Phantom Network Model Research Articles

Brauer diagram model of suppositional phantom network in double-network gel undergoing mechanochemical coupling

Complex coupling of brittle and stretchable networks in double-network (DN) gel has been a hot research topic to understand working principle of ultra-high mechanical toughness, in comparison with the conventional gels. In this study, both the percolation theory and phantom network model were employed to formulate a Brauer diagram model, which was used to describe the suppositional phantom double networks and explore their coupling effects in DN gel. The mechanochemical coupling was investigated using the proposed Brauer diagram model, in which three critical parameters of coupling fraction, surviving probability and phantom degree, were proposed to formulate a constitutive relationship among rubber elasticity, mechanochemical coupling and stress-elongation ratio. Finally, effectiveness of the Brauer diagram model was verified using analytical data from the finite element analysis and molecular dynamics, as well as experimental results of DN gels reported in the literature.

Gaussian network model revisited: effects of mutation and ligand binding on protein behavior

The coarse-grained Gaussian network model (GNM), considers only the alpha carbons of the folded protein. Therefore it is not directly applicable to the study of mutation or ligand binding problems where atomic detail is required. This shortcoming is improved by including all atom pairs within the coordination shell of each other into the Kirchoff adjacency matrix. Counting all contacts rather than only alpha carbon contacts diminishes the magnitude of fluctuations in the system. But more importantly, it changes the graph-like connectivity structure, i.e., the Kirchoff adjacency matrix of the protein. This change depends on amino acid type which introduces amino acid specific and position specific information into the classical coarse-grained GNM which was originally modeled in analogy with the phantom network model of rubber elasticity. With this modification, it is now possible to explain the consequences of mutation and ligand binding on residue fluctuations, their pair-correlations and mutual information shared by each pair. We refer to the new model as ‘all-atom GNM’. Using examples from published data we show that the all-atom GNM gives B-factors that are in better agreement with experiment, can explain effects of mutation on long range communication in PDZ domains and can predict effects of GDP and GTP binding on the dimerization of KRAS.

Adding the Effect of Topological Defects to the Flory-Rehner and Bray-Merrill Swelling Theories.

The Flory-Rehner and Bray-Merrill swelling theories are venerable theories for calculating the swelling of polymer networks and are widely applied across polymer materials. Here, these theories are revised to include cyclic topological defects present in polymer networks by using a modified phantom network model. These closed-form equations assume defect contributions to the swelling elasticity to be linear and additive and allow different assumptions regarding prestrain of larger loops to be incorporated. To compare to the theories, swelling experiments are performed on end-linked poly(ethylene glycol) gels in which the topological defects (primary and secondary loops) have been previously measured. Gels with higher loop densities exhibit higher swelling ratios. An equation is derived to compare swelling models independent of knowledge of the Flory-Huggins χ parameter, showing that the revised swelling models for loop defects are more accurate than both the phantom network model that neglects loops and the Bray-Merrill equation.

Water Swelling Behavior of Poly(ethylene glycol)-Based Polyurethane Networks.

Defects in a polymer network complicate an accurate calculation of structural parameters such as the molar mass between cross-links Mc, typically obtained from experimental swelling data. In this paper the formation and structure of poly(ethylene glycol) (PEG)-based polyurethane networks containing PEG-mono methyl ether dangling chains are studied. The phantom network model can describe the swelling behavior of these networks only when a composition-dependent interaction parameter is used and the formation of allophanates is accounted for. A clear transition in the network formation is found at the PEG network precursor molar mass at which entanglements are formed in the melt. Correction factors based on structure calculations using the Miller–Macosko–Vallés probability approach are proposed and validated for an accurate calculation of the Mc of these defect-containing networks. This provides a new approach for studies that requires an accurate estimate of the Mc, only based on experimentally straightforward swelling experiments.

Molecular simulation of networks formed by end-linking of tetra-arm star polymers: Effects of network structures on mechanical properties

The structures and the mechanical properties of networks formed by end-linking of two types of tetra-arm star polymers are studied by a coarse-grained molecular dynamics simulation. The formation efficiency of elastically effective chains is improved by increasing the length of the arms and the volume fraction of the star polymers at the preparation stage. When the uniaxial elongation is applied to the networks in the as-prepared state, the shear modulus is proportional to the number density of the elastically effective chains and indicates an intermediate value between the phantom and affine network models. In the equilibrium swollen state, the shear moduli of the networks prepared at high volume fractions are significantly enhanced by trapped entanglements. The networks formed from the tetra-arm star polymers show higher number densities of the elastically effective chains than networks consisting of telechelic functional polymers and cross-linkers.

Connectivity dependence of gelation and elasticity in AB-type polymerization: an experimental comparison of the dynamic process and stoichiometrically imbalanced mixing.

To control various physical properties of polymer gels, it is important to control the connection probability between functional groups of network structures (connectivity). In this study, we compare two methodologies tuning the connectivity in AB-type polymerization: one is stopping the reaction intentionally at a certain conversion, and the other is mixing two prepolymers in a stoichiometrically imbalanced ratio. By experimentally examining the relationships between elastic modulus and connectivity, we find that the relationships are almost the same for these two methodologies. However, the critical connectivity for gelation is different. These results are well reproduced by a kind of phantom network model whose structural parameters are estimated by using a mean-field approximation.

Thermodynamic Analysis of Polymer Gel Elasticity

Elastomer and polymer gel have a three-dimensional polymer network structure and the characteristic of being soft and well extended (rubber elasticity). Many researches on rubber elasticity have been carried out mainly with elastomer since 1950's. Entanglement of network chains dominates the physical properties because elastomer contains almost no solvent, thus it is difficult to investigate the correlation between the structure and the physical properties in detail. On the other hand, polymer gel is a dilute system coexisting with solvent, thus it is possible to design various network structures with controlled entanglement. Recently, we developed Tetra-PEG gel, which is formed by AB-type polycondensation of tetra-armed prepolymers. Tetra-PEG gel has a highly homogeneous network structure, which is suitable for verifying the theory of polymer gel. In the previous study, Phantom network model, which is a rubber elasticity model, was verified. Phantom network model assumes that the crosslinks fluctuate with deformation, and elastic modulus (G) is described as G = ξk B T Here, k B is the Boltzmann constant, T is absolute temperature, and ξ is the number densities of independent cycles surrounded by the mesh chain. In tree-like approximation, ξ can be calculated from polymer concentration, polymer molecular weight, and connection probability between prepolymers. This equation indicates that one cycle has the influence of the thermal energy k B T on the elasticity. As a result of the verification using Tetra-PEG gel, it was confirmed for the first time in the world that the connection probability dependence of the elastic modulus and the calculated ξ (ξ th) was the same. This indicates that Phantom network model can describe the elasticity of polymer gel. However, the polymer concentration (C) dependence of them was different as shown in Figure 2. Here, the polymer concentration was normalized by the overlapping concentration (C*). The polymer concentration is a concept inherent to polymer gel, and it has not been considered so much in conventional researches. Therefore, we returned to thermodynamics, which is the origin of rubber elasticity models, and experimentally verified it. The elastic modulus from Helmholtz free energy is described as the following equation. G = U” + S”T The first term (U”) is referred to as energy elasticity, and the second term (S”T) as entropy elasticity. From this equation, it is shown that elastic modulus can be separated into energy elasticity and entropy elasticity by measuring temperature dependence of elastic modulus. Classical rubber elasticity theory ignores the energy elasticity and predicts that the elastic modulus is proportional to temperature (G ~ T). It has been experimentally confirmed that the proportional relationship is roughly correct in the case of elastomer. On the other hand, the elastic modulus of Tetra-PEG gel was not proportional to temperature, and the negative energy elasticity whose value cannot be ignored was measured (Figure 1). This result may indicate an essential difference between elastomer and polymer gel. Therefore, we further investigated the energy elasticity and entropy elasticity using Tetra-PEG gel with various network structures. When only the connection probability was changed, energy elasticity was proportional to entropy elasticity (U”~ S”T). Since in previous study it was shown that the connection probability dependence of the elastic modulus and ξ was the same, we found that the dependence of the entropy elasticity and the energy elasticity was also the same as ξ. Furthermore, the entropy elasticity had the same polymer concentration dependence as ξ, which shows that the entropy elasticity follows Phantom network model except the value. Therefore, this result also shows the different concentration dependence of the elastic modulus and ξ is derived from the energy elasticity (Figure 2). In the poster presentation, I will talk about the cause of energy elasticity and the effect of solvent quality of polymer gel, too. Figure 1

Synthesis of methacrylate pendant polyelectrolyte as water-soluble multifunctional cross-linker

A new cationic polyelectrolyte, constituting with quaternary poly(4-vinyl pyridine) (PVPy) backbone and methacrylate pendant groups was prepared by spontaneous polymerization methodology. Thus, action of 2-bromoethyl methacrylate (BEMA) on 4-vinyl pyridine resulted in spontaneous polymerization of vinyl pyridine monomer, without initiator, while methacrylate double bonds remaining unreacted. The resulting water-soluble polymer (QMEP); poly(2-methacryloyloxyethyl 4-vinyl pyridinium bromide) was demonstrated to undergo rapid gelation in concentrated aqueous solutions via self-crosslinking of the methacrylate pendant groups by radical polymerization (The abbreviation QMEP stands for quaternary methacrylate functional polymer). Crosslinking densities of QMEP hydrogels estimated by “Phantom Network Model” using equilibrium swelling values, revealed 62.5% of the residual methacrylate double bonds. QMEP was also employed as multifunctional crosslinker for synthesis of highly swelling hydrogels derived from a set of monomers, such as acrylic acid, acrylamide, N-vinyl formamide and N-vinyl pyrrolidinone.

Demonstration of thermo-sensitive tetra-gel with implication for facile and versatile platform for a new class of smart gels

A tertiary branched poly(N-isopropylacrylamide) with controlled molecular weight, distribution and the end amino-functionalization (tetra-PNIPAAm-NH2) was studied for the ability to form a gel via in situ chain-end reaction with a counterpart tertiary branched poly(ethyleneglycol) bearing N-hydroxysuccinimide end groups (tetra-PEG-NHS), a well-documented class of building block to yield the tetra-gel. Some of these polymers, both comparable and distinct (relative to the counterpart) extended chain length pairs, provided a self-standing and macroscopically homogeneous gel, which was capable of undergoing thermo-sensitive and reversible change in hydration in line with the nature of PNIPAAm. Phantom network model based calculation indicated that a half molar fraction of the polymer chains in the network remained unreacted, revealing further room for optimizing the reaction condition. Since such tetra-PNIPAAm based motif can be readily tailored to a variety of other physicochemical stimuli-responsive analogues, our finding may give important insight into a platform for ‘smart’ tetra-gels with exceptional mechanical properties and potentially highly controllable molecular cut-off capability.

Transport features of nano-hydroxylapatite (n-HA) embedded silicone rubber (SR) systems: influence of SR/n-HA interaction, degree of reinforcement and morphology.

We report the transport characteristics of silicone rubber/nano-hydroxylapatite (SR/n-HA) systems at room temperature with reference to the effects of n-HA loading, morphology and penetrant nature, using toluene, xylene, ethyl acetate and butyl acetate in the liquid phase and methanol, ethanol, 1-propanol, 2-propanol and butanol in the vapour phase as probe molecules. The interaction between the n-HA particles and SR matrix has been confirmed by FTIR analysis. As the n-HA content in the SR matrix increased, the penetrant uptake has been found to decrease. The observations have been correlated with the density and void content of the systems. Scanning electron microscopy images have been found to be complementary to the observed transport features. The reinforcement effect of n-HA particles on the SR matrix has been verified by Kraus equation. Molecular mass between the cross links has been observed to decrease with an increase in n-HA loading. The results have been compared with affine, phantom network, parallel, series and Maxwell models. The transport data have been complemented by observations on biological fluid uptake with urea, d-glucose, KI, saline water, phosphate buffer and artificial urine as the media.

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 (\( {\overline{M}}_n \)) were employed, of which the shortest chain had \( {\overline{M}}_n \) 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 (Ge) 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 Ge 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 Ge 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.

Transition between Phantom and Affine Network Model Observed in Polymer Gels with Controlled Network Structure

The elastic moduli of elastomeric materials are predicted by the affine or phantom or junction affine network models. Although these models are often used, we do not know the requirement conditions for each model or even the validity of each model. The validation of these models is difficult because of the network heterogeneity. In this study, we tried to evaluate these models using Tetra-PEG gel, which has extremely homogeneous network structure. We performed the stretching and tearing tests, and for the first time, observed the transition between the phantom and affine network models around the overlapping concentration of prepolymers.

Examination of the Theories of Rubber Elasticity Using an Ideal Polymer Network

We evaluate the homogeneity of Tetra-PEG gel and examine the models predicting rubber elasticity. Infrared spectroscopy revealed the near absence of dangling chains. Concentration dependence of the number of elastically effective chains revealed the near absence of elastically ineffective loops and the validity of the phantom network model. These data suggest that Tetra-PEG gel is close to an ideal polymer network and that the phantom network model is the true model describing the contribution of the chemical cross-link to the rubber elasticity of a swollen polymer network.

Synthesis and network parameters of hydrophobic poly(N‐[3‐(dimethylaminopropyl)]methacrylamide‐co‐lauryl acrylate) hydrogels

AbstractHydrophobic poly(N‐[3‐(dimethylaminopropyl)]methacrylamide‐co‐lauryl acrylate) [P(DMAPMA‐co‐LA)] hydrogels with different LA content were synthesized by free‐radical crosslinking copolymerization of corresponding monomers in water by using N,N‐methylenebis(acrylamide) as the crosslinker, ammonium persulfate as the initiator, and N,N,N′,N′‐tetramethylethylenediamine as the activator. The swelling equilibrium of the hydrogels was investigated as a function of temperature and hydrophobic comonomer content in pure water. An interesting feature of the swelling behavior of the P(DMAPMA‐co‐LA) hydrogels with low LA content was the reshrinking phase transition where the hydrogels swell once and collapse as temperature was varied in the range of 30–40°C. The average molecular mass between crosslinks (M̄c) and polymer–solvent interaction parameter (χ) of the hydrogels were calculated from equilibrium swelling values. The enthalpy (ΔH) and entropy (ΔS) changes appearing in the χ parameter for the hydrogels were determined by using the Flory–Rehner theory based on the phantom network model of swelling equilibrium. The positive values for ΔH and ΔS indicated that the hydrogels had a positive temperature‐sensitive property in water, that is, swelling at a higher temperature and shrinking at a lower temperature. © 2006 Wiley Periodicals, Inc. J Appl Polym Sci 101: 4159–4166, 2006

Effect of the amount and type of the crosslinker on the swelling behavior of temperature-sensitive poly(N-tert-butylacrylamide-co-acrylamide) hydrogels

Temperature-sensitive poly(N-tert-butylacrylamide-co-acrylamide) [P(NTBA-co-AAm)] hydrogels were synthesized by free-radical copolymerization in a water–methanol mixture using three types of crosslinkers: 1,2-ethyleneglycol dimethacrylate, N,N-methylenebisacrylamide, and 1,3-butandiol dimethacrylate. These thermosensitive hydrogels were swollen to equilibrium in water at 20°C and examined by gravimetric measurements. The influence of type and content of crosslinkers on the swelling ratio, the polymer–solvent interaction parameter (χ), the average molecular mass between crosslinks \({\left( {\overline{M} _{c} } \right)},\) and the effective crosslinking density (νE) of the hydrogels were reported and discussed. The swelling process in water was found to be non-Fickian diffusion. The enthalpy (ΔH) and entropy (ΔS) changes appearing in the χ parameter for the hydrogels were determined by using the Flory–Rehner theory based on the phantom network model of swelling equilibrium. Negative values for ΔH and ΔS indicated that the hydrogels had a negative temperature-sensitive property in water; that is, swelling at a lower temperature and shrinking at a higher temperature. The temperature-reversibility and on–off switching properties of the P(NTBA-co-AAm) hydrogels may be considered as good candidates for designing novel drug-delivery systems.

Synthesis and network structure of ionic poly( N, N-dimethylacrylamide-co-acrylamide) hydrogels: Comparison of swelling degree with theory

At four different charge densities, ionic hydrogels based on N, N-dimethylacrylamide (DMAAm), acrylamide (AAm), and itaconic acid (IA) were synthesized by free-radical cross-linking copolymerization in water with N, N-methylenebis(acrylamide) (BAAm) as the cross-linker, ammonium persulfate (APS) as the initiator, and N, N, N′, N′-tetramethylenediamine (TEMED) as the activator. The swelling behaviors of these hydrogels were analyzed in buffer solutions at various pH. It was observed that the swelling behavior of cross-linked ionic poly( N, N-dimethylacrylamide-co-acrylamide) [P(DMAAm-co-AAm)] hydrogels at different pHs agreed with the modified Flory–Rehner equation based both on the phantom network and affine network models and the ideal Donnan theory. In addition, the kinetics of swelling of the hydrogels was studied in pH 2, 5 and 9 buffer solutions. The swelling curves exhibited the characteristic features of transport process, apparently the Fickian diffusion of fast rates.

Swelling of crosslinked polyacrylates in isotropic and anisotropic solvents

AbstractThe purpose of this study was to examine the swelling and deswelling of photochemically crosslinked poly(n‐butylacrylate) networks in isotropic and anisotropic solvents. The phase diagrams were established in terms of composition and temperature for five isotropic solvents, acetone, cyclohexane, methanol, tetrahydrofuran, and toluene, and two low‐molecular‐weight nematic liquid crystals, 4‐cyano‐4′‐n‐pentyl‐biphenyl and an eutectic mixture of cyanoparaphenylenes. Networks were formed by ultraviolet curing in the presence of 0.5 wt % difunctional monomer (hexane diol‐di‐acrylate) and 0.5 wt % photoinitiator (Darocur 1173). Immersion in excess solvent allowed us to measure the solvent uptake by weight and to determine the size increase by optical microscopy in terms of temperature. We calculated weight and diameter ratios considering the swollen‐to‐dry network states of the samples. Phase diagrams were analyzed with the phantom network model according to the Flory–Rehner theory of rubber elasticity, and for the anisotropic solvents, modeling was supplemented with the Maier–Saupe theory of nematic order for free energy. The polymer–solvent interaction parameter was deduced as a function of temperature, but the values were in discrepancy with Fedors's model of solubility parameters, which overestimated the interaction. © 2003 Wiley Periodicals, Inc. J Appl Polym Sci 91: 1–9, 2004

Effect of Cross-Linking Density on Swelling Behavior of NIPA Gel Particles

Submicrometer-sized N-isopropylacrylamide(NIPA) gel particles with various composition of cross-linker are prepared by precipitation polymerization and their swelling behaviors are examined by using photon correlation spectroscopy(PCS) technique. Theoretically, Hu et al.'s double-lattice model has been modified by introducing a new universal constant and simplifying the expression of the Helmholtz energy of mixing. The modified double-lattice model(MDL) describes very well the lower critical solution temperature (LCST) behavior of linear PNIPA/water system. For cross-linked NIPA gel particles in water, we have combined MDL theory with Flory and Erman's theory of elasticity, which takes into account the nonaffine displacements of network junctions under high degree of swelling. Energy parameters obtained from the linear PNIPA/water system were directly used to predict swelling equilibria for the NIPA gel/water system. Our results represented that an affine network model explained the swelling behavior at high swelling ratio more accurately than that of the phantom network model. To correct the deviation of the continuous volume transition temperature, we have considered the interaction energy parameter between cross-linker segment and solvent molecule. The corrected model was applied to the swelling equilibria of gel particles with various cross-linking density and agreed well with their experimental data with no adjustable model parameters.

Swelling of sodium chloride filled polybutadiene networks in water, water/ acetone and water/THF mixtures

The swelling of polybutadiene (PB) networks of different cross-link densities filled with finely divided sodium chloride (NaCI) particles are studied in water, water/acetone and water/THF mixtures. Swelling over periods of 2 months are recorded. The degree of swelling was observed to increase continuously in water during this period, whereas in water/acetone mixtures it reached a peak followed by deswelling and it converged to a maximal value in water/THF mixtures. The highly cross-linked samples exhibited unusually high degrees of swelling, and calculations indicated full stretching of the network chains. The enthalpic contribution to the chemical potential of the networks in equilibrium with solvent was determined using the phantom network model of rubber elasticity theory. The swollen network-salt system was approximated by the lattice model of polymer solutions. The positive contributions of water and acetone to the chemical activity of the swollen network were offset by the negative contribution of dissociation of salt in water, resulting in a small negative enthalpy of mixing. The highest rate of diffusion of solvent was observed for the PB/water/THF ternary system. Diffusion rates of pure water and water/acetone mixture were much smaller and of comparable magnitude. Mechanical measurements of stress and strain at break, and dynamic mechanical measurements of storage and loss moduli of filled and unfilled samples showed that filling with NaCl did not change the mechanical properties of the networks significantly, in spite of the dramatic change in the swelling behaviour.

Self-Diffusion of Constituent Cationic Surfactants in Threadlike Micelles

Self-diffusion mechanisms of a surfactant molecule in entangled threadlike micelles formed from a mixture of cetyltrimethylammonium bromide (CTAB) and sodium salicylate (NaSal) in aqueous solution (CTAB:NaSal/W) are discussed. The self-diffusion coefficient (Dself) for the cetyltrimethylammonium (CTA+) cation in the micelle was calculated on the basis of a phantom network model. Two formulations for the self-diffusion coefficient are derived for two separate regimes with respect to time and space. When the relaxation time (τm) for the micellar system is short, Dself is represented as 〈ξ2〉{6τmP(Le,τm)}, where 〈ξ2〉 is the mean square of the entanglement spacing which depends on the CTA+ concentration and P(Le,τm) is a probability for the CTA+ cation to remain in the initial part of the threadlike micelle with the average contour length of Le between two entanglements. On the other hand, when τm is long, Dself is expressed as (2Q/3)(Dlat/2τm)1/2, where Q and Dlat are the persistence length of the threadlike ...