Thickness, chain mean-square

Specimen thickness over which occurs the transition from fC]" to Mean squared end-to-end distance of unperturbed chain Mean squared end-to-end distance of a chain of molar mass Me Velocity of the void surface between the fibrils in a craze Displacement... 

The adsorption transition also shows up in the behavior of the chain linear dimension. Fig. 6(a) shows the mean-square gyration radii parallel, i gl, and perpendicular, to the adsorbing plate. While these components do not differ from each other for e for e > ej i g strongly increases whereas Rh decreases. In the first case (non-adsorbed chain) oc oc N as a dilute solution in a good solvent in the bulk. For adsorbed chains R /N 0 for oo because the thickness is finite it is controlled by the distance e- e from the adsorption threshold, but does not diverge as N oo. The adsorbed chain follows in fact a... 

Fig. 10. The root-mean-square layer thickness tms as a function of the square root of chain length r for four values of polymer volume fraction 0 . Hexagonal lattice,...

Hesselink39,40 derived relations between the root-mean-square thickness and the loop or tail size. For the case in which no intrasegment interaction exists, he derived the segment distribution p4(z) for a single loop of size i by considering all posable configurations of the chain that starts at the interface and returns at its end to the interface. His expression for p4(z) reads... 

The mean square displacement A of the centre of mass of a chain (thick solid line) and the mean square displacement Ajg/2 of the central particle are measured in units of the intermediate length . The curves are calculated according to formulae (5.5) and (5.13) for the values of the parameters B = 100 x = 10-2. The displacement of the centre of mass does not depend on parameter ip, but the mean square displacement of the internal particles does. The values of parameter ip are shown at the curves for Ajg/2- The picture demonstrates the existence of the universal intermediate scale . Adapted from the papers of Pokrovskii and Kokorin (1985) and Kokorin and Pokrovskii (1990). 

As observed for block copolymers in the bulk [190-193], there is a detailed interplay between the order in the system and the chain linear dimension chains like to stretch out somewhat in the direction perpendicular to the lamellae. If one studies the mean square gyration radius component RgZ in the z-direction perpendicular to the walls, one hence observes an oscillatory dependence on thickness D for 1/T>0.5,since only for n ib=D (n=l,2,3,...) the parallel morphology fits nicely into the film, cf. Sect. 2.3, and the dumbbell -like block copoly-... 

The steric interaction between two approaching surfaces appears when the film thickness becomes of the order of, or smaller than 2L, where L is the mean-square end-to-end distance of the hydrophilic portion of the chain. If the chain were entirely extended, then L would be equal to M with / the length of a segment however, due to the Brownian motion L < Nl. For an anchored chain, such as that depicted in Figure 5.25a, in a theta solvent, L can be estimated as ... 

If the side chain liquid crystalline polymers goes into the smectic A phase from the nematic phase, the backbone chain is confined between two successive smectic layers, occasionally jumping into the neighboring layer gap. See Figure 2.30 where the cylinders denote side groups and thick lines represent backbones. The mean square end-to-end distances parallel and perpendicular to the director differs more than that in the nematic phase. 

One important area in which SANS has yielded information not otherwise available is determining chain dimensions in bulk polymers. Samples can be a couple of millimeters in thickness, with up to 50% of the incident neutrons scattered. The mean-square radius of gyration and can be determined for a deuterated version of a polymer mixed with its unlabeled analog. No Zimm analysis is required, despite overlap and interpenetration of the chain molecules. The only requirement is that both isotopic species have the same My and MWD. A model-independent analysis of the scattering used the Gunier approximation... 

However, for bottlebrush polymers where the backbone chain and the arms are flexible, the chain stiffness is a consequence of chain thickness. The simulations give rather clear evidence [61, 66, 70] that the chains do get stiffer with increasing chain lengths of the side chains (Fig. 13a). However, there is a monotonic increase of the mean square end-to-end distance of the backbone with backbone chain length Ab, from the rod-like behavior at small Np, where we find oc ... 

The permeability of small penetrant molecules through an organic matrix is determined by the solubility and diffusivity of the small molecule in the matrix as well as by the mean-square displacement (total path length traveled) divided by the sample thickness. In principle, the addition of a filler in the polymer matrix is expected to affect the solubility and diffusivity of a penetrant molecule, especially in the vicinity of the filler (i.e in the filler-polymer interfacial region and at least one polymer Rg away from the filler surface). Also, it is expected that fillers will affect the path tortuosity (mean-square displacement of penetrant versus film thickness) directly, when penetrants are forced to travel around impermeable fillers, and indirectly, when fillers induce polymer chain aUgnment or alignment and modification of polymer crystallites. ... 

Figure 9.16 Interaction curves for (a) incompressible film, (b) film of terminally adsorbed chains, (c) film of random co-polymer chains, at equal root-mean-square film thicknesses, calculated from Equations 9.9 and 9.10 using a/Z = 5, 9 = 1.0, A/kT = 12.5. From [40] with permission.

Figure 4. The influence of the molecular weight of the PEG side chains on the coefficient of friction (squares y-axis on the left-hand-side) and lubricant film thickness (circles y-axis on the right-hand-side) was measured as a function of speed by means of MTM and ultra-thin-film interferometry. The test lubricants were aqueous buffer solutions containing either PLL(20)-g[3.4]-PEG(2) (black symbols) or PLL(20)-g[3.4]-PEG(5) (white symbols). The lines between data points serve as a guide for the eye. Ball = stainless steel (19 mm in diameter), substrate = silica, buffer solution = 10mM HEPES (pH 7.4), polymer concentration - 0.25mg/ml, load = ION, T = 25°C.

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