Thickness, chain

Fig. 31 Trajectories of seven crystallizing chains (thick lines) selected at random from 640 chains during crystallization at 330 K a at 0.128 ns, b at 6.4 ns, c at 12.8 ns, and d at 19.2 ns. Also shown are the growing crystalline domains (thin parallel lines). Pictures are all side view along the x-axis...

The minimum in F(L) is observed to be near L/ q — 9 for aU chain lengths examined. It is to be noted that this minimum is the global minimum and the barrier between this state and other thicker lamellae increases prohibitively as the thickness increases. These simulations strongly suggest that a lamellar thickness that is much smaller than the extended chain thickness is actually an equilibrium result. 

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). 

Prussian Blue was tentatively deposited onto isolated polycations by the well-known layer-by-layer (or multiple sequential adsorption) technique. The stepwise adsorption of hexacyanoferrate anions and ferric (or ferrous) cations was actually effective in fabricating PB films [12], In this work, the first adsorption step of hexacyanoferrate anions along positively charged PMB chains were successful as confirmed by the increase of the chain thickness by approximately 0.7 nm (Fig. la). 

As shown in Fig. 34.1, if the three fatty acids in the TAG are saturated and approximately of the same length, each of the layers in the fat bilayer will be approximately two fatty acid chains in height. If the fatty acid on the number 2 glycerol carbon is appreciably shorter than those on the 1 and 3 carbons, each of the layers will be three chains thick. If the TAG is symmetrical, with the 2 position glycerol carbon or both the 1 and 3 carbons... 

Fig. 6. A rough surface of die wall, where the stagnant (thin) chains allow the unbound chains (the middle thick chain without dot) to entangle and adsorb at the effective interface (dashed line). Adsorbed chains (thick chains with dots) are also present. Such an interface can also produce a stick-slip transition upon a coil-stretch transition involving the thick chains. See Ref. [27]. For clarity, we only draw three tethered chains (two adsorbed and one entangled) besides the stagnant chains in one valley. The chains not drawn here, of course, fill up all the space away from the rough wall...

When the chain length (i.e. x) increases, factor bo rapidly becomes unity. Therefore, if the chain thickness d both at moderate and high x is taken into account this does not introduce any changes into the birefringence theory. 

Polymer adsorption is, therefore, a sharp transition with chain thickness changing rapidly in the small interval 6kT of monomer-surface interaction... 

The persistence length model therefore describes the whole spectrum from the more rodlike oligomers (small j) to the well-developed coils (large y). However, the model ignores the finite thickness of the chain, and so it only holds strictly for unperturbed coils. The error due to the finite thickness may be neglected when the persistence length is much greater than the chain thickness. 

The Yamakawa-Fujii theory [2, 3] was developed by using the Kirkwood-Riseman formalism with the effect of chain thickness approximately taken into account. The following remarks may be in order. The Oseen interaction tensor was preaveraged. Force points were distributed along the centroid of the wormlike cylinder (not over the entire domain occupied by the cylinder). The no-slip hydrodynamic condition was approximated by equating the mean solvent velocity over each cross-section of the cylinder to the velocity of the cylinder at that cross-section (Burgers approximate boundary condition). 

Chain, (thick at ends and at changes of direction, thin elsewhere)... 

Fig. 14.1. Illustration of semicrystalline polyethylene on the left, consisting of crystalline lamellae and non-crystalline material (not necessarily indicative of actual morphology). The simulation box in the middle shows two crystalline lamellae and the interlameUar phase in between, which consists of loops thin solid lines), bridge chains thick solid lines), and tails dashed lines). The polyethylene unit cell grey box, middle figure) with coordinate axes (a, b, c) is tilted by an angle Qtnt with respect to the surface normal and the coordinate system x, y, z). The bars with label I indicate the extended interface between crystalline and non-crystalline material, while B denotes regions with bulk-like properties, as explained in the text. On the right, snapshots are shown, viewing along the x- and j/-direction, respectively...

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 ... 

It is observed that Q is independent of the chain thickness, c, and it is proportional with the length of tensioned chain. For the poly(vinyl alcohol) or poly(acrylic acid) in water, where =... 

In the vicinity of the 0 temperature, both x(T) and v(T) vary linearly as a function of relative deviation z= (T-0)/T5 1 from the 0 temperature. The second virial coefficient v can be approximated as v Vqt, where Vo is geometrical (bare) exduded-volume parameter, v—>Vo at T>0. Assuming that the effective chain thickness is on the order of the monomer unit length a, one finds Vo = a. For semiflexible polymers, the second virial coefficient for a statistical segment can be estimated at p = f/fl S 1 as Vp s zi a=za p. ... 

Hat resistors are not useful for radio-frequency designs because of selfresonance effects. In the process chain, thick-film resistors are always fired in the last 850°C sintering cycle to avoid resistance changes due to refirings. 

---