Star polymers in a good solvent

Simulations, both MC and MD, have been used to test these scaling predictions and to determine other properties of a star polymer, including the static structure factor in the dilute limit. At present, it is not possible to simulate a melt or even a semi-dilute solution of many-arm star polymers due to the long relaxation times. For few-arm stars f 12) MC methods are clearly most efficient, while for large number of arms, MD methods work very well. For small /, the density of monomers of the star is low almost everywhere and static MC methods in which one generates the chains by constructing walks can be Using this method, 

Batoulis and Kremer were able to make very accurate estimates of the exponent 7 as well as p r) and (R ) for/ 6 in a good solvent. Dynamic MC also works well in this limit, particularly if one invokes nonlocal moves, such the pivot algorithm.However as /increases, the interior becomes very dense and many of these methods fail or become inefficient. In this case, one can use either MD methods or a local stochastic MC method such as the bond fluctuation method on a lattice or a simple off-lattice MC in which one attempts to move one monomer at a time. It is also possible to use nonlocal moves in the dilute, outer regions of the star and local moves near the interior, though this has not been done to the best of our knowledge. 

In a good solvent, the results from various groups agree very 

Experimental results for polystyrene and polyisoprene are also shown. Note that by normalizing by R g ) single arm, the results for 

Kremer for 3 / 6. Using a SCF model, Dan and Tirrell and Wijmans and Zhulina found that p(r) scaled as predicted by eq. (9.15). As seen in Fig. 9.7, the scaling is valid even at very short distances from the center indicating that, at least for these stars, the simulations clearly exhibit the scaling predicted for a swollen star and there is no need to consider the core region. This core region is important for micellar stars and for chains grafted onto a small colloidal particle. 

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Calculate the density profile in an /-arm star polymer in a good solvent as a function of the distance from the branch point. Ho y does the size of the polymer depend on the number of arms / the degree of polymerization N, monomer size b, and the excluded volume v > 0 ... 

Fig. 9.9 Form factor P q) versus qu for four-star polymers in a good solvent with/=10-50 for JV = 50 simulated by MD for a purely repulsive Lennard-Jones interaction between nonbonded monomers from Ref. 96. Also shown are the results for two linear polymers with 50 and 100 monomers. The data have been offset for clarity.

Fig. 4.13 Comparison of experimental gas-liquid coexistence binodals (data) compared to GFVT (curves). Left panel, spherical colloids mixed with polymer chains in a -solvent for q = 0.84 (open triangles, [20]), 1.4 (stars, [21]) and 2.2 (crosses, [21]). Right panel, colloidal spheres plus polymers in a good solvent for q = 0.67 (open squares, [20]), 0.86 (inverse filled triangle, [54]) and 1.4 (pluses, [20])...

Figure 3 The blob model of a star polymer in dilute solution under good or 0 solvent conditions (a), semidilute solution of star polymers (b), and star polymer in a poor solvent (c).

Sikorski and Romiszowski455 study confined branched star polymers by on-lattice MC simulation. Attractive forces are excluded and only excluded volume accounted for, thus making the simulations relevant for chains in a good solvent. Contrary to expectation, they find that the diffusion constant is very similar for either moderate or highly confined chains and scales approximately as A 1, though a more accurate representation is suggested by... 

The box-like cell model of a PE star can be considered as a generalization of a classical mean-field Flory approach, which was first suggested to describe the swelling of a polymer chain in a good solvent [90], The Flory approach estimates the equilibrium dimensions of a macromolecule, as a function of its parameters, by balancing the free energy of intramolecular (repulsive) interactions with the conformational entropy loss of a swollen chain. Within the box-like approximation, the star is characterized by the radius of its corona, R (end-to-end distanee of the arms), or by the average intramolecular concentration of its monomers ... 

The scaling theory captures the essential features of star polymer in both good and 0 solvent conditions and predicts power-law dependences for the overall star size Rsai on number of branches/and the degree of polymerization N. These scaling predictions were tested by molecular dynamics and Monte Carlo simulations " and experimentally. " Although certain discrepancies were detected (see, e.g., the discussion in Reference 68), a simple bloh model remains an important theoretical tool to interpret experimental data on nonionic star macromolecules. 

In the neighborhood of their c, monodisperse multiarm stars in a good solvent are expected to crystallize. This is a consequence of the concentration-induced enhanced osmotic pressure which outbalances the elastic energy of the sUetched arms. A variety of crystalline phases have been predirted from an effective interactions approach using V r) of eqn [ 18]. At higher concentrations c c, the crystal shoirld melt due to the enhanced osmotic pressure, hence recovering the semidilute polymer solution behavior. However, these prediaions have not been supported convincingly by experimental evidence. It has been... 

Fig. 9.12 Typical configurations of systems of 50 chains of length N =50 grafted on a line at linear density (a) pm = 0.38, (b) 3.14, (c) 6.28 in a good solvent. The chains are projected on the plane perpendieular to the grafting lines. Note that these are not star polymers. (Results are from Ref. 104.)...

Comparing the radii of gyration determined in a good solvent is not necessarily the appropriate solution because branched and linear polymers swell differently. Indeed, the expansion coefficient (a) varies with the type of structure and branched < linear- Ii the casc of Star polymers, the closer to the core, the more stretched are the branch segments and at the same time a star polymer as a whole cannot swell as much as its linear equivalent. 

Figure 5 Zero-concentration form factors of star polymers with different functionalities. Data obtained in a good solvent, methylcyclohexane-di4 with SANS. From bottom to top /= 8 (polyisoprene),/=18 (polyisoprene),/= 32 (polybutadiene),/= 64 (polybutadiene), and/= 128 (polybutadiene). The data are offset vertically for clarity. The solid line represents Eq. (27). Arrows indicate Q=the onset of the asymptotic regime in which scattering is caused by the swollen blobs in the corona. (From Ref 31.)...

Elastic and quasi-elastic (NSE) neutron scattering experiments were performed on dilute solutions of linear poly(isoprene) (PIP) polymers and of PIP stars (f = 4,12,18) [150]. In all cases the protonated polymers were dissolved in d-benzene and measured at T = 323 K, where benzene is a good solvent. Figure 50 shows the results of the static scattering profile in a scaled Kratky representation. In this plot the radii of gyration, obtained from a fit of the... 

Star polymers, with a structure where chains of different molecular weight and/or chemical nature radiate from a common junction point, are expected to have different solution behavior compared to regular or symmetric star polymers. By forcing chemically different arms to be joined, expansion of the molecular dimensions is predicted (see Sect. 3.1.1) as a consequence of the increase in the number of heterocontacts in good and theta solvents or new kinds of structures can be formed in selective solvents. 

A more detailed explanation of Eqs 3 and 4 shall be given in section 2. For our present purposes, the main conclusion is that Jc is not dependent on X for linear chains. But, without any calculation, we can expect that a branched object will enter much more painfully in a narrow tube. Thus permeation studies may provide an interesting assessment of branching. In section 3, we discuss briefly the case of star polymers [11] which is relatively easy to visualise. Then in section 4, we study the morerealistic problem of branched objects [12]. All our discussion is very crude restricted to polymers in very good (athermal) solvents and restricted to the level of scaling laws. 

Studies on two symmetrical star polymers have been reported by Schrag, Ferry et al. for dynamic properties at infinite dilution. One is for a four armed polybutadiene in two good solvents, a-chloronaph-thalene and chlorinated diphenyl (Aroclor 1232) (101). The polymer... 

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