## 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, *R*_{i}, or from the relative distance between the beads, *l*_{i}:

*H*_{l} ({*l*_{i}}) = *C*_{l}
∑_{i=1}^{N} (*l*_{i} - *l*_{0})^{2}
/ 2

= *C*_{l}
∑_{i=1}^{N} (*|R*_{i} - *R*_{i-1}| - *l*_{0})^{2
}/ 2

where one typically chooses the constants *C*_{l},, *l*_{0} and *l*_{max} 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

*C*_{l} = 3 k_{B}T / <*l*^{2}>

where <*l*^{2}> 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** =

*C*

_{l}· Δ

**= [3 k**

*r*_{B}

*T*/ <Δ

*r*^{2}>] · Δ

**= [3 k**

*r*

_{B}

*T*/ N

*l*

_{0}

^{2}] · Δ

**= [3 k**

*r*_{B}

*T*/

*L*

_{c}l_{0}] · Δ

*r*where <Δ*r*^{2}> is the mean squared end-to-end distance of the entire chain
and *L*_{c} 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

- Gert Strobl,
*The Physics of Polymers*, 3^{rd}Edition, Heidelberg 2007