New theoretical considerations in polymer rheology: elastic breakdown of chain entanglement network

Recent experimental evidence has motivated us to present a set of new theoretical considerations and to provide a rationale for interpreting the intriguing flow phenomena observed in entangled polymer solutions and melts [P. Tapadia and S. Q. Wang, Phys. Rev. Lett. 96, 016001 (2006); 96, 196001 (2006); S. Q. Wang et al., ibid. 97, 187801 (2006)]. Three forces have been recognized to play important roles in controlling the response of a strained entanglement network. During flow, an intermolecular locking force f(iml) arises and causes conformational deformation in each load-bearing strand between entanglements. The chain deformation builds up a retractive force f(retract) within each strand. Chain entanglement prevails in quiescence because a given chain prefers to stay interpenetrating into other chains within its pervaded volume so as to enjoy maximum conformational entropy. Since each strand of length l(ent) has entropy equal to k(B)T, the disentanglement criterion is given by f(retract)>f(ent) approximately k(B)Tl(ent) in the case of interrupted deformation. This condition identifies f(ent) as a cohesive force. Imbalance among these forces causes elastic breakdown of the entanglement network. For example, an entangled polymer yields during continuous deformation when the declining f(iml) cannot sustain the elevated f(retract). This opposite trend of the two forces is at the core of the physics governing a "cohesive" breakdown at the yield point (i.e., the stress overshoot) in startup flow. Identifying the yield point as the point of force imbalance, we can also rationalize the recently observed striking scaling behavior associated with the yield point in continuous deformation of both shear and extension.     

Segmental orientation in entangled chains

Slip-link model of an entangled chain is used to calculate average orientation of chain segments. The results in the asymptotic regime of very long chains prove linear dependence of optical anisotropy on stress despite complex stress-strain relation. The linear stress-optical law is predicted both for a single chain and a model network subjected to uniaxial deformation. The calculated stress exhibits non-linearity in Mooney-Rivlin plot. Effects due to entanglements are proportional to assumed number of slip-links per chain.     

“Belt-loop” model of chain entanglement

A simple model of an entangled chain is proposed. Statistical properties of the model are examined based on the partition function derived to include geometrical constraints imposed by entanglements. The model chain statistics results, for long chains, in a modified Gaussian function. The new statistics applied in the affine network theory yield stress-strain dependence, which qualitatively agrees with experimental data obtained for uniaxial extension and compression. Non-linear Mooney-Rivlin plots with a maximum appearing in the compression region are predicted for unswollen networks. With increasing swelling, non-linearity decreases. The proposed explanation of these phenomena is based on the restraints imposed on entangled chains, rather than on network junctions, unlike in the Flory-Erman theory. No arbitrary parameters are involved in the model.     

Influence of uniaxial deformation of colored polyvinyl alcohol films on their thermal-oxidative destruction

The thermal-oxidative destruction of films made of polyvinyl alcohol colored by brilliant yellow and subjected to uniaxial deformation was studied by the thermogravimetric method. The intensive dehydration of polyvinyl alcohol begins at temperatures higher than 150°C in non-deformed films and higher than 220°C in samples subjected to the uniaxial extension. The dye practically does not influence thermal-oxidative destruction of non-oriented samples. The stretching and treatment by a ??cross-linking?? agent during the uniaxial deformation of the film increases its thermal stability.     

Monte Carlo Studies on the Deformation Behavior of Entangled Polymer Chains

In this work we present a continuous three‐dimensional bond‐fluctuation model (CBFM) in combination with the method of confined self‐avoiding walks (CSAW). This method enables us to analyse both the compression and the stretching regime of the deformation process of two entangled polymer chains. We studied the deformation behavior in respect to the number of entanglements, the distance of the end monomers on the confining surfaces and the orientation of the entanglements to the deformation axis. Our analysis of the behavior of the force and the structural properties of the systems during the deformation process leads us to the assumption that the entanglements act as one topological crosslink with variable strength.     

Flow,high-elastic (recoverable) deformation,and rupture of uncured high molecular weight linear polymers in uniaxial extension

The behavior of narrow molecular weight distribution polymers has been investigated under uniaxial extension at constant deformation rate and at constant stress. It has been established that up to rupture these polymers behave as linear viscoelastic bodies. A detailed investigation of the rupture phenomenon has shown that the rupture of fluid polymers is due to their transition to the rubbery state at critical deformation rates, with the result that they disintegrate like quasi-cured rubbers. The effect of the temperature and the molecular weight on the critical conditions of rupture has been described in terms of viscoelastic relaxation.     

Mechanism of shear deformation of a coiled myosin coil: Computer simulation

Polymer coiled coils are composed of entangled linear chains in a helical conformation. Their mechanical characteristics are interesting because these structures are involved in the composition of natural fibrillar structures. The method of molecular dynamics is used for the simulation of stretching at a constant rate for a superhelical fragment of myosin protein composed of two identical α helices containing 126 amino acid residues in each helix. The case of shear deformation of a molecule is considered (the load is applied to the N terminus of one chain and to the C terminus of another chain). In this case of loading, slippage of chains with respect to each other can occur. Deformation of a molecule proceeds in several stages. At the initial stage, the superhelix is unfolded and there is a gradual unfolding of end fragments of individual α helices; this process is accompanied by their displacement with respect to the helical fragment of the neighboring chain. In this case, the reaction force increases. At the second stage of stretching, the process passes to the mechanism of deformation when, in the central part of the molecule, α-helical fragments of both chains unfold. In this region, the reaction-force-extension curve shows a plateau region. Between unfolded fragments, new hydrophobic contacts and hydrogen bonds are formed, and fragments of the β structure emerge. Once all turns of α helices in the central parts of the molecule are unfolded, the mechanism of deformation changes and further extension of a molecule proceeds via straightening of previously unfolded central fragments, a process that is accompanied by an increase in the reaction force. When chains achieve their limiting extension, slippage commences with an accompanying decrease in the reaction force.     

Uniaxial and biaxial stretching of silicone putty

Two novel experimental methods are used. Vertical uniaxial stretching is obtained by attaching a perspex rod to the lower end of a silicone putty cylinder; the rod then descends into water of constant depth. The stress and rate of extension change little during each test, but the rate of extension may be varied from 0.005 to 0.10 s⁻¹ by modifying the experimental conditions. Biaxial stretching is acchieved by placing a disc of silicone putty across the top of an open glass cylinder which is lightly pressurized. The sample expands as a spherical cap, the height of the centre above the cylinder being timed. The stress in the cap passes through a shallow minimum as it expands (at constant pressure) and the slowly varying rate of biaxial extension may be readily determined. This lies in the range 0.003–0.06 s⁻¹. For low rates of uniaxial or biaxial extension, it is possible to plot the extension against time and to show how the extensional viscosity varies with the strain rate (or principal extension ratio). For high rates of extension, a ‘single point’ determination of the extensional viscosity may be made, with the stress and strain rate averaged at the mid-point of the sample's extension. The temperature is 26.5 ± 1.5 °C. The following is shown under the experimental conditions:(a) the extensional viscosity (uniaxial or biaxial) is in the range 1.0 × 105 to 3.0 × 10⁵ Pa s;(b) for extensional strain rates between 0.01 and 0.04 s⁻¹, the uniaxial and biaxial extensional viscosities are of comparable value;(c) both forms of the extensional viscosity tend to decrease with increased extensional strain rate, the biaxial extensional viscosity falling more rapidly and being higher than the uniaxial viscosity at low strain rates and lower at high strain rates;(d) there are no signs of rupture in uniaxial extension (principal extension ratios up to 1.8 and extensional strain rate up to 0.1 s⁻¹);(e) in biaxial extension, the sample tends to rupture more easily as the strain rate is increased. (The sample fails at the principal extension ratio of 2.0 at an extensional strain rate of 0.02 s⁻¹ and fails at a principal extension ratio of 1.3 at an extensional strain rate of 0.07 s⁻¹.)     

A highly coarse-grained model to simulate entangled polymer melts

We introduce a highly coarse-grained model to simulate the entangled polymer melts. In this model, a polymer chain is taken as a single coarse-grained particle, and the creation and annihilation of entanglements are regarded as stochastic events in proper time intervals according to certain rules and possibilities. We build the relationship between the probability of appearance of an entanglement between any pair of neighboring chains at a given time interval and the rate of variation of entanglements which describes the concurrence of birth and death of entanglements. The probability of disappearance of entanglements is tuned to keep the total entanglement number around the target value. This useful model can reflect many characteristics of entanglements and macroscopic properties of polymer melts. As an illustration, we apply this model to simulate the polyethylene melt of C(1000)H(2002) at 450 K and further validate this model by comparing to experimental data and other simulation results.     

高分子熔体和浓溶液流变性能的研究

基于多重缠结网络结构模型和高分子链上缠结点在流动中可进行动态解缠和再缠结的多重蠕动机理,用统计力学和动力学相结合的方法,分别计算出了缠结链组的末端距分布函数;处于缠结状态下高分子链构象统计分布函数;受力下聚合物熔体粘弹性形变自由能和解除外力下高分子挤出体可回复性粘弹性形变自由能,提出了高分子挤出体可回复形变的粘弹性分子理论。推导出的高分子熔体的回忆函数、简单剪切流下的本构方程和物料函数,并采用一种新的方法测定出物料的四种参数： η0、 GN0、 n′和 a。对于高分子挤出体,可回复性粘弹性形变由快速弹性形变和慢速粘弹性形变两者组成,当把两种形变量的复合结构参数－分子链的反式构象分数引入两种形变自由能表达式后,就从理论上得到了可回复形变量同挤出胀大比间的定量表达式,从而建立起一个具有分子链结构参数的新的挤出胀大比方程,可回复形变量同挤出条件（温度、挤出速率和量以及口模长径比不同的挤出机）和树脂结构特征（分子量及分布）的关系式以及在特殊情形下的简化表达式,并用几种高分子熔融体系的挤出胀大比和可回复性形变量的实验数据对理论进行验证,理论方程同实验数据较好的符合。     

Brittle‐ductile transition in uniaxial compression of polymer glasses

We carried out a large set of tests to establish a correlation between the molecular (network) structure (influenced by molecular weight, molecular weight distribution, and melt predeformation) and mechanical responses of several glassy polymers to uniaxial compression at different temperatures and different compression speeds. The experimental results show that to have ductile responses there must be an adequate chain network, afforded by the interchain uncrossability among sufficiently long chains. Specifically, polystyrene (PS) and poly(methyl methacrylate) of sufficiently low molar mass do not have chain network and are found to be very brittle. Binary PS mixtures are brittle at room temperature when the volume fraction of the high‐molecular‐weight component is sufficiently low (e.g., at and below 27.5%). Moreover, sufficiently melt‐stretched PS mixtures show brittle fracture when compressed along the same direction, along which melt stretching was made. All the experimental findings confirm that a robust chain network is also a prerequisite for yielding and ductile cold compression of polymer glasses, as is for extension. © 2019 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2019 , 57, 758–770     

Effect of chain stiffness and entanglements on the elastic behavior of end-linked elastomers

The effect of chain stiffness and entanglements on the elastic behavior and microscopic structure of cross-linked polymer networks was studied using Monte Carlo simulations. We investigated the behavior of entangled and entanglement-free networks at various degrees of chain stiffness and densities. Based on previous results that indicated that trapped entanglements prevent strain-induced order-disorder transitions in semiflexible chain networks, we prepared the entangled networks by end-linking the chains in very dilute conditions so as to minimize the extent of trapped entanglements. We also considered the entanglement-free case by using a "diamond" structure. We found that the presence of even a very small amount of trapped entanglements is enough to prevent a discontinuous strain-induced transition to an ordered phase. In these mildly entangled networks, a nematiclike order is eventually attained at high extensions but the elastic response remains continuous and the cross-links remain uniformly distributed through the simulation box. The entanglement-free diamond networks on the other hand show discontinuities in their stress-strain data. Networks at higher densities exhibit a more stable ordered phase and show an unusual staircaselike stress-strain curve. This is the result of a stepwise extension mechanism in which the chains form ordered domains that exclude the cross-links. Extension is achieved by increasing the number of these ordered domains in the strain direction. Cross-links aggregate in the spaces between these ordered domains and form periodic bands. Each vertical upturn in the stress-strain data corresponds to the existence of an integer number of ordered domains. This stepwise elastic behavior is found to be similar to that exhibited by some tough natural materials.     

Entanglement junctions in melts and concentrated solutions of flexible-chain polymers: Macromodeling

The effect of deformation on the behavior of intermolecular entanglements in melts and in concentrated solutions of polymers is simulated with the use of knots formed by long flexible threads. Two cases are discussed: the behavior of individual junctions and the pattern of change in the entanglements in a network of statistically entangled threads. It is shown that, at low strain rates, the junctions disentangle and threads slip out of the entanglements. This situation simulates an irreversible flow of macromolecules in the classical tube model. At high rates, the junctions tighten, a phenomenon that simulates the impossibility of irreversible motion of macromolecules. The transition from slipping to tightening of entanglements simulates the change in the pattern of behavior of the melt, i.e., from flow to rubberlike deformations. A generalized dependence of elastic deformations on the Weissenberg number is constructed; it shows that the transition occurs at Weissenberg numbers on the order of 3–5, which correspond to the characteristic lifetimes of an entanglement junction. At high strain rates, redistribution and local accumulation of entanglements are observed.     

The molecular kink paradigm for rubber elasticity: numerical simulations of explicit polyisoprene networks at low to moderate tensile strains

Based on recent molecular dynamics and ab initio simulations of small isoprene molecules, we propose a new ansatz for rubber elasticity. We envision a network chain as a series of independent molecular kinks, each comprised of a small number of backbone units, and the strain as being imposed along the contour of the chain. We treat chain extension in three distinct force regimes: (Ia) near zero strain, where we assume that the chain is extended within a well defined tube, with all of the kinks participating simultaneously as entropic elastic springs, (II) when the chain becomes sensibly straight, giving rise to a purely enthalpic stretching force (until bond rupture occurs) and, (Ib) a linear entropic regime, between regimes Ia and II, in which a force limit is imposed by tube deformation. In this intermediate regime, the molecular kinks are assumed to be gradually straightened until the chain becomes a series of straight segments between entanglements. We assume that there exists a tube deformation tension limit that is inversely proportional to the chain path tortuosity. Here we report the results of numerical simulations of explicit three-dimensional, periodic, polyisoprene networks, using these extension-only force models. At low strain, crosslink nodes are moved affinely, up to an arbitrary node force limit. Above this limit, non-affine motion of the nodes is allowed to relax unbalanced chain forces. Our simulation results are in good agreement with tensile stress vs. strain experiments.     

Drawing of microshear deformation zones in glassy polymers: Poly(phenylene oxide) (PPO)

The topography of the microscopic shear deformation zones (SDZ) in the glassy polymer PPO was studied by using atomic force microscopy (AFM) and was used to analyze the growth and breakdown of the SDZ. It was found that the local stress and strain are almost constant within the deformation zones but higher than those in the elastic regions. The maximum strain rate during stretching was found to always locate near the SDZ boundaries, indicating that most drawing took place there. With both the local stress and strain obtained for every point within the SDZ, it is possible to construct a full stress-strain curve for the drawing of the tiny local deformation zones. The stress-strain curve clearly demonstrates a yield point in the beginning of microyielding where the tensile modulus was found to be much lower than that in the elastic regime. Some strain hardening, however, took place at larger deformation. Moreover, we found that for each microscopic region participated in the microdrawing the local strain rate increased with local strain until a critical strain around 0.65 was reached, after which the strain rate decreased with strain. This critical strain may be related to the chain entanglement network structure because it shifted to 0.75 when PS diluents were blended into PPO, indicating that strain hardening was delayed by the increase of chain entanglement mesh size. © 1996 John Wiley & Sons, Inc.     

Characterization of polyethylene crystallization from an oriented melt by molecular dynamics simulation

Molecular dynamics is used to characterize the process of crystallization for a united atom model of polyethylene. An oriented melt is produced by uniaxial deformation under constant load, followed by quenching below the melting temperature at zero load. The development of crystallinity is monitored simultaneously using molecular-based order parameters for density, energy, and orientation. For crystallization temperatures ranging from 325 to 375 K, these simulations clearly show the hallmarks of crystal nucleation and growth. We can identify multiple nucleation events, lamellar growth up to the limit imposed by periodic boundaries of the simulation cell, and lamellar thickening. We observe a competition between the rate of nucleation, which results in multiple crystallites, the rate of chain extension, which results in thicker lamellae, and the rate of chain conformational relaxation, which is manifested in lower degrees of residual order in the noncrystalline portion of the simulation. The temperature dependence of lamellar thickness is in accord with experimental data. At the higher temperatures, tilted chain lamellae are observed to form with lamellar interfaces corresponding approximately to the [201] facet, indicative of the influence of interfacial energy.     

Melt drawing as a route to high performance polyethylene

Well established routes for obtaining stiff and strong polyethylene (PE) involve solid state drawing either of solution crystallized gel films or melt crystallized spherulitic PE. The aim of this work is to show the potential of melt deformation as an alternative route for obtaining highly oriented products. Our previous work on the melt deformation route showed that oriented PE fibers could be directly extruded under appropriately controlled conditions [8,9]. Here, we show that PE films (or filaments) can also be melt drawn in the temperature window 130–160 °C, thus yielding oriented products. The advantage of melt drawing over direct melt extrusion is that it allows a wider operational latitude and thus does not require such carefully controlled conditions.The morphology produced by melt deformation is different from solid state deformation and consists of extended chain fibrils with platelet overgrowths. The relative amount of fibrils and platelets depends on operating parameters. The temperature window of PE melt drawing is identified with the regime where some flow induced crystallization takes place. The conditions for melt drawability are of wider generality for crystallizable flexible chain polymers. They are: (i) adequate strain rate to overcome entropie resistance to chain extension, (ii) but not high enough to activate the elastic response of the transient networks in the entangled system, (iii) sufficient strain to fully extend the chain, (iv) appropriate temperature for flow-induced crystallization and strain hardening, and (v) cooling to freeze the oriented structure.Ultra high molecular weight PEs were not the most suitable for melt drawing due to their high recoverable elongation in the melt (melt elasticity) in addition to added limitations imposed by their nascent grain systeme. Our work suggests that an optimum molecular weight for melt drawing is¯M _w(400–900)×10³ with further possibilities for improvement through multimodal distributions.     

A non-Gaussian model for the chemo-mechanical coupling behavior of largely deformed hydrogels

A non-Gaussian model is developed to precisely describe the chemo-mechanical coupled large deformation of responsive hydrogels. In this model, the free-energy density is composed of two parts, including the elastic energy due to stretching of cross-linked polymer networks described by Gent hyperelastic model, and the mixture energy of polymer network and described by Flory–Huggins theory. The effects of junction functionality and chain entanglements are investigated by analyzing free swelling of a cubic hydrogel and constrained swelling of a blanket layer of the gel. The present model is found to exhibit obvious hardening characteristic during large deformation of the hydrogel, and the considerations of functionality of junctions and chain entanglements are essential in the coupled chemo-mechanical deformation analysis of hydrogels.