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Schematic representation of the polymer networks. Polymer chains are connected at junctions that have tetrafunctionality and fixed at the surface.
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We consider the spatiotemporal fluctuation of slip-link positions via the implementation of elastic slip-links. The level of description is similar to our previously proposed slip-link model, wherein we use the entanglement position in space as dynamic variables, and the number of Kuhn steps between entanglements. However, since it is a mean-field,...
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... sketch of the model is shown in Fig. 1. A fixed point of the chain on the surface of the sample is shown by an "." In both cases, these are deformed affinely. We denote the position in space of such a node by R k 0 . A cross-link is shown by a circle "." In case A, these are free to fluctuate at all times from Brownian motion. In case B, these are ini- tially fixed at their ...
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Citations
... It is generally accepted that the 2/ f factor is associated with the presence of fluctuations in the strand size, which are also proportional to this factor (see Equation (22)). In fact, fluctuations have nothing to do with it [15]: an exact elastic modulus of Gaussian phantom network is different from that of the perfect network model and coincides with the elastic modulus of the classical non-fluctuating grid, in which each strand is replaced by a corresponding elastic thread with the same elastic stiffness coefficient [16]. This coincidence is due to affine deformation of the average distances between the cross-links in such networks. ...
... is the density of gel monomers, see Equations (16) and (15). Therefore, G = 0 at p cµ − p cγ ∼ 1/ √ P (0) , in accordance with the result of numerical simulations [26]. ...
A review of the main elasticity models of flexible polymer networks is presented. Classical models of phantom networks suggest that the networks have a tree-like structure. The conformations of their strands are described by the model of a combined chain, which consists of the network strand and two virtual chains attached to its ends. The distribution of lengths of virtual chains in real polydisperse networks is calculated using the results of the presented replica model of polymer networks. This model describes actual networks having strongly overlapping and interconnected loops of finite sizes. The conformations of their strands are characterized by the generalized combined chain model. The model of a sliding tube is represented, which describes the general anisotropic deformations of an entangled network in the melt. I propose a generalization of this model to describe the crossover between the entangled and phantom regimes of a swollen network. The obtained dependence of the Mooney-Rivlin parameters C 1 and C 2 on the polymer volume fraction is in agreement with experiments. The main results of the theory of heterogeneities in polymer networks are also discussed.
... [3][4][5] Our understanding of chain dynamics within the confining tube under equilibrium conditions has recently been considerably enriched by methodologies capable of extracting the PP network from lower-level simulations. Both entanglement statistics [6][7][8] and their dynamics [9][10][11][12][13][14][15] have been analyzed using such methods, and the information has been passed over to slip-link 16,17 and multiscale approaches. [18][19][20][21] As a continuation of their proposed framework at equilibrium, DE obtained a constitutive equation for concentrated polymer melts and concentrated solutions for arbitrary homogeneous flow fields. ...
The complete kinetic theory model for concentrated polymer solutions and melts proposed by Curtiss and Bird is solved for shear flow: (a) analytically by providing a solution for the single-link (or configurational) distribution function as a real basis spherical harmonics expansion and then calculating the materials functions in shear flow up to second order in the dimensionless shear rate and, (b) numerically via the execution of Brownian dynamics simulations. These two methods are actually complementary to each other as the former is accurate only for small dimensionless shear rates where the latter produces results with increasingly large uncertainties. The analytical expansions of the material functions with respect to the dimensionless shear rate reduce to those of the extensively studied, simplified Curtiss-Bird model when ε′ = 0, and to the rigid rod when ε′ = 1. It is known that the power-law behavior at high shear rates is very different for these two extremal cases. We employ Brownian dynamics simulation to not only recover the limiting cases but to find a gradual variation of the power-law behaviors at large dimensionless shear rates upon varying ε′. The fact that experimental data are usually located between these two extremes strongly advocates the significance of studying the solution of the Curtiss-Bird model. This is exemplified in this work by comparing the solution of this model with available rheological data for semiflexible biological systems that are clearly not captured by the original Doi-Edwards or simplified Curtiss-Bird models.
... For instance, it has been extensively shown that single-chain temporary network models that include Brownian forces, but not motor forces, satisfy the uctuation-dissipation theorem. 38,42 It is common for active gels to contain permanent passive cross-links such as biotin or a-actinin in actomyosin networks. In those cases myosin contracts F-actin into dense foci around the permanent cross-links. ...
... Similar calculations are oen performed for single-chain models of temporary networks, to demonstrate thermodynamic consistency (FDT compliance). 38,42 A. Dynamic modulus of active gels from a Green-Kubo formula ...
... If the uctuation-dissipation theorem were obeyed the two moduli would be equal. 38,42,47 We nd that for the active dumbbell model there is a frequency-dependent discrepancy between the moduli obtained from the two different methods. This discrepancy is in general determined by the attachment and detachment rates, as well as by the friction coefficients. ...
We have developed a single-chain theory to describe the dynamics of active gels. Active gels are networks of semiflexible polymer filaments driven by motor proteins that convert chemical energy from the hydrolysis of adenosine triphosphate to mechanical work and motion. In active gels molecular motors create active cross-links between the semiflexible filaments. We model the semiflexible filaments as bead-spring chains; the active interactions between filaments are accounted for using a mean-field approach, in which filaments have prescribed probabilities to undergo a transition from one motor attachment state into the other depending on the state of the probe filament. The level of description of the model includes the change in the end-to-end distance of the filaments, the attachment state of the filaments, and the motor-generated forces, as stochastic state variables which evolve according to a proposed differential Chapman-Kolmogorov equation. The motor-generated forces are drawn from a stationary distribution of motor stall forces that can be measured experimentally. The general formulation of the model allows accounting for physics that is not possible, or not practical, to include in available models that have been postulated on coarser levels of description. However, in this introductory manuscript, we make several assumptions to simplify the mathematics and to obtain analytical results, from which insight into the microscopic mechanisms underlying the dynamics of active gels can be gained. We treat the filaments as one-dimensional dumbbells, approximate the elasticity of the semiflexible filaments with a Hookean spring law, and assume that the transition rates are independent of the tension in the filaments. We show that even in this simplified form, the model can predict the buckling of individual filaments that is thought to be the underlying mechanism in the self-contraction of non-sarcomeric actin-myosin bundles [Lenz et al., Phys. Rev. Lett., 2012, 108, 238107]. The active dumbbell model can also explain the violation of the fluctuation-dissipation theorem observed in microrheology experiments on active gels [Mizuno et al., Science, 2007, 315, 370-373].
... We agree with Likhtman [21] that this situation is undesirable and that efforts should instead be directed toward a multi-scale approach, in which reptation models are derived, with clear assumptions, from more-detailed single-chain or multi-chain models. In a recent paper [22], such a coarsegraining procedure was applied to the mobile slip-link model (MSM) proposed by Schieber and Horio [23]. This resulted in a generalization of the chain free energy for the reptation model of Read et al [20]. ...
... Qin et al [56] extended this model with a semiflexibility correction to contour-length fluctuations, which improved agreement with molecular dynamics simulations in terms of the tangent correlation function along the PP. ESFs were also included, on more detailed levels of description, in the slip-spring simulation of Likhtman [57] and the MSM of Schieber and Horio [23]. Here we use a recently proposed version of the MSM, in which the ESFs become anisotropic in flow [58]. ...
... Here we use a recently proposed version of the MSM, in which the ESFs become anisotropic in flow [58]. We refer to the original model of Schieber and Horio [23] as the isotropic MSM and to the new model as the anisotropic MSM. At equilibrium, where the statistics of the two models are equivalent, we simply use the term MSM. ...
We present a method to map the full equilibrium distribution of the primitive-path (PP) length, obtained from multi-chain simulations of polymer melts, onto a single-chain mean-field ‘target’ model. Most previous works used the Doi-Edwards tube model as a target. However, the average number of monomers per PP segment, obtained from multi-chain PP networks, has consistently shown a discrepancy of a factor of two with respect to tube-model estimates. Part of the problem is that the tube model neglects fluctuations in the lengths of PP segments, the number of entanglements per chain and the distribution of monomers among PP segments, while all these fluctuations are observed in multi-chain simulations. Here we use a recently proposed slip-link model, which includes fluctuations in all these variables as well as in the spatial positions of the entanglements. This turns out to be essential to obtain qualitative and quantitative agreement with the equilibrium PP-length distribution obtained from multi-chain simulations. By fitting this distribution, we are able to determine two of the three parameters of the model, which govern its equilibrium properties. This mapping is executed for four different linear polymers and for different molecular weights. The two parameters are found to depend on chemistry, but not on molecular weight. The model predicts a constant plateau modulus minus a correction inversely proportional to molecular weight. The value for well-entangled chains, with the parameters determined ab initio, lies in the range of experimental data for the materials investigated.
... For example, the steady-state viscosity decreases with decreasing friction of the junctions [91]. However, the mobile junctions are not connected with the background network by virtual springs, as often assumed in the study of junction fluctuations [41,65,92]. A lack of virtual springs enhances the deviation of mobile junctions from affine motion.) ...
... However, since it contains more fluctuations, it has a somewhat smaller front factor. ESFs were incorporated in a more recent version of the slip-link model by Schieber and Horio [92]. However, they showed analytically that the plateau modulus was the same as for the original slip-link model, with affine motion of entanglements. ...
... In the detailed model, our level of description is the number of strands Z, the number of Kuhn steps N i in each strand, the connector vector Q i between two slip-links, the vector X i pointing from the center of the confinement potential (the anchor) to the slip-link, and a dimensionless tensor n i describing the strength and shape of the confinement potential: Figure 2 illustrates the notation for the vectors used. The tensors {n i } are new compared with earlier work [92]. The free energy on this level of description is given by ...
We consider four criteria of acceptability for single-chain mean-field entangled polymer models: consistency with a multi-chain level of description, consistency with non-equilibrium thermodynamics, consistency with the stress-optic rule, and self-consistency between Green--Kubo predictions and linear viscoelastic predictions for infinitesimally driven systems. Each of these topics has been considered independently elsewhere. However, we are aware of no molecular entanglement model that satisfies all four criteria simultaneously. Here we show that an idea from Ronca and Allegra [J. Chem. Phys. bf 1975, it 63, 4990--4997], generalized to arbitrary flows, can be implemented in a slip-link model to create a model that does satisfy all four criteria. Aside from the direct benefits of agreement, the result modifies the relation between the measured plateau modulus $G_N^0$ and the entanglement molecular weight $M_mathrme$. If this implementation is correct, current estimates for $M_mathrme $ would require modification that brings their values more in line with estimates based on topological analysis of molecular dynamics simulations.
... The DSM is a single-chain mean-field mathematical model, which was first proposed by Schieber et al. (2003) and further developed by Khaliullin and Schieber (2009, 2010). ...
The discrete slip-link model (DSM) was developed to describe the dynamics of flexible polymer melts. The model is able to predict linear viscoelasticity of monodisperse linear, polydisperse linear, and branched systems. The model also shows good agreement with dielectric relaxation experiments, except for the single data set available for bidisperse linear systems with a small volume fraction of long chains. In this work, both shear and elongational flow predictions obtained using the DSM without parameter adjustment are shown. Model predictions for shear flow agree very well with experimental results. The DSM is able to capture the transient response as well as the steady-state viscosity. However, for elongational flow, agreement is unsatisfactory at large strains. The DSM captures the onset of strain hardening, but after a Hencky strain between 2 and 3, it predicts transient strain softening, whereas experiments show only monotonic growth. We explore a number of assumptions and approximations of the model and their effect on flow predictions. The approximations are related to the neglect of these phenomena, which are expected to be more sensitive in elongational flow: Finite extensibility, convective constraint release, and deformation of dangling ends. We indeed find that shear flow predictions are insensitive to these approximations, but elongational flow is affected. However, none of these effects is able to bring prediction in line with experiments. We conclude that the currently accepted view of entanglement dynamics is incomplete. 2013 The Society of Rheology. [http://dx.doi.org/10.1122/1.4788909]
... Our arguments in this paper are based on a single-chain description of polymer networks assuming that a single probe chain in the network can capture the important physics necessary for rheological modeling. As usual in single-chain treatments Panyukov, 1997, 2002;Likhtman, 2005;Schieber and Horio, 2010;Indei and Takimoto, 2010), we implement the fluctuations of network junctions by introducing harmonic virtual springs that connect the points on the chains which are supposed to be network junctions and the anchor points fixed at the macroscopic elastic background (i.e., matrix) that deforms affinely. By this method, it will become possible to investigate how the plateau modulus and the frequency profile of the dynamic modulus are affected by the fluctuations of junctions. ...
... Recently, Schieber and Horio (SH) studied the effects of junction fluctuations on the plateau modulus of entangled polymer melts on the basis of virtual work argument for a single-chain slip-link model (Schieber and Horio, 2010). A polymer melt is a network physically cross-linked by entanglements that can be created/annihilated by diffusion of the polymer chain along its primitive path (Doi and Edwards, 1986). ...
... 2), after briefly reviewing several multichain and single-chain models in terms of junction fluctuation, we discuss why this difference of the front factor occurs. An important point is that the plateau modulus derived by us (Schieber and Horio, 2010;Indei and Takimoto, 2010) includes all of the contribution from virtual springs in the stress. ...
We study the effects of fluctuations of crosslinking points, or junctions, on the dynamic mechanical and viscoelastic properties of polymer networks formed by multisticker associating polymers on the basis of a single-chain approach. Fluctuations of junctions are implemented by introducing virtual springs. We consider two possible cases for the treatment of virtual springs: the direct contribution from virtual springs is either neglected or included in the stress. We show that, if neglected, the fluctuation of junctions decreases (or softens) the dynamic modulus over a wide range of frequencies. This result agrees qualitatively with the result of several multichain models that predicts the decrease of the static or plateau modulus.We also show that the fluctuation accelerates the associative Rouse mode at low frequencies originating from the association/ dissociation process of stickers. These results are apparently reasonable, but it is expected that there are some errors arising from thermodynamical inconsistency due to the neglect of virtual-spring contribution from the total stress against the virtual work principle and the second law of thermodynamics. On the other hand, if the direct contribution from the virtual springs is included in the stress, thermodynamics is satisfied but the plateau modulus does not change, contrary to the multichain prediction. The softening occurs only at the low-frequency regime. Thus, each method has both merits and demerits, and hence the treatment of junction fluctuations in the framework of single-chain approaches requires care and further investigation.
... In this section, we consider this assumption and study how the motion of junctions around the affine motion changes the dynamic modulus by following the analysis by Schieber and Horio. 34 We will consider the time regime where the ͑segmental͒ Rouse relaxation time can be regarded as 0. This condition is attained by taking the low friction limit of free stickers. ...
... Free stickers are not attached to such virtual springs. This approach was recently developed in Ref. 34 for the purpose of investigating the effects of fluctuations in entanglement points of homopolymer melts by the slip-link model. The mathematical descriptions of the slip-link model and the present model for associating polymers are similar, and hence some of the following equations are common to those in Ref. 34. ...
... This approach was recently developed in Ref. 34 for the purpose of investigating the effects of fluctuations in entanglement points of homopolymer melts by the slip-link model. The mathematical descriptions of the slip-link model and the present model for associating polymers are similar, and hence some of the following equations are common to those in Ref. 34. ...
We have developed a single-chain theory that describes dynamics of associating polymer chains carrying multiple associative groups (or stickers) in the transient network formed by themselves and studied linear viscoelastic properties of this network. It is shown that if the average number N of stickers associated with the network junction per chain is large, the terminal relaxation time τ(A) that is proportional to τ(X)N(2) appears. The time τ(X) is the interval during which an associated sticker goes back to its equilibrium position by one or more dissociation steps. In this lower frequency regime ω<1/τ(X), the moduli are well described in terms of the Rouse model with the longest relaxation time τ(A). The large value of N is realized for chains carrying many stickers whose rate of association with the network junction is much larger than the dissociation rate. This associative Rouse behavior stems from the association/dissociation processes of stickers and is different from the ordinary Rouse behavior in the higher frequency regime, which is originated from the thermal segmental motion between stickers. If N is not large, the dynamic shear moduli are well described in terms of the Maxwell model characterized by a single relaxation time τ(X) in the moderate and lower frequency regimes. Thus, the transition occurs in the viscoelastic relaxation behavior from the Maxwell-type to the Rouse-type in ω<1/τ(X) as N increases. All these results are obtained under the affine deformation assumption for junction points. We also studied the effect of the junction fluctuations from the affine motion on the plateau modulus by introducing the virtual spring for bound stickers. It is shown that the plateau modulus is not affected by the junction fluctuations.
... For example, Likhtman has proposed two different expressions for stress, and it is not yet clear whether either of these is correct. We recently modified the slip-link model to allow such fluctuations [20], but utilized ideas from Rubenstein and Panyukov [19] to remove the contributions of the virtual springs to the stress tensor. We only mention the conclusions of that paper here, that we find that the static properties and stress-tensor expression of the model are largely unchanged, and that thermodynamic consistency can be recovered. ...
... Using some of the results from the paper of the previous section [20], we have derived a free energy for a slip-link model that is more like a tube model. In fact, the expression reduces to a free energy [17] that is used to justify the GLaMM tube model [5]. ...
