Helfand studies

It also appears, as we shall discuss presently, that some macromolecules, such as polyvinylidene fluoride noted above, have exceptional interaction properties, in which the segments may behave rather differently than the chemical monomer units would imply. For these and many related reasons, the studies of E. Helfand at Bell Laboratories on theoretical concepts of interfaces in polyphase systems of macromolecules give us a keen sense of the scope of future discoveries that are possible in this field. 

Weber and Helfand were among the first to propose an MD-based approach to the study of the structure of amorphous polymers. In particular, they modeled polyethylene at 425 °C by employing a box of edge length 18.2 A. The box contained 200 carbons. 

Despite these complications, there are now numerous evidences that the tube model is basically con-ect. The signatory mark that the chain is trapped in a tube is that the chain ends relax first, and the center of the chain remains unrelaxed until relaxation is almost over. Evidence that this occurs has been obtained in experiments with chains whose ends are labeled, either chemically or isotopically (Ylitalo et al. 1990 Russell et al. 1993). These studies show that the rate of relaxation of the chain ends is distinctively faster than the middle of the chain, in quantitative agreement with reptation theory. The special role of chain ends is also shown indirectly in studies of the relaxation of star polymers. Stars are polymers in which several branches radiate from a single branch point. The arms of the star cannot reptate because they are anchored at the branch point (de Gennes 1975). Relaxation must thus occur by the slower process of primitive-path fluctuations, which is found to slow down exponentially with increasing arm molecular weight, in agreement with predictions (Pearson and Helfand 1984). 

Here Qa is the mean value of property Q averaged over basin a (at energy ), and (X) is the spectral weight in the continuum limit of the modes with exponential decay constant X. If 2(0 in fact has the stretched exponential form, then (X) will be proportional to the Laplace transform F(X), for which both numerical (Lindsey and Patterson, 1980) and analytical (Helfand, 1983) studies are available. In the simple exponential decay limit= 1, F(X) reduces to an infinitely narrow Dirac delta function but it broadens as p decreases toward the lower limit to involve a wide range of simple exponential relaxation rates. 

Because of the softness of interactions in block copolymers (here we restrict our consideration to flexible molten blocks above the glass transition temperature), thermal fluctuation in these systems is expected to be significant, especially near the order-disorder transition temperatures (Fredrickson and Helfand, 1987). In addition, the long relaxation times, due to the slowness of the motion of polymers, often lead to metastable and other kinetic states. Thus, full understanding of the self-assembly in block copolymers requires understanding of the nature of fluctuation, metastability, and kinetic pathways for various transitions. Most of this article is focused on theoretical studies of these issues in the simpler AB block copolymers. A key concept that emerges from these studies is the concept of anisotropic fluctuations first, these fluctuations determine the stability limit of an ordered phase second, they are responsible for the emergence of new structures... 

In a previous publication 17, we compared the experimental anisotropies for dilute solutions of labeled polyisoprene in hexane and cyclohexane to several theoretical models. These results are shown in Table II. The major conclusions of the previous study are 1) The theoretical models proposed by Hall and Helfand, and by Bendler and Yaris provide good fits to the experimentally measured correlation function for both hexane and cyclohexane. The model suggested by Viovy, et al. does not fit as well as the other two models. 2) Within experimental error, the shape of the correlation function is the same in the two solvents (i.e, the ratio of t2/ti is constant). 3) The time scale of the correlation function decay scales roughly with the solvent viscosity. 

Fredrickson and Helfand incorporated coupling terms in the transport equation and showed that "these terms provide mobility to molecules that are absorbed into microvoids, even if the molecules have an intrinsic diffusion coefficient in the hole phase that vanishes." These investigators proposed experiments that could provide information on the size and topological connectivity of microcavities (51). Related studies have also been made by Chern, Koros, et al. (52) and by Barrer (53). However, Petropoulos (54) has expressed the opinion that the treatment of Fredrickson and Helfand (51 ) as well as that of Barrer (53) "introduces more diffusion parameters than can reasonably be expected to be measurable on the basis of past experience". Petropoulos also showed that these treatments impose certain limitations on the physical meaning of the diffusion parameters. In the same study, Petropoulos has examined the modification of his dual-mode sorption model ( 5) which are necessary if the Langmuir domains are sufficiently extensive to constitute a macroscopically recognizable phase rather than scattered individual sites (or microcavities) (54). ... 

Fredrickson and Helfand [58] used a selfconsistent Hartree approximation in Eq. (199) to study the Gaussian fluctuations around the solution, Eq, (200). While for ordinary second order transition the local function <4>2(r) > — << >>2 stays small even at Tc, this local fluctuation diverges here as T - Tc The reason is that normally the phase space for critical fluctuations is only the vicinity of a point in reciprocal space (the surroundings of q = 0 for a ferromagnet, the surroundings of a few discrete points qB at the Brillouin zone boundary for antiferromagnets, etc.), while here it is the vicinity of a sphere ( q = q ). Fluctuations lead here to a divergence of the mean square displacement of 4> similar as it happens due to phonons in one-dimensional crystals. 

As seen in Part 4.2, several theoretical approaches have been proposed for the description of the interfacial phenomena. The lattice theories by Helfand, Roe, Noolandi and their collaborators are based on the study of conformation and molecular environment. The derived relations are written in terms of the binary thermodynamic interaction parameter %i2 and the lattice constants. The theories do agree that the interfacial tension coefficient is a function of but the predicted functional dependencies are different Vj %"l2> with exponent n = 1/2 to 3/2, depending on the assumptions. 

The same idea - of having a third species that has less unfavourable interactions with either of the polymers forming the interface, in the hope that the third species will segregate to the interface and lower the interfacial energy - can also be applied to a homopolymer additive. Thus, if one has a homopolymer C such that xac and %bc are less than %ab, one can expect the homopolymer C to segregate to the A/B interface. One case that has been studied both theoretically (Helfand 1992) and experimentally (Faldi et al. 1995) occurs when the polymer C is completely miscible with A but not with... 

Numerous studies were completed on block copolymers in the SSL long before a theory of the detail inherent in the Helfand and Wasserman approach... 

Helfand 1975a). The theory was further developed by Hong and Noolandi (1981). Helfand also applied this theoretical method to the study of microdomain structures in block copolymer systems (Helfand 1975b), and tried to make precise computations of the phase diagrams by numerical methods (Helfand and Wasserman 1976). 

Zuniga et al. performed fully atomistic MD simulations of isolated polyethylene chains in a vacuum [42]. Chains of 50 and 100 carbon atoms were investigated. The results of their study were in qualitative agreement with Helfand s BD simulations of polyethylene. While cooperative transition irs most often involved second-neighbors, overall most of the transitions were isolated. 

Many NMR dynamics studies attempt to correlate the observed data with models of molecular motion. Some of the more successful are the Hall-Helfand [189,190], the Dejean-Laupretre-Monnerie (DEM) [189,191, 192], and the Williams-Watts [189,194,195] models. They predict the relaxation times and NOEs expected from specific types of motion such as unrestricted diffusion or discrete jumps. Very often, distinct parts of the molecule are be best modeled by different functions. For a poly(isobutyl methacrylate) solu-... 

Helfand, M. and Clayton, C. R., "Variable Angle XPS Studies of Passive Ni-P-Cr Metallic Glasses, Corrosion, Electrochemistry and Catalysis of Metastable and Intermetallic, C. R. Clayton and K. Hashimoto, Eds., The Electrochemical Society, Pennington, NJ, 1993. 

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