Experimental evidence for trapped chain entanglements: their influence on macroscopic behaviour of networks

The swelling and elastic properties of poly(dimethyl siloxane) networks prepared by end-linking and subsequently swollen in heptane and toluene have been investigated as a function of the volume fraction, v_c, at which networks are generated. Increases in both swelling degree and shear modulus with v_c demonstrate the increase in number of trapped entanglements with v_c. The results may be described by a simple scaling law approach.     

Elastic properties of swollen real polymer networks

Summary In this paper a new set of stress-strain relations (uniaxial, equi-biaxial, unequibiaxial extensions and pure shear) for swollen networks were presented. Which are derived from the molecular theory of rubber elasticity with constraints of junctions and trapped entanglements and with crosslinks, trapped entanglements and carbon black-polymer interactions. They successed in relating the elastic equation of state to the volume fraction of polymer in swollen networks by three molecular parameters C₁, C₂ and C₃. The relation of stress-strain for uniaxial extension was verified by experiments. It is shown that this relation can successfully predict the dependence of V2 on the C₁, C₂ and C₃, and the contribution of modulus for swollen networks from the trapped entanglement which it shows that role of entanglements can only approach to a limited value, never to zero.     

Polymer Network Mobility and Environmental Stress Cracking Resistance of High Density Polyethylene

Zero shear viscosity and molecular weight between entanglements (M _e ) are determined from dynamic oscillatory shear experiments. Lower M _e value means higher number of entanglements in the system and is associated with increasing strain hardening stiffness. With the understanding that strain hardening is related to environmental stress cracking resistance (ESCR) of high density polyethylene (HDPE), M _e is then related to the ESCR of several resins in this study. The inversely proportional relationship between M _e and ESCR indicates that low network mobility due to an increasing number of chain entanglements increases the ESCR of HDPE.     

Entanglement networks of 1,2-polybutadiene cross-linked in states of strain. V. Relaxation phenomena and calculations of entanglement trapping

Linear 1,2-polybutadiene is cross-linked near its glass transition temperature by γ-irradiation while strained in simple extension with a stretch ratio λ_o. After release, the sample retracts to a state of ease (λ_s). From λ_o, λ_s, and stress-strain measurements in extension from the state of ease, the concentrations of network strands terminated by trapped entanglements (v_N) and by cross-links (v_x) can be calculated. For v_x/v_N ≡ R′_o > 1, retraction to the state of ease is rapid. For R ? 0.3 or less, retraction is slow and extends over many logarithmic decades of time scale. When an eased sample is stretched to λ_o where the cross-links do not contribute to stress, the subsequent stress relaxation of the entanglement network toward equilibrium is also very slow if R′_o is small. The slow timedependent processes are attributed to a high proportion of untrapped entanglements on dangling branched structures. The concentration of trapped entanglement strands, v_N, can also be calculated from the equilibrium stress at λ_o. The fraction of trapped entanglements agrees rather well with the predictions of the theory of Langley.     

Mechanistic studies in tackified acrylic emulsion pressure sensitive adhesives

Twenty‐three wt % aqueous tackifier dispersion based on glycerol ester abietic acid (T_g = 64°C, M_w = 940) was added to emulsion polymer 50/32/15/3 poly(2‐ethyl hexyl acrylate‐co‐vinyl acetate‐co‐dioctyl maleate‐co‐acrylic acid) pressure sensitive adhesive (PSA). From these latices, 25 μm thick films were cast. The films were dried at 25°C for 24 h or at 121°C for 5 min. Dynamic mechanical analysis (DMA) of the films included measuring elastic modulus (G′) and damping factor (tan δ). Under the above drying conditions, the films did not produce significant differences in their DMA and PSA properties as measured by loop tack, peel, and shear holding power. DMA of the tackified acrylic film showed thermodynamic miscibility between the tackifier and polymer regardless of the drying conditions. Microgels formed during emulsion polymerization of the acrylic PSA brought inherent weakness to the tackified film properties. In the neat acrylic PSA film, these discrete networks entangled with the uncrosslinked chains while in the tackified film, these networks could not form entanglements due to the increased molecular weight between entanglements for the uncrosslinked chains. This lack of network entanglements caused shear holding power of the tackified acrylic PSA film to be 4× lower than that of the neat acrylic PSA film. © 2000 John Wiley & Sons, Inc. J Appl Polym Sci 76: 1965–1976, 2000     

Effect of crosslink density on molecular relaxations in diepoxide-diamine network polymers. Part 2. The rubbery plateau region

A comparison of the prediction of the theory of rubber elasticity with the experimentally observed variation of the shear storage modulus, G, as a function of crosslink concentration shows that deviations occur when the network strand concentration in diepoxide-diamine polymers exceeds approximately 1.5 mole kg^?1. The rapid rise in G above this level is accounted for in terms of the increasing importance of non-Gaussian chain statistics and steric interactions. It is also established that the contribution from entanglements is significant and the behavior over the entire crosslink density range can be described by the following equation where v and ?T_e are the concentrations of elastically active strands which orginate from fixed points and entanglements respectively, ψ is an empirical constant related to the importance of the non-ideal behavior, and ? is the so-called “front factor”. This latter constant is found to depend on the functionality of the network junctions, varying from 0.9, for a system with tetrafunctional junctions, to an average of 0.53 for those networks with trifunctional junctions.     

Entanglement networks of 1,2-polybutadiene crosslinked in states of strain. IV. States of ease and stress–strain behavior

Linear 1,2-polybutadiene, glass transition temperature (T_g) ?18°C, is crosslinked at ?10°C, to ?20°C by γ irradiation while strained in simple extension, with extension, ratios (λ₀) from 1.2 to 2.7. After release, the sample retracts to a state of ease (λ_s) at room temperature. From equilibrium stress–strain measurements up to a stretch ratio relative to the state of ease (Λ) of 1.2, together with λ₀ and λ_s, the concentration of network strands terminated by trapped entanglements (νN) is calculated. For this purpose, a three-constant Mooney–Rivlin formulation is used, in which the entanglement network is described by Mooney–Rivlin coefficients C_1N and C_2N, whereas the crosslink networks is described by the coefficient C_1x only. The ratio ψ_N = C_2N/(C_1N + C_2N) is estimated from parallel studies of nonlinear stress relaxation of the uncrosslinked polymer, taking into account the thermal history before and during irradiation. For substantial degrees of crosslinking, i.e., for R_0′ = ν_x/ν_N > 0.4 (where ν_N is the concentration of network strands terminated by crosslinks), and for λ₀ < 1.8, C_2N agrees rather well with the value obtained from stress relaxation of the uncrosslinked polymer in the range of time scale where it is nearly independent of time (1.87 X 10⁵ pascals). The corresponding value of ν_N is 2.3 × 10^?4 moles/cm³, in good agreement with that obtained from viscoelastic measurements of the uncrosslinked polymer in the plateau zone (2.5 × 10^?4). However, for R_0′ ? 0.2, smaller values of C_2N and ν_N are obtained, indicating that for low degrees of crosslinking the entanglements are not completely trapped. Also, for higher values of λ₀, C_2N and ν_N turn out to be somewhat smaller. Similar, less extensive results were obtained previously on a 1,2-polybutadiene with somewhat higher vinyl content and a higher T_g. Crosslinked samples of both these polymers were subjected to equilibrium stress–strain measurements in simple elongation from the state of ease at higher strains up to Λ = 1.7. The results agreed closely with calculations from the three-constant Mooney–Rivlin theory.     

Solute diffusion in swollen membranes. IV. Theories for moderately swollen networks

A theoretical model was developed to describe solute transport through moderately swollen networks. The model is an extension of the previous analysis of Peppas and Reinhart (1983), and it describes the normalized effective diffusion coefficient of the solute through the network as a function of the equilibrium degree of swelling, Q, the hydrodynamic radius of the solute, r_s, the number average molecular weight between crosslinks, M¯_c, and a function f(ξ) of the mesh size ξ, which takes into consideration barriers due to crosslinks, entanglements, etc. For the development of this model, the Cohen–Turnbull (1958) free volume theory was modified to incorporate topological and mobility characteristics from de Gennes' analysis (1979).     

Modelling tie chains and trapped entanglements in polyethylene

A Monte Carlo random walk model was developed to simulate the chain structure of amorphous layers in polyethylene. The chains emerging from the orthorhombic crystal lamellae were either folding back tightly (adjacent re-entry) or performing a random walk (obeying phantom chain statistics) forming statistical loops or tie chains. A correct amorphous density (ca. 85% of the crystalline density) was obtained by controlling the probability of tight folding. Important properties like fracture toughness depend on the number of chains covalently linking together the crystalline regions. The model structure was analysed with a novel numerical topology algorithm for calculating the concentration of tie chains and trapped entanglements. The numerical efficiency of the algorithm allowed molecular cubic systems with a side length of 100 nm to be readily analysed on a modern personal computer. Simulations showed that the concentration of trapped entanglements was larger than the concentration of tie chains and that the thickness of the amorphous layer (L_a) had a greater impact than the crystal thickness (L_c) on the tie-chain concentration. In several other commonly used models, such as the Huang–Brown model, the influence of trapped entanglements and the effect of the L_a/L_c ratio are neglected. Simulations using as input the morphology data from Patel generated results in agreement with experimental rubber modulus data.     

Characterization of nonideal networks by stress-strain measurements at large extensions

It is shown how a characterization of unfilled, amorphous rubber networks can be evaluated from uniaxial stress-strain measurement data. Beside the cross-linking density, the relative scission probability during the curing procedure is evaluated, which determines the amount of dangling free chain ends and the number of trapped entanglements. These values are found from the C₂ term of the Mooney-Rivlin equation by using the predictions of a tube model. A necessary requirement for applying stress-strain measurements at large extensions is the consideration of the finite extensibility component of the reduced stress. It is taken into account by using a numerical procedure, which derives from a series expansion of the inverse Langevin approximation. The experimental results found at natural rubber networks cross-linked with thiuram (TMTD) and peroxid (DCP) show that network defects cannot be neglected in the DCP networks. They are assumed to be connected to the worse tensil strength properties compared to the TMTD networks. © 1993 John Wiley & Sons, Inc.     

Viscoelasticity of randomly crosslinked EPDM networks

The network structure of plasticized EPDM compounds, crosslinked with resol at different concentrations, was studied by means of rheological methods consisting in oscillatory shear tests, to determine the equilibrium modulus G_e, and long-time relaxation tests in compression followed by strain recovery (a protocol that also yielded values of the compression set of the samples). G_e results were analyzed with respect to the phenomenological model of Langley and Graessley which takes into account the contribution of crosslinks and trapped entanglements to the shear equilibrium modulus. A correction was introduced in order to take into account the presence of plasticizer. The measurement of the soluble polymer fraction in the different samples allowed a more detailed characterization of the networks to be carried out, following a molecular approach by Pearson and Graessley. This method enabled to calculate the crosslink density and trapping factor, but also to compute the probability ψ₁ for an un-crosslinked polymer unit to belong to a dangling chain. This probability was shown to increase as resol concentration, and then crosslink density, decreased. The empirical Chasset-Thirion equation was used to model the long-time relaxation data for each sample. Chasset-Thirion parameters were interpreted by Curro and Pincus within a theoretical framework based on the idea that the longest relaxation times are associated with the pendent chains of the network. The relaxation times, obtained from the fitting of experimental relaxation moduli, dramatically increased as the crosslink density decreased. This result corroborates the evolution of ψ₁: both tend to demonstrate that in the present compounds, the decrease of crosslink density is accompanied by an increase of the number and length of the dangling chains, leading to increasing relaxation times. The large soluble fraction and long pendent chains of samples showing the lowest crosslink densities were responsible for their poor elastic recovery. The relaxation data were used to model the elastic recovery of the compounds and predict their compression set profiles. Very satisfactory agreement was obtained between experimental data and computations.     

Fabrication and apparent kinetic study of poly(methyl methacrylate)/poly(octyl acrylate) and poly(octyl acrylate)/poly(methyl methacrylate) latex interpenetrating polymer networks

Two latex interpenetrating polymer networks (LIPNs) were synthesized with methyl methacrylate (MMA) and octyl acrylate (OA) as monomers, respectively. The apparent kinetics of polymerization for the LIPNs was studied. This demonstrates that network II does not have a nucleus formation stage. The monomers of network II were diffused into the latex particles of network I and then formed network II by in situ polymerization. It indicates that the polymerization of network I obeys the classical kinetic rules of emulsion polymerization. But the polymerization of network II only appears a constant‐rate stage and a decreasing‐rate stage. The apparent activation energies (E_a) of network I and network II of PMMA/POA were calculated according to the Arrhenius equation. The E_a values of POA as network I (62 kJ/mol) is similar to that of POA as network II PMMA/POA (60 kJ/mol). However, the E_a value of PMMA as network II POA/PMMA (105kJ/mol) is higher than that of PMMA as network I (61 kJ/mol). Results show that the E_a value of the network II polymerization is related to the properties of its seed latex. © 2010 Wiley Periodicals, Inc. J Appl Polym Sci, 2010     

Diffuse shear banded zones of blends of polystyrene and poly(2,6-dimethyl-1,4-phenylene oxide)

The formation, mechanical properties, thermal characteristics, and density of diffuse shear banded zones of polystyrene, poly(2,6-dimethyl-1,4-phenylene oxide) (PPO), and their miscible blends were studied. A significant increase in density of 0.2 to 0.3 percent was found for the diffuse shear banded zones. Differential scanning calorimetry results revealed a volume recovery process that occurs below T_g for the diffuse shear banded zones. The post-yield-stress drop, anelastic shear strain within the zone, and anelastic tensile strain were all found to decrease with increasing PPO content in an identical manner. The sharp shear band to diffuse shear banded zone transition was related to chain mobility, molecular packing, and free energy as manifested in the post-yield-stress drop. The decrease in anelastic shear strain with increasing PPO content for the blends is possibly related to the beta transition and length between entanglements.     

Mechanical degradation of high molecular weight polymers in dilute solution

The mechanical shear degradation of polydisperse polyisobutene and monodisperse polystrene in oils of different viscosities in the concentration range of 0.1% to 1% was studied using a high-shear concentric cylinder viscometer under laminar and uniform well-defined shear field conditions. Molecular weight distributions (MWDs) were measured by gel permeation chromatography (GPC). Degradation of polydisperse polyisobutene solutions narrows the distributions principally through the breaking down of large molecules. Degradation of monodisperse polystyrene broadens the distributions at lower shear stress. At higher shear stresses, the distributions do not broaden as much but are still broader than those of the original polymer. The final M_w/M_n ratios are considerably different from the value of 2 expected for random degradation. Hence, the degradation is likely a nonrandom process. It was found that the extent of degradation has a negative concentration dependence coefficient at relatively high molecular weight and a positive concentration dependence at lower molecular weight. Competing mechanisms of “stretching” and “entanglements” for degradation were postulated to explain the results. The degradation data indicate that the shear stress is the controlling parameter, not the shear rate. The shear degradation is independent of initial molecular weight and viscosity of the solvent.     

Second order non-linear optical interpenetrating polymer networks based on polyurethane or poly(methyl methacrylate) and epoxy polymer

Two interpenetrating polymer networks (IPNs), (one pseudo-IPN consisting of a linear polyurethane/epoxy-based polymer network and one full-IPN consisting of a poly(methyl methacrylate)/epoxy-based polymer network) have been synthesized and characterized. Both IPNs showed only one T_g; hence a homogeneous phase morphology is suggested. The second-order non-linear optical coefficients (d₃₃) of the pseudo-IPN and the full-IPN were measured and found to be 2.78 × 10⁻⁷ esu and 1.86 × 10⁷ esu, respectively. The study of temporal stability at room temperature and elevated temperature (120 °C) indicates that the full-IPN is more efficient at improving the orientational stability of the non-linear optical chromophores than the pseudo-IPN, because of the permanent entanglements of the two component networks in the full-IPN. © 1999 Society of Chemical Industry     

Evaluation of rheological properties of non-newtonian fluids in internal mixers: An alternative method based on the power law model

A new method was developed to evaluate rheological properties of polymer melts such as shear stress, shear rate, apparent viscosity and other rheological parameters in internal mixers. It is based on the classical power law model where the power law index n is directly evaluated from a set of data containing speed S, torque M, and the consistency index m is indirectly determined by an empirical relation. The method is based on a model with only one geometrical parameter (α), which involves a chamber radius R₂ and an equivalent radius R_e. It is assumed that the measuring head consists of two adjacent sets of coaxial cylinders. This method has advantages over previously reported models that use two parameters and do not propose a straightforward method to evaluate m. The pseudoplastic nature of the polymer melt decreases as the rate of loss of structural points, i.e., molecular entanglements and network junctions, increases, which produces greater mobility of the melt. A relationship between α and C(n) is found, which is simpler than other models previously reported. These results further demonstrate the feasibility of evaluating rheological properties of polymer melts in internal mixers.     

The Effect of the Density and Nature of Spatial Network in Elastomers on Their Viscoelastic Characteristics

Abstract

We have studied the effect of the density of the networks formed by fluctuating entanglements and chemical crosslinking on the relationships between the circular frequency ω and the storage and loss moduli, G' and G”, for polybutadienes of narrow and wide molecular weight distributions (the ratio M_w/M_n varied from 1.1 to 3.35) and different microstructure. Polybutadienes were crosslinked by thermal, radiation, and sulphur vulcanization. With increasing density v of a network of chemical crosslinks, which is characterized by the average molecular weight of a chain length (M_e ), pseudo-equilibrium plateau extends to the side of low frequencies with a certain small increase of its level. This increase becomes noticeable when M_e is approximately equal to the average molecular weight M_e of the chain length between the fluctuating entanglements of an uncrosslinked elastomer. At the same time the maxima on the curves of G”(ω) are smoothed out and the losses reduce to negligibly small values with decreasing frequency.     

The Effects of Network Imperfections on the Small-Strain Moduli of Polydimethylsiloxane Elastomers Having High Functionality Cross Links

Abstract

Poly(dimethylsiloxane) networks of high cross-link functionality have been prepared by end linking vinyl-terminated chains with multifunctional poly(methylhydrosiloxane) chains. They covered a wide range in the extent of reaction, P_vi , of the vinyl end groups. At small strains, these networks had elongation moduli that significantly exceeded the values predicted by the Flory-Erman theory. Neglected in such standard analyses, however, is the fact that the segments between cross links along the junction precursor molecule can themselves act as short network chains, contributing to the modulus and giving a strongly bimodal distribution of both network chain lengths and cross-link functionalities. As would be expected, an unmistakable transition is observed in values of the shear modulus G toward the phantom limit of deformation as the crosslink density increases. Calculations based on recognition of such short chains give results in much better agreement with experiment. The results so obtained showed strong dependence of the elastomeric properties on the extents of reaction and the inherent network imperfections. Such imperfections have a pronounced effect on the equilibrium modulus, more specifically on the empirical constant 2C ₂. The dependence of 2C ₂ on the volume fraction of the elastically “effective” chains is thus established. Moreover, the results unambiguously demonstrate that the empirical constant 2C ₂ is essentially a topological contribution and contains no contributions from the chemical network.     

Crosslinked epoxies: Network structure characterization and physical–mechanical properties

A homologous series of epoxy resins, based on the digylcidyl ether of bisphenol-A, were reacted with stoichiometric quantities of m-phenylenediamine (mPDA) to form networks with varying crosslink densities. Infrared spectroscopic analyses revealed that the networks were formed predominantly by the epoxy—amine addition reactions with little or no OH-etherification. The ultimate glass transition temperatures were inversely proportional to the epoxy chain mol wt, as predicted by a model based on the assumptions of additivity and redistributivity of free volume. The average mol wt between crosslinks, M?_c, determined from the dynamic mechanical shear moduli, agreed to within 15 to 18% with the values predicted by the reaction stoichiometry. The glassy-state modulus did not exhibit any M?_c dependence above room temperature, however, the modulus—temperature plot for all the networks exhibited a discontinuity near ?50°C with a steeper slope below ?50°C than above it. The change in slope occurred over the same temperature interval as the β-transition temperature of the networks. The discontinuity in the modulus—temperature plot can be attributed to the freezing of the localized motions of the molecular groups responsible for the β-transition. © 1992 John Wiley & Sons, Inc.     

An empirical model relating the molecular weight distribution of high-density polyethylene to the shear dependence of the steady shear melt viscosity

An empirical model has been developed to relate molecular weight distribution to the shear dependence of the steady shear viscosity in high-density polyethylene melts. It uses a molecular weight, M_c, which partitions molecular weights into two classes; those below M_c contribute to the viscosity as they do at zero shear, and those above M_c contribute to the viscosity as though they were of molecular weight M_c at zero shear. Each individual molecular weight species contributes on the basis of its weight fraction. M_c is proposed to be a unique function of the shear rate. Using this method of treating the molecular weight distribution, and the zero shear relation for relating η0 to molecular weight, the calculated steady shear viscosities at various shear rates for polyethylene samples of widely varying polydispersities agree well with experimental results. The model makes no judgment on the existence or importance of entanglements in non-Newtonian behavior since it has no specific parameters involving an entanglement concept. Use of the model suggests that for the samples studied, only the upper portion of the molecular weight distribution contributes toward the experimentally observed decrease of steady shear viscosity with shear rate for shear rates of up to 10,000 sec^?1. The lower molecular weight species are assumed to behave in a Newtonian manner.