Order parameter similarity

Here, V is vector notation for the set of all component energies Vy, and A, j gives the coefficient of Vy in the ith run. The Ay, without subscript i, indicate the values of A in the target ensemble. The histograms collected in the runs are multidimensional in that they are tabulated as functions of the component energies as well as the order parameter . Similarly, the final result of the WHAM calculation is a multidimensional probability distribution in V J and . 

However, it is well known that a mechanical deformation of a conventional, isotropic polymer network causes anisotropy. Under deformation the chain segments become oriented according to the symmetry of the external field and the state of order of the network can be characterized by an order parameter similar to that of nematic liquid crystals. Very early mechanical experiments on nematic polydomain elastomers actually demonstrate that a uniaxial deformation of a nematic elastomer converts the polydomain structure into a macroscopically xmi-formly ordered monodomain network [44]. This is shown in Fig. 2, where the opaque polydomain becomes optically transparent and converts into a monodomain... 

An even coarser description is attempted in Ginzburg-Landau-type models. These continuum models describe the system configuration in temis of one or several, continuous order parameter fields. These fields are thought to describe the spatial variation of the composition. Similar to spin models, the amphiphilic properties are incorporated into the Flamiltonian by construction. The Flamiltonians are motivated by fiindamental synnnetry and stability criteria and offer a unified view on the general features of self-assembly. The universal, generic behaviour—tlie possible morphologies and effects of fluctuations, for instance—rather than the description of a specific material is the subject of these models. 

Fig. 3. Order parameter as a function of temperature for -methoxybeiizylidene-/) - -butylariiline (MBBA), a room temperature nematic Hquid crystal. S(T) is determined from tbe polarization of tbe absorption (dicbroism) of small quantities of a dye molecule of similar stmcture (/n / .f-dimetby1aminonitrosti1bene) wbicb bas been dissolved in tbe Hquid crystal bost (1).

Another principal difficulty is that the precise effect of local dynamics on the NOE intensity cannot be determined from the data. The dynamic correction factor [85] describes the ratio of the effects of distance and angular fluctuations. Theoretical studies based on NOE intensities extracted from molecular dynamics trajectories [86,87] are helpful to understand the detailed relationship between NMR parameters and local dynamics and may lead to structure-dependent corrections. In an implicit way, an estimate of the dynamic correction factor has been used in an ensemble relaxation matrix refinement by including order parameters for proton-proton vectors derived from molecular dynamics calculations [72]. One remaining challenge is to incorporate data describing the local dynamics of the molecule directly into the refinement, in such a way that an order parameter calculated from the calculated ensemble is similar to the measured order parameter. 

Here we review the properties of the model in the mean field theory [328] of the system with the quantum APR Hamiltonian (41). This consists of considering a single quantum rotator in the mean field of its six nearest neighbors and finding a self-consistent condition for the order parameter. Solving the latter condition, the phase boundary and also the order of the transition can be obtained. The mean-field approximation is similar in spirit to that used in Refs. 340,341 for the case of 3D rotators. 

To start the review of the PIMC results [328], we note that the detailed study of the quantum APR model (Eq. (41)) was partly motivated by the strong changes in shape of the orientational order parameter as a function of temperature as the rotational constant was increasing from its classical value 0 = 0 (see Fig. 3 in Ref. 327). For small enough 0 it was found that the order parameter decays monotonically with increasing temperature, similarly to the classical case. This is qualitatively different for larger 0, where T ) becomes a non-monotonic function of temperature. 

Recently, a formalism has been developed to determine the second and the fourth order parameters of films using polarized total internal reflection fluorescence (TIRF) [71]. Similarly to IR-ATR spectroscopy (Section 4), the experiment makes use of p- and s-polarized excitation, but the fluorescence emission (analyzed either in p- or s-direction) is detected normal to the substrate. Two approaches are developed based on the measurements of two intensity ratios. In the first one, the S angle has to be known experimentally or theoretically, and the order parameters (P2) and (P4) can be determined. In the second one, the order parameter (R ) is obtained by another technique, for instance IR-ATR spectroscopy, which allows deducing the order parameter (P4) and (cos2<5). 

Metadynamics defines coarse-grained variables which are assumed to be slow coordinates of the system. Those coordinates are similar to the order parameters considered earlier in this chapter. The coarse variables are evolved independently following a steepest-descent equation. In the case of a single variable, Laio and Parrinello [34] use... 

This probability distribution is similar to the probability distribution from (7.28), except that here we average only over configurations that have evolved from configurations in. (/ a time t earlier. By integrating this probability distribution over all order parameter values corresponding to region 23 we obtain the total probability that a system initially in 2 is in 23 at time t later. This is nothing other than the time correlation function C(t) and we can write... 

The systems undergoing phase transitions (like spinodal decomposition) often exhibit scaling phenomena [ 1—4] that is, a morphological pattern of the domains at earlier times looks statistically similar to a pattern at later times apart from the global change of scale implied by the growth of L(f)—the domain size. Quantitatively it means, for example, that the correlation function of the order parameter (density, concentration, magnetization, etc.)... 

Many conformations were sampled by the usual MC procedure. The result is of course that there is no preferred orientation of the molecule. Each conformation can, however, be characterised by an instantaneous main axis this is the average direction of the chain. Then this axis is defined as a director . This director is used to subsequently determine the orientational order parameter along the chain. The order is obviously low at the chain ends, and relatively high in the middle of the chain. It was found that the order profile going from the centre of the molecules towards the tails fell off very similarly to corresponding chains (with half the chain length) in the bilayer membrane. As an example, we reproduce here the results for saturated acyl chains, in Figure 10. The conclusion is that the order of the chains found for acyl tails in the bilayer is dominated by intramolecular interactions. The intermolecular interactions due to the presence of other chains that are densely packed around such a chain,... 

From the density profiles one cannot really judge the effect of the double bonds the density profiles for membranes of saturated lipids are very similar to those of unsaturated ones. Therefore it is necessary to consider some of the conformational characteristics of the tails. It is possible to compute the order parameter profile for both the saturated and the unsaturated chains. The order parameter profile for the saturated chain closely follows the results presented in Figure 17 for DMPC membranes for both the SCF and the MD predictions. The order parameter profiles for the unsaturated chain closely follows the MC predictions, as discussed in Figure 9. A pronounced dip is found near the cis double bond. For this reason, we choose here to present complementary data about the conformational properties of the acyl chains. 

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