2864
ZOLTANG.Soos AND ARTHURV. TOBOLSKY
A New Function for Dipole Orientation and Rubber Elasticity by Zoltih G . Soos and Arthur V. Tobolsky Department of Chemistry, Frick Chemical Laboratory, Princeton University, Princeton, New JeTSey 08640 (Received January 9, 1969)
A new function for dipole orientation is derived to describe a collection of fixed, randomly oriented molecules with a dipole moment fp, achievable by an intramolecular motion such as inversion. This model should be applicable to tertiary amines in the glassy state or imbedded in a glassy matrix. The new function, called the Y function, is similar to the Langevin function at low fields. However at very high fields, the Y function predicts an average dipole moment of p / 2 rather than the value of p obtained from the Langevin function. The Y function has also been applied to obtain a new equation of state for rubber elasticity at high strains, in the same manner that had previously been achieved starting with the Langevin function (freely rotating model). This new equation of state for rubber elasticity is superior to that based on the Langevin equation in that it predicts a finite rather than an infinite strain energy.
The average dipole moment of an electric dipole p e along a static electric field 2 is given classically by the Langevin formula
Here L is the Langevin function, k is the Boltzmann constant, and T is the absolute temperature. Identical results hold for the average moment of a classical magnetic dipole p m in a magnetic field I? or for the average length of a polymer segment lo in a uniform force field p . The latter gives the forceextension relation for a freely rotating polymer chain and is the basis for treatments of network elasticity.’J The quantum result for a magnetic moment differs from eq 2 because the moments are quantized. Thus for a magnetic moment 1/2, with allowed projections * p e along the field H,the result is easily obtained by Boltzmann statistics
PI/% =
e (erzH/kT
- e-rsH/loT)/(ernH/kT
+ e-~aH/kT
)
(3) For a general magnetic moment J, with J an integer or half an integer
(4) where pz is the maximum allowed projection along the field and BJ is the Brillouin f u n c t i ~ n . ~For a fixed moment, the classical result is regained as J -t m , since then B,(z) -t L(x). The Journal of Physical Chemistry
We consider here the modifications of the classical results when the electric dipole moment, or each polymer segment, is constrained to a finite number of rotational conformers. The simple case of two f conformers describes the state of such molecules as tertiary amines. I n the gaseous state all possible orientations of these molecules are allowed, and the Langevin treatment is valid. Consider, however, such molecules in the glassy state, or possibly even the liquid state. The molecules us u whole are assumed to be randomly oriented but unable to align with the field. However, internal isomerization (inversion) between two conformers can occur, with reversal of the dipole moment pe, and this inversion is affected by the field. The average moment per molecule in an electric field 2 is
This result differs from the classical treatment, eq 2, because only two orientations (conformers) are permitted for any dipole. It differs from the quantum result, eq 3, since here the axis of quantization is random, reflecting the assumed random orientations of the molecules, whereas magnetic spins are quantized along the applied field. The function Y ( z ) defined above is almost identical with the Langevin function for small arguments: Y ( z ) = 4 3 O(z2) for z
