Force curve associated with squeezing a worm-like chain (WLC) between two parallel plates

Question

Let's say I have a polymer, of contour length $L_p$ and persistence length $P$, positioned between two parallel plates separated by a distance $z$. I slowly squeeze the plates together until only two-dimensional diffusion is observed. Under the worm-like chain (WLC) model, how will the force curve, $F(z)$, scale as a function of the distance between the plates?

As a sort of addendum... I'm curious how compression of the polymer between two parallel plates contrasts with the force curve associated with packing the polymer in a spherical cage? In the latter case of the spherical cage, I'm not exactly sure that the WLC model is relevant when the cross-sectional dimensions of the cage approach the polymer's persistence length.

Update - To make a prediction, the problem of squeezing a WLC or FJC polymer between two plates is probably more straightforward than that of 'crushing' a polymer into a spherical ball (which could perhaps be experimentally accomplished by placing a hydrophilic polymer into increasingly 'bad'/non-polar solvent) since it avoids having to estimate any bending energies. So, focusing on the 3D --> 2D scenario, my guess is that the energy to accomplish the task of squeezing the polymer until only two dimensional diffusion is observed would be similar to that of reducing one degree of diffusional freedom of a monoatomic gas - i.e. (Number of persistence length units in polymer) * $\frac{1}{2}(kB)(T)$. I would also predict that the force curve is non-linear, since the probability density for the location of a particular persistence length segment of a polymer is probably not uniform across the gap between the plates. However, that's simply a guess.

Answer 1

Under the isothermic condition, the force acting on each plate equals the derivative of the free energy w.r.t the distance between the plates. That is, one has to compute the partition sum of the system as a function of the interplate distance $z$. In the case of FJC model with the link length $d$ much shorter than $z$ the calculation is straightforward. Indeed, the partition function can be expressed as a path integral, which in turn boils down to a diffusion equation with zero boundary condition on the plates. Although no explicit formula for all $z$ can be written, some asymptotics can be calculated. For instance, in the case $d\ll z \ll \sqrt{L_p d}$, the free energy scales as the biggest eigenvalue of the diffusion operator, that is like $1/z^2$ - therefore, the force grows as $1/z^3$.

The case of WLC model could probably be treated analogously. The main difference is in the path integral formula. Since the polymer has finite stiffness, the quantity in the diffusion equation should be a 2-tensor (average tensor square of the direction of the chain), instead of scalar (average density of the chain). Also, the boundary condition will be zero for the component of the tensor normal to the plate. This will change the asymptotics at $z \approx P$. I do not see right now the answer, but it should be not difficut to compute it. Basically, you have two distinct diffision coefficients (one for the traceless part of the tensor, the other for the trace), and the boundary condition which mixes two parts. At the first glance, there should be a finite limit for the free energy as $z\to 0$, hence the force should tend to zero (thus reaching a maximum somewhere around $z \approx P$.