Order Parameters

Distinguishable Ends

The heteronuclear diatomic might be H-Cl. This diatomic has a permanent dipole moment that is colinear with the bond. The dipole moment can be represented by a vector, v. The anisotropy is represented by an order parameter which is used to examine any property of a polymer that can be represented by a vector.

The order parameter can be used to analyze the static structure of a system, such as the tendency for orientation along a particular axis, denoted here as the z axis. The order parameter is defined as

(v dot z)_average = (cos theta_z)_average

where v denotes the unit vector we are investigating, z is a unit vector along the z axis, and theta_z is the angle between the vector and the z axis. The limits for this order parameters are -1 to +1. If there is no correlation of v with the z axis, the order parameter has a value of 0.

This order parameter might be used to examine any tendency for order in the side chains in polypropylene or polystyrene at a surface, taking advantage of the fact that the two ends of the C-C bond are distinguishable in this case. One carbon atom is in the backbone, and the other carbon atom is in the methyl group of the polypropylene, or in the para position of the ring of the polystyrene. theta_z is the angle between this bond vector and the normal (z-axis) to the surface. An example for polystyrene can be found in Figure 2 of J. Phys. Chem. B 2001, 105, 11493. This example shows that the styrene rings in the surface region tend to be oriented so that they stick outward from the surface.

The order parameter can also be used to monitor the dynamics of a system. For this purpose, it is defined as

<v_t dot v₀> / <v²> = (cos theta_z)_average

where v_t denotes the vector at time t, theta_z is the angle between the vector at time t and the same vector in initial orientation at t = 0, and the denominator on the left-hand side normalizes the order parameter to a value of 1 at zero time. At infinite time, this order parameter has a value of:

• 1 if there is no motion, so that the vector at time t has a perfect alignment with its initial orientation.
• 0 if all orientations are equally probable at time t, and all correlation with the initial orientation is lost.
• -1 if there is perfect alignment of the vector along the z axis, with the vector pointing in the -z direction.

Indistinguishable Ends

The homonuclear diatomic might be H-H, with indistinguishable ends. Rotation of 180^o about an axis perpendicular to the bond does not lead to a distinguishable state. The anisotropy of the system is represented by an order parameter which is used to examine the tendency for order in bonds with indistinguishable ends, such as the C-C bonds in polyethylene. The order parameter is defined as

(1/2) [3 (cos² theta_z)_average - 1]

This order parameter has a value of:

• 1 if there is perfect alignment (both directions accepted, because they are indistinguishable) along the z axis.
• 0 if all orientations are equally probable (the average value of cos² is 1/3 under this condition).
• -1/2 if there is perfect alignment of the bond perpendicular to the z axis.

Time scales for reorientation

In an isotropic system, both order parameters will be time dependent, decaying from 1 at zero time to 0 at infinite time. The time scales of their decays are often of interest in studies of the dynamics of polymers.

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