A small-angle light-scattering (SALS) technique is performed to investigate the phase separation in the films of flexible polymer (polystyrene, PS) mixed with low molecular weight thermotropic liquid crystal LC (4-cyano-4‘-n-octylbiphenyl, 8CB). The growth of isotropic (polymer) domains is studied in both isotropic and anisotropic (nematic or smectic phase) LC matrices, as a function of time, film thickness, and film composition. The size of the domains, L(t), grows algebraically with time as L(t) ∼ tβ. For 70/30 wt % of 8CB/PS, we found the diffusion growth in isotropic (β = 0.25) and smectic (β = 0.28) matrices, independent of the film thickness. In the nematic matrix, β changes from 0.33 to 0.47 as we change the thickness of the sample from 120 to 10 μm. We think that the change of β is due to the attractive forces between the polymer domains which follow from the elastic deformations of the nematic matrix caused by the glass surfaces and polymer domains. In every matrix and for different thicknesses of a sample, scaling is observed in this growth regime. At longer times, there is a crossover from the diffusion growth to the hydrodynamic fast-mode growth, characteristic for systems in which one of the components wets the confining walls. In this regime, we do not observe the scaling; i.e., there is more than one characteristic length scale in the system, and β ranges from 1 to 3/2. Considering extremely viscous systems (50/50 wt % of 8CB/PS), we also find the diffusion growth but with smaller exponent β < 0.2. In this case, we observe two peaks in the scattering intensity S(q,t). One of them is the surface and one the bulk peak. For systems with small amount of a polymer (90/10 wt % of 8CB/PS), the process of growth is very fast, and β = 0.4. In this case, the bulk peak is very quickly covered by the peak coming from the growth of the domains at the surface. To perform the measurements in anisotropic LC systems, we had to eliminate the multiple scattering of light coming from the large difference in ordinary and extraordinary refractive indexes of LC.
The model for the scattering amplitudes is applied to a quantitative analysis of the scattering spectra of the n-block copolymer systems. The method allows interpretation of the X-ray diffraction patterns in systems of the multiblock copolymers forming multiply continuous triply periodic structures (MCTPS). We give simple formulas for the scattering intensity for structures of different topology and symmetry appearing in block copolymers. The straightforward fitting procedure presented in this article allows determination of the multicontinuous architecture adopted by a multiblock copolymer system, the volume fractions of the continuous domains and the width of the interdomain intefaces. The model is robust, i.e., applicable to n-block linear copolymers, and should be helpful in the quantitative analysis of the experimental data.
The X-ray scattering amplitudes are given for the multicontinuous P, D, G, and I-WP cubic phases formed in block copolymer systems. The formulas can be used to retrieve the scattering amplitudes for an arbitrary n-block copolymer system adopting one of the above multiply continuous triply periodic structures (MCTPS).
In this chapter the authors discuss systems with internal surfaces, ordered surfaces, topological transformations, and dynamical scaling. In Section II the authors show specific examples of mesoscopic systems with special attention devoted to the surfaces in the system—that is, periodic surfaces in surfactant systems, periodic surfaces in diblock copolymers, bicontinuous disordered interfaces in spinodally decomposing blends, ordered charge density wave patterns in electron liquids, and dissipative structures in reaction–diffusion systems. Section III covers the theory of morphological measures, including the Euler characteristic and the Gaussian and mean curvatures.
We discuss the influence of dissipation at a system boundary (film-meniscus interface) on the dynamics of dislocation loops inside a smectic film. This dissipation induces a strong coupling between dislocations—effectively independent of their separation—leading to their nontrivial dynamics. Because of these dynamics, the effective “dynamical” radius of nucleation can be 10 times larger than the usual static critical radius.
We apply de Gennes model (Langmuir, 1990, 6, 1448−1450) for presmectic ordering to a freely suspended smectic film coupled to the meniscus. We show that the quartic term in the expansion of the free energy in power of the smectic order parameter cannot be neglected in the model in order to explain the full behavior of the contact angle, θm, between the film and the meniscus. As shown previously (Phys. Rev. E2001, 63, 021705-1−021705-9), this contact angle is nonzero due to the disjoining pressure caused by the enhanced smectic ordering at the film surfaces. The temperature dependence of θm with a maximum above the bulk temperature of the nematic-smectic phase transition, TNA, is qualitatively reproduced by the model for all film thicknesses h = Nd, where N is the number of layers and d the smectic period, although some discrepancy between theory and experiment for the temperature at the maximum still remains. The apparent change of slope of the contact angle at TNA observed experimentally in thin films cannot be reproduced by the mean-field treatment of the model and presumably critical fluctuations in the meniscus and their influence on the behavior of the surface tension close to TNA have to be taken into account.
We discuss the evolution of the distribution function of distances between point defects of opposite signs distributed on a line of finite length interacting via a potential attractive at short distances and repulsive at large distances. The standard deviation of the distribution grows quickly at short times, attains maximum, and decreases logarithmicaly at longer times. The distance between the defects increases monotonically and at equilibrium is about two times larger than the distance at which the repulsive force attains maximum. The distance dependent viscosity does not change qualitatively these conclusions, but only increases the time scale of evolution by one order of magnitude.
The method presented in the preceding article is applied in this paper to the analysis of X-ray spectra of cubic phases formed in the following systems: 1,2-dilauoryl-sn-glycero-3-phosphoethanolamine, 1,2-dielaidoyl-sn-glycero-3-phosphoethanolamine, 1,2-dioleoyl-sn-glycero-3-phosphocholine, and didodecyldimethylammonium bromide lipids in water; glycerolmonooleate amphiphilic molecules with Polaxamer P407 in water; R6FΣEO2 fluorinated surfactants in aqueous solutions and polimerized system formed in the cetyltrimethylammonium chloride with tetraethyl orthosilicate additives. We show that our analysis allows determination of the width of the layer decorating the periodic surface in the cubic phase, the composition of the cubic phase in the system with excess water, the type of the cubic phase (direct or inverse), surface area per head of the amphiphilic molecule, the volume fractions of the coexisting cubic phases, the epitaxial relations between the coexisting phases, and the kinetic pathways in the phase transitions between ordered phases.
Bragg reflection amplitudes are examined for the structures P, D, G, C(P), C(D), I-WP, and F-RD in amphiphilic systems. For these ordered phases, a very simple analytical formula for the scattering amplitudes is given. The formula allows determination of the scattering amplitudes for any cross-sectional density profile of the membrane decorating the minimal surface. Within this approximation an analytical solution for the Debye−Waller factor is presented. Finally we propose a simplified model that can greatly facilitate examination of the experimental scattering patterns.
The maximum entropy principle is applied to study the morphology of a phase ordering two-dimensional system below the critical point. The distribution of domain area A is a function of ratio of the area to contour length L, R=A/L(A), and is given by exp(−λRμ) with exponent μ=2, which follows from the Lifshitz-Cahn-Allen theory. A and L are linked through the relation L∼Aν. We find two types of domain in the system: large of elongated shape (ν=0.88) and small of circular shape (ν=0.5). A crack pattern in broken glass belongs to the same morphology class with μ=1 and ν=0.72.
We perform light scattering and direct optical experiments on a homopolymer blend of poly(methylphenylsiloxane) (PMPS) with polystyrene (PS). The system is subjected to the three-step process. The system is first quench to low temperature (T1) and allowed to separate for 5 h; next it is heated to a high temperature (T0) to the one-phase region where it mixes for a couple of minutes (1−10 min) and then quenched back to T1 and observed for 5 h. We note that annealing at T0 can be quantitatively studied by the analysis of the scattering intensity summed over a linear array of photodiodes. This quantity is very sensitive to the structure exisiting in the system. If the system is properly annealed, it has a noisy behavior and while the structure inside the system persists, it behaves very regularly. Moreover, one can observe the differences in the scattering intensity between the first and the second quench at very short wavevectors, indicating that large domains survived the annealing process for short annealing time (less than 4 min). However, the average area of the domains per unit volume is the same as obtained from the tail of the scattering intensity, indicating that small domains dominating in the system do not survive the mixing process even if it is very short (2 min). Finally, the direct observation under the microscope reveals that they dissolve in such a way that their size changes at the end of the process of dissolution, when as we suspect the size of the interface becomes comparable to the size of the dissolving domain. Domains inside the domains are also observed at short times after the second quench. In general, our methods allow the quantitative estimate of the annealing time for polymer mixtures and thus can save a lot of time, especially if we have to repeat the same measurements many times and we need to anneal the samples between measurements.
The order-disorder transition is studied in a system of a scalar nonconserved order parameter. We use this well studied system to show that the application of the methods of topology and geometry reveals that our knowledge of the kinetic pathways by which the order-disorder transition proceeds is far from being complete. We show that in two-dimensional (2D) and 3D systems there are three dynamical regimes in the evolution of the system: early, intermediate, and late. In the intermediate regime two length scales govern the behavior of the system, whereas in the early and intermediate regime there is only one length scale. The size distribution of the domain area indicates the pathway by which the domains change their size. There are only two types of domains in a 2D system: circular and elongated with well defined characteristics (scaling of the area with the contour length) which in the late regime do not depend on time after rescaling by the average area and contour in the system. The elongated domains continuously change into circular domains reducing in this way the overall dissipation in the system. In order to reach a Lifshitz-Cahn-Allen (LCA) late stage regime the number of elongated domains must be strongly reduced. In the intermediate regime the number of elongated domains is large and simple LCA scaling does not hold. In a 3D symmetric system we always have a bicontinuous structure that evolves by cutting small connections. The late stage regime seems to be associated with the appearance of the preferred nonzero mean curvature. The early-intermediate regime crossover is associated with the saturation of the order parameter inside the domains, while the intermediate-late stage regime crossover is related to the global breaking of the ± order parameter symmetry (marked by the appearance of the nonzero mean curvature but still zero average magnetization). The times for the occurrence of these crossovers do not depend on the size of the system.