Freely movable chain

The freely movable chain ( English freely jointed chain or ideal chain , also Gauss chain or ideal tangle ) is the simplest model with which a polymer can be described. The model neglects interactions between the monomers , so that they can rotate around their two ends at will, which mathematically corresponds to a random walk . The model of the worm-like chain represents an improvement , in which the monomers are subject to restrictions in their mobility, so that even polymers can be described with stiffness.

properties

A polymer is represented in this model as a chain of stiff pieces of length , the so-called Kuhn length - the maximum length is thus through ${\displaystyle N}$ ${\displaystyle l}$ ${\displaystyle L}$

${\displaystyle L=Nl}$

given. The parts are freely movable, comparable to a hinge (but three-dimensional here). This results in a random walk with the step length and the number of steps . The central limit theorem applies to large . ${\displaystyle l}$ ${\displaystyle N}$ ${\displaystyle N}$

In this approach, no interactions between the monomers are assumed, the energy of the polymer is assumed to be independent of its shape. This means that all conceivable configurations are equally probable in thermodynamic equilibrium , the polymer goes through them all in the course of time - the fluctuations are described by the Maxwell-Boltzmann distribution .

Let be the end-to-end vector of the ideal chain and the vectors to individual monomers. These randomly distributed vectors have three components in the x , y and z directions. We assume that the number of monomers N is large, so that the Central Limit Theorem applies. The figure below shows the sketch of a short ideal chain: ${\displaystyle {\vec {R}}}$ ${\displaystyle {\vec {r}}_{1},\ldots ,{\vec {r}}_{N}}$

The ends of the chain do not coincide, but since they fluctuate freely the following applies of course to the mean ( expected value ):

${\displaystyle \langle {\vec {R}}\rangle =\Sigma _{i=1}^{N}\langle {\vec {r}}_{i}\rangle ={\vec {0}}~}$

Since are statistically independent, it follows from the central limit theorem that there are normally distributed: more precisely, in 3D and according to a normal distribution of the mean value 0 with variance : ${\displaystyle {\vec {r}}_{1},\ldots ,{\vec {r}}_{N}}$ ${\displaystyle {\vec {R}}}$ ${\displaystyle R_{x},R_{y},}$ ${\displaystyle R_{z}}$

${\displaystyle \sigma ^{2}=\langle R_{x}^{2}\rangle -\langle R_{x}\rangle ^{2}=\langle R_{x}^{2}\rangle -0}$

${\displaystyle \langle R_{x}^{2}\rangle =\langle R_{y}^{2}\rangle =\langle R_{z}^{2}\rangle =N\,{\frac {l^{2}}{3}}}$

To characterize a freely moving chain, the middle square of : ${\displaystyle \langle R^{2}\rangle }$ ${\displaystyle {\vec {R}}}$

${\displaystyle \langle R^{2}\rangle =\langle R_{x}^{2}\rangle +\langle R_{y}^{2}\rangle +\langle R_{z}^{2}\rangle =Nl^{2}}$

The force-distance curve of the freely moving chain is:

${\displaystyle f={\frac {k_{B}T}{l}}{\mathcal {L}}^{-1}\left({\frac {R}{(N-1)l}}\right),}$

where f is the force, l the bond length, N the number of monomers in the chain (with N-1 bonds), the inverse Langevin function , the Boltzmann constant, T the temperature and R the end-to-end distance . ${\displaystyle {\mathcal {L}}^{-1}}$ ${\displaystyle k_{B}}$

Individual evidence

1. ↑ James E. Mark: Physical Properties of Polymers Handbook , Springer, 2007. ISBN 9780387690025 . P. 68f. (Book preview) .
2. ↑ Gabriele Cruciani: Short textbook physical chemistry . John Wiley & Sons, 2006, ISBN 978-3-527-31807-0 ( limited preview in Google Book Search).
3. ↑ Meyer B. Jackson: Molecular and Cellular Biophysics . Cambridge University Press, 2006. ISBN 9780521624411 . P. 60f. (Book preview) .