Order parameter at the surface

If one assumes that no polymer segments are adsorbed onto the surfaces, the boundary condition commonly used [48,39,11,49] for the polymer order parameter at the surface is... 

The relation between W and Ws, the domain wall widths in the bulk and at the surface, can be seen in Figure 8. The effect of the surface relaxation is clearly visible as the order parameter at the surface Qs never reaches the bulk value Qo- The distribution of the square of the order parameter at the surface shows the structure that some of the related experimental works have been reported (Tsunekawa et al. 1995, Tung Hsu and Cowley 1994), namely a groove centred at the twin domain wall with two ridges, one on each side. 

The change in the sign of the order parameter at the surface has been observed (for ferroelectricty) by using a model of imaging developed for the detection of static surface charge (Saurenbach and Terris 1990). For ferroelastics, this corresponds to the profile of the lateral reactive force. The SFM non-contract dynamic mode images (Lithi et al 1993) would correspond to the distribution of the normal reactive force. The divergence of the lateral force distribution away from the centre of the wall can be attributed to the simulated infinite extension of the lattice. In the simulated array, the lateral component of the force reached a finite value between two adjacent domain walls. 

In general, the ellipticity coefficient is temperature dependent because of the temperatm e dependence of the correlation length (T) and the surface order parameter 5q(T). It increases by approaching the isotropic-nematic phase transition from above. By measuring ps(T) one can therefore directly determine the product T)Sq T), and, if we assume a power law dependence of (T), the temperature dependence of the nematic order parameter at the surface Sq T) can be extracted. 

The extension to dynamics is again based on equations similar to Eqs. 2, and 4 however, in addition to an equation similar to Eq. 7, one now has to satisfy two boundary conditions resulting from the surfaces. Since the local order parameter at the surface is not a conserved quantity, Eq. 25 leads to [16]... 

During the early stages, the fast relaxation of the nonconserved order parameter at the surfaces provides a boundary condition for the phase of the concentration waves which grow in the thin film. While in the bulk the random orientations and phases of these growing waves do not yield a time evolution of the average order parameter, these surface-directed concentration waves add up to an average oscillatory concentration profile near the surfaces of a thin film [16]. However, for a quantitative description, extensive numerical calculations solving Eqs. 2, 4, 5, 6, 7, 27, and 28 are required [16]. 

Fig. 10.5 Calculated order parameter at the surface So as a function of temperature. The numbers at the curves corresponds to different surface potential in dimensionless units W = 0 (1), 0.0056 (2), 0.008 (3), 0.01 (4), Wc = 0.01078 (5), 0.012 (6), 0.017 (7). Note that at Wc the discontinuity of the first order N-lso phase transition disappears (adapted from [7])...

Equation (35) was advanced by Ginzburg and Sobyanin for superfluidity in confined finite systems [135]. This theory will be applied herein for the onset of superfluidity of He confined in a sphere of radius Rq. Adopting the step function approximation, the boundary condition for the order parameter at the free surface is taken as /(/ o) = 0. Eor low values of / the first-order linear form of Eq. (30) is... 

The above results demonstrate that ultrathin depositions of LC molecules similar to a 2D gas can be formed at the inner surfaces of Anopore membranes. Their orientational order parameter is rather large, negative, and changes gradually fi-om 5sat —0.14 for the case when the depositions coexist with bulk (saturated depositions) to S = —0.5 for extremely diluted depositions. The rather strong dependence of the orientational order parameter on the surface coverage and a surface diffusion with a rate similar to bulk LC indicate a collective, 2D liquid-like behavior. 

Some solid surfaces induce disorder in nematic liquid crystals. It means that the order parameter at the interface is lower than the bulk value. For instance, evaporated SiO layers of a certain thickness due to their roughness decrease the order parameter of MBBA from the bulk value Sb 0.6 down to So 0.1-0.2. In some cases, the surface order parameter may be equal to zero (surface melting). 

Powder XR diffraction spectra confirm that all materials are single phase solid solutions with a cubic fluorite structure. Even when 10 mol% of the cations is substituted with dopant the original structure is retained. We used Kim s formula (28) and the corresponding ion radii (29) to estimate the concentration of dopant in the cerium oxide lattice. The calculated lattice parameters show that less dopant is present in the bulk than expected. As no other phases are present in the spectrum, we expect dopant-enriched crystal surfaces, and possibly some interstitial dopant cations. However, this kind of surface enrichment cannot be determined by XR diffraction owing to the lower ordering at the surface. 

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