A Description of the Derivation of the Langevin Equation for Equilibrium Polarization

George Collins, Visiting Scientist
Medical Device Concept Laboratory, New Jersey Institute of Technology

Molecules are three dimensional atomic frameworks over which electronic charge is distributed. The binding effects of the shared electrons hold this framework together. Different rolex replica uk atomic components of the molecule attract or release electrons to different extents depending on the particular atomic structure. If a molecule does not replica watches have a center of symmetry, the electrons will be non-uniformly distributed over the molecular framework. The consequence of this non-uniform distribution is the presence of a permanent dipole, which represents the separation of the more negative, electron rich, regions of the molecular structure from the more breitling replica positive, electron deficient, regions. The permanent dipole gives an electrostatic directionality to the molecule, which is represented in terms of the dipole moment. The dipole moment is a vector quantity that is expressed as the magnitude of the charge multiplied by the vector that describes the distance and relative direction of the charge separation. This quantity is often represented by the symbol, p. Also, it is often graphically represented as an arrow with a tip that points to the positive charge.

The permanent electric dipole allows the molecule to interact with electric fields. In the presence of a static electric field, a torque is exerted on the dipole that tends to rotate the dipole into the direction of the field. The dynamics of this process are that the molecule will physically move from its initial orientation to an orientation that is parallel to the field.

In order to quantify the behavior of replica watches uk dipoles in an electric field, it is instructive to start with an isolated dipole structure suspended in an electric field as illustrated in Figure 1.

Figure 1

The dipole moment, p, is the product of a the charge at one end of the dipole times the separation vector, d. The units of the dipole moment are Coulomb-meter. The electric field, E, is a uniform potential gradient in space, also represented as vector because of its directional properties. The direction of the dipole moment makes an angle q with the direction of the field. The field acts on the two poles of the dipole to produce equal forces in opposite directions. The total external force is zero so that the center of mass of the dipole is not accelerated from its initial position. The torque, t , however, is not balanced and results in an angular acceleration that is proportional to q ,

        eqn. 1

The dipole tends to align with the field because in any position other than q =0, there is an angular acceleration into the direction of the field.

The potential energy of the dipole in the field relates to the work required to bring the dipole into the field from a position infinitely far away and place it at the angle q . This potential energy is given by the quantity,

        eqn. 2

The relative alignment to the field can be quantified in terms of the magnitude of the component of the dipole vector that is in the direction of the field. That component is called the polarization and is straightforwardly given by,

            eqn. 3

Since no typical physical system is composed of a single dipole, it is necessary to increase the complexity of the description by considering a collection or ensemble of dipoles. Any physically plausible system would contain a very high number of dipoles. Since the individual orientation of each dipole is impossible to describe, the behavior of the system would have to be done statistically.

The description of the ensemble can be considerably simplified by making two assumptions, which define the attributes of the model system,

1. The dipoles are sufficiently widely separated that there are no electrostatic interactions.

2. The centers of mass of the dipoles do not occupy fixed positions on a lattice. They are free to translate and collide with each other with energy determined by the temperature.

Normal gases are physical systems that nominally conform to these assumptions.

Consider a unit of volume containing an ensemble of a large number, N, of dipolar structures that satisfy the conditions above. For conceptual clarity, there are two conditions of the system that must be kept in mind. In the absence of a field, the thermal collisions insure that the net orientation of the dipoles is completely random; there is no preferred direction. When an electric field is imposed the ensemble, the resulting torque on each dipole structure tends to rotate it into the direction of the field. The field tends to align the dipoles; there is a preferred direction, but the multiple thermal collisions prevent the dipoles from fully achieving that alignment.

In general, the polarization per unit volume can be calculated as in equation 3 if the angle q can be determined for each dipole,

        eqn. 4

This equation, however, does not fully represent the behavior of the system, because each individual dipole does not have a single value of q i. Instead, each dipole has a statistically averaged value determined by the energy of the collisions that the dipole experiences. Since all the dipoles are the same, and the energy distribution is the same for all the dipoles, a single statistical average of q describes the ensemble of dipoles in the field. Equation 4 can be expressed as,

        eqn. 5

where is the average value of cosq that describes the statistically averaged angle of alignment of the dipoles to the field.

The analysis of the polarization of the dipole ensemble reduces to the calculation of the average value for cosq . In its most rigorous form, this analysis would involve a complete description of the statistical mechanical elements that must be invoked to formulate a solution. The solution, however, can be intuitively understood in a manner that is mathematically consistent.

Each dipole in the collection is considered to be identical, and the system of dipoles has a fixed energy at constant temperature. A collection that satisfies these conditions is called a microcanonical ensemble. The potential energy of each dipole will contribute to the potential energy of the total system such that,

        eqn. 6

where ui is the potential energy of an individual dipole and Ni is the number of dipoles with ui potential energy.

The statistics of systems of this type can be described in terms of the Boltzman distribution. Ignoring any possible degeneracy, the probability of a particular potential energy state is characterized by the population of that state,

        eqn. 7

Using this expression, the average potential energy can be found,

        eqn. 8

Equation 2 describes the potential of the individual dipoles in the field, so that,

            eqn. 9

with the sum taken over all the dipoles.

Considering that q can vary continuously in three dimensions, equation 9 can be rewritten in terms of the integral of the continuously varying solid angle, W ,

            eqn. 10

As indicated in Figure 2, the solid angle can be described in terms of two components, the angle of inclination with respect to the field, q , and the azimuthal angle, d .

Figure 2

For the solid angle, W ,

        eqn. 11

The angle of inclination is what determines the component of the dipole vector that is in the field direction and contributes to polarization. Polarization will depend on the number of dipoles with a particular angle, q . The number of dipoles in an incremental angle element, dq , will be proportional to dq rotated fully through the azimuthal angle, d , so that,


With this, equation 10 becomes,

            eqn. 13

Once the problem has been expressed as in equation 13, many physics texts leave the solution as an exercise. There two keys to this problem that make solution fairly straightforward. It took me some time to get to the solution, even after I knew what the keys were. I found it worth the effort to work through the details because by doing that, the final solution didnt seem quite so foreign.

The first key is to recognize that,


This allows equation 13 to be expressed in a simpler form when the following substitutions are made,

        eqns. 15

Using the property of definite integrals,

        eqn. 16

equation 13 becomes,

        eqn. 17

The second key is to recognize that the numerator has a form that is suited for integration by parts using the form,


With the substitutions,

        eqns. 19

the numerator in equation 13 can be evaluated using the form of equation 18,





                   eqn. 20


The denominator can be evaluated directly,


                          eqn. 21

The numerator from equation 20 and the denominator from equation 21 provide the solution for equation 13,


                   eqn. 22


From the hyperbolic function definitions,

        eqn. 23

Replacing the substitution for x from equations 15, the final expression is the Langevin equation, L(x)

            eqn. 24

Figure 3 shows the plot of L(x) versus x. At room temperature, the plot provides a profile of the average dipole orientation as function of increasing field strength. It shows that at very high fields the orientation increases and will level off because of fewer unoriented dipoles are available. This leveling off is the condition of orientation saturation.

Figure 3

The Langevin expression is rarely applied to physical systems in the form that it appears in equation 24. In typical gases, electric field on the order of 107 V/m can cause dielectric breakdown, which means that fields of that strength are the highest that can be used to accomplish dipolar orientation. A single unit of electric charge diplaced through 0.6 Angstoms produces a dipole moment of 10-29 Coulomb-meter. At 20 C (293 K), the value of x can be calculated,

         eqn. 25

where, in terms of energy equivalence, 1 V-C = 1 J.

Since physical materials breakdown with increasing field strength well before saturation is reached, only the initial linear portion of the graph in Figure 3 is required to describe polarization. The hyperbolic cotangent can be expressed as the Taylor series expansion.

         eqn. 26

where Bn are Bernoulli Numbers.

For small values of x, the higher order terms do not make a substantial contribution, and the function can be approximated with the first two terms. Using this approximation in equation 24,

            eqn. 27

Now from equation 5 and equations 15, the total polarization can be described by the expression,

            eqn. 28

Equation 28 is typically the form that is used when calculating equilibrium polarization.