MOLECULAR MECHANISM QF RATE PROCESSES IN SOLIDS A. RELAXATION PROCESSES INTRODUCTORY PAPER BY C. J. F. BOTTCHER This introduction may be started with a few remarks concerning the title and the three sub-titles of the present discussions. In defining a rate process we usually limit the term to a system proceeding from one equilibrium state to another, under the influence of a change of the external parameters, in such a way that energy dissipation, and hence entropy production, takes place. If one thinks of the problem in terms of potential barriers, then it is first neces- sary that certain particles receive sufficient energy from their surroundings to overcome a barrier and then-on reaching a new condition-they must loose kinetic energy, this loss being experimentally represented by the appearance of energy dissipation as heat.Whether such a picture of potential barriers is appro- priate for the particular problem or not, there must still be, in any rate process, the element of an experimentally observable time, in which energy received from the external disturbance is dissipated as heat. Concerning the mechanism of such a dissipation process a few remarks shouid be made. The molecular models used for its description have always the dis- advantage that macroscopic concepts are used on a microscopic scale. A striking example is Debye’s model of a molecular dipole, rotating in a viscous continuum, to which Stokes’s law is applied. In essence this model is merely a very small edition of the well-known mechanical Voigt or Kelvin model, which is, in this case, so to speak, electrically actuated.Unfortunately such a use of macroscopic concepts in molecular models seems to be almost inevitable at the present state of knowledge of the statistical inter- pretation of energy dissipation and entropy production. Even in the derivation in Brinkman and Schwarzl’s paper 1-although in many respects more realistic from the molecular point of view-concepts like friction and diffusion had to be introduced. Another example is that sometimes heat conduction is used as the underlying mechanism of energy dissipation. In that case the notion of two intermixed sub-systems of different temperatures is introduced, these being, for instance, in the acoustic relaxation of gases the translational and the internal degrees of freedom.Similarly Casimir and du Pre, in their theory of paramagnetic relaxation,2 intr0duce.d the idea of different spin and lattice temperatures. As a last example of such a use of macroscopic concepts I might mention the model of the scattering of an “ ordered ” elastic wave at dislocations as described in the paper of Seeger and collaborators.3 In this model the energy of the ordered wave ultimately disappears into the continuum of the “ disordered ” elastic waves, determining the thermodynamic state of the crystal. Such an order-disorder interpretation may well be closer to a real molecular theory, but quantitative treatments in specific cases will be necessary before one can judge how fruitful such an approach will be. In the programme, section A has been described as “ relaxation processes ” and the sections B and C as “ steady-state processes ” without and with lattice rearrangement, respectively.I think this classification does not represent the 78 INTRODUCTION actual differences between the papers of group A and those of B and C. Before discussing this I should first remark that there seems to be no single definition for a steady-state process. A chemical engineer would say it is a process in which after a transient period all external conditions of the essential part of the system are maintained constant. A mathematical physicist, however, would refcr to a periodic form of solution of his equations in which transient terms have dis- appeared. In Ubbelohde’s paper 4 the term “ steady-state ” refers to the case where after the build-up of an activated state of the solid a steady migration is established. By either of these three definitions, and perhaps by any other definition of steady-state processes, only part of the experiments of the sections B and C are of a steady-state type.Moreover there is not a border line between relaxation and steady-state processes ; as in most relaxation phenomena it is possible to work under steady-state conditions, provided the developed heat is taken away. There is, however, another way to distinguish between section A and the other sections : in the sections B and C a net translational migration is essential, whereas in A this is only the case in the conductivity measurements of Granicher and col1aborators.s After these general remarks I might return to my task to introduce section A on relaxation processes.In order to demonstrate some characteristic features of relaxation phenomena I choose the example of a polar dielectric between the plates of an ideal plane condenser to which a uniform static field is applied. We imagine two fundamental experiments, represented in fig. 1 and fig. 2. E D P t -- t FIG. 1 .-Dielectric relaxation under prescribed E. D E FIG. 2.-Dielectric relaxation under prescribed D. In experiment I we keep the condenser plates at a constant potential difference during a certain period. Before and after that period we take care that the poten- tial is zero. Thus the electrical field E is prescribed as shown in fig. 1. The re- sulting curve for the dielectric displacement D is shown at the right-hand side: the electronic and atomic polarization are established immediately ; the dipoles, however, need some time before their new orientational equilibrium distribution is reached.When the potential drops to zero it takes some time before a random orientation of the dipoles is re-established. In experiment I1 we do not prescribe the potential, but the true charge of the plates and hence D. As the contribution of the dipole orientation to the field E has the opposite sign to the contribution of the true charges, the field E gradually decreases during the first period until the new equilibrium distribution of the dipoles is reached. When the true charges are suddenly taken away, for instance by removing the condenser plates, a field of the opposite direction remains, due toc.J . F . B ~ T T C H E R 9 the dipole polarization. It gradually disappears while the random orientation of the dipoles is re-established. At room temperature such time-lags are usually of an order of magnitude of 10-10 sec, but in very viscous materials, and in particular at low temperatures they can be much larger, even seconds or minutes. It should be mentioned that if the D-curve of fig. 1 would be forced, the E-curve of fig. 1 would result, according to the superposition principle. We can compare the first parts of these two experiments with two ways of extending a fibre : the first part of experiment I corresponds to an elongation of the fibre under constant stress : the first part of experiment I1 shows the stress as a function of time for a prescribed constant elongation.In such mechanical examples the two cases are always distinguished by applying the term retardation to the first part of experiment I and the term relaxation to the first part of experiment 11. In electrical problems, however, only the term relaxation is used although most experiments correspond with what in mechanical terms is called retardation. As it would be very difficult to change this generalized use of the term relaxation, I would suggest it is accepted as the general name for all after-effects and to dis- tinguish, where necessary, between (i) relaxation under prescribed stress, or retardation ; (ii) relaxation under prescribed strain, or relaxation in the strict sense of the In many experimental studies of relaxation phenomena an alternating external disturbance is applied.In that case usually a distinction is made between resonance and relaxation, for instance in the paper of Seeger and collaborators. Strictly speaking such a sharp division is not necessary because the cases where the term relaxation is used could also be described as resonance with over-critical damping. The other extreme would be the ideal case of resonance without energy dissipation. In announcing this conference the stress had been laid on the molecular mechan- ism of the rate processes. Consequently, in this introduction to section A, par- ticular attention must be paid to this question: to what extent can results of relaxation experiments lead to conclusions concerning the underlying molecular mechanism? In this respect it must be emphasized that from relaxation measure- ments on one single system hardly any conclusion can be drawn about the mole- cular mechanism.A wide variety of mechanisms can account for the same macroscopic behaviour. A striking example is that of dielectric relaxation where it is even impossible to distinguish between the frequency-dependent part of ionic conductivity losses and dipolar losses. Another example of the same limitation is the fact that from relaxation measurements on a single system only, one cannot de- cide whether the relaxation is caused by a chemical reaction or by a physical process. Some indication, however, can be gained from the magnitude of the relaxa- tion time or times, calculated from the slope of the curves in non-periodic cases and from the frequency dependence of the phenomenon for a periodical disturbances and also from the activation free energy, determined from the ex- ponential temperature dependence of the relaxation time.The experimental values of these two quantities, relaxation time and activation energy, can be used to test a hypothetical molecular mechanism in the few cases where it is possible to predict their approximate values from the model. If no large deviations are found the model niight be accepted as one of the possible explanations. The most satisfactory way, however, to arrive at more or less definite conclusions concerning the molecular mechanism is to examine experimentally how the phenomenon depends on the physical and chemical structure; in other words, physical and chemical modulation of the solid must be an essential element of the experiments.word.10 INTRODUCTION This final remark is not restricted to section A but applies to all the rate pro- cesses. In solids it means that we must examine how the rate processes depend on crystal structure and lattice imperfections, on the chemical composition of the solid-not overlooking the impurities-and, in some, on the chemical structure of the particles. Even then a profound knowledge of other physical properties of the solids is required. A classical example is the well-known work of Snoek 7 on the mechanical relaxation of cc-iron crystals, with carbon as an impurity. Snoek and his collaborators showed that in order to arrive at a definite con- clusion about the molecular mechanism it was necessary to vary both the main component and the impurity by replacing, for instance, carbon by nitrogen and part of the iron by manganese. Needless to say an accurate knowledge of the crystal structure was indispensable in this case too. The papers of our conference present many similar examples and it might even be concluded that the striking common element of most of these papers is that they show that variation of the physical and chemical structure of the solid is the essential experimental condition for obtaining conclusive information concerning the molecular mechanism of rate processes. 1 Brinkman and Schwarzl, this Discussion. 2 Casimir and du Prk, Physica, 1938,5, 507. 3 Seeger, Donth and Pfaff, this Discussion. 4 Ubbelohde, this Discussion. 5 Granicher, Jaccard and Steinemann, this Discussion. 6 see, e.g., Bottcher, Theory of electric polarisation (Amsterdam, Elsevier Publ. Co., 7 Snoek, Physica, 1939, 6, 591 ; 1941, 8,711,734. 1952), chap. 10.