Problem: Explain why for a random walk in \(\mathbf R^d\), the probability distribution of the random vector sum \(\mathbf x:=\mathbf x_1+…+\mathbf x_N\) of \(N\) i.i.d. random vectors \(\mathbf x_i\) each with identical mean \(\boldsymbol{\mu}:=\langle\mathbf x_i\rangle\) and identical covariance matrix \(\sigma^2:=\langle\mathbf x_i^{\otimes 2}\rangle-\boldsymbol{\mu}^{\otimes 2}\) in the large \(N\gg 1\) limit is of the form:
\[p(\mathbf x)=\frac{1}{(2\pi N)^{d/2}\det\sigma}e^{-(\mathbf x-N\boldsymbol{\mu})^T\sigma^{-2}(\mathbf x-N\boldsymbol{\mu})/2N}\]
Solution: Central limit theorem.
Problem: Explain what an ideal polymer chain is.
Solution: A model of polymer chains lacking a concept of excluded volume. The word “ideal” here is strictly weaker than e.g. the “ideal” in “ideal gas” (where it really means no interactions of any kind); here the monomers are permitted to have short-range interactions (e.g. the covalent bonds holding them together, steric hindrance, etc.) but no long-range interactions (in particular, if a later segment of the chain happens to fold back onto an earlier segment, this is allowed within the ideal chain paradigm due to the lack of excluded volume). In particular, one can think of ideal polymer chains as the universality class of polymer chains whose expected end-to-end distance scales as \(\sqrt{\langle r^2\rangle}\sim N^{1/2}\).
Problem: Consider the freely-jointed chain model of a polymer in \(\mathbf R^3\) which consists of \(N\gg 1\) monomers \(\mathbf x_i\) each of fixed length \(\ell:=|\mathbf x_i|\) connected to each other but are otherwise mutually non-interacting and are also isolated from any external solvent, etc. For the ideal chain, calculate the distribution of the end-to-end vector \(\mathbf x:=\sum_{i=1}^N\mathbf x_i\) of the polymer chain.
Solution: In this case, one has \(\boldsymbol{\mu}=\mathbf 0\) and writing \(\mathbf x:=\ell(\cos\phi\sin\theta,\sin\phi\sin\theta,\cos\theta)\) for \((\cos\theta,\phi)\in [-1,1]\times[0,2\pi]\) uniformly distributed, the covariance matrix is proportional to the identity \(\sigma^2=\ell^2/3\) (e.g. \(\langle z^2\rangle=\frac{1}{2\times 2\pi}\int_{-1}^1d\cos\theta\int_0^{2\pi}d\phi (\ell\cos\theta)^2=\ell^2/3\)). Thus:
\[p(\mathbf x)=\left(\frac{3}{2\pi N\ell^2}\right)^{3/2}e^{-3|\mathbf x|^2/2N\ell^2}\]
which implies the radial distribution function for \(r:=|\mathbf x|\):
\[p(r)=\left(\frac{3}{2\pi N\ell^2}\right)^{3/2}4\pi r^2e^{-3r^2/2N\ell^2}\]
with the expected \(\langle r^2\rangle=\int_0^{\infty}drp(r)r^2=N\ell^2\)
(aside: for certain kinds of ring polymers that close back on themselves, the end-to-end distance is trivially \(0\). In that case, an alternative metric for quantifying the polymer chain size is the radius of gyration \(r_g^2:=\frac{1}{N}\sum_{i=1}^N|\mathbf r_i-\mathbf R|^2\) where each \(\mathbf r_i\) refers to the position of (say, the center of mass, but in the coarse-grained limit it doesn’t matter) the \(i^{\text{th}}\) monomer, and \(\mathbf R:=\frac{1}{N}\sum_{i=1}^N\mathbf r_i\) is the center of mass of the polymer chain. By Lagrange’s identity, this can be rewritten as the average distance between pairs of monomers on the chain:
\[r_g^2=\frac{1}{2N^2}\sum_{i=1}^N\sum_{j=1}^N|\mathbf r_i-\mathbf r_j|^2=\frac{1}{N^2}\sum_{i<j}|\mathbf r_i-\mathbf r_j|^2\]
This form is easier to work with because (assuming \(i<j\)) one can express the displacement \(\mathbf r_j-\mathbf r_i=\frac{\mathbf x_i+\mathbf x_j}{2}+\sum_{k=i+1}^{j-1}\mathbf x_{i+1}\) (assuming each monomer’s center of mass is at its midpoint, but this is irrelevant in the \(N\gg 1\) limit). The \(N\gg 1\) coarse-grained result for a freely-jointed chain is:
\[\langle r_g^2\rangle=\frac{\langle r^2\rangle}{6}=\frac{N\ell^2}{6}\]
is smaller than the average end-to-end distance by a factor of \(\sqrt{6}\)).
Problem: An improvement to the freely-jointed chain model can be made if one assumes that \(\mathbf x_i\cdot\mathbf x_{i+1}=\ell^2\cos\theta\) for some fixed bond angle \(\theta\) (strictly speaking this is supplementary to what chemists usually call the “bond angle” \(\theta_{\text{chemist}}=\pi-\theta\)). However, the torsion angle \(\phi\in[0,2\pi]\) is assumed to be unrestricted (this model is called the freely rotating chain). In this case, show that for \(N\gg 1\):
\[\langle r^2\rangle\approx N\ell^2\frac{1+\cos\theta}{1-\cos\theta}\]
(note: \(\frac{1+\cos\theta}{1-\cos\theta}=\cot^2\theta/2\) is monotonically decreasing from a singularity at \(\theta=0\) to \(1\) at \(\theta=\pi/2\) to \(0\) at \(\theta=\pi\)). If furthermore the torsion angle \(\phi\) is non-uniform \(p(\phi)\neq 1/2\pi\) but instead subject to an achiral potential \(V(-\phi)=V(\phi)\) so that \(p(\phi)\sim e^{-\beta V(\phi)}\) (e.g. trans conformations being preferred over gauche), show that \(\langle r^2\rangle\) receives an additional correction factor of the same form as the above but with \(\cos\theta\mapsto\langle\cos\phi\rangle:=\int_0^{2\pi}d\phi p(\phi)\cos\phi\):
\[\langle r^2\rangle\approx N\ell^2\frac{1+\cos\theta}{1-\cos\theta}\frac{1+\langle\cos\phi\rangle}{1-\langle\cos\phi\rangle}\]
(this is called the hindered rotation model of the polymer chain).
Solution: The key lemma is to convince oneself that the average of any vector dotted with a circle of unit vectors in an arbitrary plane is zero. Once one realizes this, it is straightforward to check \(\langle\mathbf x_i\cdot\mathbf x_j\rangle=\ell^2\cos^{|i-j|}\theta\), and the geometric series may be summed and terms simplified in the \(N\gg 1\) limit to obtain the desired result.
Note that one can rewrite the \(2\)-point correlator:
\[\langle\mathbf x_i\cdot\mathbf x_j\rangle=\ell^2\cos^{|i-j|}\theta=\ell^2e^{-|i-j|/N_p}\]
where \(N_p:=-1/\ln\cos\theta\) is the persistence number, and \(\ell_p:=\ell N_p\) is the persistence length. For \(\theta\ll 1\), \(N_p\approx 2/\theta^2\) in which case one can the Kratky-Porod limit \(\theta,\ell\to 0\) in such a way that \(\ell_p=2\ell/\theta^2\) is fixed. This effectively replaces the freely rotating model with a worm-like chain model in which the polymer is parameterized by a smooth trajectory \(\mathbf x(s)\) of its arc length \(s\) with unit tangent vector \(d\mathbf x/ds\) obeying:
\[\biggr\langle\frac{d\mathbf x}{ds}\cdot\frac{d\mathbf x}{ds’}\biggr\rangle=e^{-|s-s’|/\ell_p}\]
One can estimate \(\ell_p\) by modelling the worm-like chain as an Euler-Bernoulli beam! In that case, the polymer Hamiltonian functional is quadratic:
\[H[\theta(s)]=\frac{1}{2}EI\int_0^Lds\left(\frac{d\theta}{ds}\right)^2\]
hence by the equipartition theorem, the variance \(\langle\theta^2\rangle\) in the angular fluctuations is \(\frac{1}{2}EI\frac{\langle\theta^2\rangle}{\ell}=2\times\frac{1}{2}k_BT\), so equating \(\langle\theta^2\rangle=\theta^2\) from the deterministic freely rotating model, one finds the persistence length is set by the ratio of flexural rigidity to thermal energy:
\[\ell_p=\frac{EI}{k_BT}\]
and the expected end-to-end distance is now \(\langle r^2\rangle=\int_0^L ds’\int_0^L ds e^{-|s-s’|/\ell_p}=2\ell_p^2(e^{-L/\ell_p}-1+L/\ell_p)\) which in the limit \(\ell_p\ll L\) results in a Kuhn length \(\ell_{\text{eff}}=2\ell_p\) twice the persistence length (in the opposite limit \(\ell_p\gg L\), one has the obvious \(\langle r^2\rangle\to L^2\)).
For the hindered rotation model, the math is similar, just need to remember that \(\langle\sin\phi\rangle=0\) because \(V(-\phi)=V(\phi)\) is even.
Problem: Define the Kuhn length \(\ell_{\text{eff}}\) of an ideal polymer chain.
Solution: The Kuhn length of an ideal polymer chain is defined by:
\[\ell_{\text{eff}}:=\frac{\langle r^2\rangle}{L}\]
with \(L=N\ell\) the total chain length. This ensures that even if the polymer chain is not freely jointed, it may be approximated by an equivalent freely jointed chain whose “Kuhn monomers” are of length \(\ell_{\text{eff}}>\ell\). For instance, \(\ell_{\text{eff}}=\ell\cot(\theta/2)\) for the freely rotating chain.
Problem: Describe Flory’s mean-field theory.
Solution: The free energy is \(F=\langle E\rangle-TS\). The energy \(E\) due to collisions of monomers with excluded volume \(V_{\text{ex}}\) can be modelled by a pseudopotential \(V(\mathbf x)=V_{\text{ex}}k_BT\delta^3(\mathbf x)\) as this can be checked (after Taylor expanding the Mayer \(f\)-function \(f(\mathbf x)=e^{-V_{\text{ex}}\delta^3(\mathbf x)}-1\approx -V_{\text{ex}}\delta^3(\mathbf x)\)) to have the same second virial coefficient \(\tilde{B}_2(T)=-\int d^3\mathbf x f(\mathbf x)=V_{\text{ex}}\) as a corresponding hard-sphere monomer of excluded volume \(V_{\text{ex}}=8V_{\text{monomer}}\). Then for number density \(n(\mathbf x)\) of monomers:
\[E=\frac{1}{2}\sum_{i\neq j}V(\mathbf x_i-\mathbf x_j)\approx \frac{V_{\text{ex}}k_BT}{2}\int d^3\mathbf x d^3\mathbf x’n(\mathbf x)n(\mathbf x’)\delta^3(\mathbf x-\mathbf x’)\]
\[=\frac{V_{\text{ex}}k_BT}{2}\int d^3\mathbf x n^2(\mathbf x)\]
Thus, the ensemble average \(\langle E\rangle\) is evaluated by a mean-field approximation:
\[\langle E\rangle=\frac{V_{\text{ex}}k_BT}{2}\int d^3\mathbf x \langle n^2(\mathbf x)\rangle\]
\[\approx \frac{V_{\text{ex}}k_BT}{2}\int d^3\mathbf x \langle n(\mathbf x)\rangle^2\]
If one furthermore assumes monomers \(\langle n(\mathbf x)\rangle=N/V\) are uniformly distributed, then this simplifies to \(E(N,r)=V_{\text{ex}}N^2k_BT/2V=3V_{\text{ex}}N^2k_BT/8\pi r^3\). Meanwhile, the entropy is simply (up to some irrelevant baseline \(S_0\)) the log of the normal distribution \(p(\mathbf x)\) for the freely-jointed Gaussian chain from earlier:
\[S(N,r)=S_0-\frac{3k_Br^2}{2N\ell^2}\]
Thus, by setting \(\partial F/\partial r:=0\), one obtains the equilibrium relation:
\[r=\left(\frac{3V_{\text{ex}}\ell^2}{8\pi}\right)^{1/5}N^{3/5}\]
By fortunate cancellation, it turns out in a good solvent this simple mean-field exponent \(\nu=3/5\) is close the true exponent \(\nu\approx 0.588\) arising from a detailed RG flow analysis.
Problem: In the naive analysis above, it was assumed that the excluded volume \(V_{\text{ex}}>0\) is a literal volume, and must therefore be positive. What’s wrong with that?
Solution: In practice this is not the case, and indeed one can run through the standard van der Waals correction (this time including some long-range potential such as Lennard-Jones on top of the hard sphere excluded volume) and find a \(T\)-dependent effective excluded volume which is just another way of saying the second virial coefficient:
\[\tilde B_2(T)=V_{\text{ex}}\left(1-\frac{T_B}{T}\right)\]
where the interaction-dependent Boyle temperature \(T_B\) is also called the “theta temperature” in this context. It is conventional to define the dimensionless Curie-like Flory interaction parameter \(\chi(T):=T_B/2T\) in terms of which \(\tilde{B}_2(T)=V_{\text{ex}}(1-2\chi(T))\). Thus, there exist \(3\) regimes:
- \(T>T_B\) means \(B_2(T)>0\) and \(\chi(T)<1/2\) in which case the polymer chain swells as \(r\propto N^{3/5}\) (“good solvent”).
- \(T=T_B\) means \(B_2(T)=0\) and \(\chi(T)=1/2\) in which case the polymer chain behaves as if ideal/freely-jointed \(r\propto N^{1/2}\) (“theta solvent”).
- \(T<T_B\) means \(B_2(T)<0\) and \(\chi(T)>1/2\) in which the polymer coil transitions into a globule \(r\propto N^{1/3}\) (“poor solvent”).
Problem: State the Flory-Huggins free energy density per lattice site \(\Delta F_{\text{mix}}(\phi)/N\) as a function of the polymer volume fraction \(\phi:=N_mN_p/N\) (i.e. each of the \(N_p\) polymer chains contains \(N_m\) monomer subunits, and all \(N=N_s+N_mN_p\) lattice sites are assumed to be occupied by either one of the \(N_mN_p\) monomers or one of the \(N_s\) solvent molecules).
Solution: The Flory interaction parameter for coordination number \(z\in\mathbf Z^+\) is given by \(\chi(T):=z\Delta E/k_BT\) where \(\Delta E:=E_{sm}-(E_{ss}+E_{mm})/2\):
\[\Delta F_{\text{mix}}(\phi)/N=k_BT\left(\frac{\phi}{N_m}\ln\phi+(1-\phi)\ln(1-\phi)+\chi\phi(1-\phi)\right)\]
(aside: how is this definition of \(\chi\) related to the previous one that was used when discussing the coil-to-globule transition? After all, this one seems like it can be positive or negative but the other one can only be positive right…).
If instead of polymers dissolved in a solvent, it was one polymer (of chain length \(N_m\)) dissolved in another polymer (of chain length \(N’_m\)) (known as a polymer blend), the Flory-Huggins free energy density would naturally be modified to:
\[\Delta F_{\text{mix}}(\phi)/N=k_BT\left(\frac{\phi}{N_m}\ln\phi+\frac{1-\phi}{N’_m}\ln(1-\phi)+\chi\phi(1-\phi)\right)\]
If in addition the polymer blend is in fact a polymer melt (i.e. even though \(N_m\neq N’_m\), the monomer subunits are chemically identical), then \(\chi=0\) and hence the polymers behave like random walks \(r\sim N_m^{1/2}\). Think hot glue or \(3\)D printer filament immediately after extrusion.
Problem: Compute the osmotic pressure \(p_{\text{os}}\) and the chemical potential \(\mu\) from the Flory-Huggins interaction parameter.
Solution: Take the relevant partial derivatives…
Problem: The dilute limit is \(\phi\to 0\), whereas the polymer melt is the limit \(\phi\to 1\). It turns out one can define a critical volume fraction \(\phi^*\in [0,1]\) representing the threshold for transition from the dilute regime \(\phi<\phi^*\) to a “semi-dilute” regime \(\phi^*<\phi\ll 1\). Show that \(\phi^*\propto N_m^{-4/5}\) in a good solvent.
Solution: Similar to Bose-Einstein condensation, the critical point occurs when the “phase space density” \(N^*_pr^3/V\propto 1\Rightarrow N^*_p/N\propto r^{-3}\propto N_m^{-9/5}\). The corresponding critical volume fraction \(\phi^*=N_mN^*_p/N\propto N_m^{-4/5}\).
Problem: Compute the spinodal curve \(\chi(\phi)\) and state its physical significance.
Solution: The spinodal curve is given by \(\partial^2(\Delta F_{\text{mix}}/N)/\partial\phi^2=0\) which amounts to:
\[\chi(\phi)=\frac{1}{2}\left(\frac{1}{N_m\phi}+\frac{1}{1-\phi}\right)\]
This is maximized \(\partial\chi/\partial(\phi=\phi_c)=0\) at:
\[\phi_c=\frac{1}{1+\sqrt{N_m}}\]
with corresponding Flory interaction parameter slightly larger than \(\chi_c>1/2\):
\[\chi(\phi_c):=\chi_c=\frac{1}{2}\left(1+\frac{1}{\sqrt{N_m}}\right)^2\]