A 4-component clay-polymer-salt-water system was studied by neutron scattering. The clay-salt-water system consisted of n-butylammonium vermiculite, n-butylammonium chloride and heavy water, and the volume fraction of clay in the system was held constant, at r = 0.01. Three polymers in the molecular weight range 10,000 to 30,000 were studied, poly(vinyl methyl ether) (PVME), poly (ethylene oxide) (PEO) and poly(acrylic acid) (PAA), at a polymer volume fraction of v = 0.01. The addition of PAA suppressed the clay swelling, irrespective of the salt concentration, c. The addition of the neutral polymers had no effect on the phase transition temperature, Tc, between the gel and tactoid phases of the system, its value remaining at 14 °C for c = 0.1 M and 30 °C for c = 0.01 M. At c = 0.01 M, the neutral polymers also had a negligible effect on the lattice constant d along the swelling axis of the clay colloid, but at c = 0.1 M, the d-value was significantly lower than in the system without added polymer. For a PVME sample of molecular weight 18,000, both d and Tc were measured as a function of ν, for volume fractions between 0 and 0.04. The addition of polymer, up to v = 0.04, had no effect on Tc. However, even for v values as low as 0.001, the vermiculite layers in the gel phase were more parallel and more regularly spaced than in the system without added polymer. In the gel phase, d decreased exponentially as a function of v, from 12 nm at v = 0 to 8 nm at v = 0.04. In the tactoid phase, at T < 14 °C, the d-value in the crystalline regions was equal to 1.94 nm at v = 0 and v = 0.04, showing that the spacing between the vermiculite layers is not affected by the added polymer when they are collapsed by an increase in temperature. The addition of a PVME sample of molecular weight 110,000, at v = 0.001, had no noticeable effect on either d or Tc.

The swelling of n-butylammonium vermiculite in water was investigated as a function of the sol concentration (r), the salt concentration (c) and the temperature (T).

The interlayer spacing in the gel phase was investigated as a function of r and c by neutron diffraction and by laboratory experiments which measured how many times its own volume a crystal would absorb. The salt concentration was found to be the stronger variable with the interlayer spacing decreasing proportional to c0.5, which is consistent with previous results and with the Coulombic attraction theory. The sol concentration was found to affect the swelling for two reasons, the salt fractionation effect and the trapped salt effect. Both of these cause the salt concentration in the supernatant fluid to be greater than that originally added to the crystals and so reduce the swelling.

A new method was used for extracting the solution from inside the gels by collapsing the gels by the addition of potassium hydrogen carbonate. The Volhardt titration was carried out on the extracted and supernatant solutions from about 250 gels. The ratio of the external to the internal chloride concentration was found to be approximately constant across the range of salt concentrations. Its average value was equal to 2.6, again in agreement with Coulombic attraction theory and showing the surface potential to be constant at about 70 mV.

The (r, c, T) boundary of the two phase colloid region was investigated by three methods. A plot of log c against Tc was linear within experimental accuracy, with a gradient of 0.077 K−1 or 13 K per log unit. This shows that the surface potential varies by only 1 mV per decade in the salt concentration. The system is therefore governed by the Dirichlet boundary condition and not by the Nernst equation.

We consider a model of binary opinion dynamics where one opinion is inherently “superior” than the other, and social agents exhibit a “bias” toward the superior alternative. Specifically, it is assumed that an agent updates its choice to the superior alternative with probability α > 0 irrespective of its current opinion and opinions of other agents. With probability $1-\alpha$, it adopts majority opinion among two randomly sampled neighbors and itself. We are interested in the time it takes for the network to converge to a consensus on the superior alternative. In a complete graph of size n, we show that irrespective of the initial configuration of the network, the average time to reach consensus scales as $\Theta(n\,\log n)$ when the bias parameter α is sufficiently high, that is, $\alpha \gt \alpha_c$ where αc is a threshold parameter that is uniquely characterized. When the bias is low, that is, when $\alpha \in (0,\alpha_c]$, we show that the same rate of convergence can only be achieved if the initial proportion of agents with the superior opinion is above certain threshold $p_c(\alpha)$. If this is not the case, then we show that the network takes $\Omega(\exp(\Theta(n)))$ time on average to reach consensus.

The Ginzburg–Landau functional is a phase transition model which is suitable for classification type problems. We study the asymptotics of a sequence of Ginzburg–Landau functionals with anisotropic interaction potentials on point clouds Ψn where n denotes the number data points. In particular, we show the limiting problem, in the sense of Γ-convergence, is related to the total variation norm restricted to functions taking binary values, which can be understood as a surface energy. We generalize the result known for isotropic interaction potentials to the anisotropic case and add a result concerning the rate of convergence.

The consistency of a non-local anisotropic Ginzburg–Landau type functional for data classification and clustering is studied. The Ginzburg–Landau objective functional combines a double well potential, that favours indicator valued functions, and the p-Laplacian, that enforces regularity. Under appropriate scaling between the two terms, minimisers exhibit a phase transition on the order of ɛ = ɛn, where n is the number of data points. We study the large data asymptotics, i.e. as n → ∝, in the regime where ɛn → 0. The mathematical tool used to address this question is Γ-convergence. It is proved that the discrete model converges to a weighted anisotropic perimeter.

The phase transition behavior of [011]- and [001]-oriented 0.24PIN–0.43PMN–0.33PT single crystals was investigated through dielectric measurement in the process of heating and direct current (DC) bias. The phase transformation sequence in the [011]-oriented crystals is rhombohedral (R) → monoclinic (MB) → orthorhombic (O) → monoclinic (MC) → tetragonal (T) → cubic (C). The phase transition temperatures of R to MB$\left( {{{\rm{T}}_{{\rm{R}} - {{\rm{M}}_{\rm{B}}}}}} \right)$ and MB to O $\left( {{{\rm{T}}_{{{\rm{M}}_{\rm{B}}} - {\rm{O}}}}} \right)$ decrease; meanwhile, the transition temperatures of O to MC$\left( {{{\rm{T}}_{{\rm{O}} - {{\rm{M}}_{\rm{C}}}}}} \right)$ phase, MC to T $\left( {{{\rm{T}}_{{{\rm{M}}_{\rm{C}}} - {\rm{T}}}}} \right)$ phase, and T to C (TT–C) phase increase with the increase of DC bias. The phase transformation sequence in the [001]-oriented crystals is R → T → C. As DC bias increases, the transition temperature TR–T of R to T phase declines and the transition temperature TT–C rises. Intermediate phases MB, O, and MC are only found in the [011]-oriented crystal. The phase transition characteristics of the [011]-oriented crystals are rather more complex than those along [001] direction. The micro–macro domain transition at Tf is related to crystal orientation and DC bias voltage. The phase diagram in terms of temperature and bias voltage is established for [011]- and [001]-oriented 0.24PIN–0.43PMN–0.33PT crystals. The DC bias dependent dielectric properties and phase transition characteristics are also compared with the crystals along [111] direction.

Phase transitions in the BaAl2O4–SrAl2O4 solid solution have been analysed as a function of temperature and composition using infrared (IR) powder absorption spectroscopy. The improper ferroelectric phase transition P6322 → P63 (2A superstructure) in the BaAl2O4 end-member can be detected through a change in slope of the wavenumbers of hard modes at ∽450 K. A change in line widths at ∽520 K appears to correlate with the development of diffuse intensity in a*-b* planes of electron diffraction patterns reported elsewhere in the literature. The same shift in wavenumber of hard modes is not observed in spectra from samples with compositions corresponding to 60, 80 and 90% of BaAl2O4 component, but a change in line widths at ∽500 K has been tentatively explained in terms of a different phase transition, from a P6322 parent structure to a P63 (√3A superstructure) product. Strain analysis of published high-temperature lattice parameter data suggests that the hexagonal → monoclinic transition in Sr-rich members of the solid solution may consist of two discrete transitions, and a sequence P6322 → C2 → P21 is suggested. The second transition could be related to instabilities in the hexagonal solid solution. Autocorrelation analysis of the IR spectra reveals a large positive deviation from linear behaviour across the solid solution, which is interpreted in terms of microscopic strain effects. These microscopic strains are probably responsible for the different transformation behaviour shown by samples with different compositions across the solid solution.

The crystal structure of the high-pressure phase-II of cristobalite has been solved by neutron diffraction (space group P21/c, a = 8.3780(11) Å, b = 4.6018(6) Å, c = 9.0568(13) Å, β = 124.949(7)°, at P = 3.5 GPa). This phase corresponds to a distortion of the high-temperature cubic β-phase, rather than of the ambient temperature and pressure tetragonal α-phase.

Synchrotron X-ray powder diffraction data have been collected on three synthetic leucite analogues with the general formula Cs2XSi5O12(X = Cd, Cu, Zn) between 295 and 1173 K. All three samples have the orthorhombic Pbca leucite structure at room temperature with ordered framework T-site cations. The sample with X = Cd retains the Pbca structure over the whole of the investigated temperature range. The sample with X = Cu also retains the Pbca structure, but there is a transition to a less distorted structure with a larger unit-cell volume at ∼333 K. The sample with X = Zn shows evidence for a transition to a previously unknown Pa cubic structure, with some T-site cation disorder, at 566 K, on heating. This transition is reversible on cooling to 633 K.

The high-pressure behaviour (up to 30 kbar) of dehydrated analcime, Na[AlSi2O6), has been studied in detail using polarized microscopy and Raman micro-spectroscopy. Samples were compressed using a diamond anvil cell in a quasi-hydrostatic medium (glycerol or water-ice). Two transitions at 3.7 and 11 kbar were observed.

At the first transition, phase II is observed under cross polarized light as a contrasting black zone, moving from the edge to the centre of the sample. At this transition the strong Raman doublet at ∼480 and 500 cm−1 transforms discontinuously to a singlet, which is similar to that of quasi-cubic natural analcime. This transition, with an increase of effective symmetry, seems unusual on increasing pressure.

The analcime framework may be thought of as an array of four-, six-, or eight-membered interconnected rings. Phase transitions in the analcime group are considered in the literature in terms of deformations of the six-membered rings. However, the present work shows that these phase transitions should be associated with deformations of the four-membered rings (as the minimum size secondary building units) by rotation of rigid TO4 units. The strongest Raman band frequency is correlated with the mean T-O-T angle inside the four-membered ring: a rate of 4.5 cm−1/degree was found for analcimes and leucites. Using this correlation, one can estimate from Raman data the possible deformation of four-membered rings at the phase transitions.

In situ, high-temperature energy-dispersive X-ray powder diffraction (EDXRD) data have been collected on synthetic and a natural sample of mercury sulphide (HgS). These measurements were made between temperatures of 295 and 798 K. Synthetic samples of HgS were prepared by reaction between sulphide and mercury in aqueous solution. In a subsequently dried and aged synthetic HgS sample, heated in vacuo, there is a change from a poorly crystalline pseudocubic material into a well crystalline cubic material in the temperature region 583–623 K. At higher temperature (748 K), there is evidence for a partial phase transition to the high temperature hypercinnabar HgS structure. In a neoformed synthetic sample, heated in a sealed Ti container, the initial ‘pseudocubic’ metacinnabar phase partially transforms to a previously unknown phase (XHgS) in the temperature range 467–522 K. This phase disappears at 527 K, and the metacinnabar phase changes to a well crystalline cubic phase; cinnabar develops at 542 K. The proportion of cinnabar continues to increase up to 647 K. Both metacinnabar and cinnabar phases are retained on cooling. No phase transitions were observed for the natural cinnabar sample.