Schulz-Zimm Distribution Research Articles

Effects of excluded volume and polydispersity on solution properties of lentinan in 0.1 M NaOH solution

Seven lentinan fractions of various weight-average molecular weights (M(w)), ranging from 1.45 x 10(5) to 1.13 x 10(6) g mol(-1) were investigated by static light scattering and viscometry in 0.1M NaOH solution at 25 degrees C. The intrinsic viscosity [eta] - M(w) and radius of gyration s(2)(z) (1/2) - M(w) relationships for lentinan in 0.1M NaOH solution were found to be represented by [eta] = 5.1 x 10(-3)M(w) (0.81) cm(3) g(-1) and s(2)(z) (1/2) = 2.3 x 10(-1)M(w) (0.58) nm, respectively. Focusing on the effects of the M(w) polydispersity with the Schulz-Zimm distribution function, the data of M(w), s(2)(z) (1/2), and [eta] was analyzed on the basis of the Yoshizaki-Nitta-Yamakawa theory for the unperturbed helical wormlike chain combined with the quasi-two-parameter (QTP) theory for excluded-volume effects. The persistence length, molecular weight per unit contour length, and the excluded-volume strength were determined roughly to be 6.2 nm, 980 nm(-1), and 0.1, respectively. Compared with the theoretical value calculated by the Monte Carlo model, the persistence length is longer than that of the single (1 --> 3)-beta-(D)-glucan chain. The results revealed that lentinan exists as single-stranded flexible chains in 0.1M NaOH solution with a certain degree of expansion due to the electrostatic repulsion from the interaction between the OH(-) anions and lentinan molecules.

Distribution of molecular weight and composition in diblock copolymers

Abstract The distribution of molecular weight (MW) and composition of diblock copolymers is considered theoretically. Assuming that the chain end of each block is coupled randomly, the weight-average MW of the block copolymers is given by Mw = w1 (Mw,1 + Mn,2) + w2 (Mw,2 + Mn,1), irrespective of the shape of the distribution of each block, where wi is the weight fraction, and Mn,i and Mw,i are the number- and weight-average MW of each block. In copolymer chains, the chemical compositions as well as the MWs cannot be identical for all polymers, and there exists a bivariate distribution of MW and composition. When the MW distribution (MWD) of both blocks follows the Schulz-Zimm distribution, the bivariate distribution can be obtained analytically. In addition to the bivariate distribution, the full MWD, the average composition as a function of MW, the composition distribution of copolymers having a specified MW, and the overall composition distribution are obtained. The composition distribution, as well as the average composition, becomes independent of MW under the condition σ1/Mn,1 = σ2/Mn,2, where σ i is a parameter indicating the narrowness of the Schulz-Zimm distribution. The present theoretical analysis provides new insight into the design of diblock copolymers.

Characterization of poly(ethylene oxide)-b-poly(L-lactide) block copolymer by matrix-assisted laser desorption/ionization time-of-flight mass spectrometry

A poly(ethylene oxide)-b-poly(L-lactide) diblock copolymer (PEO-b-PLLA) is characterized by matrixassisted laser desorption/ionization time-of-flight mass spectrometry (MALDI-TOF MS) and a block length distribution map is constructed. Although the MALDI-TOF mass spectrum of PEO-b-PLLA is very complicated, most of the polymer species were identified by isolating the overlapped isotope patterns and by fitting the overlapped peaks to the Schulz-Zimm distribution function. Reconstructed MALDI-TOF MS spectrum was nearly identical to the measured spectrum and this method shows its potential to be developed as an easy and fast analysis method of low molecular weight block copolymers.

Crosslinking Index, Molecular Weight Distribution and Rubber Equilibrium Shear Modulus During Polyfunctional Crosslinking of Existing Polymer, 6. Primary Polymer with a Schulz–Zimm Distribution of Molecular Weights

A network model for the crosslinking of already existing polymer molecules with a so-called. Schulz-Zimm distribution of their molecular weights is presented. It is an extension of previously developed statistical network models applied to the crosslinking of primary polymers with several other molecular weight distributions and with crosslinks of any functionality. The model results in the possibility to obtain more insight into the structure of polymers, especially those with narrow distributions of the molecular weight. In more detail. the model can give a perspective on structural network parameters such as the weight fractions of ideal network. of dangling polymer ends, and of those molecules not connected to the network, i.e., the soll fraction, the number of crosslinks in which a polymer molecule is bound. the functionality of the crosslinks, or the average molar mass of the polymer molecules in between the crosslinks M c . Results of calculations are shown for a hypothetical crosslinking process of polymers with various molecular weight distributions. Moreover, the dependency of the network parameters on the polydispersity index and the type of molecular weight distribution is shown. Finally the increase of the functionality of the crosslinks during the ageing process of a 9.9% poly(vinyl chloride) gel as a function of the polydispersity index of the molecular weight distribution is presented.

Investigations on ternary metallocene-based catalyst systems

The influence of the cocatalyst triisobutyl aluminium (TiBA) on the two step reaction of the ternary activation of the metallocene precursor Ph2C(CpFlu)ZrCl2 with TiBA and the activator N,N-dimethylanilinium-tetrakis(pentafluorophenyl)borat (DMAB) was investigated by means of NMR spectroscopy. More than 5 equivalents of TiBA are required for a total conversion of Ph2C(CpFlu)ZrCl2. The reaction exclusiveley leads to the monoalkyl complex Ph2C(CpFlu)ZrClBui, independent of the Al / Zr ratio. The reaction of the Ph2C(CpFlu)ZrCl2 / TiBA mixtures, using 10, 20 and 50 equivalents of TiBA with DMAB, generates two cationic metallocene species. Using 100 equivalents of TiBA exclusiveley leads to the formation of the cationic heterodinuclear metallocene complex [Ph2C(CpFlu)Zr-(μ-H)-(μ-C4H7)-AlBu2i]+. In no case was the degradation or complexation of [B(C6F5)4]− detected, whereas the direct reaction between TiBA and DMAB gives AlBui3−x(C6F5)x compounds. A mechanism was postulated to explain the formation of [Ph2C(CpFlu)Zr-(μ-H)-(μ-C4H7)-AlBui2]+. The catalyst system Ph2C(CpFlu)ZrCl2 / TiBA / DMAB was used for continuous high pressure polymerizations of ethylene at 150 MPa and 180°C. To investigate the influence of additional TiBA and triethyl aluminium (TEA) in the reactor, their concentrations were varied over a wide range. Highest productivities and molecular weights were obtained with low concentrations of TiBA in the reactor. Up to a concentration of 30 molppm Al in the reactor, unimodal polymers were formed with Mw / Mn between 2 and 3. With higher aluminium concentrations, the products formed contained small amounts of waxes. This was due to oligomerization catalyzed by the aluminium alkyl compounds. Using the Schulz-Zimm distribution all MWDs were analyzed with regard to the amount of waxes produced by ethylene oligomerization and with regard to the influence of chain transfer reactions to the aluminium. The rate constants of chain transfer to aluminium in relation to the rate constants of insertion of ethylene were estimated.

Influence of Aluminium Alkyl Compounds on the High-Pressure Polymerization of Ethylene with Ternary Metallocene-Based Catalysts. Investigation of Chain Transfer to the Aluminium

The influence of aluminium alkyl compounds on metallocene-catalyzed high pressure polymerizations of ethylene has been investigated at 150 MPa and 180°C in a continuously operated autoclave. The catalysts were based on the metallocenes bis(cyclopentadienyl)zirconium dichloride (Cp2ZrCl2) and diphenylmethylene (cyclopentadienylfluorenyl)zirconium dichloride (Ph2C-(CpFlu)ZrCl2), which were preactivated outside the reactor with triisobutylaluminium (TiBA) and N,N-dimethylanilinium tetrakis(pentafluorophenyl)borate (DMAP, [PhNHMe2][B(C6F5)4]). The concentrations of triisobutylaluminium (TiBA) and triethylaluminium (TEA) in the reactor were varied over a wide range, using a separate dosing for these two aluminium alkyl compounds. Productivity and polymer properties strongly depended on the type and the concentration of the aluminium alkyl compound used. Highest productivities and molecular weights were obtained with low concentrations of TiBA in the reactor. Up to a concentration of 30 molppm Al in the reactor, unimodal polymers were formed with M̄w/M̄n between 2 and 3. With higher aluminium concentrations the products formed contained small amounts of waxes, due to oligomerization catalyzed by the aluminium alkyl compounds. The molecular weight distributions (MWDs) of these products could be described as a superimposition of two Schulz-Zimm distributions. All MWDs were analyzed with regard to the amount of waxes produced by ethylene oligomerization and with regard to the influence of chain transfer reactions to the aluminium. The rate constants of chain transfer to aluminium, in relation to the rate constants of insertion of ethylene, were estimated.

Influence of Aluminium Alkyl Compounds on the High-Pressure Polymerization of Ethylene with Ternary Metallocene-Based Catalysts. Investigation of Chain Transfer to the Aluminium

The influence of aluminium alkyl compounds on metallocene-catalyzed high pressure polymerizations of ethylene has been investigated at 150 MPa and 180°C in a continuously operated autoclave. The catalysts were based on the metallocenes bis(cyclopentadienyl)zirconium dichloride (Cp2ZrCl2) and diphenylmethylene (cyclopentadienylfluorenyl)zirconium dichloride (Ph2C-(CpFlu)ZrCl2), which were preactivated outside the reactor with triisobutylaluminium (TiBA) and N,N-dimethylanilinium tetrakis(pentafluorophenyl)borate (DMAP, [PhNHMe2][B(C6F5)4]). The concentrations of triisobutylaluminium (TiBA) and triethylaluminium (TEA) in the reactor were varied over a wide range, using a separate dosing for these two aluminium alkyl compounds. Productivity and polymer properties strongly depended on the type and the concentration of the aluminium alkyl compound used. Highest productivities and molecular weights were obtained with low concentrations of TiBA in the reactor. Up to a concentration of 30 molppm Al in the reactor, unimodal polymers were formed with Mw/Mn between 2 and 3. With higher aluminium concentrations the products formed contained small amounts of waxes, due to oligomerization catalyzed by the aluminium alkyl compounds. The molecular weight distributions (MWDs) of these products could be described as a superimposition of two Schulz-Zimm distributions. All MWDs were analyzed with regard to the amount of waxes produced by ethylene oligomerization and with regard to the influence of chain transfer reactions to the aluminium. The rate constants of chain transfer to aluminium, in relation to the rate constants of insertion of ethylene, were estimated.

A segregation approach to molecular weight control in controlled degradation of polymers

In a reactive extrusion process a new approach - degrading only a portion of the input followed by a mixing process with unreacted portion - has been proposed as a method to control the molecular weight and molecular weight distribution (MWD). The number average molecular weight, M n , and the weight average molecular weight, M w , are studied through the Schulz-Zimm molecular weight distribution. The results indicate that M n , and M w can be regulated separately during degradation. M w and the polydispersity index (PDI) of a segregated degradation process are higher than the corresponding results of a uniform system. In the later stage of a segregated degradation the PDI can be higher than that of the original polymers. Starting from a narrow unimodal MWD it is possible to produce a bimodal MWD.

Postgel properties in the statistical crosslinking of heterochains. I. Systems withN types of chains

The heterochain crosslinking model describes nonrandom crosslinking of polymer chains and is an extension of the classical Flory/Stockmayer gelation theory. We consider the postgelation relationship for the system consisting of N types of polymer chains, in which the probability that a crosslink point on an i-type chain is connected to a j-type chain is explicitly given by pij. The analytical solutions for the weight fraction of the sol, the number-average and weight-average molecular weights within the sol fraction, and the crosslinking density within the sol and gel fractions are derived for the systems, with each type of chain conforming to the Schulz–Zimm distribution. Illustrative calculations are shown for the systems consisting of two and three types of chains, and the obtained results agree with those from the Monte Carlo method. © 2000 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 38: 2333–2341, 2000

Dynamical Monte Carlo study of equilibrium polymers: Static properties

We report results of extensive dynamical Monte Carlo investigations on self-assembled equilibrium polymers (EP) without loops in good solvent. (This is thought to provide a good model of giant surfactant micelles.) Using a novel algorithm we are able to describe efficiently both static and dynamic properties of systems in which the mean chain length 〈L〉 is effectively comparable to that of laboratory experiments (up to 5000 monomers, even at high polymer densities). We sample up to scission energies of E/kBT=15 over nearly three orders of magnitude in monomer density φ, and present a detailed crossover study ranging from swollen EP chains in the dilute regime up to dense molten systems. Confirming recent theoretical predictions, the mean-chain length is found to scale as 〈L〉∝φαexp(δE) where the exponents approach αd=δd=1/(1+γ)≈0.46 and αs=1/2[1+(γ−1)/(νd−1)]≈0.6,δs=1/2 in the dilute and semidilute limits respectively. The chain length distribution is qualitatively well described in the dilute limit by the Schulz-Zimm distribution p(s)≈sγ−1 exp(−s) where the scaling variable is s=γL/〈L〉. The very large size of these simulations allows also an accurate determination of the self-avoiding walk susceptibility exponent γ≈1.165±0.01. As chains overlap they enter the semidilute regime where the distribution becomes a pure exponential p(s)=exp(−s) with the scaling variable now s=L/〈L〉. In addition to the above results we measure the specific heat per monomer cv. We show that the average size of the micelles, as measured by the end-to-end distance and the radius of gyration, follows a crossover scaling that is, within numerical accuracy, identical to that of conventional monodisperse quenched polymers. Finite-size effects are discussed in detail.

Modeling molecular weight distribution of comb-branched graft copolymers

Grafting one type polymer onto a different polymer type may yield a comb-branched copolymer. The branching density has a significant effect on its overall molecular weight distribution. A general model is derived to describe the bivariate distribution of molecular weight and branching density for such comb copolymers. The model is applicable for various grafting mechanisms provided the side chains are randomly grafted onto the backbone. The determining parameters are the molecular weight distributions of backbone and side chains, and the branching density. Analytical expressions are obtained for the cases of the side chains having uniform and Schulz-Zimm distributions.

Change in Molecular Weight Distribution During Formation and Degradation of Star Polymers

ABSTRACT A general model has been derived to describe the formation of star polymers via various polymerization mechanisms. The model gives the bivariate distributions of molecular weight and number of branches of the star polymers. Key parameters determining the distributions include the distribution of primary polymers and the branching density. Analytical distribution functions are obtained for the systems having branches of uniform, random, or Schulz-Zimm distributions. The distribution polydispersity of the star polymers approaches unity for sufficiently high branching densities regardless of the distribution of primary chains. The model, has also been extended to give the change in the average molecular weights when star polymers undergo degradation. The scheme of disconnecting branches from star cores is considered. Polydispersity of polymer mixture shows a maximum value at an intermediate degree of degradation. †Present address: PMC Film Canada Inc., Ten Srimati Place, Tottenham, Ontario, Canada L0G 1 W0 †Permanent address: Department of Polymer Science and Engineering, Jiangsu Institute of Petrochemical Technology, Changzhou, Jiangsu, P.R. China.

Solutions of the general equations for the radiation dose dependence of the molecular weights of irradiated polymers with an initial Schulz-Zimm distribution

The general equations for the radiation dose dependence of irradiated polymer molecular weights have been solved exactly. For an initial most probable molecular weight distribution (σ=1), the solutions are analytical and exact. For the general case (σ≠1) the solutions are numerical and exact. The present approach has resulted in the solutions for both σ=1 and σ≠1 being incorporated into a group of FORTRAN computer programs which will solve experimental data for scission and crosslinking yields by both minimization and exact treatments. Simulated data treated using these FORTRAN programs are give. The FORTRAN programs are available from the authors.

Sonication, Fractionation, and Molecular Weight Distribution of Alternating Polydeoxyribonucleotides: Poly(dG-dC) · Poly(dG-dC) and Poly(dA-dT) · Poly(dA-dT)

To obtain well-defined rodlike, low-molecular-weight nucleic acid samples with the narrow distribution, the sonication of Poly(dG-dC) · Poly(dG-dC) and Poly(dA-dT) · Poly(dA-dT) was carried out at 105 W and at 0°C for totally 10 min. Each sample was fractionated into five to six fractions by Sephacryl S-500HR gel-filtration. The molecular weight distribution and weight-average molecular weight, Mw, of the unirradiated, irradiated, and fractionated samples were determined by the gel-permeation-chromatography/low-angle-laser-light-scattering method. Values of Mw were reduced from 5.7×105 to 1.6×105 for Poly(dG-dC) · Poly(dG-dC) and from 7.7×105 to 1.1×105 for Poly(dA-dT) × Poly(dA-dT) by sonication. The degrees of polydispersity, Mw/Mn, of sonicated but unfractionated samples were 1.3 and 1.8, respectively, differing from 2, the value expected for random scission. Mw/Mn of sonicated samples was further narrowed to a 1.04—1.42 range by the gel filtration. Compared with the measured molecular weight distributions, the Schulz–Zimm distribution function was adequate to describe sonicated but unfractionated samples, but the logarithmic-normal distribution represented the experimental distribution of fractionated samples. The double-helical structures of fractionated samples were found to be intact in all cases by various treatments.

R&D Note: On the Calculation of Molecular Weight Distribution from the Moments Using Laguerre Polynomials

ABSTRACTThe method of moments has shown its versatility to calculate various average molecular weights based on the kinetic schemes especially for nonlinear polymerizations. About 40 years ago, Bamford and Tompa (1953, 1954) proposed a numerical method to calculate the molecular weight distribution profile from its moments by application of the Laguerre polynomials. In this paper, the applicability of this method is reinvestigated using some of the representative distribution functions in polymer science such as the Schulz-Zimm distribution (Schulz, 1939; Zimm, 1948) and the logarithmic normal distribution. It was found that the series does not converge fast and even severe oscillatory behaviour is observed depending on the breadth and the shape of distribution functions. The approximation using Laguerre polynomials has shown fairly limited applicability especially for broad distributions that are of significant use in nonlinear polymerizations.

Phase diagrams of quasi-binary polymer systems with LCST/UCST spinodals and hour-glass cloud-point curves

A generalized Flory-Huggins theory is utilized to investigate liquid-liquid phase diagrams of quasi-binary polymer solutions and blends, in which one component may be polydisperse. A temperature and concentration dependent χ parameter is employed. It is illustrated that a polymer solution, in which the polymeric component has a Schulz-Zimm molecular weight distribution, may exhibit combined lower critical solution temperature/upper critical solution temperature spinodals and hour-glass cloud-point curves (CPCs). This is a direct consequence of the critical points being offset from the extrema of the CPCs and spinodals for quasi-binary systems, in contrast to true binary systems (such as solutions and blends of monodisperse polymers). Thus, for quasi-binary systems, it may be hazardous to infer the behaviour of spinodals from observations of CPCs or vice versa.

QELSS diffusion data for moderately broad molar mass distribution polymer samples compared with data from classical gradient techniques

The differences between the QELSS and classical diffusion coefficient of a polydisperse polymer resulting from distinct definitions of experimentally accessible average values are calculated for two assumed specific forms of molar mass distributions. Predicted deviations are compared with the experiment using NBS 706 standard polystyrene. QELSS Dz of this sample relates within 2-4% to the classical diffusion coefficient, if the Schulz-Zimm molar mass distribution is assumed to be valid. In general, differences between the height-area and QELSS diffusion coefficient of about 20% may be found for Mw/Mn ~ 2, and this value may increase above 35%, if strongly tailing molar mass distribution pertains to the sample.

The effect of polymolecularity on the radius of gyration of stiff polymer chains

The z-average of the unperturbed mean-square radius of gyration, 〈S2〉o,z, is calculated for polymolecular Kratky-Porod worm-like chains, commonly used to represent stiff polymer chains. Polymolecularity correction factors for the Schulz-Zimm and logarithmic normal distribution functions are tabulated. By contrast with those for unperturbed random coils, they depend significantly on the chain length.

Evaluation of polydispersity index Mw/Mn by quasielastic light scattering

The response of quasielastic light scattering to the polydispersity of scattering objects has been investigated. A new method of the polydispersity index determination has been suggested, suitable for the range 1.02 ⪬ Mw/Mn ⪬ 2.0 and consisting in the measurement of the dependence of the apparent decay time on the correlator sampling time. The polydispersity index can be determined by comparing these dependences with the theoretical ones obtained using correlation curves simulated for various values of the polydispersity index, assuming lognormal and Schulz-Zimm distributions of molecular weights. The test measurements on polystyrene standards having molecular weights in the range 9 103 – 20.6 106 give polydispersity index values Mw/Mn that are in a good agreement with those given by the manufacturer. The polydispersity index for polystyrene having the molecular weight Mw = 20.6 106 thus determined was Mw/Mn = 1.35.

Comparison of different polymolecularity correction formulae for the burchard‐stockmayer‐fixman relationship

Polymolecularity correction of the original and the modified procedure according to Burchard, Stockmayer and Fixman, especially useful for the determination of the unperturbed dimensions of macromolecules, is derived via the general “principle of corresponding averages” and compared with the polymolecularity correction methods published by Sutter et al. and Rosenvasser et al. It is shown that the polymolecularity correction factor to be used with the mass-average molar mass in the Burchard-Stockmayer-Fixman relationship is identical to that derived from the corresponding averages of this relationship. Numerical calculations with series of hypothetical monomolecular and polymolecular (exhibiting Schulz-Zimm distributions of the molar mass) polymer-homologous polymer samples in three different thermodynamically good solvents also demonstrate the superiority of the correction method according to Sutter et al. as compared to the correction method of Rosenvasser et al.