End to end vector square average

In accordance with the random walk model, the end to end vector average vanishes due to symmetry considerations. Therefore, in order to get an estimate of the polymer size, we turn to the end to end vector variance: ${\displaystyle \left\langle {\vec {R}}^{2}\right\rangle =Nl^{2}}$ with the end to end vector defined as: ${\displaystyle \textstyle {\vec {R}}\equiv \sum _{n=1}^{N}{\vec {r}}_{n}}$.

Thus, a first crude approximation for the polymer size is simply ${\displaystyle R_{0}\equiv {\sqrt {\left\langle {\vec {R}}^{2}\right\rangle }}={\sqrt {N}}l}$.

End to end vector probability distribution

As mentioned, we are usually interested in statistical features of the polymer configuration. A central quantity will therefore be the end to end vector probability distribution:

${\displaystyle \Phi ({\vec {R}},N)=\left({\frac {3}{2\pi Nl^{2}}}\right)^{\frac {3}{2}}\exp \left(-{\frac {3{\vec {R}}^{2}}{2Nl^{2}}}\right)}$

Note that the distribution depends only on the end to end vector magnitude. Also, the above expression gives non-zero probability for sizes larger than ${\displaystyle Nl}$, clearly an unreasonable result which stems from the limit taken ${\displaystyle N\rightarrow \infty }$ for its derivation.

Governing differential equation

Taking the limit of a smooth spatial contour for the polymer conformation, that is, taking the limits ${\displaystyle N\rightarrow \infty }$ and ${\displaystyle l\rightarrow 0,}$ under the constraint ${\displaystyle Nl=const}$ one comes to a differential equation for the probability distribution:

${\displaystyle {\frac {\partial \Phi }{\partial N}}={\frac {l^{2}}{6}}\nabla ^{2}\Phi }$

With the laplacian ${\displaystyle \textstyle \nabla ^{2}}$ taken in respect to actual space. One way to derive this equation is via Taylor expansion to ${\displaystyle \Phi ({\vec {R}},N}$) and ${\displaystyle \Phi ({\vec {R}},N+\Delta N).}$

One might wonder why bother with a differential equation for a function already analytically obtained, but as will be demonstrated, this equation can also be generalized for non-ideal circumstances.

Path integral expression

Under the same assumption of a smooth contour, the distribution function can be expressed using a path integral:

${\displaystyle \Phi ({\vec {R}},N)=\int _{0,0}^{{\vec {R}},N}\exp \left\{-\int _{0}^{N}L_{0}d\nu \right\}{\mathcal {D}}{\vec {R}}(\nu )}$

Where we defined ${\displaystyle \textstyle L_{0}={\frac {3}{2l^{2}}}\left({\frac {d{\vec {R}}}{d\nu }}\right)^{2}.}$

Here ${\displaystyle \nu }$ acts as a parametrization variable for the polymer, describing in effect its spatial configuration, or contour.

The exponent is a measure for the number density of polymer configurations in which the shape of the polymer is close to a continuous and differentiable curve.^[3]

Dust

Consider a space filled with small impenetrable particles, or "dust". Denote the fraction of space excluding a monomer end point by ${\displaystyle f({\vec {R}})}$ so its values range: ${\displaystyle 0\leq f({\vec {R}})\leq 1}$.

Constructing a Taylor expansion for ${\displaystyle \Phi ({\vec {R}},N+\Delta N).}$, one can arrive at the new governing differential equation:

${\displaystyle {\frac {\partial \Phi }{\partial N}}={\frac {l^{2}}{6}}\nabla ^{2}-f\Phi }$

For which the corresponding path integral is:

${\displaystyle \Phi ({\vec {R}},N)=\int _{0,0}^{{\vec {R}},N}\exp \left\{-\int _{0}^{N}[L_{0}+f({\vec {R}})]d\nu \right\}{\mathcal {D}}{\vec {R}}(\nu )}$

Walls

To model a perfect rigid wall, simply set ${\displaystyle \textstyle {\frac {f({\vec {R}})}{l^{2}}}\rightarrow +\infty }$ for all regions in space out of reach of the polymer due to the wall contour.

The walls a polymer usually interacts with are complex structures. Not only can the contour be full of bumps and twists, but their interaction with the polymer is far from the rigid mechanical idealization depicted above. In practice, a polymer will often be "absorbed" or condense on the wall due to attractive intermolecular forces. Due to heat, this process is counteracted by an entropy driven process, favoring polymer configurations that correspond to large volumes in phase space. A thermodynamic adsorption-desorption process arises. One common example for this are polymers confined within a cell membrane.

To account for the attraction forces, define a potential per monomer denoted as: ${\displaystyle \textstyle V({\vec {R}})}$. The potential will be incorporated through a Boltzmann factor. Taken for the entire polymer this takes the form:

${\displaystyle \exp \left\{-\beta \sum _{j=0}^{N}V({\vec {R}}_{j})\right\}\cong \exp \left\{-\beta \int _{0}^{N}V({\vec {R}}(\nu ))\right\}}$

Where we used ${\displaystyle \beta =(k_{b}T)^{-}1}$ with ${\displaystyle T}$ as Temperature and ${\displaystyle k_{b}}$ the Boltzmann constant. In the right hand side, our usual limits ${\displaystyle N\rightarrow \infty \quad \&\quad L\rightarrow 0}$ were taken.

The number of polymer configurations with fixed endpoints can now be determined by the path integral:

${\displaystyle Q_{V}({\vec {R}}_{N},N|{\vec {R}}_{0},0)=\int _{{\vec {R}}_{0},0}^{{\vec {R}}_{N},N}\exp \left\{-\int _{0}^{N}[L_{0}]d\nu \right\}{\mathcal {D}}{\vec {R}}(\nu )}$

Similarly to the ideal polymer case, this integral can be interpreted as a propagator for the differential equation:

${\displaystyle {\frac {\partial f}{\partial N}}={\frac {l^{2}}{6}}\nabla ^{2}f-\beta V({\vec {R}})f}$

This leads to a bi-linear expansion for ${\displaystyle Q_{V}({\vec {R}}_{N},N|{\vec {R}}_{0},0)=\sum _{n}f_{n}({\vec {R}}_{N})f_{n}^{*}({\vec {R}}_{0})\exp(-E_{N}N)}$ in terms of orthonormal eigenfunctions and eigenvalues:

${\displaystyle \left[{\frac {l^{2}}{6}}\nabla ^{2}f-\beta V({\vec {R}})\right]f_{n}({\vec {R}}_{n})=E+nf_{n}({\vec {R}}_{n})}$

and so our absorption problem is reduced to an eigenfunction problem.

For a typical well like (attractive) potential this leads to two regimes for the absorption phenomenon, with the critical temperature ${\displaystyle T_{c}}$ determined by the specific problem parameters ${\displaystyle l,V({\vec {R}})}$ :

In high temperatures ${\displaystyle T>T_{c}}$, the potential well has no bound states, meaning all eigenvalues are positive and the corresponding eigenfunction takes the asymptotic form ${\displaystyle <(x\rightarrow \infty )}$:

${\displaystyle f_{n}\cong A_{n}\sin({\sqrt {6\lambda _{n}/L^{2}}}x)+B_{m}\cos({\sqrt {6\lambda _{m}/L^{2}}}x)}$ with ${\displaystyle \lambda _{n}}$ denoting the calculated eigenvalues.

The result is shown for the x coordinate after a separation of variables and assuming a surface at ${\displaystyle x=0}$. This expression represents a very open configuration for the polymer, away from the surface, meaning the polymer is desorbed.

For low enough temperatures ${\displaystyle T<T_{c}}$, there exist at least one bounded state with a negative eigenvalue. In our "large polymer" limit, this means that the bi-linear expansion will be dominated by the ground state, which asymptotically ${\displaystyle (x\rightarrow \infty )}$ takes the form:

${\displaystyle f(x_{0})\cong A_{0}\exp(-{\sqrt {6|\lambda _{0}|/l^{2}}}x)}$

This time the configurations of the polymer are localized in a narrow layer near the surface with an effective thickness ${\displaystyle \textstyle {\frac {l}{\sqrt {6|\lambda _{0}|}}}}$

A wide variety of adsorption problems boasting a host of "wall" geometries and interaction potentials can be solved using this method. To obtain a quantitatively well defined result one has to use the recovered eigenfunctions and construct the corresponding configuration sum.

For a complete and rigorous solution see.^[11]

Excluded volume

Another obvious obstruction, thus far blatantly disregarded, is the interactions between monomers within the same polymer. An exact solution for the number of configurations under this very realistic constraint has not yet been found for any dimension larger than one.^[3] This problem has historically came to be known as the excluded volume problem. To better understand the problem, one can imagine a random walk chain, as previously presented, with a small hard sphere (not unlike the "specks of dust" mentioned above) at the endpoint of each monomer. The radius of these spheres necessarily obeys ${\displaystyle r<l/2}$, otherwise successive spheres would overlap.

A path integral approach affords a relatively simple method to derive an approximated solution:^[12] The results presented are for three dimensional space, but can be easily generalized to any dimensionality. The calculation is based on two reasonable assumptions:

1. Statistical characteristics for the volume excluded case resemble that of a polymer without excluded volume but with a fraction ${\displaystyle f({\vec {R}})}$ occupied by small spheres of an identical volume to the hypothesized monomer sphere.
2. These aforementioned characteristics can be approximated by a calculation of the most probable chain configuration.

In accordance with the path integral expression for ${\displaystyle \textstyle Q_{V}({\vec {R}}_{N},N|{\vec {R}}_{0}.0)}$ previously presented, the most probable configuration will be the curve ${\displaystyle {\vec {R}}^{*}(\nu )}$ that minimizes the exponent of the original path integral:

${\displaystyle S[{\vec {R}}(\nu )]\equiv \int _{0}^{N}\left\{{\frac {3}{2l^{2}}}\left({\frac {d{\vec {R}}}{d\nu }}\right)^{2}+f({\vec {R}})\right\}d\nu }$

To minimize the expression, employ calculus of variations and obtain the Euler–Lagrange equation:

${\displaystyle {\frac {3}{l^{2}}}{\frac {d^{2}{\vec {R}}^{*}}{d\nu ^{2}}}=\nabla f({\vec {R}}^{*})}$

We set ${\displaystyle R\equiv R^{*}}$.

To determine the appropriate function ${\displaystyle f({\vec {R}})}$, consider a sphere of radius ${\displaystyle R}$, thickness ${\displaystyle dR}$ and profile ${\displaystyle 4\pi R^{2}}$ centered around the origin of the polymer. The average number of monomers in this shell should equal ${\displaystyle \textstyle {\frac {4\pi R^{2}}{(4/3)\pi r^{3}}}f(R)dR}$.

On the other hand, the same average should also equal ${\displaystyle \textstyle d\nu =\left({\frac {dR}{d\nu }}\right)^{-1}}$ (Remember that ${\displaystyle \nu }$ was defined as a parametrization factor with values ${\displaystyle 0\leq \nu \leq N}$). This equality results in:

${\displaystyle f({\vec {R}})={\frac {(4/3)\pi r^{3}}{4\pi }}R^{2}\left({\frac {dR}{d\nu }}\right)^{-1}}$

We find ${\displaystyle S[{\vec {R}}(\nu )]}$ can now be written as:

${\displaystyle S[{\vec {R}}(\nu )]=\int _{0}^{N}\left\{{\frac {3}{2l^{2}}}\left({\frac {dR}{d\nu }}\right)^{2}+{\frac {(4/3)\pi r^{3}}{4\pi }}R^{2}\left({\frac {dR}{d\nu }}\right)^{-1}\right\}d\nu }$

We again use variation calculus to arrive at:

${\displaystyle \left\{{\frac {3}{l^{2}}}+{\frac {2(4/3)\pi r^{3}}{4\pi }}R^{2}\left({\frac {dR}{d\nu }}\right)^{-3}\right\}{\frac {d^{2}R}{d\nu ^{2}}}+4{\frac {(4/3)\pi r^{3}}{4\pi }}R^{2}\left({\frac {dR}{d\nu }}\right)^{-1}=0}$

Note that we now have an ODE for ${\displaystyle R(\nu )}$ without any ${\displaystyle f({\vec {R}}^{*})}$ dependence. Although quite horrendous to look at, this equation has a fairly simple solution:

${\displaystyle R(\nu )=\left({\frac {3\pi }{(4/3)\pi r^{3}L^{2}}}\right)^{-1/5}\left({\frac {3}{5}}\right)^{-3/5}\nu ^{3/5}}$

We arrived at the important conclusion that for a polymer with excluded volume the end to end distance grows with N like:

${\displaystyle R\cong \left({\frac {3\pi }{(4/3)\pi r^{3}L^{2}}}\right)^{-1/5}N^{3/5}}$, a first departure from the ideal model result: ${\displaystyle R\sim {\sqrt {N}}}$.

Conformational distribution

So far, the only polymer parameters incorporated into the calculation were the number of monomers ${\displaystyle N}$ which was taken to infinity, and the constant bond length ${\displaystyle l}$. This is usually sufficient, as that is the only way the local structure of the polymer affects the problem. To try and do a bit better than the "constant bond distance" approximation, let us examine the next most rudimentary approach; A more realistic description of the single bond length will be a Gaussian distribution:^[13]

${\displaystyle \psi ({\vec {r}})=\left({\frac {3}{2\pi l^{2}}}\right)^{3/2}\exp \left(-{\frac {3{\vec {r}}^{2}}{2l^{2}}}\right)}$

So like before, we maintain the result: ${\displaystyle \langle {\vec {r}}^{2}\rangle =l^{2}}$. Note that although a bit more complex than before, ${\displaystyle \psi ({\vec {r}})}$ still has a single parameter - ${\displaystyle l}$.

The conformational distribution function for our new bond vector distribution is:

${\displaystyle {\begin{aligned}\Psi (\left\{{\vec {r}}_{n}\right\})&=\prod _{n=1}^{N}\psi ({\vec {r}}_{n})\\&=\prod _{n=1}^{N}\left({\frac {3}{2\pi l^{2}}}\right)^{3/2}\exp \left[-{\frac {3{\vec {r}}_{n}^{2}}{2l^{2}}}\right]\\&=\left({\frac {3}{2\pi l^{2}}}\right)^{3N/2}\exp \left[-\sum _{n=1}^{N}{\frac {3({\vec {R}}_{n}-{\vec {R}}_{n-1})^{2}}{2l^{2}}}\right].\end{aligned}}}$

Where we switched from the relative bond vector ${\displaystyle {\vec {r}}_{n}}$ to the absolute position vector difference: ${\displaystyle ({\vec {R}}_{n}-{\vec {R}}_{n-1})}$.

This conformation is known as the Gaussian chain. The Gaussian approximation for ${\displaystyle \psi ({\vec {r}})}$ does not hold for a microscopic analysis of the polymer structure but will yield accurate results for large-scale properties.

An intuitive way to construe this model is as a mechanical model of beads successively connected by a harmonic spring. The potential energy for such a model is given by:

${\displaystyle U_{0}(\{{\vec {R}}_{n}\})={\frac {3}{2l^{2}}}k_{b}T\sum _{n=1}^{N}({\vec {R}}_{n}-{\vec {R}}_{n-1})}$

At thermal equilibrium one can expect the Boltzmann distribution, which indeed recovers the result above for ${\displaystyle \Psi (\left\{{\vec {r}}_{n}\right\})}$.

An important property of the Gaussian chain is self-similarity. Meaning the distribution for ${\displaystyle {\vec {R}}_{n}-{\vec {R}}_{m}}$ between any two units is again Gaussian, depending only on ${\displaystyle l}$ and the unit to unit distance ${\displaystyle (n-m)}$:

${\displaystyle \phi ({\vec {R}}_{n}-{\vec {R}}_{m}|n-m)=\left({\frac {3}{2\pi l^{2}|n-m|}}\right)^{3/2}\exp \left[-{\frac {3({\vec {R}}_{n}-{\vec {R}}_{m})^{2}}{2|n-m|l^{2}}}\right]}$

This immediately leads to ${\displaystyle <({\vec {R}}_{n}-{\vec {R}}_{m})^{2}>=|n-m|l^{2}}$.

As was implicitly done in the section for spatial obstructions, we take the suffix ${\displaystyle n}$ to a continuous limit and replace ${\displaystyle {\vec {R}}_{n}-{\vec {R}}_{m}}$ by ${\displaystyle \partial {\vec {R}}_{n}/\partial n}$. So now, our conformational distribution is expressed by:

${\displaystyle \Psi (\left\{{\vec {r}}_{n}\right\})=\left({\frac {3}{2\pi l^{2}}}\right)^{3N/2}\exp \left[-{\frac {3}{2l^{2}}}\int _{0}^{N}dn\left({\frac {\partial {\vec {R}}_{n}}{\partial n}}\right)^{2}\right].}$

The independent variable transformed from a vector into a function, meaning ${\displaystyle \Psi [{\vec {R}}(n)]}$ is now a functional. This formula is known as the Wiener distribution.

Chain conformation under an external field

Assuming an external potential field ${\displaystyle U_{e}({\vec {R}})}$, the equilibrium conformational distribution described above will be modified by a Boltzmann factor:

${\displaystyle \Psi (\left\{{\vec {r}}_{n}\right\})=\left({\frac {3}{2\pi l^{2}}}\right)^{3N/2}\exp \left[-{\frac {3}{2l^{2}}}\int _{0}^{N}dn\left({\frac {\partial {\vec {R}}_{n}}{\partial n}}\right)^{2}-\beta \int _{0}^{N}dnU_{e}[{\vec {R}}(n)]\right].}$

An important tool in the study of a Gaussian chain conformational distribution is the Green function, defined by the path integral quotient:

${\displaystyle G({\vec {R}},{\vec {R}}';N)\equiv {\frac {\displaystyle \int _{{\vec {R}}_{0}={\vec {R}}'}^{{\vec {R}}_{N}={\vec {R}}}{\mathcal {D}}{\vec {R}}(n)\exp \left[-{\frac {3}{2l^{2}}}\displaystyle \int _{0}^{N}dn\left({\frac {\partial {\vec {R}}_{n}}{\partial n}}\right)^{2}-\beta \displaystyle \int _{0}^{N}duU_{e}[{\vec {R}}(n)]\right]}{\displaystyle \int d{\vec {R}}'\displaystyle \int d{\vec {R}}\displaystyle \int _{{\vec {R}}_{0}={\vec {R}}'}^{{\vec {R}}_{N}={\vec {R}}}{\mathcal {D}}{\vec {R}}_{n}\exp \left[-{\frac {3}{2l^{2}}}\displaystyle \int _{0}^{N}dn\left({\frac {\partial {\vec {R}}_{n}}{\partial n}}\right)^{2}\right]}}}$

The path integration is interpreted as a summation over all polymer curves ${\displaystyle {\vec {R}}(n)}$ that start from ${\displaystyle {\vec {R}}_{0}={\vec {R}}'}$ and terminate at ${\displaystyle {\vec {R}}_{N}={\vec {R}}}$.

For the simple zero field case ${\displaystyle U_{e}=0}$ The Green function reduces back to:

${\displaystyle G({\vec {R}}-{\vec {R}}';N)=\left({\frac {3}{2\pi l^{2}N}}\right)^{3/2}\exp \left[-{\frac {3({\vec {R}}-{\vec {R}}')^{2}}{2Nl^{2}}}\right]}$

In the more general case, ${\displaystyle G({\vec {R}}-{\vec {R}}';N)}$ plays the role of a weight factor in the complete partition function for all possible polymer conformations:

${\displaystyle Z=\int d{\vec {R}}~d{\vec {R}}'~G({\vec {R}}-{\vec {R}}';N).}$

There exists an important identity for the Green function that stems directly from its definition:

${\displaystyle G({\vec {R}},{\vec {R}}';N)=\int d{\vec {R}}''G({\vec {R}},{\vec {R}}'';N-n)G({\vec {R}}'',{\vec {R}}';N),\quad (0<n<N).}$

This equation has a clear physical significance, which might also serve to elucidate the concept of the path integral:

The product ${\displaystyle \textstyle G({\vec {R}},{\vec {R}}'';N-n)G({\vec {R}}'',{\vec {R}}';N))}$ expresses the weight factor for a chain which starts at ${\displaystyle R'}$, passes through ${\displaystyle R''}$ in ${\displaystyle n}$ steps, and ends at ${\displaystyle R}$ after ${\displaystyle N}$ steps. The integration over all possible midpoints ${\displaystyle R''}$ gives back the statistical weight for a chain starting at ${\displaystyle R'}$, and terminating at ${\displaystyle R}$. It should now be clear that the path integral is simply a sum over all possible literal paths the polymer can form between two fixed endpoints.

With the help of ${\displaystyle G({\vec {R}},{\vec {R}}';N)}$ the average of any physical quantity ${\displaystyle A}$ can be calculated. Assuming ${\displaystyle \textstyle A}$ depends only on the position of the ${\displaystyle n}$-th segment, then:

${\displaystyle \left\langle A({\vec {R}}_{n})\right\rangle ={\frac {\displaystyle \int d{\vec {R}}_{N}~d{\vec {R}}_{n}~d{\vec {R}}_{0}~G({\vec {R}}_{N},{\vec {R}}_{n};N-n)G({\vec {R}}_{n},{\vec {R}}_{0};n)A({\vec {R}}_{n})}{\displaystyle \int d{\vec {R}}_{N}~{\vec {d}}R_{0}~G({\vec {R}}_{N},{\vec {R}}_{0};N)}}}$

It stands to reason that A should depend on more than one monomer. assuming now it depends on ${\displaystyle {\vec {R}}_{m}}$ as well as ${\displaystyle {\vec {R}}_{n}}$ the average takes the form:

${\displaystyle \left\langle A({\vec {R}}_{n},{\vec {R}}_{m})\right\rangle ={\frac {\displaystyle \int d{\vec {R}}_{N}~d{\vec {R}}_{n}~d{\vec {R}}_{m}~d{\vec {R}}_{0}~G({\vec {R}}_{N},{\vec {R}}_{n};N-n)G({\vec {R}}_{n},{\vec {R}}_{m};n-m)A({\vec {R}}_{n},{\vec {R}}_{m})}{\displaystyle \int d{\vec {R}}_{N}~{\vec {d}}R_{0}~G({\vec {R}}_{N},{\vec {R}}_{0};N)}}}$

With an obvious generalization for more monomers dependence.

If one imposes the reasonable boundary conditions:

${\displaystyle {\begin{aligned}&G({\vec {R}},{\vec {R}}';N<0)=0\\&G({\vec {R}},{\vec {R}}';0)=\delta ({\vec {R}}-{\vec {R}}')\\\end{aligned}}}$

then with the help of a Taylor expansion for ${\displaystyle G({\vec {R}},{\vec {R}}';N+\Delta N)}$, a differential equation for ${\displaystyle G}$ can be derived:

${\displaystyle \left({\frac {\partial }{\partial N}}-{\frac {l^{2}}{6}}{\frac {\partial ^{2}}{\partial {\vec {R}}^{2}}}+\beta U_{e}({\vec {R}}))\right)G({\vec {R}},{\vec {R}}';N)=\delta ^{3}({\vec {R}}-{\vec {R}}')\delta (N).}$

With the help of this equation the explicit form of ${\displaystyle G({\vec {R}},{\vec {R}}';N)}$ is found for a variety of problems. Then, with a calculation of the partition function a host of statistical quantities can be extracted.