The present contribution focuses on the thermodynamically consistent mechanical modeling of the strain-induced crystallization in unfilled polymers. This phenomenon is of particular importance for the mechanical properties of polymers as well as for their manufacturing and the application. The model developed uses the principle of the minimum of dissipation potential and assumes two internal variables: the deformations due to crystallization and the regularity of the network. In addition to the dissipation potential necessary for the derivation of evolution equations, the well-established Arruda-Boyce model is chosen to depict the elastic behavior of the polymer. Two special features of the model are the evolution direction depending on the stress state and the distinction of crystallization during the loading and unloading phase. The model has been implemented into the finite element method and applied for numerical simulation of the growth and shrinkage of the crystal regions during a cyclic tension test for samples with different initial configurations. The concept enables the visualization of the microstructure evolution, yielding information that is still inaccessible by experimental techniques.
Crystallization within the discrete spheres of a block copolymer mesophase was studied by time-resolved x-ray scattering. The cubic packing of microdomains, established by self-assembly in the melt, is preserved throughout crystallization by strong interblock segregation even though the amorphous matrix block is well above its glass transition temperature. Homogeneous nucleation within each sphere yields isothermal crystallizations which follow first-order kinetics, contrasting with the sigmoidal kinetics normally exhibited in the quiescent crystallization of bulk polymers.
We have studied the structure and free energy landscape of a semi-flexible lattice-polymer in the presence of a surface. At low temperatures coexistence of two-dimensional integer-folded crystals is observed. As the temperature is increased there is a transition from these crystalline configurations to a disordered coil adsorbed onto the surface. The polymer then gradually develops three-dimensional character at higher temperatures. We compute the free energy as a function of increasing crystallinity and compare with the free energy profiles assumed by the Lauritizen-Hoffman surface nucleation…
In this paper, we report dynamic Monte Carlo simulation results on the crystallization of crystalline/crystalline (A/B) symmetric binary polymer blend, wherein the melting temperature of A-polymer is higher than B-polymer. Crystallization of A-polymer precedes the crystallization of B-polymer upon cooling from a homogeneous melt. The morphological development is controlled by the interplay between crystallization and macrophase separation. With increasing segregation strength, smaller and thinner crystals form with lesser crystallinity. During the process, A-polymer crystallizes first followed…
Polymer blends offer an exciting material for various potential applications due to their tunable properties by varying constituting components and their relative composition. Our simulation results unravel an intrinsic relationship between the phase behavior and crystallization characteristics with the relative composition of A- and B-polymer in the system. We report simulation results for non-isothermal and isothermal crystallization with weak and strong segregation strength to elucidate the composition dependent crystallization behavior. With increasing composition of low melting B-polymer,…
In the present work, crystallization in melts and poor-solvent solutions of semiflexible polymers with different concentration was studied by means of dissipative particle dynamics simulation technique. We use a coarse-grained polymer model trying to catch general principles of crystallization in such systems on large time and length scales. We observe the crystallization process starting from an initial randomly prepared system with different polymer volume fractions in a poor solvent. Because the solvent is very poor, the macrophase polymer-solvent separation takes place very fast and is acc…
The work introduces a 3D cellular automaton model for the spatial and crystallographic prediction of spherulite growth phenomena in polymers at the mesoscopic scale. The automaton is discrete in time, real space, and orientation space. The kinetics is formulated according to the Hoffman-Lauritzen secondary surface nucleation and growth theory for spherulite expansion. It is used to calculate the switching probability of each grid point as a function of its previous state and the state of the neighboring grid points. The actual switching decision is made by evaluating the local switching probab…
I develop a kinetic mechanism to explain chain folding in polymer crystallization which is based on the competition between the formation of stems, which is due to frequent occupations of trans states along the chains in the supercooled polymer melt, and the random coil structure of the polymer chains. Setting equal the average formation time of stems of length $% d_l$ with the Rouse time of a piece of polymer of the same arc length $d_l$ yields a lower bound for the thickness of stems and bundles. The estimated lamellar thickness is inversely proportional to the supercooling. The present appr…
Investigating thermodynamic properties of liquid-solid transitions of flexible homopolymers with elastic bonds by means of multicanonical Monte Carlo simulations, we find crystalline conformations that resemble ground-state structures of Lennard-Jones clusters. This allows us to set up a structural classification scheme for finite-length flexible polymers and their freezing mechanism in analogy to atomic cluster formation. Crystals of polymers with "magic length" turn out to be perfectly icosahedral.
We study the possibility of the spinodal decomposition in the induction period of the polymer crystallization. This phenomenon was first reported in an X-ray scattering experiment, and has still been controversial due to various experiments and theories that support or deny the phenomenon. In this article, we explain the condition for the spinodal decomposition to occur in polymer melts by deriving a Ginzburg-Landau model of the free energy as a functional of the density and the orientation of the segments, where we introduce the excluded volume and the nematic interactions through a combinati…
We report Monte Carlo simulations of a lattice-polymer model that can account for both polymer crystallization and liquid-liquid demixing in solutions of semiflexible homopolymers. In our model, neighboring polymer segments can have isotropic interactions that affect demixing, and anisotropic interactions that are responsible for freezing. However, our simulations show that the isotropic interactions also have a noticeable effect on the freezing curve, as do the anisotropic interactions on demixing. As the relative strength of the isotropic interactions is reduced, the liquid-liquid demixing t…
Based on classical nucleation theory, we propose a couple of theoretical models for the nucleation of polymer crystallization, i.e. one for a single chain system (Model S) and the other for a multi-chain system (Model M). In these models, we assume that the nucleus is composed of tails, loops and a cylindrical ordered region, and we evaluate the conformation entropy explicitly by introducing a transfer matrix. Using these two models, we evaluate the occurrence probability of critical nucleus as a function of the polymer chain stiffness. We found that the critical nucleus in Model M is easier t…
The present contribution deals with the thermomechanical modeling of the strain-induced crystallization in unfilled polymers. This phenomenon significantly influences mechanical and thermal properties of polymers and has to be taken into consideration when planning manufacturing processes as well as applications of the final product. In order to simultaneously capture both kinds of effects, the model proposed starts by introducing a triple decomposition of the deformation gradient and furthermore uses thermodynamic framework for material modeling based on the Coleman--Noll procedure and minimu…
Molecular process of crystallization from an oriented amorphous state was reproduced by molecular dynamics simulation for a realistic polyethylene model. Initial oriented amorphous state was obtained by uniaxial drawing an isotropic glassy state at 100 K. By the temperature jump from 100 K to 330 K, there occurred the crystallization into the fiber structure, during the process of which we observed the developments of various order parameters. The real space image and its Fourier transform revealed that a hexagonally ordered domain was initially formed, and then highly ordered crystalline stat…
The isothermal, isobaric spontaneous crystallization of a supercooled polymer melt is investigated by MD simulation of an ensemble of fully-flexible linear chains. Frustration is introduced via two incommensurate length scales set by the bond length and the position of the minimum of the non-bonding potential. Marked polymorphism with considerable bond disorder, distortions of both the local packing and the global monomer arrangements is observed. The analyses in terms of: i) orientational order parameters characterizing the global and the local order and ii) the angular distribution of the ne…
While the $Θ$-collapse of single long polymers in bad solvents is usually a continuous (tri-critical) phase transition, there are exceptions where it is preempted by a discontinuous crystallization (liquid $\leftrightarrow$ solid) transition. For a version of the bond-fluctuation model (a model where monomers are represented as $2\times 2\times 2$ cubes, and bonds can have lengths between 2 and $\sqrt{10}$) it was recently shown by F. Rampf {\it et al.} that there exist distinct collapse and crystallization transitions for long but {\it finite} chains. But as the chain length goes to infinity,…
Effect of Pre-Shear and Dispersity on Crystallization of a Model Polymer with Soft Pair Interactions using Molecular Dynamics Simulations
University of Illinois, Urbana–Champaign（伊利诺伊大学厄巴纳-香槟分校）；The Grainger College of Engineering, University of Illinois Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校格雷恩格工程学院）
Recent experiments in some polymer melts quenched below the melting temperature have reported spinodal kinetics in small-angle X ray scattering before the emergence of crystalline structure. To explain these observations we propose that the coupling between density and chain conformation induces a liquid-liquid binodal within the equilibrium liquid--crystalline solid coexistence region. A simple phenomenological theory is developed to illustrate this idea, and several experimentally testable consequences are discussed. Shear is shown to enhance the kinetic role of the hidden binodal.
By means of multicanonical computer simulations, we investigate thermodynamic properties of the aggregation of interacting semiflexible polymers. We analyze a mesoscopic bead-stick model, where nonbonded monomers interact via Lennard-Jones forces. Aggregation turns out to be a process, in which the constituents experience strong structural fluctuations, similar to peptides in coupled folding-binding cluster formation processes. In contrast to a recently studied related proteinlike hydrophobic-polar heteropolymer model, aggregation and crystallization are separate processes for a homopolymer wi…
We present a kinetic model of crystal growth of polymers of finite molecular weight. Experiments help to classify polymer crystallization broadly into two kinetic regimes. One is observed in melts or in high molar mass polymer solutions and is dominated by nucleation control with $G \sim \exp(1/T ΔT)$, where $G$ is the growth rate and $ΔT$ is the super-cooling. The other is observed in low molar mass solutions (as well as for small molecules) and is diffusion controlled with $G \sim ΔT$, for small $ΔT$. Our model unifies these two regimes in a single formalism. The model accounts for the accum…
We prepared a buoyancy matched binary mixture of polydisperse polystyrene microgel spheres of size ratio 0.785 and at a volume fraction of 0.567 just below the kinetic glass transition. In line with theoretical expectations, a eutectic phase behavior was observed, but only a minor fraction of the samples crystallized at all. By adding a short non-adsorbing polymer we enforce inter-species fractionation into coexisting pure component crystals, which in turn also shows signs of intra-species fractionation. We show that in formerly inaccessible regions of the phase diagram binary hard sphere phys…
In this paper, the response of single three-dimensional phantom and self-avoiding polymers to localized step strains are studied for two cases in the absence of hydrodynamic interactions: (i) polymers tethered at one end with the strain created at the point of tether, and (ii) free polymers with the strain created in the middle of the polymer. The polymers are assumed to be in their equilibrium state before the step strain is created. It is shown that the strain relaxes as a power-law in time $t$ as $t^{-η}$. While the strain relaxes as $1/t$ for the phantom polymer in both cases; the self-avo…
We develop a simple coarse-grained bead-spring polymer model exhibiting competing crystallization and glass transitions. For quench rates slower than the critical nucleation rate $|\dot{T}|_{crit}$, systems exhibit a first-order crystallization transition below a critical temperature $T=T_{cryst}$. Such systems form close-packed crystallites of FCC and/or HCP order, separated by domain walls, twin defects, and an amorphous interphase. The size of amorphous regions grows continuously as the quench rate $|\dot{T}|$ increases, producing nearly amorphous structure for $|\dot{T}|>|\dot{T}|_{crit}$.…
Simulations are used to examine the microscopic origins of strain hardening in polymer glasses. While traditional entropic network models can be fit to the total stress, their underlying assumptions are inconsistent with simulation results. There is a substantial energetic contribution to the stress that rises rapidly as segments between entanglements are pulled taut. The thermal component of stress is less sensitive to entanglements, mostly irreversible, and directly related to the rate of local plastic arrangements. Entangled and unentangled chains show the same strain hardening when plotted…
Through a combination of rheological characterization and temperature-variable imaging methods, a novel gelation pathway in dilute solutions of a semiconducting polymer to achieve interconnected, crystalline networks with hierarchical porosity is reported. Upon rapid cooling, solutions of regioregular poly(3-hexylthiophene) (RR-P3HT) in ortho-dichlorobenzene formed thermoreversible gels. Temperature-variable confocal microscopy revealed cooling-induced structural rearrangement to progress through viscoelastic phase separation. The phase separation process arrested prematurely during the format…
We map out the solid-state morphologies formed by model soft-pearl-necklace polymers as a function of bending stiffness $k_b$ spanning the range from fully flexible to rodlike chains. The ratio of Kuhn length to bead diameter ($l_K/r_0$) increases monotonically with increasing $k_b$ and yields a one-parameter model that relates chain shape to bulk morphology and yields insights into the packing of anisotropic particles. In the flexible limit, monomers occupy the sites of close-packed crystallites while chains retain random-walk-like order. In the rodlike limit, nematic chain ordering typical o…
While proteins have been treated as particles with a spherically symmetric interaction, of course in reality the situation is rather more complex. A simple step towards higher complexity is to treat the proteins as non--spherical particles and that is the approach we pursue here. We investigate the phase behavior of enhanced green fluorescent protein (eGFP) under the addition of a non--adsorbing polymer, polyethylene glycol (PEG). From small angle x-ray scattering we infer that the eGFP undergoes dimerization and we treat the dimers as spherocylinders with aspect ratio $L/D-1 = 1.05$. Despite …
Going beyond the classical Gaussian approximation of Einstein's fluctuation theory, Ruppeiner gave it a Riemannian geometric structure with an entropic metric. This yielded a fundamental quantity - the Riemannian curvature, which was used to extract information on the nature of interactions between molecules in fluids, ideal gases and other open systems. In this article, we examine the implications of this curvature in a non-equilibrium thermodynamic system where relaxation is sufficiently slow so as not to invalidate the local equilibrium hypothesis. The non-equilibrium system comprises of a …
We solve a two dimensional model for polymer chain folding in the presence of mechanical pulling force ($f$) exactly using equilibrium statistical mechanics. Using analytically derived expression for the partition function we determine the phase diagram for the model in the $f$-temperature ($T$) plane. A square root singularity in the susceptibility indicates a second order phase transition from a folded to an unfolded state at a critical force ($f_c$) in the thermodynamic limit of infinitely long polymer chain. Surprisingly, the temperature dependence of $f_c$ shows a reentrant phase transiti…
Constitutive equations are derived for the viscoelastic response of rubbery polymers at finite strains. A polymer is thought of as a network of long chains connected to temporary junctions. At a random time, a chain detaches from a junction, which is treated at transition from its active state to the dangling state. A dangling chain randomly captures a new junction in the vicinity of its free end and returns to its active state. Breakage and reformation of long chains are modeled as thermo-mechanically activated processes. Stress-strain relations for a rubbery polymer are developed using the l…
The effects of entanglement and chain orientation on strain hardening in glassy polymers are separated by examining mixtures of chains with different lengths. Simulations show that the orientation of a molecule of a given chain length is the same in monodisperse systems and bidisperse mixtures, even when entangled and unentangled chains are mixed. In addition, the stress in mixtures is equal to the weighted average of the stresses in monodisperse systems. These results indicate that chains contribute independently to strain hardening, that chain orientation is determined by local interactions …
In a recent letter, Govaert et al. examined the relationship between strain hardening modulus $G_r$ and flow stress $σ_{flow}$ for five different glassy polymers. In each case, results for $G_r$ at different strain rates or different temperatures were linearly related to the flow stress. They suggested that this linear relation was inconsistent with simulations. Data from previous publications and new results are presented to show that simulations also yield a linear relation between modulus and flow stress. Possible explanations for the change in the ratio of modulus to flow stress with tempe…
We demonstrate that nascent polymer crystals (i.e., nuclei) are anisotropic entities, with neither spherical nor cylindrical geometry, in contrast to previous assumptions. In fact, cylindrical, spherical, and other high symmetry geometries are thermodynamically unfavorable. Moreover, post-critical transitions are necessary to achieve the lamellae that ultimately arise during the crystallization of semicrystalline polymers. We also highlight how inaccurate treatments of polymer nucleation can lead to substantial errors (e.g., orders of magnitude discrepancies in predicted nucleation rates). The…
Simulations are used to examine the microscopic origins of strain hardening in polymer glasses. While stress-strain curves for a wide range of temperature can be fit to the functional form predicted by entropic network models, many other results are fundamentally inconsistent with the physical picture underlying these models. Stresses are too large to be entropic and have the wrong trend with temperature. The most dramatic hardening at large strains reflects increases in energy as chains are pulled taut between entanglements rather than a change in entropy. A weak entropic stress is only obser…
We present a theoretical framework for the linear and nonlinear visco-elastic properties of reversibly crosslinked networks of semiflexible polymers. In contrast to affine models where network strain couples to the polymer end-to-end distance, in our model strain rather serves to locally distort the network structure. This induces bending modes in the polymer filaments, the properties of wich are slaved to the surrounding network structure. Specifically, we investigate the frequency-dependent linear rheology, in particular in combination with crosslink binding/unbinding processes. We also deve…
The formation of chain-folded structures from the melt is observed in molecular dynamics simulations resembling the lamellae of polymer crystals. Crystallization and subsequent melting temperatures are related linearly to the inverse lamellar thickness. Analysis of the single chain conformations in the crystal shows that most chains reenter the same lamella by tight backfolds. Simulations are performed with a mesoscopic bead-spring model including a specific angle bending potential. They demonstrate that chain stiffness alone, without an attractive inter-particle potential, is a sufficient dri…
Oriented block copolymers exhibit a buckling instability when submitted to a tensile test perpendicular to the lamellae direction. In this paper we study this behavior using a coarse grained molecular dynamics simulation approach. Coarse grained models of lamellar copolymers with alternate glassy rubbery layers are generated using the Radical Like Polymerization method, and their mechanical response is studied. For large enough systems, uniaxial tensile tests perpendicular to the direction of the lamellae reveal the occurrence of the buckling instability at low strain. The results that emerge …
The porous PLA foams potential for tissue engineering usage are prepared by a modified solvent casting/particulate leaching method with different crystallinity. Since in typical method the porogens are solved in the solution and flow with the polymers during the casting and the crystallinity behavior of PLA chains in the limited space cannot be tracked, in this work the processing is modified by diffusing the PLA solution into a steady salt stack. With a thermal treatment before leaching while maintaining the stable structure of the porogens stack, the crystallinity of porous foams is made pos…
We image semi-flexible polymer networks under shear at the micrometer scale. By tracking embedded probe particles, we determine the local strain field, and directly measure its uniformity, or degree of affineness, on scales of 2-100 micron. The degree of nonaffine strain depends on polymer length and crosslink density, consistent with theoretical predictions. We also find a direct correspondence between the uniformity of the microscale strain and the nonlinear elasticity of the networks in the bulk.
The dynamics of structural relaxation in a model polymer glass subject to spatially-homogeneous, time-periodic shear deformation is investigated using molecular dynamics simulations. We study a coarse-grained bead-spring model of short polymer chains below the glass transition temperature. It is found that at small strain amplitudes, the segmental dynamics is nearly reversible over about $10^4$ cycles, while at strain amplitudes above a few percent, polymer chains become fully relaxed after a hundred cycles. At the critical strain amplitude, the transition from slow to fast relaxation dynamics…
Super-compressible foam-like carbon nanotube films have been reported to exhibit highly nonlinear viscoelastic behaviour in compression similar to soft tissue. Their unique combination of light weight and exceptional electrical, thermal and mechanical properties have helped identify them as viable building blocks for more complex nanosystems and as stand-alone structures for a variety of different applications. In the as-grown state, their mechanical performance is limited by the weak adhesion between the tubes, controlled by the van der Waals forces, and the substrate allowing the forests to …
We perform molecular-dynamics simulations for polymer melts of the coarse-grained polyvinyl alcohol model that crystallizes upon slow cooling. To establish the properties of its high temperature liquid state as a reference point, we characterize in detail the structural features of equilibrated polymer melts with chain lengths $5\le N \le 1000$ at a temperature slightly above their crystallization temperature. We find that the conformations of sufficiently long polymers with $N >50$ obey essentially the Flory's ideality hypothesis. The chain length dependence of the end-to-end distance and the…
We use molecular dynamics simulations to investigate the microscopic and macroscopic response of model polymer networks to uniaxial elongations. By studying networks with strands lengths ranging from $N_s=20$ to 200 we cover the full crossover from cross-link to entanglement dominated behavior. Our results support a recent version of the tube model which accounts for the different strain dependence of chain localization due to chemical cross-links and entanglements.
We analyze plane strain cold rolling processes, at large strains but slow strain rates, by finite element modeling. At low temperatures, slow strain rates, and moderate thickness reductions during rolling (at which Bauschinger effect can be neglected for the particular class of polymeric films studied here), the task of material modeling is greatly simplified, and enables us to deploy a computationally efficient, yet accurate, finite deformation rate-independent elastic-plastic material behavior (with inclusion of isotropic-hardening). The finite deformation elastic-plastic material behavior b…
The polymer relaxation dynamic of a sample, stretched up to the stress hardening regime, is measured, at room temperature, as a function of the strain $λ$ for a wide range of the strain rate $\dotγ$, by an original dielectric spectroscopy set up. The mechanical stress modies the shape of the dielectric spectra mainly because it affects the dominant polymer relaxation time $τ$, which depends on $λ$ and is a decreasing function of $\dotγ$. The fastest dynamics is not reached at yield but in the softening regime. The dynamics slows down during the hardening, with a progressive increase of $τ$. A …
The non-linear stress-strain relation for crosslinked polymer networks is studied using molecular dynamics simulations. Previously we demonstrated the importance of trapped entanglements in determining the elastic and relaxational properties of networks. Here we present new results for the stress versus strain for both dry and swollen networks. Models which limit the fluctuations of the network strands like the tube model are shown to describe the stress for both elongation and compression. For swollen networks, the total modulus is found to decrease like (V_0/V)^{2/3} and goes to the phantom …
Spin coating is an out-of-equilibrium technique for producing polymer films and colloidal crystals quickly and reproducibly. In this review, we present an overview of theoretical and experimental studies of the spin coating of colloidal suspensions. The dynamics of the spin coating process is discussed first, and we present insights from both theory and experiment. A key difference between spin coating with polymer solutions and with monodisperse colloidal suspensions is the emergence of long range (centimeter scale) orientational correlations in the latter. We discuss experiments in different…
Numerous models have been developed in the literature to simulate the thermomechanical behavior of amorphous polymer at large strain. These models generally show a good agreement with experimental results when the material is submitted to uniaxial loadings (tension or compression) or in case of shear loadings. However, this agreement is highly degraded when they are used in the case of combined load cases. A generalization of these models to more complex loads is scarce. In particular, models that are identified in tension or compression often overestimate the response in shear. One difficulty…
Using coarse-grained molecular-dynamics simulations, a generic styrene-(block)-butadiene-(block)-styrene (SBS) triblock copolymer under lamellar conformation is used in order to investigate the mutual entanglement evolution when a structure of alternating glassy (S)/rubbery (B) layers is submitted to an imposed deformation. By varying the amount of loop chains between each phase, i.e. noncrossing chains, it is possible to generate different types of S/B interface definitions. A specific boundary driven tensile strain protocol has been developed in order to mimic "real" experiments and measure …
We derive a large-strain plate model that allows to describe transient, coupled processes involving elasticity and solvent migration, by performing a dimensional reduction of a three-dimensional poroelastic theory. We apply the model to polymer gel plates, for which a specific kinematic constraint and constitutive relations hold. Finally, we assess the accuracy of the plate model with respect to the parent three-dimensional model through two numerical benchmarks, solved by means of the finite element method. Our results show that the theory offers an efficient computational framework for the s…
The rheological properties of highly concentrated suspensions of hard-sphere particles are studied with particular reference to the rheological response of shear induced crystals. Using practically monodisperse hard spheres, we prepare shear induced crystals under oscillatory shear and examine their linear and non-linear mechanical response in comparison with their glassy counterparts at the same volume fraction. It is evident, that shear-induced crystallization causes a significant drop in the elastic and viscous moduli due to structural rearrangements that ease flow. For the same reason the …
