Deformation of a stretched polymer knot

The static properties of a knotted polymer under a stretching force f are s tudied by Monte Carlo simulations. Chain lengths up to N = 82 and knot type s 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 sing le linear chain under traction force still hold for the knotted type polyme rs. That is, the average knot size under a force f scales as [R-f] similar to R(F)(2)f at weak tension forces while for strong forces [R-f] similar to R(F)(1/v)f((1/v)-1), where R-F similar to N(v)p(-4/5), v approximate to 3/ 5 is the usual self-avoiding avoiding walk exponent and p is a topological invariant representing the aspect ratio (length to diameter) of a knotted p olymer at its maximum inflated state. Our results also show that the elasti c modulus of a knotted polymer is larger compared to an equal-length linear chain. More complex knots are in general stiffer. A simple composite sprin g model is employed to derive the increase in stiffness of knots relative t o the linear chain, and the results agree well with the simulation data. Se gregation of the crossings into a small tight region of the knot structure at strong forces is also observed.