Molecular weight relations for crosslinking of chains with length and site distribution

Many polymer networks are formed by crosslinked polymer chains through reactive sites distributed along the chains. How these sites are distributed as well as the chain length distribution can have a significant effect on properties like the gel conversion and molecular weight. Previous treatments have used simplifying approximations. In this paper we eliminate these approximations and derive computational formulae for weight average molecular weight and gel point for polymer chains of any length and reactive site distribution. Three types of crosslinking are considered: direct coupling of chains (homopolymerization), direct coupling through propagation, and coupling through copolymerization with small monomers.     

Molecular weight distribution in random crosslinking of polymer chains

The molecular weight distribution (MWD) of crosslinked polymer molecules formed during polymeric network formation is the sum of the fractional MWDs containing 0, 1, 2, 3, … crosslinkages. The MWD for polymer molecules containing ?? crosslinkages is investigated for the random crosslinking of polymer chains whose initial MWD is given by the Schulz-Zimm distribution. For a very narrow initial MWD, each fractional MWD with ?? = 0, 1, 2, … is independent and a multimodal distribution is obtained for the whole distribution. When the initial MWD is uniform, the average crosslinking density within the polymer fraction whose degree of polymerization is r, ρ_r is simply given by ρ_r = ρ_gel,c – 2/r irrespective of the extent of crosslinking reaction where ρ_gel,c is the crosslinking density within gel fraction at the gel point. On the other hand, the MWDs with ?? crosslinkages overlap each other with different ?? values significantly for the broader initial distributions, and ρ_r increases with the progress of crosslinking reactions. The value of ρ_r increases with increasing r but levels off asymptotically at large r. The average crosslinking density of polymer molecules containing ?? crosslinkages ρ_?? is an increasing function of k but soon reaches a plateau; sooner for the broader initial MWDs. For ?? ≥ 1, ρ_?? is always larger than the average crosslinking density of the whole reaction system ρ in the pregelation period, i.e., in terms of the crosslinking density, the difference between polymer molecules with and without crosslinkage is most significant. In general, the average crosslinking density ρ, which is convenient to use in describing the nature of the whole reaction system, cannot be considered as a characteristic degree of crosslinking for polymer molecules containing at least one crosslinkage. Consideration of the bivariate distribution of r and k reveals important aspects of the polymeric network formation that have been obscured in the conventional theories in which the averages including linear polymers are solely considered. © 1995 John Wiley & Sons, Inc.     

Thermoreversible crosslinking of polyelectrolyte chains

Thermoreversible crosslinking of polyelectrolyte chains via short-range attractions such as hydrogen bonding induced by uncharged or charged particles is studied within the Flory model of ideal association. Electrostatic interactions between the charges at different linking fractions are taken into account by using a generalized random phase approximation approach which includes the network connectivity. We find that at certain concentration of linking agents an infinitely large polymer network is formed. We calculate the structural gelation lines for linkers of different charges and functionalities.     

Bivariate chain length and long chain branching distribution for copolymerization of olefins and polyolefin chains containing terminal double-bonds

Polyolefins containing long chain branches can be synthesized using certain metallocene catalysts such as Dow Chemical's constrained geometry catalyst. These polyolefins combine the excellent mechanical properties of polymers with narrow molecular weight distribution with the easy processability of polymers containing long chain branches. A mathematical model for the chain length distribution for these novel polyolefins was derived from basic principles and an analytical solution for the chain length distributions of the populations containing different number of long chain branches per polymer molecule was obtained. The analytical solution agrees with the direct solution of the population balances and with a Monte-Carlo simulation model. It is also shown that this solution applies for copolymers using pseudo-kinetic rate constants and Stockmayer's bivariate distribution.     

Hydroxyl functionalized thermosensitive microgels with quadratic crosslinking density distribution

N-isopropylacrylamide (NIPA) based uniform thermosensitive microgels were synthesized by dispersion polymerization by using relatively hydrophilic crosslinking agents with hydroxyl functionality. Glycerol dimethacrylate (GDMA), pentaerythritol triacrylate (PETA) and pentaerythritol propoxylate triacrylate (PEPTA) were used as crosslinking agents with different hydrophilicities. A protocol was first proposed to determine the crosslinking density distribution in the thermosensitive microgel particles by confocal laser scanning microscopy (CLSM). The microgels were fluorescently labeled by using hydroxyl group of the crosslinking agent. The CLSM observations performed with the microgels synthesized by three different crosslinking agents showed that the crosslinking density exhibited a quadratic decrease with the increasing radial distance in the spherical microgel particles. This structure led to the formation of more loose gel structure on the particle surface with respect to the center. Then the use of hydrophilic crosslinking agents in the dispersion polymerization of NIPA made possible the synthesis of thermosensitive microgels carrying long, flexible and chemically derivatizable (i.e., hydroxyl functionalized) fringes on the surface by a single-stage dispersion polymerization. The microgels with all crosslinking agents exhibited volume phase transition with the increasing temperature. The microgel obtained by the most hydrophilic crosslinking agent, GDMA exhibited higher hydrodynamic diameters in the fully swollen form at low temperatures than those obtained by PETA and PEPTA. Higher hydrodynamic size decrease from fully swollen form to the fully shrunken form was also observed with the same microgel.     

Development and validation of COMPASS force field parameters for molecules with aliphatic azide chains

To establish force-field-based (molecular) modeling capability that will accurately predict condensed-phase thermophysical properties for materials containing aliphatic azide chains, potential parameters for atom types unique to such chains have been developed and added to the COMPASS force field. The development effort identified the need to define four new atom types: one for each of the three azide nitrogen atoms and one for the carbon atom bonded to the azide. Calculations performed with the expanded force field yield (gas-phase) molecular structures and vibrational frequencies for hydrazoic acid, azidomethane, and the anti and gauche forms of azidoethane in good agreement with values determined experimentally and/or through computational quantum mechanics. Liquid densities calculated via molecular dynamics (MD) simulations were also in good agreement with published values for 13 of 15 training set compounds, the exceptions being hydrazoic acid and azidomethane. Of the 13 compounds whose densities are well simulated, nine have experimentally determined heats of vaporization reported in the open literature, and in all of these cases, MD simulated values for this property are in reasonable agreement with the published values. Simulations with the force field also yielded reasonable density estimates for a series of 2-azidoethanamines that have been synthesized and tested for use as hydrazine-alternative fuels.     

Ab initio modelling of crosslinking in polymers. A case of chains with furan rings

A quantum chemical model (Abinitio HF-MO and DFT-B3LYP) of polyfurfuryl alcohol crosslinking is shown. Two pathways were considered: (a) a Diels–Alder (D–A) reaction between a dihydrofuran moiety and a furan ring; (b) the addition reaction of electrophilic specimens on conjugated double bonds. The ability for the formation of D–A adducts (dienes and dienophiles) was investigated by the frontier molecular orbital (FMO) theory. Energy gaps between the FMO's of diene and dienophile structures suggest that the occurrence of the D–A reaction is possible. The results suggest that the carbons of the exo double bonds attached to dihydrofuran rings are the sites most likely to be attacked by electrophilic species, while the C₃ and C₄ atoms display the opposite tendency due to their low HOMO electron densities. Results of a thermochemical approach to these reactions at the MP2 level was in agreement with reactivity modelled by MO predictions.     

Light-fueled dynamic covalent crosslinking of single polymer chains in non-equilibrium states

While polymer synthesis proceeds predominantly towards the thermodynamic minimum, living systems operate on the reverse principle – consuming fuel to maintain a non-equilibrium state. Herein, we report the controlled formation of 3D macromolecular architectures based on light-fueled covalent non-equilibrium chemistry. In the presence of green light (525 nm) and a bivalent triazolinedione (TAD) crosslinker, naphthalene-containing polymers can be folded into single chain nanoparticles (SCNPs). At ambient temperature, the cycloaddition product of TAD with naphthalene reverts and the SCNP unfolds into its linear parent polymer. The reported SCNP is the first example of a reversible light triggered folding of single polymer chains and can readily be repeated for several cycles. The folded state of the SCNP can either be preserved through a constant supply of light fuel, kinetic trapping or through a chemical modification that makes the folded state thermodynamically favored. Whereas small molecule bivalent TAD/naphthalene cycloaddition products largely degraded after 3 days in solution, even in the presence of fuel, the SCNP entities were found to remain intact, thereby indicating the light-fueled stabilization of the SCNP to be an inherent feature of the confined macromolecular environment.

Synthetic polymers consume green light as fuel for intramolecular crosslinking, yielding non-equilibrium single chain nanoparticles that can be light-stabilised, kinetically and chemically trapped, or else unfold in the absence of light fuel.     

Convergence of the distribution functions for wormlike chains

The Daniels-type distribution functions of the end-to-end distance of three-dimensional and two-dimensional wormlike chains are obtained to terms of order t^?10, by an operational method with use of a digital computer, where t is the ratio of the total chain contour length to the Kuhn segment length in three-dimensional cases and of the contour length to the persistence length in two-dimensional cases. The convergence of the ring-closure probability and the mean reciprocal distance is examined on the basis of these distribution functions. A similar study of the moment-based distribution functions is also made.     

The thermomechanical method for evaluation of the crosslinking parameters in irradiated polyethylene

The measurement of penetration deformation above the melting point was used for determination of viscoelastic parameters, compliance and retardation time, in γ-irradiated polyethylene systems. The time-dependent penetration deformation was followed over two creep and recovery cycles. The five-element mechanical model equation was used for calculation of amplitudes, time factors and remained deformations. The modulus of elasticity, derived from the equilibrium compliance was used to obtain M_c, the value of the average molecular weight between crosslinks.     

Moments and distribution function of polyelectrolyte chains

It is demonstrated that the moments R(2k) of the end-to-end distance distribution function f(r) of charged wormlike chains with excluded volume effects in solution with added salt, in the Debye-Huckel approximation, obey a remarkably simple relation. It is shown that the R(2k) can be expressed as weighted sums of the corresponding moments of ideal wormlike chains. As an application of this method, we show that the Fourier transform of f(r) can also be represented by a superposition of distribution functions of ideal chains. The quantities so calculated are compared with the results of Monte Carlo simulations. Excellent agreement between them is observed.     

Moments and distribution functions for polypeptide chains. Poly-L-alanine.

Statistical mechanical averages of vectors and tensors characterizing the configurations of polypeptides have been calculated for poly-L-alanines (PLA) of xu = 2-400 peptide units. These quantities are expressed in the reference frame of the first peptide unit, the X axis being situated along the virtual bond, the Y axis in the plane of the peptide unit. The persistence vector a identical to (r) converges rapidly with chain length to its limit a infinity which lies virtually in the XZ plane. Configurational averages of Cartesian tensors up to the sixth rank formed from the displacement vector p = r-a have been computed. For xu greater than 50 the even moments of fourth and sixth rank formed from the reduced vector p for the real chain are well repreented by the freely jointed chain with 21.7 virtual bonds equivalent to one of the model. The moments of p display assymmetry for xu less than 50. Density distribution functions Wa(p), evaluated from the three-dimensional Hermite series truncated at the term in the polynomial involving the tensors of p of sixth rank, display no obvious symmetry for xu less than 50. Approximate spherical symmetry of the distribution of p about a is observed only for xu greater than or equal to 100.     

Postgel properties in the statistical crosslinking of heterochains. I. Systems with N 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 p_ij. 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     

Peroxide crosslinking of linear low-density polyethylenes with homogeneous distribution of short chain branching

Extraction and swelling experiments were carried out in order to measure the crosslinkability by dicumyl peroxide (DCP) of new homogeneous linear low-density polyethylenes (LLDPEs) with different molecular weight, branch content, and length of branches. Considering that the tertiary radicals are most probable macroradicals, our results indicate that tertiary carbons in LLDPEs could be either sites for scission reaction of effective coupling, depending upon degree of branching and to a lesser extent on degree of crosslinking. In the range of branching between 16 and n CH₃/1000 C (n ? 31 for low DCP concentration and decreases with DCP content approaching the value of 23 for the range above ca. 6 DCP moles per number average PE mole), the tertiary carbons seem to be rather the sites for effective coupling. In this range of branching an increase of crosslinkability with branch concentration and molecular weight of PE is due to the increase of crosslinking efficiency, but the scission probability is very low. At branch content exceeding the value of n CH₃/1000 C, the dominant reaction on tertiary carbons is scission, leading to a small decrease of crosslinkability with increasing branch content. The possible mechanisms favoring tertiary carbons for either coupling or scission are discussed. © 1995 John Wiley & Sons, Inc.     

Conformation and normal frequency distribution of polymer chains

It is argued on the basis of a simple model for a polymer chain that characteristics of conformation can be determined from the analysis of the frequency distribution of backbone vibrations. In a chain with one degree of freedom per repeat unit, a fold tends to uncouple the motions on each side of the fold. This transfers highest and lowest modes towards the mean value. Conversely, a local unfolding of a random chain enhances the coupling and produces out-of-band modes. From the ordering of these modes, statistical aftereffects can be detected. The Green's function method allows a calculation of the single localized mode as a function of angular parameters. The dynamical matrix elements of the Kirkwood-Pitzer model are calculated for a folded polyethylene backbone chain. The eigenvalue distribution is determined with the Givens-Sturm procedure. It indicates a one-to-one correspondence between modes in the vicinity of 670 cm.^?1 in the stretch-bend gap and chain folds, for low concentration of folds. We conclude that conformations should have two characteristic parameters distinctively reflected in the frequency distribution, namely the range of allowed dihedral angles and the ordering of configurations along the chain. The frequency distribution for a given conformation depends however on the force constant ratios when the degree of freedom is greater than one. Interpretation of the frequency spectrum is then made by adjustment of dynamical and geometrical parameters. These are seen to be dependent variables with respect to the spectrum.     

Determination of the kinetic parameters of pseudoliving radical polymerization by linearizing the chain length distribution of macromolecules

A new approach is proposed for determination of quantitative kinetic parameters of pseudoliving radical polymerization proceeding in the mode of reversible termination, which include the reversible-termination rate constant, the length of the elementary step (number of growth events beginning from chain initiation to termination), and the number of steps that compose the chain during its growth. The method is based on the linearization of the chain length distribution of macromolecules and constitutes further development of the previously known approach to the determination of the chain transfer constant in nonliving polymerization. The applicability of the method was theoretically substantiated and experimentally verified using, as an example, some systems with reversible inhibition by nitroxides.