Asif Raza and Debasish RoyComputational Mechanics Laboratory, Department of Civil Engineering, Indian Institute of Science, BangaloreSanhita Das‡Department of Civil and Infrastructure Engineering, Indian Institute of Technology Jodhpur(Dated: March 26, 2024)Stimulus-induced volumetric phase transition in gels may be potentially exploited for various bio-engineering and mechanical engineering applications. Since the discovery of the phenomenon in the 1970s, extensive experimental research has helped in understanding the phase transition and related critical phenomena. Yet, little insight is available on the evolving microstructure. In this article, we aim at unravelling certain geometric aspects of the micromechanics underlying discon tinuous phase transition in polyacrylamide gels. Towards this, we use geometric thermodynamics and a Landau-Ginzburg type free energy functional involving a squared gradient, in conjunction with Flory-Huggins theory. We specifically exploit Ruppeiner’s approach of Riemannian geometry enriched thermodynamic fluctuation theory that has been previously employed to investigate phase transitions in van der Waals fluids and black holes. The framework equips us with a scalar curvature that relates to the microstructural interactions of a gel during phase transition and at critical points. This curvature also provides an insight into the universality class of phase transition and the nature of polymer-polymer interactions.
It is important to characterize the critical point and determine c, v2c and Tc so that all three projections of the phase diagram may be represented in terms of reduced dimensionless quantities. One direct advantage of such a representation is that the critical exponents associated with the phase transition may be readily calculated. Another signicant advantage is the clear identication of the stable, unstable, and metastable phase boundaries, i.e. the spinodal and coexistence curves.
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The phase diagram projection v2 is redrawn using the reduced pressure = mod/c, reduced volume fraction v2 = v2/v2c and reduced temperature T = T /Tc and shown in Fig. 2. The critical point is more distinctly illustrated in Fig. 2 and so are the stable and unstable zones. For isotherms T < 1, the part of the isotherm where it is parallel to the horizontal axis, is the region where transition from swollen to shrunken phase takes place. In Fig. 2, the coexistence curve is represented by the black dashed curve while the spinodal curve is represented by the black dash-dotted curve. Fig. 3 shows the T v2 projection of the phase diagram. The solid black curve denotes the isobar /c = 0. Coexistence and spinodal curves may be seen in v2 projection as well. The swollen limb of the coexistence curve where predominately swollen phase exists is repSH 1.02 Supercritical Gel 1 Critical Point SW SH Coexistence SW Curve 0.98 MSW MSH Coexistence SH Curve
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Unstable SW + SH 0.96 0.94 Spinodal Curve 0 0.5 1 1.5 2 FIG. 3: T v2 phase diagram denoting the various phases. The red and blue solid curves represent the swollen and shrunken limbs of the coexistence curve respectively. The black dot denotes the critical point and the black solid curve represents the isobar at = 0. The dotted horizontal line at T = 0.976 corresponds to the phase transition temperature and the black dash-dotted curve represents the spinodal curve. resented by the solid red curve while the shrunken limb of the coexistence curve is represented by the solid blue curve in Fig. 3. Finally, the
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T projection is shown in Fig. 4. In this gure, the entire coexistence or the phase transition region described in the earlier diagrams (Fig. 2 and Fig. 3) appears as the solid red line. The shrunken and swollen phases outside the phase transition region are also shown in the gure. Also shown is the critical point and the supercritical gel region. The spinodal curve has also been depicted by the black dash-dotted curve in Fig. 3 and Fig. 4. For further analysis of phase transition mechanism, we formulate a method for approaching equilibrium in a system where surface eects, as characterized by a gradient energy given by Eq. (23), play a signicant role. In line with the work of , the continuity equation corresponding to the free energy of the system as in Eq. (24) may be written as, v2 t = { Dv2(1 v2)
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