Lecture 4

1. Lecture 4 • Statistical-mechanical approach to dielectric theory. • Kirkwood-Fröhlich's equation. • The Kirkwood correlation factor. • Applications: pure dipole liquids, mixtures, dipolar solids.

2. (4.1) The methods of statistical mechanics provide a way of obtaining macroscopic quantities when the properties of the molecules and the molecular interactions are known. All statistical-mechanical theories of the dielectric constant start from the consideration that the polarization P, given by: is equal to the dipole density P, when the influence of higher multipole densities may be neglected. By definition, we may write the dipole densityP of a homogeneous system: whereV is the volume of the dielectric under consideration an <M> is its average total (dipole) moment (the brackets < > denote a statistical mechanical average). If we assume the system to be isotropic, we also have

3. (4.2) (4.3) Thus we find: Since the dielectric is isotropic, <M> will have the same direction as E and it will be sufficient to calculate the average component of M in the direction of E. Using eto denote a unit vector in the direction of the field, we may therefore rewrite Eqn (4.2) in scalar form: Another way of writing Eqn.(4.3) results form the fact that Pand <M> contain in general also terms in higher powers of Ethan the first (see eqn.(1.19)). Thus, (-1)E/4 is the first term in a series development of Pin powers of E, and must be set equal to the term linear in E of the series development of <Me>/V in powers of E. Since variations in V due to electrostriction do not appear in the leaner term we may develop <Me> in a Taylor series, finding:

4. (4.4) (4.5) Rewriting with the external field Eoinstead of the Maxwell field Eas the independent variable we obtain: For the average of a quantity like Me, which is a function of the positions and orientations of all molecules, one can write: where X denotes the set of position and orientation variables of all molecules. This expression for the average can be obtained from the general ensemble average by integrating over the momenta. The integration over the moment results in a weight factor, which is contained in our notation dX. For example, if we take spherical coordinates r,, for the position of a molecule and integrate over the conjugated momenta, we obtain a weight factor r2sin, in this integration over the coordinates of that molecule. Thus, in this case dX=r2sindrdd, which is the expression for a volume element in spherical coordinates.

5. (4.6) Before the expression for the dielectric constant in terms of molecular quantities can be obtained two problems have to be solved. • E, the Maxwell field, has to be expressed as a function of the external field Eo. In some special cases electrostatic theory leads to a simple relation between Eoand E, which can be used immediately. In general case, however,Eis given by the sum of the external field and the average field due to all molecules of the dielectric. It must be remarked that even for homogeneous Eo the Maxwell field Ewill not be homogeneous. 2.<Me>, expressed as a function of Eo, has to be calculated as the average of the sum of the dipole moments of all molecules for a given value for the external field. When the total number of particles in the dielectric is N, and the instantaneous dipole moment of the i-th particle is mi, we may write for the instantaneous total momentM:

6. Let us take a region with N molecules which are treated explicitly; the remaining N-N molecules are considered to form a continuum and are treated as such. The approximations in this method can be made as small as necessary by taking N sufficiently large. If this value of N is still manageable in the calculations, the method can be used to introduce the molecular interactions into the calculation of the dielectric constant of polar liquids. Non-polarizable molecules (rigid dipoles) Let us consider the idealized case that the polarizability of the molecules can be neglected, so that only the permanent dipole moments have to be taken into account. We are taking a sphere of volume V, containing N molecules. For convenience in the calculations we suppose that it is embedded in its own material, which extends to infinity. The material outside the sphere can be treated as a continuum with dielectric constant .In this case the external field working in the sphere is the cavity field (eqn.2.14):

7. (4.7) (4.8) where N=N/V is the number density, and the tensor Aplays the role of a polarizability; Because we defined that the (4.9) where Eis the Maxwell field in the material outside the sphere. We can substitute (4.6) in the general expression for the dielectric constant of homogeneous, isotropic dielectric: molecules are non polarizable, <eAe>o=0. In this case after substitution of (4.7) into (4.8), we obtain: The average of the square of the total moment can be calculated as follows in the case of non-polarizable molecules:

8. (4.10) (4.11) so that we may write: In this equation the superscript N to dX to emphasize that the integration is performed over the positions and orientations of N molecules. The integration in the numerator of eqn.(4.11) can be carried out in two steps. Since iis a function of the orientation of the i-th molecule only, the integration over the positions and orientations of all other molecules, denoted as N-i, can be carried out first. In this way we obtain (apart from a normalizing factor) the average moment of the sphere in the field of the i-th dipole with fixed orientation. The averaged moment, denoted by Mi*, can be written as:

9. (4.12) (4.13) (4.14) The average moment Mi* is a function of the position and orientation of the i-th molecule only. Expression (4.12) for Mi*can be substituted into eqn. (4.11). Denoting the position and orientation coordinates of the i-th molecule by Xiand using a weight factor p(Xi) We obtain: since after integration over the positions and orientations of moleculei, the resulting expression will not depend on the value of i.

10. (4.15) (4.16) (4.17) Before we substitute the expression for <M2>o into eqn.(4.9) , we note that it is possible to rewrite expression (4.14) in suggestive form. According to eqn.(4.12) and (4.13) Mi* can be written as the sum of momentsj, averaged with the orientation of the i-th dipole held fixed. denoting the angle between the orientation of thei-th and the j-th dipole by ij,this leads to: and thus to: This expression for <M2>o can be abbreviated by introducing the average of cosij, defined as:

11. (4.18) (4.19) (4.20) We then write: If we now substitute Eqn. (4.14) or its another form (4.18) into (4.9) we find after some rearrangements, and using N=N/V for the number density: Let us compare this expression with (3.34) from the previous lecture, for a special case of a pure dipole liquid with non-polarizable molecules(=0): This expression was derived in previous lecture with the help of the continuum approach, in which a sphere containing only one moleculeis used. Therefore the (4.20) is a special case of (4.19) when the sphere containing Nmolecules is restricted to one, Mi* =iand cosijis equal to 1.

12. (4.21) When the sphere contains more than one molecule, the value of Mi* can be different from i . When the number of molecules included in the sphere increases, Mi* reaches a limiting value, so that Mi* will be independent of N as long as N exceeds a certain minimum value. The dipole moment Mof a sphere in the field of an arbitrary charge distribution within it, is given by: where mis the dipole moment of the charge distribution. Since this expression forM does not depend on the radius of the sphere, the dipole moment of a spherical shell in the field of a point dipole within the inner sphere must be zero. This conclusion will also hold if the sphere is not in vacuum, but embedded in a dielectric, even with same dielectric constant as the sphere itself.

13. Thus the addition of a number of molecules contained in a spherical shell to original number N will not change the moment of the sphere as long as the spherical shell can be treated macroscopically. From this argument we conclude that the deviations of Mi* from the value i are the result of molecular interactions between the i-th molecule and its neighbors. It is well known that liquids are characterized by short-range order and long-range disorder. The correlations between the orientations (and also between positions) due to the short-range ordering will lead to values of Mi* differing from i. This is the reason that Kirkwood introduced a correlation factor g which accounted for the deviations of from the value 2:

14. (4.22) (4.23) With the help of this definition, eqn. (4.19) may be written as: When there is no more correlation between the molecular orientations than can be accounted for with the help of the continuum method, one has g=1 we are going to Onsager relation for the non-polarizable case, for rigid dipoles with =1. An approximate expression for the Kirkwood correlation factor can be derived by taking only nearest-neighbors interactions into account. In that case the sphere is shrunk to contain only the i-th molecule and its z nearest neighbors. We then have:

15. (4.24) (4.25) (4.26) with N=z+1. Substitution this into (4.22) and using the fact that the material is isotropic, we obtain: Since after averaging the result of the integration will be not depend on the value of j, all terms in the summation are equal and we may write: Since cosijdepends only on the orientation of the two molecules, all other coordinates can be integrated out and we may write:

16. where iandjdenote the orientation coordinates of thei-th and the j-th molecules, and is a rotational intermolecular interaction energy, averaged over all positions and the orientations of all other molecules. (4.27) It is clear from eqn. (4.27) that gwill be different from1when <cosij>0,i.e. when there is correlation between the orientations of neighboring molecules. When the molecules tend to direct themselves withparallel dipole moments, <cosij> will be positive and g>1. When the molecules prefer an ordering withanti-parallel dipoles, g <1.

17. (4.28) (4.29) Polarizable Molecules Let as consider the system of N identical molecules with permanent dipole strength and scalar polarizabilities . The i-th molecule is located at a point with radius vector ri and has an instantaneous dipole moment mi. This dipole moment will be given by: where (E1)i, the local field at the position of the i-th molecule for a specified configuration of the other molecules, is given by: In this equation Eois the external field and -Tij · mjis the field at ri due to a dipole moment mjat ri. We can also define the 33-dimensional dipole-dipole interaction tensor Tij, connected with the molecules iand j.

18. (4.30) (4.31) In the case of polarizable molecules the total moment of the sphere in Kirkwood approximation will be given by: where pi is the induced moment of the i-th molecule. The induced moment piis a function of the positions and orientations of all other molecules. Taking into account (4.28) we can write: The local field depends on the positions and orientations of all other molecules. Therefore it is not possible to perform the integrations in <M2>o in two steps , as we did in the non-polarizable case.

19. (4.32) Approximation of Fröhlich For the representation of a dielectric with dielectric permittivity , consisting of polarizable molecules with a permanent dipole moment, Fröhlich introduced a continuum with dielectric constant  in which point dipoles with a moment d are embedded. In this model each molecule is replaced by a point dipole d having the same non-electrostatic interactions with the other point dipoles as the molecules had, while the polarizabilityof the molecules can be imagined to be smeared out to form a continuum with dielectric constant . Let us also split polarization Pin two parts: the induced polarization Pin and the orientation polarization Por The induced polarization is equal to the polarization of the continuum with the , so that we can write

20. (4.33) (4.34) (4.35) The orientation polarization is given by the dipole density due to the dipoles d. If we consider a sphere with volume V containing N dipoles (as we did in non-polarizable case), we can write: where: <Md·e>, the average component in the direction of the field, of the moment due to the dipoles in the sphere, is given by an expression :

21.  (4.36) Here U is the energy of the dipoles in the sphere. This energy consists of three parts: • the energy of the dipoles in the external field • the electrostatic interaction energy of the dipoles • the non-electrostatic interaction energy(London-Van der Waals energy) between the molecules which is responsible for the short-range correlation between orientations and positions of the molecule. The external field in this model is equal to the field within a spherical cavity filled with a continuum with dielectric constant , while the cavity is situated in a dielectric with dielectric constant (Fröhlich field EF ). The field of this cavity field with dielectric will be:

22. (4.33) (4.37) (4.32) Thus the energy of the dipoles in the external field can be written as -Md·EF. Taking in consideration that the first derivative of P with respect to Ewas identified with (-1)/4 we can write: after substituting (4.32) and (4.33) and rearrangement we can obtain: We now rewrite with EFinstead of E as the independent variable: This equation has the same form as eqn (4.4). The differentiation with respect to EFin expression (4.37) gives:

23. (4.38) (4.39) (4.40) (4.41) With the help of (4.36) for EF we can write : or, after rearrangement: The average in (4.39) and (4.40) can be evaluated in the same way as has been done for <M2 >o in (4.9). Instead of eqn. (4.29) we now obtain: The moment dcan be connected with the moment of the molecule in the gas phase in the following way.

24. (4.42) (4.43) (4.44) In terms of the simplified model, evaporation consists in the disengagement of small spheres with dielectric constant  and a permanent dipole moment d in the center. The momentmof such a sphere in vacuum consists of the permanent moment d and the moment induced by d in the surrounding dielectric: Obviously, m must be set equal to the moment of the molecule in the gas phase. In this case we find: Substituting eqn.(4.43) into eqn.(4.41), we obtain after simple rearrangement: Equation (4.44)is called theKirkwood-Fröhlich equation.

25. This equation gives the relation between , dielectric permittivity, , the dielectric permittivity of induced polarization, the temperature, the density, and the permanent dipole moment, for those cases where the intermolecular interactions are sufficiently well known to calculate Kirkwood correlation factor g. If there is no specific correlations one has g=1. If the correlations are not negligible, detailed information about the molecular interactions is required for the calculations of g. For associating compounds, where the occurrence of hydrogen bonds makes relevant the assumption that only certain specific angles between the dipoles neighboring molecules are possible the molecular interactions may be represented by simplified models. Let us derived this extended equation for those compounds where the molecules or the polar segments form clusters of limited size. Consider each kind of polymers or multimers as a separate compound and neglecting any specific correlation between the total dipole moments of the polymers or multimers we can try the general equation for the polar fluids:

26. (4.45) where index o refers to the non-polar solvent, and index n refers to the polymers or multimers containing n polar units. The average is taken over all conformations of the polymer or multimer. The upper limit of the summation can be extended to infinity since Nn, the number of n-mers per cm3, becomes zero when nbecomes large. The polarizability o can be calculated from the Clausius-Mossotti equation for the pure solvent. (4.46) (4.47) Combining the Clausius-Mossotti equation and the Onsager approximation for the radius of the cavity, we have

27. (4.48) (4.49) (4.50) (4.51) We assume that the polarizabilities and the molecular volumes of the n-mers are proportional to n, so that: where Here, dis the density and is the dielectric constant of induced polarization of the polar compound in the pure state. In the same way we find, using Onsager's approximation for the radius of the cavity: We now substitute equations (4.46) - (4.50) into (4.45) and divide both members by (2+1), obtaining:

28. The average can be calculated as (4.52) (4.53) wheren,i is the moment of thei-th dipolar unit of the n-mer. By taking the i-th dipolar unit as the representative unit and averaging over all possible positions of the i-th unit in the chain, we may write eq.(4.52) as follows: where has been used to denote the dipole strength of a single unit. The factor gnrepresents the average value of the ratio between the component of the moment of the whole n-mer in the direction of the permanent moment of an arbitrary segment and the dipole strength of the segment. Since we assumedthat there is no correlation between different chains, gn represents the total correlation between the permanent moment of a segment in a n-mer and its surroundings.

29. (4.54) (4.55) (4.56) To obtain the Kikwood correlation factorg, we must average gn over all values of n, with a weight factor equal to the chance that a segment forms part of a n-mer. This chance is given by if Nn is the number of n-mers per cm3. Thus, we find with the help of eqn. (4.53): We now use molar fractions xoand xp for the nono-polar component respectively, regarding each segment as a separate molecule:

30. (4.57) (4.58) In these equations =(xoMo+xpM1)/d denotes the molar volume of the mixture. Substituting eqns. (4.54)-(4.56) into eqn.(4.51) we find: From this it follows: This equation makes possible the calculation of the Kirkwood correlation factor g from experimental data for solutions of associating or polymeric compounds in non-polar solvents.