3480
J . Phys. Chem. 1992,96, 3480-3485
R (A) BULK
62
36
18
-194 1
2 ' .00. d l . 0 2 .0J .04 .05 . 0 6
1/R
'
(Ae1)
Figure 1. Pore size dependence of the experimental T, of the studied molecular liquids confined to porous silica glasses. The Tgvalues for bulk liquids are also given (1 /R = 0): A, isopropylbenzene; V, glycerol; W, di-n-butyl phthalate; 0 , ten-butylbenzene;0, n-butyl acetate.
Since the depression of the glass transition temperature of the liquids in the pores is relatively small, in this limited range Aa, ACp, and Au can be assumed to be constant, and eq 3 gives us a linear relation between the depression of the glass transition temperature and the inverse of pore radius. This agrees with our experimental results for the depression of the glass transition temperature of all molecular liquids studied, as shown in Figure 1 which plots T , vs the reciprocal of the radius R . Figure 1 shows that the decrease of the glass transition temperature of confined liquids compared to the glass transition temperature in the bulk changes linearly with the inverse of the pore radius R. It is worth mentioning that linear regression gives the following correlation coefficients for the plots shown in Figure 1: 0.973,0.992,0.993,0.993,and 0.969. It is interesting to compare the behavior of the glass transition temperature of confined liquids to that observed for the freezing
points. One notes that the depression ratio of the glass transition, AT/T,, is relatively small (below 5% for the smallest pore size) compared with the freezing point depression ratio, AT/Tf (above 20% for the same pore size). These results show that the confinement effect on the glass transition of confined liquids is much weaker than the confinement effect on crystallization of confined liquids. In order to give a specific example, we note that Strange et a1.*' reported that the melting point of cyclohexane is lowered by about 40 K and the solid-solid phase transition by about 20 K when cyclohexane is confined to porous silica glass with a pore diameter of 90 A. The idea that the depression of glass transition points is the result of the capillary effect on the confined liquids can be tested by calculating the AT from the previous data. For example, according to the experimental data given in ref 39,the interfacial tension of glycerol, Au, is estimated to be 70 dyn/cm above the glass transition point. The pressure decrease of the glycerol confined to a 36-Asize pore, AP,is about 0.4kbar obtained by using the Kelvin equation. It is also known that dT,/dP = 4 K/kbarm for glycerol, so the depression of the glycerol glass transition temperature, AT, is determined to be approximately 2 K. This agrees surprisingly well with our experimental results (see Table I or Figure 1) in spite of the approximative nature of our interpretation. The main result of our experimental study using DSC is the finding that the glass transition temperature, T,, of several molecular liquids is lowered when the liquid is confined with a porous silica glass. It is also interesting to note that the effects of confinement on T are much smaller when compared to the analogous effects on the freezing point temperature or on the solidsolid phase transition?' The experimental observationz4of a linear dependence of the glass transition temperature, T,, and the inverse of the pore radius ( 1 / R ) was phenomenologically interpreted in terms of the Kelvin equation and the Ehrenfest relation. Acknowledgment. This work was supported in part by the National Science Foundation Grant NSF DMR 89-20538. Our thanks are due to P. G. Wolynes for bringing this problem to our attention and to N. Goldenfeld and G. B. McKenna for helpful comments. (39) Glycerol; Miner, C. S.,Dalton, N. N., Eds.;Reinhold: New York, 1953. (40) DiMarzio, E. A.; Gibbs, J. H.; Fleming, P. D., 111; Sanchez, 1. C. Macromolecules 1976, 9, 763.
Dipole Relaxation near Boundaries M. Urbakh* and J. Klafter* School of Chemistry, Tel- Aviu University, 69978, Ramat- Aviu, Tel- Aviv, Israel (Received: November 1 1 , 1991)
The influence of a solid-liquid interface on the relaxation of a point dipole embedded in the liquid side is discussed. The dielectric friction of a dipole as a function of its distance from a boundary is calculated. The calculations are carried out assuming a nonlocal dielectric function of the liquid, characterized by a typical correlation length which may depend on temperature. The corrections to the relaxation of a dipole due to the presence of a boundary are shown to be small. Larger correctionscan be introduced by postulatingstructural changes in the nature of the liquid near the boundary. As a demonstration we apply the proposed formalism in the study of temperature and pore size dependence of a dipole relaxation.
1. Introduction A large number of experimental and theoretical studies have
dressed, i.e., changes in freezing and viscosity as well as chemical reactions and energy-transfer AIS0 traIlSlatiOna1
been devoted to the understanding of the role of spatial restrictions in modifying the properties of embedded liquids and molecules. Both thermodynamical and dynamical questions have been ad-
( I ) Klafter, J., Drake, J. M., Eds. Molecular Dynamics in Restricted Geometries; Wiley: New York, 1989.
0022-3654/92/2096-3480503.00/00 1992 American Chemical Society
Dipole Relaxation near Boundaries and rotational diffusion of probe molecules have been shown to be influenced by the presence of b0undaries.l It is by now well established that liquids in small confinements differ significantly from bulk liquids. The behavior of molecular systems under geometrical restrictions is of importance in a wide range of problems related to catalysis, microemulsions, polymer solutions, biological membranes, and more. In this paper we concentrate on the question of the rotational relaxation of a dipole in a liquid close to a boundary. This problem has been investigated experimentally, among others, by Awschalom et al.2 using optical birefringence and by Zinsli3 and more recently by Drake et al.4 using optical depolarization. The common observation has been that in the vicinity of the boundary there is a pronounced change of molecular reorientation time. Awschalom et a1.,2 who conducted temperature dependence studies near the liquid freezing point, have inferred from their studies the temperature and pore size dependence of a molecule's rotational time. A detailed microscopic picture for the observed phenomena is still missing. We will study the rotational relaxation of a point dipole within the continuum approach based on the dielectric friction6in a liquid which is characterized by a nonlocal dielectric function. The effect of an interface is introduced through the concept of additional boundary conditions' which leads to a generalization of the image charges picture." The nonlocal nature of the liquid defines a length scale, A, which is a measure of spatial correlations in the liquid. For instance, in aqueous solutions the correlation length, A, is of the order of the extension of local hydrogen-bonded clusters. This correlation length enables one to introduce temperature into observable quantities at least phenomenologically and creates dipoleboundary distance and pore size dependencies which do not appear in the case of a local dielectric function. Corrections to the bulk polarizability and to the dielectric friction due to the boundary are calculated but are shown to be small for most of the relevant parameters. In order to mimic the possibility that the liquid itself changes its dielectric behavior due to interaction with the b o ~ n d a r y ? ~we~ assume ~ J ~ a region of modified liquid near the boundary. This allows for more significant changes in the dipole relaxation and concurs with experimental observationsa2 We end with an example of a possible temperature behavior of the relaxation time arising as a consequence of the temperature dependence of the correlation length, A, of the liquid nonlocal dielectric function. 2. The Model
We now consider a model for the relaxation of a time-dependent point dipole embedded in a liquid near a nonmetallic substrate. We assume that the substrate is characterized by a local dielectric function, CSub(O), and the liquid is described by the nonlocal dielectric function"
(2) Awschalom, D. D.; Warnock, J. In ref 1 . (3) Zinsli, P. E. J . Phys. Chem. 1979, 83, 3223. (4) Drake, J . M.; Klafter, J . Phys. Today 1990, 43, 46. (5) Liu, G.;Li, Y.-Z.; Jonas, J . J . Chem. Phys. 1989, 90, 5881. (6) Bottcher, C . J . F.; Bordewijk, P. Theory of Electric Poloriration; Elsevier: Amsterdam, 1979. (7) Agranovich, V . M.; Ginzburg, V. L. Spatial Dispersion in Crystal Optics and the Theory of Excitons; Interscience: New York, 1976. (8) van der Zwan, G.; Mazo, R. M . J . Chem. Phys. 1985, 82, 3344. (9) See, e&: The Chemicol Physics of Solvation; Dogonadze, R. R., Kalman, E., Kornyshev, A. A., Ulstrup. J., Us.; Elsevier: Amsterdam, 1988; Parts A and C. (10) (a) Israelachvili, J. N. Intermolecular and Surfoce Forces with Applications IO Colloidal and Biological Systems; Academic Press: London, 1985. (b) Davis, H. T.; Somers, S.A.; Tirrell, M.; Bitsanis, 1. In Dynamics in Small Confining Systems. Extended Abstract of the 1990 Fall Meeting of the MRS;Drake, J. M., Klafter, J., Kopelman. R., Eds.; p 73. (c) Lupkowski, M.; van Swol, F. In Dynamics in Small Confining Systems. Extended Abstract of 1990 Fall Meeting of the MRS; Drake, J . M., Klafter. J., Kopelman, R., Eds.; MRS: Pittsburgh, 1990; p 19.
The Journal of Physical Chemistry, Vol. 96, No. 8, 1992 3481 Here e,(o) is the short-wavelength dielectric constant of a bulk liquid, tb(@) = tb(k=O,w) is the local dielectric function, and k is a wave vector. For the usual Debye form of cb(w) we have
where T is the relaxation time. A similar expression is used for e.(w). In the low-frequency range which we consider in this paper the relationship lel(o)l ( 5 )
The dipole induces in the liquid side two waves, transverse and longitudinal, reflected from the substrate. Correspondingly the potential of these waves can be written in the form $.Jind(Z$k]) = A exp(-klz) + B exp(-rz), z > 0 (6) The total potential created by a dipole in the liquid is q5p In the substrate the potential has the form
+
&ub(Z&l) = D exp(klz), z < 0 (7) exp(klz) is the solution of Laplace equation, which goes to zero a t z --. The prefactors, A , B, and D are obtained from the following boundary conditions at z = 0: (a) continuity of the potential
-
( 4 p + 6ind)z-0 = +sublz=O
(8)
(b) continuity of the normal component of the displacement D,(Z,kI 1
(c) additional boundary condition due to the nonlocal nature of the liquid. In this paper we will assume that on the liquid side the nonlocal part of the polarization (or the polarization current density) must vanish at the boundary This assumption is based on the premise that the solvent dipoles which carry the polarization cannot penetrate into the substrate.I2 Within the method of additional boundary conditions (ABC) used here the interface is supposed to be sharp and the solution of the complete spatial nonhomogeneous problem is constructed ( I 1 ) Kornyshev, A . A. In The Chemical Physics of Solvation; Dogonadze,
R.R., Kalman, E., Kornyshev, A . A., Ulstrup, J., Eds.;Elsevier: Amsterdam, 1985; Part A , p 77. (12) van der Zwan, G.;Hynes, J. T . Physica 1983, 121A, 227.
3482 The Journal of Physical Chemistry, Vol. 96, No. 8, 1992 by matching the bulk solutions known for the medium on either side of the interface.'qi3 The usual boundary conditions a and b are sufficient to solve the problem of reflection and transmission of the usual transversal waves. In the nonlocal media new types of bulk waves (longitudinal modes) can arise. Therefore, ABC, condition c, has to be added to make the problem well-defined. In the framework of linear electrodynamics, the general type of the ABC is quite clear. These conditions should linearly interlink the fields E and D and their derivatives on the surface;' of course, the relationship should be a homogeneous equation (surface sources of fields are absent). Hence, if derivatives are disregarded, the ABC can be written as D ~ ( Z = O , ~+J
3
c rijEj(z=o,kl) = o
j-1
(11)
Here rij is a tensor which depends on the properties of the boundary. The relation 10 is the particular case of (1 1). It can be shown that the results are weakly dependent on the form of rij(see ref 14). The boundary condition (10) was used previously15for the calculation of the effective dipole polarizability near a metal described by the nonlocal dielectric function. The solution of eqs 8-10 gives the following expression for the induced potential, eq 6, in the liquid
C12exp(-klzo) 2fsub
+ C,C2exp(-rzo)
exp(-k,z) -
+ Cl'hb + C2%ubkl/r C2exp(-rzo) -2tsub
C22exp(-rzo) +
Cltsub
Urbakh and Klafter
= -j/zF(zo,w)p,f (17) Equation 16 constitutes the central result in the study of the influence of a boundary on the dipole properties. It provides the induced field input in the calculation of the dielectric friction,12J6 FD(w). The net friction is the sum of the bulk, .#(w), and the surface, # ( w ) , terms, and #(o) can be expressed through F(zo,w) as follows E'(r0)
where I is the moment of inertia of the dipole. For low frequencies such that w 7 > A, namely f >> 1; this implies F(z0,w) =
2cIc2
F(zo,w) =
(: 11
- F(zo,w)
(21) where we have assumed that JZ,, and hence E', lie in the x-z plane. The solution of eq 21 can be written in the form
+
2Csubkl[(Cl2exp(-2k,zo) c,?exp(-2rzo) + exp[-(r + kl)zOl\/I1 + CIhb + C2Csubkl/r)l) ( l 5 ) Introducing the dimensionless variables, u = klzo,and I' = r z 0 = [u2 + we obtain
7R(U),
(19) can be defined
(20) Generally, in deriving the rotational relaxation in real systems a mechanical friction term has to be included.6+16We, however, will concentrate on the dielectric contribution. The influence of a surface on the optical absorption of the molecule is also determined by the induced field, eq 16. The net field acting on the dipole is the sum of the applied field Eo and induced field E(ro). Correspondingly the dipole moment p of a dipole with isotropic frequency-dependent polarizability, ao(w), is equal 7R(W)
-X
+ C,C2exp(-klzo)
a(w)
1 1 fb(@)- €sub 420) fb(w)
€b(w)
+ €sub
(24)
which is the classical result obtained8*" for a local representation of the liquid in contact with a substrate and is due to the effect of image charges. 2. zo > Ic.(w)l (which typically holds) 1 1 c*(w) - €sub F(z0,w) = - 420) e*(w) c*(W) + €sub
(25)
This result originates from our nonlocal description of the liquid and is essentially insensitive to the details of the boundary. (16) Nee, Tsu-Wei; Zwanzig, R. J . Chem. Phys. 1970,52, 6353. (17) Chance, R. R.; Prock, A,; S i l k y , R. In Aduances in Chemical Physics; Prigogine, I., Rice, S . A,, Eds.;Wiley: New York, 1978; Vol. 37,
P
1.
The Journal of Physical Chemistry, Vol, 96, No. 8, 1992 3483
Dipole Relaxation near Boundaries
Figure 2. Dipole inside the modified liquid layer.
where R is the radius of cavity. According to our model for the bulk dielectric function, eq 1,
Y I 1
L 3
L 5
l - 2 7 9 1
1
‘0
Figure 1. Dependence of the normalized dielectric friction at a given frequency, ( r ~ I w / p 2 ) & )on , the distance between the dipole and the boundary. Curves 1 correspond to the correlation length, A = 3 A; curves 2 - A = 1 A and curves 3 - A = 0 (local limit). (a, top) The values of the parameters are % = 80, WT = 0.1, = 5 , C. = 3. At the same values of parameters the dielectric friction in the bulk liquid, (R31w/p2)&), equal to 1.8 X IO-’ at R / A = m ; 25 X lo-’ at R / A = 3; 31 X IO-’ at R / A = I . (b, bottom) The values of the parameters are cb = 80, W T = 0.1, club = 2, 6. = 6. At the same values of parameters the dielectric friction in the bulk liquid, (R3Iw/p2)# is equal to 1.8 X IO-’ at R/A = 0 ;15 X lo-’ at R / A = 3; 18 X lo-’ at R/A = 1.
(Boundary conditions (1 1) as well as (10) lead to eq 25.) We see that at small distances from the substrate the function F(z0,w) has the same “image force” form as for large distances, eq 24, but with a reduced effective dielectric constant, t.. The leading terms of F(z0,w) in the two limits, (24) and (25), are 1/4ego3 and 1/4t.z? correspondingly. In qualitative terms we may conclude that the structure of the liquid, as reflected in the spatial correlation of polarization fluctuations, does not allow the liquid to respond fully at characteristic structure distances. Only beyond these distances are limits of macroscopic electrodynamics reached. In the local model8 of the liquid for Itb(U)l> Gub (a case which is the most interesting for us) the presence of the boundary increases the bulk dielectric friction and correspondingly the rotation time. The effect of nonlocality leads to some important qualitative consequences. Thus, for 1c.l < (1 + 4 2 ) < lal, the boundary correction to the bulk dielectric friction, &z0,w), changes sign as the distance from the substrate, zo, changes (Figure la). The net friction decreaw (compare to the bulk value) in the immediate vicinity of the boundary, zo < A In Ibb/t*l, and increases at larger distances from the boundary. Under this condition the effect of nonlocality leads to the decreasing of the dielectric friction compared to the result of the local model at all distances from the boundary. For (1 + 42)c,b < 1e.l < ltbl the boundary correction to the dielectric friction is positive in the whole range of distances from the surface and is larger than the result obtained in the local model (Figure lb). In order to elucidate the relative role of the boundary correction to the bulk value of the friction, #, we have to evaluate the bulk friction, ,#. Following the cavity approach1L16J8the bulk friction, #), can be approximated by the expression (see Appendix)
Equation 29a corresponds to the “traditional” behavior of the dielectric frictioni6 and holds only for distances R >> A. In the opposite limit a similar relation holds, but with the high-frequency dielectric constant, tr. Comparing eqs 24,25, and 29 we can c ~ n c l u d ethat ’ ~ in both limiting cases, zo < A and zo > A, the boundary corrections to the dielectric friction in the liquid bulk are small, being of the relative order of R3/zo3. The cavity size, R, is estimated to be of the order of a few molecule radii which limits the contribution of the boundary to the dielectric friction and consequently to the rotational time. However, me expects the contribution to be significant for macromolecules near an interface where R is larger than the radius of the liquid molecules.* For such cases a more realistic approach is required in order to account of the detailed charge distribution in molecules.
3. Substrate Modification There is some experimental and numerical evidence that a liquid near a substrate may have modified p r o p e r t i e ~ . ~Close ~ ~ ~ to J~ the interface the liquid may develop local structure due to interaction with the substrate. Such a structure will be characterized by different diekwtric properties and can be observed, as proposed by Awschalom et al.2 and by Zinsli,’ using rotational relaxation studies. Here we extend our results of the previous section and describe the liquid side by two dielectric functions which correspond to t h e modified liquid in the close vicinity of the boundary, e,(w), and the bulk solution, cb(w), (see Figure 2). Following the same type of calculations and boundary conditions as in previous section, eqs 8 and 9, we arrive at the following (19) Here we assume that the distance of the dipole from the boundary,
larger than the cavity radius, R. In principle, in the opposite limit the conclusion may be different but in that case it is necessary to take into account the modification of the dielectric properties of the cavity by the substrate. A similar effect is considered in the next section. zo. is
(18) Gersten, J.; Nitzan, A. J . Chem. Phys. 1991, 95, 686.
3484
The Journal of Physical Chemistry, Vol. 96, No. 8, 1992
Urbakh and Klafter
expression for F(zo,o) for a dipole in the bulk liquid F ( z o ~ =) rl2 + r23 exp(-2ud/zO) x m d uu2 exp(-2u) (30) 1 r12r23 exp(-2ud/zo) cb(w)z:
+
where d is the layer thickness and e,
- 6,
61
+ €1
ri, = -, i , j = 1, 2, 3 and
tl
= tb,
t2
= tS,t 3 =
tsub
(31) The equation similar to eq 30 were used previously17 for the description of radiation decay of the dipole near the surface covered by a film. In the case of d > zo we obtain a similar expression for F(ro,w) but with the dielectric function of the layer replacing the substrate one. A more interesting situation arises for a dipole located within the surface layer. Now (32) F(zO,w) = Fb(W) + Fs(zO,a) where
and FS(Z,WO) = 3 x m d u u2[r23exp(-2u) tSZ0
111 + 2r12r23 exp(-2ud/zO)l/[l
- r I 2exp[-2u(d/zo
+ r12r23
exp(-2ud/zO)l
-
(34)
The first term, Fb(@),in eq 32 is the “bulk” contribution which determines the difference between dielectric frictions in two bulk liquids with different dielectric functions tb(W) and c,(w). Usually due to the structuring effect of the substrate9*I0there is the lowering of the permittivity of the liquid near the boundary and ltsl P
A = A o e x p ( LT) - T *
(36)
Here T1 is the depressed transition temperature in pores which is found to be lower than the bulk transition temperature, Tb*. The amount of the depression has been observed2 to scale with l/L, L being the pore radius, so that Tb* - T* = g/L (37) In what follows we ignore the boundary effect and concentrate on the bulk dielectric friction, eqs 27 and 28. From eqs 27, 28, and 35, 36 we derive an expression for the dielectric friction, #, for different pore radii, L.
(35) This correction may be large enough to observe experimentally. The second term, Fs(zo,w),takes into account the influence of the boundaries. The dependence of the dielectric friction, #, on the position of the dipole inside the modified liquid layer are shown in Figure 3. The existence of two layers and their boundaries leads to a distribution of characteristic rotational times. However, ignoring the boundary contributions (see previous section) we obtain two relaxation times which, if sufficiently separated, would be experimentally observed. 4. Temperature and Pore Size Dependence: An Example All our considerations until now have been under the assumption of a fixed temperature, high enough to imply the liquid state. There are, however, experimental investigations2 of rotational relaxation near the liquid freezing point in small pore systems which point toward a sharp increase in the rotational time of a dipole as the freezing point is approached. It has also been observed that there is a depression in the freezing point in small pores so that the relaxation time at a given temperature, near freezing, is shorter for smaller pores. The high-temperature experiments show almost no effect of the pore size on the relaxation time which agrees with our conclusion on the weak effect of the boundary on the dipole relaxation (unless the liquid in the surface vicinity is structured). Depression has also been observed in
Here $6”) is the bulk value of dipole friction corresponding to the “high”-temperature limit ( R >> A), eq 29a, and R2
r
2To
1
with
It is clear from eqs 38 and 39 that in the narrow temperature (20)Jackson, C. L.; Mckenna, G.B. In Dynamics in Small Confining Systems. Extended Absfrocf of 1990 Fall Meeting of the MRS; Drake, .I. M., Klafter, J., Kopelman, R.. Eds.; MRS: Pittsburgh, 1990; p 31.
3485
J. Phys. Chem. 1992, 96, 3485-3489 4
r lr:l\ ' \ ' \ ' \
3 1
Appendix In order to calculate the dielectric friction in the bulk liquid we will follow the cavity approach.12J6q'8 Consider a dielectric sphere of radius R and dielectric constant el at the center of which a dipole is located. Outside the sphere there is a liquid with dielectric function e(k,w), eq 1. For this model the electrostatic potential, 4, can be written by expanding the solution of the Laplace equation in spherical coordinates. The potential in the liquid, r > R , should vanish at infinite distance, and the homogeneous part of the potential inside the sphere, r < R , has to be analytic at the origin. The nonhomogeneous part of the potential in the sphere arises from the point dipole with the dipole moment p at the center. Thus we have
'
T
Figure 4. Temperature dependencies of the normalized rotation relaxation time, 7 i Rat a given frequency; y = 2R'kTIw/p2. The temperatures TIL&)and T2*(L2) are the transition temperatures in eq 36 for the pores with different sizes, L, > L2;To= I , cI = 1, cb = 80,w i = 0.1, c. = 3. Curves 1 and 1' are obtained at R / & = IO; curves 2 and 2' at R / & = 4.
interval around the transition point, T*(L),the dielectric friction and correspondingly the orientational relaxation time change from the value €6") to the value €6")' determined by eq 29b. The temperature and size dependencies of the rotational relaxation time at different values of parameters are shown in Figure 4. It should be emphasized, however, that eqs 36-39 present an example rather than an analysis of a real system. A different temperature dependence will be obtained using the same methodology but assuming different behaviors of A and T . In conclusion, we have studied the role of a boundary in modifying the relaxation behavior of a dipole embedded in a liquid. The formalism presented here which is based on the continuum approach extends previous works by introducing a nonlocal dielectric description of the liquid. Although the relaxation may be nonexponential in time we have calculated the dielectric friction for a given frequency as a function of zo, the distance of the dipole from the boundary. The effect of the boundary has been shown to be small unless the properties of the liquid itself are drastically changed due to the presence of the interface. The nonlocal nature of dielectric function modifies the dielectric friction derived within the local approximation and allows to introduce temperature dependence into the relaxation process through the characteristic length,
4=
('42) The coefficients A, B, and C are obtained from the boundary conditions at r = R which are similar to the conditions 8,9, and 10 used for the plane interface. As a result the field induced at the surface of dielectric sphere, r = R, has the form
€I )
4R/ A) I I)
(A3 1
withf(R/A) given by eq 28. A similar approach for the calculation of the dipole damping in the liquid (with the same boundary conditions) was adopted by van der Zwan and Hynes in ref 12. Our results, however, differ from theirs.
Refractive Index and Density Variations in Pure Liquids. A New Theoretical Relation A. Proutiere,* E. Megnassan, and H. Hucteau Laboratoire de Spectrochimie des Ions, UniversitE de Nantes, 2, rue de la Houssinigre, 44072 Nantes Cedex 03, France (Received: July 17, 1991)
A new theoretical expression of the density fluctuation of the refractive index (dn2/ddp or DFRI has been deduced from a general expression of the Lorentz-Lorenz equation and from the calculation of the partial derivatives of n2. The square of the refractive index in each point of the liquid has been considered as a function of the number N of molecules per volume unit, of the temperature T and of the pressure P. The final expression of the DFRI is the sum of the large isothermal term (an2/adP,, and of the smaller one (an2/ar),,/(dd/dT),. Seventy-five organic solvents have been tested and the results have been compared to the experimental values deduced from (dn/dT), and (dd/dT), measurements. This new DFRI expression gives numerical results in better agreement with experimental data than the empirical Eykman's rule.
Introduction From the values of the refractive index n and of the density d of a pure liquid the Lorentz-Lorenz (L-L) expression (n2 l)/(n2 + 2) = NAda/3Mq, gives the mean molecular polarizability This equation is based on the classical internal field ap( I ) Glasstone, S. Texr Book of Physical Chemistry; Van Nostrand: Princeton, NJ, 1940. (2) Partington, J. R. An Aduanced Treatise on Physical Chemiswy; Longmans, Green: London, 1953;Vol. 1V.
proximation (Lorentz field) and it is commonly used for the calculation of molecular polarizabilities at ordinary temperature." Unfortunately, the values of dn2/dd which are deduced from this equation are in disagreement with the experimental results. The (3) Le Fevre, C. G.; Le Fevre, R. J. W. Reo. Pure Appl. Chem. 1955,5, 261. (4) Bottcher, C. J. F. Theory o/ dielectric polarization; Elsevier: Amsterdam, 1973,Vol. I; 1975,Vol. 2. ( 5 ) ProutiZre, A. J . Chim. Phys. 1976,73, 665.
0022-3654/92/2096-3485%03.~0/0 0 1992 American Chemical Society
