Molecular size distribution

Expressions for molecular size distributions in multifunctional condensation leading to three-dimensional polymers are derived (Flory, 1941, 1946, 1953 Stock-mayer, 1943, 1952, 1953) by following much the same approach as for linear polymers, though with much more difficulty. Only the results of these derivations will, however, be considered here. The derivations are based on three simplifying assumptions of ideal network formation (1) all functional groups of the same type are equally reactive and independent of size of molecules to which they are attached (2) all groups react independently of one another and (3) no intramolecular reactions (cyclization) occur. 

The simplest possible type of three-dimensional polymer is that formed by stepwise homopolymerization of a multifunctional monomer, such as etherifica- 

The number- and weight-average degrees of polymerization in polymer systems of the above type are given (Ghosh, 1990) by 

Problem 5.29 The reaction of phenol with formaldehyde to phenolic resins is very com- 

The simplest possible type of three-dimensional polymer is that formed by stepwise homopolymerization of a multifunctional monomer, snch as etheri cation of pentaerythritol, or by stepwise copolymerization of equimolar amounts of two monomers having the same fnnctionality /, snch as the condensation of a trihydric alcohol with an equimolar proportion of a tribasic acid, all three functional groups on each monomer being equally reactive. For such polymerizations, the number Nx, the number or mole fraction %, and weight fraction Wx of x-mer molecules in the reaction system are given (Ghosh, 1990 Odian, 1991), respectively, by 

---

Morris K F and Johnson C S Jr 1993 Resolution of discrete and continuous molecular size distributions by means of diffusion-ordered 2D NMR spectroscopy J. Am. Chem. See. 115 4291-9... 

A further feature of anionic polymerisation is that, under very carefully controlled eonditions, it may be possible to produee a polymer sample which is virtually monodisperse, i.e. the molecules are all of the same size. This is in contrast to free-radical polymerisations which, because of the randomness of both chain initiation and termination, yield polymers with a wide molecular size distribution, i.e. they are said to be polydisperse. In order to produce monodisperse polymers it is necessary that the following requirements be met ... 

When applied to the SEC column, the calibrated polydisperse polymer solution provides a large number of data points in a single run. Use of a standard with a molecular size distribution that encompasses the full separation range for the column allows the entire separation range to be calibrated in a single run (Fig. 2.4). 

Light wave technologies provide a number of special challenges for polymeric materials. Polymer fibers offer the best potential for optical communications in local area network (LAN) applications, because their large core size makes it relatively cheap to attach connectors to them. There is a need for polymer fibers that have low losses and that can transmit the bandwidths needed for LAN applications the aciylate and methacrylate polymers now under study have poor loss and bandwidth performance. Research on monomer purification, polymerization to precise molecular-size distributions, and weU-controlled drawing processes is relevant here. There is also a need for precision plastic molding processes for mass prodnction of optical fiber connectors and splice hardware. A tenfold reduction in the cost of fiber and related devices is necessaiy to make the utilization of optical fiber and related devices economical for local area networks and tlie telecommunications loop. 

Flory, P.J. Molecular size distribution in three-dimensional polymers I. Gelation, J. Am. Chem. Soc., 63, 3083, 1941. 

G(S) and G(X) have been estimated by quantifying the effect on molecular size distributions inferred from sedimentation velocity, gel permeation chromatography, and dynamic light-scattering measurements [58]. 

The susceptibility of polyesters and polyamides to interchange reactions, such as may occur in the former between a terminal hydroxyl of one molecule and an interunit ester group of another, was discussed in Chapter III. These interchange processes do not decrease the number of molecules, and hence do not affect Mnj but they might permit some molecular species to be formed in preference to others. In other words, they may conceivably bring about an alteration of the molecular size distribution. 

It follows from considerations of this nature that the form of the molecular size distribution obtained in bifunctional condensation is unaffected by such variations as portion-wise addition of one reactant to the other, polymerization followed by partial degradation, and so forth. Virtually all conceivable polymerization schemes involving reactions... 

Exploration of the molecular size distribution is conveniently carried out by selecting an unreacted A group at random, and then proceeding in a manner which parallels the procedure applied above... 

The other case which we consider is that of a most probable primary distribution. The molecular size distribution after random cross-linking must correspond exactly to that which would be obtained by random condensation of a mixture of bifunctional and tetrafunctional units. This follows as an extension of the correspondence between these two cases considered in the discussion of the critical condition given in the preceding section. The equations developed there are applicable to this case. 

H. K. Pant, D. Vaughan, and A. C. Edwards, Molecular size distribution and enzy-... 

Parisi, L. and von Hunolstein, C., Determination of the molecular size distribution of Haemophilus influenzae type b-tetanus toxoid conjugate vaccines by size-exclusion chromatography, /. Chromatogr. A, 847, 209, 1999. 

A complete theory predicting retention times or volumes as function of molecular size has not yet been formulated for gel permeation chromatography. A specific column or set of columns is calibrated empirically to give such a relationship, by means of which a plot of amount of solute versus retention volume (the chromatogram figure below) can be changed into a molecular-size-distribution curve. Generally commercially available narrow... 

Characterization data for dendrigraft-po y(styr ene)-graft-po y(tert-buty methacrylate) copolymers synthesized with either short (Mw 5000, PtBMA5) or long (Mw 30000, PtBMA30) branches are provided in Table 9.7. The low apparent polydispersities (MJMn = 1.13-1.20) obtained for the graft polymers are indicative of a narrow molecular size distribution. 

Flory, P. J., Molecular size distribution in linear condensation polymers,... 

Whiteway S. G, Smith I. B. and Masson C. R. (1970). Theory of molecular size distribution in multichain polymers. Can. Jour. Chem., 48 33-45. 

Size exclusion chromatography (SEC) polymer elution profiles yield information regarding the molecular size distributions of polydisperse macromolecules. Polymer molecular weight distribution (MWD) represents an intrinsic property which provides direct correlation with many end-use physical properties and a universal criterion for polymer characterization (1). In order to convert elution profiles or chromatograms into MWD information proper calibration methods are required. SEC molecular weight calibration techniques represent experimental approaches for transformation of polymer elution profiles into MWD information and are dependent upon instrumentation, columns, and the polymer/solvent system under study. 

The samples of crude and distillate for GPC analysis were prepared by dissolving the sample in dry additive-free THF to obtain a 25% solution and the solution was filtered through micro-pore filters (Millipore, 0.5 micrometer size). A solution containing both the calibration standard and the sample was used to determine the molecular size distribution of the sample. 

The road asphalt used in this study was obtained from the road as a fresh sample. The road asphalt is composed of asphaltenes (GPC peak at lOOA and petroleum residual oils (15) (GPC peak at n-C QHgo). The GPC of road asphalt is shown in Figure 9. Since petroleum asphaltenes cannot be separated by a lOOA pore size gel column, the asphaltene appears without any separation at the total size exclusion limit of the column. But the nonasphaltene components are separated showing a peak at n-C QHg2. The performance of the road asphalt depends on the asphaltene content as well as on the molecular size distribution of the nonasphaltenic fraction. 

The transition between c and d is rather abrupt, suggesting that we may indeed be witnessing contributions from two more or less discrete populations of molecules with regard to molecular size distributions. However, in view of the extremely complex interplay between intramolecular and intermolecular, as well as intraparticle and extraparticle, parameters in the experiments described here, the possible presence of a continuum of molecular sizes can certainly not be ruled out at this point. 

The resulting chromatogram is then a reflection of the molecular size distribution. The relationship between molecular size and molecular weight depends on the conformation of the polymer in solution. As long as the polymer conformation remains constant, which is... 

The molecular size distribution functions for three-dimensional polymers are derived in a manner analogous to those for linear polymers, but with more difficulty. The derivations have been discussed elsewhere [Flory, 1946, 1953 Somvarsky et al., 2000 Stockmayer,... 

Hence, in principle, we could explain the discrepancies between predicted and observed BAFt values by the reduction of the bioavailability of the compounds caused by sorption to colloidal organic matter present in the culture media. We should note, however, that depending on the nature of the dissolved organic matter (i.e., molecular size distribution, aromaticity, polarity, etc.), the KiDOC value of a compound may vary considerably (see Section 9.4). 

The statistical treatment of random stepwise crosslinking reactions (e.g. polycondensation) neglecting ring formation originates from Stock-mayer and Flory and is explained in Flory s book (55) on a number of examples. Using simple probability statistics, it is possible to calculate the molecular size distribution in the sol and in the gel, fractions of sol and gel, the crosslinking density and the fraction of free functionalities in... 

Stockmayer, W. H. Molecular size distribution in high polymers. In Advancing fronts in chemistry I, p. 47. New York Reinhold 1945. 

---