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Physics LibreTexts

6.6: Polymers

  • Page ID
    18774
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        at template()
        at (Bookshelves/Thermodynamics_and_Statistical_Mechanics/Book:_Thermodynamics_and_Statistical_Mechanics_(Arovas)/06:_Classical_Interacting_Systems/6.06:_Polymers), /content/body/p/span, line 1, column 23
    
    \)
    \( \newcommand\Vnu
    ParseError: invalid DekiScript (click for details)
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        at (Template:MathJaxArovas), /content/body/div/p[1]/span[19], line 1, column 1
        at template()
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    \)
    \( \newcommand\Vxi
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    \)
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    \)
    \( \newcommand\Vpi
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    \)
    \( \newcommand\Vvarpi
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    \)
    \( \newcommand\Vrho
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    \)
    \( \newcommand\Vvarrho
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    \)
    \( \newcommand\Vsigma
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    \)
    \( \newcommand\Vvarsigma
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    \)
    \( \newcommand\Vtau
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    \)
    \( \newcommand\Vupsilon
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    \)
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    \)
    \( \newcommand\Vvarphi
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    \)
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    \)
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    \)
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    \)
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    \)
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    \)
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    Basic concepts

    Linear chain polymers are repeating structures with the chemical formula \((A)\ns_x\), where \(A\) is the formula unit and \(x\) is the degree of polymerization. In many cases (polystyrene), \(x\gtwid 10^5\) is not uncommon. For a very readable introduction to the subject, see P. G. de Gennes, Scaling Concepts in Polymer Physics.

    Quite often a given polymer solution will contain a distribution of \(x\) values; this is known as polydispersity. Various preparation techniques, such as chromatography, can mitigate the degree of polydispersity. Another morphological feature of polymers is branching, in which the polymers do not form linear chains.

    clipboard_effa7ad7d832f2de7dffa4679b5ec3fb0.png
    Figure \(\PageIndex{1}\): Some examples of linear chain polymers.

    Polymers exhibit a static flexibility which can be understood as follows. Consider a long chain hydrocarbon with a \(-\RC-\RC-\RC-\) backbone. The angle between successive \(\RC-\RC\) bonds is fixed at \(\theta\approx 68^\circ\), but the azimuthal angle \(\vphi\) can take one of three possible low-energy values, as shown in the right panel of Figure \(\PageIndex{2}\). Thus, the relative probabilities of gauche and trans orientations are

    \[{{Prob}\,({gauche})\over{Prob}\,({trans})} = 2\,e^{-\RDelta \ve/\kT}\]

    where \(\RDelta \ve\) is the energy difference between trans and gauche configurations. This means that the polymer chain is in fact a random coil with a persistence length

    \[\ell\ns_\Rp = \ell\ns_0 \, e^{\RDelta\ve/\kT}\]

    where \(\ell\ns_0\) is a microscopic length scale, roughly given by the length of a formula unit, which is approximately a few Ångstroms (see Figure \(\PageIndex{2}\)). Let \(L\) be the total length of the polymer when it is stretched into a straight line. If \(\ell\ns_\Rp > L\), the polymer is rigid. If \(\ell\ns_\Rp \ll L\), the polymer is rigid on the length scale \(\ell\ns_\Rp\) but flexible on longer scales. We have

    \[{\ell\ns_\Rp\over L}={1\over N}\,e^{\RDelta\ve/\kT}\quad ,\]

    where we now use \(N\) (rather than \(x\)) for the degree of polymerization.

    In the time domain, the polymer exhibits a dynamical flexibility on scales longer than a persistence time. The persistence time \(\tau\ns_\Rp\) is the time required for a trans-gauche transition. The rate for such transitions is set by the energy barrier \(B\) separating trans from gauche configurations:

    \[\tau\ns_\Rp=\tau\ns_0\,e^{B/\kT}\]

    where \(\tau\ns_0\sim 10^{-11}\,\Rs\). On frequency scales \(\omega\ll \tau_\Rp^{-1}\) the polymer is dynamically flexible. If \(\RDelta\ve\sim\kT \ll B\) the polymer is flexible from a static point of view, but dynamically rigid. That is, there are many gauche orientations of successive carbon bonds which reflect a quenched disorder. The polymer then forms a frozen random coil, like a twisted coat hanger.

    clipboard_e7289268ec0f8e57de012fa6fc99de648.png
    Figure \(\PageIndex{2}\): Left: trans and gauche orientations in carbon chains. Right: energy as a function of azimuthal angle \(\vphi\). There are three low energy states: trans (\(\vphi=0\)) and gauche (\(\vphi=\pm\vphi_0\)).

    Polymers as random walks

    A polymer can be modeled by a self-avoiding random walk (SAW). That is, on scales longer than \(\ell\ns_\Rp\), it twists about randomly in space subject to the constraint that it doesn’t overlap itself. Before we consider the mathematics of SAWs, let’s first recall some aspects of ordinary random walks which are not self-avoiding.

    We’ll simplify matters further by considering random walks on a hypercubic lattice of dimension \(d\). Such a lattice has coordination number \(2d\), there are \(2d\) nearest neighbor separation vectors, given by \(\Bdelta = \pm a\,\ehat\ns_1 \, , \, \pm a\,\ehat\ns_2 \, , \, \ldots \, , \, \pm a\,\ehat\ns_d\) , where \(a\) is the lattice spacing. Consider now a random walk of \(N\) steps starting at the origin. After \(N\) steps the position is

    \[\BR\ns_N=\sum_{j=1}^N\Bdelta\ns_j\]

    where \(\Bdelta\ns_j\) takes on one of \(2d\) possible values. Now \(N\) is no longer the degree of polymerization, but somthing approximating \(L/\ell\ns_\Rp\), which is the number of persistence lengths in the chain. We assume each step is independent, hence \(\langle \delta_j^\alpha\,\delta_{j'}^\beta\rangle=(a^2/d)\,\delta\ns_{jj'}\delta^{\alpha\beta}\)  and \(\blangle \BR_N^2\brangle=N a^2\). The full distribution \(P\ns_N(\BR)\) is given by

    \[\begin{split} P\ns_N(\BR)&=(2d)^{-N}\sum_{\Bdelta\ns_1} \cdots\sum_{\Bdelta\ns_N} \delta\ns_{\BR,\sum_j\Bdelta\ns_j} \\ &= a^d\!\!\! \int\limits_{-\pi/a}^{\pi/a}\! \!{dk\ns_1\over 2\pi} \cdots\!\!\! \int\limits_{-\pi/a}^{\pi/a}\! \!{dk\ns_d\over 2\pi} \>e^{-i\Bk\cdot\BR} \> \Bigg[ {1\over d} \sum_{\mu=1}^d \cos(k\ns_\mu a) \Bigg]^N \\ &=a^d\!\int\limits_{\HOmega}\!\!{d^d\!k\over (2\pi)^d} \> e^{-i\Bk\cdot\BR} \, \exp\!\Bigg[ N\ln\!\bigg( 1 - {1\over 2d} \,\Bk^2 a^2 + \ldots \bigg) \Bigg] \\ &\approx \bigg({a\over 2d}\bigg)^{\!d}\int\!\! d^d\!k \> e^{-N\Bk^2 a^2/2d}\, e^{-i\Bk\cdot\BR} = \bigg({d\over 2\pi N}\bigg)^{\!d/2}\, e^{-d\BR^2/2Na^2}\quad . \label{polydist} \end{split}\]

    This is a simple Gaussian, with width \(\blangle \BR^2 \brangle = d\ncdot(Na^2/d)=Na^2\), as we have already computed. The quantity \(\BR\) defined here is the end-to-end vector of the chain. The RMS end-to-end distance is then \(\langle\BR^2\rangle^{1/2}=\sqrt{N} a\equiv R\ns_0\).

    A related figure of merit is the radius of gyration, \(R\ns_\Rg\) , defined by

    \[R_\Rg^2={1\over N}\Big\langle \sum_{n=1}^N\big(\BR\ns_n-\BR\ns_{\ssr{CM}}\big)^2 \Big\rangle \quad,\]

    where \(R\ns_{\ssr{CM}}={1\over N}\sum_{j=1}^N \BR\ns_j\) is the center of mass position. A brief calculation yields

    \[R_g^2=\big(N+3-4N^{-1}\big)\,a^2\sim {Na^2\over 6}\quad,\]

    in all dimensions.

    The total number of random walk configurations with end-to-end vector \(\BR\) is then \((2d)^N P\ns_N(\BR)\), so the entropy of a chain at fixed elongation is

    \[S(\BR,N)=\kB\ln\Big[ (2d)^N P\ns_N(\BR)\Big] = S(0,N) - {d\kB \BR^2\over 2 Na^2}\quad .\]

    If we assume that the energy of the chain is conformation independent, then \(E=E\ns_0(N)\) and

    \[F(\BR,N)=F(0,N) + {d\kT \BR^2\over 2 Na^2}\quad .\]

    In the presence of an external force \(F\ns_{ext}\), the Gibbs free energy is the Legendre transform

    \[G(\BF\ns_{ext},N)=F(\BR,N)-\BF\ns_{ext}\cdot\BR\quad ,\]

    and \(\pz G/\pz\BR=0\) then gives the relation

    \[\blangle \BR(\BF\ns_{ext},N)\brangle = {Na^2\over d\kT}\,\BF\ns_{ext}\quad .\]

    This may be considered an equation of state for the polymer.

    Following de Gennes, consider a chain with charges \(\pm e\) at each end, placed in an external electric field of magnitude \(E=30,000\,\RV/{cm}\). Let \(N=10^4\), \(a=2\,\)Å, and \(d=3\). What is the elongation? From the above formula, we have

    \[{R\over R\ns_0} = {eE R\ns_0\over 3\kT} = 0.8 \ ,\]

    with \(R\ns_0=\sqrt{N} a\) as before.

    clipboard_e4f4f7b5bb11b2d88995f90a414966b0f.png
    Figure \(\PageIndex{3}\): The polymer chain as a random coil.

    Structure factor

    We can also compute the structure factor,

    \[ \begin{align*} S(\Bk) &={1\over N}\,\Big\langle\sum_{m=1}^N\sum_{n=1}^N e^{i\Bk\cdot(\BR\ns_m-\BR\ns_n)}\Big\rangle\\[4pt] &=1+{2\over N}\,\sum_{m=1}^N\sum_{n=1}^{m-1} \Big\langle e^{i\Bk\cdot(\BR\ns_m-\BR\ns_n)}\Big\rangle \end{align*}\]

    For averages with respect to a Gaussian distribution,

    \[\blangle e^{i\Bk\cdot(\BR\ns_m-\BR\ns_n)} \brangle = \exp\bigg\{\!-{1\over 2}\>\Big\langle \big(\Bk\cdot(\BR\ns_m-\BR\ns_n)\big)^2 \Big\rangle\bigg\}\quad .\]

    Now for \(m>n\) we have \(\BR\ns_m-\BR\ns_n=\sum_{j=n+1}^m \Bdelta\ns_j\) , and therefore

    \[\Big\langle \big(\Bk\cdot(\BR\ns_m-\BR\ns_n)\big)^2 \Big\rangle=\sum_{j=n+1}^m \blangle (\Bk\cdot \Bdelta\ns_j)^2\brangle = {1\over d}\,(m-n)\,\Bk^2 a^2\quad,\]

    since \(\langle \delta_j^\alpha\,\delta_{j'}^\beta\rangle=(a^2/d)\,\delta\ns_{jj'}\delta^{\alpha\beta}\) . We then have

    \[S(\Bk)=1+{2\over N}\sum_{m=1}^N\sum_{n=1}^{m-1} e^{-(m-n) \,\Bk^2 a^2/2d}\\ ={N\,(e^{2\mu_\Bk}-1) -2\, e^{\mu_\Bk}\, (1-e^{-N\mu_\Bk})\over N\big(e^{\mu_\Bk}-1\big)^2}\quad ,\]

    where \(\mu_\Bk=\Bk^2 a^2/2d\). In the limit where \(N\to\infty\) and \(a\to 0\) with \(Na^2=R_0^2\) constant, the structure factor has a scaling form, \(S(\Bk)=N f(N\mu_\Bk)=(R\ns_0/a)\,f(k^2R_0^2/2d)\) , where

    \[f(x)={2\over x^2}\,\big(e^{-x}-1+x\big)= 1 - {x\over 3} + {x^2\over 12} + \ldots\ .\]

    Rouse model

    Consider next a polymer chain subjected to stochastic forcing. We model the chain as a collection of mass points connected by springs, with a potential energy \(U=\half k \sum_n \big(\Bx\nd_{n+1}-\Bx\nd_n\big)^2\). This reproduces the distribution of Equation \ref{polydist} if we take the spring constant to be \(k=3\kT/a^2\) and set the equilibrium length of each spring to zero. The equations of motion are then

    \[M{\ddot\Bx}\nd_n + \gamma\,{\dot\Bx}\nd_n=-k\big(2\Bx\nd_n-\Bx\nd_{n-1}-\Bx\nd_{n+1}\big)+\Bf\nd_n(t)\quad,\]

    where \(n\in\{1,\ldots,N\}\) and \(\{f^\mu_n(t)\}\) a set of Gaussian white noise forcings, each with zero mean, and

    \[\blangle f^\mu_n(t)\,f^\nu_{n'}(t')\brangle=2\gamma\kT\,\delta\nd_{nn'}\,\delta^{\mu\nu}\,\delta(t-t')\quad.\]

    We define \(\Bx\nd_0\equiv \Bx\nd_1\) and \(\Bx\nd_{N+1}\equiv\Bx\nd_N\) so that the end mass points \(n=1\) and \(n=N\) experience a restoring force from only one neighbor. We assume the chain is overdamped and set \(M\to 0\). We then have

    \[\gamma\,{\dot\Bx}\nd_n=-k\sum_{n'=1}^N A\nd_{nn'}\,\Bx\nd_{n'} + \Bf\nd_n(t)\quad,\]

    where

    \[A=\begin{pmatrix} 1 & -1 & 0 & 0 & \cdots & 0 \\ -1 & 2 & -1 & 0 & \cdots & 0 \\ 0 & -1 & 2 & -1 & \cdots & 0 \\ 0 & 0 & -1 & \ddots & \cdots & \vdots \\ \vdots & & \ddots & \ddots & 2 & -1 \\ 0 & \cdots & \cdots & 0 & -1 & 1 \end{pmatrix}\quad.\]

    The matrix \(A\) is real and symmetric. Its eigenfunctions are labeled \(\psi\nd_j(n)\), with \(j\in\{0,\ldots,N-1\}\):

    \[\begin{split} \psi\nd_0(n)&={1\over\sqrt{N}}\\ \psi\nd_j(n)&=\sqrt{2\over N}\,\cos\!\bigg({(2n-1)j\pi\over 2N}\bigg) \quad,\quad j\in\{1,\ldots,N-1\} \end{split}\]

    The completeness and orthonormality relations are

    \[\sum_{j=0}^{N-1}\psi\nd_j(n)\,\psi\nd_j(n')=\delta\nd_{nn'}\qquad,\qquad\sum_{n=1}^N\psi\nd_j(n)\,\psi\nd_{j'}(n)=\delta\nd_{jj'}\quad,\]

    with eigenvalues \(\lambda\nd_j=4\sin^2\!\big(\pi j/2N\big)\). Note that \(\lambda\nd_0=0\).

    We now work in the basis of normal modes \(\{\eta^\mu_j\}\), where

    \[\eta^\mu_j(t)=\sum_{n=1}^N\psi\nd_j(n)\,x^\mu_n(t)\qquad,\qquad x^\mu_n(t)=\sum_{j=0}^{N-1}\psi\nd_j(n)\,\eta^\mu_j(t)\quad.\]

    We then have

    \[{d\Beta\nd_j\over dt}=-{1\over\tau\nd_j}\,\Beta\nd_j + \Bg\nd_j(t)\quad, \label{RouseNM}\]

    where the \(j^{th}\) relaxation time is

    \[\tau\nd_j={\gamma\over 4k\sin^2\!\big(\pi j/2N\big)}\]

    and

    \[g^\mu_j(t)=\gamma^{-1}\sum_{n=1}^N\psi\nd_j(n)\,f^\mu_n(t)\quad.\]

    Note that

    \[\blangle g^\mu_j(t)\,g^\nu_{j'}(t')\brangle=2\gamma^{-1}\kT\,\delta\nd_{jj'}\,\delta^{\mu\nu}\,\delta(t-t')\quad.\]

    Integrating Equation \ref{RouseNM}, we have for, \(j=0\),

    \[\Beta\nd_0(t)=\Beta\nd_0(0)+\int\limits_0^t\!\!dt'\>\Bg\nd_0(t')\quad.\]

    For the \(j>0\) modes,

    \[\Beta\nd_j(t)=\Beta\nd_j(0)\,e^{-t/\tau\nd_j}+\int\limits_0^t\!\!dt'\>\Bg\nd_j(t')\,e^{(t'-t)/\tau\nd_j}\quad.\]

    Thus,

    \[\begin{split} \blangle\eta^\mu_0(t)\,\eta^\nu_0(t')\brangle\nd_\Rc&=2\gamma^{-1}\kT\,\delta^{\mu\nu}\,\textsf{min}(t,t')\\ \blangle\eta^\mu_j(t)\,\eta^\nu_j(t')\brangle\nd_\Rc&=\gamma^{-1}\kT\,\delta^{\mu\nu}\,\tau\nd_j\,\Big( e^{-|t-t'|/\tau\nd_j}-e^{-(t+t')/\tau\nd_j}\Big)\quad, \end{split}\]

    where the ‘connected average’ is defined to be \(\langle A(t)\,B(t')\rangle\nd_\Rc\equiv \langle A(t)\,B(t')\rangle-\langle A(t)\rangle\langle B(t')\rangle\). Transforming back to the original real space basis, we then have

    \[\blangle x^\mu_n(t)\,x^\nu_{n'}(t')\brangle\nd_\Rc = {2\kT\over N\gamma }\,\delta^{\mu\nu}\textsf{min}(t,t') + {\kT\over \gamma}\delta^{\mu\nu}\sum_{j=1}^{N-1} \tau\nd_j\,\psi\nd_j(n)\,\psi\nd_{j}(n')\,\Big( e^{-|t-t'|/\tau\nd_j}-e^{-(t+t')/\tau\nd_j}\Big)\quad.\]

    In particular, the ‘connected variance’ of \(\Bx\nd_n(t)\) is

    \[\textsf{CVar}\big[\Bx\ns_n(t)\big]\equiv \blangle \big[\Bx\nd_n(t)\big]^2\brangle\nd_\Rc={6\kT\over N\gamma}\,t + {3\kT\over\gamma}\sum_{j=1}^{N-1} \tau\nd_j\,\big[\psi\nd_j(n)\big]^2\,\big(1-e^{-2t/\tau\nd_j}\big)\quad. \label{Rousevar}\]

    From this we see that at long times, when \(t\gg\tau\ns_1\) , the motion of \(\Bx\nd_n(t)\) is diffusive, with diffusion constant \(D=\kT/N\gamma\propto B^{-1}\), which is inversely proportional to the chain length. Recall the Stokes result \(\gamma=6\pi\eta R/M\) for a sphere of radius \(R\) and mass \(M\) moving in a fluid of dynamical viscosity \(\eta\). From \(D=\kT/\gamma M\), shouldn’t we expect the diffusion constant to be \(D=\kT/6\pi\eta R\propto N^{-1/2}\), since the radius of gyration of the polymer is \(R\nd_\Rg\propto N^{1/2}\) ? This argument smuggles in the assumption that the only dissipation is taking place at the outer surface of the polymer, modeled as a ball of radius \(R\ns_\Rg\). In fact, for a Gaussian random walk in three space dimensions, the density for \(r<R\ns_\Rg\) is \(\rho\propto N^{-1/2}\) since there are \(N\) monomers inside a region of volume \(\big(\sqrt{N}\,\big)^3\). Accounting for Flory swelling due to steric interactions (see below), the density is \(\rho\sim N^{-4/5}\), which is even smaller. So as \(N\to\infty\), the density within the \(r=R\ns_\Rg\) effective sphere gets small, which means water molecules can easily penetrate, in which case the entire polymer chain should be considered to be in a dissipative environment, which is what the Rouse model says – each monomer executed overdamped motion.

    A careful analysis of Equation \ref{Rousevar} reveals that there is a subdiffusive regime12 where \(\textsf{CVar}\big[\Bx\ns_n(t)\big]\propto t^{1/2}\). To see this, first take the \(N\gg 1\) limit, in which case we may write \(\tau\ns_j=N^2\tau\ns_0/j^2\), where \(\tau\ns_0\equiv \gamma/\pi^2 k\) and \(j\in\{1,\ldots,N-1\}\). Let \(s\equiv (n-\half)/N \in [0,1]\) be the scaled coordinate along the chain. The second term in Equation \ref{Rousevar} is then

    \[S(s,t)\equiv {6\kT\over\gamma}\cdot{\tau\ns_1\over N}\sum_{j=1}^{N-1} {\cos^2(\pi j s)\over j^2}\,\big(1-e^{-2 j^2 t/\tau\ns_1}\big)\quad. \label{RouseS}\]

    Let \(\sigma\equiv (t/\tau\ns_1)^{1/2}\). When \(t\ll\tau\ns_1\) , \(\sigma\ll 1\), we have

    \[S(s,t)\simeq {6\kT\over\gamma}\cdot{\tau\ns_1\over N}\>\sigma\!\!\int\limits_0^{N\sigma} \!\!du\>{\cos^2(\pi u s/\sigma)\over u^2}\,\big(1-e^{-2u^2}\big)\quad.\]

    Since \(s/\sigma\gg 1\), we may replace the cosine squared term by its average \(\half\) . If we further assume \(N\sigma\gg 1\), which means we are in the regime \(1\ll t/\tau\ns_0\ll N^2\), after performing the integral we obtain the result

    \[S(s,t)={3\kT\over\gamma}\,\sqrt{2\pi\tau\ns_0 t\,}\quad, \label{SSTRouse}\]

    provided \(s=\CO(1)\) , the site \(n\) is not on either end of the chain. The result in Equation \ref{SSTRouse} dominates the first term on the RHS of Equation \reF{Rousevar} since \(\tau\ns_0\ll t\ll\tau\ns_1\). This is the subdiffusive regime.

    When \(t\gg\tau\ns_1=N^2\tau\ns_0\), the exponential on the RHS of Equation \ref{RouseS} is negligible, and if we again approximate \(\cos^2(\pi j s)\simeq\half\), and we extend the upper limit on the sum to infinity, we find \(S(t)=(3\kT/\gamma)(\tau\ns_1/N)(\pi^2/6)\propto t^0\), which is dominated by the leading term on the RHS of Equation \ref{Rousevar}. This is the diffusive regime, with \(D=\kT/N\gamma\).

    Finally, when \(t\ll\tau\ns_0\), the factor \(1-\exp(-2t/\tau\ns_j)\) may be expanded to first order in \(t\). One then obtains \(\textsf{CVar}\big[\Bx\ns_n(t)\big]=(6\kT/\gamma)\,t\), which is independent of the force constant \(k\). In this regime, the monomers don’t have time to respond to the force from their neighbors, hence they each diffuse independently. On such short time scales, however, one should check to make sure that inertial effects can be ignored, that \(t\gg M/\gamma\).

    One serious defect of the Rouse model is its prediction of the relaxation time of the \(j=1\) mode, \(\tau\nd_1\propto N^2\). The experimentally observed result is \(\tau\nd_1\propto N^{3/2}\). We should stress here that the Rouse model applies to ideal chains. In the theory of polymer solutions, a theta solvent is one in which polymer coils act as ideal chains. An extension of the Rouse model, due to my former UCSD colleague Bruno Zimm, accounts for hydrodynamically-mediated interactions between any pair of ‘beads’ along the chain. Specifically, the Zimm model is given by

    \[{dx^\mu_n\over dt}=\sum_{n'} H^{\mu\nu}(\Bx\nd_n-\Bx\nd_{n'})\Big[k\big(x^\nu_{n'+1}+x^\nu_{n'-1}-2x^\nu_{n'}\big)+f^\nu_{n'}(t)\Big]\quad,\]

    where

    \[H^{\mu\nu}(\BR)={1\over 6\pi\eta R}\big(\delta^{\mu\nu}+\HR^\mu\HR^\nu\big)\]

    is known as the Oseen hydrodynamic tensor (1927) and arises when computing the velocity in a fluid at position \(\BR\) when a point force \(\BF=\Bf\,\delta(\Br)\) is applied at the origin. Typically one replaces \(H(\BR)\) by its average over the equilibrium distribution of polymer configurations. Zimm’s model more correctly reproduces the behavior of polymers in \(\theta\)-solvents.

    Flory Theory of Self-Avoiding Walks

    What is missing from the random walk free energy is the effect of steric interactions. An argument due to Flory takes these interactions into account in a mean field treatment. Suppose we have a chain of radius \(R\). Then the average monomer density within the chain is \(c=N/R^d\). Assuming short-ranged interactions, we should then add a term to the free energy which effectively counts the number of near self-intersections of the chain. This number should be roughly \(Nc\). Thus, we write

    \[F(\BR,N)=F\ns_0 + u(T)\,{N^2\over R^d} + \half d \kT\,{R^2\over Na^2}\quad .\]

    The effective interaction \(u(T)\) is positive in the case of a so-called ‘good solvent’.

    The free energy is minimized when

    \[0={\pz F\over\pz R}= -{dvN^2\over R^{d+1}} + d\kT\,{R\over Na^2}\quad ,\]

    which yields the result

    \[R\ns_\RF(N)=\bigg({u a^2\over\kT}\bigg)^{\!\!1/(d+2)}N^{3/(d+2)}\propto N^\nu\quad .\]

    Thus, we obtain \(\nu=3/(d+2)\). In \(d=1\) this says \(\nu=1\), which is exactly correct because a SAW in \(d=1\) has no option but to keep going in the same direction. In \(d=2\), Flory theory predicts \(\nu=\frac{3}{4}\), which is also exact. In \(d=3\), we have \(\nu\ns_{d=3}=\frac{3}{5}\), which is extremely close to the numerical value \(\nu=0.5880\). Flory theory is again exact at the SAW upper critical dimension, which is \(d=4\), where \(\nu=\half\), corresponding to a Gaussian random walk13. Best. Mean. Field. Theory. Ever.

    How well are polymers described as SAWs? Figure \(\PageIndex{4}\) shows the radius of gyration \(R\ns_\Rg\) versus molecular weight \(M\) for polystyrene chains in a toluene and benzene solvent. The slope is \(\nu=d\ln R\ns_\Rg/d\ln M=0.5936\). Experimental results can vary with concentration and temperature, but generally confirm the validity of the SAW model.

    For a SAW under an external force, we compute the Gibbs partition function,

    \[Y(\BF\ns_{ext},N)=\int \! d^d\!R\>P\ns_N(\BR)\,e^{\BF\ns_{ext}\cdot\BR/\kT} = \int\!d^d\!x\>f(x)\>e^{s\nhat\cdot\Bx}\ ,\]

    where \(x=R/R\ns_\RF\) and \(s=\kT/R\ns_\RF F\ns_{ext}\) and \(\nhat={\hat\BF}\ns_{ext}\). One than has \(R(F\ns_{ext})=R\ns_\RF\,\Phi(R\ns_\RF/\xi)\), where \(\xi=\kT/F\ns_{ext}\) and \(R(F\ns_{ext})=F\ns_{ext}R^2_\RF/\kT\). For small values of its argument one has \(\Phi(u)\propto u\). For large \(u\) it can be shown that \(R(F\ns_{ext})\propto (F\ns_{ext}R\ns_\RF/\kT)^{2/3}\).

    clipboard_efa3133c25792ff82399992e5124dfba0.png
    Figure \(\PageIndex{4}\): Radius of gyration \(R_\Rg\) of polystyrene in a toluene and benzene solvent, plotted as a function of molecular weight of the polystyrene. The best fit corresponds to a power law \(R_\Rg\propto M^\nu\) with \(\nu=0.5936\). From J. Des Cloizeaux and G. Jannink, Polymers in Solution: Their Modeling and Structure (Oxford, 1990).

    On a lattice of coordination number \(z\), the number of \(N\)-step random walks starting from the origin is \(\ROmega\ns_N=z^N\). If we constrain our random walks to be self-avoiding, the number is reduced to

    \[\ROmega^{\ssf{SAW}}_N=\CC\,N^{\gamma-1}\,y^N\quad ,\]

    where \(\CC\) and \(\gamma\) are dimension-dependent constants, and we expect \(y \ltwid z-1\), since at the very least a SAW cannot immediately double back on itself. In fact, on the cubic lattice one has \(z=6\) but \(y=4.68\), slightly less than \(z-1\). One finds \(\gamma\ns_{d=2}\simeq\frac{4}{3}\) and \(\gamma\ns_{d=3}\simeq \frac{7}{6}\). The RMS end-to-end distance of the SAW is

    \[R\ns_\RF=a\,N^\nu\quad ,\]

    where \(a\) and \(\nu\) are \(d\)-dependent constants,with \(\nu\ns_{d=1}=1\), \(\nu\ns_{d=2}\simeq\frac{3}{4}\), and \(\nu\ns_{d=3}\simeq\frac{3}{5}\). The distribution \(P\ns_N(\BR)\) has a scaling form,

    \[P\ns_N(R)={1\over R^d_\RF}\,f\bigg({R\over R\ns_\RF}\bigg) \qquad (a \ll R \ll Na)\quad .\]

    One finds

    \[f(x)\sim\begin{cases} x^g & x\ll 1 \\ \exp(-x^\delta) & x \gg 1\ , \end{cases}\]

    with \(g=(\gamma-1)/\nu\) and \(\delta=1/(1-\nu)\).

    Polymers and Solvents

    Consider a solution of monodisperse polymers of length \(N\) in a solvent. Let \(\phi\) be the dimensionless monomer concentration, so \(\phi/N\) is the dimensionless polymer concentration and \(\phi\ns_\Rs=1-\phi\) is the dimensionless solvent concentration. (Dimensionless concentrations are obtained by dividing the corresponding dimensionful concentration by the overall density.) The entropy of mixing for such a system is given by Equation 2.352. We have

    \[S\ns_{mix}=-{V\kB\over v\ns_0}\cdot\bigg\{ {1\over N}\phi\ln\phi + (1-\phi)\ln(1-\phi) \bigg\} \ ,\]

    where \(v\ns_0\propto a^3\) is the volume per monomer. Accounting for an interaction between the monomer and the solvent, we have that the free energy of mixing is

    \[{v\ns_0\,F_{mix}\over V\kT}={1\over N}\,\phi\ln\phi+(1-\phi)\ln(1-\phi)+\xhi\,\phi(1-\phi)\ .\]

    where \(\xhi\) is the dimensionless polymer-solvent interaction, called the Flory parameter. This provides a mean field theory of the polymer-solvent system.

    The osmotic pressure \(\Pi\) is defined by

    \[\Pi=-{\pz F\ns_{mix}\over \pz V}\bigg|_{N\ns_\Rp} \ ,\]

    which is the variation of the free energy of mixing with respect to volume holding the number of polymers constant. The monomer concentration is \(\phi=N N\ns_\Rp v\ns_0/V\), so

    \[{\pz\over\pz V}\bigg|_{N_\Rp}=-{\phi^2\over N N_\Rp\, v\ns_0}\,{\pz\over\pz\phi}\bigg|_{N_\Rp}\ .\]

    Now we have

    \[F_{mix}=N N_\Rp\, \kT\left\{ {1\over N}\,\ln\phi+(\phi^{-1}-1)\ln(1-\phi) +\xhi\,(1-\phi)\right\}\ ,\]

    and therefore

    \[\Pi ={\kT\over v\ns_0}\Big[(N^{-1}-1)\,\phi-\ln(1-\phi)-\xhi\,\phi^2\Big]\ .\]

    In the limit of vanishing monomer concentration \(\phi\to 0\), we recover

    \[\Pi={\phi\,\kT\over N v\ns_0}\ ,\]

    which is the ideal gas law for polymers.

    For \(N^{-1}\ll\phi\ll 1\), we expand the logarithm and obtain

    \[\begin{split} {v\ns_0\,\Pi\over\kT}&={1\over N}\phi + \half(1-2\xhi)\,\phi^2+\CO(\phi^3)\\ &\approx\half(1-2\xhi)\,\phi^2\ . \label{osmopoly} \end{split}\]

    Note that \(\Pi>0\) only if \(\xhi<\half\), which is the condition for a ’good solvent’.

    In fact, Equation \ref{osmopoly} is only qualitatively correct. In the limit where \(\xhi\ll\half\), Flory showed that the individual polymer coils behave much as hard spheres of radius \(R\ns_\RF\). The osmotic pressure then satisfies something analogous to a virial equation of state:

    \[\begin{split} {\Pi\over\kT}&={\phi\over Nv\ns_0} + A\,\bigg({\phi\over N v\ns_0}\bigg)^{\!\!2}R_\RF^3 + \ldots \\ &={\phi\over N v\ns_0}\,h(\phi/\phi^*)\ . \end{split}\]

    This is generalized to a scaling form in the second line, where \(h(x)\) is a scaling function, and \(\phi^*=Nv\ns_0/R_\RF^3\propto N^{-4/5}\), assuming \(d=3\) and \(\nu=\frac{3}{5}\) from Flory theory. As \(x=\phi/\phi^*\to 0\), we must recover the ideal gas law, so \(h(x)=1+\CO(x)\) in this limit. For \(x\to\infty\), we require that the result be independent of the degree of polymerization \(N\). This means \(h(x)\propto x^p\) with \(\frac{4}{5} p = 1\), \(p=\frac{5}{4}\). The result is known as the des Cloiseaux law:

    \[{v\ns_0\Pi\over\kT}=C\,\phi^{9/4}\ ,\]

    where \(C\) is a constant. This is valid for what is known as semi-dilute solutions, where \(\phi^*\ll\phi\ll 1\). In the dense limit \(\phi\sim 1\), the results do not exhibit this universality, and we must appeal to liquid state theory, which is no fun at all.