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Dielectric constant and density of states of the Drude-Lorentz model of a nonpolar fluid
PHYSICA ELSEVIER
Physica A 241 (1997) 6-11
Dielectric constant and density of states of the Drude-Lorentz model of a nonpolar fluid B. Cichocki a, B.U. Felderhof b'* aInstitute of Theoretical Physics, Warsaw University, Hoza 69, 00-618 Warsaw, Poland blnstitut fiir Theoretische Physik A, R. W.T.H. Aachen, Templergraben 55, 52056 Aachen, Germany
Abstract The absorption spectrum and density of states of the Drude-Lorentz model of a nonpolar fluid are studied on the basis of a statistical theory. In the dilute limit the shift and broadening of the spectral line and of the density of states, due to dipolar interactions, are described by universal scaling functions, which have been evaluated in a self-consistent ring approximation. At higher densities the lineshapes are influenced by structural effects. These are taken into account in a pair approximation. The theory agrees well with simulation data for semidilute fluids. PACS: 78.20.Cc; 05.20.-y Keywords: Nonpolar fluids; Dielectric constant; Density of states; Drude-Lorentz model
1. Introduction In the Drude-Lorentz model o f the atom a spectral line is represented as an harmonic oscillator resonance broadened by radiation damping [1]. The corresponding spectral line o f a collection of identical atoms can be represented by a set of harmonic oscillators coupled by the radiation field. The interaction causes broadening o f the spectral line. In simplest approximation the field generated by each atom is calculated in the electrostatic dipole approximation, with neglect of retardation. The absorption lineshape can be found from the linear response of the set o f coupled oscillators to an applied oscillating field. The emission lineshape follows by application o f Kirchhoff's law. In model calculations on fluids, Doppler line broadening due to atomic motion is left out o f consideration. Then the lineshape is determined entirely by interactions, and can in principle be found from an average over the probability distribution o f positions, as given, for example, by a thermal equilibrium ensemble. The microstructure o f the fluid enters via partial distribution functions o f low order. The latter may be regarded as known. * Corresponding author. Fax: +492418888188; e-mail; [email protected]. 0378-4371/97/$17.00 Copyright (~) 1997 Elsevier Science B.V. All rights reserved PII S0378-4371 (97)00053-8
B. Cichocki, B.U. Felderhof/Physica A 241 (1997) 6-11
7
Even for known microgeometry an exact calculation of the linear response of a disordered many-atom system is out of question. One must have recourse to approximation schemes, which allow partial evaluation of the many contributions. Some time ago, we performed numerical simulations of the dielectric response of a system of DrudeLorentz oscillators coupled by electrostatic dipole interaction, and distributed with hard sphere statistics [2]. Later, the complex dielectric constant of a system of oscillators distributed according to an equilibrium ensemble of atoms with Lennard-Jones interactions was found by numerical simulation [3]. At the time there was no satisfactory theory of the absorption spectra found in the simulations. A similar problem arises for the density of states of the Drude-Lorentz model [4]. This is relevant for the van der Waals binding energy of the fluid [5]. The density of states may be found from a related linear response property, the so-called selfsusceptibility. The density of states has been determined by computer simulation for fluids with hard sphere statistics [4], as well as for Lennard-Jones fluids [3]. Recently, elaborating ideas developed by Leegwater and Mukamel [3,6] and by Schvaneveldt and Loring [7], we proposed an approximation scheme which gives a fair description of the simulation data for the density states of semi-dilute fluids [8]. The same scheme can be employed for a calculation of the dielectric constant and the corresponding absorption lineshape. Again reasonably good agreement with the simulations is found [9]. This suggests that the approximation scheme captures important features of the problem, and can be extended to the description of dense systems.
2. Dilute systems and scaling limit It is somewhat surprising that even dilute systems pose a nontrivial problem. It has been suggested by Leegwater and Mukamel [3,6] on the basis of their computer simulation of a system with Lennard-Jones statistics that the density of states and the absorption spectrum have a nontrivial universal shape even in the dilute gas limit. Their argument for this is not entirely convincing, since it involves only the behavior with distance of the dipolar interaction, but not the strength of the interaction. We have argued [8] that in the dilute limit the self-susceptibility and the dielectric constant show universal scaling behavior as a function of density n and atomic polarizability cc Each dipolar interaction carries a factor ~/R 3, where R is the pair distance. If distances are scaled with the mean interparticle distance n-1/3, then this factor is nCt/r 3, where r is the dimensionless distance in scaled units. In the center of the resonance the polarizability is large. We therefore consider the limit n ~ 0, ~ ~ co, keeping the product net constant. In this limit the linear response properties have scaling behavior, and depend only on the product net. The limit is universal in the sense that the scaling behavior does not depend on details of the geometrical microstructure. The universal scaling behavior is important, because one may expect that its functional dependence exerts an influence similar to that of a critical point in thermodynamics. In the scaling limit the system still has complicated many-body properties,
8
R Cichocki, R U. Felderhof/Physica A 241 (1997) 6-11
but the details of the microstructure become irrelevant. It may be expected that the universal properties are relevant also at higher density, though then the microstructure can be felt, and structural corrections must be calculated. Even though in the scaling limit structural features disappear, and the statistics reduces to that of an ideal gas, except for a non-overlap condition with infinitesimal hard core, the linear response properties remain difficult to calculate. So far, the properties have been evaluated in a self-consistent ring approximation [8,9], in an extension of earlier work by Schvaneveldt and Loring [7], who considered rings of at most three particles. It tums out that the density of states and the absorption spectrum tend to a nontrivial lineshape in the scaling limit. Both calculated lineshapes are asymmetric, in agreement with the simulations. The self-susceptibility Zs is defined from the response of a selected atom, labeled 1, to an applied electric field E01 acting only on that atom. For an isolated atom the induced dipole moment would be Pl = ~E01. Due to interactions with surrounding atoms the average dipole moment in a dense system is modified to ~ = ~'Eot with a polarizability ~' which depends upon the bare polarizability ~, density n and microstructure. The self-susceptibility is defined as Zs = n~'. If ~(~o) is the harmonic oscillator response (e2/m)/(o) 2 -092) of the Drude-Lorentz model, then Imgs(~O) is proportional to the density of states, i.e. to the number of normal mode frequencies of the coupled system per frequency interval. The dielectric constant c is defined from the relation between the average polarization and the average Maxwell field. It depends on the bare polarizability ~, density n and the microstructure of the system. If ~(e)) is the harmonic oscillator response of the Drude-Lorentz model, then ~olmc(~o) is proportional to the absorption spectrum. The self-susceptibility Ks may be expressed as [8] xs-
l -
Zo 4 zo,t '
(1)
where Xo = nu is the bare susceptibility, and the coefficient 2 accounts for the effect of interactions and geometrical structure. The dielectric constant E may be expressed as = 1+
4~Zs
(2)
1 - 4 z (1 +
where p is a second coefficient accounting for interactions and microstructure. Both Eqs. (1) and (2) are exact, and merely represent a reformulation of the problem. If the coefficients 2 and # are put equal to zero, then Eq. (2) reduces to the ClausiusMossotti formula for the dielectric constant. Thus, the reformulation incorporates the knowledge that the Lorentz local field is a good first approximation to the actual local field. The coefficients 2 and/~ were first defined in a multiple scattering analysis [10]. It was shown that the coefficients may be expressed in terms of a cluster expansion 2=
2s, s--2
/~ =
¢~s, s
2
(3)
B. CichockL B.U. FelderhoflPhysica A 241 (1997) 6-11
9
where 2s and #s are given by cluster integrals involving the solution of an s-particle problem and integration over an s-body correlation function. In the scaling limit the coefficients 2, # tend to values 2,,(~(0), #u()~0) depending only on the product g o = n e t . The functions 2~(Zo),#u(Z0 ) can be expressed as sums of well-defined corresponding cluster integrals 2~(Z0), #us(Z0). The pair integrals 2u2(~(0), #~2(Z0) are quite simple and easily evaluated. Their values are 2g •~u2(Z0) = ~-i,
2 #~2(g0)= - ~ ln2,
(4)
for Im Z0 > 0. For Im Z0 < 0 the coefficient 2~2 has the opposite sign. If all higher order cluster integrals are neglected, then the approximate values for 2,,#~ lead to shifted and broadened Lorentzian lines for the density of states and the absorption spectrum. The higher order universal cluster integrals 2us, #us for s ~>3 are difficult to calculate. In the self-consistent ring approximation a large class of the contributing many-body scattering processes is summed. As a first step one evaluates ring integrals 2es(Z0) and #Rs(ZO) as a contribution of ring diagrams. Summing over all s~>3 one finds corresponding functions 2R(Z0),#R(Z0). As a second step the self-susceptibility Zs is approximated as the solution of the self-consistent equation Zo Zsu = 1 -
4gZ0[~u2 +
2R(ZSu)] "
(5)
An analysis of the multiple scattering expansion shows that this implies summation of a large class of scattering sequences. The imaginary part of the solution Zs~ of Eq. (5) provides the self-consistent ring approximation to the exact universal density of states. As a third step one derives a corresponding approximation to the dielectric constant, by substitution of the approximate value # R ( Z S ) into the exact relation Eq. (2). This yields as an approximation to the universal dielectric constant 47rZs~ e~ = 1 + 1 - 47tXSu[1 + #u2 ÷ #R(ZSu)]
(6)
The imaginary part of the self-susceptibility Zsu given by Eq. (5) leads to a distortion and shift of the Lorentzian density of states found with only the two-particle cluster integral. The absorption lineshape found from the imaginary part of dielectric constant cu given by Eq. (6) is also a distorted and shifted Lorentzian.
3. Influence of microstructure
The universal scaling functions Zsu(){o) and eu(Zo) depend only on the product The functions are given approximately by Eqs. (5) and (6). At low density n the exact self-susceptibility Zs and dielectric constant e of any fluid of atoms with polarizable point dipoles tend to the universal values Zsu and eu, but at higher density the microstructure of the fluid makes itself felt. This can be seen from the Zo=n~.
10
B. Cichocki, B . U . F e l d e r h o f / P h y s i c a
A 241 ( 1 9 9 7 ) 6 - 1 1
exact expressions for the cluster integrals 2s and ]2s. For example the pair integral 22 is given exactly by [8,11]
r6 - 2Z2 4zz)y(n-l/3r)r2 dr
)~2 = 6Z0
(r 6 _ ~02~~ -
,
(7)
2an I 3
where 9(R) is the radial distribution function, and a is the hard core radius. Similarly, the pair integral ]22 is given exactly by
]22 = 6Z2
f
F3 ( r6 -- Z2)( r6 -- 4Z2) y ( n o I/3r) r2 dr.
(8)
2an ~ 3
Evidently, the dependence of these integrals on the bare susceptibility Z0 is influenced by the fluid structure via the radial distribution function. The higher order cluster integrals 2s, ]2s involve higher order partial distribution functions. The pair integrals 22 and ]22 can be evaluated numerically without difficulty for a given radial distribution function 9(R). For semi-dilute fluids the function 9(R) may be approximated by the Boltzmann factor exp[-/~v(R)], where v(R) is the pair interaction. The universal two-particle cluster integrals 2u2 and #u2, obtained by taking the limit n ~ 0 in Eqs. (7) and (8), are particular examples of ring integrals. The self-consistent ring approximation may be corrected for structural effects at the two-particle level by the replacement of the universal pair values }~u2 and ]2u2, as given by Eq. (4), by the exact pair integrals 22(Z0) and ]22(Z0). Correspondingly the self-consistent equation, Eq. (5), for the self-susceptibility is replaced by Zo zs = 1 - %~0[)~:(zs)
(9) + ,~R(zs)] '
and Eq. (6) for the dielectric constant is replaced by
47tZs e = 1 + 1 - 34-~zs[1 + ] 2 2 ( z s ) + ] 2 R ( z s ) ]
(10)
We have compared the imaginary part of the solution )~s of Eq. (9) with the density of states found by computer simulation for a hard sphere fluid [4] at volume fraction ¢p=0.1 and with that for a Lennard-Jones fluid [3] at volume fraction ~ ~ 0.05, and found fairly good agreement [8]. The corresponding values of the dielectric constant, as given by Eq. (10), are also in fair agreement with the simulation data [9].
4. Discussion
We have developed a scheme which allows approximate calculation of the complex dieleetric constant and self-susceptibility of a fluid consisting of atoms with a polarizable point dipole at their center. At low fluid density the dependence on frequency of the linear response properties is dominated by universal scaling functions.
B. Cichocki, B.U. Felderhof/Physica A 241 (1997) 6-11
11
At higher densities the response is influenced by structural effects. The universal scaling functions are defined exactly as sums of cluster integrals, but can be evaluated only approximately. A self-consistent ring approximation, corrected for structural effects on a pair level, yields fairly good agreement of the predicted response properties with simulation data for semi-dilute fluids. The Drude-Lorentz harmonic oscillator model, combined with the electrostatic dipole approximation, provides a benchmark system in the study of the electrical response of polarizable fluids. A successful approximation scheme developed for this model can be extended without great difficulty to more complicated systems. For example, by including higher order multipoles one could apply the scheme to the calculation of the effective dielectric constant of a suspension of uniform spheres. The absorption spectrum corresponds to the spectral density in the Bergrnan representation of the effective dielectric constant [12]. Simulation data for the spectral density exist for hard sphere suspensions in a range of volume fractions [13]. It would be of interest to elaborate the approximation scheme for this model and compare with simulation data. The theory developed here is of importance beyond the study of electrical response properties. Many physical phenomena can be understood as the linear response of a many-particle system to an applied force, with coupling between particles communicated by a propagating field. As examples we mention sedimentation and effective viscosity of fluid suspensions, and the effective elastic properties of solid suspensions. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
L. Rosenfeld, Theory of Electrons (North-Holland, Amsterdam 1951) p. 59. B. Cichocki and B.U. Felderhof, J. Chem. Phys. 90 (1989) 4960. J.A. Leegwater and S. Mukamel, J. Chem. Phys. 99 (1993) 6062. B. Cichocki and B.U. Felderhof, J. Chem. Phys. 92 (1990) 6104. B. Cichocki and B.U. Felderhof, J. Chem. Phys. 92 (1990) 6112. J.A. Leegwater and S. Mukamel, Phys. Rev. A 49 (1994) 146. S.J. Schvaneveldt and R.F. Loring, J. Chem. Phys. 101 (1994) 4133. B. Cichocki and B.U. Felderhof, J. Chem. Phys. 104 (1996) 3013. B. Cichocki and B.U. Felderhof, submitted for publication. B. Cichocki and B.U. Felderhof, J. Stat. Phys. 53 (1988) 499. B.U. Felderhof, G.W. Ford and E.G.D. Cohen, J. Stat. Phys. 28 (1982) 649. D.J. Bergman, Phys. Rep. 43 (1978) 377. K. Hinsen and B.U. Felderhof, Phys. Rev. B 46 (1992) 12 955.
Physica A 241 (1997) 6-11
Dielectric constant and density of states of the Drude-Lorentz model of a nonpolar fluid B. Cichocki a, B.U. Felderhof b'* aInstitute of Theoretical Physics, Warsaw University, Hoza 69, 00-618 Warsaw, Poland blnstitut fiir Theoretische Physik A, R. W.T.H. Aachen, Templergraben 55, 52056 Aachen, Germany
Abstract The absorption spectrum and density of states of the Drude-Lorentz model of a nonpolar fluid are studied on the basis of a statistical theory. In the dilute limit the shift and broadening of the spectral line and of the density of states, due to dipolar interactions, are described by universal scaling functions, which have been evaluated in a self-consistent ring approximation. At higher densities the lineshapes are influenced by structural effects. These are taken into account in a pair approximation. The theory agrees well with simulation data for semidilute fluids. PACS: 78.20.Cc; 05.20.-y Keywords: Nonpolar fluids; Dielectric constant; Density of states; Drude-Lorentz model
1. Introduction In the Drude-Lorentz model o f the atom a spectral line is represented as an harmonic oscillator resonance broadened by radiation damping [1]. The corresponding spectral line o f a collection of identical atoms can be represented by a set of harmonic oscillators coupled by the radiation field. The interaction causes broadening o f the spectral line. In simplest approximation the field generated by each atom is calculated in the electrostatic dipole approximation, with neglect of retardation. The absorption lineshape can be found from the linear response of the set o f coupled oscillators to an applied oscillating field. The emission lineshape follows by application o f Kirchhoff's law. In model calculations on fluids, Doppler line broadening due to atomic motion is left out o f consideration. Then the lineshape is determined entirely by interactions, and can in principle be found from an average over the probability distribution o f positions, as given, for example, by a thermal equilibrium ensemble. The microstructure o f the fluid enters via partial distribution functions o f low order. The latter may be regarded as known. * Corresponding author. Fax: +492418888188; e-mail; [email protected]. 0378-4371/97/$17.00 Copyright (~) 1997 Elsevier Science B.V. All rights reserved PII S0378-4371 (97)00053-8
B. Cichocki, B.U. Felderhof/Physica A 241 (1997) 6-11
7
Even for known microgeometry an exact calculation of the linear response of a disordered many-atom system is out of question. One must have recourse to approximation schemes, which allow partial evaluation of the many contributions. Some time ago, we performed numerical simulations of the dielectric response of a system of DrudeLorentz oscillators coupled by electrostatic dipole interaction, and distributed with hard sphere statistics [2]. Later, the complex dielectric constant of a system of oscillators distributed according to an equilibrium ensemble of atoms with Lennard-Jones interactions was found by numerical simulation [3]. At the time there was no satisfactory theory of the absorption spectra found in the simulations. A similar problem arises for the density of states of the Drude-Lorentz model [4]. This is relevant for the van der Waals binding energy of the fluid [5]. The density of states may be found from a related linear response property, the so-called selfsusceptibility. The density of states has been determined by computer simulation for fluids with hard sphere statistics [4], as well as for Lennard-Jones fluids [3]. Recently, elaborating ideas developed by Leegwater and Mukamel [3,6] and by Schvaneveldt and Loring [7], we proposed an approximation scheme which gives a fair description of the simulation data for the density states of semi-dilute fluids [8]. The same scheme can be employed for a calculation of the dielectric constant and the corresponding absorption lineshape. Again reasonably good agreement with the simulations is found [9]. This suggests that the approximation scheme captures important features of the problem, and can be extended to the description of dense systems.
2. Dilute systems and scaling limit It is somewhat surprising that even dilute systems pose a nontrivial problem. It has been suggested by Leegwater and Mukamel [3,6] on the basis of their computer simulation of a system with Lennard-Jones statistics that the density of states and the absorption spectrum have a nontrivial universal shape even in the dilute gas limit. Their argument for this is not entirely convincing, since it involves only the behavior with distance of the dipolar interaction, but not the strength of the interaction. We have argued [8] that in the dilute limit the self-susceptibility and the dielectric constant show universal scaling behavior as a function of density n and atomic polarizability cc Each dipolar interaction carries a factor ~/R 3, where R is the pair distance. If distances are scaled with the mean interparticle distance n-1/3, then this factor is nCt/r 3, where r is the dimensionless distance in scaled units. In the center of the resonance the polarizability is large. We therefore consider the limit n ~ 0, ~ ~ co, keeping the product net constant. In this limit the linear response properties have scaling behavior, and depend only on the product net. The limit is universal in the sense that the scaling behavior does not depend on details of the geometrical microstructure. The universal scaling behavior is important, because one may expect that its functional dependence exerts an influence similar to that of a critical point in thermodynamics. In the scaling limit the system still has complicated many-body properties,
8
R Cichocki, R U. Felderhof/Physica A 241 (1997) 6-11
but the details of the microstructure become irrelevant. It may be expected that the universal properties are relevant also at higher density, though then the microstructure can be felt, and structural corrections must be calculated. Even though in the scaling limit structural features disappear, and the statistics reduces to that of an ideal gas, except for a non-overlap condition with infinitesimal hard core, the linear response properties remain difficult to calculate. So far, the properties have been evaluated in a self-consistent ring approximation [8,9], in an extension of earlier work by Schvaneveldt and Loring [7], who considered rings of at most three particles. It tums out that the density of states and the absorption spectrum tend to a nontrivial lineshape in the scaling limit. Both calculated lineshapes are asymmetric, in agreement with the simulations. The self-susceptibility Zs is defined from the response of a selected atom, labeled 1, to an applied electric field E01 acting only on that atom. For an isolated atom the induced dipole moment would be Pl = ~E01. Due to interactions with surrounding atoms the average dipole moment in a dense system is modified to
l -
Zo 4 zo,t '
(1)
where Xo = nu is the bare susceptibility, and the coefficient 2 accounts for the effect of interactions and geometrical structure. The dielectric constant E may be expressed as = 1+
4~Zs
(2)
1 - 4 z (1 +
where p is a second coefficient accounting for interactions and microstructure. Both Eqs. (1) and (2) are exact, and merely represent a reformulation of the problem. If the coefficients 2 and # are put equal to zero, then Eq. (2) reduces to the ClausiusMossotti formula for the dielectric constant. Thus, the reformulation incorporates the knowledge that the Lorentz local field is a good first approximation to the actual local field. The coefficients 2 and/~ were first defined in a multiple scattering analysis [10]. It was shown that the coefficients may be expressed in terms of a cluster expansion 2=
2s, s--2
/~ =
¢~s, s
2
(3)
B. CichockL B.U. FelderhoflPhysica A 241 (1997) 6-11
9
where 2s and #s are given by cluster integrals involving the solution of an s-particle problem and integration over an s-body correlation function. In the scaling limit the coefficients 2, # tend to values 2,,(~(0), #u()~0) depending only on the product g o = n e t . The functions 2~(Zo),#u(Z0 ) can be expressed as sums of well-defined corresponding cluster integrals 2~(Z0), #us(Z0). The pair integrals 2u2(~(0), #~2(Z0) are quite simple and easily evaluated. Their values are 2g •~u2(Z0) = ~-i,
2 #~2(g0)= - ~ ln2,
(4)
for Im Z0 > 0. For Im Z0 < 0 the coefficient 2~2 has the opposite sign. If all higher order cluster integrals are neglected, then the approximate values for 2,,#~ lead to shifted and broadened Lorentzian lines for the density of states and the absorption spectrum. The higher order universal cluster integrals 2us, #us for s ~>3 are difficult to calculate. In the self-consistent ring approximation a large class of the contributing many-body scattering processes is summed. As a first step one evaluates ring integrals 2es(Z0) and #Rs(ZO) as a contribution of ring diagrams. Summing over all s~>3 one finds corresponding functions 2R(Z0),#R(Z0). As a second step the self-susceptibility Zs is approximated as the solution of the self-consistent equation Zo Zsu = 1 -
4gZ0[~u2 +
2R(ZSu)] "
(5)
An analysis of the multiple scattering expansion shows that this implies summation of a large class of scattering sequences. The imaginary part of the solution Zs~ of Eq. (5) provides the self-consistent ring approximation to the exact universal density of states. As a third step one derives a corresponding approximation to the dielectric constant, by substitution of the approximate value # R ( Z S ) into the exact relation Eq. (2). This yields as an approximation to the universal dielectric constant 47rZs~ e~ = 1 + 1 - 47tXSu[1 + #u2 ÷ #R(ZSu)]
(6)
The imaginary part of the self-susceptibility Zsu given by Eq. (5) leads to a distortion and shift of the Lorentzian density of states found with only the two-particle cluster integral. The absorption lineshape found from the imaginary part of dielectric constant cu given by Eq. (6) is also a distorted and shifted Lorentzian.
3. Influence of microstructure
The universal scaling functions Zsu(){o) and eu(Zo) depend only on the product The functions are given approximately by Eqs. (5) and (6). At low density n the exact self-susceptibility Zs and dielectric constant e of any fluid of atoms with polarizable point dipoles tend to the universal values Zsu and eu, but at higher density the microstructure of the fluid makes itself felt. This can be seen from the Zo=n~.
10
B. Cichocki, B . U . F e l d e r h o f / P h y s i c a
A 241 ( 1 9 9 7 ) 6 - 1 1
exact expressions for the cluster integrals 2s and ]2s. For example the pair integral 22 is given exactly by [8,11]
r6 - 2Z2 4zz)y(n-l/3r)r2 dr
)~2 = 6Z0
(r 6 _ ~02~~ -
,
(7)
2an I 3
where 9(R) is the radial distribution function, and a is the hard core radius. Similarly, the pair integral ]22 is given exactly by
]22 = 6Z2
f
F3 ( r6 -- Z2)( r6 -- 4Z2) y ( n o I/3r) r2 dr.
(8)
2an ~ 3
Evidently, the dependence of these integrals on the bare susceptibility Z0 is influenced by the fluid structure via the radial distribution function. The higher order cluster integrals 2s, ]2s involve higher order partial distribution functions. The pair integrals 22 and ]22 can be evaluated numerically without difficulty for a given radial distribution function 9(R). For semi-dilute fluids the function 9(R) may be approximated by the Boltzmann factor exp[-/~v(R)], where v(R) is the pair interaction. The universal two-particle cluster integrals 2u2 and #u2, obtained by taking the limit n ~ 0 in Eqs. (7) and (8), are particular examples of ring integrals. The self-consistent ring approximation may be corrected for structural effects at the two-particle level by the replacement of the universal pair values }~u2 and ]2u2, as given by Eq. (4), by the exact pair integrals 22(Z0) and ]22(Z0). Correspondingly the self-consistent equation, Eq. (5), for the self-susceptibility is replaced by Zo zs = 1 - %~0[)~:(zs)
(9) + ,~R(zs)] '
and Eq. (6) for the dielectric constant is replaced by
47tZs e = 1 + 1 - 34-~zs[1 + ] 2 2 ( z s ) + ] 2 R ( z s ) ]
(10)
We have compared the imaginary part of the solution )~s of Eq. (9) with the density of states found by computer simulation for a hard sphere fluid [4] at volume fraction ¢p=0.1 and with that for a Lennard-Jones fluid [3] at volume fraction ~ ~ 0.05, and found fairly good agreement [8]. The corresponding values of the dielectric constant, as given by Eq. (10), are also in fair agreement with the simulation data [9].
4. Discussion
We have developed a scheme which allows approximate calculation of the complex dieleetric constant and self-susceptibility of a fluid consisting of atoms with a polarizable point dipole at their center. At low fluid density the dependence on frequency of the linear response properties is dominated by universal scaling functions.
B. Cichocki, B.U. Felderhof/Physica A 241 (1997) 6-11
11
At higher densities the response is influenced by structural effects. The universal scaling functions are defined exactly as sums of cluster integrals, but can be evaluated only approximately. A self-consistent ring approximation, corrected for structural effects on a pair level, yields fairly good agreement of the predicted response properties with simulation data for semi-dilute fluids. The Drude-Lorentz harmonic oscillator model, combined with the electrostatic dipole approximation, provides a benchmark system in the study of the electrical response of polarizable fluids. A successful approximation scheme developed for this model can be extended without great difficulty to more complicated systems. For example, by including higher order multipoles one could apply the scheme to the calculation of the effective dielectric constant of a suspension of uniform spheres. The absorption spectrum corresponds to the spectral density in the Bergrnan representation of the effective dielectric constant [12]. Simulation data for the spectral density exist for hard sphere suspensions in a range of volume fractions [13]. It would be of interest to elaborate the approximation scheme for this model and compare with simulation data. The theory developed here is of importance beyond the study of electrical response properties. Many physical phenomena can be understood as the linear response of a many-particle system to an applied force, with coupling between particles communicated by a propagating field. As examples we mention sedimentation and effective viscosity of fluid suspensions, and the effective elastic properties of solid suspensions. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
L. Rosenfeld, Theory of Electrons (North-Holland, Amsterdam 1951) p. 59. B. Cichocki and B.U. Felderhof, J. Chem. Phys. 90 (1989) 4960. J.A. Leegwater and S. Mukamel, J. Chem. Phys. 99 (1993) 6062. B. Cichocki and B.U. Felderhof, J. Chem. Phys. 92 (1990) 6104. B. Cichocki and B.U. Felderhof, J. Chem. Phys. 92 (1990) 6112. J.A. Leegwater and S. Mukamel, Phys. Rev. A 49 (1994) 146. S.J. Schvaneveldt and R.F. Loring, J. Chem. Phys. 101 (1994) 4133. B. Cichocki and B.U. Felderhof, J. Chem. Phys. 104 (1996) 3013. B. Cichocki and B.U. Felderhof, submitted for publication. B. Cichocki and B.U. Felderhof, J. Stat. Phys. 53 (1988) 499. B.U. Felderhof, G.W. Ford and E.G.D. Cohen, J. Stat. Phys. 28 (1982) 649. D.J. Bergman, Phys. Rep. 43 (1978) 377. K. Hinsen and B.U. Felderhof, Phys. Rev. B 46 (1992) 12 955.