4 Linear Debye Relaxation

[401.2.2.1] This section provides some background material and definitions for the discussion of dielectric relaxation in glass-forming liquids. [401.2.2.2] In the linear phenomenological theory of irreversible processes one assumes that the displacements resulting from the application of (generalized) forces are linear [36]. [401.2.2.3] Let $x_{i}(t)$ denote the observable displacement [page 402, §0] or current at time instant $t$ corresponding to a force $p_{i}(t)$ . [402.1.0.1] If there is no time delay between the application of the forces and the response of the currents or displacements then the linear theory postulates

$x_{i}(t)-x^{{\mathrm{eq}}}_{i}=\sum _{j}L_{{ij}}p_{j}(t)$

(11)

where $x^{{\mathrm{eq}}}_{i}$ is defined as the value of $x_{i}$ for vanishing force $p_{i}=0$ , and $L_{{ij}}$ are the so called kinetic coefficients.

[402.1.1.1] When the time variation of the forces becomes too fast the response of the displacements or currents generally starts to lag behind. [402.1.1.2] This experimental fact is the basis of memory effects. [402.1.1.3] By linearity the delayed effect of the forces must be superposed to obtain the current value of the displacements. [402.1.1.4] Causality requires that only the effects from the past enter in the linear superposition. [402.1.1.5] This leads to the generalized relation

$x_{i}(t)-x^{{\mathrm{eq}}}_{i}=\int _{{-\infty}}^{t}\sum _{j}\left[\chi^{\infty}_{{ij}}\delta(t-s)+\chi _{{ij}}(t-s)\right]p_{j}(s)\;\mathrm{d}s$

(12)

between forces and displacements (or currents). [402.1.1.6] Here $\delta(x)$ denotes the degenerate $\delta$ -distribution. [402.1.1.7] The first term describes the instantaneous response while the second describes the delayed response (aftereffect). [402.1.1.8] The kernel function $\chi _{{ij}}(t)$ is called the response function. [402.1.1.9] In writing eq. (12) one also assumes homogeneity in time, i.e. that the response of the system does not depend on the origin of time.

[402.1.2.1] The dynamic susceptibility (also called generalized compliance, complex admittance, etc.) is defined as

$\chi _{{ij}}(\omega)=\chi^{\infty}_{{ij}}+\int _{0}^{\infty}\exp({\mathrm{i}}\omega t)\chi _{{ij}}(t)\;\mathrm{d}t=\chi^{\infty}_{{ij}}+\mathscr L\{\chi _{{ij}}(t)\}(u)$

(13)

in terms of the Laplace transform of $\mathscr L\{\chi _{{ij}}(t)\}(u)$ of the response function where $u=-{\mathrm{i}}\omega=-2\pi{\mathrm{i}}\nu$ where $\nu$ is the frequency. [402.1.2.2] In this paper a conveniently normalized dynamical susceptibility will be used. [402.1.2.3] It is defined as

$\hat{\chi}_{{ij}}(u)=\frac{\chi _{{ij}}(\omega)-\chi^{\infty}_{{ij}}}{\chi _{{ij}}(0)-\chi^{\infty}_{{ij}}}$

(14)

where

$\chi _{{ij}}(0)=\chi^{\infty}_{{ij}}+\int _{{0}}^{\infty}\chi _{{ij}}(t)\;\mathrm{d}t=\chi^{\infty}_{{ij}}+f_{{ij}}(0)$

(15)

by virtue of (13) and (16).

[402.2.1.1] The response function is closely related to the so called relaxation function defined by the relation

$f_{{ij}}(t)=\int _{t}^{\infty}\chi _{{ij}}(s)\mathrm{d}s.$

(16)

[402.2.1.2] Hence one has

$\chi _{{ij}}(t)=-\frac{\mathrm{d}}{\mathrm{d}t}f_{{ij}}(t).$

(17)

[402.2.1.3] The relaxation function $f_{{ij}}(t)$ describes the relaxation of the observable $x_{i}$ when an applied force $p_{j}$ of unit magnitude is switched off abruptly.

[402.2.2.1] In the following subscripts will be suppressed to simplify the notation. [402.2.2.2] Using equation (17) one finds

$\hat{\chi}(u)=1-u\mathscr L\{\hat{f}(t)\}(u)$

(18)

in terms of the Laplace transform of the normalized relaxation function $\widehat{f}(t)=f(t)/f(0)$ .

[402.2.3.1] There are many relaxation phenomena in nature whose relaxation function obeys the simple approximate equation

$\tau\frac{\mathrm{d}}{\mathrm{d}t}\hat{f}(t)+\hat{f}(t)=0.$

(19)

[402.2.3.2] An example occurs in dielectric relaxation where eq. (19) is known as the Debye type relaxation equation. [402.2.3.3] For dielectric relaxation phenomena the force $p$ is the electric field and the displacement $x$ is the dielectric displacement or polarisation. [402.2.3.4] The equilibrium value $x^{{\mathrm{eq}}}$ vanishes (except for ferro-electrics). [402.2.3.5] The dynamical susceptibility $\chi$ becomes the complex dielectric function. [402.2.3.6] The solution of eq. (19) is the normalized exponential Debye-relaxation function

$\hat{f}(t)=\exp(-t/\tau)$

(20)

with relaxation time $\tau$ . [402.2.3.7] The corresponding normalized susceptibility (dielectric function) is the Debye susceptibility

$\hat{\chi}(u)=\frac{1}{1+u\tau}.$

(21)

Authors:	R. Hilfer und N.B. Wilding
originally published in:	Chemical Physics, Vol. 284, pages 399-408 (2002)
originally submitted:	December 7th, 2001