Chain order parameter

All results were obtained from chains of 200 backbone atoms, averaging 100 chains to obtain M2, axial extensions, end-to-end distances, and segmental and chain order parameters and directors. 

Figure 6 Results for lipid raft systems with varying amounts of POPC (palmitoyl-oleoyl-phosphatidylcholine), SM, and cholesterol (42), which shows plots for one leaflet from above. The top row for the system is a 1 1 1 mixture of POPC, SM, and cholesterol. The middle row shows data for Sg (2 1 1 for POPC SM cholesterol), and the bottom row shows data for Sq (62 1 1 for POPC SM cholesterol). The first column on the left shows data for the acyl chain order parameter (Sqq) averaged over selected carbons in POPC and SM chains. The second and third columns depict the in-plane electron densities of cholesterol and the selected chain carbons, respectively.

These results mean a variation of the hydrocarbon chain order parameter profile with the concentration. However, the monotonously decreasing values for higher oil contents give a regularity to the pattern and the values for 10% of oil were chosen to illustrate the variation of order parameter with the position of the carbon atom (Fig. 4). 

A = chain ordering parameter fixed = 2 for a random distribution of chain segments in space. 

Whenever La-Hu transitions are observed thermotropically, the La phase is always the lower temperature phase. The fact that the Hu phase, which has two-dimensional symmetry, occurs at higher temperatures than the one-dimensional La phase indicates that the lattice entropy is not the dominant entropy contribution within the system. Instead, the dominant entropy sources are the large number of configurations available to the lipid hydrocarbon chains and (probably to a lesser extent) to the water. The shape of the mean molecular volume available to the lipid hydrocarbon chains changes dramatically at the transition, which suggests that the hydrocarbon entropy increases at the transition. This hypothesis is supported by recent deuterium NMR studies of the chain order parameter profile that indicate that the Hu chains are more disordered than the La chains (19, 20). 

The lateral pressure concept provides useful insight into the spatial distribution of forces and is a sum over terms corresponding to layers of finite thickness (dz). Quantitative measurements of the LP remain difficult and even the concept has been criticized. Chain order parameters, measured by H-NMR studies on amphiphiles with specifically deuterium-labelled methylene units in the tails, ° gave a similar view of the distribution of the chain attractions/repulsions as a function of position in the tails. Increasing quad-rupole splittings, indicated by an increasing distance between the two peaks in the H-NMR spectrum, are associated with increasing chain order. 

Also included in Table 1 are the average areas and Ay obtained from the entire chains (i.e. order parameters accounts for the fact that half thickness <(> [19,20,8]. The problems encountered when correlating chain order parameters and the average interfacial areas per chain in liquid crystalline aggregates have been recently discussed by Nagle [21]. 

Likewise, the acyl chain order parameters of phos-phatidylethanolamines collectively decrease when going from the L to the Hu phase, due to the greater motional freedom of the chains in the inverted hexagonal structure [27,28,20]. Geometrical models have been recently proposed that relate chain order parameters with the cylinder curvature in a Hi.rphase [20,29]. It has been noted that similar considerations may not be valid for the Hi phase due to the possibility of chain backbending [20]. 

When the cholesteric mesophase is formed by mesogenic groups added to the polymer main chain, order parameter S is slightly lower than in low-molecular-weight liquid crystals [82], which should result in a slight increase in the pitch. At the same time, rotation of the mesogenic side groups around their... 

The summation runs over all carbon atoms in the chain. is the angle between the bilayei normal and the molecular axis, as discussed above. is the field strength this may be parametrised to reproduce appropriate experimental data such as the deuterium NMR order parameters or it may be obtained by a self-consistent protocol, as described below. In his work on lipid bilayers Marcelja used a slightly different expression for i jjisp which... 

Two physically reasonable but quite different models have been used to describe the internal motions of lipid molecules observed by neutron scattering. In the first the protons are assumed to undergo diffusion in a sphere [63]. The radius of the sphere is allowed to be different for different protons. Although the results do not seem to be sensitive to the details of the variation in the sphere radii, it is necessary to have a range of sphere volumes, with the largest volume for methylene groups near the ends of the hydrocarbon chains in the middle of the bilayer and the smallest for the methylenes at the tops of the chains, closest to the bilayer surface. This is consistent with the behavior of the carbon-deuterium order parameters,. S cd, measured by deuterium NMR ... 

Figure 14 Measures of disorder m the acyl chains from an MD simulation of a fluid phase DPPC bilayer, (a) Order parameter profile of the C—H bonds (b) root-mean-square fluctuation of the H atoms averaged over 100 ps.

The reversible aggregation of monomers into linear polymers exhibits critical phenomena which can be described by the 0 hmit of the -vector model of magnetism [13,14]. Unlike mean field models, the -vector model allows for fluctuations of the order parameter, the dimension n of which depends on the nature of the polymer system. (For linear chains 0, whereas for ring polymers = 1.) In order to study equilibrium polymers in solutions, one should model the system using the dilute 0 magnet model [14] however, a theoretical solution presently exists only within the mean field approximation (MFA), where it corresponds to the Flory theory of polymer solutions [16]. 

Figure 1. Crossover scaling plot for tlie order parameter ( m > = ( ( ia - Bl / (<1>a + B)> of a symmetrical polymer mixture simulated by tlie bond fluctiiatioii model on tlie simple cubic lattice, with a concentration (jiv = 0.5 of vacant sites. Here N " ( m > is plotted vs. N t, and chain lengths from N = 32 to N = 512 are...

In the previous section we assumed that disorder results in random fluctuations of the order parameter around some average value A<). Such an approach is, essentially, a mean field treatment of the lattice. It requires sufficiently strong interchain interactions, whose role is to establish a coherence between the phases of the order parameter in different chains. 

Figure 3-9. The numerically obtained order parameter along a polyucciylcne chain of 80 pairs of carbon atoms tor one panicular (but typical) realization of oil-diagonal disorder. The thin line does not account for lattice relaxation, the thick line does (adapted from Ref. 30 )...

While for strong interchain interactions large deviations of the order parameter from its average value are unlikely, for weak interactions the minimal-energy lattice configuration of disordered chains contains a finite density of kinks and anti-... 

Fig. 18. The dependence of the second rank orientational order parameters, P2 , for the X-C bond on its position along the alkyl chain containing four (A), six ( ) and eight ( ) atoms...

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