Calculation of the orientational linear and nonlinear

Calculation of the orientational linear and nonlinear correlation factors of polar liquids from the rotational Dean-Kawasaki equation P. M. Déjardin, Y. Cornaton, P. Ghesquière, C. Caliot, and R. Brouzet

Citation: The Journal of Chemical Physics 148, 044504 (2018); View online: https://doi.org/10.1063/1.5010295 View Table of Contents: http://aip.scitation.org/toc/jcp/148/4 Published by the American Institute of Physics

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THE JOURNAL OF CHEMICAL PHYSICS 148, 044504 (2018)

Calculation of the orientational linear and nonlinear correlation factors of polar liquids from the rotational Dean-Kawasaki equation 1 Y. Cornaton,1 P. Ghesquiere, ´ ` 1 C. Caliot,2 and R. Brouzet1 P. M. Dejardin, 1 Laboratoire

de Math´ematiques et Physique, Universit´e de Perpignan Via Domitia, 52, Avenue Paul Alduy, F-66860 Perpignan Cedex, France 2 Laboratoire Proc´ ed´es, Mat´eriaux et Energie Solaire, PROMES-CNRS, 7 Rue du Four Solaire, F-66120 Font-Romeu-Odeillo-Via, France

(Received 24 October 2017; accepted 8 January 2018; published online 25 January 2018) A calculation of the Kirkwood and Piekara-Kielich correlation factors of polar liquids is presented using the forced rotational diffusion theory of Cugliandolo et al. [Phys. Rev. E 91, 032139 (2015)]. These correlation factors are obtained as a function of density and temperature. Our results compare reasonably well with the experimental temperature dependence of the linear dielectric constant of some simple polar liquids across a wide temperature range. A comparison of our results for the linear dielectric constant and the Kirkwood correlation factor with relevant numerical simulations of liquid water and methanol is given. Published by AIP Publishing. https://doi.org/10.1063/1.5010295

I. INTRODUCTION

The Kirkwood-Fr¨ohlich theory of the linear dielectric constant ε 1 is the most successful one to explain a number of experimental data as far as the probing DC electric field amplitude is weak. By “weak,” it is meant that the fieldinduced orientational energy per dipole ξ = µE/ (kT ) 1,  2 56λ while for the positive sign βUm (z, z 0) = λzz 0, they become gK(+) (λ)

(+) (−) FIG. 1. The Kirkwood correlation factors gK and gK as a function of λ. Solid line: exact calculation from Eqs. (51) and (60). Dashed line: asymptotic expansions of these quantities for large λ. Dotted line: small λ expansion.

λ    1 − , λ > 1,  2 16λ

4λ    , λ > 1.  2 28λ

(64)

of the Yvon-Born-Green hierarchy generated by the averaged equilibrium rotational Dean-Kawasaki equation, i.e., Eq. (15) for a typical model interaction potential with minima representing the parallel and antiparallel alignments of the dipole pairs, Eq. (36), as equilibrium states. Moreover, specifying this interaction potential has allowed us to calculate the various correlation factors involved in the linear dielectric constant of polar liquids as well as in the cubic nonlinear dielectric increment. Since the approximations we have made are related to the first term in the virial expansion, they are of practical value at weak densities only. However, since the various quantities technically can be calculated for all densities, we believe that it is interesting to examine the temperature and density dependences of all the correlation factors that have been calculated here. They are shown in Figs. 1 and 2. Regarding the Kirkwood correlation factor (Fig. 1), we see that 0 < gK(+) ≤ 1, while gK(−) ≥ 1. This is indeed as expected because the plus sign corresponds to a tendency for dipoles to align antiparallel, while the minus sign corresponds to a tendency to align parallel, and this is the physics which is encoded in the model interaction potential, Eq. (36), inspired by the literature in the area of ferromagnetism, where the exchange constant in the Heisenberg Hamiltonian is positive for the parallel alignment and negative for the antiparallel one.32 Both types of factors mainly display monotonic behavior at low λ

(65)

IV. RESULTS AND DISCUSSION

We have just demonstrated how the first term in the virial expansion of the pair distribution function may be recovered via the Dean-Kawasaki formalism. This derivation is relatively simple and relies on the method of successive approximations applied to the Ornstein-Zernike relation, interpreted as a Fredholm integral equation for the orientational pair correlation function. Thus, we were able to solve the first member

FIG. 2. The Piekara-Kielich correlation factor as a function of λ. The same as in Fig. 1, save the exact calculation is achieved from Eqs. (51) and (61).

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and are constant at large values of λ, 0.5 for the antiparallel alignment and 3.5 for the parallel alignment. Those saturation values are doubtless a consequence of the virial approximation that we have implicitly made in order to avoid solving the Ornstein-Zernike equation. Some examples of the temperature dependence of the Kirkwood correlation factor can be found in the literature in both cases.2,33 Nevertheless, the large values found in supercooled glycerol (gK ≈ 6)34 prevent our calculations being applied to the supercooled state in general. This was indeed expected from the above general discussion. The temperature and density behavior of the nonlinear cubic correlation factor is shown in Fig. 2 for both parallel and antiparallel alignments. The most interesting feature is the anomalous behavior found in the anti-parallel case, where the Piekara-Kielich factor RP(+) changes sign at low densities and then saturates to the value 0.5 at large densities. This resembles to what happens, for example, in solutions of nitrobenzene in nonpolar solvents2 at low and large densities, where the nonlinear dielectric increment switches from positive to negative values, so that at weak concentrations the anomalous saturation effect dominates the normal one. The fact that the dipolar arrangement in nitrobenzene is antiparallel is corroborated by its experimental Kirkwood correlation factor which is less than 1,35 which was measured recently. For parallel alignment, such effect does not exist as is apparent from Fig. 2: RP(−) maintains its (negative) sign so that the trend to parallel alignment always leads to the normal saturation effect in the context of our calculations. V. COMPARISON WITH EXPERIMENT

To reinforce of our calculations, comparisons of them with experimental data concerning the temperature behavior of the linear dielectric constant of some polar liquids with already well-known dielectric properties are required. To accomplish this, we must recall that molecules may also have a molecular polarizability determined by the linear complex permittivity at high frequencies ε ∞ , defined as the dielectric constant at frequencies where the orientational mechanism has ceased to operate. This definition allows one to alter slightly ε ∞ for fitting experimental data if required. In order to compare our calculations with experimental data, we also recall that the dipole moment µ occurring in Eq. (3) is not the dipole moment in the gas phase µg .2,36 Instead, it is that of the dipole in the liquid phase, which is related to that of the gas phase by the equation2 ε∞ + 2 µg . µ= 3 Since in our calculations, gK depends on the interaction parameter λ defined by Eq. (37), we can define λ g by the equation 4π ρ0 β µg λg = 3

√ gK(−) (λ) =

TABLE I. Values of the parameters used in Eq. (67) for liquid water and liquid methanol together with the temperature range within which that equation may be applied.

a (kg m 3 ) Water 0.143 95 Methanol 54.566

b 0.0112 0.233 211

c (K)

d

649.727 0.051 07 513.16 0.208 875

Temperature range of validity (K) 273 < T < 648 181 < T < 513

so that the relation between λ and λ g is (ε ∞ + 2)2 λg. 9 Bearing the latter in mind, we rewrite Eq. (3) in the KirkwoodFr¨ohlich form,36 viz., (ε − ε ∞ ) (2ε + ε ∞ ) λ g = gK (λ) . (66) 3 ε(ε ∞ + 2)2 λ=

We do not attempt a detailed comparison with experiment because we neglect the temperature dependence of ε ∞ . However, we account for the temperature dependence of the density of the liquid under consideration and select this dependence from the DIPPR (Design Institute for Physical Properties) 105 equation,37 yielding the volumic mass M of a liquid as a function of temperature. This equation is a , (67) M (T ) = d 1+(1− Tc ) b where a, b, c, and d are parameters fitted to experimental data, which are tabulated for quite a number of substances.38 The parameters a, b, c, and d for liquid water and methanol are given in SI units in Table I. The number of molecules per unit volume at a given temperature T is calculated as ρ0 (T ) =

M (T ) NA , Mmol

where M mol is the molar mass of a molecule of the liquid and N A is Avogadro’s number, i.e., N A = 6.02 × 1023 mol 1 . Equation (67) well reproduces the experimental values of the density of liquids in the entire temperature range where they are in the liquid state, with the possible exception near the dilatometric anomaly of water, where the value rendered by Eq. (67) differs from the experimental one by an order of at most 5% in relative error. Therefore, in the Kirkwood-Fr¨ohlich formula, we may consider that using Eq. (67) accounts for the experimental values of the density as a function of temperature. The two examples chosen are the temperature dependence of pure liquid water 39 and liquid methanol,40 where in Eq. (66) we use the analytical expression of gK(−) (λ) that we have obtained, and so it represents a tendency for the dipoles to orient in the parallel direction. This expression is

√  2πλ 3/2 (15 + 4λ) erfi 2λ + 4 (2 − λ) e2λ − 16 (3 + 5λ)  √   , √ 60 λ − λe2λ + 2πλ 3/2 erfi 2λ

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where erfi (z) is the error function of imaginary argument given by z 2 2 erfi (z) = √ et dt. π 0

The comparison between our calculations and experimental measurements of the dielectric constant of water and methanol is shown in Fig. 3. For liquid water, as illustrated by Fig. 3, the agreement is very good across the whole temperature range, taking a (temperature-independent) mean optical refractive index n = 1.35 and ε ∞ = n2 . This agreement is even surprising because of the complexity of intermolecular interactions in such a liquid and also because the microstructure of water does not appear explicitly in our calculations. Here, we suppose that the information concerning the microstructure is hidden in the density of the liquid as a function of temperature, where in essence the intermolecular interactions are properly accounted for since the density used in the Kirkwood-Fr¨ohlich formula is the experimental density. The rest of the agreement may be explained either on the basis that the higher order terms in the virial expansion of the orientational pair distribution function are ignored in comparison to the term given by Eq. (33) or by compensation of large terms in the virial expansion. Also, as is often the case, this may be explained by compensation of errors. Despite these caveats, our analytical expression for gK(−) (λ) remains acceptable in future applications involving the linear static dielectric constant of water since no fitting parameter is required in order to obtain the presented agreement between theory and experiment. The fact that the microstructure of water is not explicitly involved in the calculation of its linear dielectric constant using our model encouraged us to compare our calculations with the results for the experimental dielectric constant of some other polar liquids. Thus we attempted a comparison with another well-studied polar liquid, viz., methanol, where we have taken n = 1.45, i.e., slightly above the experimental value n = 1.32, yielding ε ∞ ≈ 1.18n2 . Between 40 ◦ C and 60 ◦ C, the agreement is just as good as for water, possibly for the same reasons as the ones we just discussed regarding our comparison with water data. However, a noticeable discrepancy appears below

FIG. 3. The temperature variation of the dielectric constant. Curve (1): liquid water;38 curve (2): liquid methanol.39 Solid line: Kirkwood-Fr¨ohlich equation (−) (λ). Dots: experimental data. (66) with gK = gK

40 ◦ C. Here, either the temperature dependence of ε ∞ may

be invoked in order to fit low-temperature data2,35 or we may conclude that the interaction potential (36) or (37) is no longer appropriate for the description of these data. Also by defining ε ∞ as the value of the permittivity at a frequency where the orientational mechanism has ceased to operate, in general, makes ε ∞ weakly temperature-dependent so that it exceeds the square of the refractive index. Then, since the dielectric constant determined by the Kirkwood-Fr¨ohlich equation is extremely sensitive to the value of ε ∞ which is used, the foregoing ansatz may be of use in reconciling our calculations with experimental data. Since we have also derived an expression for the Kirkwood correlation factor for antiparallel orientation of the dipoles, comparison with experimental data is also in order. We have chosen acetic acid which has a dielectric constant ε = 6.20 at ambient conditions with a dipole moment µg = 1.74 D. Now the formula we have derived for the Kirkwood-Fr¨ohlich correlation factor for the (preferred) antiparallel alignment is √  √ 2πλ erfi 2λ + 4λ + 1 *.8 − 4λ + √  / . gK(+) (λ) = √ 2λ + 2πλ erfi 12 1 − e 2λ , Thus, using the relation between the dipole moment in the liquid and the gas phases, ε ∞ = n2 with n = 1.37, we obtain ε = 4.35 at 293 K and a Kirkwood correlation factor gK = 0.45. Now, slightly altering ε ∞ (as just alluded to the above, this tuning corresponds to the experimental definition of this quantity as the value of the dielectric constant at a frequency where the orientational mechanism has ceased to operate, once more no fitting parameter is involved), i.e., ε ∞ = 2.575 ≈ 1.37n2 , we obtain the desired value for the dielectric constant of acetic acid, namely, ε = 6.206, and a Kirkwood correlation factor gK = 0.47 so that latter is slightly smaller than the experimental value gK = 0.64 found by Parthipan et al.,41 and ε ∞ being slightly larger. The value of the latter is again mostly explained by dipolar arrangement, while the microstructure arguments do not explicitly appear in our calculation of the dielectric constant of this particular polar liquid. This argument also holds for substances such as acetonitrile and CH2 Cl2 where the present calculations may be fitted to experiment with ε ∞ as an adjustable parameter. Now, certain limitations to the present calculations exist because for some substances they do not compare favorably at all well with experimental data, e.g., we have attempted to compare our calculations with the linear dielectric constant of glycerol. Here, the calculations markedly disagree with experiment, even at room temperature, predicting either a too small (antiparallel) or a too large (parallel) dielectric constant. Now, we could also have fitted ε ∞ for the antiparallel case to experiment; however, this would have rendered a Kirkwood correlation factor that is far too small with respect to its experimental value. Here, we believe that it is the form of the interaction potential, Eq. (36), chosen which is unable to capture the underlying physics of the dipoles of glycerol molecules. Thus, we believe that dipolar order in glycerol is more subtle than just being the simple parallel/antiparallel

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alignment of pairs of dipoles at equilibrium, even in the liquid state. The calculations proposed here with the potential given by Eq. (36) or (37) also do not agree either for polar substances where the Kirkwood-Fr¨ohlich factor is about 1 irrespective of the temperature. In order to justify this value of gK , a model for the interaction energy which differs from Eq. (36) must be proposed. For example, this situation occurs in some monohydroxy-alcohols, where this value of gK is attributed to the coexistence of polar chains and nonpolar rings (see Ref. 8 for a review). Another well-known polar liquid where this occurs is pure dimethyl sulfoxide (DMSO) for which Onsager’s equation [i.e., our Eq. (66) with gK = 1] works extremely well at all temperatures where this substance remains in liquid form.42 With the present calculations, the sole possibility of attaining a Kirkwood correlation factor near unity is to assume that only one molecule exists in the cavity so that our model interaction potential37 is replaced by zero (this is also evident on Fig. 1). Although in this case there is no preferred orientation for a pair of dipoles, this ansatz also means that the dipoles do not interact. This is clearly an absurd result for dense polar liquids. Therefore, in using the Dean-Kawasaki formalism, a theoretical potential appropriate to this situation has still to be developed, and this is left for future investigations. We do not compare our expressions for the nonlinear dielectric increments with experimental data because an exact theory of the dielectric increments at arbitrary order in the field strength has still to be developed.2,3 Nevertheless, some physical insight may be gained from the investigation of the existing theoretical correlation factors. As it stands, our calculations show that it is sometimes possible to obtain analytical expressions for the various correlation factors involved in the linear and nonlinear dielectric susceptibilities without invoking the detailed microstructure of the liquid under study and that quantitative agreement with experiment can sometimes be obtained without explicitly appealing to this microstructure. This is so in spite of the fact that this microstructure may be significant even for a “simple” associated liquid such as water, where a priori the contribution of nearest and next-nearest neighbors must be accounted for in order to obtain agreement with experiment. However, this does NOT mean that we exclude the contribution of the microstructure of the liquid on the dielectric constant. Instead, we believe that in certain situations, explicit account of the microstructure is not crucial in the calculation of the dielectric constant of polar liquids. We have ignored dynamical aspects of the problem, which requires the calculation of the complex permittivity of polar liquids. Considering the dynamics requires a kinetic equation for W2 (u, u 0, t) which will greatly increase the mathematical complexity because that equation must be solved in conjunction with Eq. (13). Thus, although the dynamical effects may be treated via the Dean-Kawasaki formalism, this poses a far more complicated problem that will be dealt with elsewhere. VI. COMPARISON WITH NUMERICAL SIMULATIONS

We also compare our results for the dielectric constant with molecular dynamics simulations which were accomplished for various force fields.

J. Chem. Phys. 148, 044504 (2018) TABLE II. Comparison between the dielectric constant of water obtained from the molecular dynamics numerical simulations of Zhang and Galli22 and those of the present work.

T T T T T T

= 240 K = 260 K = 280 K = 300 K = 350 K = 400 K

ε, Ref. 22

ε, this work

89.90 ± 0.41 85.19 ± 0.54 76.44 ± 0.40 73.16 ± 0.29 57.16 ± 0.29 48.82 ± 0.08

96.95 89.27 82.69 76.99 65.57 57.01

We compare first our results with the numerical simulations of Zhang and Galli.22 They have used the SPC/E (Simple Point Charge/Extended) potential for rigid, nonpolarizable molecules and have calculated the dielectric constant of water for several temperatures. The results are given in Table II. We note that the temperatures 240, 260, and 400 K are irrelevant for the liquid state of water, where our calculations are not valid. However, our values always exceed those computed by Zhang and Galli because these authors did not include the polarizability of the molecules in any way22 in their simulations. Instead, Zhang et al.21 have recently proposed a new algorithm for computing the dielectric constant and Kirkwood correlation factor of liquid water from SPC/E molecular dynamics in particular. This algorithm attempts to account for distortional (or displacement) polarization and a fortiori includes polarizability effects. Their study is accomplished at T = 298 K only. They found for the dielectric constant ε = 71. We find at this temperature that ε = 77.45, slightly higher, but also in better agreement with experimental data (the experimental value is ε = 78.339 ). We infer that this slight disagreement between their simulation and our result originates in the two distinct ways in which the polarizability of the molecules is included, for example, in one way via our analytical formula and in the other by their simulation (with zero electric displacement) which are not exactly equivalent. Nevertheless, our calculated dipole moment in the liquid phase of water is µ = 2.35 D, in agreement with usual SPC/E molecular dynamics simulations.21 Finally, comparison between our results and the various potentials used in molecular dynamics simulations, including the Watanabe-Klein (WK) simulation is summarized in Table III. They are computed at T = 298 K at ordinary pressures. The relevant Kirkwood correlation factors were computed by van der Spoel et al.23 for a different temperature, i.e., T = 303 K. Comparison with our results is meaningful at ordinary pressures only since this is the range within which our calculations are valid. At this temperature, our analytical formula for the dielectric constant yields ε = 75.9, in TABLE III. Comparison between the dielectric constant obtained from various interaction potentials as used in molecular dynamic simulations and our analytical formula at T = 298 K.

ε

SPC/E

SPC

TIP4P

WK

This work

Experiment

70 ± 9

72 ± 7

61 ± 7

80 ± 8

77.45

78.3

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TABLE IV. Comparison between the Kirkwood correlation factors obtained from various interaction potentials used in molecular dynamic simulations of (−) water and our formula for gK at T = 303 K.

gK

SPC/E

SPC

TIP4P

This work

2.53

2.39

1.95

3.34

agreement with experiment. Comparison of the various gK values are summarized in Table IV. As expected, gK nearest to ours is yielded by SPC/E simulations. This is also the numerical simulation method that yields a Kirkwood correlation factor closest to the experimental values. Numerical simulations of methanol have also been undertaken by Fonseca and Ladanyi using molecular dynamics24 and by Deb et al.25 using Density Functional Theory (DFT) calculations with Becke’s third functional for the exchange and Lee, Yang, and Parr’s functional for the correlation (B3LYP). Again, the calculations of Refs. 24 and 25 are compared in Ref. 25 at T = 298 K, while Deb et al.25 have also calculated the temperature dependence of the dielectric constant of methanol. Comparison of the values yielded by the present work with those of Fonseca and Ladanyi and Deb et al. are shown in Tables V and VI. Thus our calculation of gK and ε substantially agrees with the experimental values for methanol and water and is generally better than numerical simulations as long as the liquid phase of these substances is considered. Now, as already explained by Scaife9 and Evans et al.,36 it is extremely difficult to disentangle the various polarization mechanisms in Eq. (3) (this is also substantiated by Fonseca and Ladanyi24 ). In particular, as expressed by our temperature-dependent KirkwoodFr¨ohlich equation (66), the Kirkwood correlation factor is itself affected by the distortional polarization mechanism and thus by the temperature dependence of the density of the

liquid. Therefore, postulating that the distortional polarization mechanism has been completely separated out in Eq. (3) is not correct, as hM 2 i0 includes them in a nontrivial way. This is also true here, where because of the transcendental nature of the error function of imaginary argument, the separation in Eq. (66) is also incomplete. However, one should recall that by using the asymptotic expansions for large λ in Eqs. (62) and (64) and then by replacing the dipole moment by that in the liquid state in Eq. (3), separation is possible although it is not exact. VII. CONCLUSION

Here, we have shown how linear and non-linear static orientational correlation factors can be calculated analytically via the Dean-Kawasaki approach without explicitly invoking the detailed microstructure of some simple polar liquids. Nevertheless, we do not mean that the microstructure of the liquid is no longer relevant for the calculation of the dielectric constant of polar liquids. Rather, we believe that its contribution is mostly concealed in the temperature dependence of the density of the liquid and, to some degree, in the dielectric constant ε ∞ . Our calculations have been compared with experimental data for some simple associated polar liquids, and provided the potential is suitably chosen, quantitative agreement between theory and experiment may sometimes be obtained with minimal effort across a wide temperature range. Our results can also be applied to orientational correlations of magnetic dipole moments in blocked ferrofluids, so leading to the linear and nonlinear dynamic susceptibilities of polar fluids and assemblies of magnetic nanoparticles. ACKNOWLEDGMENTS

We are grateful to Dr. F. Ladieu, Professor F. van Wijland, Professor W. T. Coffey, and Professor Yu. P. Kalmykov for helpful conversations. 1 J.

TABLE V. Comparison of gK and ε at 298 K for liquid methanol from Ref. 24 and from the present work.

Reference 24 This work Experiment

gK

ε

1.9 ± 0.2 3.10 2.94

23 32.25 32.6

TABLE VI. Temperature dependence of gK and ε for liquid methanol from Ref. 25 and from the present work. T (K) gK (Ref. 25) gK (this work) ε (Ref. 25) ε (this work) ε (expt.) 288 293 298 303 308 313 318

1.87 1.87 1.87 1.87 1.87 1.87 1.87

3.13 3.12 3.10 3.08 3.08 3.07 3.06

28.49 27.83 27.18 26.57 25.97 25.39 24.84

33.91 33.07 32.25 31.45 30.68 29.93 29.19

34.05 33.60 32.60 31.64 30.68 30.60 28.87

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