Past exam of the quantum condensed matter course of the University of Cambridge / 2008 / Section A / A4

Take the same segment length and a large common number of steps. The ideal walks have size proportional to $N^{1/2}$; excluded volume swells a three-dimensional chain to approximately $N^{0.59}$ and a two-dimensional chain to $N^{3/4}$. Projection changes an isotropic walk's prefactor, rather than its exponent. The projected three-dimensional ideal walk has a smaller planar size than an otherwise equal two-dimensional ideal walk. Therefore the asymptotic ordering isThe following subparts make the size and projection factors explicit. Comparing root-mean-square sizes gives the same ordering as comparing mean distance magnitudes here.

For the planar random walk with independent steps of length $b$, cross terms vanish and $⟨R^2⟩=N{b}^2$. Thus $R_{\mathrm{rms}}=b{N}^{1/2}$. At large $N$, the planar Gaussian end distribution gives $⟨R⟩\simeq b\sqrt{\pi N}/2$.

A planar self-avoiding walk cannot revisit its occupied sites. In the good-solvent scaling regime, the two-dimensional Flory approximation gives $R\sim b{N}^{3/4}$, larger at large $N$ than either ideal walk and the projected three-dimensional self-avoiding walk. This is a scaling statement, with microscopic prefactors dependent on the walk model.

For an isotropic three-dimensional ideal walk, each Cartesian component contributes $N{b}^2/3$ to the mean square. Keeping two components gives $⟨R_{\mathrm{proj}}^2⟩=2N{b}^2/3$. The planar projection of an isotropic random vector therefore has $R_{\mathrm{proj},\mathrm{rms}}=b\sqrt{2N/3}$, smaller than the unprojected planar ideal walk by $\sqrt{2/3}$. The corresponding two-dimensional Gaussian mean distance has the same ratio.

A three-dimensional self-avoiding walk has swollen size $R\sim b{N}^{\nu }$, with $\nu \simeq 0.59$; the three-dimensional Flory approximation gives the nearby estimate $3/5$. Isotropic projection gives $⟨R_{\mathrm{proj}}^2⟩=(2/3)⟨R^2⟩$, preserving this exponent. For the mean distance rather than its root mean square, the orientation average instead gives $⟨R_{\mathrm{proj}}⟩=(\pi /4)⟨R⟩$. Neither prefactor changes its position between the ideal $N^{1/2}$ walks and the planar swollen $N^{3/4}$ walk.

The Kuhn length $l_K$ is the segment length of an equivalent freely jointed chain chosen to match contour length $L$ and the unperturbed mean-square end-to-end distance: $⟨R^2⟩=L{l}_K$. Since $R\le L$, its maximum finite-chain value is $l_K=L$, attained by a straight rigid rod. Independent steps of length $b$ have $L=Nb$ and $⟨R^2⟩=N{b}^2$, soFor a long freely rotating chain with fixed angle $\alpha$ between successive forward bond vectors, orientational correlations decay as $c^s$ with $c=\cos \alpha$. Summing them gives the Kuhn length of a freely rotating chain:Positive persistence increases the statistical segment length, while negative bond correlations can reduce it. A chemical internal bond angle is supplementary to this forward-vector angle, so the cosine convention matters. Near the perfectly aligned limit the finite-chain bound $l_K\le L$ replaces the long-chain asymptotic formula.

Write $\ell$ for a segment length and $v_2$ for the excluded-area coefficient, so that the interaction term has the correct dimensions. For a coil of planar size $R$ and area of order $R^2$, the two-dimensional Flory approximation isThe first term is the entropic stretching cost and the second counts repulsive monomer contacts. Minimization gives $2R/(N{\ell }^2)-2v_2{N}^2/R^3=0$, henceFor $v_2$ of order ${\ell }^2$ this becomes $R\sim \ell N^{3/4}$. The parameter multiplying $N^2/A$ in a two-dimensional interaction energy must have dimensions of area; it cannot simultaneously denote an unsquared bond length.

The polymer overlap concentration is reached when the number of chains per coil area is of order one. If $n_c$ is the chain number per unit interfacial area, then $n_c^{\ast }{R}^2\sim 1$. Equivalently, the monomer area fraction is ${\varphi }^{\ast }\sim N{\ell }^2/R^2$, assuming an occupied monomer area of order ${\ell }^2$. Below this scale coils are mostly separate; above it they interpenetrate.

For the planar good-solvent coil, $R\sim \ell N^{3/4}$. The two-dimensional polymer overlap fraction is thereforeThis is a scaling estimate; the monomer geometry and choice of coil radius set order-one numerical factors.

Here the osmotic pressure is a two-dimensional surface pressure, with units of energy per area. In the dilute regime, independent whole chains provide the translational entropy, soIn the semidilute polymer solution, use a correlation blob containing $g$ monomers. Within it the chain still follows the planar good-solvent law $\xi \sim \ell g^{3/4}$. Its area fraction is $\varphi \sim g{\ell }^2/{\xi }^2\sim g^{-1/2}$, so $g\sim {\varphi }^{-2}$ and $\xi \sim \ell {\varphi }^{-3/2}$. There is about one independent thermal contribution per blob area. The semidilute osmotic pressure of a two-dimensional polymer solution is thusIt is independent of total chain length once each chain contains many blobs. The two estimates agree at ${\varphi }^{\ast }\sim N^{-1/2}$, where each gives $(k_{B}T/{\ell }^2)N^{-3/2}$.