Knotted Polymer Research Articles

Deformation of a stretched polymer knot

The static properties of a knotted polymer under a stretching force f are studied by Monte Carlo simulations. Chain lengths up to $N=82$ and knot types of ${0}_{1},$ ${3}_{1},$ ${4}_{1},$ ${5}_{1},$ ${6}_{1},$ and ${8}_{1}$ are considered. Our simulation data show that the scaling laws proposed by de Gennes and Pincus for a single linear chain under traction force still hold for the knotted type polymers. That is, the average knot size under a force f scales as $〈{R}_{f}〉\ensuremath{\sim}{R}_{F}^{2}f$ at weak tension forces while for strong forces $〈{R}_{f}〉\ensuremath{\sim}{R}_{F}^{1/\ensuremath{\nu}}{f}^{(1/\ensuremath{\nu})\ensuremath{-}1},$ where ${R}_{F}\ensuremath{\sim}{N}^{\ensuremath{\nu}}{p}^{\ensuremath{-}4/15},$ $\ensuremath{\nu}\ensuremath{\approx}3/5$ is the usual self-avoiding walk exponent and p is a topological invariant representing the aspect ratio (length to diameter) of a knotted polymer at its maximum inflated state. Our results also show that the elastic modulus of a knotted polymer is larger compared to an equal-length linear chain. More complex knots are in general stiffer. A simple composite spring model is employed to derive the increase in stiffness of knots relative to the linear chain, and the results agree well with the simulation data. Segregation of the crossings into a small tight region of the knot structure at strong forces is also observed.

Evolution of Fragments Formed at the Rupture of a Knotted Alkane Molecule

Common experience tells us that a knot significantly weakens the polymer strand in which it is tied, which, in turn, leads to more facile chain rupture under tensile loading. Using first-principles molecular dynamics calculations we describe the dynamical evolution of the radicals that form after chain rupture of a single knotted alkane molecule in its very early stages of life. They are able to recombine, to form cyclic alkanes, and to undergo disproportionation phenomena with nearby chain segments. The breaking of a single knotted polymer chain under mechanical loading is thus predicted to reveal phenomena falling in the domain of ultrafast spectroscopy.

Tight Knots in Polymers

Tight Knots in Polymers

Topological Effects of Knots in Polymers

We present a phenomenological theory of the static and dynamic effects of knots in polymers. The theory is tested by Monte Carlo simulations using the kink-jump-crankshaft and Berg-Foerster-Aragao de Carvalho-Caracciolo-Froehlich algorithms. A long time relaxation mode is discovered in the dynamics of knotted polymers that is not predicted by the Rouse model. This mode is not present in the trivial knot and appears to be the result of purely topological interactions.