Quasicrystals, rational approximants and quasicrystalline tilt boundaries : Theory and experiments

Abstract

A range of NaBr concentrations (0.02 < X < 0.21) can be examined in an experimentally accessible (Tx - T) range, and there is good temporal stability in Tx also. The cloud point curves for different concentrations of NaBr were determined through visual observation, and the coexistence surface was generated from these. The line of critical points (Tu and Ti) and the critical concentration of MP, (xmp)c over a wide range of X (0.004 < X < 0.35) were determined. The mismatch between the extremum and the critical concentration of the cloud point curves, even in the limit of vanishing ?T, is pointed out as quite acute. It is also established that there is a distinct difference between the coexistence surfaces of the two closely related systems MP + W + NaBr and MP + W + NaCl. Chapter 4, the core of this thesis, presents the results of extensive light scattering measurements carried out in the one-phase region below Tx on samples with their X varying from 0.02 to 0.21 to deduce the value of the critical exponent ? for osmotic susceptibility (?T). It was found that the light scattering data could not be described by pure Ising behaviour in the case of any of the samples. Ising behaviour (? = 1.24) with one correction-to-scaling term was observed for X < 0.08. An increase in the value of ? above its Ising value was seen in the case of the lowest NaBr concentration (X = 0.025) studied. This increase is due to the fact that the size of the loop (?T = 71.1 K) is not very large. For 0.08 < X < 0.16, a distinct crossover from Ising asymptotic behaviour to mean-field (? = 1.00) regime was seen when moving away from Tx. The crossover becomes more pronounced and the crossover temperature shifts closer to Tx as X increases. Typically, when X = 0.08, the crossover to mean-field behaviour occurs at (Ti - T) ~ 9.5 K, and when X = 0.1396, the crossover occurs at (Tx - T) ~ 4 K. Two correction-to-scaling terms were found to be necessary to fit the data for X > 0.08. Moreover, the amplitude of the first correction-to-scaling term is found to be negative, and its magnitude increases systematically as X increases. The negative value of the first correction-to-scaling amplitude implies that the crossover is nonmonotonic in nature. In the case of monotonic crossover, the Ising value of ? (= 1.24) is approached from below, whereas in nonmonotonic crossover the Ising value of ? is approached from above 1.24. Analysis of data employing an effective susceptibility exponent, ?eff [= - t d ln?T/dt where t = (Tx - T)/Tx] clearly demonstrates the nature of crossover to be nonmonotonic. This has been independently confirmed through analysis of the data using a crossover model with two independent crossover parameters developed by Anisimov et al. [Phys. Rev. Lett. 75, 3146 (1995)]. The crossover behaviour has been attributed to the presence of an additional length scale (?p) in the system due to the presence of NaBr. Data analysis using the two-parameter model suggested a divergence of ?p, and the shift of the crossover temperature to very close to Ti at X ~ 0.165. For 0.165 < X < 0.21, the light scattering data indicated a remarkable shrinkage of the critical region. Moreover, in this range, reproducible results were obtained only when the samples were allowed to equilibrate for ~24 hours before the light scattering measurements. In the t-range investigated (10?³ < t < 2 × 10?²), complete mean-field behaviour (? = 1.00) is observed with no indication of a crossover to the Ising regime. With X > 0.165, the phase separation is accompanied by the appearance of a solid-like phase on the meniscus and in the lower phase. The predominance of mean-field behaviour coupled with the appearance of the third phase for X > 0.165 suggests the possibility of a multicritical (Lifshitz or Tricritical) point in this system which could lead to the observed crossover phenomenon. In summary, this chapter provides compelling evidence for crossover from Ising to mean-field critical behaviour in an apparently simple system. Chapter 5 offers experimental evidence adduced from excess molar volume (vE), and SAXS measurements for an additional length scale in the system. Precise density measurements were made with a vibrating tube densimeter on samples with 0 < X < 0.16 in order to calculate vE. In a plot of X vs. vE, a distinct change in the character of the plot can be seen at X ~ 0.08, which provides gross evidence for structuring in MP + W + NaBr for X > 0.08. In an effort to examine whether the additional length scale is dependent on X, and to gather further evidence for structuring-specifically MP-rich clusters induced by NaBr-SAXS measurements were made in this system. These measurements were obtained at room temperature (298 K) in the one-phase region (below the relevant Tx) at different values of X (i.e., X = 0.02 - 0.17). The details of the experimental procedures adopted and the difficulties encountered are discussed. Cluster size distributions, estimated on the assumption that the clusters are spherical, show systematicity in that the peak of the distribution shifts towards larger values of cluster radius as X increases. The largest spatial extent of the clusters (45 Å) is seen at X = 0.17. Data analysis assuming arbitrary shapes and sizes of clusters gives a limiting value of cluster size (45 Å) that is not very sensitive to X. For X > 0.17 SAXS measurements could not be performed due to the close proximity of Tx to the temperature (298 K) at which the experiments were done. Chapter 6 summarizes the major findings of the work reported in the thesis. Some relevant problems that can be addressed profitably in the interests of advancing and deepening our understanding of crossover are also presented.