Mean-Square
dipole moment of
molecular chains
Tutorial 5
Introduction
(From Kremer – Schönhals book)

After the concept of the macromolecule has
been established in the mid-l920s, it became
clear that some properties of polymers, such as
their anomalous viscoelastic behavior, were
dependent on the internal degrees of freedom
of the molecular chains.
Introduction

Kuhn, Guth and Mark made the first attempts for the
mathematical description of the spatial conformations of
flexible chains.

The skeletal bonds were considered steps in a random walk of
three dimensions, the steps being uncorrelated one to another.

A more realistic approach to the description of the
conformation-dependent properties of molecular chains,
resting on the rotational isomeric states (RIS) model,

It was developed in large measure by Volkenstein and others in
the late 1950s and early 1960s.
Introduction



The model takes into account skeletal bond lengths and angles,
rotational angles associated with each skeletal bond, and their
probabilities, as well as the contribution of each skeletal bond,
to the property to be measured.
It was rationalized by Flory and coworkers in the 1960’s.
The model has proved to be suitable for calculation of
conformation-dependent properties at equilibrium, such as the
mean-square end-to-end distances, the mean-square dipole
moments, the molar Kerr constants, optical configuration
parameters, etc., as a function of the chemical structure
DIPOLE MOMENTS OF GASES

The molar polarization, P, of a gas has two
components: the orientation polarization,
Po and the induced polarization Pd ,
 

 1 M 4
P  Po  Pd 
  NA 
 
 2  3
 3kBT

2
P
=o+d
Permanent dipole moment of the molecule
Dipole moment of gases

The induced polarizability is governed by the
strength with which nuclear charges prevent the
distortion of the electronic cloud by the applied
field.

This parameter increases with the atomic
number, atomic size, and low ionic potential of
the atoms



Therefore, induced polarizability results from the electronic
polarizability, e, arising from the distortion of the
electronic cloud by the action of the electric field, and the
atomic polarizability, a, is caused by small displacements
of atoms and groups of atoms in the molecule by the effect
of the electric field.
The magnitude of e can directly be obtained by making
μ=0 and considering the Maxwell relationship (λ)=n2(λ).
Because d corresponds to a static electric field, the index of
refraction should be obtained at different wavelengths and
its value extrapolated to 1/ λ→ 0 .
Clausius Mossoti equation


The atomic polarizability cannot be determined
directly, but its value is small and often negligible.
Debye equation was found to hold for a variety of
gases and vapors at ordinary pressures
Permanent dipole of the molecule
Refraction index
(Electronic polarizability)
Static permittivity
(Total polarizability)
DIPOLE MOMENTS OF LIQUIDS
AND POLYMERS

Since the Debye equation can only be used to determine the
dipole moments of gases, its extension to measurements of the
polarity of liquids requires measuring in conditions such that
these substances may be considered to behave like gases

This situation can be achieved if the molecules of liquids are
sufficiently separated one from another by nonpolar molecules.

Thus the interactions between the permanent dipole moments
is reduced.

In a solution containing n1 molecules of nonpolar solvent and n2 molecules
of solute of molecular weights M1 and M2 respectively, the total molar
polarization can be written as
 is the polarizability of the solution,
where x1 and x2 are respectively the molar fraction of solvent and solute.

For very dilute solutions (x2 →0), intermolecular interactions between the
molecules of solute will be negligible and  will be the average of the
polarizabilities of the solute 2 and solvent 1

Since the molar polarization of the solvent is given by

the expression for the molar polarization of the solute is:

At very low concentrations, the density and the dielectric permittivity
of the solution can be expanded into a series, giving
f
1 2 f 2
f ( x)  f (0)  x 
x  ...
x
2! x 2
1 – Solution
2 - Solute

Assuming that ρ→ρ1 and →1 when w2→0 Halverstadt and
Kumler equation obtained

ν and νl are the specific volume of the solution and solvent.

The molar electronic polarization of the solute, Pe2, can be
obtained taking into account that, at very high frequencies,
1=nl2 and =n2, where n and nl are the index of refraction of the
solution and the solvent. Accordingly,

The molar orientation polarization PO2 of the solute is given by:

PO2= P2- Pe2 -Pa2

In most systems, the molar atomic polarization Pa2 amounts to
only 5-10% of the molar electronic polarization. Therefore, this
contribution is often neglected in the calculation of dipole
moment.

This expression is often used for the experimental
determination of the dipole moments of molecules
without internal degrees of freedom.




Flexible molecules are continuously changing their spatial
conformations, and, because the dipole moment associated with
each conformation is generally different, the dipole moments
that are measured are average values.
Then, the expression of the should be written as
By defining a fictitious atomic polarizability for the solute as
where a2 is the polarizability of the solvent and V2 and V1 are
the molar volume , of the solute and the solvent, the application
of the Debye equation to solutions leads to

In principle, the atomic polarizability of nonpolar solvents
(μp=0) can be obtained by means of the Debye equation
Actually

Experimental findings in the determination of the dielectric
permittivity and the index of refraction of nonpolar solvents
show that Pal is 10% and even less of Pel .

There is no reason to believe that Pa for polar substances is
larger than the molar polarization for nonpolar ones.
Guggenheim-Smith equation

In this Debye-based equations the dipole-dipole interactions
are eliminated by progressive dilution (Intramolecular dipoledipole correlations are not considered).

Models developed by Kirkwood and Fröhlich, allow to take into
account the interaction of surrounding dipoles by correlation
function treatment.

Despite that K-F method would be more appropriate, their
application introduces difficulties and computations that are
often rather arbitrary.

Many dipole moments obtained for oligomers and polymers
using Debye-type equations ( Halverstadt-Kumler and
Guggenheim-Smith) show consistency among them,
presumably because intramolecular dipole-dipole interactions
in flexible chains fade away for dipoles separated by four or
more flexible skeletal bonds.
EFFECT OF THE ELECTRIC FIELD ON
THE MEAN-SQUARE DIPOLE MOMENT

Let us consider a macromolecular system under an external electric field
acting along the x axis.

The energy VF associated with a given conformation of a molecular chain is
the result of the energy of that conformation in the absence of an electric
field (V), plus the interactions of the permanent and induced dipole
moments of the conformation with the electric field,

where F is the effective electric field


The component of the polarizability tensor in the direction of the field, ’xx
can be neglected for polar systems, so that
VF = V - μxF
Effective Electric field
Mean square moment
without electric field
Mean square moment
with applied electric field
Since the interactions between the electric field and the dipole
moments decrease the energy of the system, those conformations
with higher energy are favored by the field effect.
Dipole moment of a polymer chain

Dipole moments can rigidly be attached to the skeletal bonds or
associated with flexible side groups.

In the former case, dipoles can be parallel or perpendicular to the chain
contour, and, according to Stockmayer's notation, these dipoles are of
type A and B respectively .
Cl

Cl
Cl
Cl
Cl
Dipoles located in flexible side
groups are of type C.
O
O
O
O
O
O
O
O

Some polar polymers, such as poly (propylene
oxide), characterized for not having the repeat
unit appropriate symmetry elements display
dipole moments with components parallel and
perpendicular to the chain contour, and these
chains are of type AB.

Dipole moments and end-to-end distance are
uncorrelated for chains with dipoles of types B
and C, and therefore the mean-square dipole
moment of these chains should not exhibit
excluded volume effects.

However, the dipole moments for chain type A
and AB are correlated with the end-to-end
distance of the chains, r, and present an
excluded volume effect.
O
O
O
Dipole Autocorrelation Coefficient in
Polymers


As occurs with the molecular dimensions, mean
square dipole of polymers increases with
molecular weight.
It is convenient to express this quantity as the
dimensionless parameter g, also called the
dipolar autocorrelation coefficient
g

2

2
0
fj
Mean square dipole moment of the
freely jointed chain


2
n
n
 m · m
i 1
i
j 1
n
  mi 
2
j
i 1
 m ·m
i j
i
j
j
=0
For a freely jointed chain the dipoles associated with the
j and i bonds are uncorrelated, that is, any angle among
them between 0 and 2π has the same probability of
occurrence, and the average of its cosine vanishes

n
2
  mi  nm
2
fj
i 1
2
g

2
nm
2
Number of skeletal bonds
Mean square
dipolar moment
of the bond

For short chains, the dipole autocorrelation coefficient, is
molecular weight dependent.

The same situation it’s found for the dimension autocorrelation
coefficient or characteristic ratio (C=<r2>o / n·l2 , where <r2>o
mean-square end-to-end distance and l2 are respectively the
and the average of the squares of the skeletal bond lengths)

However, the values of these quantities remain nearly constant,
independent of the molecular weight, for long chains

Dipole moments present some advantages for the
study of conformation dependent properties of
polymer chains

These include the following:

1 Dipole moments can be measured for chains of any length,
whereas the unperturbed dimensions can only be
experimentally obtained for long chains

2 Dipole moments of most polymer chains do no present
excluded volume effects

3 Since skeletal bonds change more in polarity than they do
in length, dipole moments are usually more sensitive to
structure than unperturbed dimensions
Experimental examples of
determination of the Dipole moment
From Macromolecules, 1978, 11, 956 – 959 (Riande E, Mark J. E.)

The polyethers are a class of macromolecules having C-O-C bonds in
the chain backbone.

They are polar material.

One of the most important and interesting types of polyether are the
polyoxides, which have the repeat unit [(CH2)yO-].

Another important class of polyethers, very similar in chemical structure
to the polyoxides, are the polyformals [CH2O(CH2)yO-].

As in the case of the polyoxides, the properties of these polymers vary
markedly with the number of methylene groups in the repeat unit.

In this work poly(l,3-dioxolane) (PXL) [CH2O(CH2)2O-] is used
O
*





O
Experimental part:
1 – The molecular weight of the polymer was
estimated from measurements of the intrinsic
Pravikova et all,
Polym. Sci. USSR (Engl.
viscosity [η] of each of the samples in
Transl.), 12, 658 (1970).
chlorobenzene at 25 ºC.
2 - Dielectric Constants and Refractive Indices.
The dielectric constant was were carried out using a
capacitance bridge (General Radio type 1620A) at a
frequency of 10 kHz (at which the dielectric
constant  is to a good approximation the static
value).
Values of the index of refraction n of the solutions
were obtained using a Brice-Phoenix differential
refractometer.
*
<μ2>/nm2 n=number of skeletal bonds (5M/Mo), Mo= molecular weight of the structural unit
m2=dipole moment of the structure unit = 1/5 (4m2c-o + m2c-c)
mc-o=1,07 D; mc-c=0 D

According to the experimental results, the mean
square dipole moment increases with the temperature.

In our systems, there is not interaction between
dipoles (dilute system)

The increase in the mean square dipole moment could
be interpreted as some conformational change in the
polymer chain, from one conformation with low
dipole moment to another with high dipole moment.

In the case of solids, we also must to take into
account the interaction between dipoles.
The temperature dependence of the mean square dipole, give an idea about
the temperature dependence of the conformational states in a polymer
chain.
d ln  2 / nm2
dT
Comparison of Theory and Experiment

The rotational isomeric state model adopted for the PXL chain
rotational states located at 0º (trans, t), 120º (gauche positive, g+), and 120º (gauche negative, g-).
g=t+1.4kcal/mol
g=t+0.4kcal/mol
g=t+0.9kcal/mol
g=t+1.4kcal/mol
g=t+0.9kcal/mol
-1,2 kcal/mol
Rotational transition
g – t,
Bond type a and e
In simplest molecular terms, the PXL chain has a very small dipole
moment ratio.
It’s caused by its preference for gauche states of low dipole moment.
Its dipole moment increases markedly with increase in temperature,
(increase in the number of alternative rotational states, of higher
energy and larger dipole moment).

Dependence of the dipole moment ratio, at
25ºC, on the number of skeletal bonds.

The dipole moment ratio reaches an asymptotic
limit at relatively low chain length.
Summary



Mean square dipole moment depend on the
conformation of the polymer chain.
Rotational isomeric states model it is proposed for the
evaluation of the possible conformational states in
polymer chains.
Two different equation are proposed to estimate the
mean square dipole (Based on the Debye equation):
Halverstadt and Kumler
Guggenheim-Smith
Summary

Both equation are restricted at condition of
applicability of the Debye equation, that is:
Polar molecules are in an nonpolar solvent, in order to
eliminate dipole – dipole interaction
 Systems are dilute


By mean of the determination of the dielectric
permittivity of dilute solutions, and the refraction
index it’s possible to estimate the mean square dipole
of the molecule.
Summary


Comparison of experimental results with
theoretical ones based on the Rotational
Isomeric States shows a good agreement with
each other.
Determination of the mean square dipole
moment it’s a good way to evaluate the
conformational configuration of the polymer
chain.