Dielectric Spectroscopy: Choosing the Right Approach - Pharmaceutical Technology

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Dielectric Spectroscopy: Choosing the Right Approach
This tutorial paper is meant to aid in dielectric-sensor selection

Pharmaceutical Technology
Volume 9, Issue 32, pp. 8293

In a rigorous and exhaustive approach to model multianalyte concentration variations, the authors consider the material under test to be a homogeneous mixture of the analyte, other constituent chemicals, and air. By varying the proportion of this mixture, we can vary the relative concentrations of the materials. The effect of variation in proportion of the mixture on its dielectric property is nonlinear and requires considerable effort to model. The static (frequency-independent) relative dielectric permittivity for an isotropic ensemble of N equivalent molecules each having a dipole moment, μ, and contained in a spherical volume, V, is given by the following equation:

in which M(0) is the instantaneous dipole moment of the macroscopic sphere at the arbitrary time t=0. The equilibrium value of the scalar product ‹M(0)M(0)› taken over all the complexions of the ensemble will be independent of time for a stationary thermodynamic system. Considering a system of N constituents in the mixture of volume V

in which P i L (0) is the instantaneous liquid dipole moment of the ith constituent of the mixture. Detailed analysis of mixtures and their dielectric properties can be found in References 33–37.

Obtaining wideband spectroscopic measurements can be lengthy process because the time available for inline measurements is limited. Alternatives for such measurements are single-frequency or selected-frequency excitations. In this case, the response of a sample at a single or selected frequency can be used to monitor changes in physical properties. It should be noted that the observation of dielectric spectra at just selected frequencies could result in measurement artifacts being misinterpreted as dielectric phenomena; therefore, such an approach is possible if there is some prior knowledge of the sample response at the chosen frequency. Off-line studies can establish such characteristics of the sample. To illustrate a single- and selected-frequency approach, data presented in Figure 3 is analyzed further to establish a dependency between capacitance and coating thickness. Ideally the mapping between measured capacitance, C, and the coating thickness, T would be of the form

The first term of this equation represents the exponential decrease in the effect of the core layers as the thickness increases. The second term is a function that accounts for the escaping electronic fields. Because these are mostly owing to the zero-order field lines and their dispersion depends on the material property of the coating, this loss also can be written as a function of the thickness, χ(T); α and β are constants. To obtain thickness as a function of capacitance, the previous equation can be rearranged, expanded as a series in T, and the higher order terms neglected. This gives a polynomial fit. Because there are only four data points and at least one degree of freedom is needed to validate the fit, a quadratic expansion is chosen. Thus, the thickness of the coating can be obtained from the measured capacitance using the equation


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