Mean-Square Mean-Square dipole moment of dipole moment of molecular chainsmolecular chains

Tutorial 5Tutorial 5

IntroductionIntroduction After the concept of the macromolecule After the concept of the macromolecule

has been established in the mid-l920s, it has been established in the mid-l920s, it

became clear that some properties of became clear that some properties of

polymers, such as their anomalous polymers, such as their anomalous

viscoelastic behavior, were dependent on viscoelastic behavior, were dependent on

the internal degrees of freedom of the the internal degrees of freedom of the

molecular chains. molecular chains.

(From Kremer – Schönhals book)

IntroductionIntroduction Kuhn, Guth and Mark made the first attempts for the Kuhn, Guth and Mark made the first attempts for the

mathematical description of the spatial conformations of mathematical description of the spatial conformations of

flexible chains.flexible chains. The skeletal bonds were considered steps in a random walk of The skeletal bonds were considered steps in a random walk of

three dimensions, the steps being uncorrelated one to another.three dimensions, the steps being uncorrelated one to another.

A more realistic approach to the description of the A more realistic approach to the description of the

conformation-dependent properties of molecular chains, conformation-dependent properties of molecular chains,

resting on the resting on the rotational isomeric states (RIS)rotational isomeric states (RIS) model, model, It was developed in large measure by Volkenstein and others It was developed in large measure by Volkenstein and others

in the late 1950s and early 1960s.in the late 1950s and early 1960s.

IntroductionIntroduction

The model takes into account The model takes into account skeletal bond lengthsskeletal bond lengths and and

anglesangles, , rotational anglesrotational angles associated with each skeletal associated with each skeletal

bond, and their bond, and their probabilitiesprobabilities, as well as the contribution , as well as the contribution

of each skeletal bond, to the property to be measured.of each skeletal bond, to the property to be measured.

It was rationalized by Flory and coworkers in the 1960’s.It was rationalized by Flory and coworkers in the 1960’s.

The model has proved to be suitable for calculation of The model has proved to be suitable for calculation of

conformation-dependent properties at equilibrium, such conformation-dependent properties at equilibrium, such

as the as the mean-square end-to-end distancesmean-square end-to-end distances, the , the mean-mean-

square dipole momentssquare dipole moments, the molar , the molar Kerr constantsKerr constants, optical , optical

configuration parameters, etc., as a function of the configuration parameters, etc., as a function of the

chemical structurechemical structure

DIPOLE MOMENTS OF GASES

TThe molar polarization, he molar polarization, PP, of a gas , of a gas has two components: the orientation has two components: the orientation polarization, polarization, PPoo and the induced and the induced polarization polarization PPd d , , 21 4

2 3 3P

o d AB

MP P P N

k T

=o+d

Permanent dipole moment of the molecule

Dipole moment of gasesDipole moment of gases

The The induced polarizabilityinduced polarizability is governed by is governed by

the strength with which the strength with which nuclear chargesnuclear charges

prevent the distortion of the prevent the distortion of the electronic electronic

cloudcloud by the applied field. by the applied field. This parameter increases with the atomic This parameter increases with the atomic

number, atomic size, and low ionic number, atomic size, and low ionic

potential of the atomspotential of the atoms

Therefore, induced polarizability results from the electronic Therefore, induced polarizability results from the electronic

polarizability, polarizability, ee, , arising from the distortion of the electronic arising from the distortion of the electronic

cloud by the action of the electric field, and the atomic cloud by the action of the electric field, and the atomic

polarizability, polarizability, aa, , is caused by small displacements of atoms is caused by small displacements of atoms

and groups of atoms in the molecule by the effect of the and groups of atoms in the molecule by the effect of the

electric field. electric field.

The magnitude of The magnitude of ee can directly be obtained by making can directly be obtained by making μμ==0 0

and considering the Maxwell relationship and considering the Maxwell relationship ((λλ)=n)=n22((λλ)). .

Because Because dd corresponds to a static electric field, the index of corresponds to a static electric field, the index of

refraction should be obtained at different wavelengths and refraction should be obtained at different wavelengths and

its value extrapolated to its value extrapolated to 1/ 1/ λλ→→ 00 . .

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

Static permittivity(Total polarizability)

Refraction index (Electronic polarizability)

Permanent dipole of the molecule

DIPOLE MOMENTS OF LIQUIDS AND POLYMERS

Since the Debye equation can only be used to determine the Since the Debye equation can only be used to determine the

dipole moments of gases, its extension to measurements of dipole moments of gases, its extension to measurements of

the polarity of liquids requires measuring in conditions such the polarity of liquids requires measuring in conditions such

that these substances may be considered to behave like gases that these substances may be considered to behave like gases

This situation can be achieved if the molecules of liquids are This situation can be achieved if the molecules of liquids are

sufficiently separated one from another by nonpolar sufficiently separated one from another by nonpolar

molecules.molecules.

Thus the interactions between the permanent dipole moments Thus the interactions between the permanent dipole moments

is reduced.is reduced.

In a solution containing In a solution containing nn11 molecules of nonpolar solvent andmolecules of nonpolar solvent and nn22

molecules of solute of molecular weights molecules of solute of molecular weights MM11 and and MM22 respectively, respectively,

the total molar polarization can be written as the total molar polarization can be written as

is the polarizability of the solution, is the polarizability of the solution,

where where xx1 1 and and xx22 are respectively the molar fraction of solvent and are respectively the molar fraction of solvent and

solute. solute.

For very dilute solutions For very dilute solutions ((xx22 →→00), ), intermolecular interactions intermolecular interactions

between the molecules of solute will be negligible and between the molecules of solute will be negligible and will be will be

the average of the polarizabilities of the solute the average of the polarizabilities of the solute 22 and solvent and solvent 11

Since the molar polarization of the solvent is given by Since the molar polarization of the solvent is given by

the expression for the molar polarization of the solute is:the expression for the molar polarization of the solute is:

At very low concentrations, the density and the dielectric At very low concentrations, the density and the dielectric permittivity of the solution can be expanded into a series, permittivity of the solution can be expanded into a series, givinggiving

1 – Solution2 - Solute

22

2

1( ) (0) ...

2!

f ff x f x x

x x

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 PThe molar orientation polarization PO2O2 of the solute is of the solute is given by:given by:

PPO2O2= P= P22-- PPe2 e2 -P-Pa2a2

In most systems, the molar atomic polarization In most systems, the molar atomic polarization PPa2a2 amounts to only amounts to only 5-10%5-10% of the molar electronic of the molar electronic polarization. Therefore, this contribution is often polarization. Therefore, this contribution is often neglected in the calculation of dipole moment.neglected in the calculation of dipole moment.

This expression is often used for the This expression is often used for the experimental determination of the dipole experimental determination of the dipole moments of molecules moments of molecules without internal degrees without internal degrees of freedomof freedom..

Flexible molecules are continuously changing their Flexible molecules are continuously changing their

spatial conformations, and, because the dipole moment spatial conformations, and, because the dipole moment

associated with each conformation is generally different, associated with each conformation is generally different,

the dipole moments that are measured are average the dipole moments that are measured are average

values.values.

Then, the expression of the should be written asThen, the expression of the should be written as

By defining a fictitious atomic polarizability for the By defining a fictitious atomic polarizability for the

solute as solute as

where where a2a2 is the polarizability of the solvent and is the polarizability of the solvent and VV22 and and

VV11 are the molar volume , of the solute and the solvent, are the molar volume , of the solute and the solvent,

the application of the Debye equation to solutions leads the application of the Debye equation to solutions leads

toto

In principle, the atomic polarizability of nonpolar In principle, the atomic polarizability of nonpolar

solvents (solvents (μμpp=0=0) can be obtained by means of the Debye ) can be obtained by means of the Debye

equation Actuallyequation Actually

Experimental findings in the determination of the Experimental findings in the determination of the

dielectric permittivity and the index of refraction of dielectric permittivity and the index of refraction of

nonpolar solvents show that nonpolar solvents show that PPalal is is 10%10% and even less of and even less of

PPelel . .

There is no reason to believe that There is no reason to believe that PPaa for polar for polar

substances is larger than the molar polarization for substances is larger than the molar polarization for

nonpolar ones. nonpolar ones.

Guggenheim-Smith equation

In this Debye-based equations the dipole-dipole interactions are eliminated by progressive dilution (Intramolecular dipole-dipole 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

Mean square moment with applied electric field

Mean square moment without electric field

Effective Electric field

Since the interactions between the electric field and the Since the interactions between the electric field and the dipole moments decrease the energy of the system, those dipole moments decrease the energy of the system, those conformations with higher energy are favored by the conformations with higher energy are favored by the field effect.field effect.

Dipole moment of a polymer chain

Dipole moments can rigidly be attached to the skeletal bonds Dipole moments can rigidly be attached to the skeletal bonds

or associated with flexible side groups. or associated with flexible side groups.

In the former case, dipoles can be parallel or perpendicular to In the former case, dipoles can be parallel or perpendicular to

the chain contour, and, according to the chain contour, and, according to StockmayerStockmayer's notation, 's notation,

these dipoles are of type these dipoles are of type AA and and BB respectively . respectively .

Dipoles located in flexible side Dipoles located in flexible side

groups are of type groups are of type CC..

Cl Cl Cl Cl Cl

OO

OO

O O OO

Some polar polymers, such as poly Some polar polymers, such as poly

(propylene oxide), characterized for not (propylene oxide), characterized for not

having the repeat unit appropriate having the repeat unit appropriate

symmetry elements display dipole symmetry elements display dipole

moments with components parallel and moments with components parallel and

perpendicular to the chain contour, and perpendicular to the chain contour, and

these chains are of type these chains are of type ABAB..

Dipole momentsDipole moments and and end-to-end distanceend-to-end distance

are are uncorrelateduncorrelated for chains with dipoles for chains with dipoles

of types of types BB and and CC, and therefore the mean-, and therefore the mean-

square dipole moment of these chains square dipole moment of these chains

should not exhibit excluded volume should not exhibit excluded volume

effects.effects.

However, the dipole moments for chain However, the dipole moments for chain

type type AA and and ABAB are are correlatedcorrelated with the with the

end-to-end distanceend-to-end distance of the chains, of the chains, rr, , and and

present an excluded volume effect.present an excluded volume effect.

O

O

O

Dipole Autocorrelation Dipole Autocorrelation Coefficient in PolymersCoefficient in Polymers

As occurs with the molecular As occurs with the molecular dimensions, mean square dipole of dimensions, mean square dipole of polymers increases with molecular polymers increases with molecular weight.weight.

It is convenient to express this It is convenient to express this quantity as the dimensionless quantity as the dimensionless parameter parameter gg, also called the dipolar , also called the dipolar autocorrelation coefficientautocorrelation coefficient

2

02

fj

g

Mean square dipole moment of the freely jointed chain

2 2

1 1 1

· ·i

n n n

i j i ji j i i j j

m m m m m

For a freely jointed chain the dipoles associated For a freely jointed chain the dipoles associated with the with the jj and and ii bonds are uncorrelated, that is, bonds are uncorrelated, that is, any angle among them between any angle among them between 00 and and 22ππ has the has the same probability of occurrence, and the average same probability of occurrence, and the average of its cosine vanishesof its cosine vanishes

= 0

2 2 2

1i

n

fji

m nm

2

2g

nm

Number of skeletal bonds

Mean square dipolar moment of the bond

For short chains, the dipole autocorrelation coefficient, For short chains, the dipole autocorrelation coefficient,

is molecular weight dependent. is molecular weight dependent.

The same situation it’s found for the dimension The same situation it’s found for the dimension

autocorrelation coefficient or characteristic ratio autocorrelation coefficient or characteristic ratio

(C(C=<r=<r22>>o o / n·l/ n·l22 , where , where <r<r22>>oo mean-square end-to-end mean-square end-to-end

distance and distance and ll22 are respectively the and the average of are respectively the and the average of

the squares of the skeletal bond lengths) the squares of the skeletal bond lengths)

However, the values of these quantities remain nearly However, the values of these quantities remain nearly

constant, independent of the molecular weight, for constant, independent of the molecular weight, for

long chainslong chains

Dipole moments present some advantages for Dipole moments present some advantages for

the study of conformation dependent the study of conformation dependent

properties of polymer chains properties of polymer chains These include the following:These include the following:

1 1 Dipole moments can be measured for chains of Dipole moments can be measured for chains of

any length, whereas the unperturbed dimensions any length, whereas the unperturbed dimensions

can only be experimentally obtained for long chainscan only be experimentally obtained for long chains

22 Dipole moments of most polymer chains do no Dipole moments of most polymer chains do no

present excluded volume effectspresent excluded volume effects

33 Since skeletal bonds change more in polarity than Since skeletal bonds change more in polarity than

they do in length, dipole moments are usually more they do in length, dipole moments are usually more

sensitive to structure than unperturbed dimensionssensitive to structure than unperturbed dimensions

Experimental examples of Experimental examples of determination of the Dipole determination of the Dipole

momentmomentFrom 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

Experimental part:Experimental part: 1 – 1 – The molecular weight of the polymer

was estimated from measurements of the intrinsic viscosity [η] of each of the samples in 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.

* O

O

*

Pravikova et all, Polym. Sci. USSR (Engl. Transl.), 12, 658 (1970).

<μ2>/nm2 n=number of skeletal bonds (5M/Mo), Mo= molecular weight of the structural unitm2=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 According to the experimental results, the mean square dipole moment increases with mean square dipole moment increases with the temperature.the temperature.

In our systems, there is not interaction In our systems, there is not interaction between dipoles (dilute system)between dipoles (dilute system)

The increase in the mean square dipole The increase in the mean square dipole moment could be interpreted as some moment could be interpreted as some conformational change in the polymer chain, conformational change in the polymer chain, from one conformation from one conformation with lowwith low dipole dipole momentmoment to another with to another with high dipole momenthigh dipole moment..

In the case of solids, we also must to take In the case of solids, we also must to take into account the interaction between dipoles.into account the interaction between dipoles.

The temperature dependence of the mean square dipole, The temperature dependence of the mean square dipole, give an idea about the temperature dependence of the give an idea about the temperature dependence of the conformational states in a polymer chain. conformational states in a polymer chain.

2 2ln /d nm

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+1.4kcal/mol

g=t+0.9kcal/mol

g=t+0.4kcal/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.

SummarySummary Mean square dipole moment depend on the Mean square dipole moment depend on the

conformation of the polymer chain.conformation of the polymer chain. Rotational isomeric states Rotational isomeric states model it is proposed model it is proposed

for the evaluation of the possible for the evaluation of the possible conformational states in polymer chains.conformational states in polymer chains.

Two different equation are proposed to Two different equation are proposed to estimate the mean square dipole (Based on the estimate the mean square dipole (Based on the Debye equation): Debye equation):

Guggenheim-Smith

Halverstadt and Kumler

SummarySummary Both equation are restricted at condition of Both equation are restricted at condition of

applicability of the Debye equation, that is:applicability of the Debye equation, that is: Polar molecules are in an nonpolar solvent, in Polar molecules are in an nonpolar solvent, in

order to eliminate dipole – dipole interactionorder to eliminate dipole – dipole interaction Systems are diluteSystems are dilute

By mean of the determination of the By mean of the determination of the dielectric permittivity of dilute solutions, and dielectric permittivity of dilute solutions, and the refraction index it’s possible to estimate the refraction index it’s possible to estimate the mean square dipole of the molecule.the mean square dipole of the molecule.

SummarySummary

Comparison of experimental results Comparison of experimental results with theoretical ones based on the with theoretical ones based on the Rotational Isomeric States shows a Rotational Isomeric States shows a good agreement with each other.good agreement with each other.

Determination of the mean square Determination of the mean square dipole moment it’s a good way to dipole moment it’s a good way to evaluate the conformational evaluate the conformational configuration of the polymer chain.configuration of the polymer chain.