Branched polymers dilute solution

Two basic theoretical models exist to describe the degree of long-chain branching in dilute solution. According to the nondraining model depicted by Zimm and Kilb (25) and Kilb (26), the ratio of intrinsic viscosities of branched and linear polymers is given by ... 

With appropriate caUbration the complex characteristic impedance at each resonance frequency can be calculated and related to the complex shear modulus, G, of the solution. Extrapolations to 2ero concentration yield the intrinsic storage and loss moduH [G ] and [G"], respectively, which are molecular properties. In the viscosity range of 0.5-50 mPa-s, the instmment provides valuable experimental data on dilute solutions of random coil (291), branched (292), and rod-like (293) polymers. The upper limit for shearing frequency for the MLR is 800 H2. High frequency (20 to 500 K H2) viscoelastic properties can be measured with another instmment, the high frequency torsional rod apparatus (HFTRA) (294). 

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. 

Other dilute solution properties depend also on LCB. For example, the second virial coefficient (A2) is reduced due to LCB. However, near the Flory 0 temperature, where A2 = 0 for linear polymers, branched polymers are observed to have apparent positive values of A2 [35]. This is now understood to be due to a more important contribution of the third virial coefficient near the 0 point in branched than in linear polymers. As a consequence, the experimental 0 temperature, defined as the temperature where A2 = 0 is lower in branched than in linear polymers [36, 37]. Branched polymers have also been found to have a wider miscibility range than linear polymers [38], As a consequence, high MW highly branched polymers will tend to coprecipitate with lower MW more lightly branched or linear polymers in solvent/non-solvent fractionation experiments. This makes fractionation according to the extent of branching less effective. 

The core first method has been applied to prepare four-arm star PMMA. In this case selective degradation of the core allowed unambiguous proof of the star structure. However, the MWD is a little too large to claim that only four-arm star polymers are present [81], Comb PMMAs with randomly placed branches have been prepared by anionic copolymerization of MMA and monodisperse PMMA macromonomers [82], A thorough dilute solution characterization revealed monodisperse samples with 2 to 13 branches. A certain polydispersity of the number of branches has to be expected. This was not detected because the branch length was very short relative to the length of the backbone [83]. Recently, PMMA stars (with 6 and 12 arms) have been prepared from dendritic... 

The synthesis of well-defined LCB polymers have progressed considerably beyond the original star polymers prepared by anionic polymerization between 1970 and 1980. Characterization of these new polymers has often been limited to NMR and SEC analysis. The physical properties of these polymers in dilute solution and in the bulk merit attention, especially in the case of completely new architectures such as the dendritic polymers. Many other branched polymers have been prepared, e.g. rigid polymers like nylon [123], polyimide [124] poly(aspartite) [125] and branched poly(thiophene) [126], There seems to be ample room for further development via the use of dendrimers and hyperbran-... 

In this article I review some of the simulation work addressed specifically to branched polymers. The brushes will be described here in terms of their common characteristics with those of individual branched chains. Therefore, other aspects that do not correlate easily with these characteristics will be omitted. Explicitly, there will be no mention of adsorption kinetics, absorbing or laterally inhomogeneous surfaces, polyelectrolyte brushes, or brushes under the effect of a shear. With the purpose of giving a comprehensive description of these applications, Sect. 2 includes a summary of the theoretical background, including the approximations employed to treat the equifibrium structure of the chains as well as their hydrodynamic behavior in dilute solution and their dynamics. In Sect. 3, the different numerical simulation methods that are appHcable to branched polymer systems are specified, in relation to the problems sketched in Sect. 2. Finally, in Sect. 4, the appHcations of these methods to the different types of branched structures are given in detail. 

Nj,=N/f is the number of beads per branch or arm). For larger chains, however, the solvent can penetrate in outer regions of the star and the situation within these regions is more Hke a concentrated solution or a semi-dilute solution. These portions of the arms constitute a series of blobs, whose sizes increase in the direction of the arm end. The surface of a sphere of radius r from the star center is occupied by f blobs. Then the blob size is proportional to rf. Most internal blobs are placed in conditions similar to concentrated solutions and, consequently, their squared size is proportional to the number of polymer units inside them as in an ideal chain. This permits one to obtain the density of units inside the blob, as a function of r ... 

Most important, however, was the discovery by Simha et al. [152, 153], de Gennes [4] and des Cloizeaux [154] that the overlap concentration is a suitable parameter for the formulation of universal laws by which semi-dilute solutions can be described. Semi-dilute solutions have already many similarities to polymers in the melt. Their understanding has to be considered as the first essential step for an interpretation of materials properties in terms of molecular parameters. Here now the necessity of the dilute solution properties becomes evident. These molecular solution parameters are not universal, but they allow a definition of the overlap concentration, and with this a universal picture of behavior can be designed. This approach was very successful in the field of linear macromolecules. The following outline will demonstrate the utility of this approach also for branched polymers in the semi-dilute regime. 

The preceding sections have demonstrated that dendrimers of lower generation are akin to branched polymeric structures. It is therefore to be expected that their flow behavior in dilute solution may be described in terms of the well-known concepts of dilute polymer solutions [14, 15]. Hence, dissolved dendrimers should behave like non-draining spheres. From an experimental comparison of and the immobilization of solvent inside the den-drimer can be compared directly since in this case the dendrimer may be approximated by a homogeneous sphere. Therefore R = 3/5 Rl where Ry, denotes the hydrodynamic radius of the dendrimer. This has been found experimentally [19]. 

Branching Parameter g from. SEC/LALLS. The effect of polymer branching upon the dilute solution configuration of polymers is conveniently expressed as the ratio of intrinsic viscosities of branched and linear polymers of the same chemical composition and molecular weight (35), i.e.. 

Stockmayer and Fixman (2) summarised the state of knowledge of the dilute solution properties of branched polymers in 1953. Dexheimer and co-workers (10) have given a comprehesive survey of the literature up to 1968, including the effects of branching (both short and long) on properties. Nagasawa and Fujimoto (11) have reviewed the results of work on rationally synthesised branched polymers (mostly polystyrenes) up to 1973, with particular reference to their viscoelastic properties. 

Dilute Solution Properties of Model Branched Polymers... 

Fig. 5. Diffusion of C PsCl (open symbols) and cispolyisoprene (filled symbols) in solution at SO °C, from dilute solution to the melt. Left linear polymer, M =- 10 right eight-armed star-branched polymer, M = 4x 10 (Ref.32 >, with permission).

This work examines the effect of long-chain branching on the low-shear concentrated solution viscosity of polybutadienes over a broad range of molecular weights and polydispersity. It will show that the reduction in molecular coil dimension arising from long-chain branching is more sensitively measured in concentrated than in dilute solutions for the polymers examined. 

It is evident that due to polymeric specificity of LC polymers most of the information on their molecular parameters, i.e. molecular mass, conformational state, polymeric chain flexibility and mobility, optical anisotropy and others, may be obtained from studies of dilute solutions of these compounds. However, taking into account that this branch of polymer science has already been reviewed 134>172-176> we will here confine our treatment only to the initial steps of LC phase formation in polymer solutions. 

This was qualitatively shown in investigations of conformational behaviour and intramolecular mobility (IMM) of cholesterol-containing polymers in dilute solutions as of a function of solvent quality 134-136,185-l88) and temperature. Polarization luminescence provides one of the most fruitful methods for the evaluation of IMM l75,176). The method permits to get direct information about rotational mobility of the macromolecule as a whole, as well as about the mobility of the main chains and side branches. This is achieved via the attachment to macromolecules of so called luminescent markers (LM) — anthracylacyloxymethane groups in the case reported. Below are shown the chain fragments with LM which give information on the mobility of main chains (LM-1) and of side groups (LM-2) ... 

---