Flory-Schulz molecular weight distribution

Jackson, C. and Yau, W. W., Computer simulation study of multidetector size-exclusion chromatography. Flory-Schulz molecular weight distribution, in Chromatographic Characterization of Polymers, Hyphenated and Multidimensional Techniques, Provder, T., Barth, H. G., and Urban, M. W., Eds., American Chemical Society, Washington, D.C., 1995, chap. 6. 

A computer simulation of size-exclusion chromatography-viscom-etry-light scattering is described. Data for polymers with a Flory-Schulz molecular weight distribution (MWD) are simulated, and the features of the different detector signals are related to the molecular weight and polydispersity of the distribution. The results are compared with previously reported simulated results using a Wesslau MWD. 

Metals such as Fe, Co, Ni, or Ru on alumina or other oxide supports convert CO and H2 to hydrocarbons. Using different catalysts and reaction conditions either CH4, liquid hydrocarbons, high molecular weight paraffins, methanol, higher alcohols, olefins, and aromatics can be obtained, though rarely (with the exception of CFL, and methanol) with high selectivity. Hydrocarbons typically exhibit a Schulz-Flory type molecular weight distribution. 

The catalytic single-step Alfen process has a good space-time yield, and the process engineering is simple. The molecular weight distribution of the olefins of the single-step process is broader (Schulz-Flory type of distribution) than in the two-step Alfen process (Poisson-type distribution) (Fig. 2). As a byproduct 2-alkyl-branched a-olefins also are formed, as shown in Table 6. About... 

For entries 3-5 the increase in molecular weight observed can be assigned to the increase in the rate of insertion and the rate of termination remains practically the same. An increase of the rate of polymerisation with the steric bulk of the ligand is usually ascribed to the destabilisation of the alkene adduct while the energy of the transition state remains the same. As a chain transfer reaction presumably P-hydride elimination takes place or traces of water might be chain transfer agents. Chain transfer does occur, because a Schulz-Flory molecular weight distribution is found (PDI 2, see Table 12.2). Shorter chains are obtained with a polar ortho substituent (OMe, entry 2) and in methanol as the solvent, albeit that most palladium is inactive in the latter case. 

With a constant temperature and constant concentration of reacting components near the active site the molecular weight distribution of the polymer formed by a single-site catalyst can be described by a Flory-Schulz equation, that can be derived easily from the polymerization mechanism ... 

Chains with monodisperse molecular weight distribution (Mw/Mn = 1.00) can occur in idealized conditions when all polymerizing centers initiate instantaneously and chain termination is absent. In these cases the catalyst is actually an initiator. These living polymerizations are quite rare among transition metal catalysts. More often, random chain termination leads to many chains formed per metal atom. A Schulz-Flory most probable distribution of polyalkene molecular weights (Mw/Mn = 2.00) is the result. In cases when more than one type of active site is present, bimodal or multimodal distributions of molecular weights result (Mw/Mn > 2.00). 

Figure 10.3. Schulz-Flory mole-fraction distribution (left) and corresponding molecular-weight distribution (right) at different degrees of fractional conversion of functional groups (adapted from Flory [27]).

The molecular-weight distribution is a Schulz-Flory distribution as in eqn 10.45. 

Figure 1. Wesslau and Flory-Schulz differential weight-fraction MWDs on a logarithmic scale, where W is the weight fraction and M is the molecular weight. Both distributions are for M = 10,000 gjmol and M /M = 2.0.

The importance of molecular weight distribution in studies of polymerization, polymer processing and the physical and mechanical properties of polymers creates a need for mathematical description of the distribution. Several models are commonly used (Flory [1], Schulz-Zimm... 

In coordination polymerization, monomer forms an adduct with a transition-metal complex, and further monomer is then successively inserted between metal and carbon. Termination occurs when the metal complex splits off from the polymer or the chain is broken intentionally by hydrogenolysis. Since the initiator is restored to its original form, the process is catalytic. The most important industrial processes are Ziegler-Natta polymerizations of a-olefins and employ solid catalysts. Most catalysts for coordination polymerization are hydride complexes of transition metals. An important example is the Shell Higher Olefin Process (SHOP) for homogeneous oligomerization of ethene with a complex nickel catalyst. The molecular-weight distribution is a Schulz-Flory distribution. The rate is first order in the catalyst metal. 

The results above are only valid for tetrafunctional crosslinking of monodisperse polymer. However, in many thermoreversible systems the crosslinks have functionalities that are much larger than four. Moreover, the polymers used are not monodisperse in general. In order to be able to calculate network parameters the present author [39—44] extended the Flory-Stockmayer model for polydisperse polymer which is crosslinked with f-functional crosslinks. It was possible to calculate network parameters for polymers of various molecular weight distributions (monodisperse polymer with D s M, /r3 = 1, a Schulz-Flory distribution with D = 1.5, a Flory distribution with D = 2, a cumulative... 

Schulz-Flory distribution with D > 1.5 and a cumulative Flory distribution with D > 2). The molecular weight distribution of many polymers may be approximated by a cumulative Flory distribution, vriiich is given by... 

For certain combinations of analytical molecular weight distributions (e.g., Schulz-Flory distribution) and fugacity-coefficient expressions, the integral in (33) can be solved analytically and no (time-consuming) summation is required. 

If the average molecular weight of the total oligomer is experimentally available, its monomer dependence allows one to decide which of the two reaction schema (set a or set b) is applicable. In either case, a normal molecular weight distribution (Schulz-Flory distribution) results. This means that the weight fraction nip of oligomer of degree of... 

---