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Equilibration of High Molecular Weight Polymer Melts: A Hierarchical Strategy
Abstract
A strategy is developed for generating equili-brated high molecular weight polymer melts described with microscopic detail by sequentially backmapping coarse-grained (CG) configurations. The microscopic test model is generic but retains features like hard excluded volume interactions and realistic melt densities. The microscopic representation is mapped onto a model of soft spheres with fluctuating size, where each sphere represents a microscopic subchain with Nb monomers. By varying Nb , a hierarchy of CG representations at different resolutions is obtained. Within this hierarchy, CG configurations equilibrated with Monte Carlo at low resolution are sequentially fine-grained into CG melts described with higher resolution. A Molecular Dynamics scheme is employed to slowly introduce the microscopic details into the latter. All backmapping steps involve only local polymer relaxation; thus, the computational efficiency of the scheme is independent of molecular weight, being just proportional to system size. To demonstrate the robustness of the approach, microscopic configurations containing up to n = 1000 chains with polymerization degrees N = 2000 are generated and equilibration is confirmed by monitoring key structural and conformational properties. The extension to much longer chains or branched polymers is straightforward.
... Schemes building on the Flory ideality hypothesis 12 superpose pre-equilibrated chain configurations with the proper large-scale random walk statistics in the simulation box and gradually remove the local overlap between monomers 9,[13][14][15][16] . This approach can be generalized to a systematic multi-scale approach, which equilibrates density fluctuations and chain conformations from the largest scales down to the monomer [16][17][18] or even atomic scale 19 . ...
... In the present context, pre-packing and push-off schemes can be generalized to a systematic multi-scale approach, which equilibrates density fluctuations and chain conformations from the largest scales down. Zhang et al. [17][18][19] represented the chains in a polymer melt via a hierarchy of soft blob models with matching invariant degrees of polymerization. Each blob model can then be fine-grained to an equivalent lower resolution model until a scale comparable to KG models is reached. ...
... However, this is a modest computational effort compared to previous methods. [17][18][19] employ a hierarchical blob-based description. For both approaches, the total computational effort for the multiscale equilibration of long-chain polymer melts is dominated by the equilibration of the local chain and melt structure during the last fine-graining step. ...
We present a computationally efficient multiscale method for preparing equilibrated, isotropic long chain model polymer melts. As an application we generate Kremer-Grest melts of $1000$ chains with $200$ entanglements and $25000$-$2000$ beads per chain, which cover the experimentally relevant bending rigidities up to and beyond the limit of the isotropic-nematic transition. In the first step, we employ Monte Carlo simulations of a lattice model to equilibrate the large-scale chain structure above the tube scale while ensuring a spatially homogeneous density distribution. We then use theoretical insight from a constrained mode tube model to introduce the bead degrees of freedom together with random walk conformational statistics all the way down to the Kuhn scale of the chains. This is followed by a sequence of simulations with carefully parameterized force-capped bead-spring models, which slowly introduce the local bead packing while reproducing the larger scale chain statistics of the target Kremer-Grest system at all levels of force-capping. Finally we can switch to the full Kremer-Grest model without perturbing the structure. The resulting chain statistics is in excellent agreement with literature results on all length scales accessible in brute-force simulations of shorter chains.
... Fortunately, these runs need not follow physically realistic dynamics, and several modern equilibration algorithms exploit this fact. [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22] The simpler algorithms fall into two basic categories: core-softening 12,13 and topology-changing. [8][9][10][11] Both approaches greatly speed up diffusive equilibration by eliminating the constraints on chains' transverse motion (and hence their slow reptation dynamics), but both have inherent limitations. ...
... Figure 5 shows how the large-scale intrachain structure and entanglement vary with n over the course of equilibration runs. Panel (a) shows data for selected systems' Kuhn lengths ℓK obtained by fitting their large-n chain statistics to Eq. (14). Here, ℓK(n) is the Kuhn length at the end of each n-step, while ℓK (12) is the Kuhn length at the end of the equilibration run. ...
... (19) and (20)], further refinement might provide a significant additional speedup. Note that our method is suitable for generating coarse-grained configurations which can be used in conjunction with configurational-backmapping methods 14,15,20 to generate equilibrated well-entangled semiflexible atomistic or united-atom-model polymer melts. ...
The widely used double-bridging hybrid (DBH) method for equilibrating simulated entangled polymer melts [R. Auhl et al., J. Chem. Phys. 2003, 119, 12718-12728] loses its effectiveness as chain stiffness increases into the semiflexible regime because the energy barriers associated with double-bridging Monte Carlo moves become prohibitively high. Here we overcome this issue by combining DBH with the use of core-softened pair potentials. This reduces the energy barriers substantially, allowing us to equilibrate melts with $N \simeq 40 N_e$ and chain stiffnesses all the way up to the isotropic-nematic transition using simulations of no more than 100 million timesteps.For semiflexible chains, our method is several times faster than standard DBH; we exploit this speedup to develop improved expressions for Kremer-Grest melts' chain-stiffness-dependent Kuhn length $\ell_K$ and entanglement length $N_e$.
... realistic dynamics, and several modern equilibration algorithms exploit this fact. [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22] The simpler algorithms fall into two basic categories: core-softening [12,13] and topology-changing. [8][9][10][11] Both approaches greatly speed up diffusive equilibration by eliminating the constraints on chains' transverse motion (and hence their slow reptation dynamics), but both have inherent limitations. ...
... [19][20], further refinement might provide a significant additional speedup. Finally, we point out that our method is suitable for generating coarse-grained configurations which can be used in conjunction with configurational-backmapping methods [14,15,20] to generate equilibrated well-entangled semiflexible atomistic or united-atom-model polymer melts. ...
... Figure 10 illustrates the evolution of S(q). lim q→0 S(q) decreases rapidly with increasing n as long-wavelength density fluctuations anneal out; this decrease is a key indicator of equilibration [7,10,14,[16][17][18]. On the other hand, for 5.0σ −1 < q < 8.0σ −1 , S(q) increases with increasing n. ...
The widely used double-bridging hybrid (DBH) method for equilibrating simulated entangled polymer melts [R. Auhl \textit{et al.}, \textit{J.\ Chem.\ Phys.}\ \textbf{2003}, 119, 12718-12728] loses its effectiveness as chain stiffness increases into the semiflexible regime because the energy barriers associated with double-bridging Monte Carlo moves become prohibitively high. Here we overcome this issue by combining DBH with the use of core-softened pair potentials. This reduces the energy barriers substantially, allowing us to equilibrate melts with $N \simeq 40 N_e$ and chain stiffnesses all the way up to the isotropic-nematic transition using simulations of no more than 100 million timesteps. For semiflexible chains, our method is several times faster than standard DBH; we exploit this speedup to develop improved expressions for Kremer-Grest melts' chain-stiffness-dependent Kuhn length $\ell_K$ and entanglement length $N_e$.
... Auhl et al [16] have proposed the so-called configuration assembly method, where the basic idea is to start with an ensemble of phantom chains and slowly introduce excluded volume interactions. Recently, powerful hierarchical backmapping strategies have been developed [18][19][20][21][22][23][24][25] enabling the equilibration of melts of unprecedented size, e.g. thousands of chains with length equivalent to several tens of entanglement lengths [18]. ...
... Recently, powerful hierarchical backmapping strategies have been developed [18][19][20][21][22][23][24][25] enabling the equilibration of melts of unprecedented size, e.g. thousands of chains with length equivalent to several tens of entanglement lengths [18]. Some of these divide-and-conquer methods are based [18][19][20][21][22] on the idea of successively refining a blob-based model [26]. ...
... thousands of chains with length equivalent to several tens of entanglement lengths [18]. Some of these divide-and-conquer methods are based [18][19][20][21][22] on the idea of successively refining a blob-based model [26]. In this approach, stepwise finegraining of configurations, first equilibrated at the lowest resolution, is employed until reaching a blob-based representation where the level of detail is sufficiently high to allow for reinsertion of microscopic features. ...
Recent theoretical studies have demonstrated that the behaviour of molecular knots is a sensitive indicator of polymer structure. Here, we use knots to verify the ability of two state-of-the-art algorithms - configuration assembly and hierarchical backmapping - to equilibrate high-molecular-weight polymer melts. Specifically, we consider melts with molecular weights equivalent to several tens of entanglement lengths and various chain flexibilities, generated with both strategies. We compare their unknotting probability, unknotting length, knot spectra, and knot length distributions. The excellent agreement between the two independent methods with respect to knotting properties provides an additional strong validation of their ability to equilibrate dense high-molecular-weight polymeric liquids. By demonstrating this consistency of knotting behaviour, our study opens the way for studying topological properties of polymer melts beyond time and length scales accessible to brute-force molecular dynamics simulations.
... A novel and very efficient methodology has recently been developed 28,43 for equilibrating large and highly entangled polymer melts in bulk described by the bead-spring model. 22,23 Through a hierarchical backmapping of CG chains described by the soft-sphere CG model 27,28 from low resolution to high resolution and a reinserting of microscopic details of bead-spring chains, finally, highly entangled polymer melts in bulk are equilibrated by molecular dynamics (MD) simulations using the package ESPResSO++. ...
... where ε (CG) w is the interaction strength between soft spheres and the walls and z and Lz − z are the vertical distances from the two walls, respectively. of Chemical Physics For the parameterization of the soft-sphere CG model, we take 15 independent and fully equilibrated bulk polymer melts of beadspring polymer chains with k θ = 1.5ε obtained from the previous works 35,42,43 as our reference systems. Using Eqs. ...
... For N b = 25(<Ne = 28), the chain to a good approximation can be described as a Gaussian chain, while this is not the case for significantly shorter chains. In this case, we can simplify several steps of hierarchical backmapping 43 to only one step of fine-graining to introduce microscopic details of subchains once a CG melt reaches its equilibrated state. ...
Equilibration of polymer melts containing highly entangled long polymer chains in confinement or with free surfaces is a challenge for computer simulations. We approach this problem by first studying polymer melts based on the soft-sphere coarse-grained model confined between two walls with periodic boundary conditions in two directions parallel to the walls. Then, we insert the microscopic details of the underlying bead-spring model. Tuning the strength of the wall potential, the monomer density of confined polymer melts in equilibrium is kept at the bulk density even near the walls. In a weak confining regime, we observe the same conformational properties of chains as in the bulk melt showing that our confined polymer melts have reached their equilibrated state. Our methodology provides an efficient way of equilibrating large polymer films with different thicknesses and is not confined to a specific underlying microscopic model. Switching off the wall potential in the direction perpendicular to the walls enables to study free-standing highly entangled polymer films or polymer films with one supporting substrate.
... hierarchical strategy for the equilibration of dense polymer melts. The hierarchical equilibration strategy comprises a recursive coarse-graining algorithm with its corresponding sequential back-mapping [41]. ...
... Hence, an effective method for decreasing the equilibration time is required. The hierarchical equilibration strategy pioneered in Ref. [41,102] is a particularly suitable way to do this. ...
... The hierarchical equilibration strategy consists of recursive coarse-graining and sequential back-mapping [41]. At first, a polymer chain, originally consisting of M monomers, is replaced by a coarse-grained (CG) chain consisting of M/N b softblobs, mapping from each subchain with N b monomers, represented as the model developed by Vettorel [103]. ...
Molecular simulation is a scientific tool used in many fields including material science and biology. This requires constant development and enhancement of algorithms within molecular simulation software packages. Here, we present computational tools for multiscale modeling developed and implemented within the ESPResSo++ package. These include the latest applications of the adaptive resolution scheme, the hydrodynamic interactions through a lattice Boltzmann solvent coupled to particle-based molecular dynamics, the implementation of the hierarchical strategy for equilibrating long-chained polymer melts and a heterogeneous spatial domain decomposition. The software design of ESPResSo++ has kept its highly modular C++ kernel with a Python user interface. Moreover, it has been enhanced by automatic scripts that parse configurations from other established packages, providing scientists with the ability to rapidly set up their simulations.
... Schemes building on the Flory ideality hypothesis 10 superpose preequilibrated chain configurations with the proper large-scale random walk statistics in the simulation box and gradually remove the local overlap between monomers 7,[11][12][13] . This approach can be generalized to a systematic multi-scale approach, which equilibrates density fluctuations and chain conformations from the largest scales down 13,14 . ...
... 3,73,74,77 In the present context, pre-packing and push-off schemes can be generalised to a systematic multi-scale approach, which equilibrates density fluctuations and chain conformations from the largest scales down. Zhang et al. 14,78,79 represented the chains in a polymer melt via a hierarchy of soft blob models with matching invariant degrees of polymerization. Each blob model can then be fine-grained to an equivalent lower resolution model until a scale comparable to KG models is reached. ...
We present a computationally efficient multiscale method for preparing very well equilibrated, isotropic long chain model polymer melts. As an application we generate Kremer-Grest melts of $1000$ chains with $200$ entanglements and $25000$-$2000$ beads per chain, which cover the experimentally relevant bending rigidities up to and beyond the limit of the isotropic-nematic transition. In the first step, we employ Monte Carlo simulations of a lattice model to equilibrate the large-scale chain structure above the tube scale while ensuring a spatially homogeneous density distribution. We then use theoretical insight from a constrained mode tube model to introduce the bead degrees of freedom together with random walk conformational statistics all the way down to the Kuhn scale of the chains. This is followed by a sequence of simulations with force-capped bead-spring models slowly introducing the local bead packing. Finally we can switch to the full Kremer-Grest model without perturbing the structure. The resulting chain statistics is in excellent agreement with literature results on all length scales accessible in brute-force simulations of shorter chains.
... In general, building an atomic model of a PNC should require two steps: (1) filling a periodic box with a compact and well-equilibrated polymer melt, and (2) inserting the reinforcement molecules within the periodic melt. Regarding the first step, several approaches have been proposed; such as Monte Carlo steps 18,19 , where monomers are linked into a growing polymer until box saturation; Hierarchical Coarse-Grained techniques 20 , where softblobs that represent polymer segments are gradually backmapped into monomers and all-atom models; and MD protocols that involve Annealing Cycles 21,22 , where polymers are arranged into a grid and then collapsed through MD simulations at different temperatures and pressures. While Monte Carlo and Hierarchical Coarse-Grained approaches efficiently reduce the computational cost of shoehorning polymers within a finite volume; those methods are not exempt of caveats: Monte Carlo insertions progressively decrease the accessible space and end up generating voids, while Hierarchical Coarse-Grained requires the availability of accurate coarse models as the blobs get close to the allatom scale. ...
... The MD workflow is still computationally expensive and requires several intermediate steps to reach a well-equilibrated system. Some of these limitations can be addressed by combining all-atom and coarse-grained simulations 20 , which can also include explicit π-π interactions in the force field. We believe that these and future studies can help to move towards a quantitative prediction and rational tailoring of nano-composite mechanics. ...
Nanocomposites built from polymers and carbon nanotubes (CNTs) are a promising class of materials. Computer modeling can provide nanoscale views of the polymer-CNT interface, which are much needed to foster the manufacturing and development of such materials. However, setting up periodic nanocomposite models is a challenging task. Here we propose a computational workflow based on Molecular Dynamics simulations. We demonstrate its capabilities and showcase its applications, focusing on two existing nanocomposite materials: polystyrene (PS) with CNT and polyether ether ketone with CNT. The models provide insights into the polymer crystallization inside CNTs. Furthermore, the PS+CNT nanocomposite models are mechanically tested and able to predict an enhancement in Young's modulus due to the addition of highly dispersed CNTs. We accompany those results with experimental tests and provide a prediction model based on Dynamic Quantized Fracture Mechanics theory. Our study proposes representative simulations of polymer-CNT nanocomposites as promising tools to guide the rational design of this class of materials.
... For long-chain polymers, the time scale to fully equilibrate the chain conformation, i.e., the longest polymer relaxation time, is well beyond the reach of brute-force MD. For this reason, the development of methods for polymer structure generation has remained an active area of research after more than three decades [4][5][6][7][8][9]. For dynamical properties and phenomena, the limited time scale accessible by MD is a more direct challenge. ...
... Not only does the chain length N has to reach at least O(1000), a large number of these chains must also be packed into the simulation cell to minimize the correlation between the periodic images of the chains. This became possible only recently with the the latest algorithmic developments for efficient amorphous chain equilibration [8,9]. For a semi-flexible FENE bead-spring model, Hsu & Kremer [49] calculated G(t) using the Green-Kubo approach for N =500, 1000 and 2000 and all three cases give nearly the same N e ≈ 28 ( fig. ...
Bottom-up prediction that links materials chemistry to their properties is a constant theme in polymer simulation. Rheological properties are particularly challenging to predict because of the extended time scales involved as well as large uncertainty in the stress output from molecular simulation. This review focuses on the application of molecular simulation in the prediction of such properties, including approaches solely based on molecular simulation and its integration with rheological models. Most attention is given to the prediction of quantitative properties, in particular, those most studied such as shear viscosity and linear viscoelasticity. Studies on the fundamental understanding of rheology are referenced only when they are directly relevant to the property prediction. The review starts with an overview of the major methods for extracting rheological properties from molecular simulation, using bead-spring chain models as a sandbox system. It then discusses materials-specific prediction using chemically-realistic models, including systematically coarse-grained models that allow the mapping between scales. Finally, integrating molecular simulation with rheological models extends the prediction to highly entangled polymers. Recent development of several multiscale predictive frameworks allowed the successful prediction of rheological properties from the chemical structure for polymers of experimentally relevant molecular weights.
... The positions of the CG particles are arranged to reproduce the structural features of the corresponding FA model in a space confined by other polymer chains. Such a sequential approach is found in some studies, 35,36 but the finest resolution of the models was at the KG level, and reverse-mapping to FA models was not conducted. We applied our scheme to the construction of polyethylene (PE), cis 1,4-polybutadiene (PB), and poly(methyl methacrylate) (PMMA). ...
We propose a method to build full-atomistic (FA) amorphous polymer structures using reverse-mapping from coarse-grained (CG) models. In this method, three models with different resolutions are utilized, namely the CG1, CG2, and FA models. It is assumed that the CG1 model is more abstract than the CG2 model. The CG1 is utilized to equilibrate the system, and then sequential reverse-mapping procedures from the CG1 to the CG2 models and from the CG2 to the FA models are conducted. A mapping relation between the CG1 and the FA models is necessary to generate a polymer structure with a given density and radius of chains. Actually, we have used the Kremer–Grest (KG) model as the CG1 and the monomer-level CG model as the CG2 model. Utilizing the mapping relation, we have developed a scheme that constructs an FA polymer model from the KG model. In the scheme, the KG model, the monomer level CG model, and the FA model are successively constructed. The scheme is applied to polyethylene (PE), cis 1,4-polybutadiene (PB), and poly(methyl methacrylate) (PMMA). As a validation, the structures of PE and PB constructed by the scheme were carefully checked through comparison with those obtained using long-time FA molecular dynamics (MD) simulations. We found that both short- and long-range chain structures constructed by the scheme reproduced those obtained by the FA MD simulations. Then, as an interesting application, the scheme is applied to generate an entangled PMMA structure. The results showed that the scheme provides an efficient and easy way to construct amorphous structures of FA polymers.
... We here apply a recently developed efficient hierarchical methodology to equilibrate the highly entangled melts of long polymer chains in bulk, 39,40 and confined and free-standing polymer films. 41 The required computer time scales linearly with system size, independent of chain length. ...
The glass transition temperature of confined and free-standing polymer films of varying thickness is studied by extended molecular dynamics simulations of bead–spring chains. The results are connected to the statistical properties of the polymers in the films, where the chain lengths range from short, unentangled to highly entangled. For confined films, perfect scaling of the thickness-dependent end-to-end distance and radius of gyrations normalized to their bulk values in the directions parallel and perpendicular to the surfaces is obtained. In particular, the reduced end-to-end distance in the perpendicular direction is very well described by an extended Silberberg model. For bulk polymer melts, the relation between the chain length and Tg follows the Fox–Flory equation. For films, no further confinement induced chain length effect is observed. Tg decreases and is well described by Keddie’s formula, where the reduction is more pronounced for free-standing films. It is shown that Tg begins to deviate from bulk Tg at the characteristic film thickness, where the average bond orientation becomes anisotropic and the entanglement density decreases.
... where u 2 = t/τ 0 , with τ 0 setting the time scale. Later on, Eq. (1) has been used in its exponentiated form [Eq. (2)] 27 not only to interpret basically all further NSE experiments on polymer melts but also simulation results 7 (and references therein 6,[34][35][36][37][38]. These experiments were able to manifest the existence of an intermediate length for polymer dynamics and provided a microscopic proof of a well-defined length scale for the topological constraints. ...
In this work, we compare the single chain dynamic structure factors for five different polymers: polyolefins (PE and PEP), poly-dienes (PB and PI), and a polyether (PEO). For this purpose, we have extended the De Gennes approximation for the dynamic structure factor. We describe the single chain dynamic structure factor in multiplying the coherent scattering functions for local reptation and Rouse motion within the Rouse blob. Important results are (i) the simple De Gennes structure factor S(Q, t)DG approximates within a few Å the outcome for the tube diameter of the more elaborate structure factor (exception PI); (ii) the extended De Gennes structure factor together with the Rouse blob describes the neutron spin echo spectra from the different polymers over the complete momentum transfer range and the full time regime from early Rouse motion to local reptation; and (iii) the representation of the scattering functions could significantly be improved by introducing non-Gaussianity corrections to the Rouse-blob dynamics. (iv) The microscopic tube step length in all cases is significantly larger than the rheological one; further tweaking the relation between tube length and entanglement blob size may indicate a possible trend toward an anisotropic lean tube with a step-length larger than the lateral extension. (v) All considered polymer data coincide after proper (Q, t) scaling to a universal behavior according to the length scale of the tube, while the relevant time scale is the entanglement time τe. (vi) In terms of the packing model, the required number of chains spanning the entanglement volume consistently is about 40% larger than that obtained from rheology.
... Hsu and Kremer [78,79] made molecular dynamics simulations of 1000 chains containing N = 500, 1000, or 2000 beads at volume fraction 0.85, using a novel scheme for equilibrating a large assembly of long chains [80]. The chains were described by a beadspring model [37] with a FENE potential between bonded atoms, a Lennard-Jones potential between non-bonded atoms, and in different simulations, a bend-bonding constant k θ of 0 or 1.5. ...
An extensive review of literature simulations of quiescent polymer melts is given, considering results that test aspects of the Rouse model in the melt. We focus on Rouse model predictions for the mean-square amplitudes ⟨(Xp(0))2⟩ and time correlation functions ⟨Xp(0)Xp(t)⟩ of the Rouse mode Xp(t). The simulations conclusively demonstrate that the Rouse model is invalid in polymer melts. In particular, and contrary to the Rouse model, (i) mean-square Rouse mode amplitudes ⟨(Xp(0))2⟩ do not scale as sin−2(pπ/2N), N being the number of beads in the polymer. For small p (say, p≤3) ⟨(Xp(0))2⟩ scales with p as p−2; for larger p, it scales as p−3. (ii) Rouse mode time correlation functions ⟨Xp(t)Xp(0)⟩ do not decay with time as exponentials; they instead decay as stretched exponentials exp(−αtβ). β depends on p, typically with a minimum near N/2 or N/4. (iii) Polymer bead displacements are not described by independent Gaussian random processes. (iv) For p≠q, ⟨Xp(t)Xq(0)⟩ is sometimes non-zero. (v) The response of a polymer coil to a shear flow is a rotation, not the affine deformation predicted by Rouse. We also briefly consider the Kirkwood–Riseman polymer model.
... Recently, we have developed an efficient hierarchical methodology to equilibrate highly entangled melts of long polymer chains [39,40] and extended this to confined and free-standing polymer films [14]. The required computer time scales linearly with system size, independent of chain length. ...
The glass transition temperature and its connection to statistical properties of confined and free-standing polymer films of varying thickness containing unentangled to highly entangled bead-spring chains are studied by molecular dynamics simulations. For confined films, perfect scaling of the thickness-dependent end-to-end distance and radius of gyrations normalized to their bulk values in the directions parallel and perpendicular to the surfaces is obtained. Particularly, the reduced end-to-end distance in the perpendicular direction is very well described by the extended Silberberg model. For bulk polymer melts, the relation between chain length and $T_g$ follows the Fox-Flory equation while $T_g$ for a given film thickness is almost independent of chain length. For films, $T_g$ decreases and is well described by Keddie's formula, where the reduction is more pronounced for free-standing films. For the present model, $T_g$ begins to deviate from bulk $T_g$ at the characteristic film thickness, where the average bond orientation becomes anisotropic and the entanglement density decreases.
... Hsu and Kremer [78,79] made molecular dynamics simulations of 1000 chains containing N =500, 1000, or 2000 beads at volume fraction 0.85, using a novel scheme for equilibrating a large assembly of long chains [80]. The chains were described by a beadspring model [37] with a FENE potential between bonded atoms, a Lennard-Jones potential between non-bonded atoms, and in different simulations a bend-bonding constant k θ of 0 or 1.5. ...
An extensive review of literature simulations of polymer melts is given, considering results that test aspects of the Rouse model in the melt. We focus on the mean-square amplitudes < (X_p(0))^{2} > and time correlation functions < X_p(0) X_p(t) > of the Rouse modes $X_p(t)$. Contrary to the Rouse model: (i) Mean-square Rouse mode amplitudes < (X_p(0))^2> do not scale as sin^{-2}(p \pi/2N), N being the number of beads in the polymer. For small p (say, p <= 3) < (X_p(0))^2> scales with p as p^{-2}$; for larger p it scales as p^{-3}. (ii) Rouse mode time correlation functions < X_p(t) X_p(0) > do not decay with time as exponentials; they instead decay as stretched exponentials exp(-a t^b)$. b depends on p, typically with a minimum near N/2 or N/4. (iii) Polymer bead displacements are not described by independent Gaussian random processes. (iv) For p not equal to q, < X_p(t) X_{q}(0) > is sometimes non-zero. (v) The response of a polymer coil to a shear flow is a rotation, not the affine deformation predicted by Rouse. Simulations thus conclusively demonstrate that the Rouse model is invalid in polymer melts. We also briefly consider the Kirkwood-Riseman polymer model.
... Hsu and Kremer [70,71] made molecular dynamics simulations of 1000 chains containing N =500, 1000, or 2000 beads at volume fraction 0.85, using a novel scheme for equilibrating a large assembly of long chains [72]. The chains were described by a bead-spring model [26] with a FENE potential between bonded atoms, a Lennard-Jones potential between non-bonded atoms, and in different simulations a bend-bonding constant k θ of 0 or 1.5. ...
The file is a Chapter from my review volume "Polymer Physics: Phenomenology of Polymeric Fluid Simulations". The chapter treats literature tests of the Rouse model, which is widely invoked as a description of polymer motion in melts. In summary: The literature conclusively demonstrates that the Rouse model does not describe polymer motion in melts. Simulations find that the temporal autocorrelation function of a single Rouse amplitude is a stretched exponential in time, not the pure exponential predicted by the Rouse model. Also, the mean-square amplitude of the Rouse modes <(X_p (0) X_p (0) > deviates from the model's prediction, at least for p > 3. Furthermore, the relaxation time of <(X_p (0) X_p (t) > depends on p, but not as predicted by the Rouse model. According to the Rouse model, bead displacements are driven by independent Gaussian random processes. Accordingly, the intermediate structure factor g(q,t) is predicted to be accurately described by the Gaussian approximation. Doob's theorem then guarantees that g(q,t) decays as a single exponential in time. Simulations show that these predictions of the Rouse model are incorrect.
... Hsu and Kremer [70,71] made molecular dynamics simulations of 1000 chains containing N =500, 1000, or 2000 beads at volume fraction 0.85, using a novel scheme for equilibrating a large assembly of long chains [72]. The chains were described by a bead-spring model [26] with a FENE potential between bonded atoms, a Lennard-Jones potential between non-bonded atoms, and in different simulations a bend-bonding constant k θ of 0 or 1.5. ...
Chapter from my review volume Polymer Physics: Phenomenology of Polymeric Fluid Simulations, reviewing tests of the Rouse model in polymer melts. The model is rejected by almost all direct tests.
... In traditional approaches, one first prepares a reasonably random initial configuration, e.g., by assembling a number of polymers with typical melt configurations, and then further relaxes it by implementing unphysical dynamics and/or Monte Carlo moves that allow chain crossing or even change chain connectivity [466][467][468][469]. In multiscale approaches [470][471][472][473][474][475][476][477], one uses CG simulations to equilibrate the melt and then reconstructs a FG configuration by increasing the level of resolution in a stepwise fashion. Tubiana et al have recently performed a systematic comparison of a traditional and multiscale equlibration scheme, focussing on topological indicators such as knot distributions [478], and found excellent agreement [479]. ...
Polymer materials have the characteristic feature that they are multiscale systems by definition. Already the description of a single molecules involves a multitude of different scales, and cooperative processes in polymer assemblies are governed by the interplay of these scales. Polymers have been among the first materials for which systematic multiscale techniques were developed, yet they continue to present extraordinary challenges for modellers. In this perspective, we review popular models that are used to describe polymers on different scales and discuss scale bridging strategies such as static and dynamic coarse-graining methods and multiresolution approaches. We close with a list of hard problems which still need to be solved in order to gain a comprehensive quantitative understanding of polymer systems on all scales.
... Recent work by Moreira et al. [40] shows how the procedure of Auhl et al. can be applied more efficiently and how the equilibration time can be shortened roughly by a factor of 6. An alternative strategy that performs even better for very long chains (again by a factor of ∼ 5) has been introduced by Zhang et al. [41]. They employ a hierarchical approach, which uses sequential backmapping from a coarsegrained representation in order to re-introduce molecular details. ...
We discuss the rejection-free event-chain Monte-Carlo algorithm and several applications to dense soft matter systems. Event-chain Monte-Carlo is an alternative to standard local Markov-chain Monte-Carlo schemes, which are based on detailed balance, for example the well-known Metropolis-Hastings algorithm. Event-chain Monte-Carlo is a Markov chain Monte-Carlo scheme that uses so-called lifting moves to achieve global balance without rejections (maximal global balance). It has been originally developed for hard sphere systems but is applicable to many soft matter systems and particularly suited for dense soft matter systems with hard core interactions, where it gives significant performance gains compared to a local Monte-Carlo simulation. The algorithm can be generalized to deal with soft interactions and with three-particle interactions, as they naturally arise, for example, in bead-spring models of polymers with bending rigidity. We present results for polymer melts, where the event-chain algorithm can be used for an efficient initialization. We then move on to large systems of semiflexible polymers that form bundles by attractive interactions and can serve as model systems for actin filaments in the cytoskeleton. The event chain algorithm shows that these systems form networks of bundles which coarsen similar to a foam. Finally, we present results on liquid crystal systems, where the event-chain algorithm can equilibrate large systems containing additional colloidal disks very efficiently, which reveals the parallel chaining of disks.
... We should note that related finite chain length effects are also observed in the static structure factor, g(q, 0), although they are much less dramatic: for infinitely long chains, 1/Ng(q, 0) drops from 1 (at q = 0) to zero at large qR G → ∞, whereas it levels off at 1/N for finite chains. In principle, this can be corrected by an appropriate backmapping procedure [60], i.e., restoring structure in the CG beads in retrospect. In the case of the dynamics, a different approach must be taken. ...
We propose a dynamic coarse-graining (CG) scheme for mapping heterogeneous polymer fluids onto extremely CG models in a dynamically consistent manner. The idea is to use as target function for the mapping a wave-vector dependent mobility function derived from the single-chain dynamic structure factor, which is calculated in the microscopic reference system. In previous work, we have shown that dynamic density functional calculations based on this mobility function can accurately reproduce the order/disorder kinetics in polymer melts, thus it is a suitable starting point for dynamic mapping. To enable the mapping over a range of relevant wave vectors, we propose to modify the CG dynamics by introducing internal friction parameters that slow down the CG monomer dynamics on local scales, without affecting the static equilibrium structure of the system. We illustrate and discuss the method using the example of infinitely long linear Rouse polymers mapped onto ultrashort CG chains. We show that our method can be used to construct dynamically consistent CG models for homopolymers with CG chain length N = 4, whereas for copolymers, longer CG chain lengths are necessary.
... Recent work by Moreira et al. [61] shows how the procedure of Auhl et al. can be applied more efficiently and how the equilibration time can be shortened roughly by a factor of 6. An alternative strategy that performs even better for very long chains (again by a factor of ∼ 5) has been introduced by Zhang et al. [96]. They employ a hierarchical The internal bead-to-bead distance R 2 /(n b 2 ) captures how well the polymers are equilibrated and has to be as close to the equilibrium result as possible because this observable equilibrates very slowly. ...
We discuss the rejection-free event-chain Monte-Carlo algorithm and several applications to dense soft matter systems. Event-chain Monte-Carlo is an alternative to standard local Markov-chain Monte-Carlo schemes, which are based on detailed balance, for example the well-known Metropolis-Hastings algorithm. Event-chain Monte-Carlo is a Markov chain Monte-Carlo scheme that uses so-called lifting moves to achieve global balance without rejections (maximal global balance). It has been originally developed for hard sphere systems but is applicable to many soft matter systems and particularly suited for dense soft matter systems with hard core interactions, where it gives significant performance gains compared to a local Monte-Carlo simulation. The algorithm can be generalized to deal with soft interactions and with three-particle interactions, as they naturally arise, for example, in bead-spring models of polymers with bending rigidity. We present results for polymer melts, where the event-chain algorithm can be used for an efficient initialization. We then move on to large systems of semiflexible polymers that form bundles by attractive interactions and can serve as model systems for actin filaments in the cytoskeleton. The event chain algorithm shows that these systems form networks of bundles which coarsen similar to a foam. Finally, we present results on liquid crystal systems, where the event-chain algorithm can equilibrate large systems containing additional colloidal disks very efficiently, which reveals the parallel chaining of disks.
... Such conformational deviations are considered small in other areas of polymer modeling, e.g., hierarchical backmapping. 14,75 The conformational deviations are somewhat larger for mesoscopic melts representing the more flexible, B = 0, and stiffer, B = 4, reference systems. Figure 5b presents δ m (s) for the mesoscopic analogues of B0d and B4c reference melts. ...
Similar to macroscopic ropes and cables, long polymers create knots. We address the fundamental question whether and under which conditions it is possible to describe these intriguing objects with crude models that capture only mesoscale polymer properties. We focus on melts of long polymers which we describe by a model typical for mesoscopic simulations. A worm-like chain model defines the polymer architecture. To describe nonbonded interactions, we deliberately choose a generic “soft” repulsive potential that leads to strongly overlapping monomers and coarse local liquid structure. The soft model is parametrized to accurately reproduce mesoscopic structure and conformations of reference polymer melts described by a microscopic model. The microscopically resolved samples retain all generic features affecting polymer topology and provide, therefore, reliable reference data on knots. We compare characteristic knotting properties in mesoscopic and microscopically resolved melts for different cases of chain stiffness. We conclude that mesoscopic models can reliably describe knots in those melts, where the length scale characterizing polymer stiffness is substantially larger than the size of monomer–monomer excluded volume. In this case, simplified local liquid structure influences knotting properties only marginally. In contrast, mesoscopic models perform poorly in melts with flexible chains. We qualitatively explain our findings through a free energy model of simple knots available in the literature.
... However, recent advances in melts equilibration techniques, see Refs. [42][43][44] , make it possible to generate very large, highly entangled well equilibrated model precursor melts for computational studies such as the present one. ...
We built randomly cross-linked model PDMS networks and used Molecular Dynamics Methods to obtain stress-strain curves. Mooney-Rivlin (MR) analysis was used to estimate the shear moduli. We applied Primitive Path analysis (PPA) and its variation, Phantom Primitive Path analysis (3PA), to estimate the entanglement and the cross-link moduli, respectively. The MR moduli estimates are in good agreement with the sum of the entanglement and the cross-link moduli, and we observe that the stress-strain data collapse to a universal form when reduced with the PPA and 3PA moduli. We studied how the MR parameters $\mathrm{C}_1$, $\mathrm{C}_2$ vary from cross-link to entanglement dominated networks. For the latter, we observed a $40\%$, $60\%$ contribution of $2\,\mathrm{C}_1$, $2\,\mathrm{C}_2$ to the shear modulus, respectively. Finally, we fitted several models to the data. While all fits are good, the estimates for the entanglement and the cross-link moduli vary significantly when compared to our PPA and 3PA benchmarks.
... As our results, especially for the mean-square internal distances, indicate that our equilibration procedure gives good but not perfect results, there is still room for improvement. In the future other equilibration methods, such as the hierarchical methods in [39,40] could be used for a faster and more precise equilibration of our polymer melts. ...
In this study, we determine the strain rate and temperature-dependent mechanical material behavior as well as the glass transition temperature and coefficient of thermal expansion of polyethylene melts using molecular dynamics simulation. In order to achieve realistic chain lengths polyethylene was simulated by three different coarse-grained models of various bead sizes. All simulations are performed by using the simulation package ESPResSo++, which we extended with a regulation procedure for the simulation of tensile tests on the micro-scale. The process-relevant observables, such as the elastic modulus, yield stress, and Poisson’s ratio are investigated at the meso-level. The chain orientation and entanglement behavior show effects that precisely illuminate the experimental stress strain response, giving important hints for production process control. Summarized, we are able to successfully reproduce the characteristic stress strain response for polyethylene as observed in experiments. Thus, we establish a closer link between microscopic and macroscopic system descriptions in order to provide a deeper understanding of material properties in their production process, i.e. for changing external conditions.
... 33 Alternatively, enhanced sampling simulations have been used to generate useful starting configurations for MD simulations. 34 In simulations of disordered polymer melts 35,36 and biological membranes, 37 multiscale approaches have proved to be very successful. Coarse-grained simulations are used to explore the space of possible arrangements. ...
Intrinsically disordered proteins (IDPs) constitute a large fraction of the human proteome and are critical in the regulation of cellular processes. A detailed understanding of the conformational dynamics of IDPs could help to elucidate their roles in health and disease. However the inherent flexibility of IDPs makes structural studies and their interpretation challenging. Molecular dynamics (MD) simulations could address this challenge in principle, but inaccuracies in the simulation models and the need for long simulations have stymied progress. To overcome these limitations, we adopt an hierarchical approach that builds on the "flexible-meccano" model of Bernadó et al. (J. Am. Chem. Soc. 2005, 127, 17968-17969). First, we exhaustively sample small IDP fragments in all-atom simulations to capture local structure. Then, we assemble the fragments into full-length IDPs to explore the stereochemically possible global structures of IDPs. The resulting ensembles of three-dimensional structures of full-length IDPs are highly diverse, much more so than in standard MD simulation. For the paradigmatic IDP α-synuclein, our ensemble captures both local structure, as probed by nuclear magnetic resonance (NMR) spectroscopy, and its overall dimension, as obtained from small-angle X-ray scattering (SAXS) in solution. By generating representative and meaningful starting ensembles, we can begin to exploit the massive parallelism afforded by current and future high-performance computing resources for atomic-resolution characterization of IDPs.
... Here we start from well-equilibrated and highly entangled polymer melts composed of weakly semiflexible bead−spring chains at a monomer density ρ = 0.85σ −3 , prepared by a new, efficient hierarchical methodology. 20,29,58 These melts are subject to strong deformation by isochoric elongation in the nonlinear rheological regime. Following this deformation, we investigate in detail the subsequent relaxation. ...
Polymer melts undergoing large deformation by elongation are studied by molecular dynamics simulations of bead–spring chains in melts. By applying a primitive path analysis to strongly deformed polymer melts, the role of topological constraints in highly entangled polymer melts is investigated and quantified. We show that the overall, large scale conformations of the primitive paths (PPs) of stretched chains follow affine deformation while the number and the distribution of entanglement points along the PPs do not. Right after deformation, PPs of chains retract in both directions parallel and perpendicular to the elongation. Upon further relaxation we observe a long-lived clustering of entanglement points. Together with the delayed relaxation time this leads to a metastable inhomogeneous distribution of topological constraints in the melts.
... 32 Alternatively, enhanced sampling simulations have been used to generate useful starting configurations for MD simulations. 33 In simulations of disordered polymer melts 34,35 and biological membranes, 36 multi-scale approaches have proved to be very successful. Coarse-grained simulations are used to explore the space of possible arrangements. ...
Intrinsically disordered proteins (IDPs) constitute a large fraction of the human proteome and are critical in the regulation of cellular processes. A detailed understanding of the conformational dynamics of IDPs could help to elucidate their roles in health and disease. However the inherent flexibility of IDPs makes structural studies and their interpretation challenging. Molecular dynamics (MD) simulations could address this challenge in principle, but inaccuracies in the simulation models and the need for long simulations have stymied progress. To overcome these limitations, we adopt an hierarchical approach that builds on the "flexible meccano" model of Bernadó et al. (J. Am. Chem. Soc. 2005, 127, 17968-17969). First, we exhaustively sample small IDP fragments in all-atom simulations to capture local structure. Then, we assemble the fragments into full-length IDPs to explore the stereochemically possible 1 All rights reserved. No reuse allowed without permission. (which was not peer-reviewed) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint. http://dx.doi.org/10.1101/731133 doi: bioRxiv preprint first posted online Aug. 9, 2019; global structures of IDPs. The resulting ensembles of three-dimensional structures of full-length IDPs are highly diverse, much more so than in standard MD simulation. For the paradigmatic IDP α-synuclein, our ensemble captures both local structure, as probed by nuclear magnetic resonance (NMR) spectroscopy, and its overall dimension, as obtained from small-angle X-ray scattering (SAXS) in solution. By generating representative and meaningful starting ensembles, we can begin to exploit the massive parallelism afforded by current and future high-performance computing resources for atomic-resolution characterization of IDPs.
... Here we start from well equilibrated and highly entangled polymer melts composed of weakly semiflexible bead-spring chains at a monomer density ρ = 0.85σ −3 , prepared by a new, efficient hierarchical methodology. 19,28,55 These melts are subject to strong deformation by isochoric elongation in the non-linear rheological regime. Following this deformation we investigate in detail the subsequent relaxation. ...
Polymer melts undergoing large deformation by uniaxial elongation are studied by molecular dynamics simulations of bead-spring chains in melts. Applying a primitive path analysis to strongly deformed polymer melts, the role of topological constrains in highly entangled polymer melts is investigated and quantified. We show that the over-all, large scale conformations of the primitive paths (PPs) of stretched chains follow affine deformation while the number and the distribution of entanglement points along the PPs do not. Right after deformation, PPs of chains retract in both directions parallel and perpendicular to the elongation. Upon further relaxation we observe a long-lived clustering of entanglement points. Together with the delayed relaxation time this leads to a metastable inhomogeneous distribution of topological constraints in the melts.
... The system was simulated in all-atom representation for 100 ns to ensure a random and uniform polystyrene configuration prior coarse-graining, while the elastic properties were computed within the coarse-grained approximation. This study aims to fine-tune the parameters of the coarse-grained potential in such a way that it can reproduce experimental observables directly, i.e., without transferring the coarse-grained model to an all-atom one, as done in some other studies [34,67]. Table I. ...
This paper presents an extended coarse-grained investigation of the elastic properties of polystyrene. In particular, we employ the well-known MARTINI force field and its modifications to perform extended molecular dynamics simulations at the $\mu$s timescale, which take slow relaxation processes of polystyrene into account, such that the simulations permit analyzing the bulk modulus, the shear modulus, and the Poisson ratio. We show that through the iterative modification of MARTINI force field parameters it turns out to be possible to affect the shear modulus and the bulk modulus of the system, making them closer to those values reported in the experiment.
... An alternative strategy involves starting with CG soft spheres that represent many beads along a chain, then using backmapping procedures to progressively increase the detail of a simulation, and maintaining correct chain configurations as the length scale of the coarse-graining decreases through local equilibration steps. 320,321 Similarly, if one were to create initial configurations of polymer chains with intermediate-resolution CG models that represent specific chemistries, one could start from energy-minimized atomistic structures to guide placement of the various bonded CG beads in a monomer and chain. 161,162 Using some or all of these ideas during system preparation can significantly decrease the amount of required equilibration time. ...
Molecular modeling and simulations are invaluable tools for the polymer science and engineering community. These computational approaches enable predictions and provide explanations of experimentally observed macromolecular structure, dynamics, thermodynamics, and microscopic and macroscopic material properties. With recent advances in computing power, polymer simulations can synergistically inform, guide, and complement in vitro macromolecular materials design and discovery efforts. To ensure that this growing power of simulations is harnessed correctly, and meaningful results are achieved, care must be taken to ensure the validity and reproducibility of these simulations. With these considerations in mind, in this Perspective we discuss our philosophy for carefully developing or selecting appropriate models, performing, and analyzing polymer simulations. We highlight best practices, key challenges, and important advances in model development/selection, computational method choices, advanced sampling methods, and data analysis, with the goal of educating potential polymer simulators about ways to improve the validity, usefulness, and impact of their polymer computational research.
We present a new simulation-guided process to create nanoporous materials, which does not require specific chemical treatment and solely relies on mechanical deformation of pure highly entangled homopolymer films. Starting from fully equilibrated freestanding thick polymer melt films, we apply a simple “biaxial expansion” deformation. Upon expansion holes form, which are prevented from growing and coalescing beyond a characteristic size due to the entanglement structure of the melt. We investigate the local morphology, the void formation upon expansion, and their stabilization. The dependence of the average void (pore) size and void fraction (porosity) on the total strain and subsequent relaxation is investigated. Furthermore, the stabilization of the porous structure of the thin expanded films through cooling below the glass transition temperature Tg is discussed.
Entropic elasticity of single chains underlies many fundamental aspects of mechanical properties of polymers, such as high elasticity of polymer networks and viscoelasticity of polymer liquids. On the other hand, single chain elasticity is further rooted in chain connectivity. Recently, mechanically interlocked polymers, including polycatenanes and polyrotaxanes, which are formed by connecting their building blocks (cyclic and linear chains) through topological bonds (e.g., entanglements), emerge as a conceptually new kind of polymers. In this work, we employ computer simulations to study linear elasticity of single linear polycatenane (or [n]catenane), in which n rings are interlocked through catenation into a chain of linear architecture. Aim of this work is to illuminate the specific role of catenation topology in the elastic moduli of linear polycatenanes by comparing with those of their [n]bonded-ring counterparts, which are formed by connecting the same number of rings but via covalent bonds. Simulation results lead to a conclusion that topological catenation makes [n]catenanes exhibit larger elastic moduli than their linear and [n]bonded-ring counterparts, i.e., larger elastic moduli in the case of [n]catenanes. Furthermore, it is revealed that those [n]catenanes composed of a smaller number of rings (smaller n) possesses larger elastic moduli than others of the same total chain lengths. Molecular mechanisms of these findings are discussed based on conformational entropy due to topological constraints.
Polymer materials are multiscale systems by definition. Already the description of a single macromolecule involves a multitude of scales, and cooperative processes in polymer assemblies are governed by their interplay. Polymers have been among the first materials for which systematic multiscale techniques were developed, yet they continue to present extraordinary challenges for modellers. In this Perspective, we review popular models that are used to describe polymers on different scales and discuss scale-bridging strategies such as static and dynamic coarse-graining methods and multiresolution approaches. We close with a list of hard problems which still need to be solved in order to gain a comprehensive quantitative understanding of polymer systems.
Scaling is one of the most fundamental concepts in theoretical polymer physics. For linear polymers in semidilute solutions under a good solvent condition, there has been a well established scaling behavior of mean-square radius of gyration of chains with the concentration, i.e., Rg2∼ϕβ, with β = −0.25, which receives strong support from experiments, computer simulations and scaling arguments. However, a clear-cut conclusion about the scaling behavior of dimensions of nonconcatenated and unknotted ring polymers in the semidilute solution with the concentration has not been achieved. Values of the corresponding scaling exponent β are diversely reported in literature, ranging from −0.25 to −0.59 and in contradiction within experimental, simulation and theoretical studies. As an endeavour for clarifying this issue, we carried out extensive molecular dynamics (MD) simulations based on the Kremer-Grest model of ring polymers with chain lengths N up to 5120 in semidilute solutions, and compared our results about ring conformations with almost all of experimental, simulation and theoretical data available in literature. Our MD simulation results lead to a conclusion that the mean-square radius of gyration of ring polymers scales with the concentration in semidilute regime as Rg2∼ϕ−0.59±0.01. This conclusion is in a good agreement with a previous finding by a lattice Monte Carlo simulation. Surprisingly, by looking into the experimental and other simulation results reported previously with sufficient caution, we tend to conclude that in reality almost all of their data can be described reasonably well by the same scaling law, Rg2∼ϕ−0.59. Furthermore, the scaling exponent obtained (−0.59) can be successfully explained by a scaling argument based on concept of correlation length. Finally, effects of the concentration on shape properties and packing behavior of ring polymers in the semidilute solutions have also been illuminated within our MD simulations.
Chain architecture effect on static and dynamic properties of unentangled polymers is explored by molecular dynamics simulation and Rouse mode analysis based on graph theory. For open chains, although they generally obey ideal scaling in chain dimensions, local structure exhibits nonideal behavior due to the incomplete excluded volume (EV) screening, the reduced mean square internal distance (MSID) can be well described by Wittmer’ theory for linear chains and the resulting chain swelling is architecture dependent, i.e., the more branches a bit stronger swelling. For rings, unlike open chains they are compact in term of global sizes. Due to EV effect and nonconcatenated constraints their local structure exhibits a quite different non-Gaussian behavior from open chains, i.e., reduced MSID curves do not collapse to a single master curve and fail to converge to a chain-length-independent constant, which makes the direct application of Wittmer’s theory to rings quite questionable. Deviation from ideality is further evidenced by limited applicability of Rouse prediction to mode amplitude and relaxation time at high modes as well as the non-constant and mode-dependent scaled Rouse mode amplitudes, while the latter is architecture-dependent and even molecular weight dependent for rings. The chain relaxation time is architecture-dependent, but the same scaling dependence on chain dimensions does hold for all studied architectures. Despite mode orthogonality at static state, the role of cross-correlation in orientation relaxation increases with time and the time-dependent coupling parameter rises faster for rings than open chains even at short time scales it is lower for rings.
We study the effect of entanglements on the glass transition of high molecular weight polymers, by the comparison of single-chain nanoparticles (SCNPs) and equilibrated melts of high-molecular weight polystyrene of identical molecular weight. SCNPs were prepared by electrospraying technique and characterized using scanning electron microscopy and atomic force microscopy techniques. Differential scanning calorimetry, Brillouin light spectroscopy, and rheological experiments around the glass transition were compared. In parallel, entangled and disentangled polymer melts were also compared under cooling from molecular dynamics simulations based on a bead-spring polymer model. While experiments suggest a small decrease in the glass transition temperature of films of nanoparticles in comparison to entangled melts, simulations do not observe any significant difference, despite rather different chain conformations.
We built randomly cross-linked model poly(dimethylsiloxane) (PDMS) networks and used molecular dynamics methods to obtain stress-strain curves. The Mooney-Rivlin (MR) analysis was used to estimate the shear moduli. We applied primitive path analysis (PPA) and its variation, phantom primitive path analysis (3PA), to estimate the entanglement and the cross-link moduli, respectively. The MR moduli estimates are in good agreement with the sum of the entanglement and the cross-link moduli, and we observe that the stress-strain data collapse to a universal form when reduced with the PPA and 3PA moduli. We studied how the MR parameters C1 and C2 vary from cross-link to entanglement-dominated networks. For the latter, we observed 40% and 60% contributions of 2C1 and 2C2 to the shear modulus, respectively. Finally, we fitted several theoretical models to our stress-strain data. While all fits are good, the estimates for the entanglement and cross-link moduli vary significantly when compared to our independently obtained PPA and 3PA benchmarks.
Membrane-based separations enable technology for reducing energy usage. Molecular simulations of polymers have the potential to describe sorption and diffusion mechanisms of moderate-size molecules in membranes and to inform new materials development for separating liquids. Advances in simulation methods have improved equilibration of chain configurations so they provide more reliable environments for simulating permeation. In situ ‘polymerization’ can generate large equilibrated cells in reasonable computation times. Reverse mapping between coarse-grained and atomistic scales can equilibrate complex polymers. Simulations of diffusion in polymers now target materials that include polyimides and PIMs. This paper reviews the current status and illustrates recent successes in equilibrating chains and simulating relevant engineering polymers, with a motivation toward applications in the fields of sorption and diffusion.
The Kremer−Grest (KG) model is a standard for
studying generic polymer properties. Here we have equilibrated
KG melts up to and beyond 200 entanglements per chain for
varying chain stiffness. We present methods for estimating the
Kuhn length corrected for incompressibility effects, for estimating
the entanglement length corrected for chain stiffness, and for
estimating bead frictions and Kuhn times taking into account
entanglement effects. These are the key parameters for enabling
quantitative, accurate, and parameter free comparisons between
theory, experiment, and simulations of KG polymer models with
varying stiffness. We demonstrate this for the chain dynamics in
moderately to highly entangled melts as well as for the shear relaxation modulus for unentangled melts, which are found to be in
excellent agreement with the predictions from standard theories of polymer dynamics.
Generating initial configurations of polymer melts above the entanglement molecular weight is a challenge in molecular dynamics and Monte Carlo simulations. In this work, we adapt an algorithm mimicking a chemical polymerization to all-atom force fields. The principle of this algorithm is to start from a bath of monomers between which bonds are created and relaxed sequentially. Our implementation is parallel and efficient. The parallelization is that of a classical molecular dynamics code and enables the user to generate large systems, up to 7 × 10⁶ atoms. The efficiency of the algorithm comes from the linear scaling between the simulation time and the chain length in the limit of very long chains. The implementation is able to produce long polymer chains, up to ∼2000 carbon atoms, with thermodynamic and local structural properties in good agreement with their experimental and numerical counterparts. Moreover, the chain conformations are close to being equilibrated right after the end of the polymerization process, corresponding to only a few hundred of picoseconds of simulation, despite a systematical drift from Gaussian-like behavior when the density of reactively available monomers decreases. Finally, the algorithm proposed in this work is versatile in nature because the bond creation can be easily modified to create copolymers, block copolymers, and mixtures of polymer melts with other material.
To study the cooling behavior and the glass transition of polymer melts in bulk and with free surfaces, a coarse-grained weakly semi-flexible polymer model is developed. Based on a standard bead spring model with purely repulsive interactions, an attractive potential between non-bonded monomers is added such that the pressure of polymer melts is tuned to zero. Additionally, the commonly used bond bending potential controlling the chain stiffness is replaced by a new bond bending potential. For this model, we show that the Kuhn length and the internal distances along the chains in the melt only very weakly depend on the temperature, just as for typical experimental systems. The glass transition is observed by the temperature dependency of the melt density and the characteristic non-Arrhenius slowing down of the chain mobility. The new model is set to allow for a fast switch between models, for which a wealth of data already exists.
The conformational statistics of ring polymers in melts or dense solutions is strongly affected by their quenched microscopic topological state. The effect is particularly strong for untangled (i.e. non-concatenated and unknotted) rings, which are known to crumple and segregate. Here we study these systems using a computationally efficient multi-scale approach, where we combine massive simulations on the fiber level with the explicit construction of untangled ring melt configurations based on theoretical ideas for their large scale structure. We find (i) that topological constraints may be neglected on scales below the standard entanglement length, $L_e$, (ii) that rings with a size $1 \le L_r/L_e \le 30$ exhibit nearly ideal lattice tree behavior characterized by primitive paths which are randomly branched on the entanglement scale, and (iii) that larger rings are compact with gyration radii $ \propto L_r⁁{2/3}$. The detailed comparison between equilibrated and constructed ensembles allows us to perform a "Feynman test" of our understanding of untangled rings: can we convert ideas for the large-scale ring structure into algorithms for constructing (nearly) equilibrated ring melt samples? We show that most structural observables are quantitatively reproduced by two different construction schemes: hierarchical crumpling and ring melts derived from the analogy to interacting branched polymers. However, the latter fail the "Feynman test" with respect to the magnetic radius, $R_m$, which we have defined based on an analogy to magnetostatics. While $R_m$ is expected to vanish for double-folded structures, the observed values of $ \propto $ provide a simple and computationally convenient measure of the presence of a non-negligible amount of local loop opening in crumpled rings.
We demonstrate that hierarchical backmapping strategies incorporating generic blob-based models can equilibrate melts of high-molecular-weight polymers, described with chemically specific, atomistic, models. The central idea is first to represent polymers by chains of large soft blobs (spheres) and efficiently equilibrate the melt on large scales. Then, the degrees of freedom of more detailed models are reinserted step by step. The procedure terminates when the atomistic description is reached. Reinsertions are feasible computationally because the fine-grained melt must be re-equilibrated only locally. We consider polystyrene (PS) which is sufficiently complex to serve method development because of stereochemistry and bulky side groups. Our backmapping strategy bridges mesoscopic and atomistic scales by incorporating a blob-based, a moderately coarse-grained (CG), and a united-atom model of PS. We demonstrate that the generic blobbased model can be parameterized to reproduce the mesoscale properties of a specific polymer – here PS. The moderately CG model captures stereochemistry. To perform backmapping we improve and adjust several fine-graining techniques. We prove equilibration of backmapped PS melts by comparing their structural and conformational properties with reference data from smaller systems, equilibrated with less efficient methods.
We use computer simulations to study the relaxation of strongly deformed highly entangled polymer melts in the nonlinear viscoelastic regime, focusing on anisotropic chain conformations after isochoric elongation. The Doi-Edwards tube model and its Graham-Likhtman-McLeish-Milner (GLaMM) extension, incorporating contour length fluctuation and convective constraint release, predict a retraction of the polymer chain extension in all directions, setting in immediately after deformation. This prediction has been challenged by experiment, simulation, and other theoretical studies, questioning the general validity of the tube concept. For very long chains we observe the initial contraction of the chain extension parallel and perpendicular to the stretching direction. However, the effect is significantly weaker than predicted by the GLaMM model. We also show that the first anisotropic term of an expansion of the 2D scattering function qualitatively agrees to predictions of the GLaMM model, providing an option for direct experimental tests.
Mean-field techniques provide a rather accurate description of single-chain conformations in spatially inhomogeneous polymer systems containing interfaces or surfaces. Intermolecular correlations, however, are not described by the mean-field approach and information about the distribution of distance between different molecules is lost. Based on the knowledge of the exact equilibrium single-chain properties in contact with solid substrates, we generate multichain configurations that serve as nearly equilibrated starting configurations for molecular dynamics simulations by utilizing the packing algorithm of [Auhl et al., J. Chem. Phys. 119, 12718 (2003)] for spatially inhomogeneous systems, i.e., a thin polymer film confined between two solid substrates. The single-chain conformations are packed into the thin film conserving the single-chain properties and simultaneously minimizing local fluctuations of the density. The extent to which enforcing incompressibility of a dense polymer liquid during the packing process is able to re-establish intermolecular correlations is investigated by monitoring intermolecular correlation functions and the structure function of density fluctuations as a function of the distance from the confining solid substrates.
Several methods for preparing well equilibrated melts of long chains polymers are studied. We show that the standard method in which one starts with an ensemble of chains with the correct end-to-end distance arranged randomly in the simulation cell and introduces the excluded volume rapidly, leads to deformation on short length scales. This deformation is strongest for long chains and relaxes only after the chains have moved their own size. Two methods are shown to overcome this local deformation of the chains. One method is to first pre-pack the Gaussian chains, which reduces the density fluctuations in the system, followed by a gradual introduction of the excluded volume. The second method is a double-pivot algorithm in which new bonds are formed across a pair of chains, creating two new chains each substantially different from the original. We demonstrate the effectiveness of these methods for a linear bead spring polymer model with both zero and nonzero bending stiffness, however the methods are applicable to more complex architectures such as branched and star polymer. Comment: 12 pages, 9 figures
Dilute or semi-dilute solutions of non-intersecting self-avoiding walk (SAW) polymer chains are mapped onto a fluid of ``soft'' particles interacting via an effective pair potential between their centers of mass. This mapping is achieved by inverting the pair distribution function of the centers of mass of the original polymer chains, using integral equation techniques from the theory of simple fluids. The resulting effective pair potential is finite at all distances, has a range of the order of the radius of gyration, and turns out to be only moderately concentration-dependent. The dependence of the effective potential on polymer length is analyzed in an effort to extract the scaling limit. The effective potential is used to derive the osmotic equation of state, which is compared to simulation data for the full SAW segment model, and to the predictions of renormalization group calculations. A similar inversion procedure is used to derive an effective wall-polymer potential from the center of mass density profiles near the wall, obtained from simulations of the full polymer segment model. The resulting wall-polymer potential turns out to depend strongly on bulk polymer concentration when polymer-polymer correlations are taken into account, leading to a considerable enhancement of the effective repulsion with increasing concentration. The effective polymer-polymer and wall-polymer potentials are combined to calculate the depletion interaction induced by SAW polymers between two walls. The calculated depletion interaction agrees well with the ``exact'' results from much more computer-intensive direct simulation of the full polymer-segment model, and clearly illustrates the inadequacy -- in the semi-dilute regime -- of the standard Asakura-Oosawa approximation based on the assumption of non-interacting polymer coils. Comment: 18 pages, 24 figures, ReVTeX, submitted to J. Chem. Phys
A recently proposed model of high-molecular-weight polymers is employed to develop a grid- based Monte Carlo method for efficient modeling of dense systems, for example, melts. The polymers are described as chains of soft spheres with fluctuating size. The spheres correspond to Gaussian density distributions of the microscopic segments and represent whole subchains. Their coordinates and radii are defined in continuum space and simple potentials keep the chain connectivity. A density functional defines the non- bonded interactions by mapping the density distributions onto a grid, without a neighbor list. The high accuracy of the scheme is demonstrated by comparing with data obtained from the standard potential-based formulation of the model. Contrary to most lattice models, the method allows for NPT simulations.
This paper outlines the derivation of an analytical pair potential in a coarse grained description of polymer melts where each chain is represented as a collection of soft spheres. Each particle is located at the center of mass of a polymer subchain, while the polymer is divided into an arbitrary number of identical chain subsections, each comprised of a large number of monomers. It is demonstrated that the soft effective pair potentials acting between these center-of-mass sites is described by a soft repulsive region at separation distances less than the average size of each coarse grained unit and a long repulsive tail, with a small attractive component. The attractive component is located at a length scale beyond the size of the coarse grained unit and its form varies with the level of interpenetration between the coarse-grained units. Consistent with numerically derived potentials, it is found that the short range features of the potential dominate the liquid structure, while the long-tail features dominate the virial-route thermodynamics of the system. It follows that the accurate determination of the effective potential in both short and large separation distances is relevant for ensuring structural and thermodynamic consistency in the coarse-grained description of the macromolecular liquid. It is further shown that due to the sensitivity of thermodynamic properties to the large-scale features of the potential, which are irrelevant to the reproducibility of structural correlations, the determination of thermodynamically accurate potentials by numerical optimization of structure alone is not a reliable strategy in the high-density regime for high levels of coarse-graining.
The joint distribution pN(ω,R1,R2) of the lengths of the vectors (R1,R2) and the angle ω between the vectors joining the center of mass of a block or bead in a chain with the centers of mass of the preceding and succeeding beads of a coarse-grained chain is investigated. Somewhat unexpectedly, the apriori and the conditional distributions for the connector length pRN(R1) and for the interconnector angle pωN(ω) can be determined in closed form for the Gaussian random walk for any number of monomers N per bead. Additionally, pωN(ω) is also obtained for an unperturbed polymethylene chain by a Monte Carlo scheme. In general, pωN(ω) is asymmetric and shifted to values greater than π/2 for all bead sizes, implying that the chain made up of the centers of mass of the beads tends to be locally more extended or spread out than a random walk. Convergence to the (analytically known) asymptotic distribution pω∞(ω) for the Gaussian random walk is given analytically and turns out to be very nearly quadratic in 1/N. For the polymethylene chain convergence is linear. The apriori and the conditional distributions are used to formulate a Monte Carlo scheme for the generation of the coarse-grained chain. The bias of pωN(ω) to angles greater than π/2 should be taken into account by any model that attempts to represent a linear chain polymer by lumping its detailed structural information in coarser units at a larger length scale.
The redesigned Extensible Simulation Package for Research on Soft matter systems (ESPResSo++) is a free, open-source, parallelized, object-oriented simulation package designed to perform many-particle simulations, principally molecular dynamics and Monte Carlo, of condensed soft matter systems. In addition to the standard simulation methods found in well-established packages, ESPResSo++ provides the ability to perform Adaptive Resolution Scheme (AdResS) simulations which are multiscale simulations of molecular systems where the level of resolution of each molecule can change on-the-fly. With the main design objective being extensibility, the software features a highly modular C++ kernel that is coupled to a Python user interface. This makes it easy to add new algorithms, setup a simulation, perform online analysis, use complex workflows and steer a simulation. The extreme flexibility of the software allows for the study of a wide range of systems. The modular structure enables scientists to use ESPResSo++ as a research platform for their own methodological developments, which at the same time allows the software to grow and acquire the most modern methods. ESPResSo++ is targeted for a broad range of architectures and is licensed under the GNU General Public License.
This work is concerned with the atomistic simulation of the volumetric, conformational and structural properties of monodisperse polyethylene (PE) melts of molecular length ranging from C78 up to C1000. In the past, polydisperse models of these melts have been simulated in atomistic detail with the end-bridging Monte Carlo algorithm [Pant and Theodorou, Macromolecules 28, 7224 (1995); Mavrantzas et al., Macromolecules 32, 5072 (1999)]. In the present work, strictly monodisperse as well as polydisperse PE melts are simulated using the recently introduced double bridging and intramolecular double rebridging chain connectivity-altering Monte Carlo moves [Karayiannis et al., Phys. Rev. Lett. 88, 105503 (2002)]. These algorithms constitute generalizations of the EB move, since they entail the construction of two trimer bridges between two properly chosen pairs of dimers along the backbones of two different chains or along the same chain. In the simulations, a new molecular model is employed which is a hybrid of the united-atom TraPPE model [Martin and Siepmann, J. Phys. Chem. B 102, 2569 (1998)] and the anisotropic united-atom model [Toxvaerd, J. Chem. Phys. 107, 5197 (1997)]. Results are first presented documenting the efficiency of the algorithm in equilibrating long-chain PE melts and its dependence on chain length and polydispersity. Simulation data concerning the volumetric, conformational and structural properties of the monodisperse PE melts, obtained with the new simulation algorithm, are found to be in excellent agreement with available experimental data. © 2002 American Institute of Physics.
We present an extremely efficient and rather general model in which whole polymer chains are represented as soft particles. The particles are characterized by their overall sizes and shapes, as given by the conformations of the underlying chains. The probability of occurrence of a particle with a given size determines its internal free energy. The density of monomers within each particle is calculated from all conformations that have the same size. The interaction between two particles is taken to be proportional to the spatial overlap of their monomer density distributions. When a large number of such particles are brought into contact, as is the case for a polymer melt, the interactions between the particles force them to shrink and modify the equilibrium size distribution. We show by simulations that this model leads to a Gaussian statistics of the chains in melt. Since the internal degrees of freedom of a chain are integrated out, a large number (of order 104) of long (e.g., N = 100) chains can be simulated within reasonable computer time on a single work-station processor. A straightforward extension of this model is used to study symmetric polymer blends. © 1998 American Institute of Physics.
We have previously proposed a method for preparing dense amorphous polymer samples which was designed to circumvent the need for long periods of relaxation to equilibrium [J. I. McKechnie, D. Brown, and J. H. R. Clarke, Macromolecules 25, 1562 (1992)]. In the current article, we examine in more detail the application of the method to the preparation of polymer melts using precise data from massively parallel simulations. We expose deficiencies in the original method and introduce a modification which improves the equilibration. The limitations of the overall procedure are discussed in detail.
We present an extensive molecular‐dynamics simulation for a bead spring model of a melt of linear polymers. The number of monomers N covers the range from N=5 to N=400. Since the entanglement length Ne is found to be approximately 35, our chains cover the crossover from the nonentangled to the entangled regime. The Rouse model provides an excellent description for short chains N<Ne, while the dynamics of the long chains can be described by the reptation model. By mapping the model chains onto chemical species we give estimates of the times and distances of onset of the slowing down in motion due to reptation. Comparison to neutron spin‐echo data confirm our mapping procedure, resolving a discrepancy between various experiments. By considering the primitive chain we are able to directly visualize the confinement to a tube. Analyzing the Rouse mode relaxation allows us to exclude the generalized Rouse models, while the original reptation prediction gives a good description of the data.
The average configuration of polymer molecules in solution is markedly influenced by the obvious requirement, ordinarily disregarded in problems relating to molecular configuration, that two elements of the molecule are forbidden from occupying the same location in space. The influence of spatial ``interferences'' between different segments of the molecule on its average configuration has been investigated by statistical and thermodynamic methods.
It is shown that if the average linear dimension of a polymer chain is to be taken proportional to a power of the chain length, that power must be greater than the value 0.50 previously deduced in the conventional ``random flight'' treatment of molecular configuration. This power should approach 0.60 for long chain molecules in good solvents. With increase in size of the solvent molecule, the influence of interference on molecular configuration diminishes, vanishing entirely in the extreme case of a solvent which is also a high polymer. The effect of a heat of interaction between solvent and polymer may also be incorporated quantitatively in the theory. A positive heat of mixing (poor solvent) tends to offset the expansive influence of interference, and the exponent referred to above tends to approach 0.50. The results are of foremost significance in the interpretation of the intrinsic viscosity and its dependence on the polymer constitution and on the solvent. It is pointed out that the spatial dimensions of the irregularly coiled polymer molecule cannot be correlated directly with hindrance to rotation about chain bonds, unless the expansion of the configuration due to interference and the effects of the heat of dilution are first of all taken into account.
We present a hierarchical approach that combines atomistic and mesoscopic simulations that can generally be applied to vinyl polymers. As a test case, the approach is applied to atactic polystyrene (PS). First, a specific model for atactic PS is chosen. The bonded parameters in the coarse-grained force field, based on data obtained from atomistic simulations of isolated PS dimers, are chosen in a way which allows to differentiate between meso and racemic dyads. This approach in principle allows to study isotactic and syndiotactic melts as well. Nonbonded interactions between coarse-grained beads were chosen as purely repulsive. The proposed mesoscopic model reproduces both the local structure and the chain dimensions properly. An explicit time mapping is performed, based on the atomistic and CG mean-square displacements of short chains, demonstrating an effective speed up of about 3 orders of magnitude compared to brute force atomistic simulations. Finally the equilibrated coarse-grained chains are back mapped onto the atomistic systems. This opens new routes for obtaining well equilibrated high molecular weight polymeric systems and also providing very long dynamic trajectories at the atomistic level for these polymers.
We report a quantitative analysis of the detailed equilibrium atomic structure of molten linear polyethylene, obtained using directed bridging Monte Carlo computer simulation techniques. The polydisperse sample has an average chain length of 6000 backbone carbon atoms, or approximately 84000 g mol−1. This molecular weight greatly exceeds that used in previous atomistic simulation studies, and is typical of commercial grades that are widely used for injection moulded articles. This large-scale simulation allows direct measurement of such properties as the chain entanglement length, the characteristic ratio, and the extended-range intermolecular packing density fluctuations which give rise to the first sharp diffraction peak.
We review a coarse-graining strategy (multiblob approach) for polymer solutions in which groups of monomers are mapped onto a single atom (a blob) and effective blob–blob interactions are obtained by requiring the coarse-grained model to reproduce some coarse-grained features of the zero-density isolated-chain structure. By tuning the level of coarse graining, i.e. the number of monomers to be mapped onto a single blob, the model should be adequate to explore the semidilute regime above the collapse transition, since in this case the monomer density is very small if chains are long enough. The implementation of these ideas has been previously based on a transferability hypothesis, which was not completely tested against full-monomer results (Pierleoni et al., J. Chem. Phys., 2007, 127, 171102). We study different models proposed in the past and we compare their predictions to full-monomer results for the chain structure and the thermodynamics in the range of polymer volume fractions Φ between 0 and 8. We find that the transferability assumption has a limited predictive power if a thermodynamically consistent model is required. We introduce a new tetramer model parametrized in such a way to reproduce not only zero-density intramolecular and intermolecular two-body probabilities, but also some intramolecular three-body and four-body distributions. We find that such a model correctly predicts three-chain effects, the structure and the thermodynamics up to Φ [similar, equals] 2, a range considerably larger than that obtained with previous simpler models using zero-density potentials. Our results show the correctness of the ideas behind the multiblob approach but also that more work is needed to understand how to develop models with more effective monomers which would allow us to explore the semidilute regime at larger chain volume fractions.
Recent years have brought exciting theoretical advances to understanding the behavior of macromolecular systems under nonequilibrium conditions. Developments in diffusion-controlled reactions of polymers are bringing molecular insights to reactive blending technologies, and improved theories relating to associating polymers should aid in the design of thickening agents and coatings. Recent progress in molecular theories of flow and deformation may facilitate the design of branched polymers with tailored rheological properties and improved adhesives.
We introduce a coarse-grained model for the simulation of the statistical properties of polymeric systems where collections of chain segments are represented by connected soft spheres of fluctuating size. A generic bead–spring model is considered as a test case for this coarse-graining approach. The effective interactions between the soft spheres and the effective free energy that governs the size distribution of soft spheres are determined using analytical calculations or empirical expressions, leading to relatively simple functional forms that facilitate implementation in the simulation code. The validity of this coarse-graining approach is tested on a variety of systems ranging from ideal chains to dilute solutions and melts. The scheme allows to vary the spatial range of coarse-graining in a flexible and consistent way, and turns out to be particularly efficient for polymer melts. The equilibrium properties of the coarse-grained model agree with those of the original bead–spring model. As the coarse graining does not conserve entanglements, the dynamics is restricted to Rouse-like and the considerably accelerated coarse-grained dynamics suggests this approach as particularly promising for the equilibration of large long-chain polymer melts and mixtures.
We present theoretical arguments and numerical results to demonstrate long-range intrachain correlations in concentrated solutions and melts of long flexible polymers, which cause a systematic swelling of short chain segments. They can be traced back to the incompressibility of the melt leading to an effective repulsion u(s) approximately s/rho R3(s) approximately c(e)/sqrt[s] when connecting two segments together where s denotes the curvilinear length of a segment, R(s) its typical size, c(e) approximately 1/rho b(e)3 the "swelling coefficient," b(e) the effective bond length, and rho the monomer density. The relative deviation of the segmental size distribution from the ideal Gaussian chain behavior is found to be proportional to u(s). The analysis of different moments of this distribution allows for a precise determination of the effective bond length b(e) and the swelling coefficient c(e) of asymptotically long chains. At striking variance to the short-range decay suggested by Flory's ideality hypothesis the bond-bond correlation function of two bonds separated by s monomers along the chain is found to decay algebraically as 1/s(3/2). Effects of finite chain length are briefly considered.
A soft ellipsoid model for Gaussian polymer chains is studied, following an idea proposed by Murat and Kremer [J. Chem. Phys. 108, 4340 (1998)]. In this model chain molecules are mapped onto ellipsoids with certain shapes, and to each shape a monomer density is assigned. In the first part of the work, the probabilities for the shapes and the associated monomer densities are studied in detail for Gaussian chains. Both quantities are expressed in terms of simple approximate formulae. The free energy of a system composed of many ellipsoids is given by an intramolecular part accounting for the internal degrees of freedom and an intermolecular part following from pair interactions between the monomer densities. Structural and kinetic properties of both homogeneous systems and binary mixtures are subsequently studied by Monte-Carlo simulations. It is shown that the model provides a powerful phenomenological approach for investigating polymeric systems on semi-macroscopic time and length scales. Comment: 17 pages, 17 figures, submitted to J. Chem. Phys
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