The calculation of Euclidean distance between points is generalized to one-dimensional objects such as strings or polymers. Necessary and sufficient conditions for the minimal transformation between two polymer configurations are derived. Transformations consist of piecewise rotations and translations subject to Weierstrass-Erdmann corner conditions. Numerous examples are given for the special cases of one and two links. The transition to a large number of links is investigated, where the distance converges to the polymer length times the mean root square distance (MRSD) between polymer configurations, assuming curvature and non-crossing constraints can be neglected. Applications of this metric to protein folding are investigated. Potential applications are also discussed for structural alignment problems such as pharmacophore identification, and inverse kinematic problems in motor learning and control.
The stretchability of polymeric materials is critical to many applications such as stretchable electronics and soft robotics, yet the stretchability of conventional cross-linked linear polymers is limited by the entanglements between polymer chains. We show using molecular dynamics simulations that cross-linked ring polymers are significantly more stretchable than cross-linked linear polymers. Compared to linear polymers, the entanglements between ring polymers do not act as effective cross-links. As a result, the stretchability of cross-linked ring polymers is determined by the maximum extens…
Using video-microscopy imaging we demonstrate the existence of a short-ranged equilibrium attraction between heavy silica colloids diffusing on soft surfaces of cross-linked polymer gels. The inter-colloid potential can be tuned by changing the gel stiffness or by coating the colloids with a polymer layer. On sufficiently soft substrates, the interaction induced by the polymer matrix leads to large-scale colloidal aggregation. We correlate the in-plane interaction with a colloid-surface attraction.
Using data from contact maps of the DNA-polymer of $E. Coli$ (at kilobase pair resolution) as an input to our model, we introduce cross-links between monomers in a bead-spring model of a ring polymer at very specific points along the chain. By suitable Monte Carlo Simulations, we show that the presence of these cross-links leads to a particular architecture and organization of the chain at large (micron) length scales of the DNA. We also investigate the structure of a ring polymer with an equal number of cross-links at random positions along the chain. We find that though the polymer does get …
We have discovered unusual behavior of polymer coils in a binary solvent (nitroethane+isooctane) near the critical temperature of demixing. The exceptionally close refractive indices of the solvent components make the critical opalescence relatively weak, thus enabling us to simultaneously observe the Brownian motion of the polymer coils and the diverging correlation length of the critical fluctuations. The polymer coils exhibit a collapse-reswelling-expansion-collapse transition upon approaching the critical temperature. While the first stage (collapse-reswelling) can be explained by the theo…
We investigate the role of fluctuations in single molecule measurements of torque-link ($t-lk$) curves. For semiflexible polymers of finite persistence length (i.e. polymers with contour length $L$ comparable to the persistence length $L_P$), the torque versus link curve in the constant torque (isotorque) ensemble is distinct from the one in the constant link (isolink) ensemble. Thus, one encounters the conceptually interesting issue of a ``free energy of transition'' in switching ensembles while making torque-link measurements. We predict the dependence on the semiflexibility parameter $β= L/…
The selective solute partitioning within a polymeric network is of key importance to applications in which controlled release or uptake of solutes in a responsive hydrogel is required. In this work we investigate the impact of cross-links on solute adsorption in a swollen polymer network by means of all-atom, explicit-water molecular dynamics simulations. We focus on a representative network subunit consisting of poly($N$-isopropylacrylamide) (PNIPAM) and $N$,$N'$-methylenebisacrylamide (BIS/MBA) cross-linker types. Our studied system consists of one BIS-linker with four atactic PNIPAM chains …
We present a numerical study of the slip link model introduced by Likhtman for describing the dy- namics of dense polymer melts. After reviewing the technical aspects associated with the implemen- tation of the model, we extend previous work in several directions. The dependence of the relaxation modulus with the slip link density and the slip link stiffness is reported. Then the nonlinear rheolog- ical properties of the model, for a particular set of parameters, are explored. Finally, we introduce excluded volume interactions in a mean field such as manner in order to describe inhomogeneous s…
We investigate the influence of a non-ionic surfactant like polymer on phospholipid vesicles. Our results from cryogenic transmission electron microscopy (cryo-TEM), dynamic light scattering (DLS), small angle neutron and X-ray scattering (SANS/SAXS), identifies the existence of multilayer vesicles and an increase in size of the vesicles in presence of the polymers. We present a generalized model to obtain the bending rigidity from neutron spin echo spectroscopy (NSE) data for multilayer vesicles. We demonstrated that polymers are trapped in the lipid bilayer, causing a partial disruption in t…
We explore the effect of random permanent cross-links on a system of directed polymers confined between two planes with their end-points free to slide on them. We treat the cross-links as quenched disorder and we use a semimicroscopic replica field theory to study the structure and elasticity of this system. Upon increasing the cross-link density, we get a continuous gelation transition signaled by the emergence of a finite in-plane localization length. The distribution of localization length turns out to depend on the height along the preferred direction of the directed polymers. The gelation…
The Kremer-Grest (KG) polymer model is a standard model for studying generic polymer properties in Molecular Dynamics simulations. It owes its popularity to its simplicity and computational efficiency, rather than its ability to represent specific polymers species and conditions. Here we show, that by tuning the chain stiffness it is possible to adapt the KG model to model melts of real polymers. In particular, we provide mapping relations from KG to SI units for a wide range of commodity polymers. The connection between the experimental and the KG melts is made at the Kuhn scale, i.e. at the …
Low-density "equilibrium" gels that consist of a percolated, kinetically arrested network of colloidal particles and are resilient to aging can be fabricated by restricting the number of effective bonds that form between the colloids. Valence-restricted patchy particles have long served as one archetypal example of such materials, but equilibrium gels can also be realized through a synthetically simpler and scalable strategy that introduces a secondary linker, such as a small ditopic molecule, to mediate the bonds between the colloids. Here, we consider the case where the ditopic linker molecu…
Cross-linked polymer networks with orientational order constitute a wide class of soft materials and are relevant to biological systems (e.g., F-actin bundles). We analytically study the nonlinear force-extension relation of an array of parallel-aligned, strongly stretched semiflexible polymers with random cross-links. In the strong stretching limit, the effect of the cross-links is purely entropic, independent of the bending rigidity of the chains. Cross-links enhance the differential stretching stiffness of the bundle. For hard cross-links, the cross-link contribution to the force-extension …
In this paper we study a system of entangled chains that bear reversible cross-links in a melt state. The cross-links are tethered uniformly on the backbone of each chain. A slip-link type model for the system is presented and solved for the relaxation modulus. The effects of entanglements and reversible cross-linkers are modelled as discrete form of constraints that influence the motion of the primitive path. In contrast to a non-associating entangled system the model calculations demonstrate that the elastic modulus has a much higher first plateau and a delayed terminal relaxation. These eff…
Recent experiments have demonstrated that the nonlinear elasticity of in vitro networks of the biopolymer actin is dramatically altered in the presence of a flexible cross-linker such as the abundant cytoskeletal protein filamin. The basic principles of such networks remain poorly understood. Here we describe an effective medium theory of flexibly cross-linked stiff polymer networks. We argue that the response of the cross-links can be fully attributed to entropic stiffening, while softening due to domain unfolding can be ignored. The network is modeled as a collection of randomly oriented rod…
Tuning the heat flow is fundamentally important for the design of advanced functional materials. Here, polymers are of particular importance because they provide different pathways for the energy transfer. More specifically, the heat flow between two covalently bonded monomers is over 100 times faster than between the two non-bonded monomers interacting via van der Waals (vdW) forces. Therefore, the delicate balance between these two contributions often provide a guiding tool for the tunability in thermal transport coefficient k of the polymeric materials. Traditionally most studies have inves…
Motivated by the structure of networks of cross-linked cytoskeletal biopolymers, we study the orientationally ordered phases in two-dimensional networks of randomly cross-linked semiflexible polymers. We consider permanent cross-links which prescribe a finite angle and treat them as quenched disorder in a semi-microscopic replica field theory. Starting from a fluid of un-cross-linked polymers and small polymer clusters (sol) and increasing the cross-link density, a continuous gelation transition occurs. In the resulting gel, the semiflexible chains either display long range orientational order…
In-vivo DNA organization at large length scales ($\sim 100nm$) is highly debated and polymer models have proved useful to understand the principle of DNA-organization. Here, we show that $<2$% cross-links at specific points in a ring polymer can lead to a distinct spatial organization of the polymer. The specific pairs of cross-linked monomers were extracted from contact maps of bacterial DNA. We are able to predict the structure of 2 DNAs using Monte Carlo simulations of the bead-spring polymer with cross-links at these special positions. Simulations with cross-links at random positions along…
We study theoretically situations where competition arises between an interdiffusion process and a cross-linking chemical reaction at interfaces between pieces of the same polymer material. An example of such a situation is observable in the formation of latex films, where, in the presence of a cross-linking additive, colloidal polymer particles initially in suspension come at contact as the solvent evaporates, and, optimally, coalesce into a continuous coating. We considered the low cross-link density situation in a previous paper (A. Aradian, E. Raphael, P.-G. de Gennes, Macromolecules 33, 9…
Polydisperse polymer networks with irregular topologies: Mechanics of cross-link distributions
Materials & Manufacturing Directorate, Air Force Research Laboratory（空军研究实验室材料制造局）；National Research Council (NRC) Research Associateship Programs, The National Academies of Sciences, Engineering, and Medicine（国家科学院、工程院和医学院国家研究委员会（NRC）研究助理项目）；Sandia National Laboratories（桑迪亚国家实验室）
Indentation is a common experimental technique to study the mechanics of polymeric materials. The main advantage of using indentation is because this provides a direct correlation between the microstructure and the small-scale mechanical response, which is otherwise difficult within the standard tensile testing. Here, majority of studies have investigated hydrogels, microgels and/or elastomers. However, a lesser investigated system is the indentation in highly cross-linked polymer (HCP) networks, where the complex network structure plays a key role in dictating their physical properties. In th…
We study theoretically a model system of a transient network of microemulsion droplets connected by telechelic polymers and explain recent experimental findings. Despite the absence of any specific interactions between either the droplets or polymer chains, we predict that as the number of polymers per drop is increased, the system undergoes a first order phase separation into a dense, highly connected phase, in equilibrium with dilute droplets, decorated by polymer loops. The phase transition is purely entropic and is driven by the interplay between the translational entropy of the drops and …
We use molecular dynamics simulations to investigate the microscopic and macroscopic response of model polymer networks to uniaxial elongations. By studying networks with strands lengths ranging from $N_s=20$ to 200 we cover the full crossover from cross-link to entanglement dominated behavior. Our results support a recent version of the tube model which accounts for the different strain dependence of chain localization due to chemical cross-links and entanglements.
Polymer rings in solution are either permanently entangled or not. Permanent topological restrictions give rise to additional entropic interactions apart from the ones arising due to mere chain flexibility or excluded volume. Conversely, entangled polymer rings systems may be formed by closing randomly entangled flexible linear chains. The dependance of linking numbers between randomly entangled rings on the chain length, more specifically the second topological moment $<n^2>$, i.e. the average squared linking number, may be determined. In this paper, an approach recently discussed in mathemat…
We investigate the network topologies of an ensemble of telechelic polymers. The telechelic polymers serve as links between nodes, which consist of aggregates of their telechelic endgroups. Our analysis shows that the degree distribution is bimodal and consists of two Poissonian distributions with different average degrees. The number of nodes in each of them as well as the distribution of links depends on temperature. By comparing the eigenvalue spectra of the simulated network gels with those of reconstructed networks, the most likely ttopology at each temperature is determined.
Understanding and controlling self-assembly processes at multiple length scales is vital if we are to design and create advanced materials. In particular, our ability to organise matter on the nanoscale has advanced considerably, but still lags far behind our skill in manipulating individual molecules. New tools allowing controlled nanoscale assembly are sorely needed, as well as the physical understanding of how they work. Here, we report a new method for the production of highly anisotropic nanoparticles with controlled dimensions based on a morphological transformation process (MORPH for sh…
Molecular dynamics simulations confirm recent extensional flow experiments showing ring polymer melts exhibit strong extension-rate thickening of the viscosity at Weissenberg numbers $Wi<<1$. Thickening coincides with the extreme elongation of a minority population of rings that grows with $Wi$. The large susceptibility of some rings to extend is due to a flow-driven formation of topological links that connect multiple rings into supramolecular chains. Links form spontaneously with a longer delay at lower $Wi$ and are pulled tight and stabilized by the flow. Once linked, these composite object…
Extensible links in a broad class of single polymer chain models
Sandia National Laboratories（桑迪亚国家实验室）；Air Force Research Laboratory（空军研究实验室）；National Research Council (NRC) Research Associateship Programs, The National Academies of Sciences, Engineering, and Medicine（美国国家科学院、工程院和医学院国家研究委员会研究助理项目）
A fundamental theory is presented for the mechanical response of polymer networks undergoing large deformation which seamlessly integrates statistical mechanical principles with macroscopic thermodynamic constitutive theory. Our formulation permits the consideration of arbitrary polymer chain behaviors when interactions among chains may be neglected. This careful treatment highlights the naturally occurring correspondence between single-chain mechanical behavior and the equilibrium distribution of chains in the network, as well as the correspondences between different single-chain thermodynami…
We develop topological methods for characterizing the relationship between polymer chain entanglement and bulk viscoelastic responses. We introduce generalized Linking Number and Writhe characteristics that are applicable to open linear chains. We investigate the rheology of polymeric chains entangled into weaves with varying topologies and levels of chain density. To investigate viscoelastic responses, we perform non-equilibrium molecular simulations over a range of frequencies using sheared Lees-Edwards boundary conditions. We show how our topological characteristics can be used to capture k…
Semiflexible polymers such as filamentous actin play a vital role in the mechanical behavior of cells, yet the basic properties of cross-linked F-actin networks remain poorly understood. To address this issue, we have performed numerical studies of the linear response of homogeneous and isotropic two-dimensional networks subject to an applied strain at zero temperature. The elastic moduli are found to vanish for network densities at a rigidity percolation threshold. For higher densities, two regimes are observed: one in which the deformation is predominantly affine and the filaments stretch an…
Shape transformations in driven and damped molecular chains are considered. Closed chains of weakly coupled molecular subunits under the action of spatially homogeneous time-periodic external field are studied. The coupling between the internal excitations and the bending degrees of freedom of the chain modifies the local bending rigidity of the chain. In the absence of driving the array takes a circular shape. When the energy pumped into the system exceeds some critical value the chain undergoes a non-equilibrium phase transition: the circular shape of the aggregate becomes unstable and the c…
Simulations in which a globular ring polymer with delocalized knots is separated in two interacting loops by a slipping link, or in two non-interacting globuli by a wall with a hole, show how the minimal crossing number of the knots controls the equilibrium statistics. With slipping link the ring length is divided between the loops according to a simple law, but with unexpectedly large fluctuations. These are suppressed only for unknotted loops, whose length distribution shows always a fast power law decay. We also discover and explain a topological effect interfering with that of surface tens…
Writhe-Based Polymer Link Classification Using Machine Learning
School of Physics and Astronomy, The University of Edinburgh（物理与天文学学院，爱丁堡大学）；School of Mathematics, The University of Edinburgh（数学学院，爱丁堡大学）；MRC Human Genetics Unit, Institute of Genetics and Cancer, University of Edinburgh（人类遗传学单位，遗传与癌症研究所，爱丁堡大学）
Using canonical Monte Carlo simulations, we introduce a new numerical procedure for comparing the entropic exponents of polymers with different constraints and/or topologies. Setting up competitions between polymer segments which can exchange monomers according to their free energies, we obtain the universal exponents of partition functions, independently of any knowledge of the non-universal part. The method is successfully tested for closed polymer loops decorated with sliding rings. We also investigate the entropic exponents of loops with a fixed knot type, in which case we are limited by s…
Shape memory polymers (SMPs) are materials with a great potential for future use in smart materials and structures. When heated from cold state (below the transformation temperature, which can either be the glass transition temperature or the melting temperature of the polymer) to hot state (above the transformation temperature) they undergo transformation which can be compared with martensitic transformation of shape memory alloys. This process induces great changes of the mechanical properties and some shape memory phenomenon can be observed. This study is an experimental evaluation of the m…
Molecular dynamics simulations are used to study structure formation in simple model polymer chains that are subject to excluded volume and torsional interactions. The changing conformations exhibited by chains of different lengths under gradual cooling are followed until each reaches a state from which no further change is possible. The interactions are chosen so that the true ground state is a helix, and a high proportion of simulation runs succeed in reaching this state; the fraction that manage to form defect-free helices is a function of both chain length and cooling rate. In order to dem…
The statistical mechanics of single polymer knots is studied using Monte Carlo simulations. The polymers are considered on a cubic lattice and their conformations are randomly changed with the help of pivot transformations. After each transformation, it is checked if the topology of the knot is preserved by means of a method called pivot algorithm and excluded area (in short PAEA) and described in a previous publication of the authors. As an application of this method the specific energy, the radius of gyration and heat capacity of a few types of knots are computed. The case of attractive shor…
We report extensive numerical simulations of different models of 2D polymer rings with internal elasticity. We monitor the dynamical behavior of the rings as a function of the packing fraction, to address the effects of particle deformation on the collective response of the system. In particular, we compare three different models: (i) a recently investigated model [Gnan \& Zaccarelli, Nat. Phys. 15, 683 (2019)], where an inner hertzian field providing the internal elasticity acts on the monomers of the ring, (ii) the same model where the effect of such a field on the center of mass is balanced…
We successfully extend a multiscale simulation (MSS) method to nonisothermal well-entangled polymer melt flows between two coaxial cylinders. In the multiscale simulation, the macroscopic flow system is connected to a number of microscopic systems through the velocity gradient tensor, stress tensor and temperature. At the macroscopic level, in addition to the momentum balance equation, we consider the energy balance equation, where heat generation plays an important role not only in the temperature distribution but also in the flow profile. At the microscopic level, a dual slip-link model is e…
Using a bead-spring model of bacterial DNA polymers of {\em C. crescentus} and {\em E. coli} we show that just $33$ and $38$ effective cross-links at special positions along the chain contour of the DNA can lead to the large-scale organization of the DNA polymer, where confinement effects of the cell walls play a key role in the organization. The positions of the $33$ cross-links along the chain contour are chosen from the contact map data of {\em C. crescentus}. We represent $1000$ base pairs as a coarse-grained monomer in our bead-spring flexible ring polymer model of the DNA. Thus a $4017$ …
Star polymer undergoes a transformation from a group of loosely coupled chains at low number of arms to a dense hairy colloid at their high number. This change affects solubility, aggregation and rheological behavior and is of much practical interest. We study the range of size and shape properties of the star molecule and of its individual arms upon this transformation. Theoretical calculations are based on a continuous chain model and are performed in the first order in $ε=4-d$. Computer simulations are done by the dissipative particle dynamics and in part by Monte Carlo method and the resul…
Limited-valency colloidal particles can self-assemble into polymeric structures analogous to molecules. While their structural equilibrium properties have attracted wide attention, insight into their dynamics has proven challenging. Here, we investigate the polymerization dynamics of semiflexible polymers in two dimensions (2D) by direct observation of assembling divalent particles, bonded by critical Casimir forces. The reversible critical Casimir force creates living polymerization conditions with tunable chain dissociation, association and bending rigidity. We find that unlike dilute polyme…
We study by Monte Carlo simulations and scaling analysis two models of pairs of confined and dense ring polymers in two dimensions. The pair of ring polymers are modelled by squared lattice polygons confined within a square cavity and they are placed in relation to each other to be either unlinked or linked in the plane. The observed rich phase diagrams of the two models reveal several equilibrium phases separated by first order and continuous phase boundaries whose critical nature depend on this reciprocal placements. We estimate numerically the critical exponents associated with the phase bo…
We link the Brownian non-Gaussian diffusion of a polymer center of mass to a microscopic cause: the polymerization/depolymerization phenomenon occurring when the polymer is in contact with a monomer chemostat. The anomalous behavior is triggered by the polymer critical point, separating the dilute and the dense phase in the grand canonical ensemble. In the mean-field limit we establish contact with queuing theory and show that the kurtosis of the polymer center of mass diverges alike a response function when the system becomes critical, a result which holds for general polymer dynamics (Zimm, …
For optimal processing and design of entangled polymeric materials it is important to establish a rigorous link between the detailed molecular composition of the polymer and the viscoelastic properties of the macroscopic melt. We review current and past computer simulation techniques and critically assess their ability to provide such a link between chemistry and rheology. We distinguish between two classes of coarse-graining levels, which we term coarse-grained molecular dynamics (CGMD) and coarse-grained stochastic dynamics (CGSD). In CGMD the coarse-grained beads are still relatively hard, …
We study the Directed Polymer model subject to a particular form of disorder, $η(x,t)=η_X(x) η_T(t)$, recently proposed in biological applications. We find that two new universality classes arise, depending on the the lattice geometry. Using an intermediate model linking the two different orientations continuously, we find that there is a phase transition separating two distinct scaling phases. For both phases we get a reasonable understanding of the nature and values of the exponents, corroborated with numerical results.
We explore the effect of an attractive interaction between parallel-aligned polymers, which are perpendicularly grafted on a substrate. Such an attractive interaction could be due to, e.g., reversible cross-links. The competition between permanent grafting favoring a homogeneous state of the polymer brush and the attraction, which tends to induce in-plane collapse of the aligned polymers, gives rise to an instability of the homogeneous phase to a bundled state. In this latter state the in-plane translational symmetry is spontaneously broken and the density is modulated with a finite wavelength…
We define the local periodic linking number, LK, between two oriented closed or open chains in a system with three-dimensional periodic boundary conditions. The properties of LK indicate that it is an appropriate measure of entanglement between a collection of chains in a periodic system. Using this measure of linking to assess the extent of entanglement in a polymer melt we study the effect of CReTA algorithm on the entanglement of polyethylene chains. Our numerical results show that the statistics of the local periodic linking number observed for polymer melts before and after the applicatio…
We investigate theoretically the effect of polymer tension on the collective behavior of reversibly binding cross-links. For this purpose, we employ a model of two weakly bending wormlike chains aligned in parallel by a tensile force, with a sequence of inter-chain binding sites regularly spaced along the contours. Reversible cross-links attach and detach at the sites with an affinity controlled by a chemical potential. In a mean-field approach, we calculate the free energy of the system and find the emergence of a free-energy barrier which controls the reversible (un)binding. The tension affe…
