Distribution of End-to-End Distances in Linear Substituted Polymethylenes

This Demonstration shows the dependence of the end-to-end distance distribution of a linear substituted polymethylene chain on energy barriers, temperature and chain length. The parameter values are automatically reset to force and .

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The conformational energy of hindered rotation in a classical molecular mechanics simulation in the general case is approximated by the equation [1]:
.
The parameters and describe a force field with torsion angle .
This Demonstration uses a modified version of the potential:
where is the energy barrier (kJ/mol) and and are the energy difference gauche(-)/trans and gauche(+)/trans conformers, respectively (kJ/mol).
The effect of excluded volume is neglected in the calculation.
Using the Monte Carlo method, 500 macromolecular chains are generated. For a chain of length , torsion angle values are generated. To simplify the calculation, selection of the torsion angle is performed taking into account the statistical weight based on the cumulative distribution function at temperature , divided into 36 equal intervals with step size 10°. This avoids the calculation of the Boltzmann factor for individual chains:
.
For each chain, the end-to-end distance is calculated. The distance is expressed in relative units of carbon-carbon bond length. For visualization, a smoothed function is used to exclude statistical scatter; values are the result of smoothing and should be ignored.
Based on the distribution, the mean , root mean square , end-to-end distance and standard deviation are calculated:
,
,
.
Reference
[1] M. Bachmann, Thermodynamics and Statistical Mechanics of Macromolecular Systems, Cambridge, UK: Cambridge University Press, 2014.
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