Applications of a two-network model for crosslinks and trapped entanglements

The dependence of stress and birefringence on strain in uniaxial extension of crosslinked rubbers can accurately be described by a model in which the properties of a network of crosslinks and a network of trapped entanglements are additive. The crosslink network is neo-Hookean and the entanglement network can conveniently be described by the Mooney-Rivlin equation. When the crosslinks are introduced in a state of strain near the glass transition temperature, the two networks have different reference undeformed states; they can be distinguished by appropriate measurements in the state of ease where their associated stresses are equal and opposite and in the state of deformation where the cross-links were introduced and make no contribution to the stress. When partial relaxation is permitted before crosslinking, trapping probabilities can be calculated for both relaxed and unrelaxed entanglements and compared with the Langley theory. The results are consistent with the terminal mechanism of relaxation in the tube theory of Doi and Edwards.     

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.     

Toughness and fracture energy of PDMS bimodal and trimodal networks with widely separated precursor molar masses

End-linked PDMS bimodal and trimodal networks display enhanced mechanical properties in uniaxial extension over those of unimodal networks with similar modulus when the molar masses of their precursor chains are widely separated. These multimodal networks have optimal mechanical properties when the short chains are near their overlap concentration and sustain most of the load, but the volume of the system is still dominated by the ductile long chain component. Such elastomers can be stretched to large elongations before fracture while displaying an upturn in stress at high strain. Improvement in fracture energy of pre-cut bimodal and trimodal networks over that of unimodal networks is much less pronounced and appears to be dictated by the average molar mass of the effective elastic strands in each network.     

Rheological predictions of network systems swollen with entangled solvent

The mechanical properties of a cross‐linked polydimethylsiloxane (PDMS) network swollen with nonreactive entangled PDMS solvent was previously studied experimentally. In this article, we use the discrete slip‐link model to predict its linear and nonlinear rheology. Model parameters are obtained from the dynamic modulus data of pure solvent. Network rheology predictions also require an estimate of the fraction and architecture of dangling or inactive strands in the network, which is not directly measurable. The active strand fraction is estimated from dynamic modulus measurements, and the molecular weight is adjusted to fit the dynamic modulus data. Then, the nonlinear rheology can be predicted without adjustments. These successful predictions strongly suggest that the observed rheological modification in the swollen blend arises from the constraint dynamics between the network chains and the dangling ends. © 2014 American Institute of Chemical Engineers AIChE J, 60: 1372–1380, 2014 We dedicate this article to Prof. R. Byron Bird on the occasion of his 90th birthday. We attempt to follow the twin exhortations of Bob to be both mathematically rigorous and practically relevant.     

Damping elastomer with broad temperature range based on irregular networks formed by end‐linking of hydroxyl‐terminated poly(dimethylsiloxane)

Irregular networks based on the condensation reaction of hydroxyl‐terminated poly(dimethylsiloxane) with cross‐linkers were investigated. The networks have excellent damping properties in a wide temperature range utilizing the viscoelastic relaxation of the irregular network with dangling chains. NMR Cross‐link Density Spectroscopy was used to explore the weight fraction of pendant chains and elastic chains in the elastomer. The transverse relaxation time for the elastomer was studied to explore the influence of pendant chains. The effects of the structure of cross‐linkers and the molecular weight of precursors were studied in detail. Elastomers cross‐linked by tetra‐functional cross‐linker (TEOS) have higher damping properties than the elastomers cross‐linked by tri‐functional cross‐linkers (MTMS and OTMS). A damping elastomer based on irregular networks with effective damping (tanδ > 0.3) temperature range of more than 250°C (from lower than ?60°C to 190°C) was prepared. POLYM. ENG. SCI., 56:97–102, 2016. © 2015 Society of Plastics Engineers     

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.     

Interpenetrating polymer networks of poly(dimethylsiloxane): 1. Preparation and characterization

Model networks of ,ω-dihydroxy-poly(dimethylsiloxane) (PDMS) were prepared by tetrafunctional crosslinking agent tetraethyl orthosilicate (TEOS) and the catalyst stannous 2-ethylhexanoate. Hydroxylterminated chains of PDMS having molecular weights 15 × 10³ and 75 × 10³ g mol⁻¹ were used in the crosslinking reaction. Bimodal networks were obtained from a 50% (w/w) mixture of PDMS chains with M_n = 15 × 10³ and 75 × 10³ g mol⁻¹. A sequential interpenetrating network of these PDMS chains was also prepared. Physical properties of the elastomers were determined by stress-strain tests, swelling and extraction experiments, and differential scanning calorimetry measurements.     

Stress-strain relationships in uniaxial extension of high cis-1,4-polyisoprene networks with different molecular weight precursors

Equilibrium stress-strain relationships in uniaxial extension for high cis-1,4-polyisoprene (Shell IR 307) networks were obtained by extrapolation of relaxation measurements to infinite time through a BKZ constitutive equation. Three series of networks were investigated, each series being characterized by its polymer precursor molecular weight. Influence of crosslinking density was studied through varying amounts of dicumyl peroxide as crosslinking agent. These results were used to test Flory and Erman's recent molecular elasticity theory of imperfect networks with constraints on junctions. It was shown that this later theory treating entanglements as restrictions on junction fluctuations could be reasonably used to characterize network topology. A universal value of 0.50 for the interpenetration parameter I is confirmed and an interpretation of parameter ζ in terms of network inhomogeneity is tentatively given.     

Non-equilibrium mechanical properties of model networks

A large number of specially prepared model networks with different types of network defects have been produced since de Gennes proposed the reptation motion for linear chains. The viscoelastic properties of these model networks show clearly that branched chains and dangling chains are the cause of the gradual and extremely slow relaxations in lightly crosslinked networks. These experimental results have been followed by theoretical developments that point towards a complete molecular theory for the non-equilibrium properties of lightly crosslinked networks. It is proposed that the presence of dangling chains in endlinked networks may be revealed by measurements of the viscoelastic properties of such networks near the transition zone.     

Unimodal and bimodal networks of physically crosslinked polyborodimethylsiloxane: viscoelastic and equibiaxial extension behaviors

Unimodal and bimodal networks of physically crosslinked polyborodimethylsiloxane (PBDMS) were prepared by end-linking hydroxy-terminated polydimethylsiloxane (PDMS) with boric acid. Their viscoelastic and equibiaxial extension behaviors were investigated. Three PDMS precursors with different number-average molecular weight ( M ¯ n $ {\overline{M}}_n $ ) were employed, of which the shortest chain had M ¯ n $ {\overline{M}}_n $ lower than the entanglement molecular weight. Bimodal networks were prepared from the mixture of the shortest and the longer PDMS chains. Linear viscoelastic behavior of unimodal network of the shortest chain gave the best fit to the Maxwell model with single relaxation time of 1.59 s, and equilibrium elastic modulus (G _e ) of the network was well-explained by phantom network model. The unimodal networks from the other two long chain precursors, however, showed multi-relaxation behavior with the longest relaxation times of 1.00–1.26 s. Moreover, their G _e was close to affine model and deviated from the phantom model with trapped entanglement factors of ~ 0.13. The bimodal networks with high mole percentage of short chains gave G _e values approximate to the predicted values of phantom model. Such bimodal networks showed an extremely large increase in modulus at high biaxial extension, attributed by the limited extensibilities of short chains and un-relaxed crosslinked junctions.     

Reversible modulus reinforcement of end-linked polydimethylsiloxane ionomer networks

We report the mechanical behavior and swelling properties of covalently end-linked networks of polydimethylsiloxane with tailored number of monomers between side carboxyl groups and number of carboxyl groups along the chain. Imperfect carboxyl networks are reinforced by neutralization of the carboxyl groups with gallium ions via conversion of pendent chains into elastically active strands due to formation of inter-molecular ionic cross-links. These networks swell in non-polar solvents to a similar degree as unmodified end-linked PDMS networks of comparable moduli. For unannealed samples, the modulus reinforcement is reversed by a polar THF:water solution that breaks the ionic cross-links. Neutralization with transition metal cations such as cobalt causes no initial reinforcement due to weak intra-molecular interactions. Reinforcement in annealed gallium, cobalt or carboxyl networks is not readily reversible.     

Linear viscoelastic relaxation modulus of polydisperse poly(dimethylsiloxane) melts containing unentangled chains

This work analyzes the relationship between the shear relaxation modulus of entangled, linear and flexible homopolymer blends and its molecular weight distribution (MWD) when a fraction of the sample contains chains with molecular weight M lower than the effective critical molecular weight between entanglements Mc_eff. This effective critical parameter is defined in terms of the critical molecular weight between entanglements M_c of the bulk polymer that forms the physical network and the effective mass fraction Wc_eff of the unentangled chains. In the terminal zone of the linear viscoelastic response, the double reptation mixing rule for blended entangled chains and a modified law for the relaxation time of chains in a polydisperse matrix are considered, where the effect of chains with M<Mc_eff is included. Although chain reptation with contour length fluctuations and tube constraint release are still the relevant mechanisms of chain relaxation in the terminal zone when the polydispersity is high, it is found that the presence of a fraction of molecules with M<Mc_eff modifies substantially the tube constrain release mode of chain relaxation. In this sense, a modified relaxation law for polymer chains in a polydisperse entangled melt that includes the effect of the MWD of unentangled chains is proposed. This law is validated with rheometric data of linear viscoelasticity for well-characterized polydimethylsiloxane (PDMS) blends and their MWD obtained from size exclusion chromatography. The short time response of PDMS, which involves the glassy modes of relaxation, is modeled by considering Rouse diffusion between entanglement points of chains with M>Mc_eff. This mechanism is independent from the MWD. The unentangled chains with M<Mc_eff occluded in the polymer network also follow Rouse modes of relaxation although they exhibit dependence on the MWD.     

Network characterisation of end-linked poly(dimethylsiloxane) by 1H-NMR-spin-spin relaxation

Summary Above the glass transition temperature ¹H-NMR-spinspin relaxation curves of elastomers usually consist of two main components, a Gauss-like decay and a longer exponential decay. They can be attributed to inter-crosslink chains and free dangling chain ends, respectively, in accordance with basic ideas of the dynamics of polymer chains. By using an increasing amount of end-linking agent during the network synthesis of poly(dimethyl siloxane), the portion of the long decay is considerably reduced while the shape and the decay times remain approximately constant. This gives experimental evidence of the common classification of the decay components for the first time. Finally, the theoretical model for network dynamics used, which includes common phantom network properties and a concept of two independent motion scales, provides two mean correlation times and an average molecular mass of inter-crosslink chains.     

Preparation of All Polyester‐Based Semi‐IPN Elastomers Containing Self‐Associative or Non‐Associative Guest Chains via Post‐Blending Cross‐Linking

Mechanical properties of semi‐interpenetrating polymer network (semi‐IPN) elastomers consisting of chemical networks and self‐associative/non‐associative guest chains are demonstrated. Amorphous low T_g polyesters with thiol side groups (PE‐SH) are first synthesized by melt polycondensation. PE‐SH are then converted to polyesters containing COOH side groups (PE‐COOH) and amide side groups (PE‐amide) through Michael addition reaction of thiol groups with acrylic acid and acrylamide, respectively. Homogeneous semi‐IPN elastomers are obtained by thermal cross‐linking for bulk mixtures of PE‐COOH and PE‐amide in the presence of diepoxy cross‐linkers, where COOH and epoxy groups are reacted to form chemical cross‐links while the amide units form self‐complementary hydrogen bonds. Another sample containing non‐associative chains is also prepared by using polyester with N,N‐dimethylamide units, instead of PE‐amide. Dynamic mechanical analysis reveals that guest chain incorporation systematically brings plateau modulus reduction and a unique relaxation with higher tan δ value depending on the fraction and nature of guest chains. Tensile properties are also affected by the fraction and nature of guest chains; the incorporation of hydrogen bonded chains are beneficial to enhance breaking elongation and toughness without the sacrifice of maximum stress. The knowledge found in this work will be thus beneficial for creating tough soft materials with damping applications.     

The effect of annealing on the nonlinear viscoelastic response of isotactic polypropylene

Three series of tensile relaxation tests are performed on isotactic polypropylene at room temperature in the vicinity of the yield point. In the first series of experiments, injection‐molded samples are used without thermal pre‐treatment. In the second and third series, the specimens are annealed at 130°C for 4 and 24 hours, respectively. Constitutive equations are derived for the time‐dependent response of semicrystalline polymers at isothermal loading with small strains. A polymer is treated as an equivalent temporary network of macromolecules bridged by junctions (physical cross‐links, entanglements and crystalline lamellae). Under loading, junctions slide with respect to their positions in the bulk material (which reflects the viscoplastic behavior), whereas active strands separate from their junctions and dangling strands merge with the network at random times (which reflects the viscoelastic response). The network is thought of as an ensemble of meso‐regions (MRs) with various activation energies for detachment of chains from temporary nodes. Adjustable parameters in the stress‐strain relations are found by fitting the observations. The experimental data demonstrate that the relaxation spectrum (characterized by the distribution of MRs with various potential energies) is independent of mechanical factors, but is altered at annealing. For specimens not subjected to thermal treatment, the growth of longitudinal strain does not affect the volume fraction of active MRs and the attempt rate for detachment of chains from their junctions. For annealed samples, the concentration of active MRs increases and the attempt rate decreases with strain. These changes in the time‐dependent response are attributed to broadening of the distribution of strengths of lamellae at annealing.     

Silicone Rubber Tack I: Relation to Network Structure

The influence of the network structure on the poly(dimethyl siloxane) (PDMS) rubber is described. The telechelic polymers have crosslinking sites restricted to the ends of polymer chains only, giving well-defined networks. Rubber-rubber tack is related to the network structure, mostly to the amount of loose polymer chains not linked into the network, that can diffuse through the interface. The amount of mobile chains can be adjusted by varying crosslinker amount and functionality. By controlled variation of the degree of crosslinking of the telechelic silicone rubbers, various levels of tack are induced, which may be related to the network topology.     

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.     

Model networks of end-linked polydimethylsiloxane chains. XII. Dependence of ultimate properties on dangling-chain irregularities

Elastomeric networks were prepared by end-linking vinyl-terminated polydimethylsiloxane (PDMS) chains having number-average molecular weights of 11.3 × 10³ g mol^?1. The tetra-functional end-linking agent, Si[OSi(CH₃)₂H]₄, was used in varying amounts smaller than that corresponding to a stoichiometric balance between its active hydrogen atoms and the chain vinyl groups. The number of dangling-chain irregularities thus introduced into the networks was directly determined by iodometric titration for unreacted vinyl groups. The (unfilled) PDMS networks thus obtained were studied in elongation to their rupture points at 25°C (a temperature sufficiently high to prevent complications from strain-induced crystallization), and in swelling equilibrium in benzene at room temperature. Small to moderately large proportions of dangling chains were found to have less of an effect on the elongation modulus than might be expected, and similarly a relatively small effect on the degree of equilibrium swelling. Most importantly, comparisons of constant values of the high deformation modulus show that dangling-chain irregularities decrease both the maximum extensibility of a network and its ultimate strength.     

Semicrystalline Shape‐Memory Elastomers: Effects of Molecular Weight,Architecture, and Thermomechanical Path

Poly(caprolactone) networks are well‐studied shape‐memory polymers owing to their high fixity and recovery, their ability to store large amounts of elastic energy, and their tunable shape‐triggering temperature. To elucidate the influence of network structure on shape‐memory features, poly(caprolactone) networks are prepared by reacting different molecular weight diacrylate prepolymers with trifunctional (trimethylolpropane tris(3‐mercaptopropionate), 3T ) or tetrafunctional (pentaerythritol tetrakis(3‐mercaptopropionate), 4T ) crosslinkers. Networks from 4T crosslinkers generally exhibit higher gel fractions, more elastically active strands, and superior shape‐memory properties compared with networks from 3T . Melted elastomers exhibit stress–strain behavior well described by the neo‐Hookean model. How the state of crystallization during the cold‐drawing process has a large effect on the draw stress, the network's shape fixity, and its elastic storage capacity is shown. Finally, the working strain range of networks is evaluated. Cured elastomers prepared from prepolymers with different molecular weights can store and release large amounts of elastic energy (>2 MJ m⁻³), over different ranges of tensile strain.     

Rheological behaviors of randomly crosslinked low density polyethylene and its gel network

Dicumy peroxide (DCP) modification of low density polyethylene (LDPE) below gel point (GP) produces modified LDPE (mLDPE) with wide molecular weight distributions while the random crosslinking beyond GP yields crosslinked LDPE (XLPE). Molecular weight distributions of mLDPE below GP and the sol-fractions of XLPE above GP were investigated. The sol-fractions in XLPE were extracted to prepare XLPE gels of different crosslinking densities. Both XLPE and the gels were subjected to shear action for studying dynamic rheology and relaxation modulus in the linearity region for revealing the role of entanglements and dangling chains to viscoelasticity of the randomly crosslinked network. The results revealed that the sol-fractions with molecular weight much higher than entanglement molecular weight contributed to equilibrium modulus in addition to the gel network, which lowered the DCP dosage for appearance of critical gel behavior of XLPE in comparison with the XLPE gel. However, rheological method yielded critical DCP dosages for appearance of apparent gel behaviors far beyond the chemical GP from the extraction experiment. The sol-fraction residing in the network gave rise to additional contribution to relaxation modulus of XLPE, shortening network relaxation time and improving equilibrium modulus.