Proc. Nati. Acad. Sci. USA Vol. 75, No. 10, pp. 4664-4665, October 1978 Chemistry Hysteresis effects of C1H4 and C2H4 in the regions of the solid-phase transitions (thermodynamics/metastability/phase transitions/solid state) SHAO-MU MA, S. H. LIN*, AND HENRY EYRING. Department of Chemistry, University of Utah, Salt Lake City, Utah 84112 Contributed by Henry Eyring, August 10, 1978

ABSTRACT The thermodynamic model of hysteresis in the deviation from the normal equilibrium condition due to the phase transitions based on the regular solution theory previ- hysteresis effect. Expanding the right-hand side in terms of AX ously developed, was applied to the solid phases of C 1H4 and C2H4. The width and height of the hysteresis loops for these and neglecting the higher order terms, we obtain systems were calculated and compared with the experimental = + - + [3] data; agreement was satisfactory. G GO- AIAAX (2kT F)AX2 4/3kTAX4, in which C0 = '/2(!'A + ,UB) + '/4F - kT In 2 and AA = ILA In a previous paper (1) a thermodynamic model of hysteresis - AB. in phase transitions based on the regular solution theory was When the system is at equilibrium, we set OG/OAX =0. The developed. Hysteresis in this model is derived from invoked result is metastable states and from an interaction term F, which de- scribes the net interaction between systems of the two states -Ajt + 2AX(2kT-r) +'6/3kTAX3 = 0. [4] involved in the phase transition. Below the critical temperature, Eq. 4 normally yields three Theoretical studies of the phase transitions in C1H4 and C2H4 distinct roots; one describes the stable equilibrium, the second are numerous (refs. 2-5; one may refer to ref. 5 for further refs. describes the metastable equilibrium, and the third describes up to 1977). No theory, however, has attempted to describe the unstable equilibrium. At the end point of the metastability, quantitatively the observed hysteresis effects (6) in the upper Eq. 4 yields one double root and one single root (1). Thus for transition in C1H4 (20.4 K) and in the two transitions in C2H4 our purpose of calculating the width of hysteresis, we rewrite (22.1 K and 27 K). In ref. 6 the authors suggested that, if the true Eq. 4 as curves ascending and descending boundary would be measured - - = 0 by supplying energy intermittently and by heating and cooling 16/3kT(AX a)(AX ()2 [5] continuously, the width of the hysteresis loop for C2H4 would and obtain be about 0.1 K for the III -1 II transition and about 0.01 to 0.02 K for the II -k I transition. By contrast, they also observed, by (3 = -a/2 = ±1I2[(Tc/T) - 1]1/2 [6] means of the plots of heat capacity of Cp against temperature and T, that the widths of these transition regions are in the reverse 32A(33 = = + TAS°, order. -Ar -AlH [7] The object of the present work is to apply the thermodynamic in which r = 2kTc was chosen. model previously proposed to the hysteresis effects in solid C1H1 For ( = 1/2[(Tc/T) - 1]1/2, we have and C2H4. 4/3[(Tc/T) - 1]3/2 = - + k [8] THEORY AT Consider A and B to represent two solid phases whose trans- If we set T = Te + AT/2, AHl = TeAS0, then Eq. 8 be- formation exhibits hysteresis. The Gibbs free energy of the total comes system near the transition point can be written as 4/3 (r+r )3/2 AS0, ASO = 1+ G = XAAA + XBAB + rXAXB + kT(XA In XA T + [9] + k(l+ k XB lnXB), [1 \Te AT/2) in which XA, XB, IAA, and AUB are the mole fractions and chemical potentials of phases A and B, respectively; F repre- or sents the net interaction energy between A and B; k is the = 4 [10] Boltzmann constant. 2 3ASk~e[(Tc/Te) -1]3/2, If we set XA = 1/2 + AX and XB = 1/2 - AX, then in which Te refers to the equilibrium temperature. G = '/2(AA + AB) + AX(AA - IB) + F('/4 - AX2) Similarly, for (3 -=-'/2[(Tc/T) - 111/2, we obtain AT 4 kTe [(Tc/Te) - ]3/2- + kT1/2 In 1/4(1-4AX2) + AX In + 2AX] [2] [11] Notice that, at XA = XB = 1/2, one obtains the equilibrium The width of the hysteresis would be AT; i.e., condition without the hysteresis. In other words, AX represents 8 kTe AT = [12] AS~ - The publication costs of this article were defrayed in part by page 3T (cl 113/2, charge payment. This article must therefore be hereby marked "ad- vertisement" in accordance with 18 U. S. C. §1734 solely to indicate * Present address: Department of Chemistry, Arizona State University, this fact. Tempe, AZ 85281. 4664 Downloaded by guest on October 2, 2021 Chemistry: Ma et al. Proc. Nati. Acad. Sci. USA 75 (1978) 4665 Table 1. Numerical calculations AvT, K AE, J Te, K Te, K Te/Te Calc. Obs. (6) Calc. Obs. (6) C'H4, upper transition 20.4 20.22 1.01 0.067 0.04-0.06 7.3 '-5 C2H4, lower transition 22.1 21.88 1.01 0.1 0.1 9.1 -5 21.76 1.016 0.2 7.4 C2H4, upper transition 27 26.73 1.01 0.08 0.01-0.02 13.9 -5 26.89 1.004 0.02 8.9

in which AS' is chosen to be positive. from these entropy data and are given in Table 1. Notice that Now we can calculate the hysteresis loop widths of other when Tc/Te = 1.01 was used, the calculated width of the loop thermodynamic quantities. From Eq. 3, the expressions for of C'H4 and that of the lower transition of C2H4 agree well with entropy the observed values. The calculated loop width of the upper transition of C2H4 is slightly higher than the observed one. Our S= (A) _ (G) calculated heights of the loop of C1H4 and C2H4, are at least Y9TJ kaT/p 50% higher than the observed data. However, the present model = s0 - ASO AX - 2kAX2 - 4/3kAX4 [13] predicts that, in the case of C2H4, the width of the hysteresis loop of the upper transition is narrower than that of the lower and internal energy transition, in agreement with those given in ref. 6, using heating E = A + TS G. + TS and cooling processes. We feel the above agreement is quite =(AO + TSO)-(AA + TAS0)AX-PAX2 satisfactory in view of the simple model of van der Waals type that was used. Without modification, our model can only pro- =°-E0 H0AXI-zX2 [14] vide the gross features of the main hysteresis loop between two can be derived. phases. In other words, we cannot explain the existence of At equilibrium without the hysteresis effect E = f'. In the scanning loops that are often observed in the hysteresis phe- presence of the hysteresis effect, in addition to Eq. 14, we nomena. have In concluding this paper, it should be noted that, using the thermodynamic model of hysteresis, one can obtain not only E = EO +AHOAX-I'AX2 [15] the true thermodynamic data (i.e., without hysteresis) but also and the width of the hysteresis loop takes the form the interactive energy r describing the interaction between the two systems involved in the phase transition. A/E = 2AH0 AX = 2(Te,AS0°) [16] The authors acknowledge the National Science Foundation for their According to the thermodynamic model, at temperatures financial support (Grant CHE7617836A02). greater or equal to the hysteresis phenomenon should not T, 1. Knittel, D. R., Pack, S. P., Lin, S. H. & Eyring, L. (1977) J. Chem. be observed. Further, the hysteresis loop becomes narrower and Phys. 67, 134-142. shorter as T approaches T, from below. 2. Von Maue, A.-W. (1937) Ann. Phys. (Leipzig), 5th series, 30, 555-576. RESULTS 3. James, H. M. & Keenan, T. A. (1959) J. Chem. Phys. 31, 12-41. The reported (6) excess entropies for the upper transition in the 4. Nakamura, T. & Miyagi, H. (1971) J. Chem. Phys. 54, 5276- 5285. solid C1H4 and for the lower and upper transitions in the solid 5. Yamamoto, T., Kataoka, Y. & Okada, K. (1977) J. Chem. Phys. C2H4 are 0.91, 0.8, and 1.24 cal/mol-K, respectively. Choosing 66,2701-2730. Tc/Te = 1.01 and other values listed in Table 1, the width and 6. Colwell, J. H., Gill, E. K. & Morrison, J. A. (1963), J. Chem. Phys. height of the hysteresis loop of C1H4 and C2H4 are estimated 39,635-653. Downloaded by guest on October 2, 2021