Effect of truncating electrostatic interactions on predicting thermodynamic properties of water–methanol systems

Figures & data

Table 1. Force field parameters for water used in this study.

Force field	TIP3P [32]	SPCE/E [34]	OPC [81]	TIP4P/2005 [17]	TIP4P/EW [82]
${\epsilon }_{\mathrm{O}\mathrm{O}}$	76.500	78.175	107.086	93.196	81.899
${\sigma }_{\mathrm{O}\mathrm{O}}$	3.1506	3.1660	3.1666	3.1589	3.1644
$q_{\mathrm{O}}$	−0.8340	−0.8476	−1.3582	–	–
$q_{\mathrm{H}}$	0.41700	0.42380	0.67910	0.55640	0.52422
$q_{\mathrm{L}}$	–	–	–	−1.11280	−1.04844
$r_{\mathrm{OH}}$	0.9572	1.0000	0.8724	0.9572	0.9572
$r_{\mathrm{OL}}$	–	–	0.1594	0.1546	0.1250

Notes: All molecules are considered rigid. ε is reported in units of K, σ and r are reported in units of Å.

Table 2. Force field parameters for methanol used in this study.

Force field	OPLS/2016 [84]	TraPPE [83]
${\epsilon }_{\mathrm{O}\mathrm{O}}$	97.775	93.000
${\sigma }_{\mathrm{O}\mathrm{O}}$	3.1659	3.0200
${\epsilon }_{{\mathrm{CH}}_3{\mathrm{CH}}_3}$	110.450	98.000
${\sigma }_{{\mathrm{CH}}_3{\mathrm{CH}}_3}$	3.6449	3.75
$q_{\mathrm{O}}$	−0.6544	−0.70000
$q_{\mathrm{H}}$	0.49980	0.43500
$q_{{\mathrm{CH}}_3}$	0.1546	0.2650
$r_{\mathrm{OH}}$	0.9450	0.9450
$r_{{\mathrm{CH}}_3\mathrm{O}}$	1.43	1.43

Notes: All molecules are considered rigid. ε is reported in units of K, σ and r are reported in units of Å.

Figure 1. (Colour online) Relative differences in computed electrostatic energies between the Wolf method, Equation (3(3) $\begin{array}{cc}{E}^{\mathrm{Wolf}}=& \frac{1}{2}\underset{r_{iajb}<R_{\mathrm{c}}}{\sum _{i=1}^{N_{\mathrm{m}}}\sum _{a=1}^{N_{\mathrm{a}}^i}\sum _{\genfrac{}{}{0ex}{}{j=1}{j\ne i}}^{N_{\mathrm{m}}}\sum _{b=1}^{N_{\mathrm{a}}^j}}{q}_{ia}{q}_{jb}\left[\frac{\mathrm{erfc}\left(\alpha r_{iajb}\right)}{r_{iajb}}-\frac{\mathrm{erfc}\left(\alpha R_{\mathrm{c}}\right)}{R_{\mathrm{c}}}\right]\\ & +\frac{1}{2}\underset{r_{iaib}<R_{\mathrm{c}}}{\sum _{i=1}^{N_{\mathrm{m}}}\sum _{a=1}^{N_{\mathrm{a}}^i}\sum _{\genfrac{}{}{0ex}{}{b=1}{b\ne a}}^{N_{\mathrm{a}}^i}}{q}_{ia}{q}_{ib}\left[\frac{\mathrm{erfc}\left(\alpha r_{iaib}\right)}{r_{iaib}}-\frac{\mathrm{erfc}\left(\alpha R_{\mathrm{c}}\right)}{R_{\mathrm{c}}}\right]\\ & -\frac{1}{2}\sum _{i=1}^{N_{\mathrm{m}}}\sum _{a=1}^{N_{\mathrm{a}}^i}\sum _{\genfrac{}{}{0ex}{}{b=1}{b\ne a}}^{N_{\mathrm{a}}^i}\frac{q_{ia}{q}_{ib}}{r_{iaib}}\\ & -\left[\frac{\mathrm{erfc}\left(\alpha R_{\mathrm{c}}\right)}{2R_{\mathrm{c}}}+\frac{\alpha }{\sqrt{\pi }}\right]\sum _{i=1}^{N_{\mathrm{m}}}\sum _{a=1}^{N_{\mathrm{a}}^i}{q}_{ia}^2.\end{array}$(3) ), and the Ewald summation for (a) water and (b) methanol. The parameters for the Ewald summation are calculated based on relative precision of ${10}^{-6}$ [93]. The SPC/E [34] and OPLS/2016 [84] force fields were used to obtain the densities of water and methanol at T=298 K and P=1 bar. Individual configurations were obtained at constant densities of 1000 $\mathrm{kg}{\mathrm{m}}^{-3}$ and 748 $\mathrm{kg}{\mathrm{m}}^{-3}$ for water and methanol, respectively.

Figure 2. (Colour online) Coupling parameter λ to scale the interactions of fractional molecules. ${\lambda }_{\mathrm{LJ}}\in [0,1]$ is the coupling parameter used to scale the LJ interactions of the fractional molecule (Equation (10(10) $\begin{array}{rl}{u}_{\mathrm{LJ}}& \left(r,{\lambda }_{\mathrm{LJ}}\right)={\lambda }_{\mathrm{LJ}}4\epsilon \\ & \left(\frac{1}{{\left[\frac{1}{2}{\left(1-{\lambda }_{\mathrm{LJ}}\right)}^2+{\left(\frac{r}{\sigma }\right)}^6\right]}^2}-\frac{1}{\left[\frac{1}{2}{\left(1-{\lambda }_{\mathrm{LJ}}\right)}^2+{\left(\frac{r}{\sigma }\right)}^6\right]}\right),\end{array}$(10) )). At $\lambda ={\lambda }^{\ast }$, the Coulombic interactions are switched on. ${\lambda }_{\mathrm{Coul}}\in [0,1]$ is the coupling parameter used to scale the Coulombic interactions of the fractional molecule (Equation (11(11) $\begin{array}{cc}{E}_{\mathrm{Coul}}^{\mathrm{DSF}}\left(r,{\lambda }_{\mathrm{Coul}}\right)& =\frac{1}{2}\underset{r_{iajb}<R_{\mathrm{c}}}{\sum _{i=1}^{N_{\mathrm{m}}}\sum _{a=1}^{N_{\mathrm{a}}^i}\sum _{\genfrac{}{}{0ex}{}{j=1}{j\ne i}}^{N_{\mathrm{m}}}\sum _{b=1}^{N_{\mathrm{a}}^j}}{\lambda }_{\mathrm{Coul}}{q}_{ia}{q}_{jb}\left[\frac{\mathrm{erfc}\left(\alpha \left[r_{iajb}+r_{\ast }\right]\right)}{r_{iajb}+r_{\ast }}-\frac{\mathrm{erfc}\left(\alpha \left[R_{\mathrm{c}}+r_{\ast }\right]\right)}{R_{\mathrm{c}}+r_{\ast }}\right.\\ & +\left.\left(\frac{\mathrm{erfc}\left(\alpha \left[R_{\mathrm{c}}+r_{\ast }\right]\right)}{{\left(R_{\mathrm{c}}+r_{\ast }\right)}^2}+\frac{2\alpha }{\sqrt{\pi }}\frac{\exp \left(-{\alpha }^2{\left[R_{\mathrm{c}}+r_{\ast }\right]}^2\right)}{R_{\mathrm{c}}+r_{\ast }}\right)\left(r_{iajb}-R_{\mathrm{c}}\right)\right]\\ & +\frac{1}{2}\underset{r_{iaib}<R_{\mathrm{c}}}{\sum _{i=1}^{N_{\mathrm{m}}}\sum _{a=1}^{N_{\mathrm{a}}^i}\sum _{\genfrac{}{}{0ex}{}{b=1}{b\ne a}}^{N_{\mathrm{a}}^i}}{\lambda }_{\mathrm{Coul}}{q}_{ia}{q}_{ib}\left[\frac{\mathrm{erfc}\left(\alpha \left[r_{iaib}+r_{\ast }\right]\right)}{r_{iaib}+r_{\ast }}-\frac{\mathrm{erfc}\left(\alpha \left[R_{\mathrm{c}}+r_{\ast }\right]\right)}{R_{\mathrm{c}}+r_{\ast }}\right]\\ & -\frac{1}{2}\sum _{i=1}^{N_{\mathrm{m}}}\sum _{a=1}^{N_{\mathrm{a}}^i}\sum _{\genfrac{}{}{0ex}{}{b=1}{b\ne a}}^{N_{\mathrm{a}}^i}\frac{{\lambda }_{\mathrm{Coul}}{q}_{ia}{q}_{ib}}{r_{iaib}+r_{\ast }}\\ & -\left[\frac{\mathrm{erfc}\left(\alpha R_{\mathrm{c}}\right)}{2R_{\mathrm{c}}}+\frac{\alpha }{\sqrt{\pi }}\right]\sum _{i=1}^{N_{\mathrm{m}}}\sum _{a=1}^{N_{\mathrm{a}}^i}{\lambda }_{\mathrm{Coul}}{q}_{ia}^2,\end{array}$(11) )).

Figure 3. (Colour online) Radial distribution functions of water–methanol mixtures (50–50%), at T=298 K and P=1 bar, for: (a) water–water (b) water–methanol (c) methanol–methanol. The TIP4P/2005 [17] and TraPPE [83] force fields were used to compute the density of water–methanol mixtures in MD and MC simulations. The relative difference in densities obtained from MD and MC simulations was 0.2%. To compute the long-range electrostatic interactions, the Ewald and DSF methods were used in MD simulations. In the MC simulations, the Wolf and DSF methods (Equations (3(3) $\begin{array}{cc}{E}^{\mathrm{Wolf}}=& \frac{1}{2}\underset{r_{iajb}<R_{\mathrm{c}}}{\sum _{i=1}^{N_{\mathrm{m}}}\sum _{a=1}^{N_{\mathrm{a}}^i}\sum _{\genfrac{}{}{0ex}{}{j=1}{j\ne i}}^{N_{\mathrm{m}}}\sum _{b=1}^{N_{\mathrm{a}}^j}}{q}_{ia}{q}_{jb}\left[\frac{\mathrm{erfc}\left(\alpha r_{iajb}\right)}{r_{iajb}}-\frac{\mathrm{erfc}\left(\alpha R_{\mathrm{c}}\right)}{R_{\mathrm{c}}}\right]\\ & +\frac{1}{2}\underset{r_{iaib}<R_{\mathrm{c}}}{\sum _{i=1}^{N_{\mathrm{m}}}\sum _{a=1}^{N_{\mathrm{a}}^i}\sum _{\genfrac{}{}{0ex}{}{b=1}{b\ne a}}^{N_{\mathrm{a}}^i}}{q}_{ia}{q}_{ib}\left[\frac{\mathrm{erfc}\left(\alpha r_{iaib}\right)}{r_{iaib}}-\frac{\mathrm{erfc}\left(\alpha R_{\mathrm{c}}\right)}{R_{\mathrm{c}}}\right]\\ & -\frac{1}{2}\sum _{i=1}^{N_{\mathrm{m}}}\sum _{a=1}^{N_{\mathrm{a}}^i}\sum _{\genfrac{}{}{0ex}{}{b=1}{b\ne a}}^{N_{\mathrm{a}}^i}\frac{q_{ia}{q}_{ib}}{r_{iaib}}\\ & -\left[\frac{\mathrm{erfc}\left(\alpha R_{\mathrm{c}}\right)}{2R_{\mathrm{c}}}+\frac{\alpha }{\sqrt{\pi }}\right]\sum _{i=1}^{N_{\mathrm{m}}}\sum _{a=1}^{N_{\mathrm{a}}^i}{q}_{ia}^2.\end{array}$(3) ) and (5(5) $\begin{array}{cc}{E}^{\mathrm{DSF}}& =\frac{1}{2}\underset{r_{iajb}<R_{\mathrm{c}}}{\sum _{i=1}^{N_{\mathrm{m}}}\sum _{a=1}^{N_{\mathrm{a}}^i}\sum _{\genfrac{}{}{0ex}{}{j=1}{j\ne i}}^{N_{\mathrm{m}}}\sum _{b=1}^{N_{\mathrm{a}}^j}}{q}_{ia}{q}_{jb}\left[\frac{\mathrm{erfc}\left(\alpha r_{iajb}\right)}{r_{iajb}}-\frac{\mathrm{erfc}\left(\alpha R_{\mathrm{c}}\right)}{R_{\mathrm{c}}}\right.\\ & +\left.\left(\frac{\mathrm{erfc}\left(\alpha R_{\mathrm{c}}\right)}{R_{\mathrm{c}}^2}+\frac{2\alpha }{\sqrt{\pi }}\frac{\exp \left(-{\alpha }^2{R}_{\mathrm{c}}^2\right)}{R_{\mathrm{c}}}\right)\left(r_{iajb}-R_{\mathrm{c}}\right)\right]\\ & +\frac{1}{2}\underset{r_{iajb}<R_{\mathrm{c}}}{\sum _{i=1}^{N_{\mathrm{m}}}\sum _{a=1}^{N_{\mathrm{a}}^i}\sum _{\genfrac{}{}{0ex}{}{b=1}{b\ne a}}^{N_{\mathrm{a}}^i}}{q}_{ia}{q}_{ib}\left[\frac{\mathrm{erfc}\left(\alpha r_{iaib}\right)}{r_{iaib}}-\frac{\mathrm{erfc}\left(\alpha R_{\mathrm{c}}\right)}{R_{\mathrm{c}}}\right]\\ & -\frac{1}{2}\sum _{i=1}^{N_{\mathrm{m}}}\sum _{a=1}^{N_{\mathrm{a}}^i}\sum _{\genfrac{}{}{0ex}{}{b=1}{b\ne a}}^{N_{\mathrm{a}}^i}\frac{q_{ia}{q}_{ib}}{r_{iaib}}\\ & -\left[\frac{\mathrm{erfc}\left(\alpha R_{\mathrm{c}}\right)}{2R_{\mathrm{c}}}+\frac{\alpha }{\sqrt{\pi }}\right]\sum _{i=1}^{N_{\mathrm{m}}}\sum _{a=1}^{N_{\mathrm{a}}^i}{q}_{ia}^2.\end{array}$(5) ))) were used.

Figure 4. (Colour online) Excess enthalpies of mixing for water–methanol mixtures based on the TIP4P/2005 [17] and TraPPE [83] force fields at T=298 K and P=1 bar. To compute the electrostatic energies, the DSF and Ewald [92] methods were used in MD simulations. In MC simulations, the Wolf and DSF methods (Equations (3(3) $\begin{array}{cc}{E}^{\mathrm{Wolf}}=& \frac{1}{2}\underset{r_{iajb}<R_{\mathrm{c}}}{\sum _{i=1}^{N_{\mathrm{m}}}\sum _{a=1}^{N_{\mathrm{a}}^i}\sum _{\genfrac{}{}{0ex}{}{j=1}{j\ne i}}^{N_{\mathrm{m}}}\sum _{b=1}^{N_{\mathrm{a}}^j}}{q}_{ia}{q}_{jb}\left[\frac{\mathrm{erfc}\left(\alpha r_{iajb}\right)}{r_{iajb}}-\frac{\mathrm{erfc}\left(\alpha R_{\mathrm{c}}\right)}{R_{\mathrm{c}}}\right]\\ & +\frac{1}{2}\underset{r_{iaib}<R_{\mathrm{c}}}{\sum _{i=1}^{N_{\mathrm{m}}}\sum _{a=1}^{N_{\mathrm{a}}^i}\sum _{\genfrac{}{}{0ex}{}{b=1}{b\ne a}}^{N_{\mathrm{a}}^i}}{q}_{ia}{q}_{ib}\left[\frac{\mathrm{erfc}\left(\alpha r_{iaib}\right)}{r_{iaib}}-\frac{\mathrm{erfc}\left(\alpha R_{\mathrm{c}}\right)}{R_{\mathrm{c}}}\right]\\ & -\frac{1}{2}\sum _{i=1}^{N_{\mathrm{m}}}\sum _{a=1}^{N_{\mathrm{a}}^i}\sum _{\genfrac{}{}{0ex}{}{b=1}{b\ne a}}^{N_{\mathrm{a}}^i}\frac{q_{ia}{q}_{ib}}{r_{iaib}}\\ & -\left[\frac{\mathrm{erfc}\left(\alpha R_{\mathrm{c}}\right)}{2R_{\mathrm{c}}}+\frac{\alpha }{\sqrt{\pi }}\right]\sum _{i=1}^{N_{\mathrm{m}}}\sum _{a=1}^{N_{\mathrm{a}}^i}{q}_{ia}^2.\end{array}$(3) ) and (5(5) $\begin{array}{cc}{E}^{\mathrm{DSF}}& =\frac{1}{2}\underset{r_{iajb}<R_{\mathrm{c}}}{\sum _{i=1}^{N_{\mathrm{m}}}\sum _{a=1}^{N_{\mathrm{a}}^i}\sum _{\genfrac{}{}{0ex}{}{j=1}{j\ne i}}^{N_{\mathrm{m}}}\sum _{b=1}^{N_{\mathrm{a}}^j}}{q}_{ia}{q}_{jb}\left[\frac{\mathrm{erfc}\left(\alpha r_{iajb}\right)}{r_{iajb}}-\frac{\mathrm{erfc}\left(\alpha R_{\mathrm{c}}\right)}{R_{\mathrm{c}}}\right.\\ & +\left.\left(\frac{\mathrm{erfc}\left(\alpha R_{\mathrm{c}}\right)}{R_{\mathrm{c}}^2}+\frac{2\alpha }{\sqrt{\pi }}\frac{\exp \left(-{\alpha }^2{R}_{\mathrm{c}}^2\right)}{R_{\mathrm{c}}}\right)\left(r_{iajb}-R_{\mathrm{c}}\right)\right]\\ & +\frac{1}{2}\underset{r_{iajb}<R_{\mathrm{c}}}{\sum _{i=1}^{N_{\mathrm{m}}}\sum _{a=1}^{N_{\mathrm{a}}^i}\sum _{\genfrac{}{}{0ex}{}{b=1}{b\ne a}}^{N_{\mathrm{a}}^i}}{q}_{ia}{q}_{ib}\left[\frac{\mathrm{erfc}\left(\alpha r_{iaib}\right)}{r_{iaib}}-\frac{\mathrm{erfc}\left(\alpha R_{\mathrm{c}}\right)}{R_{\mathrm{c}}}\right]\\ & -\frac{1}{2}\sum _{i=1}^{N_{\mathrm{m}}}\sum _{a=1}^{N_{\mathrm{a}}^i}\sum _{\genfrac{}{}{0ex}{}{b=1}{b\ne a}}^{N_{\mathrm{a}}^i}\frac{q_{ia}{q}_{ib}}{r_{iaib}}\\ & -\left[\frac{\mathrm{erfc}\left(\alpha R_{\mathrm{c}}\right)}{2R_{\mathrm{c}}}+\frac{\alpha }{\sqrt{\pi }}\right]\sum _{i=1}^{N_{\mathrm{m}}}\sum _{a=1}^{N_{\mathrm{a}}^i}{q}_{ia}^2.\end{array}$(5) )) were used to treat the electrostatic interactions. The solid line indicates experimental values for the excess mixing enthalpy [49]. Dotted lines are a guide to the eye. Error bars are smaller than symbol sizes. Raw data are listed in Tables S1, S5 and S15 of the Supporting Information.

Figure 5. (Colour online) Excess chemical potentials of: (a) water, (b) methanol, with respect to the ideal gas phase, in water–methanol mixtures obtained from MC simulations in the CFCNPT ensemble [73], at T=298 K and P=1 bar. The Wolf and the DSF methods (Equations (3) and (5)) were used to calculate the electrostatic interactions. The TIP4P/2005 [17] and TraPPE [83] force fields were used. Error bars are smaller than symbol sizes. Raw data are listed in Tables S5 and S15 of the Supporting Information.

Figure 6. (Colour online) Activity coefficients of: (a) water, (b) methanol in water–methanol mixtures obtained from MC simulations in the CFCNPT ensemble, at T=298 K and P=1 bar. The Wolf [90] and the DSF [89] methods were used to calculate the electrostatic interactions. The TIP4P/2005 [17] and TraPPE [83] force fields were used. The line indicates experimental values for the activity coefficients [55]. Raw data are listed in Tables S5 and S15 of the Supporting Information.

Figure 7. (Colour online) Excess mixing enthalpies for water–methanol mixtures defined by: (a) TraPPE [83] and (b) OPLS/2016 [84] force fields at T=298 K and P=1 bar. The TIP3P [33], SPC/E [34], OPC [81], TIP4P/2005 [17], TIP4P/EW [82] force fields were considered for water. The DSF method (Equation (5(5) $\begin{array}{cc}{E}^{\mathrm{DSF}}& =\frac{1}{2}\underset{r_{iajb}<R_{\mathrm{c}}}{\sum _{i=1}^{N_{\mathrm{m}}}\sum _{a=1}^{N_{\mathrm{a}}^i}\sum _{\genfrac{}{}{0ex}{}{j=1}{j\ne i}}^{N_{\mathrm{m}}}\sum _{b=1}^{N_{\mathrm{a}}^j}}{q}_{ia}{q}_{jb}\left[\frac{\mathrm{erfc}\left(\alpha r_{iajb}\right)}{r_{iajb}}-\frac{\mathrm{erfc}\left(\alpha R_{\mathrm{c}}\right)}{R_{\mathrm{c}}}\right.\\ & +\left.\left(\frac{\mathrm{erfc}\left(\alpha R_{\mathrm{c}}\right)}{R_{\mathrm{c}}^2}+\frac{2\alpha }{\sqrt{\pi }}\frac{\exp \left(-{\alpha }^2{R}_{\mathrm{c}}^2\right)}{R_{\mathrm{c}}}\right)\left(r_{iajb}-R_{\mathrm{c}}\right)\right]\\ & +\frac{1}{2}\underset{r_{iajb}<R_{\mathrm{c}}}{\sum _{i=1}^{N_{\mathrm{m}}}\sum _{a=1}^{N_{\mathrm{a}}^i}\sum _{\genfrac{}{}{0ex}{}{b=1}{b\ne a}}^{N_{\mathrm{a}}^i}}{q}_{ia}{q}_{ib}\left[\frac{\mathrm{erfc}\left(\alpha r_{iaib}\right)}{r_{iaib}}-\frac{\mathrm{erfc}\left(\alpha R_{\mathrm{c}}\right)}{R_{\mathrm{c}}}\right]\\ & -\frac{1}{2}\sum _{i=1}^{N_{\mathrm{m}}}\sum _{a=1}^{N_{\mathrm{a}}^i}\sum _{\genfrac{}{}{0ex}{}{b=1}{b\ne a}}^{N_{\mathrm{a}}^i}\frac{q_{ia}{q}_{ib}}{r_{iaib}}\\ & -\left[\frac{\mathrm{erfc}\left(\alpha R_{\mathrm{c}}\right)}{2R_{\mathrm{c}}}+\frac{\alpha }{\sqrt{\pi }}\right]\sum _{i=1}^{N_{\mathrm{m}}}\sum _{a=1}^{N_{\mathrm{a}}^i}{q}_{ia}^2.\end{array}$(5) )) was used to treat the electrostatic interactions. The solid line indicates experimental values for the excess mixing enthalpy [49]. Dotted lines are a guide to the eye. Raw data are listed in Tables S2-S11 of the Supporting Information.

Figure 8. (Colour online) Activity coefficients of water and methanol in water–methanol mixtures for different combinations of water–methanol force fields, at T=298 K and P=1 bar. In subfigures (a) and (b); the TraPPE force field was used for methanol and in subfigures (c) and (d); the OPLS/2016 force field was used for methanol. The TIP3P [33], SPC/E [34], OPC [81], TIP4P/2005 [17], TIP4P/EW [82] force fields were considered for water. The DSF method (Equation (5(5) $\begin{array}{cc}{E}^{\mathrm{DSF}}& =\frac{1}{2}\underset{r_{iajb}<R_{\mathrm{c}}}{\sum _{i=1}^{N_{\mathrm{m}}}\sum _{a=1}^{N_{\mathrm{a}}^i}\sum _{\genfrac{}{}{0ex}{}{j=1}{j\ne i}}^{N_{\mathrm{m}}}\sum _{b=1}^{N_{\mathrm{a}}^j}}{q}_{ia}{q}_{jb}\left[\frac{\mathrm{erfc}\left(\alpha r_{iajb}\right)}{r_{iajb}}-\frac{\mathrm{erfc}\left(\alpha R_{\mathrm{c}}\right)}{R_{\mathrm{c}}}\right.\\ & +\left.\left(\frac{\mathrm{erfc}\left(\alpha R_{\mathrm{c}}\right)}{R_{\mathrm{c}}^2}+\frac{2\alpha }{\sqrt{\pi }}\frac{\exp \left(-{\alpha }^2{R}_{\mathrm{c}}^2\right)}{R_{\mathrm{c}}}\right)\left(r_{iajb}-R_{\mathrm{c}}\right)\right]\\ & +\frac{1}{2}\underset{r_{iajb}<R_{\mathrm{c}}}{\sum _{i=1}^{N_{\mathrm{m}}}\sum _{a=1}^{N_{\mathrm{a}}^i}\sum _{\genfrac{}{}{0ex}{}{b=1}{b\ne a}}^{N_{\mathrm{a}}^i}}{q}_{ia}{q}_{ib}\left[\frac{\mathrm{erfc}\left(\alpha r_{iaib}\right)}{r_{iaib}}-\frac{\mathrm{erfc}\left(\alpha R_{\mathrm{c}}\right)}{R_{\mathrm{c}}}\right]\\ & -\frac{1}{2}\sum _{i=1}^{N_{\mathrm{m}}}\sum _{a=1}^{N_{\mathrm{a}}^i}\sum _{\genfrac{}{}{0ex}{}{b=1}{b\ne a}}^{N_{\mathrm{a}}^i}\frac{q_{ia}{q}_{ib}}{r_{iaib}}\\ & -\left[\frac{\mathrm{erfc}\left(\alpha R_{\mathrm{c}}\right)}{2R_{\mathrm{c}}}+\frac{\alpha }{\sqrt{\pi }}\right]\sum _{i=1}^{N_{\mathrm{m}}}\sum _{a=1}^{N_{\mathrm{a}}^i}{q}_{ia}^2.\end{array}$(5) )) was used to treat the electrostatic interactions. The solid lines indicate experimental values for the activity coefficients [55]. Dotted lines are a guide to the eye. Raw data are listed in Tables S2–S11 of the Supporting Information.

Figure 9. (Colour online) Excess chemical potentials of water and methanol, with respect to the ideal gas phase, for different combinations of water–methanol force fields, at T=298 K and P=1 bar. In subfigures (a) and (b); the TraPPE force field was used for methanol and in subfigures (c) and (d); the OPLS/2016 force field was used for methanol. The TIP3P [33], SPC/E [34], OPC [81], TIP4P/2005 [17], TIP4P/EW [82] force fields were considered for water. Error bars are smaller than symbol sizes. The solid lines indicate experimental values for the chemical potentials [142–145]. Dotted lines are a guide to the eye. Raw data are listed in Tables S2–S11 of the Supporting Information.

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