Branched polymer conformation

As yet, models for fluid membranes have mostly been used to investigate the conformations and shapes of single, isolated membranes, or vesicles [237,239-244], In vesicles, a pressure increment p between the vesicle s interior and exterior is often introduced as an additional relevant variable. An impressive variety of different shapes has been found, including branched polymer-like conformations, inflated vesicles, dumbbell-shaped vesicles, and even stomatocytes. Fig. 15 shows some typical configuration snapshots, and Fig. 16 the phase diagram for vesicles of size N = 247, as calculated by Gompper and Kroll [243]. 

Accurate measurements of fluid viscosity are important in many industries for such diverse uses as monitoring syrup manufacture or studying polymer structures such as polymer branching, chain conformation, solvent interactions or polymer molecular weight (MW). Historically, the drop-time type glass capillaries, such as the Ubbelohde or Cannon and Fenske types, have been widely used to measure fluid viscosity. However, this traditional method is tedius and labor intensive, and lacks the desired speed and sensitivity to... 

Keywords. Dendrimer, Dendrimer-polymer hybrid, Conformation, Branched polymers... 

These are interactions between the polymer molecule and the solvent, chain branching, conformational factors arising from polarity, restricted rotation due to resonance, and the bulk of substituents. The above, of course, assumes that the polymer molecules are fully separated from each other. 

Constrained Polymers. The conformation of polymers constrained in various ways, for example, grafted to a flat surface ( brush ), adsorbed on a spherical colloidal particle, or tethered to a central branch point as in star polymers. All such problems involve potentially large nonideal conformational effects and also introduce additional complications associated with site inequivalence within the PRISM formalism. Progress for star polymers is briefly described in the next section. 

The polymer-related problems which can be solved by spectroscopy are many and varied. They may concern chemical aspects and chain structure, e.g. tacticity, mer sequence distribution, chain branching or structure of radicals. They may concern physical aspects, e.g. chain orientation, crystallinity, crystal thickness, miscibility of polymers, chain conformation or chain dynamics. 

The central step in RG is the selection of a specific polymer trial conformation from an entire tree of possible conformations. The essential difference between the continuous-potential RG method and the earlier schemes is that the selection of the trial conformation involves two stochastic steps the first is the selection of a subset of open branches on the tree, the second is the selection of the trial conformation among the open branches. The crucial new concept in RG is that trial directions can be either open or closed. A trial direction that is closed will never be chosen as a part of the chain. For hard-core potentials, a trial direction is closed if it leads to a configuration that has at least one hard-core overlap - otherwise it is open. Therefore, the selection of the open trial directions is deterministic rather than stochastic. In contrast, for continuous potentials, we use a stochastic rule to decide whether a trial direction is open or closed. The probability that direction i is open depends on its energy ui, hence p = p (rq). It is important to note that, in principle, this stochastic rule is quite arbitrary, the only restriction is 0 < Pi < 1 (for hard-core potentials 0 < pt < 1). However, it is useful to apply the following restrictions [112]... 

FIG. 15 Conformations of fluid vesicles for different values of the bending rigidity and pressure increment (a) branched polymer (b) inflated vesicle (c) prolate vesicle (d) stomatocyte. (From Gompper and Kroll [243]. Copyright 1995 APS.)... 

The porous materials that offer the narrowest possible pore size distribution are those that have cylindrical pores of uniform diameter penetrating the entire medium without branching. Branching gives polymer molecules in the junctions extra conformational entropy. An agglomerate of tiny pieces of these porous materials, interlaced with larger voids (much larger than the pore size), should also be chosen. 

SDIBS polymers have been characterized by a variety of methods. The average number of branches BR in our earlier publications, B here to conform with other publications) in the SDIBS core was determined by selective link destmction (see Figure 7.7) using the following equation ... 

Relationships between dilute solution viscosity and MW have been determined for many hyperbranched systems and the Mark-Houwink constant typically varies between 0.5 and 0.2, depending on the DB. In contrast, the exponent is typically in the region of 0.6-0.8 for linear homopolymers in a good solvent with a random coil conformation. The contraction factors [84], g=< g >branched/ <-Rg >iinear. =[ l]branched/[ l]iinear. are another Way of cxprcssing the compact structure of branched polymers. Experimentally, g is computed from the intrinsic viscosity ratio at constant MW. The contraction factor can be expressed as the averaged value over the MWD or as a continuous fraction of MW. 

For star polymers a value of e = 0.5 has been obtained (1, V7) and studies (18) of model comb polymers indicate a value of 1.5. Other work (191 has suggested that e is near 0.5 at low LCB frequencies. For a random LCB conformation of higher branching frequency an e value between 0.7 and 1.3 might be expected, i.e. somewhere between a star and a comb configuration. 

Freire, /. /. Conformational Properties of Branched Polymers Theory and Simulations. VoL 143, pp. 35-112. 

---