## Bead-spring Model and Gaussian Chains

One of the most popular polymer models is the so called bead-spring model. The high-molecular weight polymer is described as N + 1 beads conneceted by N massless (harmonic) springs (see Figure below). Each bead may represent a subchain that contains several repeat units. In many cases, a simple harmonic potential is assumed, that is, the virtual springs follow Hooke's law of elasticity, which implies they are infinitley extendable with a linear elastic response. The potential energy of such a chain can be calculated from the positions of each bead in space, Ri, or from the relative distance between the beads, li:

Hl ({li}) = Cl i=1N (li - l0)2 / 2

= Cl i=1N (|Ri - Ri-1| - l0)2 / 2

where one typically chooses the constants Cl,, l0 and lmax in relation to the parameters ε, σ of the Leennard-Jones potential between the effective beads (if one allows only finite extension of the beads).

The simplest case is a Gaussian probably distribution of the beads, that is, it is assumed that the beads show ideal chain behavior. For this case, the spring constant reads

Cl = 3 kBT / <l2>

where <l2> is the mean squared end-to-end distance of two beads. If the polymer molecule is stretched out by applying a force to the chain ends, the tensile force that tries to pull back the beads to its equilibrium position is

F = Cl · Δr = [3 kBT / <Δr2>] · Δr

= [3 kBT / Nl02] · Δr = [3 kBT / Lc l0] · Δr

where <Δr2> is the mean squared end-to-end distance of the entire chain and Lc is the "contour length" of the chain, i.e. the end-to-end distance of the fully stretched chain. Note, the force constant is proportional to the absolute temperature. This has the important consequence that the elastic response of a polymer increases with increasing temperature. For this reason, a rubber band under constant stress contracts when it is heated instead of expanding as most materials do.

##### References
1. Gert Strobl, The Physics of Polymers, 3rd Edition, Heidelberg 2007
• ## Summary

The high-molecular weight polymer is described as N + 1 beads connected by N massless (harmonic) springs.

• Each bead may represent a subchain that contains several repeat units.

• For a Gaussian chain the force constant is proportional to the absolute temperature.

• The bead-spring model is used in Rouse/Zimm models for polymer dynamics.

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