Critical Behavior On Approaching A Double Critical Point In A Complex Mixture
Abstract
This thesis reports the results of lightscattering measurements and visual investigations of critical phenomena in the complex mixture 1propanol (1P) + water (W) + potassium chloride (KCl) which has a special critical point (or a special thermodynamic state) known as the double critical point (DCP). The main theme of the thesis is the critical behavior on approaching a special critical point (i.e., the DCP) in a complex or associating mixture in contrast with that in simple, nonassociating mixtures. The asymptotic critical behavior in complex or associating fluids, such as polymer solutions and blends, ionic and nonionic micellar solutions, microemulsions, aqueous and nonaqueous electrolyte solutions, protein solutions, etc., is now commonly accepted to belong to the 3DIsing universality class. However, the temperature range of the asymptotic regime in these fluids, with universal behavior, has a nonuniversal width and is, in general, smaller than that in simple or nonassociating fluids. In complex mixtures, which are made up of relatively large molecules or particle clusters of mesoscopic range, the coupling between the conventional correlation length of the critical fluctuations ( ξ) and an additional length scale associated with the mesoscale structures (ξD) is known to modify the approach towards the universal nonclassical critical behavior near their critical points. Nevertheless, the generality of this approach needs to be confirmed. There are also instances of a pure classical or close to classical behavior being observed in the critical domain of complex mixtures, although recent experimental results contradict the earlier observations. Therefore, further experimental evidences than that presently available are necessary before one can say how far the analogy between simple and complex fluids can be pushed. Variations in the effective dielectric constant of a mixture have been known to affect the critical behavior. Furthermore, we anticipate the presence of special critical points in complex mixtures to cause nontrivial modifications in the approach towards the universal asymptotic critical behavior. Special thermodynamic states are characterized by critical fluctuations with exceptionally large correlation length, and are displayed by multicomponent liquid mixtures, in which there are a multitude of thermodynamic paths by which a critical point can be approached, and offers rich information about the critical phenomena. These issues are being addressed in this research work. This thesis is organized into 7 Chapters.
Chapter 1 begins with an account of the historical development of the field of critical point phenomena with a brief introduction to critical phenomena in simple fluids. Critical phenomena observed in various complex systems such as aqueous and nonaqueous ionic fluids, polymer solutions and blends, micellar and microemulsion systems, etc., are discussed, with particular attention to investigations into crossover from Ising to meanfield critical behavior observed in these systems, which are relevant to the present work. Theoretical attempts at modeling ionic criticality are cited and summarized. This is followed by a discussion of reentrant phase transitions in multicomponent liquid systems. An account of the various types of special critical points, such as double critical point, critical double point, critical inflection point, quadruple critical point, etc., highlighting the critical behavior on approaching these special critical points, and some of the models of reentrant miscibility are briefly given. The Chapter ends with a statement on the goals of the present research work.
Chapter 2 describes the instrumentation developed and the data acquisition procedures adopted for the study. Details of the thermostats and precision temperature controllers used for visual and lightscattering measurements are provided. The important design considerations relating to the achievement of a high degree of temperature stability (~ ±1 mK in the range 293383 K) are elucidated clearly. The temperature sensors used in the present experiments and their calibration procedures are discussed. The lightscattering instrumentation is discussed in depth. The problems associated with the lightscattering techniques when it is used to study critical point phenomena, and the strategies adopted to overcome them are discussed. The sample cells used for visual investigations and light scattering experiments, along with the procedure adopted for cleaning and filling of sample cells are also described.
Chapter 3 essentially deals with the characterization of the system 1P + W + KCl. It begins with a brief introduction to the critical behavior in complex mixtures, and the motivation behind choosing the present system. The phase behavior in the present mixture, the generation of the coexistence curves and the line of critical points in the mixture, and the method used for preparation of the samples are described. The criticality of the samples is judged by the equal volume phase separation criterion through visual investigations. Addition of a small amount of salt (i.e., KCl) to the 1P + W solution induces phase separation in the mixture as a result of a saltingout process. Decreasing the salt concentration has the same effect as that of increasing pressure on the liquidliquid demixing of this mixture. Therefore, KCl may be considered as an appropriate field variable analogous to pressure in this mixture. The mixture 1P + W + KCl exhibits reentrant phase transitions and has an array of lower (TL) and upper (TU) critical solution temperatures. It is found that the line of TL’s and TU’s, known as the line of critical points, merge (TU  TL = ΔT → 0) to form a special thermodynamic state known as the DCP. The DCP is approached as close as 509 mK (i.e., ΔT ~ 509 mK) in this work. An analysis of the critical line shows that it is roughly parabolic in shape, which is in consonance with the predictions of the lattice models and the LandauGinzburg theory of phase transition. In addition to the presence of a special critical point, various structure probing techniques like small angle Xray scattering (SAXS), small angle neutron scattering (SANS), etc., indicate the presence of largescale density inhomogeneities or clusters in 1P + W solution and its augmentation on adding small amount of KCl. Therefore, the present mixture provides a unique possibility to investigate the combined effects of molecular structuring as well as a special critical point on the critical behavior. Only a section of the coexistence surface of the mixture could be generated, owing to various experimental limitations and other problems inherent to the system. This limited further studies on the coexistence curves in the mixture.
Chapter 4 reports the critical behavior of osmotic susceptibility in the present mixture. The behavior of the susceptibility exponent is deduced from static lightscattering measurements, on approaching the lower critical solution temperatures (TL’s) along different experimental paths by varying t [ = (T  T TL)/ TL] from the lower onephase region. The lightscattering data analysis emphasizes the need for correctiontoscaling terms for a proper description of the data over the investigated t range. Renormalization of the critical exponents is observed as the critical line is approached along certain special paths. Experimental evidence for the doubling of the extended scaling exponent Δ1 near the DCP is shown. There is no signature of Fisher renormalization in the values of the critical exponents. The data analysis yields very large magnitudes for the correction amplitudes A1 and A2, with the firstcorrection amplitude A1 being negative, signifying a nonmonotonic crossover behavior of the susceptibility exponent in the mixture. The magnitudes of the correction amplitudes are observed to increase gradually as TL approaches the DCP. The increasing need for extended scaling in the neighborhood of special critical points has been noted earlier in several aqueous electrolyte solutions, in polymersolvent systems, etc. However, the magnitudes of the correction amplitudes were not as large as that in the present case.
Analysis of the effective susceptibility exponent γeff in terms of t indicate that, for the TL far away from the DCP, γeff displays a nonmonotonic crossover from its single limit 3D Ising value (~ 1.24) towards its meanfield value with increase in t. While for that closest to the DCP, γeff displays a sharp, nonmonotonic crossover from its nearly doubled 3DIsing value (~ 2.39) towards its nearly doubled meanfield value (~ 1.84) with increase in t. For the inbetween TL’s, the limiting value of γeff in the asymptotic as well as nonasymptotic regimes gradually increases towards the DCP. The renormalized Ising regime extends over a relatively larger t range for the TL closest to the DCP, and a trend towards shrinkage in the renormalized Ising regime is observed as TL shifts away from the DCP. Nevertheless, the crossover behavior to the meanfield limit extends well beyond t > 10¯2 for the TL’s studied. The crossover behavior is discussed in terms of the emergence of a new lengthscale ξD associated with the enhanced ioninduced clustering seen in the mixture, as revealed by various structure probing techniques, while the observed unique trend in the crossover is discussed in terms of the varying influence of the DCP on the critical behavior along the TL line. The discussion is extended to explain the observed critical behavior in various reentrant systems having other special critical points. The extended renormalized Ising regime towards the DCP is also reflected in a decrease in the correlation length amplitude (ξ0) as TL approaches the DCP. It is observed that the firstcorrection amplitude A1 corresponding to fit using two correction terms becomes more negative as TL approaches the DCP, implying an increase in the value of the parameter ū of the crossover model [by Anisimov et al., Phys. Rev. Lett. 75, 3146 (1995)] as the DCP is approached. This increase in reflected in a trend towards a relatively sharp crossover behavior of γeff as TL shifts towards the DCP, i.e., towards the high temperature critical points.
The significance of the field variable tUL in understanding different aspects of reentrant phase transitions is manifested in the present system as well. Analysis of the data in terms of tUL led to the retrieval of universal values of the exponents for all TL’s. The effective susceptibility exponent as a function of tUL displays a nonmonotonic crossover from its asymptotic 3DIsing value towards a value slightly lower than its nonasymptotic meanfield value of 1. The limited (TL _ T) range restricted such a behavior of the effective exponent (in terms of t as well as tUL) for the lowest TL. This feature of the effective susceptibility exponent is interpreted in terms of the possibility of a nonmonotonic crossover to the meanfield value from lower values in the nonasymptotic, high tUL region, as foreseen earlier in micellar systems. The effective susceptibility exponent in terms of tUL also indicates an increase in the sharpness of crossover towards the high temperature TL’s. An increase in the sharpness of crossover with polymer chain length has been observed in polymer solutions. Therefore, our results suggest the need for further composition and temperaturedependent study of molecular structuring in the present mixture. There is also a large decrease in the dielectric constant of the mixture towards the high temperature TL’s.
In Chapter 5 the lightscattering measurements are performed on approaching the DCP along the line of the upper critical solution temperatures (i.e., TU’s), by varying t [ = (T  TU )/ TU ] from the high temperature onephase region in the mixture. A trend towards shrinkage in the simple scaling region is observed as TU shifts away from the DCP. Such a trend was not visible in the data analysis of the TL’s using the correction terms, due to the varying (TL  T) ranges. The lightscattering data analysis substantiates the existence of a nonmonotonic crossover behavior of the susceptibility exponent in the mixture. As with the TL’s, for the TU closest to the DCP, γeff displays a nonmonotonic crossover from its 3DIsing value towards its nearly doubled meanfield value with increase in t. While for that far away from the DCP, γeff displays a nonmonotonic crossover from its single limit Ising value towards a value slightly lower than its meanfield value of 1 with increase in t. The limited (TL – T) range restricted such a behavior of γeff for the TL far away from the DCP, This feature of γeff in the nonasymptotic, high t region is yet again interpreted in terms of the possibility of a nonmonotonic crossover to the meanfield value from below. Unlike TL’s, the crossover behavior in the present case is pronounced and more sharp for all TU’s. However, the variation in the width of the renormalized Ising regime on approaching the DCP along the TU line is quite similar to that observed along the TL line. The crossover behavior is attributed to the strong ioninduced structuring seen in the mixture, while the observed trend in the crossover as TU shifts towards/away from the DCP is attributed to the varying influence of the DCP. The influence of the DCP on the critical behavior along the TU (or TL) line decreases as TU (or TL) shifts away from the DCP.
Our observations indicate an increase in the sharpness of crossover as the critical temperature shifts from TL towards TU, or in other words, as the critical point shifts towards higher temperatures. SANS measurements on the present mixture indicate no difference in the growth of mesoscale clusters in the lower and upper onephase regions in the mixture. Hence, the observed increase in the sharpness of crossover towards the TU’s is very puzzling. The dielectric constant of the major constituent (i.e., water, ~ 62 %) of the present mixture decreases from around 80 to 63 as the critical temperature shifts from TL towards TU. Therefore, our results suggest the need to look at the crossover phenomena probably from two perspectives, namely, the solvent or dielectric effect and the clustering effect. The increase in the sharpness of the crossover behavior on approaching the high temperature critical points is probably related to the macroscopic property of the mixture, i.e., to the decrease in the dielectric constant of the mixture, while the actual nonmonotonic character of the crossover behavior is related to the microscopic property of the mixture, i.e., to the clustering effects, the extent of which determines the width of the asymptotic critical domain. However, this conclusion is somewhat subtle and calls for rigorous theoretical and experimental efforts to unravel the exact dependence of the crossover behavior on the dielectric constant.
Analysis using the field variable tUL in lieu of the conventional variable t led to the retrieval of unique, universal exponents for all TU’s irrespective of the ΔT value. For all TU’s, the effective susceptibility exponent in terms of tUL displays a nonmonotonic crossover from its asymptotic 3DIsing value towards a value slightly lower than its nonasymptotic meanfield value of 1, as that observed in the t analysis of the effective exponent for the TU far away from the DCP. Like with the TL’s, the crossover behavior
extends over nearly the same tUL range for the TU’s studied. However, the crossover is again sharper when compared to the TL’s.
Chapter 6 reports lightscattering measurements (by heating as well as cooling) on a non phaseseparating 1P + W + KCl mixture in the vicinity of the DCP. The results indicate that despite the lack of phaseseparation or critical points, criticalphenomenalike fluctuations can still occur in homogeneous mixtures if they reside in some other direction than temperature or composition (like, pressure or salt concentration) of the phase diagram. Unlike earlier studies on non phaseseparating mixtures, our results indicate a crossover behavior of the effective susceptibility exponent, in addition to the powerlaw behavior.
Chapter 7 sums up the major findings of the work reported in this thesis. It also presents a range of open problems that need to be explored further in order to fully understand the results that are reported in this thesis, especially, regarding the exact dependence of dielectric constant of the mixture on the character of the crossover behavior.
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 Physics (PHY) [414]
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