Kinetics and dynamic properties of equilibrium polymers

The statistical mechanics and scission-recombination mechanism of self-assembling linear micelles are investigated by Brownian dynamics using a newly proposed mesoscopic model representing the micelles as equilibrium polymer chains. A semidilute concentration regime, yet dynamically unentangled, is considered over a wide range of scission/recombination rates. We focus on the analysis of short and long time behaviors of the scission and recombination mechanisms. Our results show that at time scales larger than the life time of the average chain length, the kinetics is in agreement with the mean-field kinetic model proposed by Cates and Candau [J. Phys.: Condens. Matter 2, 6869 (1990)] provided the kinetic constants are estimated as effective ones. These values do take into account through a transmission coefficient that a fraction of scission/recombination events is correlated over a short time (diffusion controlled mechanism) and thus turn out to be ineffective reactive events by annihilation effects. By studying macroscopic relaxation phenomena such as the average micelle length evolution after a T jump, the monomer diffusion, and the zero shear stress relaxation function, we confirm that the effective kinetic constants found are indeed the relevant parameters when macroscopic relaxation is coupled to the kinetics of micelles.     

End-growth/evaporation living polymerization kinetics revisited

End-growth/evaporation kinetics in living polymer systems with "association-ready" free unimers (no initiator) is considered theoretically. The study is focused on the systems with long chains (typical aggregation number N ? 1) at long times. A closed system of continuous equations is derived and is applied to study the kinetics of the chain length distribution (CLD) following a jump of a parameter (T-jump) inducing a change of the equilibrium mean chain length from N(0) to N. The continuous approach is asymptotically exact for t ? t(1), where t(1) is the dimer dissociation time. It yields a number of essentially new analytical results concerning the CLD kinetics in some representative regimes. In particular, we obtained the asymptotically exact CLD response (for N ? 1) to a weak T-jump (ε = N(0)∕N - 1 ? 1). For arbitrary T-jumps we found that the longest relaxation time t(max?) = 1∕γ is always quadratic in N (γ is the relaxation rate of the slowest normal mode). More precisely t(max?)∝4N(2) for N(0) < 2N and t(max?)∝NN(0)∕(1 - N∕N(0)) for N(0) > 2N. The mean chain length N(n) is shown to change significantly during the intermediate slow relaxation stage t(1) ? t ? t(max?). We predict that N(n)(t)-N(n)(0)∝√t in the intermediate regime for weak (or moderate) T-jumps. For a deep T-quench inducing strong increase of the equilibrium N(n) (N ? N(0) ? 1), the mean chain length follows a similar law, N(n)(t)∝√t, while an opposite T-jump (inducing chain shortening, N(0) ? N ? 1) leads to a power-law decrease of N(n): N(n)(t)∝t(-1∕3). It is also shown that a living polymer system gets strongly polydisperse in the latter regime, the maximum polydispersity index r = N(w)∕N(n) being r? ≈ 0.77N(0)∕N ? 1. The concentration of free unimers relaxes mainly during the fast process with the characteristic time t(f) ～ t(1)N(0)∕N(2). A nonexponential CLD dominated by short chains develops as a result of the fast stage in the case of N(0) = 1 and N ? 1. The obtained analytical results are supported, in part, by comparison with numerical results found both previously and in the present paper.     

A model for the plateau in the relaxation modulus for linear chain polymer melts

A simple model is considered for the free energy associated with the relaxation of a linear chain polymer melt. The relaxation of the chain configurations results in an adjustment of the locations of the interchain contacts. The change in the distribution of the positions of the contacts between a pair of relaxing chains is hindered by the presence of other nearby chains in the melt. There is less configuration space available to the relaxing chains, due to the noncrossability of the chain backbones, than would be the case for phantom chains. This results in an increase in the free energy of the relaxing system. The analysis presented indicates that, given the free energy models considered, the extent of relaxation decreases as the length scale for relaxation increases. This results in a plateau in the relaxation modulus. This qualitative prediction of a plateau does not rely on the assumption of a specific mechanism for the chain dynamics, and is relatively insensitive to the form chosen for the terms in the free energy. If reasonable assumptions are made concerning the form of the free energy, then it is shown that the plateau which results depends on monomer length, Kuhn length, the monomer density for the melt, and, for solutions, the polymer concentration in a manner consistent with experimental data.     

Theoretical analysis of polydispersity in the nematic phase of self-assembled semiflexible chains

In recent simulations of semiflexible equilibrium polymers [X. Lu and J. T. Kindt, J. Chem. Phys. 120, 10328 (2004)] a roughly biexponential distribution of chain lengths was observed in the nematic phase. In this study, we show that a theory representing the nematic phase as an equilibrium mixture of randomly oriented chains below a critical length Lc and chains above Lc obeying a Gaussian orientational distribution reproduces this distribution qualitatively. The agreement between predicted and simulated phase boundaries is improved, with a narrower isotropic-nematic coexistence region, compared with a monoexponential chain distribution. We find, furthermore, that the critical length Lc scales as the inverse of monomer concentration, irrespective of the bond strength of the chains, but that at the phase boundary, the fraction of monomers found in disordered chains peaks at a certain bond strength.     

Polymerization in the crystalline state. X. Solid-state conversion of 6-aminocaproic acid to oriented nylon 6

When single crystals of 6-aminocaproic acid (ACA) are heated about 30°C below their melting point, polycondensation to nylon 6 takes place. The polymer crystallites are biaxially oriented towards each other and the relation between their orientation and that of the parent monomer crystal has been clarified. The kinetics of the process are characterized by three stages, (a) an induction period, (b) a stage in which monomer disappears at a constant rate while polymer of relatively low molecular weight is formed, and (c) a slow polycondensation of the polyamide chains after exhaustion of the monomer. Oligomer concentrations were below detectable limits at all stages of the process. Addition of monomer to the polyamide was retarded when ACA was kept from reaching its equilibrium vapor pressure (0.12 mm Hg at 170°C) by condensation on a cool surface or when an inert gas was admitted to the system. This was interpreted as suggesting that ACA is transported through the vapor phase to the propagating polyamide. A number of surfaces catalyzed the polycondensation of ACA vapor, but nylon 6 formed in this way on KCl crystals exhibited no preferred orientation. The linear dimer and trimer of ACA were also found to condense to nylon 6 in the crystalline state, although at a slower rate than the monomer. The solid-state polycondensation of these oligomers was accelerated when they were exposed to the vapor of the monomer. Solid-state polycondensation of single crystals of the linear dimer led also to biaxially oriented nylon 6.     

Model problems in depolymerization kinetics of chain oligomers: 2. Random chain scission

Kinetic equations related to the model of random chain scission (RCS) in the initial oligomer and intermediates (radicals and biradicals) have been solved, and the time dependence of concentrations, the average chain length, and its variance have been found. Despite of the difference in the sequence of formation between the intermediates (sequential reduction of chain length in the model of successive detachment of terminal monomer units and the preferential buildup of short chains in the RCS model), the changes in the average chain length in the two models are similar; i.e., they coincide with an appropriate change of the time scale. In the absence of bimolecular recombination, both the models are ergodic and nonmixing, a property that is due to the equidistant nature of the eigenvalue spectrum. Recombination leads to mixing and transition to equilibrium depolymerization. The change of the mode is determined by the ratio between the bond rupture, recombination, and evaporation rates.     

Power-law stage of slow relaxation in solutions with spherical micelles

General (independent of models selected for surfactant molecular aggregates) analytical relations are derived to describe the initial stage of slow relaxation in micellar solutions with spherical micelles. This stage precedes the final stage of the relaxation occurring via an exponential decay of disturbances with time. The relations obtained are applicable throughout the interval of micellar solution concentrations from the first to the second critical micellization concentration. It is shown that the initial stage is characterized by power laws of variations in the concentrations of monomers and micelles with time, these laws being different for the relaxation processes proceeding from above and below toward equilibrium values of micellar solution parameters. Relations are derived for the duration of this stage, and the effect of initial conditions is studied. Characteristic times of the power-law stage are determined and compared with the characteristic time of the final exponent-law relaxation stage. The behavior of these times is investigated at surfactant solution concentrations in the vicinity of, and noticeably above, the first critical micellization concentration. On the basis of the droplet and quasi-droplet thermodynamic models of surfactant molecular aggregates, numerical solutions are found for nonlinearized equations of slow relaxation for the time dependence of surfactant monomer concentrations at all stages of the slow relaxation. Numerical results obtained from the models are compared with the results of a general analytical study.     

Soft effective interactions between weakly charged polyelectrolyte chains

We apply extensive molecular dynamics simulations and analytical considerations in order to study the conformations and the effective interactions between weakly charged, flexible polyelectrolyte chains in salt-free conditions. We focus on charging fractions lying below 20%, for which case there is no Manning condensation of counterions and the latter can be thus partitioned in two states: those that are trapped within the region of the flexible chain and the ones that are free in the solution. We examine the partition of counterions in these two states, the chain sizes and the monomer distributions for various chain lengths, finding that the monomer density follows a Gaussian shape. We calculate the effective interaction between the centers of mass of two interacting chains, under the assumption that the chains can be modeled as two overlapping Gaussian charge profiles. The analytical calculations are compared with measurements from molecular dynamics simulations. Good quantitative agreement is found for charging fractions below 10%, where the chains assume coil-like configurations, whereas deviations develop for charge fraction of 20%, in which case a conformational transition of the chain towards a rodlike configuration starts to take place.     

Generalizing Rosenbluth's Algorithm to Include Along‐the‐Chain Intramolecular Energies

We incorporate the Boltzmann factors for inter‐monomer bending energy into the monomer growth direction choice in Rosenbluth's algorithm to model chains of arbitrary nearest‐neighbor rigidity. This allows for the consideration of compact (bent state lower in energy), free (straight and bent state equal in energy), or extended chains (bent state higher). We validate against, and compare to, various other results, showing very good agreement with known results for short chains and demonstrate the ability to model chains up to 500 segments long, far beyond the length at which the normal Rosenbluth method becomes unstable for reasonable nonzero bending energies. This approach is easily generalizable both to other energies determinable during chain growth, for example, polymers composed of more than one type of monomer with differing monomer interaction energies, as well as to other chain production algorithms. © 2019 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2019 , 57, 1684–1691     

梳状聚合物电解质的力学松弛时间

以苯乙烯 /马来酸酐共聚物为骨架 ,聚乙二醇单甲醚为侧链合成了 3种不同侧链长度的梳状高分子聚合物 ,制成了锂盐络合物薄膜 .动态力学性能研究结果表明 ,本体系是主链刚性、侧链较柔软的梳状接枝聚合物 .应用时间温度等效原理 ,选择Tα 作为参考温度 ,建立了主曲线 ,得到了移动因子图和等自由体积图 .Williams、Landel和Ferry(WLF)经验方程的两个参数C1、C2 随着盐浓度增加而增大 .若T0 =50℃作为参考温度 ,平均松弛时间lgτc 与锂盐浓度C呈线性关系 .随着侧链分子量增加 ,主曲线向高频率移动 ;平均松弛时间lgτn 与侧链分子量的关系也是线性的 ;另外随着盐的不同 ,主曲线也会产生移动 ,表明盐种类对松弛时间产生影响 .     

Polymerization of actin filaments coupled with adenosine triphosphate hydrolysis: Brownian dynamics and theoretical analysis

Polymerization dynamics of single actin filaments coupled with adenosine triphosphate (ATP) hydrolysis is investigated via both theoretical analysis and Brownian dynamics simulations. Brownian dynamics simulations have been applied recently to study the growth behaviors of long filaments as a function of the free actin monomer concentrations, C(T), which is found to be in agreement with the associated experiments. In the present study, both ATP cap length and length diffusivity are studied as a function of the free ATP-actin monomer concentrations, C(T). The exact analytical expressions are found to be in perfect consistency with Brownian dynamics simulations. Likewise, we find that the length diffusion coefficient is peaked near the critical concentration, C(T,cr). It is, therefore, expected that the dependence of length diffusivity on ATP-actin monomer concentrations is utilized to analyze the surprising experiments on the length fluctuations of individual actin filaments.     

Molecular Weight Distribution in Living Polymerization Proceeding with Reshuffling of Polymer Segments due to Chain Transfer to Polymer with Chain Scission, 2. Monte Carlo Simulation of Polymer Reshuffling Proceeding with Disproportionation of Chain Functionalities

The method for analyzing the reshuffling of polymer segments developed previously has been extended to systems involving the disproportionation of chain functionalities. The effect of interchain exchange reactions of this type, leading to the redistribution of chain lengths and of the chain functionalities (redistribution of living and dead chain ends), was analyzed by means of the Monte Carlo simulations. In the systems, in which no propagation occurs (monomer concentration equal to zero), a set of polymer chains containing one living and one dead end was taken as an initial material. A series of simulations were performed for systems with differing molecular weight distributions of the starting macromolecules. Uniform (no chain length distribution polymer – all chains are of the same length), Poisson, and the most probable (geometric) distributions were taken into consideration. Although the molecular weight distributions (MWDs) of functionally different chains of the same polymer were different apart from the eventual equilibrium conditions, the overall MWD was very close to that observed in analogous systems without disproportionation. The same was observed concerning MWDs in modeled polymerization systems, in which reshuffling and disproportionation accompanied propagation. Consequently, a method of estimating the ratio of rate constants of propagation and reshuffling (i. e. k_p /k _tr) in the relevant polymerization systems, using the observed polydispersity indexes, was proposed. The extent of disproportionation can be evaluated from the determined relationships of the polydispersity index and of the monofunctional chains fraction as functions of the average number of chain transformations.     

Modeling of intramolecular reactions of polymers: an efficient method based on Brownian dynamics simulations

By the traditional approach to the Brownian dynamics simulations of intrachain reactions of polymers, the initial chain conformation is sampled from the equilibrium distribution. A dynamic trajectory is carried out until a "collision" of the reactive groups takes place, i.e., the distance between their centers becomes less that a certain reaction radius. The average length of the trajectory is equal to the mean time tauF of a diffusion-controlled reaction. In this work we propose another computational scheme. The trajectory begins at the instant of collision and is carried out until the chain is relaxed. The length of the trajectory has the order of the relaxation time taurel of the distance between the reactive groups. For polymer systems with taurel < tauF, this scheme allows the computation of tauF with considerable gain in computational time. Using the present approach, we calculated the mean time of DNA cyclization for the molecule length in the range from 50 to 500 nm.     

On the inner structure and topology of clusters in two-component lipid bilayers. Comparison of monomer and dimer Ising models

It has been shown on model and biological systems that membrane clusters can affect in-plane membrane reactions and can control biochemical reaction cascades. Clusters of two-component phospholipid bilayers have been simulated by two Ising-type lattice models: the monomer and the dimer model. In each model the plane of one layer of the bilayer is represented by a triangular lattice, each site of which is occupied by an acyl chain of either a component 1 or a component 2 lipid molecule. The dimer model assumes that pairs of acyl chains (lipid molecules) are permanently connected, forming dimers on the lattice, while in the case of the monomer model this covalent connection between acyl chains is ignored. Phase diagrams of two-component phospholipid bilayers were successfully calculated by both models. In this work, we use Monte Carlo techniques to calculate thermodynamic averages of global and local characteristics of the largest component 2 cluster (such as outer/inner perimeter, percolation, minimal linear size, and local density) and compare the results obtained by the two models. A new method is developed to characterize the inner structure of the clusters. Each point of a cluster is classified based on its shortest distance (or depth) from the cluster's outer perimeter. Then local cluster properties, such as density, are calculated as a function of the depth. The depth analysis reveals that toward the cluster interior the average density usually decreases in midsize clusters and remains constant in very large clusters. On the basis of the simulations the following typical cluster topologies are identified at different cluster sizes and cooperativity parameter values: (i) branch-like, (ii) circular, (iii) band-like, and (iv) planar.We did not find qualitative differences between the cluster structures in the dimer and monomer model. However, at the same cluster size and cooperativity parameter value the cluster of the dimer model is more compact. The cluster properties of the dimer model are different from that of the monomer model because of the lower mixing entropy and higher formation energy of an elementary inner island.     

The glass transition temperature of polymer melts

We develop an analytic theory to estimate the glass transition temperature T(g) of polymer melts as a function of the relative rigidities of the chain backbone and side groups, the monomer structure, pressure, and polymer mass. Our computations are based on an extension of the semiempirical Lindemann criterion of melting to locate T(g) and on the use of the advanced mean field lattice cluster theory (LCT) for treating the thermodynamics of systems containing structured monomer, semiflexible polymer chains. The Lindemann criterion is translated into a condition for T(g) by expressing this relation in terms of the specific volume, and this free volume condition is used to calculate T(g) from our thermodynamic theory. The mass dependence of T(g) is compared to that of other characteristic temperatures of glass-formation. These additional characteristic temperatures are determined from the temperature variation of the LCT configurational entropy, in conjunction with the Adam-Gibbs model for long wavelength structural relaxation. Our theory explains generally observed trends in the variation of T(g) with polymer microstructure, and we find that T(g) can be tuned either upward or downward by increasing the length of the side chains, depending on the relative rigidities of the side groups and the chain backbone. The elucidation of the molecular origins of T(g) in polymer liquids should be useful in designing and processing new synthetic materials and for understanding the dynamics and controlling the preservation of biological substances.     

Combined use of NMR relaxation measurements and hydrodynamic calculations to study protein association. Evidence for tetramers of low molecular weight protein tyrosine phosphatase in solution

We describe a novel method for determining weak association constants of oligomeric protein complexes formed transiently under equilibrium conditions. This type of equilibrium process is recognized as being biologically important, but generally hard to study. Heteronuclear spin relaxation rates measured at multiple protein concentrations are analyzed using relaxation rates predicted from hydrodynamic calculations, yielding equilibrium constants and structural characterization of the protein complexes. The method was used to study the oligomerization equilibrium of bovine low molecular weight protein tyrosine phosphatase. X-ray structures of monomeric and dimeric forms of the protein have been reported previously. Using longitudinal and transverse (15)N relaxation rates measured at four different protein concentrations, we detected the monomer, dimer, and a previously unknown tetramer and measured the dissociation constants of the equilibria involving these species. A comparison of experimental and predicted relaxation rates for individual backbone amide (15)N spins enabled delineation of the tetramerization interface. The results suggest a novel concept for substrate modulation of enzymatic activity based on a "supramolecular proenzyme". The fast and reversible switching of the "supramolecular proenzyme" would have obvious advantages for the regulation of enzymes involved in cell signaling pathways.     

β-Lactoglobulin as an ideal random polymer coil

Unfolding of globular proteins by breaking the bonds and interactions that shape their structure, results in more flexible macromolecular chains with a relatively high percentage of apolar side group. Such properties may be beneficial for application in coatings and adhesives. It is then essential to define the flexibility of the chains at very high protein concentrations. As a measure for the stiffness of an unfolded protein chain the average persistence length , which varies with the experimental conditions, is introduced. We demonstrate that at very high protein concentrations can be predicted by using measuring data of dilute solutions, as is required by the available analytical techniques. In this study the globular protein β-lactoglobulin is used as a model. Its secondary structure can be disrupted by applying 98% formic acid as a solvent and by additional chemical modification of the thiol groups to break the two remaining covalent S---S bonds. The conformation of both unfolded structures, with and without chemical modification, can be described according to the Kuhn formalism, leading to a value of ±2 nm This value is determined by viscosimetry and by SANS and can be held representative for very concentrated protein layers where θ-conditions are assumed.     

Model problems in depolymerization kinetics of chain oligomers: 1. Successive detachment of monomer units from the chain end

For the successive detachment of monomer units from the end of a macroradical (one of the two main depolymerization mechanisms, of which the other is random chain scission), the time dependence of the monomers concentration, the average chain length, and width of the chain-length distribution of oligomers with allowance for the reverse reactions of monomer attachment to the radicals has been revealed without the use of the quasi-steady state approximation. The conditions under which the chain scission and the monomer evaporation occur at thermal equilibrium have been found.