Statistical segment lengths

Helfand and Tagami [75,76] introduced a model which considered the probability that a chain of polymer 1 has diffused a given distance into polymer 2 when the interactions are characterised by the Flory-Huggins interaction parameter x They predicted that at equilibrium the thickness , d c, of the interface would depend upon the interaction parameter and the mean statistical segment length, b, as follows ... 

ATBN - amine terminated nitrile rubber X - Flory Huggins interaction parameter CPE - carboxylated polyethylene d - width at half height of the copolymer profile given by Kuhn statistical segment length DMAE - dimethyl amino ethanol r - interfacial tension reduction d - particle size reduction DSC - differential scanning calorimetry EMA - ethylene methyl acrylate copolymer ENR - epoxidized natural rubber EOR - ethylene olefin rubber EPDM - ethylene propylene diene monomer EPM - ethylene propylene monomer rubber EPR - ethylene propylene rubber EPR-g-SA - succinic anhydride grafted ethylene propylene rubber... 

Here, b denotes the Kuhn s statistical segment length. The network is represented by a huge chain internally cross-linked at cross-linking points where it touches and at the surfaces of Mf filler particles. The point-like local cross-hnk constraints are easy to handle and can be represented by the term... 

Here we have chosen the Kuhn statistical segment length (=2q) as the unit for measuring length. The partition function Z of the total chain is given by... 

When all lengths associated with polymers are measured in units of the Kuhn statistical segment length 2q, the thermodynamic functions AF, II, and g, given by Eqs. (19)-(21), contain two molecular parameters N = L/2q and d s d/2q and two state variables c = (2q)3 c and a. Thus, numerical solution to Eqs. (23) and (31) provides ci, cA, and a as functions of N and d. The results for the phase boundary concentrations have been found to be represented to a good approximation by the following empirical expressions ... 

Fig. 2.38 Phase diagram computed using the strong segregation limit theory of Helfand and Wasserman (1982) for the poly(ethylene oxide)-poly(butylene oxide) (PEO-PBO) diblock system. Because the ratio of statistical segment lengths aPB0/ 1, the phase diagram is asymmetric about/= 0.5 (Hamley 1997).

Another interesting driving force for surface enrichment in a blend is conformational asymmetry, described e.g. by different statistical segment lengths bA,... 

Intercalated compounds offer a unique avenue for studying the static and dynamic properties of small molecules and macromolecules in a confined environment. More specifically, layered nanocomposites are ideal model systems to study small molecule and polymer dynamics in restrictive environments with conventional analytical techniques, such as thermal analysis, NMR, dielectric spectroscopy and inelastic neutron scattering. Understanding the changes in the dynamics due to this extreme confinement (layer spacing < Rg and comparable to the statistical segment length of the polymer) would provide complementary information to those obtained from traditional Surface-Force Apparatus (SFA) measurements on confined polymers (confinement distances comparable to Rp [36]. 

Statistical segment length of a polymer chain is twice its persistent length... 

Fig. 7. (a) Model of beads the polymer chain is represented as a long flexible immaterial filament, on which interacting beads are strung, a - mean-square spatial distance between subsequent beads (b) chain composed of rods connected by flexible spacers the rods have the length 1 and the diameter d, p = 1/d > 1, a is the mean-square distance between the ends of a flexible spacer (c) flexible chain with rod-like side groups (notations are the same as in (b)) (d) persistent chain of width d and of statistical segment length 1, p = 1/d > 1... 

SonK examples of stiff-chain polymers able to form a liquid-crystalline phase in the solution are listed in Table 1 The ratio of the statistical segment length of a polymer chain, 1, to its width, d, (last rolunm of Table 1) measures the degree of chain stiffness. For flexible macromolecules dA 1 stiff-drain macrcmiolecules are those for which ( A> 1. 

In Eq. (2-12), the number of backbone bonds, n, was used in defining a random-walk step size Other choices are possible. If N, the degree of polymerization, is used instead, then one can define the statistical segment length b, or just b for short, using a formula analogous to Eq. (2-12), that is. 

There have been recent efforts to predict, or at least rationalize, the x parameters of these and other polyolefin-polyolefin blends. Bates et al. (1992) and Fredrickson et al. (1994) suggest that the x parameter is correlated to a difference in statistical segment length of the polymer molecules, on a volume-normalized basis. The volume normalization is required because the definition of the statistical segment length depends on how the monomer unit... 

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