Mean-Square dipole moment of molecular chains Tutorial 5

Introduction 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. 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. (From Kremer – Schönhals book)

Introduction Kuhn, Guth and Mark made the first attempts for the mathematical description of the spatial conformations of flexible chains. 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. 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, 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. 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. 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. 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 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, P o and the induced polarization P d, The molar polarization, P, of a gas has two components: the orientation polarization, P o and the induced polarization P d, =o+d=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. 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 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. 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)=n 2 (λ). The magnitude of  e can directly be obtained by making μ=0 and considering the Maxwell relationship  (λ)=n 2 (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. 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 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 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 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. 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. Thus the interactions between the permanent dipole moments is reduced.

In a solution containing n 1 molecules of nonpolar solvent and n 2 molecules of solute of molecular weights M 1 and M 2 respectively, the total molar polarization can be written as In a solution containing n 1 molecules of nonpolar solvent and n 2 molecules of solute of molecular weights M 1 and M 2 respectively, the total molar polarization can be written as  is the polarizability of the solution, where x 1 and x 2 are respectively the molar fraction of solvent and solute. For very dilute solutions (x 2 → 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 For very dilute solutions (x 2 → 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 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 permittivity of the solution can be expanded into a series, giving At very low concentrations, the density and the dielectric permittivity of the solution can be expanded into a series, giving 1 – Solution 2 - Solute

Assuming that ρ → ρ 1 and  →  1 when w 2 → 0 Halverstadt and Kumler equation obtained ν and ν l are the specific volume of the solution and solvent. The molar electronic polarization of the solute, P e2, can be obtained taking into account that, at very high frequencies,  1 =n l 2 and  =n 2, where n and n l are the index of refraction of the solution and the solvent. Accordingly,

The molar orientation polarization P O2 of the solute is given by: The molar orientation polarization P O2 of the solute is given by: P O2 = P 2 - P e2 -P a2 P O2 = P 2 - P e2 -P a2 In most systems, the molar atomic polarization P a2 amounts to only 5-10% of the molar electronic polarization. Therefore, this contribution is often neglected in the calculation of dipole moment. In most systems, the molar atomic polarization P a2 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. 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. 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 Then, the expression of the should be written as By defining a fictitious atomic polarizability for the solute as By defining a fictitious atomic polarizability for the solute as where  a2 is the polarizability of the solvent and V 2 and V 1 are the molar volume, of the solute and the solvent, the application of the Debye equation to solutions leads to where  a2 is the polarizability of the solvent and V 2 and V 1 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 (μ p =0) can be obtained by means of the Debye equation Actually In principle, the atomic polarizability of nonpolar solvents (μ p =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 P al is 10% and even less of P el. Experimental findings in the determination of the dielectric permittivity and the index of refraction of nonpolar solvents show that P al is 10% and even less of P el. There is no reason to believe that P a for polar substances is larger than the molar polarization for nonpolar ones. There is no reason to believe that P a 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 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 V F 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 V F = V - μ x F

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 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. 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. 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. Dipoles located in flexible side Dipoles located in flexible side groups are of type C.

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. 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. 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. 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.

Dipole Autocorrelation Coefficient in Polymers As occurs with the molecular dimensions, mean square dipole of polymers increases with molecular weight. 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 It is convenient to express this quantity as the dimensionless parameter g, also called the dipolar autocorrelation coefficient Mean square dipole moment of the freely jointed chain

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 = 0 Number of skeletal bonds Mean square dipolar moment of the bond

For short chains, the dipole autocorrelation coefficient, is molecular weight dependent. 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= o / n·l 2, where o mean-square end-to-end distance and l 2 are respectively the and the average of the squares of the skeletal bond lengths) The same situation it’s found for the dimension autocorrelation coefficient or characteristic ratio (C= o / n·l 2, where o mean-square end-to-end distance and l 2 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 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 Dipole moments present some advantages for the study of conformation dependent properties of polymer chains These include the following: 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 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 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 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 [(CH 2 ) y O-]. Another important class of polyethers, very similar in chemical structure to the polyoxides, are the polyformals [CH 2 O(CH 2 ) y O-]. 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) [CH 2 O(CH 2 ) 2 O-] 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. Pravikova et all, Polym. Sci. USSR (Engl. Transl.), 12, 658 (1970).

/nm 2 n=number of skeletal bonds (5M/Mo), Mo= molecular weight of the structural unit m 2 =dipole moment of the structure unit = 1/5 (4m 2 c-o + m 2 c-c) mc-o=1,07 D; mc-c=0 D

According to the experimental results, the mean square dipole moment increases with the temperature. 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) 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. 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. 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.

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+0.9kcal/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.

Summary Mean square dipole moment depend on the conformation of the polymer chain. 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. 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): Two different equation are proposed to estimate the mean square dipole (Based on the Debye equation): Guggenheim-Smith Halverstadt and Kumler

Summary Both equation are restricted at condition of applicability of the Debye equation, that is: 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 Polar molecules are in an nonpolar solvent, in order to eliminate dipole – dipole interaction Systems are dilute 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. 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. 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. Determination of the mean square dipole moment it’s a good way to evaluate the conformational configuration of the polymer chain.