From size scaling to chain dynamics

The previous two pages were about shape: how far a chain reaches, and how quickly it forgets its own direction. This page is the bridge to motion, and it is short. One exponent is enough to fix four others.

Write the size of a subchain of \(g\) monomers as

\[R(g) \sim b\, g^{\nu}\]

That is the only input. Everything below follows from it, which is why this page has a slider for \(\nu\) and no simulation at all.

Relaxation time

A subchain relaxes when it has diffused a distance comparable to its own size. Treat it as an object with a friction coefficient \(\zeta\) held by an entropic spring of stiffness \(k \sim k_BT/R(g)^2\). The relaxation time is the ratio,

\[\tau(g) \sim \frac{\zeta}{k} \sim \frac{\zeta R(g)^2}{k_BT}\]

so the whole question becomes what \(\zeta\) is, and there are two standard answers.

In Rouse dynamics each monomer feels its own drag from the surrounding medium and those drags simply add, so \(\zeta \sim g\). Then

\[\tau(g) \sim g \cdot g^{2\nu} = g^{2\nu+1}\]

In Zimm dynamics the solvent inside the subchain is dragged along with it. The subchain moves as a single object of size \(R(g)\), and hydrodynamics gives a Stokes-like friction \(\zeta \sim \eta R(g) \sim g^{\nu}\). Then

\[\tau(g) \sim g^{\nu} \cdot g^{2\nu} = g^{3\nu}\]

Set \(\nu = 1/2\) and you get \(\gamma = 2\) and \(3/2\). Set \(\nu = 3/5\) and you get \(2.2\) and \(1.8\). Those are the standard tabulated values, recovered from one line of algebra each.

Mean-squared displacement

Now watch a single monomer over a time \(t\). The trick is to ask which subchain has just finished relaxing at that moment: set \(\tau(g) = t\) and invert, giving \(g(t) \sim t^{1/\gamma}\). Anything smaller has already equilibrated and contributes nothing further; anything larger has not yet moved. So the monomer has explored roughly the size of that subchain, and

\[\langle \Delta r^2 \rangle \sim R(g(t))^2 \sim t^{2\nu/\gamma}, \qquad \alpha = \frac{2\nu}{\gamma}\]

Two consequences are worth pausing on, and the readouts show both.

For Rouse, \(\alpha = 2\nu/(2\nu+1)\). Substituting the fractal dimension \(d_f = 1/\nu\) turns this into \(\alpha = 2/(2+d_f)\), which is the form the exponent is usually tabulated in. They are the same statement. The panel prints both so you can watch them agree as you drag the slider.

For Zimm, \(\alpha = 2\nu/3\nu = 2/3\), independent of \(\nu\) entirely. This is why the Zimm MSD exponent does not move between an ideal chain and a self-avoiding walk. It is not a coincidence in a table, it is a cancellation.

Where the shortcut fails

Two of the four models on the reference panel are flagged, and the reasons matter more than the numbers.

The equilibrium globule relaxes by reptation. The chain is so densely surrounded by itself that it cannot move sideways and has to slide along a confining tube. The tube’s contour length grows linearly with \(s\) rather than as \(s^{\nu}\), which breaks the very first line of the argument above. Working it through instead gives \(\tau \sim s^3\) and \(\alpha = 1/4\), derived on the equilibrium globule page.

The fractal globule does not have that problem, because it is unknotted and does not need reptation. The shortcut applies directly with \(\nu = 1/3\), giving \(\gamma = 5/3\) and \(\alpha = 2/5\). Its Zimm exponent is the exception: the structure is dense enough that hydrodynamic interactions are themselves partially screened, so the clean \(2/3\) result does not survive and the accepted value near \(0.42\) comes from a fuller treatment (Tamm et al.), not from this argument.

Knowing where a scaling argument stops being valid is as useful as knowing what it predicts. Both flagged rows are places where the physics changes character rather than places where the algebra was done carelessly.