TRANSITIONS IN SOLIDS AND LIQUIDS By L. A. K. STAVELEY M.A. (FELLOW OF NEW COLLEGE OXFORD AND UNIVERSITY DEMOXSTRATOR IN CHEMISTRY) IT has long been known that many substances exist in more than one solid form such that a t constant pressure one form changes into another a t a constant temperature just as a pure substance passes isothermally from the solid to the liquid statte a t the melting point. The thermodynamic descrip- tion of such transitions is simple. At the transition point the Gibbs free energies G of equal masses of each of the two solid forms are equal the free- energy curves (at constant pressure) intersecting as shown in Fig. 1 (a). The change from the low-temperature to the high-temperature form takes place with the‘ absorption of latent heat and so is accompanied by a sudden entropy increase and there is also an abrupt volume change.The effect of pressure on the transi- tion temperature is governed by the Clapeyron-Clausius equation. dp/dl’ = AS/(VI - VII) . ( 1 ) There also exist transitions in pure solid substances which do not take place sharply but over a range of tem- perature even though the major part of the change whatever it may be is 2 FIG. 1 often concentrated into a small tern- T h e elutions ship between free energy (a) (b) second and ( c ) third order. perature region. Such transitions are and tenLPerature in transitions now known to occur in substances of widely different chemical types. They are found for example in ammonium salts in condensed gases such as methane and hydrogen bromide and in alloys. They also include Curie-point phenomena in ferromagnetic materials and the analogous effects which take place in ‘‘ Seignette-electric ” sub- substances.In addition the one known transition in a purely liquid sJrstem that between the two forms of liquid helium is of this kind. Few generalisations can be made about these transitions but they are alike in one respect of great practical and theoretical importance narnely that they are all accompanied by an anomaly in the specific heat. As the low-tem- perature form ofthe substance is heated the heat capacity cp starts to become greater t8hm would be expected from its previous trend with temperature. The rise becomes more and more rapid until a maximum value is reached 65 E 66 QUARTERLY REVIEWS (Fig. 2). Sometimes the subsequent fa,ll is continuous sometimes apparently discontinuous though even if the drop is discontinuous the " normal " curve is usually not resumed a t once.Just above the temperature a t which it reaches its maximum the heat capacity lies slightly above the extrapolated normal curve. This behaviour is described as anomalous since it is not possible for the normal heat capacity due to the progressive excitation of the lattice vibrations and of the internal vibrations in molecules or ions to decrease with rising temperature.l The coefficient of expansion in a gradual transition likewise shows an anomaly similar to that in the specific heat except that it sometimes takes the form of abnormally small ( i e . negative) values in the transition region. TPK.) FIG. 2 The variation with temperature of c (ref. 36) and c, (ref.25) in the gradual transition in ammonium chloride. There is however considerable variability in these anomalies from one substance to another. Sometimes the heat capacity a t its maximum is only a few calories above the normal (extrapolated) curve sometimes it reaches N 100 cals./mole or even immeasurably high values. The observable range of the anomaly may be restricted to a few degrees and its main part t o a fraction of a degree only ; or it may spread over more than loo" parti- cularly in transitions in alloys. In calorimetric and dilatometric studies the transition temperature is taken to be that a t which cp reaches its maximum value and the coefficient of expansion its maximum or minimum value as the case may be. R. H. Fowier and E. A. Guggenheim '' Statistical Thermodynamics " (C.U.P.1939) p. 147. STAVELEY TRANSITIONS IN SOLIDS AND LIQUIDS 67 The thermodynamic description of these gradual transitions was first attempted by P. Ehrenfest.2 At a sharp transition we have two physically distinct phases in equilibrium with eqiaal free energies G and it follows from the relations that there is a sudden alteration in entropy and volume. Ehrenfest con- sidered what the thermodynamic consequences would be if at a transition there is equality of the free energies of the two forms and of their first differential coefficients as well but now a discontinuity in their second differential coefficients. (The choice of the rather ambiguous word ‘‘ form ” in the last sentence is deliberate. We shall see that there are cogent reasons for avoiding the word “ phase ” in dealing with gradual transitions.) At constant pressure the relationship between the free energies of the two forms will then be as shown in Fig.1 (b). At a certain temperature the curves will touch with the same slope but with different curvatures. At this temperature from the relations there will be a discontinuity in specific heat and coefficient of expansion. If now a(AG)/aT and a(AG)/i3p (where AG = GIr - GI) are to remain zero for changes in T and p we have the equations (aC/aT)p = - S (aG/ap)T = V . (2) aw/aT= = - c~/T aw/arrap = av/aT . * (3) and whence from (3) and the additional relation a2G/ap2 = aV/ap we have Such transitions in which there is a discontinuity in the second differential coefficients of C can be called second-order transitions in contrast to sharp transitions which are of the first order.E. Justi and M. voii Lsue 3 raised doubts about the possibility of the existence of second-order transitions pointing out that one form would always have the lower free energy [I in Fig. 1 ( b ) ] and hence that the transition I -+ II could never be observed. As the simplest way out of this apparent difficulty they suggested that there is a sudden change not in the second but in the third differential coefficients of G so that a t constant pressure the free energy relationship would be as shown in Fig. 1 ( c ) . There would then be no discontinuity in ep or in cc (the coefficient of expansion) but the curves of these quantities plotted against temperature would show a sudden alteration in slope a t the transition temperature. Justi and von Laue’s argument has however been criticised on the grounds that it is not permissible to regard the two forms of the substance separated by the anomaly as two phases with a conceivable existence on both sides of the transition point ; in other words 2Proc.Acad. Xci. Amsterdam 1933 36 153. a Sitzungsber. Akad. Berlin 1934 237. 68 QUARTERLY REVIEWS that the two forms are such that one passes continuously into the other so that extrapolation of either beyond the transition temperature is meaningless. 4 The equations (4) can in principle be subjected to experimental test the first of them being the more convenient. Unfortunately the necessary data are only available for a few substances and moreover it is difficult in any case to obtain accurate figures for the sudden changes in heat capacity and coefficient of expansion at the transition temperature.The test has however been applied t o the transition in liquid helium. Here within the limits imposed by existing experimental technique there does appear to be a discontinuous drop in cp (of 1-9 cals./mole) at the temperature (2.19" K.) a t which the lom-temperature form of liquid helium (11) finally passes into the high-temperature form I. According to W. H. Keesom and A. P. Kee~oni,~ this fall in c certainly occurs within 0.002" and probably within 0.0002". By using experimental figures for the rate of change of the transition point with pressure (= dp/dT) and for the coefficient of expansion of liquid helium I W. H. Keesom calculated from equation (4) a value for the coefficient of expansion of helium I1 a t the transition temperature which was in very fair agreement with the experimental figure.The Ehrenfest equation has also been applied to the gradual transitions in methane 7 and amnionium but there are doubts about its applicability to either of these. For methane there is considerable uncertainty about the value of Ac at the transition,* and in fact A. Eucken and E. Bartholom6 con- sidered that the possibility of the drop in cp being bontinuous could not be excluded. For ammonium chloride as we shall see there is evidence that the transition from the low- to the high-temperature form finally reaches completion isothermally or atl least in such a small temperature range as to make an accurate evaluation of Ac and Am impossible. In some gradual tranfiitions there definitely does not appear to be an abrupt fall in the heat capacity ELS for example in ferromagnetic elements in the neighbourhood of the Curie point.At this temperature the heat capacity reaches a somewhat pointed maximum and then declines con- tinuously to normal values. This behaviour is consistent with that required of a third-order transition and specific heat and coefficient of expansion anomalies in ferromagnetic elements have in fact been treated thermo- dynamically as belonging to this class.l* The first applications of statistical mechanics to gradual transitions dealt with the so-called order-disorder phenomena in certain alloys,t of J. E. Mayor and S. F. Streeter J. Claena. Physics 1939 7 1019 ; W. H. Iieesom Proc. Acad. Sci. Atnsterdam 1933 36 147. K. CIusius and A. Perliclr 2.physikal. Chena. 1933 B 24 313. Ciiftingen Nachr. Math.-Phys. K1. 11 1936 2 51. " Helium " (Elsevier 1942) Chap. 5. Physica 1935 2 557. 8E. 0. Hall Physical Rev. 1947 71 916. lo E. F. Lype Physicat Reu. 1346 69 653. ' Sce ref. (9) p. 5.5. t See F. C. Nix and ?V. Schockley Rev. Mod. Plysics 1938 10 1 for an excellent review of this sithject. STAVELEY TRANSITIONS I N SOLIDS AND LIQUIDS 69 which /?-brass (CuZn) is a simple example. The nrcture of the transition has been established by X-ray studies. In /I-brass both above and below the transition the atoms are situated a t the points of a body-centred cubic lattice. At low temperatures the structure can be regarded zs consisting of two interpenetrating simple cubic lattices one of copper atoms the other of zinc so that each copper atom is surrounded by eight zinc atoms and vice versa.Above the transition however there is no discrimination on the part of the atoms as to which lattice points they occupy so that on the average either kind of atom has four coppers and four zincs as nearest neighbours. Certain planes in the high-temperature disordered structure resolve themselves in the low-temperature ordered form into planes alter- nately consisting of each kind of atom only with different scattering powers so that X-rays are diffracted from adjacent layers of this kind with equal intensities at high temperatures and with unequal intensities a t low tem- peratures. Diffraction from these planes at low temperatures produces lines in the diffraction pattern (" superlattice lines) due to the incomplete cancellation of out-of-phase beams which are absent from the diffraction pattern of the high-temperature form.This evidence shows that it is the ordered form in which the potential energy is least but thermal agitation tends increasingly to produce the disordered state and the degree of order at any given temperature depends on a balance between these opposing factors. The definition of the degree of order s in the treatment of the problem given by W. L. Bragg and E. J. Williams 11 may be illustrated as follows. To produce p-brass we could take a body-centred cubic lattice with each of its n lattice points occupied by a copper atom and then replace half of them by zinc atoms. To obtain the completely ordered alloy the replace- ments would have to be made a t certain definite lattice points only (" correct " positions) n/Z in number.If now when the substitutions are made the average chance that a zinc atom is placed at a '' correct " position is p then s is defined as (the actual value of p ) - (value of p for complete disorder) (value of p for complete order) - (value of p for complete disorder) _______ = ( p - $)/(1 - 3) = 2(p - $) Changing the places of a copper and a zinc atom in the completely ordered alloy will involve an increase of potential energy W, but for the completely disordered alloy no alteration in potential energy will accompany Ghe interchange. In any intermediate state $he increase in potentiad energy JV will lie between 0 and Wo and will be a function of s. In Bragg and Williams's treatment W was taken to be a lingar function of s. But there will clearly be another relation involving s W and T since in a sense the transition is analogous to a chemical reaction with s playing the part of an equilibrium constant and W that of a heat of reaction (with the peculiarity that the " heat of reaction '' varies with the extent to which the reaction has taken place).This second Hence s = 1 or 0 for complete order or disorder respectively. We now come to a most important point. 11 Proc. Roy. SOC. 1934 A 145. 609 ; 1935 A 151 540. 70 QUARTERLY REVIEWS equation can be derived by the methods of statistical mechanics and fkom the two equations involving W 2nd s the equilibrium value of the degree of order at any temperature T can be determined. It is found that s falls more and more rapidly with increasing temperature until it becomes zero a t a temperature T,.(For alloys of the type AB such as CuZn the fall to zero is continuous. For some systems of the type AB, such as CU~AU s after a continuous fall drops ubmptZy to 0 at T, so that here the final disappear- ance of order is accompanied by the absorption of latent heat.) The cal- culated values of the anomalous specific heat below T are of the right order of magnitude. The theory accounts successfully for the essential features of the transition except that it predicts that the system would have a normal specific heat immediately Tc is passed whereas in fact for a short range above T the cp values are still slightly greater than normal. The reason for this is that the degree of order as defined above is an average quantity which relates t o the crystal as a whole.At all temperatures however there is a tendency for the immediate environment of a given atom to be such that the potential energy of this atom and its neighbours is a minimum and the local order to which this tendency gives rise will in virtue of its contributory potential-energy term manifest itself in an addition to the specific heat decreasing in value as T increases. N. Bethe 1 2 and R. Peierls l3 have presented treatments of order-disorder transitions in alloys in which the degree of order CT is defined with reference to an atom and its nearest neighbours only ; CT is thus a measure of the short-range order which unlike s does not become zero at Tc. From this starting point the slightly anomalous specific heat above T can be accounted for quanbitatively. There is some similarity between Curie-point phenomena in ferro- magnetic substances and transitions in alloys.The atoms in a ferro- magnetic substance are elementary magnets with a preference for a common orientation which at sufficiently low temperatures results in a state of permanent magnetisation. As the temperature rises thermal agitation tends increasingly to overcome this common orientation. The disappearance of permanent magnetisation at the Curie point is accompanied by a maximum in the specific heat. Of all gradual transitions those occurring in molecular and ionic solids have probably the greatest interest for the chemist. The majority take place below room temperature and the discovery of many of them has been a consequence of systematic low-temperature calorimetry. Usually the compound contains hydrogen in its molecule or one of its ions and it may possess two or even three such transitions.I n finer points of detail these gradual transitions show great sensitivity to the chemical nature of the compound in which they occur. Even isotopic replacement can cause qualitative changes. The first to be definitely discovered was that in ammonium chloride by F. Simon,l* R. Ewald l5 having previously observed that the heat capacity of this substance below room temperature was laproc. Roy. XOC. 1935 A 150 552. =&Ann. Phyaik 1922 68 241. l S I b i d . 1936 A 154 207. '"bid. 1914 44 1213. STAVELEY TRANSITIONS IN SOLIDS AND LIQUIDS 71 anomalous. They have since been found to exist in numerous ammonium and other salts and in many condensed gases and organic compounds.A considerable stimulus to their experimental and theoretical investigation resulted from L. Pauling’s suggestion 16 that they might mark the onset of molecular or ionic rotation in the crystal lattice. The possibility of such rotation seems to have been first considered by P. Simon and C. von Simson l7 to account for the cubical symmetry of the high-temperature form of hydrogen chloride. Pauling applied the Schrodinger equation to a molecule the potential energy of which was assumed to be a periodic function of its orientation. At low temperatures the molecule executes torsional oscilla- tions. At high temperatures it rotates non-uniformly a t first but more and more smoothly as the temperature rises. The change from the first type of motion to the second is favoured by a small moment of inertia of the molecule or ion and by a low barrier separating two minima in the potential energy.Pauling concluded that molecules like the hydrogen halides and methane should be capable of rotation in the crystal lattice a t temperatures below the melting point whereas this would not be true of a molecule like iodine with its much larger moment of inertia. He suggested that in solid hydrogen halides the change from torsional oscillation to rotation takes place a t a transition and predicted that at these transitions the dielectric constant would show a marked increase. This prediction has since been verified,l* and similar effects observed a t transitions in many other _polar substances.l9 This association of transitions in molecular and ionic solids with the change from librational to rotational movement undoubtedly provides a simple explanation of many of the facts.20 In the first place we can easily understand why so many of these transitions are gradual.At low tempera- tures the molecules are constrained to undergo torsional oscillations by the directed nonspherical field of force acting on them. If a molecule com- mences to rotate however the resulting increase in symmetry in its own field of force weakens its orientating influence on its neighbours and makes their rotation an easier process. In other words we have here as in order- disorder transitions in alloys a change which becomes progressively less exacting in its energy requirements as it proceeds. Like the atomic re- arrangements in alloys it will be a co-operative phenomenon giving rise to abnormal physical properties over a range of temperature.In sharp transitions where there is evidence that molecular rotation sets in a t the transition point it may be that this does not occur a t all a t lower tempera- tures and that the conditions which make it possible are created by the change in crystal structure a t the transition. But it is also possible that just below the transition point a few molecules or ions begin to rofate with 16 Physical Rev. 1930 36 430. l8 ( a ) R. M. Cone G. H. Dcnnison and J. D. Kemp J . Amer. Chem. SOC. 1931 53, Is C. P. Smyth Chem. Reviews 1936 19 329 ; Faraday Society Discussion 013 ,OA. Eucken 2. Elektrochem. 1939 45 126. 2. Physik 1924 21 168. 1278; (b) C. P. Smyth and C. S. Hitchcock ibid. 1933 55 1830. Dielectrics 1946 175.72 QUARTERLY REVIEWS such an effect that the lattice is compelled to alter radically before the incipient rotation has given rise t o a detectable anomaly in such properties as specific heat and coefficient of expansion. Later we shall see that i t is unwise to regard sharp and gradual transitions as being funda8mentnlly different since some appear to have the character of both in that they commence gradually but probably reach completion isothermally. Evidence in support of Pauling's theory comes from a consideration of the temperatures a t which transitions occur in substances of different chemical types. The temperatures at which cp reaches its maximum in the gradual transitions in methane hydrogen bromide ammonium chloride and sodium nitrate are 20.42" 59-75" 242.7" and 548" K.respectively. (Hydrogen bromide has two more transitions some 25 " higher.) In methane the intermolecular action is due almost entirely t o weak London dispersion forces but in hydrogen bromide there are stronger orientating forces between the permanent dipoles. In amrrionium chloride and sodium nitrate there are still more powerful interionic effects. The nitrate ion has a much larger moment of inertia than the ammonium ion. It is therefore readily under- standable that on passing along these four substances there should be an increase in the temperatures of the transitions if these are due to the com- mencement of rotation of the molecules of the first two and the cations and anions of the third arid fourth. Two o6her interesting series are provided by water hydrogen gulphide and hydrogen selenide and ammonia phos- phjne and arsine.I n each of these groups the first substance (with the most strongly polar molecule) only exists in one solid form a t ordinary pressures. I n the other two compounds of each series the lowest transition occurs in that which has the smallest dipole moment. For gradual transitions which are seldom accompanied by more than a slight change in crystal structure the entropy change rarely exceeds 2 e.u. (cals./mole/o) whereas for a sharp transition i t is often greater. A sub- stance possessing either kind of transition however invariably has a lower entropy of fusion than a substance which is chemically similar but non- polymorphic. Thus the entropy change at the sharp transition (at 225.35" K.) in carbon tetrachloride 21 is 4.86 e.u.and that on melting (at 250.3" K.) is 2.4 e.u. while the entropy of fusion of silicon tetrachloride 22 (which has no transition) is 9.1 e.u. Sonietimes the entropy of transition is so much larger than that of fusion that the high-temperature form of the solid must already in a sense be very " liquid-like ". For cyclohe~anol,~~ for example the entropy increases by 7-44 e.u. at the transition at 263-5" K. and by only 1.37 e.u. at the melting point (297.0" K.). This state of affairs is quite comprehensible if a t the transition the molecules acquire the orientational freedom which otherwise is only gained when the solid melts. Likewise if the dielectric constant increases considerably a t a transition in a polar substance its further change on melting is usually only small.Nevertheless it is wrong to conclude that uZZ gradual tramitions in 21 J. I?. G. Hicks J. G . Hooky and C. C. Stephenson J . Amer. CJ2t.m. Soc. 1944 2 a W. M. Latimer ibid. 1922 44 90. 66 1064. 33 R. K. JCelley ibid. 1920 51 1400. STAVELEY TRANSITIONS IN SOLIDS AND LIQUIDS 73 molecular and ionic solids are associated with the change from torfiional oscillation to comparatively €ree rotation of molecules or ions even about one axis only. We shall see that experimental evidence while indicating that in some lattices these particles may have almost the same rotational freedom that they enjoy in the gaseous state can for other solids only be reconciled with the persistence of librational movement both above and below the transition so that here the increase in disorder is to be attributed not to the change from libration t o rotation but from libration about ordered axes to libration about disordered axes.Consequently the indiscriminate application of the term “ rotational transition ” to these gradual changes in the kind of solid we are discussing is not justifiable just as the expression “ second-order transition ” likewise takes too much for granted. Less objectionable terms are “ lambda-point ” and “ ammonium chloride-type transition ”. The first of these was coined by P. Ehrenfest and was sug- gested by the shape of the cp anomaly in l i q ~ i d helium. The first attempt to account quantitatively for specific heat and coefficient of expansion anomalies assumed to be due to incipient molecular rotation in crystal lattices was made by R. H. Fowler.24 His treatment was similar to that which Bmgg and Williams had applied to order-disorder transitions in alloys.A molecale wits msumed to be in a potential field of - ‘clr cos 8. An order parameter s was introduced which was regarded aR representing the average degree of non-rotation of the molecules and for the all-important dependence of JV on s Fowler assumed that they were related by the equation W = Was. The calculated values of the heat capacity refer of course to constant volume and cannot strictly be com- pared with directly determined cp values. For ammonium chloride however experimental values of cv are now available 25 which show that the anomalous rise in cv is much less pronounced than that in cp (see Fig. 21 and indeed is much what would be expected from Fowler’s theory.Actual specific-heat anomalies are usually concentrated into a smaller temperaixre range than the theory predicts for which the neglect of the effect which any volume change accompanying the transition may have on the potential energy is probably partly responsible. 2 8 T 2 7 Further theoretical work in this field has been carried out by J. G. Kirkwood 27 and K. Schafer,28 both of whom have treated the transition as involving a change from a preferred to a random molecular orientation rather than from non-rotation to rotation. An essential point in Schafer’s theory is that the unit to which statistical considerations are applied is not the whole crystal but a domain consisting of comparatively few molecules (perhaps - lOOO) so that the interaction energy of the molecules in it is a function of the size of the domain.One feature of these transitions has yet to be mentioned namely the hysteresis which often (though by no means invariably) accompanies them. 24 Proc. Roy. Soc. 1935 A 149 1. 25 A. W. Lawson Physical Rev. 1940 57 417. 26 0. B. Rice J . Chern. Physics 1037 5 492 ; cf. R. Eisenschitz Proc. Roy. Icoc. 2 7 / b i d . 1940 8 205. 2 8 Z . physikal. Cliem. 1939 B 44 127. 1938 A 168 645 ; H. Bethe and J. G. Kirkwood J . Chen~. Physics 1939 7 578. 71 QUARTERLY REVIEWS The temperature of a transition showing hysteresis is higher when approached from the low-temperature side than that at which the reverse change occurs on cooling (Fig. 3). Some very careful investigations of this phenomenon have been made from which i t appears that when it exists it shows no sign of vanishing in a reasonable period of time and that it is unaffected by the measures which normally assist the establishment of equilibriuni.(Thus the hysteresis in the transition in sulphur hexafluoride is not altered by the presence of liquid carbon tetrafluoride in which it is somewha,t soluble.) 29 Although it is by no means fully understood it is not unlikely that it is connected with the mosaic structure of the crystal. Schafer 28 has applied his theory to the problem and suggests thaf the ordered domains first formed on cooling the solid through the transition will be smaller than the larger ones produced laber on and hence have dizerent properties and a different transition temperature. Another view is that the volume changes occurring in small regions of the crystal cause strains to be set up and that the hysteresis arises from t'he effect of the resulting pressure on the transition temperature.3O Different forms of a solid a,re designated as I 11 111 ctc. I being the form stable between the melting point and the highest transition temperature I1 that existing between this and the next lower transition temperature and so on. Reference will be made to the recently developed nuclear magnetic resonance method of deciding whether or no molecules or ions rotate in crystal lattices which has been applied to compounds containing hydrogen atoms.31 This depends on the fact that the effect of an applied magnetic field on the spins of the protons is to cause them t'o adopt parallel or anti-parallel orientations with respect to the field. The energy diEerence between the two orientations varies with the Geld strength and for the strengths employed corresponds to a quantum of radiation of frequency of the order of ten metres.I n the presence of the magnetic field the substance absorbs radiation of the correct frequency but if the protons are present in a lattice in molecules or ions which are not rotating there will exist in the crystal an inhomogeneous magnetic field which superimposed on the applied field will result in the absorption of radiation over a frequency range. If the molecules or ions rotate with a frequency greater than that of the radiation the internal field will be effectively homogeneous and the range of frequencies absorbed much smaller. The change from libration to rotation will therefore be manifested by a decrease in the width of the absorption line.Ammonium Halides.-In Table I are given the absolute temperatures entropy changes A 8 (in cals./mole/") and volume changes AV (in c.c./mole) for the transitions in light and heavy ammonium halides. The entry in the column headed " f1 " gives the width of the hysteresis loop in degrees. We shall now consider some typical transitions in detail. 23A. Eucken and E. Schrocler Gotlingen Nmhr. Math.-Phys. K1. 11 1938 3 65. 30 F. C . Frank and K. TVirtz NuturuGss. 1938 42 657 ; P. Dinichert Helv. Physica 31 F. Bitter N. L. Alpert H. L. Poss C. G. Lehr and S. T. Lin Physical Rev. Acta 1944 1'7 389. 1947 71 738. STAVELEY TRANSlTIONS IN SOLIDS AND LIQUIDS 75 '' No " means that the transition does not show hysteresis. The reference numbers indicate the murce of the information in $he column above them.NH,Cl . ND,Cl . NH,Br. ND,Br . NH,I ND,I . Ref. . TABLE I T OK. 457 448 414 405 258 32 - TIT + 11. 1 IV -f 111. I AV + 5.26 - + 6.34 + 8.14 34 - - T OK. 242.7 249.3 234.4 215 233 229 32,35 A S 0.82 0.34 0.30 36 - - - AV N + 0.16 - + 0.13 N - 0.06 N - 0.17 - - 0.1 N - 0.1 35 a ,-- 0.27 NO - 0.06 ,- 0.15 N O NO 35 T OK. All of the transitions 111 + I1 and IV + IT1 are gradual. Sometimes the anomalies in specific heat and coefficient of expansion extend for as much as 30" or 40". The I and I1 forms of these salts have face-centred and body- centred cubic lattices respectively. II-NH,Cl passes into III-NH,C1 with no fundamental lattice ~hange,~7 3t3 but 111 is piezoelectric 39 and therefore of lower symmetry than 11. The very careful dilaiometric study by A.Smits and C . H. MacGillavry 35 revealcd that a considerable part of the volume change in the I11 + I1 transition occurs within a few hundredths of a degree (see Fig. 3). This important observation suggests that this transition begins gradually in one and the same phase but that eventually it is perhaps completed isothermdly. By contrast thc III + I1 transition in ND,Cl takes place continuously with no sudden volume change and without hysteresis (Fig. 3). III-NH,Br on the other hand is doubly refracting and has a tetragonal structure though only slight displacements of the ions fi-om their positions in the body-centred cubic II-form are 379 Here also much of the anomalous volume change takes phce in an extremely small temperature interval-within 0-Ol" according to A.Smits J. A. A. Ketelaar and G. J. M ~ l l e r . ~ ~ ND,Br provides one of the few examples of a qualitative 3 8 K. Clusius A. Kruis and W. Schanmr 2. anorg. C'hern. 1938 236 24. 33H. Klindhardt Ann. Physik 1927 84 167. 36 A. Smits and C. H. MrzcGillavry 2. physikal. Chem. 1933 A 166 97 (NH,Cl) ; ,4. Smits G. H. Mnller and F. A. Kroger ibid. 1937 B 38 177 (ND,CI) ; A. Smits J. A. A. Ketelaar and G. J. Muller ibid. 1936 A 175 359 (NH,Br) ; 9. Smits D. Tollonaar and F. A. Kroger ibid. 1938 B 41 215 (ND,Br) ; A. Smits and G. J. Muller ibid. 1037 B 36 140 (NH,T) ; A. Smits and D. Tollenaur ibid. 1942 B 52 222 W. Bridgman Proc. Aiizer. Acad. 1916 52 55. tND,I)- 3aF. Simon C. von Simson and M. Ruhemann ibid. 1927 A 129 339. 37 J. A. A. Iietelaar Nature 1934 134 250.38 P. Dinichert Helv. Physica Acta 1942 15 462. 39 A. Hettich 2. physikal. Chenr. 1934 A 16S 333 ; S. Bahrs and J. Engl Z. Physik ao J. Weigle and K. Saini Helv. Physica Ada 1936 9 515. 1937 105 470. 76 QUARTERLY REVIEWS difference between hydrogen and deuterium compounds in tfint8 i t possesses another transition at a stiil lower temperature a t which i t reverts to a body-centred cubic lattice.35 None of the forins of ND,Rr is piczoeleetric,al and as there is nothing to distinguish forms P I and TV Smits has called the cha'nge I11 + IV a retrograde transition and has discussed i t in terms of his theory of allotropy. This transition is accompanied hy an unusually wide hysteresis loop (- 9"). I<. Clusius 4 2 has referred t o some unpublished experiments on mixed crystals of light and heavy ammonium bromide which TPC.) Fro.3 l'he anriation with temperature of the molar volumes of NH&1 and ND,Cl (ref. 35) in the %final stages of th,e gradual transitions in these substances. showed that the supercooling which precedes the formation of IV from 111 increases with the hydrogen content cntil finally the transition can no longer be observed. The I11 = I1 transitions in NH,I and ND,I involve the same structural change as for NH,Br,35s 37 but they are more extended free from hysteresis and continuous throughout. By measuring Young's modulus and the rigidity modulus of compressed rods of ammonium chloride A. W. Lawson 25 obtained values for the isothermal compressibility #? above and through the transition region. He also determined the coefficient of expansion a and using the relation cp - c = Ta2V//3 calculated c (see Fig.2). The anomaly in c is very much less marked than that in cp (never exceeding - 4 cals./mole) and just above the transition c is 18 cals./mole. The 6 cals./mole left after deduction of 12 for the lattice vibrations of the NHf and C1' ions (which at this tem- perature make a classical contribution of R per degree of freedom) is exactly the amount to be expected if each of the ammonium ions executes torsional oscillations in three degrees of freedom whereas their free rotation would contribute 3 cals./mole to c, making a total of 15. This strong evidence against free rotation of the NH,+ ions in form I1 is supported by the nuclear l A . Smits and P. G. Mesrmann 2. plzysiknl. Chem. 1941 B 49 13. r z Z . Natr~rforsch.1946 1 142. STAVELEY TRANSITIONS 1N SOLlYS AND LIQUIDS 77 magnetic resonance method 31 the width of the broad absorption line shows no change in passing through the transition. It would therefore seem that in NH,Cl a t least both above and below the I11 s I1 transition the cations undergo torsional oscillations presumably about ordered axes in I11 and disordered axes in 11. Whether the motion more nearly approaches free rotation in the high-temperature form I remains an open question. In studying the Raman effect in ammonium chloride bromide and iodide a t temperatures down to - 150° A. C. Menzies and H. R. Mills 43 discovered that with the transformation into I11 there appears in the spectrum of the chloride (but not of the other two salts) il line of low frequency (183 cm.-l).This they attributed to asymmetric lattice vibra- tions of the cations with respect to the anions. They suggested that the equilibrium position of an ammonium ion is such that each face of the tetrahedron is perpendicular to a cube diagonal and that the low-tempera- ture form of ammonium chloride is built up by simple translation of an elementary cube. The halogens are then not symmetrically placed with respect to the ammonium ions so that the low-frequency Raman line is accounted for but the asymmetry would vanish if the cations were to undcrgo torsional oscillations of " moderate amplitude ". On the other hand by constructing the lattice from elementary cubes so that the tetrahedra are arranged antisymmetrically a group of eight would possess complete cubic symmetry.Such an arrangement in the structure of III-NH,Br and III-NH,I is consistent with the absence of a low lattice vibration frequency from their Raman spectra. (A similar hypothesis had been advanced by A. Hettich 39 from a consideration of piezoelectric properties.) Menzies and Mills showed that on this basis a plausible explana- tion could be given of the opposite sign of the volume changes a t the I11 + I1 transition in ammonium chloride and bromide. Tetrahydrides of Carbon Silicon and Germanium. Table I1 gives the absolute temperatures and heats of transition and fusion for the compounds so far investigated. The values of AH for the transitions (except that in CH444 . . CD,44 . . SiH,45 . . GeH446 . . CH,D44. . I III-+II. I I1 + I. T O x . - 15-88 22-25 13-20 - AH - 13-6 19.8 130.7 - 1' OR.20-42 23.19 27.10 63.45 76.55 a Ti 1 5 7 44.4 58.7 166 129.6 Fusion. T OK. 90-64 90.42 89.78 107-26 88-48 1 AH 224 217.5 215-7 159.6 199.7 4 3 Proc. Boy. SOC. 1935 A 148 407. 4 4 K. Cliisiris and 3,. Popp Z. physikul. G'Iwm. 1940 B 46 63 4 5 K. Clusius ibid. 1933 B 23 313. 4 6 K. Clusiim and G. Faber ibid. 1943 B 51 352. 78 QUARTERLY REVIEWS silicane) are the amounts of energy in calories required to heat one mole of the substance through a short temperature range enclosing the transition ; ie. they include the " normal '' contribution from cp for which it is difficult to make an exact allowance. For silicane the figure represents the true heat of transition. All of these transitions are gradual and all show hysteresis? except perhaps that in silicane. The hysteresis in the methane transition has been particularly carefully inv-estigated by A.Eucken and E. Barth0lom6.~ In addition to the transitions recorded in Table 11 III-GeH shows an anomaly in the heat capacity cp rises gradually to a maximum value of 21 cals./mole at 62.9" K. ar,d then falls abruptly according to K. Clusius and G. Faber,46 to 13 cals./mole. No simultaneous change can be detected optically (by means of a polarisation microscope) and the anomaly may perhaps be an example of a genuine second-order transition. An X-ray investigation of methane by H. H. Mooy 4 7 gave no evidence of any alteration in the face-centred cubic structure a t the transition. K. Clusius and his colleagues have studied all forms of the above substances with a polarisation microscope.* For the methanes forms I and I1 are all isotropic but III-CH,D and III-CD are doubly-refracting.I-SiH and I-GeH both show weak double refraction which is much more pronounced in the low-temperature forms. The I11 + I1 transition in germane is detectable in this way but only a slight change in optical behaviour accom- panies it. It seems therefore that the II - + I change in germane and silicane produces a high-temperature form of greater internal symmetry? in contrast to this transition in the methanes. Although the transition in silicane begins with an anomalous rise in cP 88(3" of the heat of transition is absorbed within lo and it may well be that the change begins gradually but reaches completion in a first-order phase change. It will be seen from Table I1 that the entropy change at the transition is greater than the entropy of fusion.The presence of two transitions in mono- and tetra-deuteromethane suggests that it is methane which is abnormal in only having one. In the lattices of these compounds almost identical intermolecular forces prevail and torsional oscillations will have the largest frequencies and hence the greatest zero-point eriergies for the lightest molecules. A potential barrier restricting rotation will have almost exactly the same absolute height in all three lattices and will therefore first be overcome by methane molecules which retain the most energy a t the absolute zero. (It is significant that the temperature of the I1 -+ I transition increases with replacement of H by D. and that AH,,, for CH is very much less than for CD,.) Probably therefore the form III is one in which methane is incapable of existing (at least at ordinary pressures) on account of its high zero-point energy.Interesting experiments were carried out by E. Bartholome G. Drikos and A. Eucken 48 on mixed crystals of CD4 and CH,. They 47 Proc. Acud. Sci. Amsterdam 1931 34 550. 4 8 2 . physikal. Ghem. 1938 B 39 371. * See ref. (46) for a summary of this work. STAVELEY TRANSITIONS M SOLIDS AND LIQUIDS 79 found that the temperature of the upper transition varies linearly with the deuterium content (a striking demonstration that a co-operative change is involved) and that the transition never becomes more diffuse than in the pure compounds. The temperature of the lower transition in tetradeutero- methane on the other hand falls more and more rapidly with increasing hydrogen content and the specific heat anomaly becomes less and less marked and finally vanishes a t about 20% CD,.In mixed crystals of methane and krypton 49 the I1 + I transition becomes less pronounced and its temperature lower with increasing krypton content and when this has reached about 30% the transition disappears. This result is readily comprehensible if the transition involves a co-operative librational-rota- tional change for replacement of tetrahedral methane molecules by the spherically symmetrical rare-gas atoms will clearly weaken the forces tending to orientate any one methane molecule and therefore reduce the height of the potential barrier and also make less acute the dependence of this height on the fraction of molecules already rotating. A. Eucken and H.Veith 49 carried out an analysis of the heat capacity of solid methane and solid solutions of methane and krypton and by skilful use of rather fragmentary data for the required compressibilities and coefficients of expansion arrived at values for the rotational heat capacity of the methane molecules in the lattice. Both for the pure substance and for the solid solutions from about 50" K. upwards these are remarkably near the of a classical three-dimensional rotator. Indeed in the mixture richest in krypton its values even a t lower temperatures (where quantisa- tion becomes important) are still very close t o those calculated for gaseous methane. This evidence for attributing rotational freedom to the methane molecules in form I is supported by the nuclear magnetic rcsonance method.31 The absorption band narrows considerably in the neighbourhood of the transition as the temperature is raised.Hydrogen and Deuterium Halides. Some facts about these substances are recorded in Table 111. The values of AH for the transitions in the bromides and iodides have the same significance as in Table 11. The only sharp transition is that in hydrogen chloride from a form of very low symmetry (11) to a cubic structure With the change I1 -+ I the dielectric constant rises abruptly l8 but from a quantitative considera- tion of the polarisation-temperature relationship for phase I G. Mettner 50 concluded that the molecules are not rotating freely in this phase. According t o a provisional report,51 this conclusion is confirmed by the nuclear magnetic resonance method the absorption line being equally broad above and below the transition point.Presumably therefore the molecules in the high- temperature form undergo torsional oscillations about disordered axes but are capable of changing their orientation sufficiently frequently to account for the high dielectric constant. The infra-red and Raman spectra 54 49 Ibid. 1936 B 34 275; 1937 B 38 393. slF. Bitter et al. M.I.T. Quarterly Progress Report Oct. 1947. 6 2 E. Lee 62. B. B. M. Sutherland and C. K. Wu Proc. Roy. Soc. 1940 A 176 493. Ann. Physik 1938 33 141. 80 QUARTEBLY REVIEWS likewise provide no evidence of molecular rotation in the solid but instead show that the condensation of gas to liquid involves a considerable change in the intramolecular vibration frequency comparable with that which the frequency associated with a hydroxyl group experiences when the group enters into hydrogen-bond formation.Thus in the condensed states of hydrogen chloride there would appear t o be marked intermolecular association. TABLE 111" I 1II+ 11. 1 II+1. I I I I 17 O K . I I H C l . . . - DC1. . . - I-IBr. . . 89.76 DBr. . . 93.5 HI . . . 70.1 DI . . . 77.3 AH T OK. A H - 98.36 284.3 - 105.03 320.1 160.1 ' 113'6211 -f E}264.5 llG.86E -+ I 169.7 120.26 303.0 146.8 125.68 359.9 175.6 128.28 386.4 Fusion. T OK. A H 158.91 476.0 158.44 473.2 186.28 675.1 185.62 574.2 222.31 686.3 221.23 684.3 not yet available. The hydmgen halides were studied by W. F. Giauque and R. Wiebe J. Amer. Chem. SOC. 1928 50 101 (HC1); ibid. p. 2193 (HBr); ibid. 1929 51 1441 (HI). Hydrogen bromide unlike any other of the halides including deuterium bromide has three transitions.In mixed crystals of these two bromides the region of existence of the extra form designated E in Table 111 diminishes with increasing deuterium content and vanishes a t 470/ of DBr.42 From the similarity between DBr and HI i t would appear to be HBr and not DBr which is abnormal. From X-ray investigations it appears that hydrogen iodide has a face-centred tetragonal lattice in all its forms but unambiguous conclusions about the structure of hydrogen bromide have not been reached in this way. Polarisation microscope studies on HBr DBr and HI have revealed that I is always isotropic and I1 and I11 aniso- tropic.42~ s3 The dielectric constant of hydrogen bromide rises very rapidly as it approaches the 111-11 transition and thereafter drops almost as rapidly to a fairly steady val.me.lsb> 64 It is not certain whether the mole- cules of hydrogen bromide and iodide can rotate freely in their lattices.They are more likely to do so than those of hydrogen chloride in view of their smaller dipole moments. The 111 + I1 transition in hydrogen bromide exhibits hysteresis which has been carefully examined both thermally 5 5 and by dielectric constant rnea~urements.~~ The width of the hysteresis loop as determined by heating s3 A. Kriiis and R. Kaischew 2. physiknl. Chem. 2935 B 41 427. ji G . DamkiihIer Asm. Physik 1938 31 76. 65 A. Eucken and W. Giittner Gottingen Nuchr. Math.-Phys. K1. 11 1936 2 167. STAVELEY TRANSITIONS IN SOLIDS AND LIQUIDS 81 and cooling curves was found to remain the same even if a sample previously warmed to a point about half-way up the hysteresis loop was inoculated with the high-temperature form or alternatively subjected to supersonic waves for six hours.The dielectric-constant study did in fact show a slight shrinking of the hysteresis loop with time but limiting conditions wcre reached the hysteresis still persisting beyond which no further change was observable. Many other substances besides those discussed above have transitions of considerable interest. They include 1iitrates,5~ perchlorates,57a and other salts 57b in which there is evidence of anion rotation ; resorcinol ; 58 “ Seignette-electric ” substances like potassium dihydrogen phosphate ; 59 and numerous organic compounds with long hydrocarbon chains where the transitions are probably connected with the onset of the rotation of the chain about its axis.6o In addition many new transitions have been observed in a variety of compounds a t high pressures.61 The nature of the transition in liquid helium which of course has nothing to do with molecular rotation is fully discussed in W.H. Keesom’s book on this element. The Reviewer wishes t o thank Dr. A. H. Cooke for helpful discussions and Mr. C. J. Mandleberg for assistance in surveying recent literature. 56F. C. ISracek J . Amer. Chenz. SOC. 1931 53 2609; F. C. Kracek E. Posnjnk and S. B. Hendricks ibid. p. 3339 ; J. B. Austin and R. H. H. Pierce ibid. 1933 55 661 ; C. Finbsk and 0. Hassel 2. physikal. Ghem. 1937 B 35 25. 67 Idem ( a ) ibid. 1936 B 32 130; ( b ) ibid. p. 433. 68 J. M. Robertson and A. R. Ubbelohde Proc. Roy. Sac. 1938 A 167 136. 6B W. G. Cady “ Piezoelectricity ” (McGraw-Hill Book Co. 1946) ; C. C. Stephenson ~t at. J . Amer. Chem. SOC. 1944 66 1397 and following papers ; J. C. Slater J . Chem. Physics 1941 9 16. soA. Miiller Proc. Roy. SOC. 1932 A 138 514; W. 0. Baker and C. P. Smytli. J . Amer. Chem. Sac. 1938,60 1229 ; J. C. Southard R. T. Milner and S. B. Hendricks J . Chem. Physics 1933 1 95. 61 P. W. Briclgman “ The Physics of High Pressures ” (G. Bell and Sons 1931) Chap. 8. F