Effects of translational and rotational degrees of freedom on properties of the Mercedes–Benz water model

Abstract

Non–equilibrium Monte Carlo and molecular dynamics simulations are used to study the effect of translational and rotational degrees of freedom on the structural and thermodynamic properties of the simple Mercedes–Benz water model. We establish a non–equilibrium steady state where rotational and translational temperatures can be tuned. We separately show that Monte Carlo simulations can be used to study non-equilibrium properties if sampling is performed correctly. By holding one of the temperatures constant and varying the other one, we investigate the effect of faster motion in the corresponding degrees of freedom on the properties of the simple water model. In particular, the situation where the rotational temperature exceeded the translational one is mimicking the effects of microwaves on the water model. A decrease of rotational temperature leads to the higher structural order while an increase causes the structure to be more Lennard–Jones fluid like.

I. INTRODUCTION

Many properties of water and aqueous solutions can be captured by simple models that lack atomic details.^1–3 One class of such models has been developed by Nezbeda and co-workers.^4–6 One of the simplest models for water is the so-called Mercedes–Benz (MB) model,⁷ originally proposed by Ben-Naim in 1971.^8,9 This is a 2-dimensional model, which captures the main aspects of water physics in a simple way: the long–ranged attractions and short-ranged repulsions are treated through Lennard–Jones interactions, and hydrogen bonding is treated through an orientation-dependent interaction. Each water molecule is modeled as a disk that interacts with other such waters through: (1) a Lennard–Jones (LJ) interaction and (2) an orientation–dependent hydrogen bonding interaction through three radial arms arranged as in the Mercedes–Benz logo. The advantages of the MB model, compared to more realistic water models, are (i) the computer simulations of thermodynamic properties can be obtained in reasonable amounts of computer time, and (ii) the underlying physical principles can be more readily explored and visualized in two dimensions. NPT Monte Carlo simulations have shown that the MB model predicts qualitatively the density anomaly, the minimum in the isothermal compressibility as a function of temperature, the large heat capacity, as well as the experimental trends for the thermodynamic properties of solvation of nonpolar solutes^7,10–12 and cold denaturation of proteins.¹³ The model was also extensively studied with analytical methods like integral equation and thermodynamic perturbation theory.^14–19

Here, we employed the MB model to study non–equilibrium situations where translational and rotational temperatures differ. In this way we demonstrated the effect of translational and rotational degrees of freedom on the structural and thermodynamic properties of the simple model of water. For example, by holding the translational temperature constant and increasing the rotational temperature, one could observe the effect of faster rotational motion on the structure and thermodynamics of the model system. As explained below, this situation is closely connected with the effect of microwaves on water and aqueous solutions. In an analogous manner, the effect of faster translational motion (at constant rotational temperature) could be investigated. While such situations are experimentally difficult to control and study, the simulations are suited particularly well for this purpose. Several previous contributions provided numerous insights on this subject:^20–26 (i) hydrogen–bonds in water break upon a rotational or translational temperature increase; (ii) increase in the water’s rotational temperature improves the hydration of cations and hydrophobes and weakens the hydration of anions; (iii) on the other hand increase in translational temperature always reduces correlations among the particles in the system. However, simulation studies in the past focused mainly on more sophisticated water models (such as SPC/E) and shorter range of parameters. Here our aim is to study the physics of a simple water model that has been shown to predict several anomalies of water. Due to the simplicity of the MB model, we could afford to study the properties of the MB model in a much wider range of parameters than has been done before.

The non–equilibrium situation where the rotational temperature exceeds the translational one is mimicking the effect of the electric field of microwaves on water and aqueous solutions (it has been theoretically demonstrated that the magnetic field of microwaves has negligible effects on water²⁷). Namely, when exposed to the microwave irradiation the polar water molecules tend to align their dipole moments with the alternating electric field of microwaves.²⁸ This results in faster rotational motion of water molecules. Excess rotational kinetic energy is subsequently dissipated to the other degrees of freedom, thus rising the temperature of the whole system. However, upon continuous microwave irradiation the excess energy cannot fully dissipate into translational and vibrational motion and formally such a situation could be described with the rotational temperature higher than the temperatures governing the other degrees of freedom. Auerbach and co-workers provided convincing experimental evidence that upon the microwave irradiation the rotational temperature may substantially exceed the translational one.²⁹ One way to theoretically study the effects of microwaves is to perform the non–equilibrium molecular dynamics simulation with the explicitly modeled electric field. Such simulations have been successfully applied to study liquid water,^30,31 ice,³¹ salt solutions,³¹ proteins,^32–35 DNA,³⁶ ionic liquids,³⁷ methane hydrates,³⁸ carbon nanotubes,³⁹ aquaporins,⁴⁰ chiral liquids,⁴¹ and zeolites.^42,43 Our work represents an alternative approach where direct interaction of the water’s electric charges with the alternating electric field of microwaves is replaced by a non–equilibrium situation with a rotational temperature higher than the translational one. While simulations with explicitly included electric field are convenient for studying the effect of field intensity and frequency, our approach enables us to separately study the effect of faster rotational (or translational) motion on the structural and dynamic properties of the system.

The paper is organized in following way. In Sec. II the MB model with the interaction potential is introduced. In Sec. III we provide details of the Monte Carlo simulations; in Sec. IV, the details of molecular dynamics simulation are explained. The results are shown and discussed in Sec. V and summarized in Sec. VI.

II. THE MODEL

An MB water-like molecule is represented as a two–dimensional Lennard–Jones disk with three arms separated by an angle of 120°.^7,8 Bonding among the arms of different MB particles mimics the formation of hydrogen bonds in water Fig. 1. The interaction potential between particles i and j depends not only on their center-to-center distance, r_ij, but also on the orientation of each particle. The potential is a sum of the distance-dependent Lennard–Jones term and a hydrogen–bonding (HB) term, which depends on both—distance and orientation,

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

where $\overrightarrow{X_i}$ denotes the vector representing the coordinates and the orientation of the ith particle. The Lennard–Jones part of the potential is calculated as

$$U_{LJ}(r_{ij})=4{\epsilon }_{LJ}\left[{\left(\frac{{\sigma }_{LJ}}{r_{ij}}\right)}^{12}-{\left(\frac{{\sigma }_{LJ}}{r_{ij}}\right)}^6\right],$$ (2)

where ${\epsilon }_{LJ}$ is the depth of the LJ potential and ${\sigma }_{LJ}$ is the distance where the LJ potential equals 0. The hydrogen bonding part of the interaction potential is a sum of interactions $U_{HB}^{kl}$ between the arms k and l of molecules i and j, respectively,

$$U_{HB}(\overrightarrow{X_i},\overrightarrow{X_j})=\sum _{k,l=1}^3{U}_{HB}^{kl}(r_{ij},{\theta }_i,{\theta }_j).$$ (3)

This arm–arm interaction is a Gaussian function in distance and angles

$$\begin{array}{cc}\hfill U_{HB}^{kl}(r_{ij},{\theta }_i,{\theta }_j)& ={\epsilon }_{HB}G(r_{ij}-r_{HB})G(\overrightarrow{i_k}\overrightarrow{u_{ij}}-1)G(\overrightarrow{j_l}\overrightarrow{u_{ij}}+1)\hfill \\ \hfill & ={\epsilon }_{HB}G(r_{ij}-r_{HB})G(\cos ({\theta }_i+\frac{2\pi }{3}(k-1))-1)\times G(\cos ({\theta }_j+\frac{2\pi }{3}(l-1))+1).\hfill \end{array}$$ (4)

Here G(x) is an unnormalized Gaussian function

$$G(x)=\exp \left(\frac{-x^2}{2{\sigma }^2}\right),$$ (5)

${\epsilon }_{HB}$ is the energy of the hydrogen bond, and r_HB is a characteristic hydrogen bond length. $\overrightarrow{u_{ij}}$ is the unit vector along $\overrightarrow{r_{ij}}$ (vector pointing from the center of particle i to the center of particle j) and $\overrightarrow{i_k}$ is the unit vector representing the kth arm of the ith particle, where ${\theta }_i$ is the orientation of the ith particle. The strongest hydrogen bond occurs when an arm of one particle is co–linear with the arm of another particle and the two arms point in opposing directions. Throughout this work we used reduced units: energies were expressed in $|{\epsilon }_{HB}|$ and lengths in r_HB. Therefore, the parameter for hydrogen-bond energy, ${\epsilon }_{HB}$, equalled −1, and for hydrogen-bond length equalled 1. Parameters for the LJ interaction potential were set to ${\epsilon }_{LJ}=0.1|{\epsilon }_{HB}|$ and ${\sigma }_{LJ}=0.7r_{HB}$.

FIG. 1.

The MB model of water. Two MB water molecules separated by a distance r₁₂. The angles that the molecules make with the intermolecular axis are ${\theta }_1$ and ${\theta }_2$.

III. MONTE CARLO SIMULATIONS

We performed Monte Carlo simulations of the 2D Mercedes–Benz water model in the NVT ensemble using Metropolis algorithm. Simulations were used to compute thermodynamic and structural properties of the model. At each step, MB molecules were chosen randomly and they were assigned new coordinates (x,y) or new orientation. To weigh the acceptance probability for a translational move, we used a translational temperature, T_trs, while for rotational moves the rotational temperature, T_rot, was used. Ratio of probabilities to attempt a translational (P_trs) and rotational (P_rot) displacement equalled to the ratio of the corresponding temperatures, i.e.,

$$\frac{P_{\text{trs}}}{P_{\text{rot}}}=\frac{T_{\text{trs}}}{T_{\text{rot}}}.$$ (6)

Therefore, higher rotational temperature means the number of attempts of rotational displacements was greater. This was done to sample more rotational phase space at higher rotation temperature. In each MC cycle each particle was translated once on average (each MC cycle has N translations of randomly chosen particle). We used periodic boundary conditions and the minimum image convention to mimic an infinite system of particles. Starting configurations were selected at random. Here are some of the simulation details: 20 000 cycles were needed to equilibrate the system. Statistics were gathered over 20 000-100 000 moves to obtain well converged results. All simulations were performed with N = 100 molecules which is equivalent to about 1000 particles in 3D, in this case 2D and 3D systems have the same box size at same density. Thermodynamic quantities such as energy and heat capacity were calculated as statistical averages over the course of the simulations.⁴⁴ The interaction potential cutoff distance was half-length of the simulation box. While all our simulations were performed in the NVT ensemble, the density differed among the runs in the following way: for a simulation run at a given T_trs the chosen density equalled to the average density obtained in the equilibrium NPT simulation at P = 0.19 and T = T_trs. It was shown in previous studies that several thermodynamic properties of the 2DMB model at P = 0.19 and T = 0.2 resemble those of water at ambient conditions.⁷ In order to verify our Monte Carlo simulations, we compared their results with those obtained from the molecular dynamics (MD) simulations. We used an in–house developed code for both types of simulations.

IV. MOLECULAR DYNAMICS SIMULATIONS

To model the non–equilibrium situation where the rotational and translational temperatures are not equal, we followed our previous work on this topic²³ and integrated separately the equations for translational and rotational motions. These have been already established by Hynninen for the Mercedes–Benz water model.⁴⁶ For their integration we used a simple velocity Verlet algorithm with a time step $\mathrm{\Delta }t={10}^{-4}$. By separately solving the equations for translational and rotational motion, we were able to set different values for the translational and rotational temperature. To keep the temperatures constant, we employed the Andersen thermostat.⁴⁵ In order to minimize the effect of this thermostat on the dynamics of system, we made a slight modification. Instead of randomizing the particles’ velocities, v, we randomized only their speeds, υ (where υ = |v|), such that they preserved the orientation of the vector v. Of course, the particles’ speeds were drawn from the Maxwell–Boltzmann distribution. In this way, we effectively reduced the effect of a high thermostatting frequency on the system’s dynamics while preserving the randomization of the particles’ speeds. We have verified (for the equilibrium system) that MD simulations with a modified thermostatting protocol give equivalent results to the MC simulations. In each time step, the probability to draw a new speed for a particle equalled to 10⁻³. All the other molecular dynamics parameters were the same as for the Monte Carlo simulations.

V. RESULTS AND DISCUSSION

A. Verification of the non–equilibrium Monte Carlo simulations

The basic information about the structure of our system is given by the radial pair distribution function g(r). In Figure 2 we compared the radial distribution functions (RDFs) from the Monte Carlo (red line) and molecular dynamics simulations (black line) at various combinations of translational and rotational temperatures. Figures 2(a)–2(d) present the results for 4 different translational temperatures, T_trs. For clarity, the radial distribution functions for different rotational temperatures (at constant T_trs) are shifted relative to each other. We obtained an excellent agreement between the results from Monte Carlo and molecular dynamics simulations. Within the error of the simulations, the RDFs from the MC and MD are overlapping. This is a clear confirmation that our Monte Carlo procedure correctly samples the non–equilibrium states where the translational and rotational temperatures differ and can be used to study the properties of the MB model.

FIG. 2.

Radial pair distribution function, g(r), obtained from the molecular dynamics (black line) and Monte Carlo simulations (red line) for different combinations of translational, T_trs, and rotational temperature, T_rot. Plots (a)–(d) present the radial distribution functions for 4 different translational temperatures. For clarity, radial distribution functions for various T_rot (at constant T_trs) are shifted relative to each other.

B. Structural properties

Upon confirming the suitability of Monte Carlo simulations for the study, we first checked structural properties of the MB model obtained with the MC simulations. Figure 3 presents RDFs for different translational and rotational temperatures in more detail. Figures 3(a)–3(d) correspond to 4 different translational temperatures. The various colors in each plot represent different rotational temperatures. From the figure, we can see that a decrease of the rotational temperature enhances the network structure of water. In particular, the peak at $r\approx 1$ which corresponds to the two hydrogen–bonded water molecules (see the inset in Figure 3(c)) increases on the account of the first peak (located at $r\approx 0.7$), which is indicative of two non–bonded MB particles in close contact (see the inset in Figure 3(c)). Also the third peak positioned at $r\approx 1.73$ and corresponding to the two MB molecules bridged by the third MB particle (see the inset in Figure 3(c)) increases upon lowering of the rotational temperature. Thus, lower T_rot promotes the network structure of the MB fluid. The effect is most pronounced at the lowest T_trs: at T_trs = 0.16 and lowest rotational temperature (T_rot = 0.1) long–range correlations among MB molecules are clearly visible. On the other hand, at the same T_rot and the highest translational temperature, only short range correlations are observable (i.e., RDFs are flat from $r\approx 2$ on). At least in part this is a consequence of the higher number density at lower T_trs. With an increase in the rotational temperature, differences in structure among different T_trs decrease.

FIG. 3.

Plots (a)–(d) present radial pair distribution functions, g(r), for 4 different translational temperatures. Different colors in each plot correspond to various rotational temperatures: T_rot = 0.1 (black), 0.2 (blue), 0.5 (green), and 2.0 (red). The insets in plot (c) schematically show the structural patterns responsible for the first three peaks in RDF.

Orientational ordering of MB particles is illustrated in Figures 4 and 5, which present the angular distribution functions $p(\theta )$ of water molecules in LJ and HB contact, respectively. Two MB particles are considered to be in LJ contact, when their center-to-center distance is around 0.7r_HB (i.e., position of the first peak in RDF) and are considered to be in HB contact when this distance is about r_HB (position of the second peak in RDF). Angle θ is defined as the angle between the arm of ith MB particle and the vector pointing from the center of the particle i to the center of the particle j (for better illustration see the inset in Figure 4). Figures 4(a)–4(d) in each figure display the results for 4 different translational temperatures as in Figure 3. Different colors are used to designate various rotational temperatures at a given T_trs. Angular distribution functions for MB particles in LJ contact, Figure 4, exhibit two maxima located at $\theta =\pm {60}^{\mathrm{°}}$ and a minimum occurring at $\theta =0^{\mathrm{°}}$. This shape indicates the tendency of MB particles in LJ contact for not to waste the hydrogen bond. The MB particle that points with an arm directly to the center of its close–contact neighbour ($\theta =0^{\mathrm{°}}$) loses the ability to form a hydrogen bond with other particles. From Figure 4 we can see that this tendency is most pronounced at low T_trs and T_rot. Distributions become almost flat at the highest rotational temperature (independently of the T_trs). Interestingly, the distributions at the highest T_trs are practically flat even at the lowest T_rot (Figure 4(d)). This is especially curious when we compare with Figure 5 which displays the same angular distributions but for the two MB particles located at a bonding distance (i.e., $r\approx r_{HB}$). The dependence of these distributions upon T_trs is much weaker than for the MB particles in LJ contact, i.e., angular distributions at $r\approx r_{HB}$ are strongly dependent only on the rotational temperature. Even though MB particles at T_trs = 0.4 and T_rot = 0.1 are more likely to lose a potential hydrogen bond by pointing with an arm directly to the center of the close–contact neighbour (Figure 4(d)), they can still form 2 hydrogen bonds with the rest of the particles. The existence of bond formation at these conditions is clearly reflected in Figure 5(d). Thus, structuring of the angular distributions for the particles in LJ contact is necessary only when the particles tend to form all three possible hydrogen bonds. If a MB particle forms on average 2 or less hydrogen bonds, the close–contact orientational ordering becomes insignificant. However, the tendency to form a hydrogen bond is still reflected in the orientational ordering at a bonding distance.

FIG. 4.

Plots (a)–(d) present angular distribution functions, $p(\theta )$, of MB water in LJ contact ($r\approx {\sigma }_{LJ}$) for 4 different translational temperatures. Different colors in each plot correspond to various rotational temperatures: T_rot = 0.1 (black), 0.2 (blue), 0.5 (green), and 2.0 (red). The inset schematically shows a definition of the angle θ.

FIG. 5.

Plots (a)–(d) present angular distribution functions, $p(\theta )$, of MB water in HB contact ($r\approx r_{HB}$) for 4 different translational temperatures. Different colors in each plot correspond to various rotational temperatures: T_rot = 0.1 (black), 0.2 (blue), 0.5 (green), and 2.0 (red).

To gain a further insight into the structural changes upon an increase in rotational temperature, we plotted the probability P_n to observe a particle forming exactly n bonds as a function of T_rot (at given T_trs). Plots 6(a) through 6(d) display P_n for n = 0 (non–bonded particle) through n = 3 (maximum number of bonds per particle). Different translational temperatures are indicated by various colors. At low translational temperature (T_trs = 0.16 and T_trs = 0.2) and low rotational temperatures, most particles form a maximum number of bonds (n = 3). Upon an increase in T_rot this fraction, P₃, drops rather quickly from about 60% to less than 10%. Consequently, fractions of particles with fewer bonds increase. Fraction of 2–bonded particles, P₂, exhibits non–monotonous behavior, as it first increases with T_rot. In this range, first 3-bonded particles start to lose their hydrogen bonds and first form only 2 instead. Upon a further rotational excitation, a fraction of 2-bonded particles starts to decrease on the account of P₁ and P₀. The situation at higher translational temperature is qualitatively similar: P₃ exhibits a monotonous decrease while P₀ steadily increases. However, the fraction of single bonded particles, P₁, exhibits a non–monotonous behavior with a maximum at $T_{\text{rot}}\approx 0.4$ while P₂ monotonically decreases. It is interesting to see that all populations have a limit at high rotational temperature which depends on the translation temperature. This means that with the increase of rotational temperature only we cannot melt out all hydrogen bonds. A similar conclusion can be achieved from the dependence of LJ and HB coordination number as a function of the rotational temperature shown in Figure 6. Increase of rotational temperature decreases the number of waters in a HB shell for all translational temperatures. For high temperatures, this has almost no effect on the number of waters in the LJ shell while at lower translational temperatures this number increases. The number of waters in both shells has a limit value upon increase of the rotational temperature which depends on the translational temperature.

FIG. 6.

Coordination number n_c for the LJ (a) and HB-shell (b) as a function of the rotational temperature. Different colors represent different translational temperatures: T_trs = 0.16 (black), 0.2 (blue), 0.32 (green), and 0.4 (red).

Similar conclusions as from ratios of waters with different number of hydrogen bonds can be obtained also from energy histograms of individual molecule. Figure 7 illustrates the changes occurring in the potential energy distribution, p(E), upon the rotational temperature increase. Figures 7(a) and 7(b) present p(E) for translational temperatures T_trs = 0.2 and T_trs = 0.32, respectively. The distribution for T_trs = 0.2 and low T_rot exhibits four characteristic peaks at energies E = −3, −2, −1, and −0.25, which correspond to MB particles with n = 3, 2, 1, and 0 bonds. With an increase in rotational temperature, bonds are being broken, which is reflected in the distribution p(E) shifting towards the higher energies. For T_rot = 2.0 the distribution exhibits only two peaks, which correspond to single- and non-bonded MB particles. At higher translational temperature (T_trs = 0.32), the distribution is shifted to the right already at the lowest T_rot. Increase in the rotational temperature produces qualitatively similar behavior as we have seen above. However, the high-energy peak is substantially more pronounced than at the lower T_trs.

FIG. 7.

Probability distribution for particle energy, p(E), at T_trs = 0.2 (a) and T_trs = 0.32 (b). Different colors indicate various rotational temperatures: T_rot = 0.1 (black), 0.2 (blue), 0.5 (green), and 2.0 (red).

Figure 8 illustrates the probabilities of finding hydrogen bonded clusters of different sizes for different translational and rotational temperatures. Upon increase of rotational temperature, it is less probable to find bigger clusters in the system. At lower temperature, there might be transition from liquid to vapor due to the increase of rotation temperature, but to confirm this we would need to obtain phase diagram data for MB mode, which currently does not exist. This is consistent with previous results for radial (Fig. 3) and angular (Figs. 4 and 5) distributions and for studies of number of hydrogen bonds (Fig. 9) in the system. Figure 10 shows snapshots of the system for different rotational temperatures. Green lines are connecting centers of molecules that form hydrogen bonds in simulation box, non-connected molecules are in replicas of the main box. It can be seen that an increase of the rotational temperature melts hydrogen bonds since we have less green lines at higher rotational temperature.

FIG. 8.

Probability p(N) for observing a cluster of size N at T_trs = 0.2 (a) and T_trs = 0.32 (b). Different colors correspond to various rotational temperatures: T_rot = 0.1 (black), 0.2 (blue), 0.5 (green), and 2.0 (red).

FIG. 9.

Probability P_n for a 2DMB particle to form exactly n bonds as a function of rotational temperature, T_rot, for n = 0 (a), n = 1 (b), n = 2 (c), and n = 3 (d). Different colors represent different translational temperatures: T_trs = 0.16 (black), 0.2 (blue), 0.32 (green), and 0.4 (red).

FIG. 10.

Representative snapshots of the MB water model at a constant translational temperature (T_trs = 0.2) and different rotational temperatures: (a) T_rot = 0.1, (b) T_rot = 0.2, (c) T_rot = 0.5, and (d) T_rot = 2.0. Green lines represent bonds between MB particles.

C. Thermodynamic properties

In the last part, we examined the dependence of some thermodynamic functions on the rotational temperature. First in Figure 11 we present the average excess energy per particle, E_ex/N, as a function of the rotational temperature. Different translational temperatures are designated by various colors. In line with the observations above, we note that for the two lowest translational temperatures the E_ex/N increases steeply with the T_rot. As we have seen above, the faster rotational motion of MB particles breaks the bonds between them, therefore increasing the average energy per particle. The rise in E_ex/N upon T_rot increase is much weaker at higher translational temperatures, as the number of bonded particles is already smaller due to the lower density and faster translational motion. Figure 12 presents the excess heat capacity at constant volume, C_{v, ex}, as a function of the rotational temperature. The heat capacity exhibits clearly distinguishable maxima at T_rot where we observed the biggest energy change (compared to Figure 11). In other words, disruption of hydrogen bonds is fastest around the rotational temperature where heat capacity exhibits maximum.

FIG. 11.

The average excess energy per particle, E_ex/N, as a function of the rotational temperature, T_rot. Different colors designate various translational temperatures: T_trs = 0.16 (black), 0.2 (blue), 0.3 (green), and 0.4 (red).

FIG. 12.

The excess heat capacity at constant volume, C_{v, ex}, as a function of the rotational temperature, T_rot. Different colors designate various translational temperatures: T_trs = 0.16 (black), 0.2 (blue), 0.3 (green), and 0.4 (red).

In Figure 13 we show the virial pressure as a function of the rotational temperature. Different colors designate various translational temperatures. The rise in energy upon a T_rot increase is accompanied with a non–monotonous behavior of virial pressure. We have demonstrated above that a decrease in rotational temperature promotes the formation of hydrogen bonds. For the low enough translational temperatures an extensive cluster of bonded particles forms (see Figure 9). MB particles in such clusters have a low local density and consequently take up more space, thereby increasing the virial pressure. On the other hand, as the rotational temperature increases this network breaks into smaller clusters of bonded particles with the average interaction among the particles being attractive. Further increase in rotational temperature weakens the average interparticle interaction resulting in increase in the virial pressure.

FIG. 13.

The virial pressure, P, as a function of the rotational temperature, T_rot. Different colors designate various translational temperatures: T_trs = 0.16 (black), 0.2 (blue), 0.3 (green), and 0.4 (red).

The excess chemical potential, ${\mu }_{\text{ex}}$, as a function of the rotational temperature is plotted in Figure 14. The excess chemical potential decreases with increase of rotational temperature.

FIG. 14.

The excess chemical potential, ${\mu }_{\text{ex}}$, as a function of the rotational temperature, T_rot. Different colors designate various translational temperatures: T_trs = 0.16 (black), 0.2 (blue), 0.3 (green), and 0.4 (red).

VI. CONCLUSIONS

We have studied the effect of rotational and translational degrees of freedom on the structural and thermodynamic properties of the simple 2D MB model of water. We have shown that the non–equilibrium Monte Carlo protocol described in this paper and non–equilibrium molecular dynamics with separate thermostats for the rotational and translational motion give equivalent results. We have shown that an increase in rotational temperature causes a significant breaking of hydrogen bonds. This is reflected in weakening of the radial and orientational ordering as well as in increase of the average excess energy per particle. We have explained the non–monotonous behavior of the constant volume heat capacity and virial pressure upon a rotational temperature increase. We have demonstrated that an increasingly high rotational temperature does not break all the hydrogen bonds, but instead leads to the structure with limiting and non–zero probabilities for a MB particle to form hydrogen bonds.