Phantom Network Research Articles

Evolution of network topology of bifunctional epoxy thermosets during cure and its relationship to thermo-mechanical properties: A molecular dynamics study

Thermoset polymers are used for a wide range of application from large airframes to microelectronics and fundamental understanding of the development of their 3D network during cure and its relationship to thermo-mechanical properties is of fundamental and applied interest. Experimental characterization of network properties, such as the density of crosslinked chains, involves the use of approximate models with uncertain parameters leading, consequently, to uncertainties in resulting properties. We use molecular dynamics to simulate the cure of two thermoset epoxy systems, characterize the evolution of its network topology and establish relationships to the stiffness, yield stress and glass transition temperature. Relating the predicted crosslink density with experimental measurements of rubber elasticity indicates that these polymers behave somewhere between an affine and phantom networks; the proportionality constant extracted between rubber modulus and crosslink density can be used for similar systems to experimentally extract network characteristics.

Time to failure prediction in rubber components subjected to thermal ageing: A combined approach based upon the intrinsic defect concept and the fracture mechanics

In this contribution, we attempt to derive a tool allowing the prediction of the stretch ratio at failure in rubber components subjected to thermal ageing. To achieve this goal, the main idea is to combine the fracture mechanics approach and the intrinsic defect concept. Using an accelerated ageing procedure for an Ethylene–Propylene–Diene Monomer (EPDM), it is first shown that the average molar mass of the elastically active chains (i.e. between crosslinks) can be used as the main indicator of the macromolecular network degradation. By introducing the time–temperature equivalence principle, a shift factor obeying to an Arrhenius law is derived, and master curves are built as well for the average molar mass as for the ultimate mechanical properties. Fracture mechanics tests are also achieved and the square root dependence of the fracture energy with the average molar mass is pointed out. Moreover, it is shown that the mechanical response could be approximated by the phantom network theory, which allows to relate the strain energy density function to the average molar mass. Assuming that the fracture of a smooth specimen is the consequence of a virtual intrinsic defect whose the size can be easily estimated, the stretch ratio at break can be therefore computed for any thermal ageing condition. The estimated values are found in a very nice agreement with EPDM experimental data, making this approach a useful tool when designing rubber components for moderate to high temperature environments.

AMA to Work With APA, Other Groups on Integrated Care

AMA to Work With APA, Other Groups on Integrated Care

A new thermodynamic model of volume changes in temperature-sensitive polymer gels

In the classical swelling theory a polymer gel is a very (infinitely) large cross-linked polymer molecule that encages the solvent molecules. Here, a new swelling model is developed. Firstly, this model permits many cross-linked polymer molecules with finite molar mass instead of a single one. The gel formation is described by a huge number of consecutive swelling equilibria incorporating a solvent molecule by solvent molecule. Thus, the gel is a polydisperse mixture of cross-linked polymer molecules encaging different amounts of solvent. The polydispersity is expressed by a gel-distribution function, the width of which decreases with increasing molar mass of the cross-linked polymer. In the limit of infinite molar mass the new swelling theory is equivalent to the classical one. Secondly, different to the classical treatment, the polymer network is considered to be finitely extensible. Both theories are applied to hydrogels of poly(N-isopropylacrylamide). The mixing contribution of the Helmholtz energy is described by a modified Flory–Huggins expression, the parameters of which are fitted to cloud-point data of a solution of a linear poly(N-isopropylacrylamide). For the elastic contribution the phantom network theory is applied. The new model calculates the swelling degree as a function of temperature nearly perfect in comparison with the experimental data. The classical theory fails in the range of high swelling degrees. Furthermore, the classical treatment predicts a first-order transition between a swollen and a shrunken gel. The new model predicts also a sharp but continuous transition.

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.

Topological versus rheological entanglement length in primitive-path analysis protocols, tube models, and slip-link models

We show that the front factor appearing in the shear modulus of a phantom network, G(ph) = (1-2/f)(ρk(B)T)/N(s), also controls the ratio of the strand length, N(s), and the number of monomers per Kuhn length of the primitive paths, N(ph)(PPKuhn), characterizing the average network conformation. In particular, N(ph)(PPKuhn) = N(s)/(1-2/f) and G(ph) = (ρk(B)T)/N(ph)(PPKuhn). Neglecting the difference between cross-links and slip-links, these results can be transferred to entangled systems and the interpretation of primitive path analysis data. In agreement with the tube model, the analogy to phantom networks suggest that the rheological entanglement length, N(e)(rheo) = (ρk(B)T)/G(e), should equal N(e)(PPKuhn). Assuming binary entanglements with f = 4 functional junctions, we expect that N(e)(rheo) should be twice as large as the topological entanglement length, N(e)(topo). These results are in good agreement with reported primitive path analysis results for model systems and a wide range of polymeric materials. Implications for tube and slip-link models are discussed.

‘Phantom Networks’ of Managed Behavioral Health Providers: An Empirical Study of Their Existence and Effect on Patients in Two New Jersey Counties

Managed care organizations often tout the availability of clinicians in their provider networks, yet their clients seeking mental healthcare may find it difficult to obtain such care in a timely and effective manner. Using comprehensive data from two counties in New Jersey, the authors examine the prevalence of phantom networks of managed care providers of behavioral health services and the effects of such networks on patients’ wait times and the availability of therapists treating children.

Influence of different alcohols on the swelling behaviour of hydrogels

The swelling equilibrium of cross-linked poly(N-isopropylacrylamide) (PNIPAAm) hydrogels in alcohol solutions as a function of temperature, alcohol concentration, kind of alcohol (C1OH–C3OH) and gel properties was investigated experimentally. Additionally, the swelling degree as a function of the alcohol concentration was modelled with the UNIQUAC-Free Volume model in combination with the Phantom Network theory. The experiments show that, in pure water, the transition temperature is between 303.15 and 308.15 K depending on the properties of the gel and hence on the polymerization conditions. The transition from a swollen to a shrunken state is caused by the polymeric network and the change of polymer chain localization. In a system with hydrogel + water + alcohol, the swelling degree decreases with increasing alcohol concentration until the shrunken state is reached and increases again by further addition of alcohol at constant temperature. With increasing carbon number of the alcohols, the transition from a swollen to a shrunken state and vice versa shifts to lower concentrations at constant temperature. The use of the UNIQUAC-Free Volume model with Phantom Network theory leads to results in good agreement with the experimental data.

Quantitative assessment of a framework for creating anatomical brain networks via global tractography

Interregional connections of the brain measured with diffusion tractography can be used to infer valuable information regarding both brain structure and function. However, different tractography algorithms can generate networks that exhibit different characteristics, resulting in poor reproducibility across studies. Therefore, it is important to benchmark different tractography algorithms to quantitatively assess their performance. Here we systematically evaluated a newly introduced tracking algorithm, global tractography, to derive anatomical brain networks in a fiber phantom, 2 post-mortem macaque brains, and 20 living humans, and compared the results with an established local tracking algorithm. Our results demonstrated that global tractography accurately characterized the phantom network in terms of graph-theoretic measures, and significantly outperformed the local tracking approach. Results in brain tissues (post-mortem macaques and in vivo humans), however, showed that although the performance of global tractography demonstrated a trend of improvement, the results were not vastly different than that of local tractography, possibly resulting from the increased fiber complexity of real tissues. When using macaque tracer-derived connections as the ground truth, we found that both global and local algorithms generated non-random patterns of false negative and false positive connections that were probably related to specific fiber systems and largely independent of the tractography algorithm or tissue type (post-mortem vs. in vivo) used in the current study. Moreover, a close examination of the transcallosal motor connections, reconstructed via either global or local tractography, demonstrated that the lateral transcallosal fibers in humans and macaques did not exhibit the denser homotopic connections found in primate tracer studies, indicating the need for more robust brain mapping techniques based on diffusion MRI data.

How the Statistical Validation of Functional Connectivity Patterns Can Prevent Erroneous Definition of Small-World Properties of a Brain Connectivity Network

The application of Graph Theory to the brain connectivity patterns obtained from the analysis of neuroelectrical signals has provided an important step to the interpretation and statistical analysis of such functional networks. The properties of a network are derived from the adjacency matrix describing a connectivity pattern obtained by one of the available functional connectivity methods. However, no common procedure is currently applied for extracting the adjacency matrix from a connectivity pattern. To understand how the topographical properties of a network inferred by means of graph indices can be affected by this procedure, we compared one of the methods extensively used in Neuroscience applications (i.e. fixing the edge density) with an approach based on the statistical validation of achieved connectivity patterns. The comparison was performed on the basis of simulated data and of signals acquired on a polystyrene head used as a phantom. The results showed (i) the importance of the assessing process in discarding the occurrence of spurious links and in the definition of the real topographical properties of the network, and (ii) a dependence of the small world properties obtained for the phantom networks from the spatial correlation of the neighboring electrodes.

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.

Thermodynamics of aqueous solutions containing poly (N-isopropylacrylamide)

Hydrogels undergo reversible and discontinuous volume changes in response to variation of solution conditions such as solvent composition, temperature, salt concentration, and pH. In this contribution we focus our attention on the experimental and theoretical investigation of these swelling equilibria of aqueous cross-linked poly (N-isopropylacrylamide) solutions as well as on the connected demixing behavior of the linear polymer dissolved in water. For the experimental study of the (liquid + liquid) equilibrium an alternative method based on refractive index measurements is suggested. In order to calculate the swelling behavior a model combining an expression for the Gibbs free energy of mixing with an expression for the elastic network is applied. As a model for the Gibbs free energy of mixing the UNIQUAC-approach and the Koningsveld–Kleintjens model are used. For the elastic network contribution again two different theories, namely the phantom network theory and the affine network theory, were applied. Whereas the type of network theory has only a small influence on the calculation results, the Gibbs free energy of mixing has a large impact. Using the UNIQUAC-approach the swelling equilibria can be correlated close to the experimental data, however, this model predicts a homogeneous mixture for linear polymer chains in water. In contrast to this situation the Koningsveld–Kleintjens model does a good job in calculating the swelling equilibria as well as the demixing curve, however, the adjustable parameter must be changed slightly.

Small‐Angle Neutron Scattering from Elastomeric Networks in which the Junctions Alternate Regularly in their Functionality

AbstractWe compute scattering form factors for SANS from labeled paths in Gaussian phantom networks in which junctions alternate regularly in their functionality (the number of chains emanating from a junction). Our calculations are based on the James‐Guth model of rubber‐like elasticity, which assumes that fluctuations are strain independent, while mean vectors transform affinely with the applied strain. Kratky plots for scattering from isotropic and uniaxially stretched bifunctional networks are computed and compared with corresponding plots for the simpler unifunctional networks. The results show the effects of the length of the labeled path, extent of deformation, direction of scattering with respect to the principal axis of the deformation and the functionalities of the network junctions.magnified image

State Fines MH Insurers Over 'Phantom' Provider Networks

State Fines MH Insurers Over 'Phantom' Provider Networks

Crosslinked DADMAC polymers as cationic super absorbents

Loosely crosslinked cationic polyelectrolytes (polyquats) with super-water-sorption abilities are presented, for the first time. Hydrogels obtained by radically initiated copolymerization of N, N-diallyl, N, N-dimethyl ammonium chloride (DADMAC) with N-vinyl 2-pyrrolidone (NVP) in presence of N, N, N′, N′-tetraallyl piperazinium dichloride (TAP) as crosslinker showed reasonably high equilibrium swelling ratios, as high as those for poly(acrylic acid)-based super absorbents. The highest swelling ratios (up to 360) were attained by fully cationic hydrogels derived from DADMAC and 0.5% TAP. The hydrogels with 0.5–5% crosslinkers exhibited rapid expansion in neutral water, so that the equilibrium swelling values were attained within 2–3 min. The crosslinking densities of the gels were estimated by Flory-type swelling model using “phantom network elasticity” for the elastic contribution. This estimation revealed low crosslinking efficiencies of TAP (0.24–0.35) due to its double cationic charge. In the study “the salt effect” and effects of the comonomer ratios and crosslinker contents on the swellings were also investigated.

PH-dependent swelling of hydrogels containing highly branched polyamine macromonomers

We examine the pH-dependent swelling of end-linked hydrogels containing high concentrations of amine-functional macromonomers. Gels are formed by end-linking of epoxide-terminated, linear poly(ethylene glycol) (PEG) to either amine-terminated poly(amidoamine) (PAMAM) dendrimers or highly branched poly(ethyleneimine) (PEI). After extraction in neutral water, the hydrogels are swollen in aqueous solutions of HCl or NH4OH to vary the external pH. Equilibrium volume swelling ratios (Qs) pass through a maximum value (Qmax) at an external pH denoted as pH∗ which is approximately 4–5 for the gels studied. The swelling behavior is modeled using Donnan equilibrium theory to describe the ion swelling pressure, with the Flory–Rehner phantom network expression representing the elastic and mixing contributions to the free energy. The model accurately predicts the maximum in swelling near pH=4–5, but overestimates Qmax for several of the gels due to neglecting the finite extensibility of the short linear PEG chains.

Non-Gaussian Model for Rubber Elasticity: (I) Finite Chain Extensibility and Tube Concept

A simple, useful model for an elastic network is developed in terms of finite chain extensibility and entanglement constraints. For the former, each chain that constitutes the network is treated as a specific non-Gaussian one and for the latter is confined within a tube that represents entanglement constraints. The model is built within the framework of the classical phantom network theory. In addition, it considers fluctuations of diameter of the tube. While the extent of suppression of junction fluctuation is also incorporated through the well-known empirical factor h, the tube diameter is assumed to be deformed under a power law including its exponent (γ) that describes the strain-dependence of spatial constraints. Two free energies due to deformations of the network and tube diameter are calculated.

Stress relaxation in polymer networks: Equilibrium behavior and dynamics

The elastic and relaxational properties of a polymer network have been calculated using a stress based formulation based on the Rouse mode expansion [W. L. Vandoolaeghe and E. M. Terentjev, J. Chem. Phys. 123, 34902 (2005)]. In this article, we propose an improved Rouse mode expansion incorporating appropriate boundary conditions. In contrast to the previous work, this improved formulation provides a smooth crossover from the classical equilibrium result of rubber elasticity to the shorter-time-scale Rouse relaxation of a polymer melt. Our results are compared with the classical phantom network approach in equilibrium, as well as both equilibrium and dynamic elongation experiments. The model captures the qualitative features of the data well and some of the quantitative aspects, such as the exponents seen in the dynamic modulus G(omega).

An experimentalist's view of the physics of rubber elasticity

An experimentalist's view of the physics of rubber elasticity