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The distance between two units separated by a sufficient number of bonds in a flexible polymer chain follows a | The distance between two units separated by a sufficient number of bonds in a flexible polymer chain follows a | ||
[[Gaussian distribution]]. Therefore, the [[polymers | polymer]] chain can be described by a | [[Gaussian distribution]]. Therefore, the [[polymers | polymer]] chain can be described by a | ||
[[Coarse graining | coarse-grained model]] of N successive Gaussian segments. A dynamical model is consistent with this description and also with [[Boltzmann distribution | Boltzmann's distribution of energies]] if it assumes that the forces between units jointed by the Gaussian segments are proportional to their distances ([[Harmonic spring approximation |Hookean springs]]). The Rouse model | [[Coarse graining | coarse-grained model]] of N successive Gaussian segments. A dynamical model is consistent with this description and also with [[Boltzmann distribution | Boltzmann's distribution of energies]] if it assumes that the forces between units jointed by the Gaussian segments are proportional to their distances ([[Harmonic spring approximation |Hookean springs]]). The Rouse model (Ref. 1) describes a polymer chain as a set of N coupled [[Harmonic spring approximation | harmonic springs]]. The mathematical treatment of this model decomposes the global motion in a set of N-1 independent "normal modes". The [[time-correlation function]] of each one of these modes decays as a single exponential, characterized by a "Rouse relaxation time". The Rouse relaxation times are proportional to <math>N^2</math> (N is proportional to the chain length or the polymer molecular weight). The first Rouse mode represents the slowest internal motion of the chain. The p-''th'' Rouse relaxation time is proportional to <math>1/p^2</math>. | ||
In dilute solution, the relaxation times obtained from the Rouse model are not correct due to the presence of hydrodynamic interactions between units, introduced by Zimm | In dilute solution, the relaxation times obtained from the Rouse model are not correct due to the presence of hydrodynamic interactions between units, introduced by Zimm (Ref. 2). With this correction, the Zimm-Rouse relaxation times are proportional to <math>N^{3/2}</math>. This result is only valid for unperturbed (or [[Theta solvent | theta]]) solvents. For solvent of good quality, excluded volume interactions have also to be accounted. However, the scheme of normal modes is maintained (Ref. 3). | ||
In non-dilute solutions and melts hydrodynamic interactions and excluded volume effects are screened out and the Rouse model is correct, though only in the scale of short times and distances. | In non-dilute solutions and melts hydrodynamic interactions and excluded volume effects are screened out and the Rouse model is correct, though only in the scale of short times and distances. | ||
==References== | ==References== | ||
#[http://dx.doi.org/10.1063/1.1699180 Prince E. Rouse, Jr. "A Theory of the Linear Viscoelastic Properties of Dilute Solutions of Coiling Polymers", Journal of Chemical Physics '''21''' pp. 1272-1280 (1953)] | |||
#[http://dx.doi.org/10.1063/1.1742462 Bruno H. Zimm "Dynamics of Polymer Molecules in Dilute Solution: Viscoelasticity, Flow Birefringence and Dielectric Loss", Journal of Chemical Physics '''24''' pp. 269-278 (1956)] | |||
#[http://www.oup.com/uk/catalogue/?ci=9780198520337 M. Doi and S. F. Edwards "The Theory of Polymer Dynamics" Oxford University Press (1988)] | |||
[[Category: Models]] | [[Category: Models]] | ||
[[category: polymers]] | [[category: polymers]] | ||