Order parameter profiles

This fomi is called a Ginzburg-Landau expansion. The first temi f(m) corresponds to the free energy of a homogeneous (bulk-like) system and detemiines the phase behaviour. For t> 0 the fiinction/exliibits two minima at = 37. This value corresponds to the composition difference of the two coexisting phases. The second contribution specifies the cost of an inhomogeneous order parameter profile. / sets the typical length scale. 

By virtue of their simple stnicture, some properties of continuum models can be solved analytically in a mean field approxunation. The phase behaviour interfacial properties and the wetting properties have been explored. The effect of fluctuations is hrvestigated in Monte Carlo simulations as well as non-equilibrium phenomena (e.g., phase separation kinetics). Extensions of this one-order-parameter model are described in the review by Gompper and Schick [76]. A very interesting feature of tiiese models is that effective quantities of the interface—like the interfacial tension and the bending moduli—can be expressed as a fiinctional of the order parameter profiles across an interface [78]. These quantities can then be used as input for an even more coarse-grained description. 

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.

For each configuration C, we average over Y and Z to get an order-parameter profile T]c ) that depends on C ... 

Such order parameters can also be obtained from deuterium NMR measurements, and therefore one frequently finds predictions for these order parameters in the literature. The order parameter profile that belongs to the tails of the lipid membrane as given in Figure 5 is shown in Figure 6. 

There is a consensus in the literature about the typical patterns found in the order parameter profile along the chain in the phospholipid bilayer. The order... 

Figure 9. A comparison of the order parameter profile as found by MC simulations [72] of model 9/10 cis unsaturated chains in a monolayer (the x-line is to guide the eye) with experimental data obtained from NMR experiments (o) on the same chains incorporated into a biological membrane. Redrawn from [72] by permission of the American Institute of Physics...

Figure 17. Order parameter profile for the two tails of DMPC bilayers. The chain closest to the head group is named sn and the other one sn2. Segments closest to the glycerol backbone are numbered t = 1, and the chain ends are t = 16. The lines are drawn to guide the eye...

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. 

Fig. 21. Measured segmental order parameter profiles of d62-DPPC (left) and the ds2-DPPC-GD (4.7 mol%) mixture (right) at selected temperatures (40, 50 and 60°C).

The second puzzling feature is the observation of more liquid crystal phase disorder at the 6 position than at either the 4 or 10 positions of DPPC (Table I). The results appear statistically significant and contrast vith the observed (35) NMR order parameter profile (constant from positions 2->10, diminished order from position 10 to the bilayer center). Further experiments vith DPPC derivatives deuterated at positions closer to the bilayer center may clarify the situation. 

Fig. 12. Schematic variation of the order parameter profile /(z) of a symmetric (f=l/2) diblock copolymer melt as a function of the distance z from a wall situated at z=0. It is assumed that the wall attracts preferentially species A. Case (a) refers to the case % %v where non-linear effects are still negligible, correlation length and wavelength X are then of the same order of magnitude, and it is also assumed that the surface "field" Hj is so weak that at the surface it only induces an order parameter 0.2 n if mb is the order parameter amplitude that appears for %=%t at the first-order transition in the bulk. Case (b) refers to a case where % is only slightly smaller than %t, such that an ordered "wetting layer" of thickness 1 [Eq. (76)] much larger than the interfacial thickness which is of the same order as [Eq. (74)] is stabilized by the wall, while the bulk is still disordered. The envelope (denoted as m(z) in the figure) of the order parameter profile is then essentially identical to an interfacial profile between the coexisting ordered phase at T=Tt for (zl). The quantitative form of this profile [234] is shown in Fig. 13. From Binder [6]...

We now turn to the case of block copolymers. Early work by Shull [20] considered a lamellar arrangement parallel to the walls only. From such calculations, one obtains a unified treatment of both the weak and the strong segregation limit within mean field theory, and a detailed description of the order parameter profiles across the thin film emerges. 

Rg 7 lattice spacings. Figure 20 shows resulting order parameter profiles... 

Fig.20. Order parameter profiles m(z)=([pA(z)-pB(z)])/([pA(z)+pB(z)]), where pA(z), pB(z) are densities of A-monomers or B-monomers at distance z from the left wall, for LxLx20 films confining a symmetric polymer mixture, polymers being described by the bond fluctuation model with N=32, ab=- aa=- bb=8 and interaction range 6. Four inverse temperatures are shown as indicated. In each case two choices of the linear dimension L parallel to the film are included. While for e/kBT>0.02 differences between L=48 and L=80 are small and only due to statistical errors (which typically are estimated to be of the size of the symbols), data for e/kBT=0.018 clearly suffer from finite size effects. Broken straight lines indicate the values of the bulk order parameters mb in each case [280]. Arrows show the gyration radius and its smallest component in the eigencoordinate system of the gyration tensor [215]. Average volume fraction of occupied sites was chosen as 0.5. From Rouault et al. [56].

Figure 26 shows typical order parameter profiles obtained across the films at various temperatures. Due to the repulsive wall-A interaction the order parameter pA(z)-pB(z) is always depressed towards negative values for z near the walls. From previous studies of the same model in the bulk [190,191] one expects an order-disorder transition to a lamellar phase for T=2 (note AB/kB=l), and consequently one sees for D=20 and D=30 well-developed lamellae oriented parallel to the walls, evidenced by the periodic variation of the order parameter across the film, with an amplitude close to the saturation value unity already. However, for D=14 the system does not develop this type of order, and for D=24 there seems to be well-developed order near the right surface but not near the left surface, which is unexpected since both surfaces are equivalent. 

Fig. 28. Averaged order parameter profiles cf>av(Z,x) plotted vs the scaled distance Z=z/2 b from the left wall at z=0 for four different scaled times T after the quench as indicated, for a scaled distance D =D/2 b=60. Choosing a rescaled distance L /2 b=600, and a discretization AX=1.5, Ax=0.05, the resulting equations are solved by the cell-dynamics method. The results shown are for parameters h1=y=4, g =-4, and averaged over 2000 independent initial conditions, corresponding to random fluctuations in a state with J( )av(Z,0)dZ=0. The parameters Iq and g were chosen such that both walls prefer A but one is still in the non-wet region of the equilibrium surface phase diagram of the corresponding semi-infinite system. From Puri and Binder [145]...

Fig. 30. Laterally averaged order parameter profiles ( )av(Z,x) vs Z for four different times, for the same model as in Fig. 28, but choosing L =400, D =300, AX=1.0, Ax=0.03, and a surface potential V(Z)=-h1/Z3, with parameters h1=y=4, g =-4. Averages were taken over 200 independent runs. From Puri et al. [228]...

Budkowski et al. and Bruder et al. [70] have tried to estimate the bulk phase diagram of polymer mixtures, using a geometry where for Tphase separation with an interface running parallel to the substrate occurs (Fig. Id). They obtained order parameter profiles across the film and associate the order parameter close to both walls with the order parameters ( ) es, ( ) es of the two coexisting phases in the bulk. As shown by the model calculations of Ref. [62], such a procedure also leads to systematic errors due to the finite film thickness D, in particular for T near Tc where qb is large. However, the accuracy with which Budkowski et al. and Bruder et al. [70] estimated (]> es, (]> ,s was rather limited, and... 

Figure 4 Order parameter profiles (39) for the lipid hydrocarbon (a) sn-1 and (b) sn-2 chains in a binary membrane mixture of DPPC and cholesterol. The cholesterol concentrations are 0mol% (open circle),...

Figure 3.26. Ordering parameter profile for carbon-deuterium segments In a model 9-10 cis-unsaturated chain In a monolayer ). The open circles refer to results from NMR. (Redrawn from Levine et al., loc. cit.)...

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). 

Fig. 2. Increased acyl chain unsaturation and increased temperature produce dissimilar increases in acyl chain disorder. (A) Difference order-parameter profiles, S(n), of deuterium NMR measurements on perdeuterated 18 0 in the sn-1 position as a function of changes in temperature and unsaturation at the 5n-2 position. A,SYm) between 18 0, 22 6 PC and 18 0, 18 1 PC O A5fn) for 18 0, 18 1 PC between 27°C and47°C. Carbon atoms are numbered beginning at the glycerol backbone. (Data from Gawrisch and Holte, 1996 used by permission of K. Gawrisch). (B) Difference in orientation probability for the fluorescent membrane probe DPH. Orientation distribution of DPH in 16 0,18 1 PC at 40°C minus that in 20°C (—), and the distribution of di22 6 PC minus that of 16 0,18 1 PC at 20°C (—).

Holte LL, Peter SA, Sinnwell TM. Gawrisch K. -H nuclear magnetic resonance order parameter profiles suggest a change of molecular shape for phosphatidylcholines containing a polyunsaturated acyl chain. Biophys J 1995 68 2396-2403. 

Fig. 35. (a) Order parameter profile 0(z) across an interface between two coexisting phases the interface being oriented perpendicular to the z-direclion. (b) The radial order parameter profile for a marginally stable critical droplet in a metastable state which is close to the coexistence curve, (c) Same as (b) but for a state close to the spinodal curve, 0sp. In (a) and (b) the intrinsic thickness of the interface is of the order of the correlation length ifeoex whereas in (c) it is of the order of the critical droplet radius / . From Binder (1984b). 

Fig. 67. Order parameter profiles m(z)/mt, associated with surface-induced disorder. The coordinate z measures the distance from the surface (z = 0). is the bulk correlation length and mu the bulk order parameter. If ease (a) persists up to the first-order transition temperature Tc, this means the surface stays ordered up to 7 c, while case (b) shows surface induced disordering a layer of thickness L gets disordered already at T <71, and as T — Tc the (delocalized) interface at mean position z = L from the surface advances into the bulk, fJ(7 ) -+ oo as T - Tc, and the surface order parameter mj = m(z = 0) then vanishes continuously, mj a (1 - T/Tc). From Dasch et al. (1988).

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