Dynamics in dilute solution

LANGEVIN THEORY OF POLYMER DYNAMICS IN DILUTE SOLUTION ... 

Langevin Theory of Polymer Dynamics in Dilute Solution (Zwanzig) Large Tunnelling Corrections in Chemical Reaction Rates (Johnston) Lattices, Linear, Reversible Kinetics on, with Neighbor Effects... 

These results were corroborated by studies of labelled polystyrene in other solvents as toluene, styrene and pure tripropionin, at different temperatures. The main conclusions of these studies of polymer dynamics in dilute solution are the following ones ... 

Zwanzig, R. Langevin theory of polymer dynamics in dilute solution. Adv. Chem. Phys. 15, 325-331 (1969). 

The two new terms (relative to Eq. (1)) describe interac ons between the particles of interest and a bath. In a stochastic simulation of local polymer dynamics in dilute solution, the polymer chain is the systmi of interest and the solvent is the bath. Bath particles are not represented explicitly. Rather, the bath damps the motion of the particles with friction terms and supplies stochastic forces Nj which mimic the effect of collisions between solvent molecules and the polymer. Energy is not conserved in the system stochastic forces exdiange energy between the bath and the chain. In principle, P should now indude a hydrodynamic interaction term [14]. In practice, this term is usually neglected in simulatiorm of local dynamics and only the intramolecular potential energy of the polymer is used to determine the force. The stochastic forces in Eq. (2) are characterized by ... 

General overview of polymer dynamics in dilute solutions ... 

W. Zhu and M. D. Ediger. Viscosity dependence of polystyrene local dynamics in dilute solution. Macromolecules, 30 (1997), 1205-1210. 

B. Zimm. Dynamics of polymer molecules in dilute solutions viscoelasticity, low birefringence and dielectric loss. J Chem Phys 24 269-278, 1956. 

The deduction adopted is due to M. Planck (Thermodynamik, 3 Aufl., Kap. 5), and depends fundamentally on the separation of the gas mixture, resulting from continuous evaporation of the solution, into its constituents by means of semipermeable membranes. Another method, depending on such a separation applied directly to the solution, i.e., an osmotic process, is due to van t Hoff, who arrived at the laws of equilibrium in dilute solution from the standpoint of osmotic pressure. The applications of the law of mass-action belong to treatises on chemical statics (cf. Mel lor, Chemical Statics and Dynamics) we shall here consider only one or two cases which serve to illustrate some fundamental aspects of the theory. 

Zimm, BH, Dynamics of Polymer Molecules in Dilute Solution Viscoelasticity, Flow Birefringence and Dielectric Loss, Journal of Chemical Physics 24, 269, 1956. 

Hamau, L., Winkler, R. G., and Reineker, P., Influence of polydispersity on the dynamic structure factor of macromolecules in dilute solution, Macromolecules, 32, 5956, 1999. 

On macroscopic length scales, as probed for example by dynamic mechanical relaxation experiments, the crossover from 0- to good solvent conditions in dilute solutions is accompanied by a gradual variation from Zimm to Rouse behavior [1,126]. As has been pointed out earlier, this effect is completely due to the coil expansion, resulting from the presence of excluded volume interactions. 

The dynamics of highly diluted star polymers on the scale of segmental diffusion was first calculated by Zimm and Kilb [143] who presented the spectrum of eigenmodes as it is known for linear homopolymers in dilute solutions [see Eq. (77)]. This spectrum was used to calculate macroscopic transport properties, e.g. the intrinsic viscosity [145], However, explicit theoretical calculations of the dynamic structure factor [S(Q, t)] are still missing at present. Instead of this the method of first cumulant was applied to analyze the dynamic properties of such diluted star systems on microscopic scales. 

Since the transition from dilute to semi-dilute solutions exhibits the features of a second-order phase transition, the characteristic properties of the single- chain statics and dynamics observed in dilute solutions on all intramolecular length scales, are expected to be valid in semi-dilute solutions on length scales r < (c), whereas for r > E,(c) the collective properties should prevail [90]. 

One tool for working toward this objective is molecular mechanics. In this approach, the bonds in a molecule are treated as classical objects, with continuous interaction potentials (sometimes called force fields) that can be developed empirically or calculated by quantum theory. This is a powerful method that allows the application of predictive theory to much larger systems if sufficiently accurate and robust force fields can be developed. Predicting the structures of proteins and polymers is an important objective, but at present this often requires prohibitively large calculations. Molecular mechanics with classical interaction potentials has been the principal tool in the development of molecular models of polymer dynamics. The ability to model isolated polymer molecules (in dilute solution) is well developed, but fundamental molecular mechanics models of dense systems of entangled polymers remains an important goal. 

In this article I review some of the simulation work addressed specifically to branched polymers. The brushes will be described here in terms of their common characteristics with those of individual branched chains. Therefore, other aspects that do not correlate easily with these characteristics will be omitted. Explicitly, there will be no mention of adsorption kinetics, absorbing or laterally inhomogeneous surfaces, polyelectrolyte brushes, or brushes under the effect of a shear. With the purpose of giving a comprehensive description of these applications, Sect. 2 includes a summary of the theoretical background, including the approximations employed to treat the equifibrium structure of the chains as well as their hydrodynamic behavior in dilute solution and their dynamics. In Sect. 3, the different numerical simulation methods that are appHcable to branched polymer systems are specified, in relation to the problems sketched in Sect. 2. Finally, in Sect. 4, the appHcations of these methods to the different types of branched structures are given in detail. 

The results of this consideration may be summarized as follows. The study of global properties of macromolecules in dilute solutions by means of static and dynamic LS and by viscometry allows the determination of the molar mass and four differently defined equivalent sphere radii, R, and (see... 

Nuclear magnetic resonance spectroscopy of dilute polymer solutions is utilized routinely for analysis of tacticlty, of copolymer sequence distribution, and of polymerization mechanisms. The dynamics of polymer motion in dilute solution has been investigated also by protoni - and by carbon-13 NMR spectroscopy. To a lesser extent the solvent dynamics in the presence of polymer has been studied.Little systematic work has been carried out on the dynamics of both solvent and polymer in the same systan. 

In dilute solution, the behavior of macromolecules is quite different to that of common low-molecular-weight molecules. For example, the shape of a macro-molecular coil is subject to permanent dynamic changes, and the coils are in a more or less swollen state when compared to their unperturbed (solid state)... 

A representative example for the information extracted from a TRMC experiment is the work of Prins et al. [141] on the electron and hole dynamics on isolated chains of solution-processable poly(thienylenevinylene) (PTV) derivatives in dilute solution. The mobility of both electrons and holes as well as the kinetics of their bimolecular recombination have been monitored by a 34-GHz microwave field. It was found that at room temperature both electrons and holes have high intrachain mobilities of fi = 0.23 0.04 cm A s and = 0.38 0.02 cm / V s V The electrons become trapped at defects or impurities within 4 ps while no trapping was observed for holes. The essential results are (1) that the trap-free mobilities of electrons and holes are comparable and (2) that the intra-chain hole mobility in PTV is about three orders of magnitude larger than the macroscopic hole mobility measured in PTV devices [142]. This proves that the mobilities inferred from ToF and FET experiments are limited by inter-chain hopping, in addition to possible trapping events. It also confirms the notion that there is no reason why electron and hole mobilities should be principally different. The fact... 

Prins P, Candeias LP, van Breemen AJJM, Sweelssen J, Herwig PT, Schoo HFM, Siebbeles LDA (2005) Electron and hole dynamics on isolated chains of a solution-processable poly (thienylenevinylene) derivative in dilute solution. Adv Mater 17 718... 

Whereas polymer transport in dilute solution has been widely studied, investigations of polymer transport in concentrated solutions have been reported only infrequently. When solutions of high molecular weight polymers are concentrated in a way that intermolecular interactions between individual molecules occur, it becomes increasingly evident that both the static and dynamic properties of the macromolecules may be markedly altered compared to their behaviour in dilute solution l). 

Fluorescence is measured in dilute solution of model compounds for polymers of 2,6-naphthalene dicarboxylic acid and eight different glycols. The ratio of excimer to monomer emission depends on the glycol used. Studies as functions of temperature and solvent show that, in contrast with the analogous polyesters in which the naphthalene moiety is replaced with a benzene ring, there can be a substantial dynamic component to the excimer emission. Extrapolation to media of infinite viscosity shows that in the absence of rotational isomerism during the lifetime of the singlet excited state, there is an odd-even effect In the series in which the flexible spacers differ in the number of methylene units, but not in the series in which the flexible spacers differ in the number of oxyethylene units. 

Information on dynamic behavior at infinite dilution has greatly increased in recent years through the development of instruments for precise determination of G (co) and G"(os) in dilute solutions (112, 113). Ferry and coworkers have published extensively on the intrinsic dynamic moduli,... 

Verdier,P.H., Stockmayer, W.H. Monte Carlo calculations on the dynamics of polymers in dilute solution. J. Chem. Phys. 36, 227-235 (1962). See also Verdier,P.H. Monte Carlo studies of lattice-model polymer chains. 1. Correlation functions in the statistical-bead model. J. Chem. Phys. 45,2118-2121 (1966). 

As is well known, dynamic properties of polymer molecules in dilute solution are usually treated theoretically by Brownian motion methods. Tn particular, the standard approach is to use a Fokker-Planck (or Smoluchowski) equation for diffusion of the distribution function of the polymer molecule in its configuration space. 

Dynamics, Polymer, Langevin Theory of, in Dilute Solution (Zwanzig) 15 325... 

In this respect, another insufficiency of Lodge s treatment is more serious, viz. the lack of specification of the relaxation times, which occur in his equations. In this connection, it is hoped that the present paper can contribute to a proper valuation of the ideas of Bueche (13), Ferry (14), and Peticolas (13). These authors adapted the dilute solution theory of Rouse (16) by introducing effective parameters, viz. an effective friction factor or an effective friction coefficient. The advantage of such a treatment is evident The set of relaxation times, explicitly given for the normal modes of motion of separate molecules in dilute solution, is also used for concentrated systems after the application of some modification. Experimental evidence for the validity of this procedure can, in principle, be obtained by comparing dynamic measurements, as obtained on dilute and concentrated systems. In the present report, flow birefringence measurements are used for the same purpose. 

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