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THE JOURNAL OF CHEMICAL PHYSICS 124, 114904 共2006兲
Theory of solvation in polar nematics
Vitaly Kapko and Dmitry V. Matyushova兲
Department of Chemistry and Biochemistry and the Center for the Study of Early Events in Photosynthesis,
Arizona State University, Tempe, Arizona 85287-1604
共Received 15 December 2005; accepted 25 January 2006; published online 20 March 2006兲
We develop a linear response theory of solvation of ionic and dipolar solutes in anisotropic, axially
symmetric polar solvents. The theory is applied to solvation in polar nematic liquid crystals. The
formal theory constructs the solvation response function from projections of the solvent dipolar
susceptibility on rotational invariants. These projections are obtained from Monte Carlo simulations
of a fluid of dipolar spherocylinders which can exist both in the isotropic and nematic phases. Based
on the properties of the solvent susceptibility from simulations and the formal solution, we have
obtained a formula for the solvation free energy which incorporates the experimentally available
properties of nematics and the length of correlation between the dipoles in the liquid crystal. The
theory provides a quantitative framework for analyzing the steady-state and time-resolved optical
spectra and makes several experimentally testable predictions. The equilibrium free energy of
solvation, anisotropic in the nematic phase, is given by a quadratic function of cosine of the angle
between the solute dipole and the solvent nematic director. The sign of solvation anisotropy is
determined by the sign of dielectric anisotropy of the solvent: solvation anisotropy is negative in
solvents with positive dielectric anisotropy and vice versa. The solvation free energy is
discontinuous at the point of isotropic-nematic phase transition. The amplitude of this discontinuity
is strongly affected by the size of the solute becoming less pronounced for larger solutes. The
discontinuity itself and the magnitude of the splitting of the solvation free energy in the nematic
phase are mostly affected by microscopic dipolar correlations in the nematic solvent. Illustrative
calculations are presented for the equilibrium Stokes shift and the Stokes shift time correlation
function of coumarin-153 in 4-n-pentyl-4⬘-cyanobiphenyl and 4,4-n-heptyl-cyanopiphenyl solvents
as a function of temperature in both the nematic and isotropic phases. © 2006 American Institute of
Physics. 关DOI: 10.1063/1.2178318兴
I. INTRODUCTION
The problem of polar solvation is one of the oldest problems of physical chemistry which yet is still a field of active
theoretical and experimental researches. The calculation of
solvation free energy is particularly complex, since it is affected by a variety of contributions including the short-range
cavity formation energy, medium-range dispersion and induction forces 共nonpolar solvation兲, and long-range electrostatic interactions 共polar solvation兲. These components often
compensate and complement each other when solvents of
different polarities are considered. As a result, calculations of
the overall free energy of solvation are challenging and often
require phenomenological parametrization. Many phenomena involving changes in the multipolar distribution of the
solute charge 共spectroscopy, redox reactions, etc.兲 are mostly
affected by nonpolar and polar solvations, while the latter
often dominates in polar solvents. Therefore, much effort
over the last 80 years, following the work of Born,1
Onsager,2 and Kirkwood,3 has focused on the understanding
and modeling of electrostatic, polar solvation.
The original Born-Onsager idea of calculating the electrostatic solvation free energy as the continuum dielectric
response to charges of the solute has found broad applicaa兲
Electronic mail: [email protected]
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tions, in particular, to solvation of large molecules often encountered in biomedical research.4,5 For smaller solutes, formal liquid-state theories, most notably integral equation
theories, have found broad application. These theories are
normally formulated either in terms of site-site6 or multipolar
interaction7 potentials. The proliferation of computer simulation techniques has helped to clarify many microscopic features of solvation as well as to test and refine the formal
models.
Most effort in the field of solvation thermodynamics and
dynamics has focused on the understanding of solvation in
isotropic solvents. Anisotropic solvation, important for
chemical reactivity in biological membranes, surfaces, and
liquid crystalline solvents, has attracted relatively little attention. Also, from the side of experiment, almost nothing is
known about thermodynamics of solvation in liquid crystals.
There is a very limited evidence on the solvatochromic shift
from spectroscopy,8 and a few solvation dynamics studies9–11
have been reported. Computer experiment on solvation in
liquid crystals virtually does not exist. Continuum models,
representing the effect of solvent anisotropy by a tensorial
dielectric constant, have been proposed.12,13 These approaches provide a very useful continuum limit since, for
instance, the Onsager problem of solvation of a spherical
dipole2 has an exact analytical solution for continuum
nematics.14,15
124, 114904-1
© 2006 American Institute of Physics
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114904-2
J. Chem. Phys. 124, 114904 共2006兲
V. Kapko and D. V. Matyushov
Despite the progress in using dielectric continuum models, a few fundamental problems still need to be resolved.
First, limits of the applicability of the continuum approximation to solvation in polar nematics have not been established.
Liquid crystals are mostly made of bulky elongated molecules, and it is a priori unclear if continuum models can be
applied to solvation of solutes of size often comparable to
the size of the solvent molecules. Second, it is not clear if
dielectric response of a liquid crystal to the solute electric
field can in principle be represented by a single quantity, the
dielectric constant, in particular, close to the isotropicnematic phase transition.
The approach we propose in this paper is based on the
recently obtained microscopic solution for dipole solvation.16
The model is based on the assumption that the solute-solvent
interaction potential is given by the interaction of the solute
charges with the solvent dipolar polarization. The solvation
free energy is then expressed through the polarization autocorrelation function of the pure solvent without any particular assumptions regarding the solvent structure. The theory is
thus applicable to an arbitrary isotropic dielectric. The goal
of this paper is to generalize this approach to the case of a
solvent with axial symmetry. For solvation in isotropic liquids, two projections of the polarization autocorrelation
function, longitudinal and transverse, are sufficient to describe the dipolar response. Lowering the symmetry of the
solvent requires a larger set of projections. We derive a formally exact expression for the free energy of ionic and dipolar solvations in Sec. II 关Eq. 共46兲兴.
The full formulation of the theory requires projections of
the polarization correlation function on rotational invariants.
These are obtained here from computer simulations of a fluid
of dipolar hard spherocylinders. The application of the
theory to experiment requires, however, a solution based on
the input parameters available from experiment. This formulation is given in Sec. III 关Eq. 共58兲兴 in the form of a linear
combination of solutions obtained in the limit of zero wave
number 共continuum兲 and infinite wave number. The relative
contribution of each component depends on the correlation
length of dipolar fluctuations in the liquid 共distinct from the
correlation length of the order parameter fluctuations in
Landau–de Gennes theory of liquid crystals17兲. One of the
principle results of this study is a very slow approach of the
solvation free energy to its continuum limit, thus invalidating
continuum approaches to solvation of small and medium-size
solutes.
In Sec. IV, we study the dependence of the free energy of
solvation on the angle between the solute dipole and the
nematic director as well as on temperature when crossing the
point of isotropic-nematic transition. The solvation free energy is shown to pass through a discontinuity at the transition
temperature becoming anisotropic in the nematic phase.
Also, the Stokes shift time correlation function changes from
a single-exponential decay in the isotropic phase to a biexponential decay in the nematic phase, with a very slow second relaxation component. Our results are summarized in
Sec. V.
II. THEORY
The linear response approximation 共LRA兲 provides a solution for the solvation free energy 共strictly speaking, the
chemical potential兲 ␮ in terms of the response function
␹共r1 , r2兲 which gives the dipolar polarization in the point of
space r1 produced by the external electric field E0共r2兲 at the
point of space r2,
P共r1兲 =
冕
␹共r1,r2兲 · E0共r2兲dr2 ,
共1兲
where the subscript “0” for the variables refers to the solute.
The solvation free energy is then
␮=−
1
2
冕
P共r1兲 · E0共r1兲dr1 .
共2兲
The dependence of ␹共r1 , r2兲 on two separate positions, instead of r1 − r2 of homogeneous solvents, reflects the inhomogeneous nature of the solvent response in the presence of
the repulsive core of the solute expelling the solvent from its
volume.
In k space, Eq. 共2兲 becomes
␮=−
1
2
冕
dk1dk2
Ẽ0共k1兲 · ˜␹共k1,k2兲 · Ẽ0共− k2兲.
共2␲兲6
共3兲
Here, the Fourier transform of the electric field is taken over
the solvent volume ⍀ excluding the space occupied by the
solute,
Ẽ0共k1兲 =
冕
⍀
E0共r兲eik·rdr.
共4兲
The solute space is made by the van der Waals repulsive
cores of its atoms. The radii of the solute atoms exposed to
the solvent are augmented by the shortest distance to the
solvent dipole, which, for cylindrically symmetric molecules,
is equal to the radius of the cylindrical part of the molecule.
Further, the second-rank tensor ˜␹ is
˜␹␣␤共k1,k2兲 =
1
具␦ P̃␣共k1兲␦ P̃␤共− k2兲典0 ,
k BT
共5兲
where ␦P̃共k̃兲 is the Fourier transform of the fluctuation of the
solvent dipolar polarization.
The LRA solution is independent of the electrostatic
field of the solute, and the subscript 0 in the angular brackets
denotes the statistical average taken at the presence of a fictitious solute with the repulsive core of the real solute but the
electrostatic solute-solvent coupling turned off.16,18 In a hypothetical case of an infinitely small solute, ˜␹ is equal to the
dipolar susceptibility of pure solvent ˜␹s which depends on
only one wave vector,
˜␹共k1,k2兲 = ␦k ,k ˜␹s共k1兲,
1 2
共6兲
where ␦k1,k2 = 共2␲兲3␦共k1 − k2兲 and the subscript “s” denotes
the solvent.
In the general case, ˜␹共k1 , k2兲 is affected by the presence
of the solute and depends on two k vectors. The effect of the
solute on the solvent response can generally be separated
into two major contributions. The repulsive core of the solute
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114904-3
J. Chem. Phys. 124, 114904 共2006兲
Theory of solvation in polar nematics
distorts the local density of the solvent around it. The spherically symmetric solute-solvent pair correlation function
h0s共r兲 is then different from the solvent-solvent pair correlation function hss共r兲. This density disturbance affects the dipolar polarization and, consequently, the response function.
Another, by far more significant, effect of the solute on the
solvent response function is related to the expulsion of the
dipolar polarization from the solute volume. In continuum
models, this effect is responsible for the surface charge at the
dielectric cavity and, when the cavity does not coincide with
the equipotential surface, results in a transverse component
in the dielectric response. Maxwell’s dielectric
displacement19 D共r兲 then differs from the external electric
field E0共r兲.
Chandler’s Gaussian approximation18,20 neglects the first
effect of alteration of the solvent polarization by the local
density profile around the solute, but takes full account of the
exclusion of the dipolar polarization from the solute volume.
It results in the following equation for the k-space response
function:16
˜␹共k1,k2兲 = ␦k ,k ˜␹s共k1兲 − ˜␹⬙共k1兲 · ˜␪0共k1 − k2兲˜␹s共k2兲. 共7兲
1 2
Here, ˜␪0共k兲 is the Fourier transform of the step function,
which equals to unity inside the solute and is zero everywhere else. Further, in Eq. 共7兲,
˜␹⬙共k兲 = ˜␹s共k兲 · 关˜␹s共k兲 − ˜␹⬘共k兲兴 ,
−1
FIG. 1. Dipolar solute in a nematic solvent. The laboratory system of coordinates is chosen to align the z axis with the nematic director n̂. m0 denotes
the direction of the solute dipole and k is the wave vector. ␤ is the angle
between the dipole moment and the long axis of the solvent molecule and ␴
is the largest diameter in the plane perpendicular to the long axis of the
solvent molecule.
ity of the longitudinal and transverse projections, the solvation free energy splits into the longitudinal 共L兲 and transverse
共T兲 parts,16
␮ = 共␹tr兲−1关␹sT共0兲␮Lh + ␹sL共0兲␮Th 兴,
where
共8兲
where
˜␹⬘共k兲 =
冕
⍀⬘
dr␹s共r兲eik·r .
共9兲
The integration in Eq. 共9兲 is over the volume ⍀⬘ formed by
excluding from the liquid, the space formed by r1 − r2 when
both r1 and r2 belong to the solute. This space makes the
volume twice as large as the solute volume for spherical
solutes, but has a rather complex shape for nonspherical solutes.
The substitution of Eq. 共7兲 into Eq. 共3兲 splits the solvation free energy into the sum of two components,
␮ = ␮h + ␮corr .
共10兲
The first term ␮h corresponds to the homogeneous response
approximation 共subscript h兲 which assumes that correlations
of dipolar polarization are not modified by the solute and
␹共k1 , k2兲 can be approximated by the dipolar susceptibility
of the pure solvent according to Eq. 共6兲. The only modification introduced by the solute is the cutoff of the electric field
inside the solute 关Eq. 共4兲兴,
− ␮h =
1
2
冕
dk
Ẽ0共k兲 · ˜␹s共k兲 · Ẽ0共− k兲.
共2␲兲3
共11兲
In isotropic solvents, the tensor ˜␹s is diagonal in a coordinate system with one axis taken along k. Its eigenvalues,
the longitudinal ˜␹sL and transverse ˜␹sT projections, are quite
different in the range k␴ ⬍ 2␲ due to the long-range nature of
the dipole-dipole interaction potential 共␴ is the diameter of
the solvent molecules兲.21 Because of the mutual orthogonal-
共12兲
1
␹tr = 共␹sL共0兲 + 2␹sT共0兲兲
3
共13兲
冕
共14兲
and
− ␮L,T
h =
1
2
dk L,T
2
␹ 共k兲兩ẼL,T
0 共k兲兩 .
共2␲兲3 s
Unfortunately, this scheme is hard to implement for liquid crystals. Since a liquid crystal has its own symmetry axis
共Fig. 1兲, ˜␹s needs to be diagonalized for each value of k
making the problem rather complex. Expansion of the solvent response function in spherical harmonics22,23 appears to
be a more straightforward way to the solution. Our solution
below is given for a spherical ion 共i兲 and spherical dipole
共d兲. Fourier transforms 关Eq. 共4兲兴 of the electric fields for
these two solutes are as follows:
j0共kR1兲
k̂,
k
共15兲
j1共kR1兲
m0 · D̂k .
kR1
共16兲
Ẽ共i兲
0 共k兲 = 4␲iq0
and
Ẽ共d兲
0 共k兲 = − 4␲
In Eqs. 共15兲 and 共16兲, q0 and m0 are the solute charge and
dipolar moment, and
D̂k = 3k̂k̂ − 1
共17兲
is the dipolar tensor. Further, R0 is the solute radius, R1
= R0 + ␴ / 2 is the distance of closest solute-solvent separation,
and k̂ = k / k. Here we also use the standard notation for the
spherical Bessel functions jl共x兲.
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114904-4
J. Chem. Phys. 124, 114904 共2006兲
V. Kapko and D. V. Matyushov
The correction term in Eq. 共10兲 is given by a double k
integral,
␮corr =
1
2
冕
dk1dk2
Ẽ0共k1兲 · ˜␹⬙共k1兲 · ˜␪0共k1
共2␲兲6
− k2兲˜␹s共k2兲 · Ẽ0共− k2兲.
Azz = −
where
␺=
共18兲
F0共r兲 =
冕
1
R31
冕
d␻k
m0 · D̂k · ˜␹⬙共k = 0兲.
4␲
=
共19兲
冦冉
arctan 冑⑀储/⑀⬜ − 1/冑⑀储/⑀⬜ − 1,
⑀储 ⬎ ⑀⬜
1,
⑀储 = ⑀⬜
共21兲
In order to calculate F0 in Eq. 共20兲, we need to obtain ˜␹⬙共k
= 0兲 from the k = 0 value of the solvent dipolar susceptibility,
which we consider next.
A. Continuum limit
We use the laboratory Cartesian system of coordinates
with z-axis parallel to the nematic director n̂ 共Fig. 1兲. The
continuum limit for solvent response function can be obtained from Maxwell’s material equations with axially symmetric dielectric constant characterized by longitudinal 共⑀储
= ⑀z兲 and transverse 共⑀⬜ = ⑀x = ⑀y兲 components,24
k̂␣k̂␤共⑀␣ − 1兲共⑀␤ − 1兲
˜␹0,xx = ˜␹0,yy =
⑀⬜ + 共⑀储 − ⑀⬜兲共k̂ · n̂兲2
,
where ␣ and ␤ denote the Cartesian components. Note that
˜␹s共k = 0兲 is an even function of k̂; therefore, according to
Eqs. 共15兲 and 共19兲, F0 = 0 for a spherical ion 关Eq. 共21兲兴.
To proceed with the dipolar solute, we first calculate the
integral
d␻k
兺 D̂k,␣␥˜␹s,␥␤共k = 0兲,
4␲ ␥
共23兲
where D̂k is given by Eq. 共17兲. The matrix A is diagonal with
the elements,
Axx = Ayy =
冋
冊
/共2冑1 − ⑀储/⑀⬜兲, ⑀储 ⬍ ⑀⬜ .
册
⑀⬜ − 1
⑀储
共2⑀储 − ⑀⬜ + 2兲 − 共⑀⬜ + 2兲␺ ,
8 ␲ 共 ⑀ 储 − ⑀ ⬜兲
⑀⬜
共24兲
冧
共26兲
冕
d␻k
˜␹s共k = 0兲.
4␲
冋
共27兲
册
⑀储 − 1
⑀储
共2⑀储 − ⑀⬜ − 1兲 − 共⑀⬜ − 1兲␺ ,
8 ␲ 共 ⑀ 储 − ⑀ ⬜兲
⑀⬜
共28兲
and
⑀储 − 1
关⑀⬜ − 1 − 共⑀储 − 1兲␺兴.
4 ␲ 共 ⑀ 储 − ⑀ ⬜兲
共29兲
When Eqs. 共22兲 and 共27兲 are used in the definition of ␹⬙ in
Eq. 共8兲, the final result for the field F共d兲 becomes
F␣共d兲 =
m0,␣
R␣ ,
R31
共30兲
where
R␣ =
共22兲
冕
储
˜␹0 ⬅ ˜␹s共k = 0兲 − ˜␹⬘共k = 0兲 =
˜␹0,zz = −
A␣␤ = −
储
Then ˜␹0 is diagonal with the elements
F共i兲 = F0 = 0.
˜ s,␣␤共k = 0兲 = 共⑀␣ − 1兲␦␣␤ −
4␲␹
1 + 冑1 − ⑀ /⑀⬜
1 − 冑1 − ⑀ /⑀⬜
Note that Axx and Azz, as well as ˜␹0,xx and ˜␹0,zz below, have
no singularities at ⑀储 = ⑀⬜ because terms in the square brackets are proportional to ⑀储 − ⑀⬜ at 兩⑀储 − ⑀⬜兩 Ⰶ 1.
In the Appendix, we prove the relation
共20兲
For a spherical ion 共see below兲,
共25兲
dz
1 + 共⑀储/⑀⬜ − 1兲z2
ln
Analytic properties of the response function ˜␹⬙ in complex k plane allow one to reduce the integration over k to the
angular integral over the directions of k.16 For a spherical
dipole, F0 = F共d兲 is constant within the solute,
F共d兲 = −
1
0
In order to convert it to a computationally tractable threedimensional 共3D兲 integral, we first introduce a direct-space
field,
dk −ik·r
Ẽ0共k兲 · ˜␹⬙共k兲.
e
共2␲兲3
冕
⑀储 − 1
关⑀⬜ + 2 − 共⑀储 + 2兲␺兴,
4 ␲ 共 ⑀ 储 − ⑀ ⬜兲
⑀␣ − n␣共⑀␣ + 2兲
.
⑀␣ − n␣共⑀␣ − 1兲
共31兲
In Eq. 共31兲, the so-called depolarization factors are given
by25
nz =
⑀储共␺ − 1兲
,
⑀⬜ − ⑀储
共32兲
1 − nz
.
nx = n y =
2
In the isotropic limit, when ⑀储 = ⑀⬜, one gets nx = ny = nz
= 1 / 3.
The field F0, defined by Eq. 共19兲, is a generalization of
the reaction field, introduced by Onsager for a point dipole,2
to an arbitrary configuration of solute charges in a solute of
arbitrary shape. We have shown here that this field reduces to
expected limits in the case of spherical ionic and dipolar
solutes. In the former case, the reaction potential created by
the polar liquid within the cavity is constant, and the reaction
field is zero. In the latter case, the field is constant and our
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114904-5
J. Chem. Phys. 124, 114904 共2006兲
Theory of solvation in polar nematics
expression in Eqs. 共30兲–共32兲 coincides with the reaction field
in an axially anisotropic dielectric previously derived for a
spherical dipole by solving the Poisson equation.15
The zero reaction field in the case of a spherical ion
eliminates the correction term in Eq. 共10兲. This means that
the solvent response is longitudinal and the dielectric displacement D is equal to the external field E0. The free energy
of solvation is then fully determined by the homogeneous
solvation term 关Eq. 共11兲兴. In the case of a dipole, the solvent
response includes a transverse component, the dielectric displacement is not equal to the external field, and the correction term is necessary,
␮corr =
1
兺 R␣m0,␣
2R31 ␣␤
冕
dk
˜␹s,␣␤共k兲˜␪0共k兲Ẽ0,␤共− k兲,
共2␲兲3
共33兲
共i兲
Ẽ0,n
共k兲 = 4␲i
1
and
共d兲
共k兲 = − 16␲2
Ẽ0,n
1
2
␮共i兲
h = − 4q0 兺
n1n2l
⫻
Equations 共11兲 and 共33兲 give the correct continuum limit
共subscript c兲 for the solvation free energy after the replacement of ˜␹s共k兲 with its value at k = 0,
␮共d兲
h =−
冉 冊
␺
q2
1−
,
2R1
⑀⬜
共35兲
m20
1
共d兲
␮共d兲
关Rx + 共Rz − Rx兲cos2 ␪0兴, 共36兲
c = − m0 · F0 = −
2
2R31
for the dipole. In Eq. 共36兲, ␪0 is the angle between the solute
dipolar moment and the director 共Fig. 1兲. In the limit ⑀储
→ ⑀⬜, Eqs. 共35兲 and 共36兲 reduce to their well-know isotropic
counterparts, the Born formula1
冉 冊
1
q2
1−
,
2R1
⑀
共37兲
n1⬘
共41兲
冕
⬁
dkj20共kR1兲␹s,n1n2l共k兲共− 1兲n1+n2
0
C共11l;0,0,0兲
C共11l;n1,n2,n1 + n2兲,
2l + 1
⫻
24m20
R21
冕
⬁
0
dkj21共kR1兲 兺 ␹s,n1n2l共k兲共− 1兲n1+n2
n1n2l
C共22l;000兲
兺 C共112;n1,n⬘,n1 + n⬘兲
2l + 1 n
⬘
m2 ⑀ − 1
.
= − 30
R1 2⑀ + 1
共38兲
Note that the cavity radius is not specified in continuum
models. However, empirical experience suggests using the
van der Waals radius R0 in place of the radius of closest
solute-solvent approach R1 appearing in microscopic solvation models.
The dependence on the orientation of the wave vector in
Eqs. 共11兲 and 共33兲 can be integrated out by expanding the
solvent dipolar susceptibility ˜␹s in spherical harmonics 关Eq.
共A4兲兴,
l
共d兲
␮corr
=−
4冑6m20
5R21
兺
n n
1 2
冕
⬁
0
共42兲
dkj21共kR1兲˜␹s,n1n22共k兲共− 1兲n2
⫻C共112;n1,n2,n1 + n2兲m̂0,n1共m̂0R兲−n1 ,
共i兲
␮corr
= 0.
␹s,n1n2l共k兲 =
冑
2l + 1
˜␹s,n n l共k兲,
1 2
4␲
共43兲
and the relations between the spherical and Cartesian components of the vector m̂ = m / m are
m̂0,0 = m̂0,z ,
m̂0,1 = − 共m̂0,x + im̂0,y兲/冑2,
共44兲
m̂0,−1 = 共m̂0,x − im̂0,y兲/冑2.
B. Microscopic theory
*
˜␹s,n n 共k兲 = 兺 ˜␹s,n n l共k兲Y l,−n
1 2
1 2
⫻C共22l;n1 + n⬘,n2 − n⬘,n1 + n2兲m̂0,n⬘m̂0,−n⬘ ,
In Eq. 共42兲,
and the Onsager formula2
␮O共d兲
2 j1共kR1兲
m0
兺 C共112;n1,n1⬘,n1
5
kR1
⫻C共112;n2,− n⬘,n2 − n⬘兲
for the ion and
␮B共i兲 = −
冑
共40兲
In Eq. 共41兲, C共l1l2l ; n1 , n2 , n兲 are the Clebsch-Gordan
coefficients.22 Using the product rule and orthogonality of
spherical harmonics22 we obtain
共34兲
␮共i兲
c =−
4␲ j0共kR1兲 *
q0
Y 1,n 共␻k兲,
1
3
k
*
n1⬘兲Y 1,n⬘共m̂0兲Y 2,n +n⬘共␻k兲.
1
1 1
+
where for a spherical solute
˜␪ 共k兲 = 4␲R3 j1共kR1兲 .
0
1
kR1
冑
1−n2
共␻k兲,
共39兲
where ␻k denotes the orientation of k̂. The spherical components of the solute electric field can be obtained for the ionic
and dipolar solutes,22
Similarly, for the second-rank tensor, one has
共m̂0R兲0 = m̂0,zRz ,
共m̂0R兲1 = − 共m̂0,x + im̂0,y兲Rx/冑2,
共45兲
共m̂0R兲−1 = 共m̂0,x − im̂0,y兲Rx/冑2.
Similar relations exist for the components of the second-rank
tensor ˜␹s共k兲.22
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114904-6
J. Chem. Phys. 124, 114904 共2006兲
V. Kapko and D. V. Matyushov
For weakly polar solvents the Cartesian components of
˜␹s共k兲 form a diagonal matrix in the laboratory system of
coordinates specified in Fig. 1. Moreover, the components
related to the axes x and y are almost equal to each other. In
this approximation the solvation free energy reduces to
4q20
3
␮共i兲
h =−
␮共d兲
h =−
共d兲
␮corr
=
冕
5R21
5R21
dkj20共kR1兲A共k兲,
0
4m20
4m20
⬁
冕
⬁
dkj21共kR1兲关B共k兲 + C共k兲cos2 ␪0兴,
0
冕
⬁
共46兲
dkj21共kR1兲关Rx␹s,xx2共k兲 − 共2Rz␹s,zz2共k兲
0
+ Rx␹s,xx2共k兲兲cos2 ␪0兴,
共i兲
␮corr
=
FIG. 2. Szz,l共k兲 共upper panel兲 and Sxx,l共k兲 共lower panel兲 projections of dipolar
structure factors for the fluid of hard spherocylinders: l = 0 共solid line兲, l
= 2 共dashed line兲, and l = 4 共dotted line兲. The dot-dashed lines refer to the
Padé approximation 关 Eq. 共58兲兴 with ⌳ = 0.4␴. The structure factors are
obtained from NVT MC simulations 共Ref. 26兲 of 800 solvent molecules with
the packing density ␩ = 0.47 and the dipole moment m2 / 共kBT␴3兲 = 1.0. The
fluid is in the nematic phase with the nematic order parameter S2 = 0.85.
0,
where
2
A共k兲 = 2␹s,xx0共k兲 + ␹s,zz0共k兲 + 共␹s,zz2共k兲 − ␹s,xx2共k兲兲,
5
1
B共k兲 = 3␹s,zz0共k兲 + 7␹s,xx0共k兲 + 共3␹s,zz2共k兲 − 10␹s,xx2共k兲兲
7
4
− 共␹s,zz4共k兲 − ␹s,xx4共k兲兲,
7
共47兲
1
C共k兲 = ␹s,zz0共k兲 − ␹s,xx0共k兲 + 共5␹s,zz2共k兲 + 16␹s,xx2共k兲兲
7
+
12
共␹s,zz4共k兲 − ␹s,xx4共k兲兲.
7
In Eqs. 共46兲 and 共47兲, ␹s,␣␣l共k兲 are the coefficients of expansion of the solvent response function in Legendre polynomials Pl共cos ␪k兲,
˜␹s,␣␣共k兲 = 兺 ␹s,␣␣l共k兲Pl共cos ␪k兲.
共48兲
l
III. RESULTS
The theory developed in the previous section requires
static dielectric constants and dipolar susceptibility of the
nematic solvent as input. Here we obtain these data from
Monte Carlo 共MC兲 simulation of hard spherocylinders with
embedded point dipoles.26 This fluid transforms from isotropic to nematic phase with decreasing density.27 NVT MC
simulations of N = 800 hard spherocylinders were carried out
in our previous study.26 The dipole moment m is parallel to
the cylinder axis, and the aspect ratio of the length L of the
cylindrical part of the molecule to its diameter ␴ is equal to
5. The thermodynamic state of this fluid is fully defined by
two parameters: the reduced dipole moment 共m*兲2
= m2 / 共kBT␴3兲 and the packing fraction ␩ = 共␲ / 6兲␳␴3共1
+ 3L / 2␴兲, where ␳ = N / V is the solvent number density. Details of the simulation protocol are given in Ref. 26.
Simulations of nematics with high magnitudes of the dipole moment m* are complicated by the tendency of neighboring dipoles to orient in a locally antiferroelectric order.
Combined with the elongated shape of the spherocylinders,
local antiparallel alignment of dipoles creates bottlenecks in
the system phase space, which are hard to explore by standard simulation techniques.27 In addition, the fluid of dipolar
spherocylinders becomes smectic at 共m*兲2 ⬎ 6.0 and ␩
= 0.47. Because of the relatively high aspect ratio of the solvent molecules, the dipole moment in the range 0 艋 共m*兲2
艋 6.0 gives a relatively small overall density of dipoles and,
therefore, a low dielectric constant. As a result, the capabilities of simple hard-core models are rather limited in exploring high-polarity nematics. The structure of real polar nematic liquids, which can demonstrate rather high dielectric
constants,28 is mostly determined by dispersion site-site interactions between elongated molecules.
For weakly polar nematics, the tensor of the dipolar susceptibility ˜␹s is nearly diagonal in the Cartesian coordinate
system specified in Fig. 1, with almost equal xx and yy projections. Therefore, the formally exact formulas in Eq. 共42兲
can be replaced by the approximate relation in Eq. 共46兲. The
solvation free energy then depends on six one-dimensional
projections ␹s,␣␣l共k兲 关Eq. 共48兲兴. The corresponding structure
factors of dipolar polarization,
S␣␣,l共k兲 = 共4␲/3y兲␹s,␣␣l共k兲,
共49兲
have been obtained here from equilibrium MC configurations
of the fluid of dipolar spherocylinders 共Fig. 2兲.26 In Eq. 共49兲,
y = 4␲m2␳/共9kBT兲
共50兲
is the standard dipole density parameter of dielectric
theories.19 The reduced dipole moment and the packing fraction of the system have values 共m*兲2 = 1 and ␩ = 0.47 共the
isotropic-nematic phase transition occurs at ␩IN ⬇ 0.407兲. In
this thermodynamic state, the nematic order parameter S2 is
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114904-7
J. Chem. Phys. 124, 114904 共2006兲
Theory of solvation in polar nematics
equal to 0.85, and the longitudinal 共⑀储, parallel to the director兲 and transverse 共⑀⬜, perpendicular to the director兲 dielectric constants are 1.89 and 1.06, respectively.
The noise in the structure factors obtained from simulations 共Fig. 2兲 arises from the fluctuations of the nematic director in the laboratory system of coordinates attached to the
simulation box. This setup is necessary to insure that the
wave vectors used to calculate S␣␣,l共k兲 are eigenvectors of
the periodic replicas of the cubic simulation cell. The calculations show that the longitudinal structure factors 共Szz,l兲 are
significantly larger than the transverse structure factors
共Sxx,l兲, as expected for nematics with longitudinal dipole moment. In addition, the magnitudes of projections decrease
rapidly with increasing index l.
Exact analytical expressions are available for the polarization structure factors in k → 0 and k → ⬁ limits. The continuum limit relates Szz,l共0兲 and Sxx,l共0兲 to the anisotropic
static dielectric constants through Eqs. 共22兲, 共48兲, and 共49兲.
To calculate the k → ⬁ limit, we note that the Cartesian components of the dipolar structure factors are given by the following expression:
S␣␤共k兲 = 共3/N兲 兺 具m̂i,␣m̂ j,␤e−ik·rij典,
共51兲
ij
where N is a number of solvent molecules. All the terms in
Eq. 共51兲 with rij ⫽ 0 vanish at k → ⬁ resulting in
S␣␤共k → ⬁兲 = 共3/N兲 兺 具m̂i,␣m̂i,␤典 = 3具m̂␣m̂␤典.
共52兲
i
This yields
Sxx共k → ⬁兲 = Syy共k → ⬁兲 = 1 − S2 P2共cos ␤兲,
Szz共k → ⬁兲 = 1 + 2S2 P2共cos ␤兲,
共53兲
Szx共k → ⬁兲 = Szy共k → ⬁兲 = Sxy共k → ⬁兲 = 0,
where ␤ is the angle between the dipole moment and the
long molecular axis of the solvent molecule 共Fig. 1兲 and
P2共x兲 is the second Legendre polynomial.
From Eqs. 共11兲, 共33兲, 共49兲, and 共53兲, we find the expression for the solvation free energy when the k → ⬁ limit is
used for the solvent susceptibility,
␮⬁共i兲 = − 3y
q20
,
2R1
and
␮⬁共d兲 = −
m20y
R31
冉
共54兲
冊
1
1 + S2 P2共cos ␤兲P2共cos ␪0兲 .
5
共55兲
The continuum 关k = 0, Eqs. 共35兲 and 共36兲兴 and short
wavelength 关k → ⬁, Eqs. 共54兲 and 共55兲兴 limits are the two
asymptotes for the solvation free energy obtained by setting,
respectively, the constant ␹s共0兲 and ␹s共⬁兲 values for the solvent susceptibility in the k integrals in Eqs. 共11兲 and 共33兲.
The advantage of these limiting expressions is their simplicity and direct connection to experimentally available properties of liquid crystalline solvents. These limits can be used to
derive a practically useful analytical formula for ␮. The larg-
FIG. 3. The length of dipolar correlations in the fluid of hard spherocylinders vs the solvent packing density ␩ 共upper panel, 共m*兲2 = 1.0兲 and vs dipolar density y 共lower panel, ␩ = 0.47兲. The data are obtained from NVT MC
simulations. The closed diamonds refer to 兵zz , l = 0其 projection and the
closed squares indicate 兵xx , l = 0其 projection. The open points in the lower
panel indicate the transverse 共T兲 and longitudinal 共L兲 correlation lengths
obtained for a fluid of dipolar hard spheres with constant density ␳␴3 = 0.8
and changing dipole moment. The dashed lines connect the simulation
points. The dotted vertical line in the upper panel indicates the density of the
isotropic-nematic phase transition, ␩IN = 0.407.
est contribution to ␮ comes from the region of k where the
squared spherical Bessel functions jl共kR1兲 in Eq. 共46兲 have a
maximum. This is the region around k = 0 for the ionic solute
and k ⬇ 2 / R1 for the dipolar solute. We need, therefore, a
continuous approximation for the structure factors that generates a weighted linear combination of k = 0 and k → ⬁ limits for the solvation free energy.
The projections of the solvent susceptibility on spherical
harmonics are smooth functions of the wave vector for
weakly polar nematics. The Cartesian components of the dipolar structure factors can, therefore, be reasonably well approximated by Padé forms interpolating between the k = 0
and k → ⬁ limits,
S␣␤,l共k兲 =
S␣␤,l共0兲 + S␣␤,l共⬁兲⌳2l k2
1 + ⌳2l k2
.
共56兲
This formula introduces a new theory parameter, the polarization correlation length ⌳l. The correlation length can be
extracted from structure factors obtained from computer
simulations by fitting the slope of S␣␤,l共k兲 vs k2 to the k
→ 0 expansion of Eq. 共56兲 共k ⬍ kmax ⯝ 2 / ␴兲. Values of ⌳l for
the fluid of dipolar hard spherocylinders depending on packing fraction ␩ and dipolar density y are shown in Fig. 3
共closed points兲. ⌳l was found to be rather weakly dependent
on ␩, even through the isotropic-nematic phase transition
共Fig. 3, upper panel兲. The dependence on y was obtained at
fixed packing fraction ␩ = 0.432 and 共m*兲2 changing from 1.0
to 6.0.
We compare these results to the correlation lengths extracted from slopes of the longitudinal structure factor SL共k兲
vs k2 and the inverse structure factor 1 / ST共k兲 vs k2. The latter
definition corresponds to the Ornstein-Zernike-Debye plot
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114904-8
J. Chem. Phys. 124, 114904 共2006兲
V. Kapko and D. V. Matyushov
used for the scattering function of liquids close to the critical
temperature when the structure factor is a decaying function
of k.29 On the contrary, the longitudinal structure factor is a
rising function of k requiring the direct expansion of SL共k兲 in
k2. The correlation lengths for the longitudinal and transverse
dipolar fluctuations in isotropic liquids are substantially different. This is because these two projections mix together
harmonics of the pair distribution function with different indices l,
FIG. 4. Nonlocality functions 关Eq. 共59兲兴 and reduced solvation free energy
␮ / ␮c vs the solute size. The dashed and solid lines refer to the spherical ion
and spherical dipole solutes, respectively.
␳
S 共k兲 = 1 + 共h̃110共k兲 + 2h̃112共k兲兲,
3
L
␳
ST共k兲 = 1 + 共h̃110共k兲 − h̃112共k兲兲,
3
共57兲
where h̃lmn共k兲 is the Hankel transform 关Eq. 共A3兲兴.
In contrast to the longitudinal and transverse projections
of isotropic fluids, the projections S␣␤,l共k兲 correspond to the
same index l. The anisotropy of the nematic phase relative to
the director is then taken out to the Legendre polynomial
Pl共cos ␪k兲 关Eq. 共48兲兴. In addition, we found that ⌳l calculated
from different harmonics with the same l are approximately
equal to each other, at least for weak polar solvents. This is
why ⌳l in Eq. 共56兲 does not include Cartesian projections
␣ , ␤. The results for l = 0 are presented in Fig. 3, while the
data for l ⬎ 0 do not converge well because of large statistical
errors. The usefulness of Eq. 共56兲 thus requires further testing on model nematics with higher dielectric constants than
those available for dipolar pherocylinders.
With the structure factors given by Eq. 共56兲, the solvation free energy is a linear combination of continuum and
large wave vector limits,
␮ = ␮c + f共R1/⌳兲共␮⬁ − ␮c兲,
共58兲
where ⌳ = ⌳0. The nonlocality function f共R1 / ⌳兲 represents
the contribution of the nonlocal solvent response, influenced
by the finite length of dipolar correlations, to the solvation
thermodynamics. For the cases of spherical ionic 共i兲 and dipolar 共d兲 solutes, this function is given by the following expressions:
in this case, the Padé approximation 关Eq. 共56兲兴 gives reasonably accurate estimates for the solvation free energy in isotropic polar solvents.30
The solvation free energy in Eqs. 共58兲 and 共59兲 requires
the following solvent parameters: dielectric constants ⑀储 and
⑀⬜, the order parameter S2, and the polarization correlation
length ⌳. In addition, the ionic charge or dipole moment,
along with the radius R1 = R0 + ␴ / 2, should be supplied. The
dielectric constants and the order parameter come from experiment. The correlation length ⌳ is not experimentally
available and, for the sake of interpreting the experiment, is,
at the moment, a theory parameter requiring fitting to some
experimental observable. Simulations of the model fluid of
hard spherocylinders suggest magnitudes of ⌳ of the order
⌳ ⯝ 0.3− 0.4␴.
IV. THEORY PREDICTIONS
The present theory allows us to make some specific predictions regarding equilibrium solvation and solvation dynamics. The electrostatic component of the solvation free
energy can be measured from the steady-state Stokes shift of
optical lines, whereas solvation dynamics is probed by the
Stokes shift time correlation function. Our discussion below
will therefore target these two properties.
f 共i兲共x兲 = 0.5关1 − e−2x兴/x,
f 共d兲共x兲 = 1.5关x2 − 1 + 共x + 1兲2e−2x兴/x3 .
共59兲
Nonlocality functions f 共i兲共R1 / ⌳兲 and f 共d兲共R1 / ⌳兲 decay
monotonically from one to zero with increasing solute size
R1 共Fig. 4兲. The ionic nonlocality function decays faster than
the dipolar function indicating that ionic solvation is better
described by continuum approximation than dipolar solvation. Notice that formula 共58兲 applies to solvation in strongly
polar nematics, because it has been derived without assuming weak polarity of the solvent 共the cross terms omitted in
Eq. 共46兲 are included in ␮c兲. Figure 5 demonstrates a satisfactory agreement between the approximate solution given
by Eq. 共58兲 and the exact microscopic theory 关Eqs.
共46兲–共48兲兴. The increase of the solvent dipole moment leads
to an oscillatory character of the response function, but even
FIG. 5. Solvation free energy of ionic 共upper panel, in q20 / R1 units兲 and
dipolar 共lower panel, in m20 / R31 units兲 solutes vs R1. Shown are the microscopic calculation 关m, Eqs. 共46兲–共48兲兴, the Padé form 关“Pade,”Eq. 共58兲兴, and
the continuum limit 关c, Eqs. 共35兲 and 共36兲兴.
Downloaded 20 Mar 2006 to 129.219.244.213. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp
114904-9
J. Chem. Phys. 124, 114904 共2006兲
Theory of solvation in polar nematics
FIG. 6. The solvation free energy of a dipole solute 共in m20 / R31 units兲 vs the
angle ␪0 between the solute dipolar moment and the nematic director 共Fig.
1兲; R0 = 1.7␴. The solid line refers to ⑀储 ⬎ ⑀⬜ 共longitudinal solvent dipole,
␤ = 0兲 and the dashed line refers to ⑀储 ⬍ ⑀⬜ 共transverse solvent dipole, ␤
= 90兲. The solvent parameters are ␩ = 0.47, S2 = 0.8, y = 0.15, and ⌳ = 0.3␴. At
␤ = 0, the dielectric anisotropy is positive, ⑀⬜ = 1.092 and ⑀储 = 2.63; at ␤
= 90°, the dielectric anisotropy is negative, ⑀⬜ = 1.746 and ⑀储 = 1.092.
A. Equilibrium solvation
Our present development highlights several issues important for the understanding of equilibrium solvation in axially symmetric solvents. 共1兲 Solvation anisotropy, i.e., the
dependence of the free energy of solvation on the orientation
of the solute dipole relative to nematic director. 共2兲 The effect of crossing the phase transition temperature on the solvation thermodynamics. 共3兲 The effect of dipolar correlations
on solvation and the applicability of continuum models of
solvation.
From Eqs. 共36兲, 共55兲, and 共58兲, the solvation free energy
of a spherical dipole ␮共d兲 is a quadratic function of cos ␪0,
where ␪0 is the angle between the solute dipole and nematic
director 共Fig. 1兲. Figure 6 shows that the solvation anisotropy,
共d兲
− ␮⬜
,
⌬␮共d兲 = ␮共d兲
储
共60兲
is negative in nematics with positive dielectric anisotropy
共⌬⑀ = ⑀储 − ⑀⬜ ⬎ 0兲 and positive otherwise 关␮共d兲
= ␮共d兲共␪0 = 0兲
储
共d兲
共d兲
and ␮⬜ = ␮ 共␪0 = ␲ / 2兲兴. This can readily be verified by expanding Eq. 共36兲 in powers of the small parameter ⌬⑀,
␮共d兲
c ⬇−
冋
册
2⌬⑀
⑀s − 1
+
P2共cos ␪0兲 ,
2R31 2⑀s + 1 5共2⑀s + 1兲2
m20
共61兲
where
⑀s = 共⑀储 + 2⑀⬜兲/3.
共62兲
In order to relate anisotropy of ␮⬁共d兲 to dielectric anisotropy, one needs a relation between the order parameter and
the dielectric constants. This connection is given by the
Maier-Meier theory,31
⑀储 − 1 = 3y
3⑀s
共1 + 2S2 P2共cos ␤兲兲,
2⑀s + 1
3⑀s
共1 − S2 P2共cos ␤兲兲.
⑀⬜ − 1 = 3y
2⑀s + 1
共63兲
From Eq. 共63兲, ⌬⑀␣ ⬀ S2 P2共cos ␤兲. This means that both
⌬␮⬁共d兲 and ⌬␮共d兲 change their sign from negative to positive
when dielectric anisotropy changes its sign from positive to
negative.
FIG. 7. Stokes shift of coumarin-153 in nematic and isotropic phases of
5CB. The upper panel shows the microscopic calculation according to
Eq. 共58兲 and the lower panel shows the continuum limit. The solid lines
refer to the parallel alignment of the solute dipole with the nematic director
and the dashed lines refer to the perpendicular alignment 共Fig. 1兲. The solute
radius and the dipole moment are R0 = 4.89 Å and ⌬m0 = 7.5 D, respectively.
The temperature dependent dielectric constants 共Refs. 39 and 41兲, order
parameter 共Ref. 41兲, and density 共Ref. 39兲 of 5CB are taken from experiment; ⌳ = 0.35␴. The dot-dashed line in the upper panel refers to the continuum isotropic result calculated with the cavity radius equal to R0.
The magnitude of solvation anisotropy ⌬␮共d兲 is strongly
affected by the size of the solute. The continuum estimate of
the solvation anisotropy parameter,
共d兲
␬ = ⌬␮共d兲/␮av
,
1
共d兲
共d兲
␮av
= 共␮共d兲
+ 2␮⬜
兲,
储
3
共64兲
gives very low anisotropies 共␬ ⯝ 0.02– 0.03兲 for a large number of nematics.28 Once the dipole correlation effects are
involved through ␮⬁共d兲, anisotropy becomes quite significant,
␬ ⯝ 0.3S2. This solvation anisotropy results in a discontinuity
of the solvation chemical potential at the point of the
isotropic-nematic phase transition.
The significant effect of dipolar correlations on the solvation thermodynamics is seen from the comparison of the
lower and upper panels in Fig. 7 presenting steady-state
Stokes shift of coumarin-153 dissolved in 4-n-pentyl4⬘-cyanobiphenyl 共5CB兲 nematogen. This chromophore is
widely used as a spectroscopic probe of solvation dynamics
and thermodynamics.32,33 Stokes shift dynamics of
coumarin-153 in the isotropic phase of a liquid crystalline
solvent11 and of coumarin-503 共Refs. 9 and 10兲 in both isotropic and nematic phases have been reported. The continuum limit in the lower panel in Fig. 7 reveals a much
weaker anisotropy in the nematic phase than the full microscopic calculation in the upper panel. The continuum calculations are also much lower in absolute magnitude, which is
normally offset by choosing the radius R0 instead of the closest approach distance R1 共Fig. 7, dot-dashed line兲. However,
for the present calculation, rescaling the cavity radius does
not fully recover the solvation energy. This result suggests
that polar nematics might produce stronger solvation than
isotropic solvents with a comparable dielectric constant.
The electrostatic field of coumarin-153 is similar to that
of a point dipole,34 which makes it a convenient system to
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114904-10
J. Chem. Phys. 124, 114904 共2006兲
V. Kapko and D. V. Matyushov
test our theory. Since the Stokes shift experiments measure
only nuclear solvation, the Stokes shift hc⌬¯␯St 共¯␯ is the wave
number, cm−1兲 was calculated according to the following expression:
hc⌬¯␯St = −
2␮共d兲
n 共m0
= ⌬m0兲.
共65兲
Here, the difference in the dipole moments in the excited and
ground states of the chromophore ⌬m0 is substituted for the
solute dipole moment. The nuclear component of solvation is
calculated in the additive approximation35 in which the overall solvation free energy in a polar/polarizable liquid is assumed to be the sum of the nuclear and electronic solvation
components. The overall solvation free energy ␮共⑀储 , ⑀⬜ , y eff兲
is calculated from the anisotropic dielectric constant in the
continuum component ␮c共⑀储 , ⑀⬜兲 and the effective dipolar
density y eff in ␮⬁共y eff兲. The effective dipolar density is defined as36
y eff = 共4␲/9kBT兲␳共m⬘兲2 + 共4␲/3兲␳␣ ,
共66兲
where ␣ is the dipolar polarizability and m⬘ is the average
dipole moment of the solvent in the liquid. The nuclear free
energy of solvation is then given by
2
␮n = ␮共⑀储, ⑀⬜,y eff兲 − ␮共n2储 ,n⬜
,y e兲,
共67兲
where n⬜,储 is the anisotropic refractive index and the density
of induced dipoles is
y e = 共4␲/3兲␣␳ .
共68兲
For coumarin-153, the radius R0 = 4.89 Å and the dipole moment change ⌬m0 = 7.53 D have been adopted.37
5CB was chosen as a typical nematogen with its physical
properties well documented in the literature: the isotropicnematic transition temperature, TIN = 308.2 K,38 dipole moment, m = 4.75 D,39 and the azimuthal angle between the dipole moment and the long axis, ␤ = 21.6° 共Ref. 40兲. The
temperature dependences of the static dielectric
constants,39,40 refraction indexes,39,41 the order parameter,41
and density39 have also been reported. For the solvent diameter, the value ␴ = 5.27 Å 共Ref. 37兲 of benzene was adopted.
The polarization correlation length was calculated from Fig.
3 as ⌳ = 0.35␴ corresponding to 共m*兲2 = 3.3– 3.6. The dipole
moment m⬘ was calculated using the Onsager approximation,
m⬘ =
共n2 + 2兲共2⑀s + 1兲
m,
3共2⑀s + n2兲
共69兲
2
兲 / 3. Note that
where ⑀s is given by Eq. 共62兲 and n2 = 共n2储 + 2n⬜
the Onsager approximation is made in the Maier-Meier
theory 关Eq. 共63兲兴 which, nevertheless, describes dielectric
properties of polar nematics reasonably well.42,43 All results
in Fig. 7 have been obtained at experimentally documented
parameters of 5CB, the gap between the Stokes shift curves
around the isotropic-nematic transition temperature reflects
the absence of experimental data in this temperature range.
B. Stokes shift dynamics
The dynamics of solvation following the photoinduced
change in the solute charge distribution is recorded by measuring the Stokes shift time correlation function,32
S共t兲 =
E共t兲 − E共⬁兲
,
E共0兲 − E共⬁兲
共70兲
where E共t兲 is the time-dependent energy of the solute. The
calculation of this function is normally accomplished within
the linear response theory. Several formulations of the theory
are available in the literature,44–46 and we adopt here the
formulation due to Wolynes44 which represents the Laplace
transform E共s兲 of the time-dependent function E共t兲 as the
equilibrium solvation energy characterized by the dielectric
constant ⑀共s兲,
E共s兲 = E共⑀储共s兲, ⑀⬜共s兲兲
2
2
,y e兲兲.
= 共␮共⑀储共s兲, ⑀⬜共s兲,y eff兲 − ␮共n2储 ,n⬜
s
共71兲
The dependence of ⌳ on the dielectric constant of a
nematic solvent is generally unknown. We will therefore assume this parameter independent of the Laplace variable s.
Within this approximation, the Stokes shift function is fully
determined by the continuum expression for the solvation
energy 关Eq. 共36兲兴. The results for S共t兲 presented here are
therefore based on the continuum solution for the dipolar
solvation energy in an anisotropic dielectric with axial symmetry.
Experimental dielectric data on nematics are well characterized by the two-exponential form,
⑀a共s兲 = ⑀⬁a + 共⑀0a − ⑀⬁a兲
g
k
,
兺
1
+
s
␶ka
k=1,2
共72兲
where g1 + g2 = 1 and a stands for 储, ⬜, or iso 共isotropic兲. This
form reflects two different relaxations: high-frequency rotation around a long molecular axis 共relaxation time ␶2兲 and
low-frequency rotation around a short molecular axis 共relaxation time ␶1兲.38 In isotropic phase, ␶2iso is about ten times
smaller that ␶1iso, and the contribution from high-frequency
rotation to dielectric loss is usually small and is rarely resolved in the dielectric experiments.28,38 With only one relaxation time ␶1iso, Eq. 共72兲 reduces to the Debye dispersion
resulting in a single-exponential Stokes shift correlation
function with the relaxation time ␶S = 关共2⑀⬁ + 1兲 / 共2⑀0
+ 1兲兴␶1iso 共Ref. 46兲.
The liquid crystalline order of the nematic phase hinders
rotations around a short molecular axis when it is perpendicular to the nematic director. The corresponding relaxation
time ␶1储 is about ten times larger than ␶1iso. In contrary, ␶1⬜ is
much smaller than ␶1iso, and can be comparable to ␶2a 共a
= 储 , iso兲.28
With the dielectric constant from Eq. 共72兲 substituted
into Eq. 共36兲 the function S共t兲 needs to be calculated numerically. Following the procedure described in Ref. 47, the function F共s兲 = −sE共s兲 was fitted to a sum of Cole-Davidson-type
functions,
F共s兲 = 共E共0兲 − E共⬁兲兲 兺
i
ai
,
共1 + s␶Si兲␥i
共73兲
with the exponents ␥i and Stokes shift relaxation times ␶Si;
the linear expansion coefficients are normalized by the con-
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114904-11
J. Chem. Phys. 124, 114904 共2006兲
Theory of solvation in polar nematics
FIG. 8. Stokes shift correlation function in the isotropic phase 共“iso”兲 and in
the nematic phase at two orientations of the solute relative to the nematic
director.
dition 兺iai = 1. Equation 共73兲 allows analytical inverse
Laplace transform,
S共t兲 = 兺 ai
i
⌫共␥i,t/␶Si兲
,
⌫共␥i,0兲
共74兲
where ⌫共␥ , t兲 is the incomplete gamma function.48
Function S共t兲 calculated for a spherical dipolar dye in
4,4-n-heptyl-cyanopiphenyl 共7CB兲 is presented in Fig. 8. Experimental data for dielectric constants, refractive indexes,
and low-frequency relaxation times are taken from Ref. 49.
No high-frequency relaxation times ␶2a have been reported
for this nematogen. S共t兲 in the isotropic phase 共line marked
“iso” in Fig. 8兲 is single exponential, as expected.
Two predictions follow from our calculations in the
nematic phase. First, S共t兲 is biexponential. The slow component with ␶S1 ⬇ 100– 500 ns and ␥1 ⬇ 0.82 is related to ⑀储,
and the fast component with parameters ␶S2 ⬇ 1 – 2 ns and
␥2 ⬇ 1.2 is related to the relaxation of ⑀⬜. Second, S共t兲 is
affected by the angle ␪0 between the solute dipole and the
nematic director. For nematics with positive dielectric anisotropy 共e.g., 7CB兲 S共t兲 decays faster for ␪0 = 90° than for ␪0
= 0°. Note that the slow component has not been detected in
the experimental studies of transient Stokes shift in
nematics,9–11 probably because of the limited experimental
time resolution 共a few nanoseconds兲. All Stokes shift relaxation times monotonically increase with lowering temperature in both the nematic and isotropic phases 共Fig. 9兲.
V. CONCLUSIONS
This article presents a microscopic theory of solvation in
solvents with axial symmetry. Although applications of the
theory considered here are limited to nematic liquid crystals,
i.e., liquids with inversion symmetry of the polar axis, the
formalism is also applicable to ferroelectric solvents with a
preferential polar direction. This conclusion follows from the
fact that the linear response approximation leads to a quadratic dependence on the solute dipole moment invariant to
the dipole flip. For ferroelectrics, however, the solvation energy gains an additional contribution, linear in the solute
dipole, from the macroscopic polarization of the solvent.
The full microscopic formulation requires k-dependent
dipolar susceptibility of the nematic solvent which needs to
be obtained from computer experiment. From the analysis of
the results of MC simulations and the microscopic formalism, we have derived a formula for the solvation chemical
FIG. 9. Temperature dependence of the Stokes shift relaxation times. The
upper panel shows the faster relaxation time in the isotropic 共␶S兲 and nematic 共␶S2兲 phases. The lower panel shows the slower relaxation time 共␶S1兲
present only below TIN. The solid lines refer to the orientation of the solute
dipole parallel to the nematic director, while the dashed lines refer to the
perpendicular orientation.
potential which is based on experimentally measurable input
parameters and a theory parameter, the length of dipolar correlations in the solvent. This correlation length was obtained
from MC simulations in the range of parameters attainable
for the model fluid of dipolar spherocylinders. The understanding of the properties of the correlation length in a
broader range of parameters, in particular, for higher dielectric constants, will require simulations of nematogens with
more realistic intermolecular potentials.
The theory provides a quantitative framework for interpreting the spectroscopic steady-state and time-resolved experiments and makes several experimentally testable predictions. We show that the equilibrium free energy in the
nematic phase is a quadratic function of cosine of the angle
between the solute dipole and the solvent nematic director.
The sign of solvation anisotropy is determined by the sign of
dielectric anisotropy of the solvent: solvation anisotropy is
negative in solvents with positive dielectric anisotropy and
vice versa. The solvation free energy is discontinuous at the
point of isotropic-nematic transition. The amplitude of this
discontinuity is strongly affected by the size of the solute
becoming less pronounced for larger solutes. The discontinuity itself and the magnitude of the splitting of the solvation
chemical potential in the nematic phase are mostly affected
by microscopic dipolar correlations in the nematic solvent.
The Stokes shift correlation function in the isotropic
phase is one exponential when dielectric relaxation is given
by the Debye form. The corresponding relaxation time is a
smooth function of temperature through isotropic-nematic
transition. In the nematic phase, the Stokes shift relaxation
becomes biexponential, with a much slower relaxation component related to rotations of polar molecules around their
short axes in the nematic potential strongly hindering such
motions.
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114904-12
J. Chem. Phys. 124, 114904 共2006兲
V. Kapko and D. V. Matyushov
ACKNOWLEDGMENTS
I = kn−3
This research was supported by the National Science
Foundation 共CHE-0304694兲. This is publication No. 646
from the ASU Photosynthesis Center.
APPENDIX: DERIVATION OF EQ. „27…
We start with the formal expression for the solvent dipolar susceptibility,23
˜␹s共k兲 = 共m2/kBT兲
冕
+ ␳共␻1兲␳共␻2兲h̃共k, ␻1, ␻2兲兴,
共A1兲
where m and ␻ are the magnitude and orientation of the
solvent dipole, respectively. Further, the pair correlation
function h̃共k , ␻1 , ␻2兲 is expanded in spherical harmonics,22,23
*
h̃nl1ln2l 共k兲Y l1,n1共␻1兲Y l2,n2共␻2兲Y l,n
共␻k兲,
1+n2
1 2
1 2
12
共A2兲
where h̃nl1ln2l 共k兲 is the Hankel transform,
1 2
h̃nl1ln2l 共k兲 = 4␲il
1 2
冕
⬁
drr2 jl共kr兲hnl1ln2l 共r兲.
0
1 2
共A3兲
Then the tensor ˜␹s can be written as
*
˜␹s,n n 共k兲 = 兺 ˜␹s,n n l共k兲Y l,−n
1 2
1 2
l
1−n2
共␻k兲,
共A4兲
where coefficients ˜␹s,n1n2l共k兲 are proportional to h̃nl1ln2l 共k兲 and
1 2
depend only on the magnitude of wave vector k.
We next prove Eq. 共27兲 in which ˜␹⬘ is given by Eq. 共9兲.
Since the function ˜␹⬘ is defined by integrating in direct space
over the volume which is twice larger than the solute volume
for spherical solutes, it depends only on the long-distance
asymptote of the solvent correlation function. We will seek
the asymptotes of hnl1ln2l 共r兲 in the form 1 / rn. Since h 共12兲
1 2
cannot decay slower than the interaction potential, n 艌 3. According to 共A3兲 and 共A4兲 the contribution from the asymptotes is proportional to
I = kn−3
冕
⬁
dx
kd
jl共x兲
,
xn−2
共A5兲
where d is an arbitrary length larger than the size of the
solvent molecule.
We first consider n = 3,
I=
冕
⬁
kd
dx
jl共x兲
.
x
dx
jl共x兲
+ kn−3
xn−2
冕
␦
kd
dx
jl共x兲
.
xn−2
共A7兲
Since kd → 0, the parameter ␦ ⬍ 1 can be chosen to be small
enough such that jl共x兲 in the second integral in Eq. 共A7兲 can
be replaced by the first term, xl / 共2l + 1兲!!, of its Taylor expansion. The first integral in Eq. 共A7兲 vanishes at k → 0 since
n ⬎ 3 and ␦ is independent of k. The second integral in Eq.
共A7兲 becomes
kn−3
冕
␦
xl−n+2
⬃ kl .
共2l + 1兲!!
共A8兲
It gives a nonzero value at k → 0 only at l = 0. Therefore, at
k → 0 in Eq. 共A4兲, harmonics with asymptote 1 / r3 will produce terms with l ⬎ 0, while the l = 0 term corresponds to
1 / rn asymptote with n ⬎ 3. Assuming that only the 1 / r3 asymptotes contribute to ˜␹⬘共k = 0兲, we arrive at Eq. 共27兲 from
Eq. 共A4兲 and the orthogonality of the spherical harmonics.
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l l ln n
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