, to solve it, correlation functions can be expanded in a complete set of orthogonal functions,^16,25,28

$$z_{\alpha \beta }^{(W)}(r_{12},{\theta }_1,{\theta }_2)=\sum _{m,j=-L}^L{z}_{\alpha \beta }^{(W)}(r,m,j)\exp [i(m{\theta }_1+j{\theta }_2)],$$ (8)

formally summation is done from minus to plus infinity; however, in most cases, smaller number of coefficients is needed, so L is usually less than 9. In this way, only one variable function needs to be solved numerically. Correlation functions are transformed with Fourier transform and expanded in the same set of orthogonal functions as in real space.¹⁶ The relations that connects correlation functions is Fourier and real space are

$${\widehat{z}}_{\alpha \beta }^{(W)}(k,m,j)=2\pi i^{m+j}{\int }_0^{\infty }{z}_{\alpha \beta }^{(W)}(r,m,j)J_{m+j}(kr)rdr$$ (9)

and

$$z_{\alpha \beta }^{(W)}(r,m,j)=\frac{1}{2\pi i^{m+j}}{\int }_0^{\infty }{\widehat{z}}_{\alpha \beta }^{(W)}(k,m,j)J_{m+j}(kr)kdk,$$ (10)

where J_p is the p-th order Bessel function of the first kind. In Fourier space and after integration with respect to angle ${\theta }_3^{\prime }$ OZ, Eq. (7) is written as

$$\begin{array}{ll}\hfill {\widehat{h}}_{\alpha \beta }(k,m,j)& ={\widehat{c}}_{\alpha \beta }(k,m,j)+\sum _{p=-L}^L\sum _{\mu \nu }{\widehat{c}}_{\alpha \mu }(k,m,-p){\rho }_{\mu \nu }{\widehat{h}}_{\nu \beta }(k,p,j).\hfill \end{array}$$ (11)

Instead of Wertheim’s original correlation functions $z_{\alpha \beta }^{(W)}$, partial correlation functions z_αβ that remain finite when decreasing temperature were used. The functions are connected by the following relation:

$$\rho z_{\alpha \beta }\rho ={\sigma }_{\mathit{ABC-\alpha }}{z}_{\alpha \beta }^{(W)}{\sigma }_{\mathit{ABC-\beta }},$$ (12)

where ρ is the number density of rose particles. In matrix form, Eq. (11) is written as

$$\widehat{\mathbf{t}}={\widehat{\mathbf{c}}}^{-}\mathit{\rho }[\widehat{\mathbf{t}}+\widehat{\mathbf{c}}],$$ (13)

if we solve it for $\widehat{\mathit{t}}$, we get

$$\widehat{\mathbf{t}}={(\mathbf{I}-{\widehat{\mathbf{c}}}^{-}\mathit{\rho })}^{-1}{\widehat{\mathbf{c}}}^{-}\mathit{\rho }\widehat{\mathbf{c}},$$ (14)

where $\widehat{\mathit{t}}=\widehat{\mathit{h}}-\widehat{\mathit{c}}$, $\widehat{\mathit{h}}$, and $\widehat{\mathit{h}}$ are the matrices of the form

$$\widehat{\mathit{z}}=\left(\begin{array}{cccccccc}\hfill {\widehat{z}}_{0,0}\hfill & \hfill {\widehat{z}}_{0,A}\hfill & \hfill {\widehat{z}}_{0,B}\hfill & \hfill {\widehat{z}}_{0,C}\hfill & \hfill {\widehat{z}}_{0,AB}\hfill & \hfill {\widehat{z}}_{0,AC}\hfill & \hfill {\widehat{z}}_{0,BC}\hfill & \hfill {\widehat{z}}_{0,ABC}\hfill \\ \hfill {\widehat{z}}_{A,0}\hfill & \hfill {\widehat{z}}_{A,A}\hfill & \hfill {\widehat{z}}_{A,B}\hfill & \hfill {\widehat{z}}_{A,C}\hfill & \hfill {\widehat{z}}_{A,AB}\hfill & \hfill {\widehat{z}}_{A,AC}\hfill & \hfill {\widehat{z}}_{A,BC}\hfill & \hfill {\widehat{z}}_{A,ABC}\hfill \\ \hfill {\widehat{z}}_{B,0}\hfill & \hfill {\widehat{z}}_{B,A}\hfill & \hfill {\widehat{z}}_{B,B}\hfill & \hfill {\widehat{z}}_{B,C}\hfill & \hfill {\widehat{z}}_{B,AB}\hfill & \hfill {\widehat{z}}_{B,AC}\hfill & \hfill {\widehat{z}}_{B,BC}\hfill & \hfill {\widehat{z}}_{B,ABC}\hfill \\ \hfill {\widehat{z}}_{C,0}\hfill & \hfill {\widehat{z}}_{C,A}\hfill & \hfill {\widehat{z}}_{C,B}\hfill & \hfill {\widehat{z}}_{C,C}\hfill & \hfill {\widehat{z}}_{C,AB}\hfill & \hfill {\widehat{z}}_{C,AC}\hfill & \hfill {\widehat{z}}_{C,BC}\hfill & \hfill {\widehat{z}}_{C,ABC}\hfill \\ \hfill {\widehat{z}}_{AB,0}\hfill & \hfill {\widehat{z}}_{AB,A}\hfill & \hfill {\widehat{z}}_{AB,B}\hfill & \hfill {\widehat{z}}_{AB,C}\hfill & \hfill {\widehat{z}}_{AB,AB}\hfill & \hfill {\widehat{z}}_{AB,AC}\hfill & \hfill {\widehat{z}}_{AB,BC}\hfill & \hfill {\widehat{z}}_{AB,ABC}\hfill \\ \hfill {\widehat{z}}_{AC,0}\hfill & \hfill {\widehat{z}}_{AC,A}\hfill & \hfill {\widehat{z}}_{AC,B}\hfill & \hfill {\widehat{z}}_{AC,C}\hfill & \hfill {\widehat{z}}_{AC,AB}\hfill & \hfill {\widehat{z}}_{AC,AC}\hfill & \hfill {\widehat{z}}_{AC,BC}\hfill & \hfill {\widehat{z}}_{AC,ABC}\hfill \\ \hfill {\widehat{z}}_{BC,0}\hfill & \hfill {\widehat{z}}_{BC,A}\hfill & \hfill {\widehat{z}}_{BC,B}\hfill & \hfill {\widehat{z}}_{BC,C}\hfill & \hfill {\widehat{z}}_{BC,AB}\hfill & \hfill {\widehat{z}}_{BC,AC}\hfill & \hfill {\widehat{z}}_{BC,BC}\hfill & \hfill {\widehat{z}}_{BC,ABC}\hfill \\ \hfill {\widehat{z}}_{ABC,0}\hfill & \hfill {\widehat{z}}_{\mathit{ABC,A}}\hfill & \hfill {\widehat{z}}_{\mathit{ABC,B}}\hfill & \hfill {\widehat{z}}_{\mathit{ABC,C}}\hfill & \hfill {\widehat{z}}_{\mathit{ABC,AB}}\hfill & \hfill {\widehat{z}}_{\mathit{ABC,AC}}\hfill & \hfill {\widehat{z}}_{\mathit{ABC,BC}}\hfill & \hfill {\widehat{z}}_{\mathit{ABC,ABC}}\hfill \end{array}\right),$$ (15)

where ${\widehat{\mathit{z}}}_{\alpha ,\beta }$ are matrices of dimensionality (2L + 1) × (2L + 1) defined as ${\widehat{\mathit{z}}}_{\alpha ,\beta }^{mj}={\widehat{z}}_{\alpha ,\beta }(k,m,j)$, ${\widehat{\mathit{c}}}^{-}$ is the matrix that has same dimensionality as $\widehat{c}$ and is defined as ${\widehat{c}}_{\alpha ,\beta }^{-}(k,m,p)={\widehat{c}}_{\alpha ,\beta }(k,m,-p)$. ρ is the matrix of partial densities defined as

$$\mathit{\rho }=\left(\begin{array}{cccccccc}\hfill \rho \mathit{I}\hfill & \hfill \rho \mathit{I}\hfill & \hfill \rho \mathit{I}\hfill & \hfill \rho \mathit{I}\hfill & \hfill \rho \mathit{I}\hfill & \hfill \rho \mathit{I}\hfill & \hfill \rho \mathit{I}\hfill & \hfill \rho \mathit{I}\hfill \\ \hfill \rho \mathit{I}\hfill & \hfill 0\hfill & \hfill \frac{x_2}{x_1^2}\rho \mathit{I}\hfill & \hfill \frac{x_2}{x_1^2}\rho \mathit{I}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{x_3}{x_1{x}_2}\rho \mathit{I}\hfill & \hfill 0\hfill \\ \hfill \rho \mathit{I}\hfill & \hfill \frac{x_2}{x_1^2}\rho \mathit{I}\hfill & \hfill 0\hfill & \hfill \frac{x_2}{x_1^2}\rho \mathit{I}\hfill & \hfill 0\hfill & \hfill \frac{x_3}{x_1{x}_2}\rho \mathit{I}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill \rho \mathit{I}\hfill & \hfill \frac{x_2}{x_1^2}\rho \mathit{I}\hfill & \hfill \frac{x_2}{x_1^2}\rho \mathit{I}\hfill & \hfill 0\hfill & \hfill \frac{x_3}{x_1{x}_2}\rho \mathit{I}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill \rho \mathit{I}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \frac{x_3}{x_1{x}_2}\rho \mathit{I}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill \rho \mathit{I}\hfill & \hfill 0\hfill & \hfill \frac{x_3}{x_1{x}_2}\rho \mathit{I}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill \rho \mathit{I}\hfill & \hfill \frac{x_3}{x_1{x}_2}\rho \mathit{I}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill \rho \mathit{I}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right),$$ (16)

where I is the identity matrix and x_i are ratios of partial densities and are calculated as

$$x_1=\frac{{\sigma }_{BC}}{{\sigma }_{\mathit{ABC}}}=\frac{{(1+s_1)}^2+s_2}{{(1+s_1)}^3+3s_2(1+s_1)+s_3},$$ (17)

$$x_2=\frac{{\sigma }_A}{{\sigma }_{\mathit{ABC}}}=\frac{1+s_1}{{(1+s_1)}^3+3s_2(1+s_1)+s_3},$$ (18)

$$x_3=\frac{{\sigma }_0}{{\sigma }_{\mathit{ABC}}}=\frac{1}{{(1+s_1)}^3+3s_2(1+s_1)+s_3}.$$ (19)

Physically, these parameters mean fractions of differently bonded molecules: x₁ is the fraction of non-bonded molecules at one particular arm, x₂ is the fraction of non-bonded molecules at two particular arms, and x₃ is the fraction of molecules not bonded at any arm. Since the arms of the rose model are equivalent, the following relations are true:

$$s_1=s_A=s_B=s_C,s_2=s_{AB}=s_{AC}=s_{BC},s_3=s_{\mathit{ABC}}.$$ (20)

Coefficients s_α for different bonding states can be obtained from Wertheim’s correlation function^26,27

$$\begin{array}{ll}\hfill s_{\alpha }& =\frac{1}{2\pi }\sum _{\genfrac{}{}{0pt}{}{\gamma \subset ABC}{\gamma \ne \varnothing }}\sum _{D\in \gamma }\int e_{LJ}(r)y_{\alpha -F,\gamma -D}^{(W)}\hfill \\ \hfill & \times (r,{\theta }_1,{\theta }_2)f_{FD}(r,{\theta }_1,{\theta }_2){\sigma }_{\mathit{ABC-\gamma }}d{\theta }_{1}d{\theta }_{2}rdr.\hfill \end{array}$$ (21)

Integration is done over both angles (dθ₁, dθ₂) and radius (dr) and $e_{LJ}=\exp \left(-U_{LJ}(r)/k_{B}T\right)$, T is the temperature, k_B is Boltzmann’s constant, y is the cavity function, and f_FD(r, θ₁, θ₂) is the Mayer function of potential between arms k and l, as F represents the kth arm of the first molecule and D represents the lth arm of the second molecule. Mayer function in Eq. (21) can be expanded using orthogonal functions,

$$f_{FD}(r,{\theta }_1,{\theta }_2)=\sum _{m,j=-2L}^{2L}{f}_{FD}(r,m,j)\exp [i(m{\theta }_1+j{\theta }_2)].$$ (22)

Note that total potential has to be slitted into nine different contributions between pair of arms and expanded only one pair of arms. Equation (21) can now be rewritten using functions that remain finite as temperature closes zero,

$$\begin{array}{ll}\hfill s_{\alpha }& =2\pi \sum _{\genfrac{}{}{0pt}{}{\gamma \subset ABC}{\gamma \ne \varnothing }}\sum _{D\in \gamma }\sum _{m,j=-L}^L\int e_{LJ}(r)\frac{\rho }{{\sigma }_{\mathit{ABC-\alpha +F}}}{y}_{\alpha -F,\gamma -D}(r,m,j)\hfill \\ \hfill & \times \frac{\rho }{{\sigma }_{\mathit{ABC-\gamma +D}}}{f}_{FD}(r,-m,-j){\sigma }_{\mathit{ABC-\gamma }}rdr.\hfill \end{array}$$ (23)

Using fractions x_i, it is possible to calculate fractions of molecules with different number of bonds y_i. First, we define partial density of differently bonded molecules ρ_α, which is connected to Wertheim’s density parameters as the following examples illustrate: σ_A = ρ₀ + ρ_A, σ_BC = ρ₀ + ρ_B + ρ_C + ρ_BC. Fractions x_i can be then also written using these partial densities as

$$x_1=\frac{{\sigma }_{BC}}{{\sigma }_{\mathit{ABC}}}=\frac{{\rho }_0+{\rho }_B+{\rho }_C+{\rho }_{BC}}{\rho },$$ (24)

$$x_2=\frac{{\sigma }_A}{{\sigma }_{\mathit{ABC}}}=\frac{{\rho }_0+{\rho }_A}{\rho },$$ (25)

$$x_3=\frac{{\sigma }_0}{{\sigma }_{\mathit{ABC}}}=\frac{{\rho }_0}{\rho }.$$ (26)

Fractions of molecules with different number of bonds y_i can then be calculated as

$$y_0=\frac{{\rho }_0}{\rho }=x_3,$$ (27)

$$y_1=\frac{{\rho }_A+{\rho }_B+{\rho }_C}{\rho }=3\frac{{\rho }_A}{\rho }=3(x_2-x_3),$$ (28)

$$y_2=\frac{{\rho }_{A}B+{\rho }_{B}C+{\rho }_{C}A}{\rho }=3(x_1+x_3-2x_2),$$ (29)

$$y_3=1-y_0-y_1-y_2.$$ (30)

To numerically solve equations, a closure condition is needed. In this work, we used polymer soft mean-spherical approximation (PSMA). According to this closure, the LJ potential is divided into a short-range reference part U₀(r) and a longer-range perturbation part U₁(r),²⁹

$$U_{LJ}(r)=U_0(r)+U_1(r),$$ (31)

$$U_0(r)=\left\{\begin{array}{cc}{U}_{LJ}(r)+{ϵ}_{LJ},\hfill & r\ge r_m,\hfill \\ 0,\hfill & r<r_m,\hfill \end{array}\right.$$

$$U_1(r)=\left\{\begin{array}{cc}-{ϵ}_{LJ},\hfill & r\ge r_m,\hfill \\ U_{LJ}(r),\hfill & r<r_m,\hfill \end{array}\right.$$

where distance r_m is the position of minimum of the LJ part of interacting potential. PSMSA closure relation is the following:

$$\begin{array}{ll}\hfill c_{\alpha ,\beta }^{(W)}(r,{\theta }_1,{\theta }_2)& =f_0(r)y_{\alpha ,\beta }^{(W)}(r,{\theta }_1,{\theta }_2)\hfill \\ \hfill & +e_{LJ}(r)\sum _{D\in \alpha }\sum _{E\in \beta }{f}_{DE}(r,{\theta }_1,{\theta }_2)y_{\alpha -D,\beta -E}^{(W)}\hfill \\ \hfill & \times (r,{\theta }_1,{\theta }_2)(1-{\delta }_{\alpha ,0})(1-{\delta }_{\beta ,0}),\hfill \end{array}$$ (32)

where e_LJ(r) = exp(−U_LJ(r)/k_BT), f₀(r) = exp(−U₀(r)/k_BT) − 1, and δ_α,0 is the delta function. If closure is expanded using orthogonal functions, the following from is obtained:

$$\begin{array}{ll}\hfill c_{\alpha ,\beta }(r,m,j)& =f_0(r)y_{\alpha ,\beta }(r,m,j)+e_{LJ}(r)\sum _{D\in \alpha }\sum _{E\in \beta }\sum _{p,q=-L}^L\hfill \\ \hfill & \times \frac{{\sigma }_{\mathit{ABC-\alpha }}}{{\sigma }_{\mathit{ABC-\alpha +D}}}\frac{{\sigma }_{\mathit{ABC-\beta }}}{{\sigma }_{\mathit{ABC-\beta +E}}}{f}_{DE}\hfill \\ \hfill & \times (r,-p+m,-q+j)y_{\alpha -D,\beta -E}(r,p,q)\hfill \\ \hfill & \times (1-{\delta }_{\alpha ,0})(1-{\delta }_{\beta ,0}),\hfill \end{array}$$ (33)

in case of α = 0 and β = 0, the closure has different from

$$c_{0,0}(r,m,j)=f_0(r)y_{0,0}(r,m,j)-(f_0(r)+1)\frac{U_1(r)}{k_{B}T}.$$ (34)

The system of equations is solved by the direct iteration method. Forward and inverse Basel–Fourier transforms were performed by the method of Talman.³⁰ To obtain orientationally averaged total correlation function, we sum (0,0) harmonics of the partial correlation functions,

$$g(r)=\sum _{\alpha ,\beta }{h}_{\alpha ,\beta }(r,0,0)+1.$$ (35)

Higher coefficients of total pair correlation function are obtained as

$$\begin{array}{ll}\hfill g(r,m,j)& =\sum _{\alpha ,\beta }{g}_{\alpha ,\beta }(r,m,j)\hfill \\ \hfill & =\sum _{\alpha ,\beta }(h_{\alpha ,\beta }(r,m,j)+{\delta }_{\alpha ,0}{\delta }_{\beta ,0}{\delta }_{m,0}{\delta }_{j,0}),\hfill \end{array}$$ (36)

finally, the pair distribution function g(r, θ₁, θ₂) is calculated as

$$g(r,{\theta }_1,{\theta }_2)=\sum _{m,j=-L}^{L}g(r,m,j)\exp [i(m{\theta }_1+j{\theta }_2)].$$ (37)

When pair distribution functions are calculated, thermodynamic properties can also be calculated. Again, as in the case of correlation functions, it is useful to expand quantities in orthogonal functions

$$\begin{array}{ll}\hfill & e_{FD}(r,{\theta }_1,{\theta }_2)U_{HB}^{kl}(r,{\theta }_1,{\theta }_2)\hfill \\ \hfill & =\sum _{m,j=-2L}^{2L}{E}_{FD}(r,m,j)\exp [i(m{\theta }_1+j{\theta }_2)].\hfill \end{array}$$ (38)

Reduced internal energy per particle U/(Nk_BT) is calculated as

$$\begin{array}{ll}\hfill \frac{U}{N{k}_{B}T}& =1+\frac{\pi \rho }{k_{B}T}{\int }_0^{\infty }{U}_{LJ}(r)g(r)rdr\hfill \\ \hfill & +\frac{\pi \rho }{k_{B}T}\sum _{\genfrac{}{}{0pt}{}{\alpha \subset ABC}{\alpha \ne \varnothing }}\sum _{D\in \alpha }\sum _{\genfrac{}{}{0pt}{}{\gamma \subset ABC}{\gamma \ne \varnothing }}\sum _{D\in \gamma }\sum _{m,j=-L}^L\hfill \\ \hfill & \times {\int }_0^{\infty }{E}_{FD}(r,-m,-j)g_{\alpha -F,\gamma -D}(r,m,j)\hfill \\ \hfill & \times \frac{\rho }{{\sigma }_{\mathit{ABC-\alpha +F}}}\frac{\rho }{{\sigma }_{\mathit{ABC-\gamma +D}}}{\sigma }_{\mathit{ABC-\alpha }}{\sigma }_{\mathit{ABC-\gamma }}rdr.\hfill \end{array}$$ (39)

Pressure was calculated via compressibility as

$$\begin{array}{ll}\hfill \frac{p^c}{\rho k_{B}T}& =1-\frac{\pi }{\rho }{\int }_0^{\infty }{(\mathit{\rho }\mathit{c}(r)\mathit{\rho })}_{L+1,L+1}rdr+\frac{1}{2\pi \rho }\sum _{k=1}^{8(2L+1)}\hfill \\ \hfill & \times {\int }_0^{\infty }\left(\frac{\frac{1}{2}{\widehat{B}}_k^2(k)}{1-{\widehat{B}}_k(k)}+{\widehat{B}}_k(k)+\log (1-{\widehat{B}}_k(k))\right)kdk,\hfill \end{array}$$ (40)

where ${\widehat{B}}_k(k)$ are eigenvalues of the matrix $\mathit{B}\to (k)={\widehat{\mathit{c}}}^{-}\mathit{\rho }$. Standard thermodynamic equations were used to calculate other thermodynamic quantities.

IV. RESULTS AND DISCUSSION

All results are presented in reduced units, reduced in relation to the strength and length of an HB interaction. In this reduction, the HB energy parameter ϵ_HB was used to normalize temperature and excess internal enthalpy ($A^{*}=\frac{A}{|{\epsilon }_{HB}|},T^{*}=\frac{k_B*T}{|{\epsilon }_{HB}|}$), while all distances were normalized with the characteristic length of the hydrogen bond r_HB ($r^{*}=\frac{r}{r_{HB}}$). All results are obtained at pressure p* = 0.19 unless written otherwise.

A. Distribution functions

We start with the investigation of structural properties of water, as the structure affect other properties (for example thermodynamic properties) of water. The most basic quantity that provides us information about the structure of the liquid is the radial distribution function. Both orientation-dependent and orientation-averaged IET enable the calculation of radial distribution function g(r*), while orientation-dependent Iet also enables the calculation of angular-dependent pair distribution function g(r*, θ₁, θ₂). In Fig. 1, the radial distribution function between water molecules modeled with rose water model is shown. The distribution functions were calculated both using orientation-averaged (blue line) and orientation-dependent (black line) IET; moreover, results from IET are compared to radial distribution functions obtained from MC simulations (red line). In Fig. 1(a), MB parametrization of the model is used, and in Fig. 1(b), real parametrization of the model is used. The ODIET reproduces the long order of pair correlation function better.

FIG. 1.

Radial distribution function between water molecules modeled with the rose model with MB parametrization. Functions are plotted for different temperatures and calculated by different methods. Red lines show results from MC simulations, blue lines show results from orientation-averaged integral equation theory, and black lines show results from orientation-dependent integral equation theory. In (a), MB parametrization of the model is used, and in (b), real parametrization is used.

In case of MB parametrization, we have several peaks that correspond to different types of interactions between water molecules. The first peak in RDF (at distance 0.7) corresponds to two molecules in direct LJ interaction, the second peak (at distance 1.0) corresponds to two molecules connected with direct hydrogen bond, and the third peak (at 1.7) corresponds to two molecules between which there is a third molecule of water and the two molecules are connected to the third molecule (which is in the middle) by hydrogen-bonding. From Fig. 1, it is seen that both versions of IET are quite successful at prediction of radial distribution functions; however, the results between versions differ. Both versions correctly predict the positions of most of the peaks in RDF, while the height of peaks is different. The height of the first peak predicted by both versions of IET is similar; however, the height is higher than that calculated using MC. On the other hand, the height of the second and all later peaks is higher when calculated with MC instead of IET. The agreement between the height of the second peak calculated using MC and IET is better in case of orientation-averaged IET, as the peak is higher than the one calculated with ODIET. However, orientation-dependent IET is better at predicting the height of more distanced peaks. The reason is in different approximation. The IET is done in ideal network approximation that neglects the formation of rings and ring of rings and reproduces the first peak better.

In case of real parametrization, peaks that correspond to hydrogen-bonding interaction and LJ interaction are joined into one peak because the length of HB and minimum of LJ potential are at the same distance. Similarly, as in case of MB parametrization, orientation-dependent IET is less successful at predicting the height of the peak corresponding to direct hydrogen bond than orientation-averaged IET. Moreover, orientation-averaged version also predicts the height of more distant peaks better than orientation-dependent version. However, when it comes to the depth of minimums in RDF, ODIET results are more in agreement with MC results.

Orientation-dependent IET provides even more information about structure of liquid in comparison to orientation-averaged version as angular distributions can also be calculated. In Figs. 2–5, the angular distribution cuts of distribution function g(r*, θ₁, θ₂) are shown at different radial lengths and fixed orientations of the first molecule at angles 0° (red line), 30° (blue line), and 60° (green line). Results from ODIET are plotted in full line, while results from simulation are plotted in dashed line.

FIG. 2.

Angular distribution cuts of g(r*, θ₁, θ₂), at r* = 0.7 (a) and (b), θ₁ =: 0° (red line), 30° (blue line), and 60° (green line). With full line, results from IET are plotted, and with dashed line, results from MC are plotted. Results for MB parametrization (at temperature 0.20) are shown.

FIG. 5.

Angular distribution cuts of g(r*, θ₁, θ₂), at r* = 2.0 (a) and (b), θ₁ ≕ 0° (red line), 30° (blue line), and 60° (green line). With full line, results from IET are plotted, and with dashed line, results from MC are plotted. In (a), results for MB parametrization (at temperature 0.20) are shown, and in (b), results for real parametrization (at temperature 0.25) are shown.

Figure 2 shows the angular distribution cut at a radial distance of 0.7 between molecules of the rose model with MB parametrization. First, we notice that the overall average height of the function is higher in case of IET, which is in agreement with RDF in Fig. 1 as the first peak in RDF is higher when calculated with IET instead of MC. Next, regardless of the orientation of the first molecule, most favorable angle of orientation of the second molecule is 0°. This means that, at distance 0.7, most favorable orientation of second molecule is when HB arm is directed away from the first molecule’s center; moreover, this peak is the highest when the orientation angle of the first molecule is also 0°. Therefore, the most favorable orientation of molecules is when arms of both molecules are directed away from each other’s centers. It is interested to note that, in case of angle 30°, the most probably orientation in MC is no longer 0°, but around 20°, in this orientation, it is higher probability for both molecules to form HBs with other molecules.

In Fig. 3, the angular distribution cut at a radial distance of 1.0 is shown for both parametrizations of the model. At this distance, ODIET predicts the angular distribution quite good, as the agreement with the MC results is semi quantitative. The most probable orientation of molecules in pair is by far the orientation when the angle of the first molecule is 0° and the angle of the second one is −60° or 60°. This configuration happens when arms of molecules are directed toward each other’s centers. Moreover, it is evident from Fig. 3 that regardless the orientation of the first molecule, the most probable orientation of the second molecule is at angle −60° or 60°; this means that orientation in which an arm of the second molecule is pointed toward the first molecule is the most favorable orientation of the molecule regardless the orientation of the other molecule in pair. This result is a consequence of the fact that, in orientational part of rose potential, each molecule has independent contribution to potential. The agreement between distributions calculated with ODIET and MC quite good. However, the shape of distribution at θ₁ = 0° around θ₂ = 0° predicted by ODIET is different than the one predicted by MC. MC predicts a local maximum at θ₂ = 0°, while ODIET predicts local minimum at θ₂ = 0° and two symmetrical local maxima at θ₂ = +/−14°. This difference between ODIET and MC result is larger in case of real parametrization. The peaks at θ₂ = 0° obtained by MC simulations of the rose model with real parametrization are much more prominent than in the case of MB parametrization that means that, at real parametrization, the configuration of molecules where first molecule’s arm is pointed toward the center of the second molecule, and the second molecule’s arm is pointed away from the first molecule is quite common.

FIG. 3.

Angular distribution cuts of g(r*, θ₁, θ₂), at r* = 1.0 (a) and (b), θ₁ ≕ 0° (red line), 30° (blue line), and 60° (green line). With full line, results from IET are plotted, and with dashed line, results from MC are plotted. In (a), results for MB parametrization (at temperature 0.20) are shown, and in (b), results for real parametrization (at temperature 0.25) are shown.

In Fig. 4, the same distributions are shown as in Fig. 3 except that the radial distance is r* = 1.73. In case of both parametrizations, the most probable orientation of two molecules at distance 1.73 is when the first molecule is at angle 30° and the second molecule is also at angle 30°. This configuration appears when the third molecule is between the first and the second one and all three molecules are connected with hydrogen bonds. At θ₁ = 30°, there is another peak that is predicted by simulations; this peak is at θ₂ = −30°; in case of real parametrization, this peak is more prominent than in the case of MB parametrization. ODIet also predicts this peak at real parametrization; however, its height is much smaller as predicted by MC.

FIG. 4.

Angular distribution cuts of g(r*, θ₁, θ₂), at r* = 1.73 (a) and (b), θ₁ ≕ 0° (red line), 30° (blue line), 60° (green line). With full line, results from IET are plotted, and with dashed line, results from MC are plotted. In (a), results for MB parametrization (at temperature 0.20) are shown, and in (b), results for real parametrization (at temperature 0.25) are shown.

In Fig. 5, the same distributions are shown as in Figs. 3 and 4 except the radial distance is 2.0. Here, ODIET qualitatively predicts distributions for both parametrizations; however, amplitudes of distributions differ between IET and MC. According to MC simulations, the most probable orientation of molecules at distance 2.0 is when both molecules are at the angle 0°. Another quite prominent peak in Fig. 5 is at angles θ₁ = 60° and θ₂ = 0°. This peak corresponds to two molecules that are on the opposite sites of hexagonal ring that is formed when six molecules are connected to each other with hydrogen bonds.

Overall, it can be said that ODIET semi-quantitatively predicts angular distributions of molecules at different radial distances. In some cases, the agreement between ODIET and MC distributions is very good, while in some cases, the theory fails. However, in all cases, ODIET predicts the most significant peaks and the order of magnitude of distributions obtained by both methods is the same.

Next, we used the orientation-dependent integral equation theory to calculate the spatial distribution of water molecules around one water molecule. In Fig. 6, these spatial distributions are shown for both parametrizations of the rose model. Brighter more yellow color indicates higher probability of finding a molecule in selected place and darker colors represent lower probability. The general shape of the spatial distribution is similar for both parametrizations. Both distributions have a circle in the middle where the probability of finding another molecule is 0. Next close to the circle are three areas with the highest probability of finding a molecule, and these are positions of rose molecules directly bonded with hydrogen bonds to the reference molecule. There are areas that are seen on the angular distribution function cut at distance 1 (Fig. 3) as the angles with the highest probability. In Fig. 6, it is seen that the probability at these areas is higher in case of MB parametrization of the model that has also been seen in Fig. 3. Between high probability areas, there are areas with lower probability; however, the probability there is still more than on average. In this position, molecules that are in LJ contact with the reference molecule are located, and these molecules are not connected with hydrogen bonds to the reference molecules; therefore, their probability is lower than the probability of those connected with hydrogen bonds. If we look more closely at this area, we notice that they are different for MB and real parametrizations. At real parametrization, at the same radial distance from the center of the reference molecule, these LJ contact areas have higher probability than in the case of MB parametrization; however, MB parametrization has a higher probability at the same polar angles but a little closer to the center of the reference molecule. This is exactly what we also saw in Figs. 2 and 3, where at θ₁ = 60° and θ₂ = 0°, both parametrizations have local maximum; however, in the case of real parametrization, this maximum is higher. On the other hand, MB parametrization has the maximum at the same angles at radial distance 0.7. This happens because LJ contact distance is pushed from radial distance 0.7 to distance 1.0 in case of real parametrization. The areas with the lowest probability (besides excluded volume of the reference molecule) are at the same angles and a little larger radial distance as the areas with the highest probability. These low probability areas appear on the edge of molecules directly bonded to the reference molecule and are consequence of excluded volume by repulsing part of LJ potential. If we look spatial distributions as a whole, we can see that there is a sort of hydration shells at different distances around the reference molecule with higher probability, and between these shells, there are shells with lower probability. The reason for this shell shape is the formation of hydrogen bonds between rose molecules; in Fig. 7(b), the ideal network of hydrogen bond is drawn onto the spatial distribution so that we can easier imagine why there is a higher probability at certain areas and lower probability at other areas. From the analysis of angular distributions and spatial distribution, it is clear that the ability of the rose water model to form half hydrogen bonds increases orientational ordering in the system in comparison to a similar MB model.

FIG. 6.

Spatial distribution function of g(r*, θ₁). In (a), results for MB parametrization (at temperature 0.20) are shown, and in (b), results for real parametrization (at temperature 0.25) are shown.

FIG. 7.

(a) Diagram of cluster of 7 2D water particles shown, and relative orientations of molecules described in the text are visualized using this diagram. (b) Ideal lattice of rose water molecules connected with hydrogen bonds drawn onto polar probability distribution of water molecules around one water molecule. Molecules connected with hydrogen bonds are drawn with thicker lines and molecules that are not connected with hydrogen bonds are drawn with thinner lines.

Using Fig. 7, we can also easily imagine and explain peaks in angular distribution functions. Here, we will connect the highest probability peaks of angular distribution functions to structures in the hydrogen bonding network. At r* = 0.7, all maximums in angular distribution Fig. 2 are at θ₂ = 0° regardless the value of θ₁. This happens due to situation similar to the one shown in Fig. 7(a) where θ₁ is angle of particle 7 and θ₂ is angle of particle 4. At r* = 1.0, the highest maximum in angular distribution in Fig. 3 is at θ₁ = 0° and θ₂ = +/−60°. This corresponds to direct hydrogen bond and is seen between particles 2 and 3 in Fig. 7(a); many such configuration can be easily located on Fig. 7(b). At r* = 1.73, the highest maximum in angular distribution in Fig. 4 is at θ₁ = 30° and θ₂ = 30°. Such orientation is between particles 1 and 3 in Fig. 7(a). In case of real parametrization, there is a quite high peak at θ₁ = 30° and θ₂ = −30°; this peak corresponds to the configuration of two molecules that are in centers of neighbor hexagons. In case of MB model,¹⁶ this peak was not present, but here it is because the rose model can form half hydrogen bonds; therefore, the orientation of molecule inside of hexagon will not be random because the molecule can form half bonds with molecules of hexagon, while molecules of hexagon do not bond to the molecule inside. At r* = 2.0, one of the maximums in angular distribution (Fig. 5) is at θ₁ = 60° and θ₂ = 0° represents orientation between particles 1 and 4 in Fig. 7. The peak at θ₁ = 0° and θ₂ = 0°, which is quite high in case of real parametrization, represents the configuration between the central molecule and molecule inside of the hexagon on the right of the central molecule in Fig. 7(b), and this is the same configuration as for the peak at θ₁ = 0° and θ₂ = +/−60°; only the orientation of molecule in hexagon is different. In case of MB parametrization, these two peaks are not as high because the molecule in hexagon can move due to additional space. Furthermore, these two peaks were not observed in MB model¹⁶ because the MB model cannot form half bonds; therefore, the orientation of MB model inside hexagon is more random. The most probable orientations at distance r* = 1.73 and r* = 2.0 are typical for hexagonal structure of hydrogen bonding network.

B. Thermodynamic properties

Before we look at thermodynamic properties of the rose model and how well orientation-dependent integral equation theory predicts them, let us investigate hydrogen bonds in the system. Hydrogen bonds are kind of bridge between structural and thermodynamic properties. The structure of the system is greatly affected by presence of hydrogen bonds in the system. Moreover, due to relatively high energy of hydrogen bonds and also due to their effect on structure of the system, hydrogen bonds also significantly affect thermodynamic properties of the system.

In Fig. 8, temperature dependence of ratios of differently bonded molecules is shown for both parametrizations of the rose model. With points, results from MC simulations are shown, and with full lines, results from ODIET are shown. The agreement between rations calculated using theory and simulations is very good. The deviation between methods at moderate and high temperatures probably appears mostly because, in simulations, hydrogen bond is not straightforward to define. The energy of hydrogen bond depends on orientation of both interacting molecules and distance between them, so if hydrogen bond is slightly bended or slightly longer, then ideally the energy of the bond is lower. Therefore, an energy cut-off of minimal energy at which the interaction is still considered hydrogen bond must be chosen. Here, this energy cut-off was set to 0.4 of ideal hydrogen bond energy. At low temperatures, where solid state appears, the methods give different results because this version of integral equation is not optimized for solid state. Temperature dependence of ratios of differently bonded molecules is similar for both parametrizations, while in case of real parametrizations, all ratios are shifted toward high temperatures in comparison to MB parametrization. The ratio of molecules that form three hydrogen bonds is monotonically decreasing from low temperatures, and at temperature 0.3 (MB parametrization) or 0.4 (real parametrization) becomes zero. The ratio of molecules with two hydrogen bonds increases with increasing temperature and reaches maximum at temperature 0.15 (MB parametrization) or 0.20 (real parametrization); after maximum, this ratio gradually decreases toward zero. Ratio of molecules with one hydrogen bond has maximum too, but at higher temperatures than ratio of once bonded molecules. At high temperatures, the ratio of molecules with two hydrogen bonds decreases, but even at temperature 1 does not reach zero. Finally, the ratio of molecules with zero hydrogen bonds monotonically increases in whole temperature range as shown in Fig. 8.

FIG. 8.

Fractions of differently bonded molecules as function of temperature. Points indicate results from MC simulations and lines results from orientation-dependent integral equation theory. In (a), MB parametrization of model is used, and in (b), real parametrization is used.

Next, we investigated different thermodynamic quantities at pressure 0.19. Starting with enthalpy as function of temperature shown in Fig. 9. The trend is similar for both parametrizations; with increasing temperature, enthalpy increases. At low temperatures, there are more hydrogen bonds and other favorable interactions; thus, at lower temperature, enthalpy is lower and with increasing temperature it rises. In Fig. 9(b), MC results also indicate melting point above temperature 0.2, as the trend of increasing enthalpy changes at this temperature. In Fig. 9(a), we do not see this change of trend because the melting point is around temperature 0.15. However, the results at temperatures below melting point are not so important for this study because versions of IET used are not optimal for solid state. Orientation-averaged version of IET correctly predicts the trend of enthalpy change with temperature, but as temperature decreases, the agreement between orientation-averaged IET and MC becomes poorer. The orientation-dependent version of IET gives much better results than averaged version, as the agreement in enthalpy calculated with ODIET and MC is very good across whole temperature range where results from ODIET were obtained. The temperature range of ODIET is limited at low temperatures because convergence of the algorithm used in calculations could no longer be reached.

FIG. 9.

Excess enthalpy as function of temperature. Red points indicate results from MC simulations, blue line indicates results from orientation-averaged integral equation theory, and black lines indicate results from orientation-dependent integral equation theory. In (a), MB parametrization of model is used, and in (b), real parametrization is used.

Figure 10 shows excess heat capacity as function of temperature for both parametrizations of the model. According to MC results, heat capacity decreases with increasing temperature after melting point. In Fig. 10(b), very high peak is seen at temperature a little above 0.2; this vast increase in heat capacity happens due to phase transition from solid to liquid state. Similarly, as in case of enthalpy, orientation-averaged IET predicts general trend of how heat capacity changes with temperature (above melting point); however, the agreement with MC results is only qualitative. Use of orientation-dependent version of IET significantly improves obtained heat capacity, as the agreement with MC results is almost quantitative at moderate and high temperatures.

FIG. 10.

Excess heat capacity as function of temperature. Red points indicate results from MC simulations, blue line results from orientation-averaged integral equation theory, and using black lines, results from orientation-dependent integral equation theory are shown. In (a), MB parametrization of model is used, and in (b), real parametrization is used.

Next, Fig. 11 shows the temperature dependence of density. In case of real parametrization, we can again see indicators of phase transition at temperature above 0.2. Similarly, as in previously discussed thermodynamic quantities, orientation-averaged version of IET qualitatively describes how density changes with temperature, and as the temperature decreases, the deviation of IET from MC results increases. Again, use of ODIET significantly improves results, as the density obtained with ODIet almost quantitatively agrees with density calculated with MC.

FIG. 11.

Density as function of temperature. Red points indicate results from MC simulations, blue line results from orientation-averaged integral equation theory, and using black lines, results from orientation-dependent integral equation theory are shown. In (a), MB parametrization of model is used, and in (b), real parametrization is used.

In Fig. 12, isothermal compressibility as a function of temperature is shown from both parametrizations of the model. After melting point, compressibility increases with temperature, and at high temperatures, it reaches a plateau. In solid state, compressibility is almost zero, as seen at low temperatures in Fig. 12(b); at melting point, there is a sudden increase in compressibility. Orientation-averaged IET is capable of qualitatively predict how compressibility changes with temperature; however, the agreement with compressibility calculated with MC is quite poor. Orientation-dependent IET is much more successful at predicting isothermal compressibility. At high temperatures, the agreement of compressibility calculated with MC and ODIET is almost perfect, and this agreement becomes worse as temperature decreases; however, it is still much better as in the case of orientation-averaged IET.

FIG. 12.

Isothermal compressibility as function of temperature. Red points indicate results from MC simulations, blue line indicate results from orientation-averaged integral equation theory, and using black lines, results from orientation-dependent integral equation theory are shown. In (a), MB parametrization of model is used, and in (b), real parametrization is used.

Thermal expansion coefficient as a function of temperature was also calculated (Fig. 13). The expansion coefficient obtained using MC simulations increases with temperature after melting point, then it reaches maximum, and after that it starts to decrease. In a case of real parametrization, the maximum appears at higher temperatures than in the case of MB parametrization. Orientation-averaged IET is capable of predicting the trend with which the expansion coefficient changes, and it correctly predicts the approximate temperature at which expansion coefficient reaches maximal value; however, the height of this maximum is quite lower than the one calculated with MC simulations. Orientation-dependent IET is much more successful at predicting the expansion coefficient than averaged version, as the height of the maximum predicted by ODIET is almost the same as the one from MC simulations. Moreover, the overall agreement is also better.

FIG. 13.

Thermal expansion coefficient as function of temperature. Red points indicate results from MC simulations, blue line indicates results from orientation-averaged integral equation theory, and using black lines, results from orientation-dependent integral equation theory are shown. In (a), MB parametrization of model is used, and in (b), real parametrization is used.

As it was shown, the orientation-dependent version of IET is much better at predicting thermodynamic properties of water modeled with the rose model than orientation-averaged version of IET. When comparing success of the theory in case of different parametrizations, the theory is more successful in case of MB parametrization. It appears that the properties of real parametrization of the rose model are more difficult to predict with IET than the properties of the model with MB parametrization.

V. CONCLUSIONS

We used orientation-dependent integral equation theory to investigate structural and thermodynamic properties of water modeled with the rose water model. In the rose water model, molecules are modeled as Lennard–Jones disks with added hydrogen bonding potential in form of rose functions. Results obtained using ODIET were compared to results previously obtained by the orientation-averaged version of IET and Monte Carlo simulations. Overall use of the orientation-dependent version of IET improves both structural and thermodynamic results in comparison to the orientation-averaged version. ODIET predicts more correctly structured RDF than the orientation-averaged version; however, the height of the peak corresponding to direct hydrogen bond is more correctly predicted by averaged version. ODIet also provides angular distribution functions. The agreement between angular distributions obtained with ODIET and MC is mostly qualitative, as the shapes of distributions obtained with different methods are more or less the same, and in some cases, the agreement is almost quantitative. Using ODIET, the spatial distribution of molecule around reference molecule can also be calculated. Because the model is two-dimensional, the visualization of spatial distribution is easy and it allows as to visualize how molecules are ordered around one central molecule. Spatial distribution also helped us to determine what structures are related to certain peaks in angular distributions. This way we also identified which peaks appear due to ability of the rose model to form half hydrogen bonds. ODIET was also used to calculate fractions of differently bonded molecules as function of temperature; here, the agreement of the theory with the simulations was very good. When using the orientation average version of IET to calculate thermodynamic properties of water, the agreement with MC results is good at high temperatures and becomes poorer as the temperature decreases. Use of the angular-dependent version of IET improves results of all thermodynamic quantities studied here. The improved of results is significant also at lower temperatures; however, at low temperatures, there is still deviation between results obtained with ODIET and MC. To conclude, integral equation theory predicts both structural and thermodynamic properties of water modeled with rose model and is computationally much faster than computer simulations. The orientation-dependent version of IET provides more accurate results than the orientation-averaged version; moreover, it also provides information on angular distributions of molecules.

AUTHOR DECLARATIONS

Conflict of Interest

The authors have no conflicts to disclose.

Author Contributions

Peter Ogrin: Investigation (equal); Methodology (equal); Software (equal); Writing – original draft (equal). Tomaz Urbic: Conceptualization (equal); Methodology (equal); Software (equal); Writing – original draft (equal).

DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.