# Miscibility and thermodynamics of mixing of different models of formamide and water in computer simulation

Bálint Kiss, Balázs Fábián, Abdenacer Idrissi, Milán Szőri, Pál Jedlovszky

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Bálint Kiss, Balázs Fábián, Abdenacer Idrissi, Milán Szőri, Pál Jedlovszky. Miscibility and thermodynamics of mixing of different models of formamide and water in computer simulation. Journal of Physical Chemistry B, 2017, 121 (29), pp.7147 - 7155. 10.1021/acs.jpcb.7b04965. hal-01593264

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1 Miscibility and Thermodynamics of Mixing of Different Models of Formamide and Water in Computer Simulation

3 Bálint Kiss,†,‡ Balázs Fábián,§,|| Abdenacer Idrissi,‡ ID Milán Szőri,† ID and Pál Jedlovszky*,⊥ ID

4 †Institute of Chemistry, University of Miskolc, Egyetemváros A/2, H-3515 Miskolc, Hungary

5 ‡Laboratoire de Spectrochimie Infrarouge et Raman (UMR CNRS 8516), University of Lille Nord de France, 59655 Villeneuve

6 d'Ascq Cedex, France

7 §Department of Inorganic and Analytical Chemistry, Budapest University of Technology and Economics, Szt. Gellért tér 4, H-1111

8 Budapest, Hungary

9 ‖Institut UTINAM (CNRS UMR 6213), Université Bourgogne Franche-Comté, 16 route de Gray, F-25030 Besançon, France

10 ⊥Department of Chemistry, Eszterházy Károly University, Leányka utca 6, H-3300 Eger, Hungary

**ABSTRACT:** The thermodynamic changes that occur upon mixing five models of formamide and three models of water, including the miscibility of these model combinations itself, is studied by performing Monte Carlo computer simulations using an appropriately chosen thermodynamic cycle and the method of thermodynamic integration. The results show that the mixing of these two components is close to the ideal mixing, as both the energy and entropy of mixing turn out to be rather close to the ideal term in the entire composition range. Concerning the energy of mixing, the OPLS/AA_mod model of formamide behaves in a qualitatively different way than the other models considered. Thus, this model results in negative, while the other ones in positive energy of mixing values in combination with all three water models considered. Experimental data supports this latter behavior. Although the Helmholtz free energy of mixing always turns out to be negative in the entire composition range, the majority of the model combinations tested either show limited miscibility, or, at least, approach the miscibility limit very closely in certain compositions. Concerning both the miscibility and the energy of mixing of these model combinations, we recommend the use of the combination of the CHARMM formamide and TIP4P water models in simulations of water–formamide mixtures.

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26 ■ INTRODUCTION

27 Liquid formamide (H–CO-NH₂) and its aqueous mixtures are
28 systems of great importance, both from the purely scientific,
29 theoretical point of view and also from that of industrial
30 applications. Formamide is a highly polar, hygroscopic liquid of
31 unusually high dielectric constant, which is fully miscible with
32 water.¹ The high dielectric constant of formamide of about
33 109² cannot simply be explained by the large molecular dipole
34 moment of 3.7 D.³ Thus, for instance, the dielectric constant of
35 its double methylated derivative, N,N-dimethylformamide, is
36 only 37,² despite the fact that the dipole moment of this
37 molecule is as large as 3.9 D.³ Instead, the large dielectric
38 constant of liquid formamide is related to the fact that its
39 molecules exhibit extensive hydrogen bonding, involving both
40 N–H- and C–H-donated hydrogen bonds. In fact, similarly to
41 water,⁴,⁵ formamide is also a network forming liquid,⁶,⁷ and
42 hence these molecules can locally substitute each other in the
43 hydrogen bond network,⁸ although such substitution may
44 change the topology of the network.⁷,⁹ Due to this, and other
45 similar properties of the two liquids, mixtures of formamide and
46 water are thought to be prototypes of the ideal mixture.¹⁰,¹¹

47 Consistently, the energy and enthalpy change occurring upon

48 mixing the two compounds in any proportion turned out to be

a very small, positive value, both experimentally¹²,¹³ and in 49
computer simulation.¹⁴ 50

Industrial importance of formamide and its aqueous mixtures 51
is due to their high dielectric constant, which makes these 52
systems very good solvents of many heavy metal salts as well as 53
of salts of alkali and alkaline-earth metals. 15 Further, formamide 54
is an important solvent for resins, and it is also used as a 55
softener for paper, in animal glues, and water-soluble gums. 56
Mixtures of formamide and water are excellent plasticizers for 57
thermoplastic starch polylactic acid blends. 16 As a chemical 58
intermediate, formamide is particularly useful in the synthesis of 59
heterocyclic compounds, pharmaceutical (e.g., vitamins), crop 60
protection agent, pesticides, and for the manufacture of 61
hydrocyanic acid. 1 Formamide may be released into the 62
environment during its manufacturing process, and the fate of 63
atmospheric formamide is mainly due to its rainout. 17 64

From the scientific point of view, the importance of 65
formamide stems from the fact that it is the smallest molecule 66
that contains a peptide bond, and also the simplest molecule 67
that is able to form N–H···O type hydrogen bonds. Thus, 68
## METHODS

**Thermodynamics of Mixing.** The thermodynamic condition that two liquid compounds, A and B, are fully miscible with each other can be formulated as the inequality$^{42}$

$$D = 1 + x_A x_B \frac{\partial^2(F^{\text{ex}}/RT)}{\partial^2 x_A} > 0 \quad (1)$$

holds in the entire composition range of $0 < x_A < 1$. Here, $x_A$ and $x_B$ denote the mole fraction of the respective components, $R$ is the gas constant, $T$ is the temperature, and $F^{\text{ex}}$ is the excess molar Helmholtz free energy with respect to the ideal mixing of the two components:

$$F^{\text{ex}} = F^{\text{mix}} - RT(x_A \ln x_A + x_B \ln x_B) \quad (2)$$

In this equation, $F^{\text{mix}}$ denotes the total Helmholtz free energy change accompanying the mixing of the neat components A and B (often simply called as their free energy of mixing).

$F^{\text{mix}}$ can be calculated by considering the following thermodynamic cycle (see Figure 1).$^{36}$ In the first step, the

<figure><img src="image_3.png" /><figcaption>**Figure 1.** Schematic diagram showing the thermodynamic cycle along which the free energy of mixing of the two components is calculated. For a detailed explanation, see the text.</figcaption></figure>

neat components are brought from the liquid to the ideal gas state under isochoric conditions. In the second step, the two components are mixed in the ideal gas state; this step is accompanied by the free energy change of the ideal mixing of $RT(x_A \ln x_A + x_B \ln x_B)$. Finally, in the third step, the mixture is brought back isochorically from the ideal gas to the liquid state. Thus, the free energy change of the net process, $F^{\text{mix}}$, can be written as

$$F^{\text{mix}} = F_{A+B} - x_A F_A - x_B F_B + RT(x_A \ln x_A + x_B \ln x_B) \quad (3)$$

where $F_A$, $F_B$, and $F_{A+B}$ are the Helmholtz free energies of the corresponding systems (i.e., neat liquid A, neat liquid B, and their mixture, respectively) with respect to their ideal gas state. It should be noted that the inequality $F^{\text{mix}} < 0$ is not a sufficient condition of miscibility, as it simply reflects that the mixture is thermodynamically more stable than the two separate neat components. However, miscibility also requires that the single phase mixture is stable also with respect to any two phase systems consisting of mixed phases of different compositions. This condition is taken into account in eq 1.

Using the same thermodynamic cycle as for the calculation of $F^{\text{mix}}$, and considering that mixing in the ideal gas state is

aqueous mixtures of formamide are often considered as model systems in studying various properties of hydrated proteins and polypeptides, e.g., how hydrogen bonds$^{18,19}$ and peptide bonds$^{20}$ are formed and broken, or how water exercises kinetic and thermodynamic control over the chemical activities of polypeptides.$^{21}$ These mixtures can also be used as reference systems to study hydrophobic and hydrophilic interactions.$^{22}$ Further, aqueous formamide solutions are thought to play a non-negligible role in prebiotic evolution. Clearly, there is an increasing body of both theoretical and experimental evidence suggesting that formamide is a potential hub in the complex network of prebiotic chemical reactions that leads from simple precursors, such as H$_2$, H$_2$O, N$_2$, NH$_3$, CO, and CO$_2$, to key biological molecules.$^{23-25}$

Due to the aforementioned reasons, both neat liquid formamide$^{6,7,20,22,26-29}$ and its aqueous mixtures$^{8,10,11,14,19,30-33}$ have been studied by computer simulation. Methods several times. However, in simulating liquid mixtures one has to face the problem that potential models that work perfectly for the respective neat liquids might be incompatible with each other, resulting in a poor reproduction of the properties of the mixtures. If the thermodynamic driving force behind the miscibility of the two compounds is small, this might even involve limited miscibility of certain models of the two fully miscible compounds.$^{34,35}$ Since the mixing of water and formamide is close to the ideal mixing,$^{10-14}$ the thermodynamic changes occurring upon their mixing are expected to be rather small (as it was already shown concerning their energy of mixing.$^{12-14}$) As a consequence, the thermodynamic driving force of this mixing must be small, which makes the above issue, concerning the compatibility of the potential models with each other and their ability to reproduce full miscibility, particularly important for mixtures of water and formamide.

In this article, we study the thermodynamics of mixing formamide and water by computer simulation, concerning five widely used formamide and three such water models. Besides calculating the Helmholtz free energy, energy, and entropy of mixing of these model combinations in the entire composition range, we also analyze which of these model pairs are fully miscible with each other, and which of them exhibit miscibility gaps. Thermodynamic changes occurring upon mixing are calculated along an appropriately chosen thermodynamic cycle$^{36}$ using the method of thermodynamic integration.$^{37,38}$ This approach has already successfully been used to study the thermodynamics of mixing of various pairs of fluids.$^{34-36,39-41}$ The performance of the different model combinations considered are evaluated (i) according to their ability to reproduce full miscibility, and (ii) by comparing the calculated energy of mixing to existing experimental data.$^{12,13}$

The rest of the article is organized as follows. In the Methods Section the applied methods, including the thermodynamic path used and the method of thermodynamic integration, are explained in detail. The simulations performed and potential models considered in this study are also described here. The obtained results are presented and discussed in detail in the Results and Discussion Section. Finally, in the Summary and Conclusions Section the main conclusions of this work are summarized.
accompanied by no energy change, the energy of mixing of the two components, U^mix, can be written as

$$167 \qquad U^{\text{mix}} = U_{A+B} - x_A U_A - x_B U_B \qquad (4)$$

where $U_{A,B}$, $U_{B,A}$ and $U_{A+B}$ denote the internal energies of the corresponding systems. Finally, the entropy of mixing of the two components, S^mix, can simply be calculated from U^mix and F^mix as

$$172 \qquad S^{\text{mix}} = \frac{U^{\text{mix}} - F^{\text{mix}}}{T} \qquad (5)$$

**Thermodynamic Integration.** While the terms $U_{A,B}$, $U_{B,A}$, and $U_{A+B}$ in eq 4 can simply be calculated in computer simulation as the total potential energy of the corresponding systems, the determination of $F_{A,B}$, $F_{B,A}$ and $F_{A+B}$ in eq 3 is not that straightforward. In general, the configurational part of the Helmholtz free energy, being proportional to the logarithm of the configurational integral (i.e., configurational part of the total partition function) on the canonical ensemble, is far more difficult to be accessed in a computer simulation than other thermodynamic quantities, such as the internal energy. This is because here the entire configurational space needs to be sampled instead of only its lowest energy domains. (The kinetic part of the Helmholtz free energy depends only on the velocities of the particles, and hence remains unchanged at constant temperature.) It can be computationally feasible, however, to calculate the free energy difference between two states, as in this case only those domains of the configurational space are needed to be sampled that are considerably different in the two states.

In the method of thermodynamic integration (TI),³⁷,³⁸ the difference of the Helmholtz free energies between states X and Y is calculated as an integral along an arbitrarily chosen path connecting the two states:

$$196 \qquad \Delta F = F_Y - F_X = \int_0^1 \left( \frac{\partial F(\lambda)}{\partial \lambda} \right) d\lambda \qquad (6)$$

where $\lambda$ is the coupling parameter that describes this path, its value being 0 in state X and 1 in state Y. Considering the fundamental relations of statistical mechanics that $F = -k_B T$ and $Q = \int \exp(-U/k_B T)dq^N$, the integrand of eq 6 can be written as

$$202 \qquad \frac{\partial F(\lambda)}{\partial \lambda} = -k_B T \frac{1}{Q(\lambda)} \frac{\partial Q(\lambda)}{\partial \lambda} = \frac{\int \frac{\partial U(\lambda)}{\partial \lambda} \exp(-\beta U(\lambda)) dq^N}{\int \exp(-\beta U(\lambda)) dq^N} \\ = \left\langle \frac{\partial U(\lambda)}{\partial \lambda} \right\rangle_{\lambda} \qquad (7)$$

where Q is the configurational integral, $dq^N$, which denotes the coordinates of all N particles in the system, represents a given point of the configurational space, $k_B$ is the Boltzmann constant, $\beta = 1/k_B T$, and the brackets $<\cdots>_{\lambda}$ denote ensemble averaging at the state corresponding to the given value of $\lambda$.

When connecting states X and Y, a polynomial path is usually chosen. Considering also that in systems where the energy is pairwise additive, and the leading term of the pair potential (i.e., steric repulsion) decays with $r^{-12}$, this polynomial has to be at least of fourth order to avoid divergence of $U(\lambda)$ at $\lambda = 0$,³⁷ this path is conventionally chosen in computer simulations as

$$214 \qquad U(\lambda) = \lambda^4 U_Y + (1 - \lambda)^4 U_X \qquad (8)$$

where $U_X$ and $U_Y$ are the energies of the corresponding states. If the Helmholtz free energy difference between the (isochoric) liquid and ideal gas states of a system needs to be calculated,⁴³⁻⁴⁷ as in the present case, eq 8 can be simplified to $U(\lambda) = \lambda^4 U_Y$, considering that the potential energy in the ideal gas state, X, is zero. In this case, using also eq 7, eq 6 can be rewritten as

$$\Delta F = \int_{0}^{1} \left\langle \frac{\partial U(\lambda)}{\partial \lambda} \right\rangle_{\lambda} d\lambda = \int_{0}^{1} 4\lambda^{3} \langle U_{Y} \rangle_{\lambda} d\lambda \qquad (9)$$

Further, the Boltzmann factor in eq 7, evaluated in each step of a Monte Carlo simulation, can be rewritten as

$$\exp(-U(\lambda)/k_B T) = \exp(-\lambda^4 U_Y/k_B T) \\ = \exp(-U_Y/k_B(T/\lambda^4)) = \exp(-U_Y/k_B T^*) \qquad (10)$$

where $T^* = T/\lambda^4$. Therefore, a simulation performed at a given $\lambda$ point along the (fictitious) path connecting states X and Y with the potential function $U(\lambda)$, needed to evaluate the ensemble average in eq 7, can be substituted with a simulation performed in state Y (i.e., at $\lambda = 1$, in the liquid state of interest) with the full potential function $U(1) = U_Y$, but at the virtual temperature $T^* = T/\lambda^4$. In other words, the path connecting the liquid and ideal gas states along $\lambda$ by gradually decreasing the potential energy to zero can be substituted by a path along which the temperature of the system is increased gradually to infinity. This way, the ideal gas state is simply defined as a state of infinite kinetic energy rather than that of zero potential energy. It should be recalled that the path connecting states X and Y is fictitious, therefore, the points along which it passes (i.e., the systems simulated at various virtual temperatures $T^*$) have no physical relevance by themselves. These fictitious states only serve to connect the two end points of the path, i.e., the two states that are of real physical relevance, the free energy difference of which is to be calculated. Having the ensemble average of eq 7 thus evaluated at several values of $\lambda$ (or $T^*$), the integral of eq 9 can be performed, and $\Delta F$ can be calculated.

The values of $F_{A,B}$, $F_{B,A}$ and $F_{A+B}$ in eq 3 have been calculated using the above-described method of thermodynamic integration. Monte Carlo simulations have been performed at six different (virtual) temperatures, corresponding to the $\lambda$ values of 0.046911, 0.230765, 0.5, 0.769235, 0.953089, and 1. The first five $\lambda$ values correspond to the 5-point Gaussian quadrature,⁴⁸ while simulations at the nonfictitious state $\lambda = 1$ have been performed in order to also evaluate $U_{A,B}$ and $U_{A+B}$. To perform the integral of eq 9, a fourth order polynomial has been fitted to the $4\lambda^3\langle U_Y \rangle_{\lambda}$ vs $\lambda$ data points, and the fitted polynomial has been integrated analytically. These points, along with the fitted polynomial, are shown in Figure 2, as obtained in selected systems.

**Monte Carlo Simulations.** The integrand of eq 9 has been evaluated by performing a Monte Carlo simulation at each $\lambda$ value considered for each system. The real temperature of the systems, corresponding to $\lambda = 1$ has been 298 K. In every simulation, 512 molecules have been placed in a cubic basic simulation box, the edge length of which, L, has been set in order to reproduce the experimental mass density⁴⁹ of the system (see Table 1). Besides the two neat systems, mixtures of nine different compositions, corresponding to the formaldehyde mole fraction ($x_{FA}$) values of 0.1, 0.2, 0.3, 0.4, 0.5, 270
<figure><img src="image_4.png" /><figcaption>**Figure 2.** Integrand of the thermodynamic integration (eq 9), obtained at six $\lambda$ points for the combination of the CHARMM model of formamide with the SPC/E (top panel), TIP4P (middle panel), and TIP4P/2005 (bottom panel) models of water (full circles), together with the fourth order polynomials fitted to these data (solid curves) at the formamide mole fraction values of 0 (black), 0.2 (red), 0.4 (green), 0.6 (dark blue), 0.8 (light blue), and 1.0 (magenta).</figcaption></figure>

<table><caption>**Table 1.** Properties of the Systems Simulated</caption><thead><tr><th>x<sub>FA</sub></th><th>N<sub>FA</sub></th><th>N<sub>wat</sub></th><th>ρ/g cm<sup>-3</sup></th><th>L/Å</th></tr></thead><tbody><tr><td>0</td><td>0</td><td>512</td><td>0.997</td><td>24.85</td></tr><tr><td>0.1</td><td>51</td><td>461</td><td>1.028</td><td>25.81</td></tr><tr><td>0.2</td><td>102</td><td>410</td><td>1.051</td><td>26.69</td></tr><tr><td>0.3</td><td>154</td><td>358</td><td>1.068</td><td>27.52</td></tr><tr><td>0.4</td><td>205</td><td>307</td><td>1.082</td><td>28.32</td></tr><tr><td>0.5</td><td>256</td><td>256</td><td>1.093</td><td>29.08</td></tr><tr><td>0.6</td><td>307</td><td>205</td><td>1.103</td><td>29.80</td></tr><tr><td>0.7</td><td>358</td><td>154</td><td>1.111</td><td>30.49</td></tr><tr><td>0.8</td><td>410</td><td>102</td><td>1.118</td><td>31.15</td></tr><tr><td>0.9</td><td>461</td><td>51</td><td>1.124</td><td>31.78</td></tr><tr><td>1</td><td>512</td><td>0</td><td>1.130</td><td>32.39</td></tr></tbody></table>

0.6, 0.7, 0.8, and 0.9 have been simulated. The characteristics of the different systems simulated are collected in Table 1.

All simulations have been performed with the program MMC.⁵⁰ In a Monte Carlo step, a randomly chosen molecule

has been randomly displaced to a maximum distance of 0.25 Å,

and randomly rotated by no more than 10°. The geometry of

all molecules has been kept unchanged in the simulations. At

least 25% of the trial moves have been successful in every case.

All interactions have been truncated to zero beyond the

center-center cutoff distance of 12.4 Å, the O and N atoms 280 being regarded as the centers of the water and formamide 281 molecules, respectively. The long-range part of the electrostatic 282 interaction has been accounted for using the reaction field 283 correction method⁵¹⁻⁵³ under conducting boundary conditions. The systems have been equilibrated through 5 × 10⁷ 284 Monte Carlo steps; the value of <U> has then been evaluated 285 in a consecutive, 10⁸ 286 Monte Carlo steps long equilibrium 287 trajectory. 288

**Potential Models.** In the present study, we consider the 289 mixing of five different models of formamide with three widely 290 used water models. Since simulations have been performed at 291 six (virtual) temperatures and nine compositions for each 292 model combination and also for the neat systems, the total 293 number of simulations performed is 858. The formamide 294 potentials considered include the united atom OPLS model,² 295 the OPLS/AA model⁵⁴ modified by Pérez de la Luz et al.²⁸ 296 (referred to here as OPLS/AA_mod), the model belonging to 297 the CHARMM force field,⁵⁵ and those proposed by Cordeiro²⁷ 298 and by Macchiagodena et al.,²⁹ referred to here as MMPB. For 299 water, we consider the three-site SPC/E⁵⁶ and four-site 300 TIP4P⁵⁷ and TIP4P/2005⁵⁸ potentials. It should be noted 301 that, without additional testing, the CHARMM force field is 302 supposed to be used in combination with the SPC/E, while 303 OPLS with the TIP4P water model. However, the present 304 study represents a far more stringent test of the model 305 combinations considered than the ones on which such “thumb 306 rules” are based, hence all possible combinations of the above 307 models are included in the following analyses. 308

All models considered are pairwise additive; the interaction 309 energy of two molecules, *i* and *j*, being within the center- 310 center cutoff distance, is calculated as 311

$$ u_{ij} = \sum_{\alpha=1}^{N_i} \sum_{\beta=1}^{N_j} \frac{1}{4\pi\epsilon_0} \frac{q_{\alpha}q_{\beta}}{r_{i\alpha,j\beta}} + 4\epsilon_{\alpha\beta} \left[ \left( \frac{\sigma_{\alpha\beta}}{r_{i\alpha,j\beta}} \right)^{12} - \left( \frac{\sigma_{\alpha\beta}}{r_{i\alpha,j\beta}} \right)^{6} \right] \quad (11) \quad 312 $$

Here indices $\alpha$ and $\beta$ run over the $N_i$ and $N_j$ interaction sites 313 of molecules *i* and *j*, respectively, $\epsilon_0$ is the vacuum permittivity, 314 $r_{i\alpha,j\beta}$ is the distance of site $\alpha$ on molecule *i* from site $\beta$ on 315 molecule *j*, $q_\alpha$ and $q_\beta$ are the fractional charges corresponding 316 to the respective interaction sites, while $\epsilon_{\alpha\beta}$ and $\sigma_{\alpha\beta}$ are the 317 energy and distance parameters, respectively, of the Lennard-Jones interaction of sites $\alpha$ and $\beta$, related to the corresponding 318 parameters of the individual sites through the Lorentz- 319 Berthelot rule,⁵³ i.e., $\epsilon_{\alpha\beta} = (\epsilon_\alpha\epsilon_\beta)^{1/2}$ and $\sigma_{\alpha\beta} = (\sigma_\alpha+\sigma_\beta)/2$. We 320 have repeated the simulations involving the OPLS or OPLS/AA_mod formamide models using also the geometric means of 321 both Lennard-Jones parameters, but this change in the potential 322 has left all of our conclusions unchanged. 323

In the water models considered, the O atom is the only 326 center of a Lennard-Jones interaction, and the H atoms bear a 327 fractional charge of $q_H$. The negative fractional charge 328 compensating them is located on the O atom in the SPC/E 329 model, but it is displaced from the O atom by $r_{OM}$ to a 330

<table><caption>**Table 2.** Geometry and Interaction Parameters of the Water Models Used</caption><thead><tr><th>model</th><th>reference</th><th>r<sub>OH</sub>/Å</th><th>r<sub>OM</sub>/Å</th><th>α<sub>HOH</sub>/deg</th><th>σ/Å</th><th>ε/kJ mol<sup>-1</sup></th><th>q<sub>H</sub>/e</th></tr></thead><tbody><tr><td>SPC/E</td><td>56</td><td>1.000</td><td>0</td><td>109.47</td><td>3.1660</td><td>0.650</td><td>0.4238</td></tr><tr><td>TIP4P</td><td>57</td><td>0.9572</td><td>0.15000</td><td>104.52</td><td>3.1540</td><td>0.649</td><td>0.5200</td></tr><tr><td>TIP4P/2005</td><td>58</td><td>0.9572</td><td>0.1546</td><td>104.52</td><td>3.1589</td><td>0.775</td><td>0.5564</td></tr></tbody></table>
331 nonatomic interaction site, M, along the bisector of the H-O-
332 H bond angle, α<sub>HOH</sub>, in the TIP4P and TIP4P/2005 models.

333 The parameters of the water models considered are collected in Table 2. The OPLS model of formamide consists of only five 334 interaction sites; the CH group is treated in this model as a 335 united atom. The OPLS/AA_mod, CHARMM, and Cordeiro 336 models have interaction sites on all six atoms, while in the 337 MMPB model the fractional charge is displaced from the O 338 atom to two nonatomic interaction sites, denoted as L, located 339 at the distance of 0.31 Å from the O atom along its two lone 340 pair directions. According to the molecular geometry of 341 formamide, all interaction sites are arranged in a planar way 342 in all of these models. The interaction parameters of the 343 formamide models considered are collected in Table 3. 344

<table><caption>Table 3. Interaction Parameters of the Formamide Models Used</caption><thead><tr><th>model</th><th>reference</th><th>site</th><th>σ/Å</th><th>ε/kJ mol<sup>-1</sup></th><th>q/e</th></tr></thead><tbody><tr><td rowspan="4">OPLS</td><td rowspan="4">2</td><td>CH</td><td>3.80</td><td>0.4815</td><td>0.500</td></tr><tr><td>O</td><td>2.96</td><td>0.8792</td><td>-0.500</td></tr><tr><td>N</td><td>3.25</td><td>0.7118</td><td>-0.850</td></tr><tr><td>H</td><td></td><td></td><td>0.425</td></tr><tr><td rowspan="5">OPLS/AA_mod</td><td rowspan="5">28</td><td>C</td><td>3.80625</td><td>0.307524</td><td>0.1398</td></tr><tr><td>O</td><td>3.00440</td><td>0.615048</td><td>-0.5283</td></tr><tr><td>H(C)</td><td>2.45630</td><td>0.043932</td><td>0.1753</td></tr><tr><td>N</td><td>3.29875</td><td>0.497896</td><td>-0.4163</td></tr><tr><td>H</td><td></td><td></td><td>0.3121<sup>b</sup><br>0.3174<sup>c</sup></td></tr><tr><td rowspan="6">CHARMM</td><td rowspan="6">55</td><td>C</td><td>3.175</td><td>0.293</td><td>0.42</td></tr><tr><td>O</td><td>2.700</td><td>0.502</td><td>-0.51</td></tr><tr><td>H(C)</td><td>2.100</td><td>0.090</td><td>0.08</td></tr><tr><td>N</td><td>2.940</td><td>0.837</td><td>-0.69</td></tr><tr><td>H</td><td>0.356</td><td>0.192</td><td>0.35</td></tr><tr><td>C</td><td>3.75</td><td>0.440</td><td>0.340</td></tr><tr><td rowspan="6">Cordeiro</td><td rowspan="6">27</td><td>O</td><td>2.96</td><td>0.880</td><td>-0.460</td></tr><tr><td>H(C)</td><td>2.75</td><td>0.159</td><td>0.120</td></tr><tr><td>N</td><td>3.25</td><td>0.712</td><td>-0.830</td></tr><tr><td>H</td><td></td><td></td><td>0.415</td></tr><tr><td>C</td><td>3.75</td><td>0.43932</td><td>0.154544</td></tr><tr><td>O</td><td>2.96</td><td>0.87864</td><td>0</td></tr><tr><td rowspan="5">MMPB</td><td rowspan="5">29</td><td>L<sup>a</sup></td><td></td><td></td><td>-0.201887</td></tr><tr><td>H(C)</td><td>2.42</td><td>0.06276</td><td>0.131902</td></tr><tr><td>N</td><td>3.25</td><td>0.71128</td><td>-0.566816</td></tr><tr><td>H</td><td></td><td></td><td>0.339036<sup>b</sup><br>0.345109<sup>c</sup></td></tr></tbody></table>

<sup>a</sup>Nonatomic interaction site. <sup>b</sup>H atom in cis relative position with the carboxylic oxygen. <sup>c</sup>H atom in trans relative position with the carboxylic oxygen.

## RESULTS AND DISCUSSION

345 The Helmholtz free energy, energy, and entropy of the water- 346 formamide mixtures of various compositions, including the two 347 neat systems, with respect to the ideal gas state are shown in 348 panels a, b, and c of Figure 3, respectively, as obtained from the 349 thermodynamic integration for the various model combinations 350 considered. These quantities show a nearly linear composition 351 dependence in most cases, suggesting that the mixing of the 352 two components is probably close to the ideal mixing. Markedly 353 different behavior was previously seen for mixtures of acetone 354 both with water<sup>34,35</sup> and methanol.<sup>40</sup> The free energy of the 355 liquid decreases with increasing formamide mole fraction. This 356

free energy decrease is clearly of energetic origin, as both the 357 energy and the entropy of the systems decrease as the 358 formamide concentration increases. It is also seen that these 359 quantities can be rather different for different model 360 combinations. This difference mainly originates from the 361 different thermodynamic properties of the formamide models 362 considered: the difference of the Helmholtz free energy, energy, 363 and entropy of neat formamide can be as large as 50, 35, and 364 20%, respectively, when different potentials are considered. 365

The Helmholtz free energy, energy, and entropy change 366 accompanying the mixing of neat liquid water and formamide 367 are plotted in panels a, b, and c of Figure 4, respectively, as 368 obtained with the different model combinations. As is clear 369 from the figure, the free energy of mixing is negative in every 370 case, indicating that the mixture is always thermodynamically 371 more stable than the two separate neat liquids. The magnitude 372 of F<sup>mix</sup> is rather small, being always below 2 kJ/mol. For 373 comparison, the value of RT is about 2.5 kJ/mol at 298 K. (The 374 value of RT/2, i.e., the average kinetic energy of the molecules 375 along one single degree of freedom, is shown for comparison in 376 Figure 4.) Similarly, small values are obtained for U<sup>mix</sup> and S<sup>mix</sup> 377 in every case, their values being always within RT/2 and R/2, 378 respectively. As a consequence, the obtained F<sup>mix</sup>, U<sup>mix</sup>, and S<sup>mix</sup> 379 values are also rather close to each other as obtained with 380 different model combinations, their difference never exceeds 381 1.3, 1.8, and 3.5 J/(mol K), respectively. The free energy of 382 mixing is the smallest in magnitude as obtained with the OPLS, 383 Cordeiro, and MMPB models of formamide (the results 384 obtained with these three models are always within error bars of 385 each other), and deepest as obtained with the OPLS/AA_mod 386 model. No marked difference is seen between the water models 387 in this respect. 388

The energy of mixing shows a qualitatively different behavior 389 when the OPLS/AA_mod model of formamide is considered 390 than what is seen with the other formamide models. Namely, it 391 is always below zero in mixtures of the OPLS/AA_mod model, 392 but it is positive in all other cases. Although the deviation from 393 the experimental values<sup>12,13</sup> is roughly of the same magnitude 394 with all formamide models considered, the experimental data 395 supports the qualitative behavior of the OPLS, CHARMM, 396 Cordeiro, and MMPB models rather than that of OPLS/ 397 AA_mod in this respect. Thus, according to the experimental 398 data,<sup>12,13</sup> the observed negative free energy of mixing of water 399 and formamide is of entropic origin, as the energy of mixing is 400 very small and positive. Consistently, the entropy of mixing is 401 found to be positive for all model combinations, being the 402 largest when the CHARMM model of formamide is considered. 403

Figure 4c also includes the entropy change corresponding to 404 ideal mixing, i.e., $-R[x_{FA} \ln x_{FA} + (1 - x_{FA}) \ln(1 - x_{FA})]$. As is 405 clearly seen from the figure, the obtained S<sup>mix</sup> values are rather 406 close to this ideal term; their difference never exceeds 2.5 J/ 407 (mol K), and, considering the OPLS, Cordeiro, and MMPB 408 models, this difference always remains below 1 J/(mol K), 409 being within the error bars of the calculation. The largest 410 entropy of mixing values are always obtained with the 411 CHARMM, while the smallest ones with the OPLS/AA_mod 412 model of formamide, the S<sup>mix</sup>(x<sub>FA</sub>) curves obtained with the 413 former model being always clearly above, and those with the 414 latter one always below the values corresponding to the entropy 415 of ideal mixing. Thus, contrary to the case of OPLS/AA_mod, 416 the relatively low free energy of mixing values obtained with the 417 CHARMM model of formamide are of entropic origin, while 418 for the other three formamide models the obtained negative 419
<figure><img src="image_5.png" /><figcaption>**Figure 3.** (a) Helmholtz free energy, (b) energy, and (c) entropy of mixtures of the SPC/E (top panel), TIP4P (middle panel), and TIP4P/2005 (bottom panel) water, and OPLS (black), OPLS/AA\_mod (red), CHARMM (green), Cordeiro (dark blue), and MMPB (light blue) formamide models of various compositions with respect to the ideal gas state. The data obtained by thermodynamic integration are shown by full circles; the lines connecting the symbols only serve as guides to the eye. Error bars are typically below ±0.2 kJ/mol and ±1 J/(mol K).</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>**Figure 4.** (a) Helmholtz free energy of mixing, (b) energy of mixing, and (c) entropy of mixing of the SPC/E (top panel), TIP4P (middle panel), and TIP4P/2005 (bottom panel) models of water with the OPLS (black), OPLS/AA\_mod (red), CHARMM (green), Cordeiro (dark blue), and MMPB (light blue) models of formamide. Error bars are typically below ±0.2 kJ/mol and ±1 J/(mol K). The calculated data are shown by full circles; the lines connecting the symbols only serve as guides to the eye. Zero value is indicated by dashed line, experimental data of the energy of mixing (ref 12, 13) is shown by a thick line in panel (b), the entropy of ideal mixing is shown by dotted line in panel (c). The average kinetic energy of the molecules along one degree of freedom of RT/2 (for the Helmholtz free energy and energy of mixing) and R/2 (for the entropy of mixing) is also shown for reference by orange bars.</figcaption></figure>

free energy of mixing with water is simply the consequence of the ideal term in the entropy of mixing. It should finally be noted that both the energy and entropy of mixing values turned out to be very close to the term corresponding to ideal mixing (i.e., 0 for $U^{\text{mix}}$ and $-R[x_{\text{FA}} \ln x_{\text{FA}} + (1 - x_{\text{FA}}) \ln(1 - x_{\text{FA}})]$ for $S^{\text{mix}}$) for every model combination and at every composition, confirming our previous suggestion, based on the nearly linear dependence the Helmholtz free energy, energy, and entropy of

the liquid phase on $x_{\text{FA}}$ (see Figure 3) that the mixing of these to compounds is nearly ideal.

As it has been emphasized previously, a negative value of $F^{\text{mix}}$ is a necessary but not sufficient condition of miscibility; the two components are fully miscible with each other if and only if the value of the D parameter, defined in eq 1, is positive in the entire composition range. To determine the $D(x_{\text{FA}})$ data for the different model combinations, we fitted the Margules equation:⁵⁹
$$F^{\text{ex}}/RT = x_{\text{FA}}(1 - x_{\text{FA}})[A_{12}x_{\text{FA}} + A_{21}(1 - x_{\text{FA}}) - (C_{12}x_{\text{FA}} + C_{21}(1 - x_{\text{FA}}))x_{\text{FA}}(1 - x_{\text{FA}}) + Bx_{\text{FA}}^2(1 - x_{\text{FA}})^2] \quad (12)$$

437

438 where A₁₂, A₂₁, C₁₂, C₂₁, and B are the Margules parameters, to

439 the calculated F<sup>ex</sup>(x<sub>FA</sub>) data, and calculated the second

440 derivative of this fitted function. The quality of the Margules

441 fit is illustrated in the inset of Figure 5 for the mixtures of the

442 OPLS/AA_mod model of formamide with the SPC/E and

443 TIP4P/2005 water models.

<figure><img src="image_7.png" /><figcaption>**Figure 5.** Composition dependence of the parameter *D*, defined by eq 1, in mixtures of the SPC/E (top panel), TIP4P (middle panel), and TIP4P/2005 (bottom panel) water, and OPLS (black), OPLS/AA_mod (red), CHARMM (green), Cordeiro (dark blue), and MMPB (light blue) formamide models. Estimated uncertainty of the data is below ±0.1. The inset shows the calculated F<sup>ex</sup> data (full circles), together with their Margules fit (eq 12, solid curves) for mixtures of the OPLS/AA_mod model of formamide with the SPC/E (orange) and TIP4P/2005 (magenta) water models.</figcaption></figure>

444

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450

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453

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455

456

457

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461

462

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466

even below the estimated uncertainty of the *D* data of about 0.1, thus, immiscibility of some of these model combinations might occur in a narrow composition range. Further, full miscibility of these model combinations might also depend on the details of the simulation (e.g., long-range correction, etc.).

On the other hand, the *D* values obtained using the OPLS/AA_mod formamide model with any of the three water models considered remains safely above zero in the entire composition range, indicating that the OPLS/AA_mod model is indeed fully miscible with all these three water models. On the other hand, as it has been pointed out previously, these model combinations are qualitatively incompatible with existing experimental data<sup>12,13</sup> in respect to the energy of mixing, as they correspond to negative rather than positive energy of mixing values in the entire composition range (see Figure 4b).

Finally, the *D*(*x*<sub>FA</sub>) curves obtained with the CHARMM model of formamide exhibit a clear minimum around the equimolar composition. The value of this minimum is around 0.2 when the SPC/E, and 0.28 when the TIP4P water model is used, but drops clearly below zero in combination with the TIP4P/2005 water model. Thus, the CHARMM and TIP4P/2005 models exhibit a narrow miscibility gap in the *x*<sub>FA</sub> range of about 0.4–0.55, while the CHARMM model of formamide is fully miscible with both the SPC/E and TIP4P water models. Further, as seen from Figure 4b, these model combinations are also qualitatively compatible with the experimental data<sup>12,13</sup> concerning the sign of the energy of mixing. Considering also the fact that the *D*(*x*<sub>FA</sub>) curve remains farther above zero when the TIP4P rather than the SPC/E water model is used, we recommend the use of the combination of the CHARMM model of formamide and TIP4P model of water when water-formamide mixtures are simulated.

Finally, it should be noted that whenever the value of *D* is found to drop below zero, it remains relatively small in magnitude, never going below about –1. This means that the thermodynamic driving force of the immiscibility of the corresponding model combinations is rather small. Therefore, in simulations of the bulk liquid phase, when demixing is also suppressed by the use of periodic boundary conditions, the occurrence of a visible phase separation might require much longer simulations than what is usually done.<sup>60</sup> However, if the liquid mixture is in contact with an anisotropic object, such as in the presence of the liquid-vapor interface, phase separation occurs rather quickly.

## ■ SUMMARY AND CONCLUSIONS

504

In this article we have studied the thermodynamic changes that occur upon mixing water and formamide as well as their miscibility in computer simulation using the combination of five formamide and three water models. The Helmholtz free energy, energy, and entropy of mixing of these model pairs have been calculated along an appropriately chosen thermodynamic cycle using the method of thermodynamic integration. To evaluate the performance of the various model combinations, we used the well-known fact that these compounds are fully miscible with each other,<sup>1</sup> and the experimental values of their energy of mixing in the entire composition range.<sup>12,13</sup>

505

506

507

508

509

510

511

512

513

514

515

The results have revealed that the mixing of the two molecules is nearly ideal, as the values of both the energy and the entropy of mixing have turned out to be close to the ideal term in the entire composition range. The free energy of mixing has been found to be always negative, indicating that the mixtures are always more stable than the two neat components.

516

517

518

519

520

521
522 However, concerning the origin of this negative free energy of
523 mixing, different models of formamide have shown different
524 behavior. Thus, in the case of the OPLS/AA_mod model, this
525 is mainly of energetic origin (as here the energy of mixing is
526 always negative, while the entropy of mixing is clearly smaller
527 than the value corresponding to ideal mixing), while in the case
528 of the other four formamide models considered, it is clearly of
529 entropic origin (as the energy of mixing is positive). Existing
530 experimental data supports this latter behavior.<sup>12,13</sup>

531 The OPLS model of formamide has been found to exhibit
532 miscibility gaps with all the three water models considered,
533 while the model combinations involving the Cordeiro or
534 MMPB formamide models very closely approach the miscibility
535 limit at a certain composition, and hence full miscibility in these
536 cases might also depend on the exact conditions under which
537 the simulations are performed. Finally, the CHARMM model
538 of formamide has shown limited miscibility with the TIP4P/
539 2005 model, but full miscibility with the SPC/E and TIP4P
540 models of water. Since the latter of these two model
541 combinations remains farther from the miscibility limit than
542 the former one, the present results allow us to recommend the
543 use of the combination of the CHARMM model of formamide
544 and TIP4P model of water in simulations of water-formamide
545 mixtures.

546 ■ AUTHOR INFORMATION

547 Corresponding Author

548 *E-mail: jedlovszky.pal@uni-eszterhazy.hu.

549 ORCID

550 Abdenacer Idrissi: 0000-0002-6924-6434

551 Milán Szőri: 0000-0003-4895-0999

552 Pál Jedlovszky: 0000-0001-9304-435X

553 Notes

554 The authors declare no competing financial interest.

555 ■ ACKNOWLEDGMENTS

556 The authors acknowledge financial support from the NKFIH

557 Foundation, Hungary (project number 119732) from the

558 Hungarian-French TéT (PHC Balaton) program under project

559 Nos. 36402ND (France) and TÉT_15-1-2016-0029 (Hungary). M.Sz. is a Bolyai János Research Scholar of the

560 Hungarian Academy of Sciences (BO/00113/15/7). This

561 research was supported by the European Union and the

562 Hungarian State, cofinanced by the European Regional

563 Development Fund in the framework of the GINOP-2.3.4.15-

564 2016-00004 project, aimed to promote the cooperation

565 between the higher education and the industry.

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