2652
T. NEDETZKA, M. REICHLE, A. MAYER, AND H. VOGEL
Thermally Stimulated Depolarization. A Method for Measuring the Dielectric Properties of Solid Substances by T. Nedetzka, M. Reichle, A. Mayer, and H. Vogel Physilc Department der Technischen Hochschule Manchen, Munich, Germany (Received February 10, 1969)
Thermally stimulated depolarization (TSD) of a polarized dielectric provides a method for measuring its dielectric properties. All mechanisms contributing to the dielectric constant which are temperature dependent can be measured separately by this method, namely orientational polarization of permanent dipoles or of dipoles induced by the electric field, space charge polarization, and electrode effects. The principal features of this method are described and a general theory of TSD is presented.
Introduction The investigation of the dielectric properties of matter by the alternating field method’-4 is in general complicated by two facts. (a) The contributions of different polarization mechanisms can be distinguished in the dispersion curve only if the corresponding relaxation times differ considerably. (b) In case of nonvanishing electric conductivity within the sample, space charge phenomena on the interfaces between sample and electrodes can simulate an additional polarization mechanism superimposed with the true dielectric dispersion. Those difficulties are effective in particular when studying the dielectric properties of hydrated lyophilized proteins. , 4 $ 5 A new promising method for the investigation of dielectric properties is presented here in which the cited complications are eliminated. The principal features of the method of “thermally stimulated depolarization”-an analogous method has been applied in thermoluminescence studies ‘ja7-are the following. The sample is mounted between the plates of a condenser to which an electric field of about 1 kV/cm is applied at room temperature so that a polarization is produced in the material. The sample is then cooled down to a temperature such that the polarization is frozen in; Le., the thermal energy is not large enough to reorientate the dipoles even after removal of the electric field. Only when the sample is subsequently heated according to a given temperature program, the frozen polarization gets liberated. This process can be followed by connecting the condenser plates through a resistance RA and measuring the voltage U , which is proportional to the time derivative of the induction charges on the condenser plates liberated by the decreasing polarization (Figure l ) , The frozen polarization will be liberated at certain temperatures, corresponding to the special type of polarization mechanism, and will appear as a voltage peak over the temperature axis (Figure 2). Position and shape of those thermally stimulated depolarization curves give information on The Journal of Physical Chemistry, Vol. 74, N o , IS,1970
activation energy and relaxation time of the respective polarization mechanism and on the magnitude of the susceptibility. In the following a general theory of thermally stimulated depolarization is presented.
Theory of Thermally Stimulated Depolarization (TSD) (This theory is based on electrostatic units.) In an electric polarizable medium several different mechanisms can in general contribute to the macroscopically observed polarization P , namely electronic and atomic displacement polarization, orientational polarization of dipoles, and space charge polarization.s P is proportional to the strength of the electric field F
P
= Pel
+ + Po,+ Pep
=
Pat
XF
(1)
x
being the macroscopic susceptibility. Electronic and atomic polarizations are not temperature dependent since they are pertinent to deformation of the electronic shell and displacement of ions, respectively. Therefore those polarization contributions cannot be frozen in and do not appear in TSD measurement. Orientational polarization of permanent dipoles or of dipoles induced by internal electric fields is temperature dependent owing to the temperature dependence of both the equilibrium polarization and the relaxation time 7.9 Space charge polarization is temperature dependent since its decrease takes place by compensation of the space charge via the resistance of the sample RI which is temperature dependent (2). Orientational and (1) W.Goebel and H . Vogel, 2.Naturforsch., 19, 292 (1964). (2) G.Brausse, A. Mayer, T. Nedetzka, P. Schlecht, and H . Vogel, J . Phys. Chem., 72, 3098 (1968) (3) N. Dannhauser, J. Chem. Phys., 48,1918 (1968). (4) T. Xedetzka, Diplomarbeit, Technische Hochschule Munchen, 1964. (5) H . P. Schwan, Phys. Tech. BWZ. Res., 6,323 (1963). (6) W. Hoogenstraten, Philips Res. Rep., 13,515 (1958). (7) L.Mader and N.Riehl, Z. Phys., 206,319 (1967). (8) A. Hippel, “Dielectric and Waves,” John Wiley and Sons, New Pork, N. Y., 1954. (9) W.Kauzmann, Reu. Mod. Phys., 14,12 (1942). I
THFRM ALLY STIMULATED DEPOLARIZATION
space-charge polarization therefore both can be frozen in if the sample is cooled down to sufficiently low temperature. The theory of TSD of both polarization mechanisms is described in the following sections.
2653 rotated from an element of solid angle dQ to another dQ’ in an interval of time dt. For the transition probability, WI (in the case of no electric field) follows from chemical rate theory
w1 = svkT -e x p ( F ) exp( -
;)=
h --T WO
exp(
-jg)
(3)
47r
Figure 1. Experimental diagram of the TSD method.
I
1
Figure 2. Depolarization peaks gained with a sample of lyophilized hemoglobin. Peaks 1 and 2 belong to orientational polarization; peak 3 belongs to space charge polarization.
1. Orientational Polarization and I t s TSD. I n substances with a large static dielectric constant es(es > 3 ) , permanent molecular dipoles largely determine the dielectric properties.9 A theory of the relaxation behavior of dipoles has been given by Debyelo on the basis of rotational diffusion. The application of this theory is limited as it has been derived under the following conditions: (a) no dipole-dipole interaction, (b) the equilibrium state is attained only by a single process (by transition over a potential barrier or rotational friction), (c) all dipoles behave in the average equally. Unfortunately, those conditions are seldom fulfilled simultaneously in practical situations. An alternative theory of dielectric relaxation is that of Kauzmann9 which is based on the theory of chemical reaction rates and which is more general than the theory of Debye. I n the following we shall therefore make use of the Kauzmann theory. I n Kauzmann’s theory the time rate of change of the macroscopic polarization is expressed in terms of the motions of the individual molecular dipoles which are
where w0is the temperature independent factor, s is a sterical factor for hindered rotation, v is a transmission factor expressing the fraction of activated dipoles which accomplish the transition. Normally v is close to unity. AS is the entropy change upon transition, E is the activation energy, k is the Boltzmann factor, and T is the absolute temperature. Calculating the dipole transitions according to Boltzmann statistics and integrating over all dipoles and orientations, the following basic equations are ~ b t a i n e d : ~equilibrium polarization
where No = number of dipoles/cma, /.teff = effective dipole moment of a molecule, F = external electric field, aF = internal elect.ric field operating on the dipolell where
and x = the dielectric suceptibility ; relaxation equation (5)
where P is the instantanous polarization; time
relaxation
Equation 5 describes the time rate of change of the macroscopic polarization P in terms of the transition probability w1of the individual dipoles and is valid for continuous and discrete possibilities of dipole orientation. I n the case of thermally stimulated depolarization we are concerned not with the time dependence of the polarization but with its temperature dependence. To keep calculations simpler in the theory and for better realization in the experiment, the relation between temperature and time is chosen linear; that means the sample is heated with constant rate q (deg/sec).
T = To
+ gt
(7)
(10) P. Debye, Phys. Be?., 15,777(1913). (11) W. Weisel, “Lehrbuch der Theoretisohen Physik,” SpringerVerlag, West Berlin, 1963.
The Journal of Physical Chemistry, Vol. 749No. 1.9, iQ7O
2654
T.NEDETZKA, M. REICHLE, A. MAYER, AND H. VOGEL
where To is the initial temperature. Equation 5 can then be rewritten with the new variable T, using eq 3 d P 4aWl _ ( p e q - P) q dT~
(8)
In fact, in practical cases the external voltage applied to the sample producing the initial polarization is by orders of magnitude larger than the voltage U, which is measured. In the limiting case 4nwlc 1 (a) 4awlc
