DIELECTRIC RELAXATION IN SOLID HYDROGEN HALIDES BY ROBERT H. COLE AND STEPHEN HAVRJLIAK, JR. Metcalf Chemical Laboratories, Brown University, Providence, R.I., U.S.A. Received 3 1st January, 1957 Dielectric constant and loss measurements of solid hydrogen chloride, and hydrogen and deuterium bromide and iodide, are summarized and discussed in relation to order- disorder phase transitions. Electrostatic calculations indicate the importance of dipole interactions for the structures and rates of molecular orientation in the low temperature phases, but specific effects show the significance of other factors as well. In the less ordered higher temperature phases, the dispersion behaviour is characteristically different, but more complete dielectric measurements and structural evidence will be needed for a proper understanding.Of the rather limited number of solids which exhibit appreciable dipole orientation in an electric field, the hydrogen halides are of considerable interest because the molecules and the solid structures, to the extent they are known, are comparatively simple. The molecular " shapes " are not far from spherical, and X-ray evidence 1 indicates that the various solid phases have halogens ar- ranged in face-centred structures reminiscent of cubic close packing of spheres. One can hope that dielectric relaxation behaviour of these systems will be more easily analysed in terms of molecular orientation and interaction than more com- plicated molecules and structures. The discussion of the experimental evidence will show that the behaviour is by no means simple; at the same time, there are features often encountered in other cases which have so far resisted molecular explan- ation. The purpose of this report is to summarize the available evidence, indicate respects in which it is incomplete, and discuss the relation to structure and theory.The different solid phases of hydrogen and deuterium halides and the degrees of dipole orientation possible are conveniently shown by plots of static dielectric constant €0 against temperature as in fig. 1. Hydrogen fluoride is omitted, because the small dielectric constant of the solid indicates essentially " frozen " orientations at all temperatures, and no other evidence of phase transitions is known to us (the relevance of X-ray structure evidence to properties of the other halides is considered later).Hydrogen chloride has a first-order solid phase transition at 99" K ; above this, the dielectric constant is large and has the inverse temperature dependence of freely orienting molecules. The low-temperature phase exhibits a much smaller but measurable dipole polarization at temperatures near the transition. In the bromides and iodides, there are two (and for hydrogen bromide three) A-type specific heat transitions, with a pronounced dielectric constant change at the lowest temperature transition in the form of a rapid rise to a large peak value (over 200 for the bromides). The low-temperature transitions clearly involve considerable changes in molec- ular order with greatly increased freedom for dipole orientation, and much of the accompanying dispersion processes has been accessible to audio and radio frequency measurements.LOW TEMPERATURE DISPERSIONS Representative results for the low temperature phases a few degrees below the transitions are shown as complex plane diagrams in fig. 2. The relation of 3132 RELAXATION I N HYDROGEN HALIDES E' to e r r is represented over considerable frequency ranges by the empirical formula 4 (1) where €0 and €1 are the static and limiting high frequency dielectric constants of the dispersion, LL) is radian frequency, TO the relaxation time, and cc an empirical parameter proportional to the depression angle of the circular arc. = (€0 - d / [ 1 + (iw70)1-~1, E* = E r - iE" ZOO I00 5 0 2 0 10 5 I I I I 8 0 1 2 0 T"K I 6 0 2( FIG.1 .-Static dielectric constants (logarithmic scale) of hydrogen and deuterium halides plotted against temperature. Data of Swensonz for HCI, of Brown3 for HBr and of S. Havriliak for the other substances. The arc function has been found to represent dielectric relaxation of a variety of otherwise quite unrelated substances; of these, the present examples seem the simplest with respect to molecular structure and arrangement. It is also of interest that the arc function is an accurate fit within the rather small limits of experi- mental error (this question has been considered in detail elsewhere for HBr). The simple arc function does not represent the entire dispersion behaviour, except possibly for the iodides. This is apparent from the fact that the high frequency limit €1 is significantly larger than the induced polarization value n2 from refractive index estimates, or any somewhat larger figure making reasonable allowance for atomic polarization.There must therefore be a further dispersion process or processes somewhere between radio and infi-a-red frequencies. Such a dispersion was observed in both HBr and DBr at temperatures sufficiently low for the frequencies to lie in the experimental range (up to 500 kclsec). These high frequency portions are illustrated in fig. 3, from which it is clear that the form is approximately a circular arc, the amplitude very much less than for the " primary ", lower frequency process, and the second high frequency limit stillR . H. COLE AND S. HAVRILIAK, J R .33 perhaps a little larger than reasonable estimates of induced polarization values. Present measurements do not extend to sufficiently high frequencies to determine whether a similar situation exists in the iodides. O L 0 :'r 0 0 P I I I 2 0 3 0 € 1 4 0 700c DBR. 78' K 25' 1.0 HI, 60.4'K E'li j - - E ; 0 1 7 O f' 3 . 1 4 - 2 12' FIG. 2.-CompIex plane diagrams for low-temperature phases. 2 E " I 0 3 4 E' FIG. 3.-High-frequency dispersion of HBr, DBr in their low-temperature phases. B34 RELAXATION I N HYDROGEN HALIDES DISPERSION IN THE HIGHER TEMPERATURE PHASES A rather limited amount of information about the dispersion processes above the low-temperature transition is available, because measurements have not yet FIG. 4.-Complex plane dia- grams for DBr, HI, DI in their intermediate solid phases.The solid curves represent the range of measurements ; the dashed curves are possible continua- tions to higher frequencies dis- cussed in the text. been extended above 500 kc/sec. The results show a distinct change in character from the low temperature behaviour, as shown in representative complex plane diagrams (fig. 4) for HI and DI. The results are quite incomplete, especially at 6 higher temperatures, but do show defi- nitely that the low-frequency part of the ; relaxation is consistent with a Debye g semi-circular locus rather than the de- $ pressed cireular arcs found below the < transition (fig. 2). 0 Another way of confirming the be- haviour when the frequency range is too low to show much dispersion is by plotting absorption conductance (pro- portional to WE") against frequency, as for Debye behaviour and WTO << 1 one has lo-' lo-' WE" (€0 - E1)T0W2.(2) The plots in fig. 5 of loglo WE" against loglof, where f = ~ / 2 7 ~ , are fitted by lines of slope 2, and therefore show that the limiting low frequency behaviour conforms to a Debye-type equation (i.e. eqn. (1) with cc = 0). I 0- ' The highest temperature phases of the F rcqoc'ncy - K.C bromides -and iodides and the high- FIG. 5.-Loga&hmic plots of absorption temperature phase of hydrogen chloride conductance (- WC") against frequency. exhibit no measurable relaxation effects The straight lines are drawn with slope two.R . H . COLE A N D S . HAVRILIAK, J R . 35 below 1 Mc/sec, and one can only conclude that the characteristic frequencies must be well above 100 Mc/sec.TEMPERATURE DEPENDENCE OF RELAXATION TIMES Characteristic relaxation times for the measured dispersions are calculated from the frequency of maximum absorption when the data are sufficiently complete. If only conductance dispersion was measurable, values of TO (sec) were estimated from eqn. (2). These require an assumption about €1 which is unknown (as indeed DBR --- /‘ / I I I 10 I2 I4 ~ o o / T”K temperature (“K). FIG. &-Plots of loglo 70 (sec) for the several dispersions against reciprocal is the validity of the Debye equation for higher frequencies); for simplicity we have set €1 = 0. This is certainly not correct as €1 must exceed n2 and may well be ~ 0 / 3 or more, but the error should not significantly affect comparison of rate plots of loglo TO against reciprocal temperature.The rate plots are shown in fig. 6, and values of activation energy E, (kcal/mole) and frequency factor A (sec-1) for the equation (3) are listed in table 1. The activation energies all fall in the range 1.5 to 3.6 kcal/mole with no obvious relation to the halogen substitution, the times are longer for the deuterium isotope in the ranges studied by factors from 1.3 to 4, and the rates decrease on cooling through the low-temperature transition by factors 1.6 to 5. loglo TO =- log10 A + EJ2.3 RT TABLE 1 .-CONSTANTS OF RATE LAW DESCRIBING THE OBSERVED DISPERSIONS range of T log10 A Ea HC1 100-63 - 13.00 2-6 HBr (1st) 89-63 - 11.2 2.7 (2nd) 89-63 - 12.1 1.6 DBr (1st) 95-63 - 16.2 1.5 (2nd) 95-63 - 13.5 3.6 111-100 - 15.4 1.5 HI 68-62 - 13.6 2.2 80-70 - 17.2 3.6 DI 75-62 - 13.3 2.2 100-78 - 15.6 3.1 (“K) (sec-1) (kcaI/mole)36 RELAXATION IN HYDROGEN HALIDES STRUCTURE EVIDENCE AND THEORY Interpretation in molecular terms of the various relaxation processes will ultimately require a better knowledge of structures and changes in structure of the solid phases than is now available.The picture of the various transitions as co-operative changes to states of increasing molecular disorder in the higher temperature phases is generally accepted, but there are no certain conclusions about the ordered structures, and the intermediate states of order for dipole orientation are quite unknown. For the highest temperature phases stable just below the melting point, there is little doubt that the structures are face-centred cubic and hence isotropic.The static dielectric constant values confirm the X-ray evidence indirectly, as the ob- served values 2, 3 are quite close to those calculated from gas dipole moments by Onsager’s formula,6 which cannot be expected to hold if there is net short-range correlation of dipole orientations, and, in fact, has no relation to the behaviour in the more ordered low-temperature phases. The high-temperature transitions of the bromides and iodides show slight changes in dielectric constant of the magnitude expected from density differences, but the large effects at the low-temperature transitions and at the first-order transition in hydrogen chloride (fig. 1) indicate very large changes in dipole order.Tisza 7 has pointed out that the shift of the lower transition to lower temperatures on going from HBr to HI and of the upper ones to higher temperatures are related qualitatively to the smaller dipole moment and larger polarizability of HI, and so suggested that dipole interaction forces dominate the lower transition and van der Waals’ polarizability forces the upper. Powles 8 has pointed out that estimates of dipole-dipole energies predict much larger changes in the transition temperatures than are observed, and came to the conclusion that anisotropy of repulsive forces, corresponding to deviations of the molecules from spherical shape, were dominant in the ordered structures. What is now known of these structures indicates, however, that neither point dipole interactions nor repulsive forces are adequate for an explanation, but that the former probably have an important role.The X-ray evidence for the low-temperature phases is not conclusive but in- dicates face-centred orthorhombic or tetragonal arrangement of the halogens. Infra-red studies of the vibrational frequencies for solid films of hydrogen chloride and bromide by Hornig and associates 9s 10 limit the possibilities for orientations of the hydrogens and molecular dipoles in the unit cell, and the simplest structures found to be compatible with the observed splittings are the two shown in fig. 7. Both have zigzag chains of dipoles in layer planes normal to the C-axis of the four molecule unit cell, the difference lying in relative placement of alternative layers of oppositely directed chains.The structure A is very similar to one of two possible structures found by X-ray determinations 11 for crystalline hydrogen fluoride,* which gives some basis for the further consideration. The structures in fig. 7 are substantially different from ones found stable for dipole-dipole forces or suggested on the basis of repulsion and molecular shape, and seem more indicative of short-range incipient hydrogen linkages. However, it should be recognized that arrays considered for calculations of point dipole interaction energies 129 13 have involved head-to-tail strings (+-+), with adjacent strings parallel or antiparallel and at various angles with cubic crystalline axes. Corresponding calculations for the structures A and B of fig.7 have been made by summing the dipole interaction energy where p is the dipole moment, the angle between dipoles i andj, and Or and Oj are the angles each makes with the connecting distance vector I-,. * The structure preferred by these workers placed the hydrogen atoms randomly in the chains (F-F-F angle l2Oo), that is, as FH-F or as F-HF, but they stated that the alternative similar to A was almost as acceptable. vi, = p2 (cos 0, - 3 cos 0, cos e j y r , 3 , (3)R . H . COLE AND S . HAVRILIAK, JR. 37 The calculations (made assuming cubic symmetry and 90" dipole angles for simplicity) show that structure B is very unfavourable for this potential, but that A has quite low energy. For interactions summed over the first eight shells of neighbours (140 molecules) the result is zqj =- 1.17 N2p2, where N is the dipole concentration.This figure is not as good as Luttinger and Tisza's most favourable value - 1.808 N2p2, but it does show that dipole interactions may play a significant role. FIG. 7.-Possible structures of ordered phases of hydrogen chloride and bromide. Open and shaded circles represent halogens in adjacent layers. Kirkwood 14 and Kruger and James 15 have considered nearest-neighbour (4) and shown that a second-order transition results from considering only the first term, which is part of the dipole potential (3), while fist- and second-order transi- tions are possible for different relative values of the coefficients A and B. None of the assumptions about the potentials accounts for the most probable structure A or the actual sequences of phase transitions, and the potential (4) has the diffi- culties of representing only part of the angular dependence of dipole and quadru- pole interactions as well as being confined to first neighbours.The consequences of the second term in cos2 8, are interesting in suggesting the importance of quadrupole interactions (if expansion in higher order poles is a workable approach), and also because such terms can give rise to a secondary minimum in the potential for rotation of a single molecule through 180" from the stable position. This possibility could lead to a potential barrier problem with depths and populations of the minimum changing with temperature as more dipoles can occupy " wrong positions ".Unfortunately, our knowledge of quadru- pole moments is too meagre to permit any estimate of the reality of the possible consequences. The available evidence on structure and states of order is evidently somewhat frustrating as a starting point for correlation with the dielectric evidence. Struc- tures similar to that of fig. 7a may well represent the ordered phases of hydrogen chloride and bromide, with the situation for hydrogen iodide in doubt, but the nature of the disordering processes can only be guessed. interaction potentials of the form Y;:~ = A cos e, + B cos2 eii, RELATION OF STRUCTURE AND RELAXATION PROCESSES One is tempted to identify the activation energies E, of the dielectric relaxation processes with heights of intermolecular potential barriers to reorientation.The magnitude of barrier which dipole forces offer to rotation of one molecule with others fixed as in fig. 7a is 1.15 kcal/mole, on using pHC1 = 1.1 D, which is38 RELAXATION I N HYDROGEN HALIDES somewhere near thz magnitude of E, observed. On the other hand, there is no indication. in the observed values of a dependence on p2, which should be pronounced if dipole forces are a dominant factor ( p ~ a = 1-12, ,UHB~ = 0.85, The occurrence of two relaxation regions in the low-temperature phases may well reflect the anisotropy of the structures. That they are anisotropic is well established, and any arrangement like fig. 7a would be expected to give large differ- ences of dielectric constants measured parallel and normal to the dipole layers.The measurements are of mosaic crystals with presumably random orientations, but different rates of relaxation of the principal dielectric constants would be resolved in time by measurements over a range of frequencies. There is suggestive evidence for this in the fact that the amplitudes of the two relaxation processes of the low-temperature phases of deuterium bromide showed hysteresis effects depending on previous thermal history, which could well result from changes in crystallite orientations from stress relaxation effects. The necessity for more than a single simple dispersion in the intermediate phases of hydrogen and deuterium iodide has already been mentioned. The continuation of the observations could lead to finding a second dispersion of the form indicated by the dashed curve (a) of fig.4, and anisotropy could be advanced 1s an explanation. On the other hand, there is precedent for a continuation of the form indicated as curve (6) or in still other ways. The need is obviously for measurements in a considerable range of frequency above 1 Mc/sec, and work is in progress toward this end. Definitive measurements at the much higher fre- quencies necessary for the disordered highest temperature solid phases are a formid- able problem, which is unfortunate because knowledge of the nature of the re- laxation processes for the simple structures would be valuable in itself and for interpreting the results at lower temperatures. The change in character of the so far accessible regions of dispersion near the low-temperature transitions are quite remarkable because of the change from circular arc loci to a dispersion which, at least in its beginnings, has the simple Debye behaviour of relaxation theories incorporating a single relaxation time. The difference is brought about by definite changes in arrangement and order of neighbouring molecules, and more knowledge of the structures ought to bring the lines of evidence to such a state that each can contribute to it better under- standing of the molecular re-orientations.PHI = 0.40 D). This work has been supported by the Office of Ordnance Research, U.S. Army, and by the Office of Scientific Research, Air Research and Development Command. One of us (R. H. C.) is indebted to the John Simon Guggenheim Memorial Founda- tion for a Fellowship in 1955-56 held at the University of Leiden, and to Prof. C. J. F. Bottcher for hospitality during his stay there. 1 Natta, Mem. Accad. Italia Chem., 1931, 2, no. 3, 5 ; Gazz. chim. ital., 1933. 2 Swenson and Cole, J. Chem. Physics, 1954, 22, 284. 3 Brown and Cole, J. Chem. Physics, 1953, 21, 1920. 4 Cole and Cole, J. Chem. Physics, 1941, 9, 341. 5 Cole, J. Chem. Physics, 1955, 23, 493. 6 Onsager, J. Amer. Chem. Soc., 1936, 58, 1486. 7 Tisza, Phase Transformations in Solids (WiIey, New York, 1951), p. 28. 8 Powles, Trans. Faraday SOC., 1952,48,430. 9 Hornig and Osberg, J. Chem. Physics, 1955, 23, 662. 10 Hiebert and Hornig, J. Chem. Physics, 1952, 20, 918. 11 Atoji and Lipscomb, Acta Cryst., 1954, 7 , 173. 12 Luttinger and Tisza, Physic. Rev., 1946, 70, 954. 13 Sauer, Physic. Rm., 1940, 57, 142. 14 Kirkwood, J. Chsm. Physics, 1940, 8, 205. 15 Kruger and James, J. Chern. Physics, 1954,22, 796.