Critical behavior of polystyrene-cyclohexane: Heat capacity and mass density Sirojiddin Z. Mirzaev, 1,2 Thomas Heimburg, 3 and Udo Kaatze 1, * 1 Drittes Physikalisches Institut, Georg-August-Universität, Friedrich-Hund-Platz 1, 37077 Göttingen, Germany 2 Heat Physics Department, Uzbek Academy of Sciences, Katartal 28, Tashkent 100135, Uzbekistan 3 Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, 2100 Copenhagen, Denmark Received 23 August 2010; revised manuscript received 22 October 2010; published 7 December 2010 At temperatures between 7.5 °C and 20 °C as well as 26 °C and 40 °C we have recorded the densities and specific heat at constant pressure for critical mixtures of polystyrene in cyclohexane. The degrees of polymer- ization were N = 288 critical temperature T c =9.77 °C and N = 6242 T c =27.56 °C, respectively. In the two-phase regime a series of reproducible events exists in the specific-heat traces, indicating the existence of nonequilibrium intermediate states as likely resulting from an oscillatory instability of droplet formation. In the one-phase region the critical contribution to the heat capacity follows power law with critical exponent  =0.11 compatible with Ising-like criticality. At larger N, however, the critical amplitude of the heat capacity is noticeably smaller than at lower degree of polymerization. This finding may be taken as an indication of different effects from competing mesoscale lengths: the radius of gyration of the polymer and the fluctuation correlation length of the mixture. The density traces reveal marginal deviations from simple linear temperature dependencies. If these deviations are analyzed in terms of critical contributions, different signs in the amplitude result, in conformity with the signs in the pressure dependence of the critical temperature. The absolute values of the amplitudes, however, are substantially larger than predicted from the critical amplitudes of the heat capacities. DOI: 10.1103/PhysRevE.82.061502 PACS numbers: 64.70.Ja, 65.20.w, 64.60.Ej I. INTRODUCTION Asymptotically close to the critical demixing point poly- mer solutions, like “simple” low-molecular-weight fluids, be- long to the three-dimensional Ising class: the spatial correla- tion length  of order parameter fluctuations has grown so large that it exceeds even the mesoscopic structures of the polymer molecules 1–3. In correspondence with simple flu- ids the individual characteristics of polymer solutions, there- fore, are largely masked by the fluctuations, so that universal near-critical behavior results. Unfortunately, the range of true asymptotic critical behavior is very small and often hardly accessible to measurements. For most binary fluids, already marginally away from the critical temperature, a trend from Ising-class critical behavior toward mean-field critical behav- ior becomes obvious. This crossover is especially important in polymer systems 4. Upon departure from the critical temperature T c , the fluctuation correlation length of the Ising model decreases according to a power law,  =  0  - ˜ . 1 Here,  = T - T c /T c 2 denotes the reduced temperature and  ˜ =0.63 is a critical exponent. Hence, at a temperature T the fluctuation correla- tion length becomes comparable to the length scale of the mesoscopic structure of the polymer system:  competes with the size of the polymer molecules. Normally the radius of gyration R g of the coiled polymer molecules is considered as a natural size parameter. Competition between  and R g changes the phase separation behavior dramatically. In the crossover region, therefore, significant differences between critical phenomena in high-molecular-weight polymer sys- tems and low-weight fluids emerge. The radius of gyration is a function of the degree of po- lymerization N of the molecules. N is thus a parameter that controls the phase behavior of polymer solutions. With in- creasing N the critical concentration c c decreases to approach zero, the critical temperature T c approaches the theta tem- perature for polymer solutions with large N, and the range of universal Ising-class critical behavior shrinks. Polymer solu- tions thus allow investigations into the competition of two characteristic mesoscale lengths by tuning the radius of gy- ration via N and by varying the fluctuation correlation length via T 2,3. Divergence of  and R g at the theta point and the coupling of both order parameters lead to mean-field-like tricriticality 5. The possibility to investigate the crossover from asymptotic Ising-class criticality to mean-field tricriticality has inspired considerable interest, both in theory 6–10 and experiment, in the properties of polymer solutions near their consolute points. However, since the pioneering work by De- bye and his colleagues 11–14, most experimental studies were by static and dynamic light scattering 15–25. Small- angle neutron scattering 26 as well as shear viscosity 27 measurements have been performed and heat capacities have been recorded to obtain phase diagrams 28,29. Surpris- ingly, however, no determinations of the exponent  in the weak specific-heat divergence near the critical point of poly- mer solutions have been reported so far. This deficiency has initiated heat-capacity measurements on polystyrene PS- cyclohexane CH mixtures near their consolute point, using two significantly different molecular weights M w of the poly- mer, in order to look for an influence of the degree of poly- merization. A careful analysis of the specific heat C p is ad- * uka@physik3.gwdg.de PHYSICAL REVIEW E 82, 061502 2010 1539-3755/2010/826/0615026 © 2010 The American Physical Society 061502-1