Thermoelasticity of networks in swelling equilibrium

A theoretical analysis of the thermoelastic behavior of polymeric networks in swelling equilibrium with excess diluent, using both the kinetic theory of elasticity and the Flory-Huggins theory of mixing, is presented. Our calculations are restricted to the special case of diluents composed of a single constituent. The results are used to obtain the ratio of the energy component of the force f_e to the total force f of rubber networks swollen in excess n-decane, and we find f_e/f to be 0.17, which compares favorably with the value 0.18 reported for the unswollen network. Furthermore, f_e/f is independent of elongation, in accordance with theory. The kinetic theory of elasticity is reasonably well obeyed in this highly swollen system although there remains a small contribution to the force from the C₂ term of the Mooney-Rivlin phenomenological elasticity equation. It is not believed that this has an appreciable effect on f_e/f.     

Formation,structure, and elasticity of loosely crosslinked epoxy-amine networks. II. Mechanical and optical properties

The mechanical and optical behavior in the dry and swollen states of loosely crosslinked epoxy networks prepared from the diglycidyl ether of bisphenol A, phenylglycidyl ether, and 4,4′-diaminodiphenylmethane was investigated, and the weight fraction of sol in the networks was determined. The crosslinking density was controlled by an excess of diamine and by the fraction of monoepoxide. The reaction proceeded to almost full conversion of epoxy groups. With increasing content of monoepoxide or with increasing excess of diamine, the main transition region is shifted to lower temperatures. The dependence of the viscoelastic modulus on temperature and the optical behavior indicate that the networks are homogeneous. In all cases, the sol fraction is adequately described by the theory of branching processes (cf. Part I). The equilibrium modulus related to the dry state is the same irrespective of whether it is obtained by measurements in the dry or swollen state. The mechanical behavior in the rubbery state can be described by the theory of phantom networks with fully suppressed fluctuations of crosslinking (front factor A = 1) or by the theory of phantom networks with fully released fluctuations of crosslinks (front factor) A = f_e?2/f_e] and contribution of trapped entanglements of the Langley-Graessley type (cf. Part I). In the analysis of the equilibrium behavior, it is advantegeous to use the plot of superimposed dependences of G_e on the gel fraction, which considerably reduces the effect of experimental inaccuracy in determination of composition and degree of conversion.     

Thermoelastic properties of networks in swelling equilibrium: Poly(vinyl alcohol)

The energy component of the stress has been determined for poly(vinyl alcohol) networks in swelling equilibrium with a series of water–ethylene glycol compositions. The data are analyzed by using the equations describing the thermoelasticity of networks in swelling equilibrium. The ratio f_e/f of the energy component to the total force, as calculated from these equations, varies systematically with diluent composition but is independent of elongation in a given diluent. For a network crosslinked by terephthalaldehyde, f_e/f varied from ?0.33 to ?0.42 as the diluent composition was changed from pure water to 20% ethylene glycol. Similar effects were found in a network crosslinked by formaldehyde. It is not yet certain whether this effect represents a real solvent dependence of f_e/f or a failure of the equation of state to account for the effect of composition changes on the force.     

Statistical thermodynamics of networks at large deformations

A general theory of non-Gaussian elasticity is presented for real polymeric chains having fixed bond angles and restricted internal rotations. The theory contains the displacement-vector distribution given by Nagai, and the Flory-Wall-Hermans procedure is used for the calculation of network properties. Whereas the treatment is valid for all types of polymer chains, it is not totally satisfactory from a practical standpoint because of a slow series convergence if the chains are stiff. It is best utilized for flexible polymers under conditions of light crosslinking. Detailed network behavior is investigated only for polyethylene type chains having uncorrelated internal rotations. In this instance the fractional contribution f_e/f of the internal energy of the total force f is found to be a function of elongation at high degress of stretching. It may decrease, or increase, depending upon the sign of f_e/f at low elongations. Furthermore, the variation of f_e/f with elongation is independent of the fixed bond angle of the chain backbone. Stress–strain behavior and energy–strain behavior are in opposition, i.e., when the non-Gaussian contribution to the stress is greatest, it is the least for the ratio f_e/f, and vice versa. The presence of correlated internal rotations would not be expected to greatly alter these general conclusions.     

Effect of molecular entanglements on craze microstructure in glassy polymers

Thin films of ten glassy polymers are bonded to copper grids and strained in tension to produce crazes, which are then examined in the transmission electron microscope. The average craze fibril extension ratio λ for each polymer is determined from microdensitometer measurements of the mass thickness contrast of the crazes. The extension ratio λ is found to increase approximately linearly with the chain contour length l_e between entanglements, as determined from melt elasticity measurements of the entanglement molecular weight of these polymers. These results are analyzed by comparing them with λ_max, the maximum extension ratio of an entanglement network in which polymer chains neither break nor reptate (i.e., permanent entanglement crosslinks are assumed). The values of λ_max are given by l_e/d where d, the entanglement mesh spacing in the unoriented glass, is computed from d = k(M_e)^1/2 with k determined either from small-angle neutron scattering results on isolated chains in the glass or from coil size measurements in dilute solutions of a θ solvent. The craze extension ratios fall somewhat below λ_max at low λ but increase to well above λ_max for polymers with high l_e. This comparison suggests a significant contribution due to chain breakage (or reptation) in the higher-λ crazes of large-l_e polymers, which may arise from the higher true stresses in the craze fibrils (which for a given applied stress increase proportionally to λ). The results also imply that a useful way to increase the “brittle” fracture stress and decrease the ductile-to-brittle transition temperature of a glassy polymer is to decrease its entanglement contour length l_e.     

Dynamics of supramolecular associative polymer networks at the interplay of chain entanglement,transient chain association,and chain‐sticker clustering

The dynamic mechanical properties of supramolecular associative polymer networks depend on the average number of entanglements along the network‐forming chains, N_e, and on their content of associative groups, f . In addition, there may be further influence by aggregation of the associative groups into clusters, which, in turn, is influenced by the chemical structure of these groups, and again by N_e and f of the polymer. Therefore, the effects of these parameters are interdependent. To conceptually understand this interdependency, we study model networks in which (a) N_e, (b) f , and (c) the chemical structure of the associative groups are varied systematically. Each network is probed by rheology. The clustering of the associative groups is assessed by analyzing the rheological data at the end range of frequency covered and by comparison of the number of supramolecular network junctions with the maximum possible number of binary transient bonds. We find that if the total number of the network junctions, which can be formed either by interchain entanglement or by interchain transient associations, is greater than a threshold of 13, then the likelihood of cluster formation is high and the dynamics of supramolecular associative polymer networks is mainly controlled by this phenomenon. © 2019 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2019 , 57, 1209–1223     

Dynamic - mechanical properties of poly(oxypropylene)di-amine-diepoxide and poly(oxypropylene)triamine-diepoxide networks and their relationship to the structure of elastically active network chains

The effect of the initial mole ratio of reactive components on the shape and position of dynamic mechanical functions in the main transition and rubbery region was investigated for two series of networks made from poly(oxypropylene)diamine (D-400)-diglycidyl ether of Bisphenol A (DGEBA) and poly(oxypropylene)-triamine (T-403)-DGEBA. The networks were prepared with an excess of amine groups up to the highest conversion of epoxy groups; the ratio ^rH = 2 [ NH₂ ]₀/ [E]₀ ranged from unity to 2,1 for networks from D-400 and from unity to 3,5 for networks from T-403. By using the theory of branching processes, structural parameters of these networks were calculated, in particular, the molecular weights of elastically active network chains (EANC's) including dangling chains, of backbone EANC's and of dangling chains. A comparison between theory and experiment led to the following conclusions: (a) the mechanical behaviour in the rubberlike region can be described either by using an affine deformation model (front factor A = 1), or by means of a phantom model (A = (f_e-2)/f_e, f_e being functionality of the crosslink) with the contribution of permanent interchain interactions; (b) the temperature and frequency position of viscoelastic functions in the main transition region is conclusively affected by the concentration of EANC's; (c) the shape of visco-elastic functions, especially of retardation spectra in the main transition and rubbery region, depends on the detailed structure of EANC's, but it cannot be decided from the result which structural parameter has the strongest effect on the shape of the functions.     

Thermoelasticity of weak polyelectrolyte networks

The thermoelastic behavior of poly(vinyl alcohol)–poly(acrylic acid) networks was evaluated in pure water and CaCl₂ solution. The ratio f_e/f of the energy component of the force to the total force, evaluated without taking into account polymer–diluent specific interactions, ranged from ?0.75 for networks swollen in pure water to ?5.7 in 0.1M CaCl₂. However, an analysis based on Flory's theory of polyelectrolyte gels yields f_e/f constant at ?1.32 when specific interactions are accounted for. In addition, the variation of In (r²)₀ with CaCl₂ concentration is 2000 times that with water. In neither pure water nor CaCl₂ solution can specific interactions be neglected.     

Charge density effects in salt‐free polyelectrolyte solution rheology

Solution rheology of 2‐vinyl pyridine and N‐methyl‐2‐vinyl pyridinium chloride random copolymers in ethylene glycol was studied over wide ranges of concentration and effective charge. The fraction of quaternized monomers α and the fraction of monomers bearing an effective charge f of these copolymers were measured using counterion titration and dielectric spectroscopy, respectively. Ethylene glycol is a good solvent for neutral poly(2‐vinyl pyridine), with very few ionic impurities. The viscosity η and relaxation time τ of dilute and semidilute unentangled solutions exhibit the scaling with concentration and effective charge expected by the Dobrynin model. Reduced viscosity data are independent of concentration in dilute solution, giving an intrinsic viscosity that depends on effective charge, and the experimental data obey the Fuoss law in the semidilute unentangled regime. Scaling concentration with the overlap concentration (c/c*) reduces these data to common curves, and c* ～ f ^?12/7 as predicted by the Dobrynin model, where f is the fraction of monomers bearing an effective charge. While the overlap concentration depends strongly on effective charge until counterion condensation occurs, the entanglement concentration c_e is surprisingly insensitive to effective charge, indicating that entanglement effects are not understood using the Dobrynin model. The terminal modulus G = η/τ depends only on the number density of chains G = ckT/N for c* < c < c_e, and G ～ c^3/2 for c > c_e independent of the effective charge. © 2006 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 44: 2001–2013, 2006     

Thermoelasticity of networks in swelling equilibrium with mixed diluents: Specific polymer-diluent interactions

Thermoelasticity studies of poly(vinyl alcohol) networks swollen in aqueous glycol indicate that the effects of specific polymer-diluent interactions diminish at high swelling, much as in the situation of poly(vinyl alcohol) in pure water reported earlier. In 20% glycol, the ratio of the energy component f_e of the force to the total force f was found to be ?0.59 at high swelling. Swelling was controlled by the degree of network crosslinking. Evidence that f_e/f could be determined under the condition of a null thermal expansion coefficient was not obtained; in fact, the contrary was indicated.     

A Branched Pore Model Analysis for the Adsorption of Acid Dyes on Activated Carbon

A new branched-pore adsorption model has been developed using an external mass transfer coefficient, K _f, an effective diffusivity, D _eff, a lumped micropore diffusion rate parameter, K _b, and the fraction of macropores, f, to describe sorption kinetic data from initial adsorbent-adsorbate contact to the long-term adsorption phase. This model has been applied to an environmental pollution problem—the removal of two dyes, Acid Blue 80 (AB80) and Acid Red 114 (AR114), by sorption on activated carbon. A computer program has been used to generate theoretical concentration-time curves and the four mass transfer kinetic parameters adjusted so that the model achieves a close fit to the experimental data. The best fit values of the parameters have been determined for different initial dye concentrations and carbon masses. Since the model is specifically applicable to fixed constant values of these four parameters, a further and key application of this project is to see if single constant values of these parameters can be used to describe all the experimental concentration-time decay curves for one dye-carbon system.The error analysis and best fit approach to modeling the decay curves for both dye systems show that the correlation between experimental and theoretical data is good for the fixed values of the four fitted parameters. A significantly better fit of the model predictions is obtained when K _f, K _b and f are maintained constant but D _eff is varied. This indicates that the surface diffusivity may vary as a function of surface coverage.     

Square‐Wave Voltammetry of Quasi‐Reversible CE Reactions at Spherical Microelectrodes

A mathematical model for the CE mechanism in which the chemical together with the electrochemical reactions are quasi‐reversible at the surface of spherical macro and micro‐electrodes is presented for the case of square‐wave voltammetry. The analysis of voltammometric responses considers the influence of rate and equilibrium constants, together with the electrode radius, and their dependence on the square‐wave frequency (f). Both kinetics and the sphericity effect act synergistically on the electrochemical response. Also, the apparent electrode sphericity and the reversibility of the chemical as well as the electrochemical reactions are jointly affected by the variation of f. Disregarding the sphericity contribution in the calculation of kinetic parameters at a microelectrode may introduce errors even higher than one order of magnitude. The model allows the analysis of a more realistic and complex electrochemical system that requires not only the dependence of experimental responses on f, but also their fit with theoretical voltammograms, in order to provide some useful mechanistic information. Finally, concentration profiles are also studied to realize how the chemical contribution is buffering the absences of oxidized species at the electrode surface, and how those profiles are modified for the case of spherical macro and micro‐electrodes.     

Chain dimensions and entanglement spacings in dense macromolecular systems

In this article, we reexamine and extend a relationship proposed earlier between entanglement density and chain dimensions in polymer melts. The power-law equation presented in the earlier work, relating the entanglement molecular weight M_e, melt chain density ρ, and the packing length p is tested with additional polymer species. Now included are additional polydienes and their hydrogenated derivatives, the isotactic forms of polypropylene and polystyrene, the essentially syndiotactic form of poly(methyl methacrylate), along with poly(tetrafluoroethylene), poly(vinylmethyl ether), various poly(methacrylates), and polymeric sulfur. We find that within experimental uncertainties, M_e/ρ and p are related through an equation (M_e/ρ = 218p³) that is insensitive to temperature (25°C ≤ T ≤ 380°C) and which seems to be universal for flexible Gaussian chains in the melt state. © 1999 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 37: 1023–1033, 1999     

Development of Entanglements in a Fully Disentangled Polymer Melt

A long‐lived metastable “new melt” state of polymers has been recently reported, where monomers exhibit different mobilities due to an unusual distribution of entanglements. We study the relaxation of (fully disentangled) globules to the entangled state by means of computer simulations, and compare our data to the scenario of de Gennes' explosion upon melting. The entanglement length N_e is measured using the primitive path analysis method. The results show that in the case of relatively short chains (N ≈ 20N_e), the relaxation of the entanglement length is very fast compared to that of the chains' size which slows down as the chain length N exceeds the equilibrium value of N_e.

     

Critical charges of simple coulomb molecular systems: One‐two electron case

We consider some Coulomb systems with several infinitely massive centers of charge Z and one or two electrons: (Z,e), (2Z,e), (3Z,e), (4Z,e), (2Z,e,e), and (3Z,e,e). It is shown that the physical, integer charges Z = 1,2,… do not play a distinguished role for the total energy and for the equilibrium configuration of a system, giving no indication of a charge quantization. By definition, a critical charge Z_cr for a given Coulomb system (nZ,e) or (nZ,e,e) is a charge which separates the domain of the existence of bound states from the domain of unbound states (the domain of stability), the continuum (the domain of instability). For all the above‐mentioned systems critical charges Z_cr as well as equilibrium geometrical configurations are found. Furthermore, an indication to a branch point singularity at Z = Z_cr with exponent 3/2 was obtained. It is demonstrated that in the domain of the existence the optimal geometrical configuration for both (nZ,e) at n = 2,3,4 and (nZ,e,e) at n = 2,3 corresponds to the Platonic body. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2012     

Linear thermoviscoelasticity and characterization of noncrystalline EPDM rubber networks

An experimental study has been made to specify how the time-temperature superposition and the linear viscoelastic characteristics vary with the degree of crosslinking for a broad class of noncrystalline peroxide-cured EPDM networks. A new, very sensitive method is applied to determine the horizontal and vertical shift functions in an independent way. All uncrosslinked samples are thermoelasticoviscously simple with horizontal shift functions a_T of the WLF type and vertical shift functions almost independent of temperature, in agreement with recent theoretical understanding. Upon crosslinking, these materials become thermoviscoelastically complex networks, but superposition can still be accomplished by assuming different temperature dependences for the relaxational strength and the equilibrium modulus. The a_T functions can be taken independent of the degree of crosslinking. The vertical shift functions b_T for the relaxational strength vary with the degree of crosslinking between theoretical predictions for uncrosslinked and perfectly crosslinked EPDM networks. The equilibrium moduli of the lightly cured networks decreases with increasing temperature, which is ascribed to the presence of interchain associations between ethylene sequences in the trans state. Upon further crosslinking, these effects gradually vanish and eventually the networks can be described as viscoelastically simple with an energy elastic contribution due to the ethylene trans-gauche transitions. The linear viscoelastic characteristics, namely the storage and loss moduli and compliances and phase-angle master curves and the relaxation and retardation spectra are discussed as a function of the degree of crosslinking. A sol/gel analysis and equilibrium swelling measurements complete the experimental characterization of three familes of five EPDM networks each.     

Thermoelastic properties of crosslinked polyurethanes

Thermoelastic (stress–temperature) measurements in elongation have been carried out on polyurethane elastomers obtained from poly(ethylene adipate), poly(ethylene maleate), poly(ethylene glycol), and 4,4′-diphenylmethane diisocyanate. The elastomers were crosslinked by an excess of diisocynate and some of them were additionally crosslinked by dicumyl peroxide. Values of the temperature coefficient d ln r /dT of the unperturbed dimensions of the network chains and the fraction f_e/f of the stress, which is due to energetic effects, were calculated from measurements of unswollen samples and samples swollen in benzene to a constant volume fraction ν₂ of about 0.6. The coefficient d ln r /dT and the ratio f_e/f for unswollen samples were positive but dependent on the extension ratio α and crosslinking density. The f_e/f ratio displayed a tendency to decrease with increasing α and this decrease was smaller for elastomers of higher crosslinking density. Additional crosslinking due to dicumyl peroxide reduced the dependence of f_e/f on α. Swollen samples had lower positive values of f_e/f; which were somewhat independent of α and crosslinking density. Departures from the molecular theories of rubberlike elasticity are explained by intermolecular effects, which, in general, are influenced by crosslinking.     

Effect of concentration of the crosslinking agent and diluent during polymerization on the viscoelastic spectra of poly(2-hydroxyethyl methacrylate) networks

The effect of concentration of the crosslinking agent (ethylene dimethacrylate) and diluent (water) during the crosslinking copolymerization on the shape and position of retardation spectra in the dry state has been investigated for poly(2-hydroxyethyl methacrylate) networks. With increasing water content during network formation, the maxima of the retardation spectra, L_m, increase and the position of the spectra is shifted toward shorter retardation times, τ. The results are in quantitative agreement with the modified Rouse–Mooney (R–M) molecular theory and suggest the influence of deformation due to the diluent during network formation on the viscoelastic behavior. With increasing content of the crosslinking agent, the retardation spectra are shifted toward longer times. At a constant reference temperature T₀ = 115°C the retardation time, τ_m, at the maxima of the spectra increases with increasing content of effective chains in the network, ν_e. However, after a correction for the effect of the monomeric frictional coefficient, ξ, τ_m/ξ decreases with increasing v_e at a rate which agrees quantitatively with the R–M theory. The slope of the retardation spectra in the main transition region and the value of their maxima decrease with increasing v_e; a comparison of these dependences with theory leads to the most probable distribution of submolecules in the chains. The contribution of long retardation times to the equilibrium compliance, J_e, of the systems under investigation was estimated; it was shown that the application of the Thirion–Chasset extrapolation method for the determination of J_e of loose networks requires a certain type of dependence of the retardation or relaxation spectra on τ.     

Thermoelasticity of networks in swelling equilibrium with single component diluents: Specific polymer-diluent interactions

Using thermoelastic measurements, specific polymer-diluent interactions have been demonstrated for atactic poly(vinyl alcohol) networks swollen in water. This was done by following a thermodynamic parameter as a function of network swelling, which was controlled by varying the extent of network crosslinking. At low crosslinking (high swelling) the ratio of the energy component f_e of the force to the total force f was found to be perhaps a little less than ?0.38 for poly(vinyl alcohol) swollen in water to less than 0.36 volume fraction of polymer. The method of evaluation is new and should be applicable to other polymer single component diluent systems.     

Mechanochemical energy conversion of neutral polymer gels

Mechanochemical energy conversion exhibited by water-swollen poly(vinyl alcohol) gels has been studied. The effective mechanical work produced as well as the chemical energy converted were measured simultaneously. It is shown theoretically and experimentally that the mechanical work developed during the swelling of the mechanochemical system increases with increase of the external load (m), the degree of cross-linking (DC) and the deswelling ratio (ϕ/ϕ_e, i.e. the ratio of the volume fraction of network polymer (ϕ) to that of the same gel in equilibrium with pure diluent (ϕ_e)). The chemical energy converted was found to be practically independent of m. The efficiency of energy conversion increased with increase of m and DC, and significantly decreased with increase of ϕ/ϕ_e. Comparison is made between theoretical predictions and experimental findings.