Kuhn Steps Research Articles

Effects of chain resolution on the configurational and rheological predictions of dilute polymer solutions in flow fields with hydrodynamic interactions

In a recent study, the resolution of a polymer chain model was shown to significantly affect rheological predictions from Brownian dynamics (BD) simulations [Kumar and Dalal, “Effects of chain resolution on the configurational and rheological predictions from Brownian dynamics simulations of an isolated polymer chain in flow,” J. Non-Newtonian Fluid Mech. 315, 105017 (2023)], even in the absence of hydrodynamic interactions (HI) and excluded volume. In this study, we investigate the effects of chain resolution in the presence of HI. Toward this, we perform BD simulations of a long polymer chain, with the discretization level varying from a single Kuhn step (bead–rod model) to several tens of Kuhn-steps (bead–spring model). The chain models were subjected to flow fields of uniaxial extension (purely stretching) and steady shear (equal rates of stretching and rotation). Broadly, our results indicate an amplification of the differences observed between the differently resolved bead–rod and bead–spring models, in the presence of HI. Interestingly, all rheological predictions qualitatively fall in two groups for extensional flow, with the predictions from the bead–spring model with HI being close to those of the bead–rod model without HI. This indicates significantly reduced sensitivity of coarser bead–spring models to HI, relative to the one resolved to a single Kuhn step. However, in shear flow, the bead–spring rheological predictions fall between those of the bead–rod model with and without HI, forming a third group. This is linked to the presence of stretched and coiled states in the ensemble for shear flow. HI effects are large for the coiled states and weak for the stretched states, thereby yielding predictions that are intermediate between those for no HI and dominant HI. Thus, quite surprisingly, the quality of predictions of the bead–spring models is strongly affected by the physics of the flow field, irrespective of the parameterization.

Fraenkel Springs as an Efficient Approximation to Rods for Brownian Dynamics Simulations and Modeling of Polymer Chains

AbstractRecent studies indicate the importance of using bead‐rod models for polymer chains, resolving to a single Kuhn step, especially in strong flows. Earlier, researchers suggested the Fene‐Fraenkel spring as a computationally efficient approximation to the rod for Brownian dynamics (BD) simulations. This analysis reveals that the Fraenkel spring is an even better alternative. The predictions from BD simulations with stiff Fraenkel springs are nearly identical to those using Fene‐Fraenkel springs and rigid rods. Significantly, such excellent agreement is obtained when the spring lengths at every time step are updated once, without any further iterations for refinement. The computational speeds for single‐step update of the spring lengths, using Fraenkel springs, is an order of magnitude faster than the usual procedure of iterating until convergence. Even when multiple iterations are allowed within each time step, the computational time remains about 20–25% lower for longer chains with Fraenkel springs than Fene‐Fraenkel springs. The merits of using highly resolved chains, using Fraenkel springs, in preaveraged models are also highlighted. The predictions of such preaveraged models agree reasonably well with BD simulations. This is particularly advantageous since the predictions for shear flows are poor, especially at higher shear rates, for well‐known models like FENE‐P.

Efficient Determination of Slip-Link Parameters from Broadly Polydisperse Linear Melts.

We investigate the ability of a coarse-grained slip-link model and a simple double reptation model to describe the linear rheology of polydisperse linear polymer melts. Our slip-link model is a well-defined mathematical object that can describe the equilibrium dynamics and non-linear rheology of flexible polymer melts with arbitrary polydispersity and architecture with a minimum of inputs: the molecular weight of a Kuhn step, the entanglement activity, and Kuhn step friction. However, this detailed model is computationally expensive, so we also examine predictions of the cheaper double reptation model, which is restricted to only linear rheology near the terminal zone. We report the storage and loss moduli for polydisperse polymer melts and compare with experimental measurements from small amplitude oscillatory shear. We examine three chemistries: polybutadiene, polypropylene and polyethylene. We also use a simple double reptation model to estimate parameters for the slip-link model and examine under which circumstances this simplified model works. The computational implementation of the slip-link model is accelerated with the help of graphics processing units, which allow us to simulate in parallel large ensembles made of up to 50,000 chains. We show that our simulation can predict the dynamic moduli for highly entangled polymer melts over nine decades of frequency. Although the double reptation model performs well only near the terminal zone, it does provide a convenient and inexpensive way to estimate the entanglement parameter for the slip-link model from polydisperse data.

Parameter-free prediction of DNA dynamics in planar extensional flow of semidilute solutions

The dynamics of individual Deoxyribonucleic acid (DNA) molecules in semidilute solutions undergoing planar extensional flow is simulated using a multiparticle Brownian dynamics algorithm, which incorporates hydrodynamic and excluded volume interactions in the context of a coarse-grained bead-spring chain model for DNA. The successive fine-graining protocol [P. Sunthar and J. R. Prakash, Macromolecules 38, 617–640 (2005); R. Prabhakar et al., J. Rheol. 48, 1251–1278 (2004)], in which simulation data acquired for bead-spring chains with increasing values of the number of beads Nb, is extrapolated to the number of Kuhn steps NK in DNA (while keeping key physical parameters invariant), is used to obtain parameter-free predictions for a range of Weissenberg numbers and Hencky strain units. A systematic comparison of simulation predictions is carried out with the experimental observations of Hsiao et al. [J. Rheol. (in press)], who have recently used single molecule techniques to investigate the dynamics of dilute and semidilute solutions of λ-phage DNA in planar extensional flow. In particular, they examine the response of individual chains to step-strain deformation followed by cessation of flow, thereby capturing both chain stretch and relaxation in a single experiment. The successive fine-graining technique is shown to lead to quantitatively accurate predictions of the experimental observations in the stretching and relaxation phases. Additionally, the transient chain stretch following a step strain deformation is shown to be much smaller in semidilute solutions than in dilute solutions, in agreement with experimental observations.

Constant interchain pressure effect in extensional flows of oligomer diluted polystyrene and poly(methyl methacrylate) melts

The constant ‘interchain pressure’ idea has been addressed, to evaluate if it is an adequate quantitative assumption to describe the fluid mechanics of oligomer diluted entangled NMMD polymer systems. The molecular stress function constitutive framework has been used with the constant interchain pressure assumption. Furthermore, the maximal extensibility based on the number of Kuhn steps in an entanglement has been used based on the relative Pade inverse Langevin function. The model predictions agree with the extensional measurements on all previously published poly(methyl methacrylate)s and almost all published oligomer diluted NMMD polystyrenes. The only deviation is on the most diluted and largest molecular weight case of an 18% 1880 kg/mol polystyrene in oligomer diluent. In this case, the maximal extensibility is not needed.

A new bead-spring model for simulation of semi-flexible macromolecules.

A bead-spring model for semi-flexible macromolecules is developed to overcome the deficiencies of the current coarse-grained bead-spring models. Specifically, model improvements are achieved through incorporation of a bending potential. The new model is designed to accurately describe the correlation along the backbone of the chain, segmental length, and force-extension behavior of the macromolecule even at the limit of 1 Kuhn step per spring. The relaxation time of different Rouse modes is used to demonstrate the capabilities of the new model in predicting chain dynamics.

Challenging Tube and Slip-Link Models: Predicting the Linear Rheology of Blends of Well-Characterized Star and Linear 1,4-Polybutadienes

We compare predictions of two of the most advanced versions of the tube model, namely the “Hierarchical model” by Wang et al. [J. Rheol. 2010, 54, 223] and the BoB (branch-on-branch) model by Das et al. [J. Rheol. 2006, 50, 207], against linear viscoelastic G′ and G″ data of binary blends of nearly monodisperse 1,4-polybutadiene 4-arm star polymer of arm molar mass 24 000 g/mol with a monodisperse linear 1,4-polybutadiene of molar mass 58 000 g/mol. The star was carefully synthesized and characterized by temperature gradient interaction chromatography and by linear rheology over a wide frequency region through time–temperature superposition. We found large failures of both the Hierarchical and BoB models to predict the terminal relaxation behavior of the star/linear blends, despite their success in predicting the rheology of the pure star and pure linear polymers. This failure occurred regardless of the choices made concerning constraint release, such as assuming arm retraction in “fat” or “skinny” tubes. Allowing for “disentanglement relaxation” to cut off the constraint release Rouse process at long times does lead to improved predictions for our blends, but leads to much worse predictions for other star/linear blends described in the literature, especially those of Shivokhin et al. [Macromolecules 2014, 47, 2451]. In addition, our blends and those of Shivokhin et al. were also tested against a coarse-grained slip-link model, the “clustered fixed slip-link model (CFSM)” of Schieber and co-workers [J. Rheol. 2014, 58, 723], in which several Kuhn steps are clustered together for computational efficiency. The CFSM with only two molecular-weight- and chain-architecture-independent parameters was able to give very good agreement with all experimental data for both of these sets of blends. In light of its success, the CFSM slip-link model may be used to address the constraint release issue more rigorously and potentially help develop improved tube models.

Flexible polyelectrolyte chain in a strong electrolyte solution: Insight into equilibrium properties and force-extension behavior from mesoscale simulation.

Macromolecules with ionizable groups are ubiquitous in biological and synthetic systems. Due to the complex interaction between chain and electrostatic decorrelation lengths, both equilibrium properties and micro-mechanical response of dilute solutions of polyelectrolytes (PEs) are more complex than their neutral counterparts. In this work, the bead-rod micromechanical description of a chain is used to perform hi-fidelity Brownian dynamics simulation of dilute PE solutions to ascertain the self-similar equilibrium behavior of PE chains with various linear charge densities, scaling of the Kuhn step length (lE) with salt concentration cs and the force-extension behavior of the PE chain. In accord with earlier theoretical predictions, our results indicate that for a chain with n Kuhn segments, lE ∼ cs (-0.5) as linear charge density approaches 1/n. Moreover, the constant force ensemble simulation results accurately predict the initial non-linear force-extension region of PE chain recently measured via single chain experiments. Finally, inspired by Cohen's extraction of Warner's force law from the inverse Langevin force law, a novel numerical scheme is developed to extract a new elastic force law for real chains from our discrete set of force-extension data similar to Padè expansion, which accurately depicts the initial non-linear region where the total Kuhn length is less than the thermal screening length.

Analytic slip-link expressions for universal dynamic modulus predictions of linear monodisperse polymer melts

The discrete slip-link model (DSM) is a robust mesoscopic theory that has great success predicting the rheology of flexible entangled polymer liquids and gels. In the most coarse-grained version of the DSM, we exploit heavily the universality observed in the shape of the relaxation modulus of linear monodisperse melts. For this type of polymer, we present here analytic expressions for the relaxation modulus. The high-frequency dynamics which are typically coarse-grained out from the DSM are added back into these expressions by using a Rouse chain with fixed ends to represent the fast motion of Kuhn steps between entanglements. We find consistency in the friction used for both fast and slow modes. We test these expressions against experimental data for three chemistries and molecular weights with good agreement. Using these analytic expressions, the polymer density, the molecular weight of a Kuhn step, MK, and the low-frequency cross-over between the storage and loss moduli, \(G^{\prime }\) and \(G^{\prime \prime }\), it is now straightforward to estimate model parameter values and obtain predictions over the experimentally accessible frequency range without performing expensive numerical calculations.

Interchain tube pressure effect in the flow dynamics of bi-disperse polymer melts

The constitutive equation as reported by Rasmussen and Huang (Rheologica Acta 53:199–208, 2014b), explaining the flow dynamics of oligomer (containing a least two Kuhn step)-diluted narrow molecular weight-distributed polymers were extended to general bi-disperse polymer melt system. It was assumed that the Rouse time of a particular polymer chain is dependent on the total number of Kuhn steps of the polymers in direct contact with the considered polymer chain. This number of Kuhn steps is proportional to the weight average molecular weight, M w , replacing the involved molecular weight in the equation as reported by Rasmussen and Huang (Rheologica Acta 53:199–208, 2014b). Two separate stretch evolution equations for the long and the short polymer respectively, were introduced to handle the two involved Rouse times. Experimentally, the bi-disperse polystyrene systems of Nielsen et al. (J Rheol 50:453–476, 2006) and polyisoprene systems as reported by Read et al. (J Rheol 56:823–873, 2012) were within quantitatively agreement with the derived model. This included both startup of extension as well as shear flow. One exception was observed. In the most diluted polyisoprene blend, the measured extensional viscosities were under predicted by the model.

Universality and speedup in equilibrium and nonlinear rheology predictions of the fixed slip-link model

The discrete slip-link model (DSM) was developed to describe the dynamics of flexible entangled polymer melts. With just three molecular-weight- and chain-architecture-independent parameters—the molecular weight of a Kuhn step MK; entanglement activity β; and Kuhn step shuffling characteristic time τK—DSM is able to predict simultaneously the linear viscoelasticity of monodisperse linear, polydisperse linear, and branched systems. Without any adjustment, DSM shows excellent agreement with shear flow experiments and elongational flows with stretch up to ∼10–20. Universality observed between entangled melts with the notable exception of high-strain elongations suggests that the average number of entanglements per chain is the primary characteristic of the system. Therefore, theoretical predictions for systems with differing numbers of Kuhn steps per chain but roughly the same number of entanglements should be equivalent when rescaled. In this work, we present a scaling of the DSM parameters, which has no significant effect on model predictions yet reduces greatly computational cost. The idea behind the scaling is clustering several Kuhn steps together. Thus, we call this implementation of the DSM the clustered fixed slip-link model (CFSM). The model is limited to at least one cluster between entanglements. The CFSM assumes two clusters between entanglements on average, which appears to be a reasonable minimum. We find that clustering results in a loss of some of the high frequency modes in G* predictions, sometimes called “longitudinal modes” in tube models. Matching the low-frequency rubbery plateau height allows us to derive Mc—the molecular weight of a cluster, as a function of the DSM parameters β, MK. We also find an empirical relationship between timescale parameters τK and τc, the characteristic time for shuffling a cluster through the entanglement. We compare G* predictions of the CFSM with predictions of the DSM for monodisperse linear, polydisperse linear, and branched systems, and observe no difference. Comparison with shear flow predictions shows that only rates with a Weissenberg number based on the strand relaxation time τe of 1 and higher are affected. For elongational flow at high strains and high rates, significant difference can occur, which is perhaps not surprising given the observation above. For the systems shown in this work, we report CFSM computational savings of several hundred times. We then apply CFSM to the shear flow of a star-branched polymer melt with molecular weight not accessible to DSM without rescaling. Excellent agreement with experimental data is observed. Additionally, we report first normal stress theoretical predictions for this system.

Velocity autocorrelation spectra in molten polymers measured by NMR modulated gradient spin-echo

The segmental dynamics in molten linear polymers is studied by the NMR method of modulated gradient spin-echo, which directly probes a spectrum of molecular velocity autocorrelation function. Diffusion spectra of mono-disperse poly(isoprene-1.4) with different molecular masses, measured in the frequency range 0.1–10 kHz at a temperature of , have a form similar to the spectrum of Rouse chain dynamics, which implicates the tube-Rouse motion as the dominant dynamic process in this frequency range. The scaling of the center-of-mass diffusion coefficient, given from the fitting parameters, changes from into at around Kuhn steps, which is less than predicted by theory and simulations, while the correlation times of the tube-Rouse mode do not follow the anticipated scaling.

Systematic Coarse-Graining of the Dynamics of Self-Attractive Semiflexible Polymers

We develop a coarse-graining procedure, using bead–rod chains with varying numbers of rods per Kuhn length (Nr,K), to model the conformational dynamics of polymer chains with bending stiffness and self-attraction. We find the conditions under which increased chain resolution leads to converged results for the effect of bending stiffness, attractive interaction strength (e), and shearing flow on the conformational dynamics. As chain stiffness increases, we find a collapse transition that becomes steeper as attractive strength increases. The collapsed state depends not only on the net dimensionless attractive strength e* = eNr,K between each Kuhn step of the chain but also on the ratio of bead diameter to Kuhn length (σ*). At high e* (>0.8/σ*), collapsed globules are produced at moderate chain diameter σ* = 1/4, while for fat chains with σ* = 1, helices are formed, and for thin ones with σ* = 1/16, tori, folded bundles, and finally globules are formed as e* increases.

Interchain tube pressure effect in extensional flows of oligomer diluted nearly monodisperse polystyrene melts

We have derived a constitutive equation to explain the extensional dynamics of oligomer-diluted monodisperse polymers, if the length of the diluent has at least two Kuhn steps. These polymer systems have a flow dynamics which distinguish from pure monodisperse melts and solutions thereof, if the solvent has less than two Kuhn steps, e.g. is not a chain. The constitutive equation is based on a phenomenological tube-based model within the methodology of the molecular stress function approach. The nonlinear dynamics have been explained as a consequence of a constant thermal interchain pressure originating from the short polymer chains (e.g. the oligomers) on the wall of the tube containing the long chains. The nonlinear dynamics are uniquely defined by the Rouse time and the maximal extensibility of the long polymer chains. Both are linked to the entanglement length. The relation between the Rouse times and entanglements have been established based on published extensional experiments on nearly monodisperse polystyrene melts. The constitutive equation has shown agreement with the experimental startup of and steady extension data from Huang et al. (Macromolecules 46:5026–5035, 2013a) based on 285 and 545 kg/mol polystyrenes diluted in styrene oligomers containing 3.3 (1.92 kg/mol) and 7.3 (4.29 kg/mol) Kuhn steps.

The missing link between the extensional dynamics of polymer melts and solutions

Based on extensional viscosities measured on narrow molecular weight distributed (NMMD) polystyrenes and polystyrene oligomer dilutions thereof, we discuss the relation between the flow physics of polymer solutions and melts. A polymer solution is here characterized as a dilution where the diluent contains less than two Kuhn steps. At the same entanglement number (e.g. concentrations) its extensional viscosities are up to about 300% higher than the corresponding viscosities for polymer blends. A blend is understood as a polymer system diluted with polymer/oligomer containing active chain in term of Kuhn steps, e.g. at least two Kuhn steps. A similar disagreement is observed between blends and pure NMMD melts, both containing the same entanglement number.

Explaining the Absence of High-Frequency Viscoelastic Relaxation Modes of Polymers in Dilute Solutions

Using multiscale modeling, including molecular dynamics simulations, with both united-atom and coarse-grained force fields, as well as Brownian dynamics simulations with still higher levels of coarse-graining, we explain the long-mysterious absence of high frequency modes in the viscoelastic spectrum of isolated polymer chains in good solvents, reported years ago by Schrag and co-workers. The relaxation spectrum we obtain for a chain of 30 monomers at atomistic resolution is, remarkably, a single exponential, while that of a coarse-grained chain of 100 monomers is well fit by only two modes. These results are very surprising in view of the many degrees of freedom possessed by these chains, and in view of the many relaxation modes present in melts of such chains. However, the result agrees perfectly with experimental observations of Schrag and co-workers. We also performed Brownian dynamics (BD) simulations in which the explicit solvent molecules are replaced by a viscous continuum, using chain models of varying degrees of resolution, both in the presence and absence of hydrodynamic interactions (HI). Although the local dynamics is suppressed by the addition of bending, torsion, side groups and excluded volume interactions (as suggested in Jain and Larson(15)), none of the BD simulations predict a single exponential relaxation for a short polymer chain. The comparison of the relaxation of the bond vectors from different models indicates an additional slow-down in the presence of explicit solvent molecules, which is critical to obtain a single exponential relaxation for short chains. Our results indicate that the chain dynamics at small length scales (down to a few Kuhn steps) are significantly different from the predictions of models based on a continuum solvent, and finally help explain the experimental results of Schrag and co-workers.

Impact of Solvent Quality on the Hysteresis in the Coil–Stretch Transition of Flexible Polymers in Good Solvents

Direct observations of long flexible polymers in elongational flow show a coil to stretch transition (CST) and hysteresis in this transition. In this article, solvent effects on the hysteresis of CST were examined using Brownian dynamics simulations of a dumbbell model with conformation dependent drag. A spring force relation that is able to capture the effects of solvent quality was used. Two methods of increasing the solvent quality were analyzed: (1) by increasing the number of Kuhn steps (NK) and keeping the excluded volume parameter (v/l3) constant, and (2) by increasing v/l3 and keeping NK constant. In the first case, the width of the hysteresis (ratio of upturn to downturn strain rates) increased with increase in NK. In the second case, the width of hysteresis decreased with increase in v/l3. This implies that molecules such as ss-DNA, which have v/l3 ≈ 1 in good solvents, will have less hysteresis than a molecule with the same NK in a Θ solvent.

Tumbling and Deformation of Isolated Polymer Chains in Shearing Flow

Using Brownian dynamics simulations of polymer chains over a wide range of resolution, we find universal scaling laws for polymer coil dimensions and tumbling time. At high shear rates γ̇, the coil thickness in the gradient direction becomes independent of chain length, scaling as NK0γ̇–1/4, where NK denotes the numbers of Kuhn steps, and the tumbling time scales as NKγ̇–3/4, correcting scaling laws presented in prior studies. We find this to be a consequence of the formation of loops whose length is limited by the time required to stretch them and derive scaling laws from a balance of convection and diffusion of monomers. We find that, in the absence of hydrodynamic interaction (HI) and excluded volume (EV), for wormlike chains, the shrinkage in chain stretch observed at ultrahigh shear rates is pushed out to arbitrarily high shear rates if the chain is resolved increasingly finely below the persistence length. Finally, scaling laws in the presence of excluded volume and hydrodynamic interactions are derived that are expected to be valid for long chains at high shear rates.

Derivation of free energy expressions for tube models from coarse-grained slip-link models

We present the free energy of a single-chain mean-field model for polymer melt dynamics, which uses a continuous (tube-like) approximation to the discrete entanglements with surrounding chains, but, in contrast to previous tube models, includes fluctuations in the number density of Kuhn steps along the primitive path and in the degree of entanglement. The free energy is obtained from that of the slip-link model with fluctuating entanglement positions [J. D. Schieber and K. Horio, J. Chem. Phys. 132, 074905 (2010)] by taking the continuous limit of (functions of) the discrete Kuhn-step numbers and end-to-end vectors of the strands between entanglements. This coarse-graining from a more-detailed level of description has the advantage that no ad hoc arguments need to be introduced. Moreover, the thermodynamic consistency of the slip-link model [J. D. Schieber, J. Non-Equilib. Thermodyn. 28, 179 (2003)] can be preserved. Fluctuations in the positions of entanglements lead to a harmonic bending term in the free energy of the continuous chain, similar to that derived by Read et al. [Macromolecules 41, 6843 (2008)] starting from a modified GLaMM model [R. S. Graham, A. E. Likhtman, T. C. B. McLeish, and S. T. Milner, J. Rheol. 47, 1171 (2003)]. If these fluctuations are set to zero, the free energy becomes purely Gaussian and corresponds to the continuous limit of the original slip-link model, with affinely moving entanglements [J. D. Schieber, J. Chem. Phys. 118, 5162 (2003)]. The free energy reduces to that of Read et al. under their assumptions of a homogeneous Kuhn-step number density and a constant degree of entanglement. Finally, we show how a transformation of the primitive-path coordinate can be applied to make the degree of entanglement an outcome of the model instead of a variable. In summary, this paper constitutes a first step towards a unified mathematical formulation of tube models. The next step will be to formulate the dynamics of the primitive-path conformation and the entanglement density along the primitive path. Now that the free energy is known, statistical mechanics can be employed for this purpose.

Multiple regimes of deformation in shearing flow of isolated polymers

Using Brownian dynamics simulations, without excluded volume and hydrodynamic interactions, on single polymer molecules represented by bead-spring models with stiff Fraenkel springs mimicking a single Kuhn step, we find multiple nonlinear regimes of deformation in shear flows that are controlled by the Peclet number (Pe), which is the shear rate times the relaxation time of a Kuhn step. We observe that, for all chain lengths investigated, the average stretch in the flow direction initially increases with increasing Pe, followed by saturation in chain stretch, as observed in previous studies. At even higher Pe, the stretch begins to decrease with increasing shear rate, in accordance with similar simulations of Sendner and Netz, Eur. Phys. J. E 30, 75–81 (2009). At these rates, the trajectories reveal “premature” recoiling of the chain before attaining a fully extended state during the phase of the “tumbling orbit” in which the chain is stretching. An increasing stretch at even higher Pe is characterized by...