Path integrals in polymer science

Early attempts at path integrals can be traced back to 1918.{{cite journal | last=Daniell | first=P. J. | title=A General Form of Integral | journal=The Annals of Mathematics | publisher=JSTOR | volume=19 | issue=4 | pages=279–294 | year=1918 | issn=0003-486X | doi=10.2307/1967495 |doi-access=| jstor=1967495 }} A sound mathematical formalism wasn't established until 1921.{{cite journal | last=Wiener | first=N. | title=The Average of an Analytic Functional | journal=Proceedings of the National Academy of Sciences | volume=7 | issue=9 | date=1 August 1921 | issn=0027-8424 | doi=10.1073/pnas.7.9.253 | pages=253–260| pmid=16576602 |doi-access=free | pmc=1084890| bibcode=1921PNAS....7..253W }} This eventually lead Richard Feynman to construct a formulation for quantum mechanics,R.P. Feynman, "[http://virgilio.mib.infn.it/~zanotti/FNSN/FNSN_files/FNSN/QFT+calculi/Feynman-thesis.pdf The Principle of Least Action in quantum Mechanics]," Pd.d Thesis, Princeton University (1942), unpublished. now commonly known as Feynman Integrals.

In the core of Path integrals lies the concept of Functional integration. Regular integrals consist of a limiting process where a sum of functions is taken over a space of the function's variables. In functional integration the sum of functionals is taken over a space of functions. For each function the functional returns a value to add up.

Path integrals should not be confused with line integrals which are regular integrals with the integration evaluated along a curve in the variable's space.

Not very surprisingly functional integrals often diverge, therefore to obtain physically meaningful results a quotient of path integrals is taken.

This article will use the notation adopted by Feynman and Hibbs,R.P. Feynman and A.R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965). denoting a path integral as:

:$\backslash int\; G[f(x)]\; \backslash mathcal\{D\}f(x)$

with $G[f(x)]$ as the functional and $\backslash mathcal\{D\}f(x)$ the functional differential.

{{main article|Ideal chain}}

Image:Ideal chain random walk.svg]]

One extremely naive yet fruitful approach to quantitatively analyze the spatial structure and configuration of a polymer is the free random walk model. The polymer is depicted as a chain of point like unit molecules which are strongly bound by chemical bonds and hence the mutual distance between successive units can be approximated to be constant.

In the ideal polymer model the polymer subunits are completely free to rotate with respect to each other, and therefore the process of polymerization can be looked at as a random three dimensional walk, with each monomer added corresponding to another random step of predetermined length.

Mathematically this is formalized through the probability function for the position vector of the bonds, i.e. the relative positions of a pair of adjacent units:

:$\backslash psi(\backslash vec\; r)=\backslash frac\{1\}\{4\backslash pi\; l^2\}\; \backslash delta(\backslash left|\backslash vec\; r\backslash right\backslash vert-l)$

With $\backslash delta()$ standing for the dirac delta. The important thing to note here is that the bond position vector has a uniform distribution over a sphere of radius $l$, our constant bond length.

A second crucial feature of the ideal model is that the bond vectors $\backslash vec\; r\_n$ are independent of each other, meaning we can write the distribution function for the complete polymer conformation as:

:$\backslash Psi(\backslash left\; \backslash \{\; \backslash vec\; r\_n\; \backslash right\; \backslash \})=\backslash prod\_\{n=1\}^N\; \backslash psi(\backslash vec\; r\_n)$

Where we assumed $\backslash textstyle\; N$ monomers and $\backslash textstyle\; n$ acts as a dummy index. The curly brackets { } mean that $\backslash Psi$ is a function of the set of vectors $\backslash vec\; r\_n$

Salient results of this model include:

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: $\backslash left\; \backslash langle\; \backslash vec\; R^2\; \backslash right\; \backslash rangle\; =\; Nl^2$ with the end to end vector defined as: $\backslash textstyle\; \backslash vec\; R\; \backslash equiv\; \backslash sum\_\{n=1\}^N\; \backslash vec\; r\_n$.

Thus, a first crude approximation for the polymer size is simply $R\_0\; \backslash equiv\; \backslash sqrt\{\backslash left\; \backslash langle\; \backslash vec\; R^2\; \backslash right\; \backslash rangle\}\; =\; \backslash sqrt\{N\}l$.

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:

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

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 $Nl$, clearly an unreasonable result which stems from the limit taken $N\backslash rightarrow\backslash infty$ for its derivation.

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

:$\backslash frac\; \{\backslash partial\; \backslash Phi\}\{\backslash partial\; N\}\; =\; \backslash frac\{l^2\}\{6\}\; \backslash nabla^2\; \backslash Phi$

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

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.

Image:Three paths from A to B.png

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

:$\backslash Phi\; (\backslash vec\; R,\; N)=\; \backslash int\_\{0,0\}^\{\backslash vec\; R,\; N\}\backslash exp\backslash left\; \backslash \{\; -\backslash int\_\{0\}^\{N\}L\_0d\backslash nu\; \backslash right\; \backslash \}\; \backslash mathcal\{D\}\backslash vec\; R(\backslash nu)$

Where we defined $\backslash textstyle\; L\_0\; =\; \backslash frac\{3\}\{2l^2\}\; \backslash left\; (\; \backslash frac\{d\; \backslash vec\; R\}\{d\backslash nu\}\; \backslash right\; )^2.$

Here $\backslash nu$ 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.

Thus far, the path integral approach didn't avail us of any novel results. For that, one must venture further than the ideal model. As a first departure from this limited model, we now consider the constraint of spatial obstructions. The ideal model assumed no constraints on the spatial configuration of each additional monomer, including forces between monomers which obviously exist, since two monomers cannot occupy the same space. Here, we'll take the concept of obstruction to encompass not only monomer-monomer interactions, but also constraints that arise from the presence of dust and boundary conditions such as walls or other physical obstructions.

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

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

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

For which the corresponding path integral is:

:$\backslash Phi\; (\backslash vec\; R,\; N)=\; \backslash int\_\{0,0\}^\{\backslash vec\; R,\; N\}\backslash exp\backslash left\; \backslash \{\; -\backslash int\_\{0\}^\{N\}[L\_0+f(\backslash vec\; R)]d\backslash nu\; \backslash right\; \backslash \}\; \backslash mathcal\{D\}\backslash vec\; R(\backslash nu)$

File:CellMembraneDrawing numbered.jpg

To model a perfect rigid wall, simply set $\backslash textstyle\; \backslash frac\; \{f(\backslash vec\; R)\}\{l^2\}\; \backslash rightarrow\; +\backslash infty$ 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: $\backslash textstyle\; V(\backslash vec\; R)$. The potential will be incorporated through a Boltzmann factor. Taken for the entire polymer this takes the form:

:$$

\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 $\backslash beta=(k\_bT)^-1$ with $T$ as Temperature and $k\_b$ the Boltzmann constant. In the right hand side, our usual limits were taken.

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

:$$

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:

:$$

\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 $Q\_V(\backslash vec\; R\_N,N\; |\; \backslash vec\; R\_0,0)=\backslash sum\_n\; f\_n(\backslash vec\; R\_N)\; f\_n^*(\backslash vec\; R\_0)\backslash exp(-E\_NN)$ in terms of orthonormal eigenfunctions and eigenvalues:

:$$

\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 $T\_c$ determined by the specific problem parameters $l,\; V(\backslash vec\; R)$ :

In high temperatures $T>T\_c$, the potential well has no bound states, meaning all eigenvalues are positive and the corresponding eigenfunction takes the asymptotic form :

:$f\_n\; \backslash cong\; A\_n\backslash sin(\backslash sqrt\{6\backslash lambda\_n/L^2\}\; x)+B\_m\backslash cos(\backslash sqrt\{6\backslash lambda\_m/L^2\}x)$

with $\backslash lambda\_n$ denoting the calculated eigenvalues.

The result is shown for the x coordinate after a separation of variables and assuming a surface at $x=0$.

This expression represents a very open configuration for the polymer, away from the surface, meaning the polymer is desorbed.

For low enough temperatures

:$$

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 $\backslash textstyle\; \backslash frac\{l\}\{\backslash sqrt\{6|\backslash lambda\_0|\}\}$

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.{{cite journal | last=Rubin | first=Robert J. | title=Comment on "Conformation of Adsorbed Polymeric Chain. II" | journal=The Journal of Chemical Physics | publisher=AIP Publishing | volume=51 | issue=10 | date=15 November 1969 | issn=0021-9606 | doi=10.1063/1.1671849 | pages=4681| bibcode=1969JChPh..51.4681R | doi-access=free }}

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. 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

A path integral approach affords a relatively simple method to derive an approximated solution:{{cite journal | last=Gennes | first=P -G de | title=Some conformation problems for long macromolecules | journal=Reports on Progress in Physics | publisher=IOP Publishing | volume=32 | issue=1 | date=1 December 1968 | issn=0034-4885 | doi=10.1088/0034-4885/32/1/304 | pages=187–205| s2cid=250861107 }} 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 $f(\backslash vec\; R)$ 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 $\backslash textstyle\; Q\_V(\backslash vec\; R\_N,N|\backslash vec\; R\_0.0)$ previously presented, the most probable configuration will be the curve $\backslash vec\; R^*(\backslash nu)$ that minimizes the exponent of the original path integral:

:$$

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:

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

We set $R\; \backslash equiv\; R^*$.

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

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

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

We find $S[\backslash vec\; R(\backslash nu)]$ can now be written as:

:$$

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:

:$$

\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^2R}{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 $R(\backslash nu)$ without any $f(\backslash vec\; R^*)$ dependence.

Although quite horrendous to look at, this equation has a fairly simple solution:

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

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

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

So far, the only polymer parameters incorporated into the calculation were the number of monomers $N$ which was taken to infinity, and the constant bond length $l$. 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:M. Doi and S.F. Edwards, The Theory of Polymer Dynamics, (Clarendon press,Oxford, 1986).

:$$

\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: $\backslash langle\backslash vec\; r^2\backslash rangle=l^2$. Note that although a bit more complex than before, $\backslash psi\; (\backslash vec\; r\; )$ still has a single parameter - $l$.

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

:$$

\begin{align}

\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{align}

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

This conformation is known as the Gaussian chain. The Gaussian approximation for $\backslash psi\; (\backslash vec\; r)$ 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:

:$$

U_0(\{\vec R_n \})= \frac{3}{2l^2}k_bT \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 $\backslash Psi(\backslash left\; \backslash \{\; \backslash vec\; r\_n\; \backslash right\; \backslash \})$.

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

:$$

\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 .

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

:$$

\Psi(\left \{ \vec r_n \right \})= \left ( \frac{3}{2\pi l^2} \right )^{3N/2}\exp \left [- \frac{3}{2l^2} \int_0^Ndn \left (\frac{\partial \vec R_n}{\partial n} \right )^2 \right ].

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

Assuming an external potential field $U\_e(\backslash vec\; R)$, the equilibrium conformational distribution described above will be modified by a Boltzmann factor:

:$$

\Psi(\left \{ \vec r_n \right \})= \left ( \frac{3}{2\pi l^2} \right )^{3N/2}\exp \left [- \frac{3}{2l^2} \int_0^Ndn \left (\frac{\partial \vec R_n}{\partial n} \right )^2 -\beta \int_0^NdnU_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:

:$$

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^Ndn \left( \frac{\partial \vec R_n}{\partial n}\right )^2 - \beta \displaystyle \int_0^NduU_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^Ndn \left ( \frac{\partial \vec R_n}{\partial n} \right )^2 \right ] }

The path integration is interpreted as a summation over all polymer curves $\backslash vec\; R(n)$ that start from $\backslash vec\; R\_0=\backslash vec\; R\text{'}$ and terminate at $\backslash vec\; R\_N=\backslash vec\; R$.

For the simple zero field case $U\_e=0$ The Green function reduces back to:

:$$

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

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

:$$

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:

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

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

The product $\backslash textstyle\; G(\backslash vec\; R,\; \backslash vec\; R$ ; N-n)G(\vec R, \vec R' ; N)) expresses the weight factor for a chain which starts at $R\text{'}$, passes through $R$ in $n$ steps, and ends at $R$ after $N$ steps. The integration over all possible midpoints $R$ gives back the statistical weight for a chain starting at $R\text{'}$, and terminating at $R$. 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 $G(\backslash vec\; R,\; \backslash vec\; R\text{'}\; ;\; N)$ the average of any physical quantity $A$ can be calculated. Assuming $\backslash textstyle\; A$ depends only on the position of the $n$-th segment, then:

\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 dR_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 $\backslash vec\; R\_m$ as well as $\backslash vec\; R\_n$ the average takes the form:

\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 dR_0 ~ G(\vec R_N, \vec R_0 ;N)}

With an obvious generalization for more monomers dependence.

If one imposes the reasonable boundary conditions:

:$$

\begin{align}

& G(\vec R, \vec R' ; N<0)=0 \\

& G(\vec R, \vec R' ;0)=\delta (\vec R- \vec R')\\

\end{align}

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

:$$

\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 $G(\backslash vec\; R,\; \backslash vec\; R\text{'}\; ;\; N)$ is found for a variety of problems. Then, with a calculation of the partition function a host of statistical quantities can be extracted.

{{main article|Polymer field theory}}

A different new approach for finding the power dependence $\backslash left\; \backslash langle\; \backslash vec\; R^2\; \backslash right\; \backslash rangle\; \backslash propto\; N^\backslash alpha$ caused by excluded volume effects, is considered superior to the one previously presented.

The field theory approach in polymer physics is based on an intimate relationship of polymer fluctuations and field fluctuations. The statistical mechanics of a many particle system can be described by a single fluctuating field. A particle in such an ensemble moves through space along a fluctuating orbit in a fashion that resembles a random polymer chain. The immediate conclusion to be drawn is that large groups of polymers may also be described by a single fluctuating field. As it turns out, the same can be said of a single polymer as well.

In analogy to the original path integral expression presented, the end to end distribution of the polymer now takes the form:

:$\backslash Phi\; (\backslash vec\; R,\; N)=\; \backslash int\_\{0,0\}^\{\backslash vec\; R,\; N\}\; e^\{-\backslash mathcal\{A\}[\backslash eta]\}\; P^\backslash eta(N,l)\; \backslash mathcal\{D\}\backslash eta$

Our new path integrand consists of:

• The fluctuating field $\backslash eta(\backslash vec\; R)$
• The action :$\backslash mathcal\{A\}[\backslash eta]=\; -\backslash frac\{1\}\{2\}\backslash int\; d\backslash vec\; R\; ~\; d\backslash vec\; R\text{'}\; ~\; \backslash eta(\backslash vec\; R)V^\{-1\}(\backslash vec\; R,\; \backslash vec\; R\text{'})\backslash eta\; (\backslash vec\; R\text{'})$ with $V(\backslash vec\; R,\; \backslash vec\; R\text{'})$ denoting the monomer-monomer repulsive potential.
• $P^\backslash eta(N,L)=\backslash int\; \backslash exp\; \backslash left\; \backslash \{-\backslash int\_0^N\; d\backslash nu\; \backslash left\; [\; \backslash frac\{M\}\{2\}\backslash dot\{\backslash vec\; R\}+\backslash eta(\backslash vec\; R(\backslash nu))\; \backslash right\; ]\; \backslash right\; \backslash \}\backslash mathcal\{D\}\backslash vec\; R$ which satisfies the Schrödinger equation:

$\backslash left\; [\; \backslash frac\{\backslash partial\}\{\backslash partial\; N\}-\backslash frac\{1\}\{2M\}\backslash nabla^2\; +\; \backslash eta(\backslash vec\; R)\; \backslash right\; ]P^\backslash eta(N,L)=\backslash delta^\{(3)\}(\backslash vec\; R\; -\; \backslash vec\; R\text{'})\backslash delta(N)$

with $M$ acting as an effective mass determined by the dimensionality and bond length.

Note that the inner integral is now also a path integral, so two spaces of function are integrated over - the polymer conformations - $\backslash vec\; R(\backslash nu)$ and the scalar fields $\backslash eta(\backslash vec\; R)$.

These path integrals have a physical interpretation. The action $\backslash mathcal\{A\}$ describes the orbit of a particle in a space dependent random potential $\backslash eta(\backslash vec\; R)$. The path integral over $\backslash vec\; R(\backslash nu)$ yields the end to end distribution of the fluctuating polymer in this potential. The second path integral over $\backslash eta\; (\backslash vec\; R)$ with the weight $e^\{-\backslash mathcal\{A\}[\backslash eta]\}$ accounts for the repulsive cloud of other chain elements. To avoid divergence, the $\backslash eta(\backslash vec\; R)$ integration has to run along the imaginary field axis.

Such a field description for a fluctuating polymer has the important advantage that it establishes a connection with the theory of critical phenomena in field theory.

To find a solution for $\backslash Phi\; (\backslash vec\; R,\; N)$, one usually employs a Laplace transform and considers a correlation function similar to the statistic average $\backslash left\; \backslash langle\; A(\backslash vec\; R\_n,\; \backslash vec\; R\_m)\backslash right\; \backslash rangle$ formerly described, with the green function substituted by a fluctuating complex field. In the common limit of large polymers (N>>1), the solutions for the end to end vector distribution correspond to the well developed regime studied in the quantum field theoretic approach to critical phenomena in many body systems.D.J. Amit, Renormalization Group and Critical Phenomena, (World Scientific Singapore, 1984.)G. Parisi, Statistical Field Theory, (Addison-Wesley, Reading Mass. 1988).

Another simplifying assumption was taken for granted in the treatment presented thus far; All models described a single polymer. Obviously a more physically realistic description will have to account for the possibility of interactions between polymers. In essence, this is an extension of the excluded volume problem.

To see this from a pictorial point, one can imagine a snap shot of a concentrated polymer solution. Excluded volume correlations are now not only taking place within one single chain, but an increasing number of contact points from other chains at increasing polymer concentration yields additional excluded volume. These additional contacts can have substantial effects on the statistical behavior of the individual polymer.

A distinction must be made between two different length scales.{{cite journal | last1 = Vilgis | first1 = T.A. | year = 2000 | title = Polymer Theory: Path Integrals and Scaling | url = http://www.sciencedirect.com/science/article/pii/S0370157399001222 | journal = Physics Reports | volume = 336 | issue = 3 | pages = 167–254 | doi = 10.1016/S0370-1573(99)00122-2 | bibcode = 2000PhR...336..167V }} One regime will be given by small end to end vector scales . At these scales the chain piece experiences only correlations from itself, i.e., the classical self-avoiding behavior. For larger scales $R\_0>\; \backslash xi$ self-avoiding correlations do not play a significant role and the chain statistics resemble a Gaussian chain. The critical value $\backslash xi$ must be a function of the concentration. Intuitively, one significant concentration can already be found. This concentration characterizes the overlap between the chains. If the polymers just marginally overlap, one chain is occupied in its own volume. This gives:

C^*=N/R_0^3 \sim N/N^{3\sigma}=N^{1-3\sigma} Where we used $R\_0\; \backslash sim\; N^\{\backslash sigma\}$

This is an important result and one immediately sees that for large chain lengths N, the overlap

concentration is very small. The self-avoiding walk previously described is changed and therefore the partition function is no longer ruled by the single polymer volume excluded paths, but by the remaining density fluctuations which are determined by the overall concentration of the polymer solution. In the limit of very large concentrations, imagined by an almost completely filled lattice, the density fluctuations become less and less important.

To begin with, let us generalize the path integral formulation to many chains.

The generalization for the partition function calculation is very simple and all that has to be done is to take into account the interaction between all the chain segments:

Z=\int \prod_{\alpha=1}^{n_p} \mathcal{D}\vec R_\alpha (\nu) \exp \{ -\beta \mathcal {H}([\vec R_\alpha (\nu)])\}

Where the weighed energy states are defined as:

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\beta \mathcal {H}([\vec R_\alpha (\nu)])= \frac{3}{2l^2} \sum_{\alpha=1}^{n_p} \int_0^{N_\alpha} \left ( \frac {\partial \vec R_\alpha}{\partial \nu} \right )^2 d \nu + \frac{1}{2}\sigma \sum_{\alpha , \beta = 1}^{n_p} \int_0^{N_\alpha}d \nu \int_0^{N_\beta}d \nu ' \delta(\vec R_\alpha (\nu) - \vec R_\beta (\nu '))

With $n\_p$ denoting the number of polymers.

This is generally not simple and the partition function cannot be computed exactly.

One simplification is to assume monodispersity which means that all chains have the same length. or, mathematically: $N\_\backslash alpha\; =\; N\_\backslash beta\; \backslash quad\; \backslash forall\; \backslash \; \backslash alpha\; ,\; \backslash beta$.

Another problem is that the partition function contains too many degrees of freedom. The number of chains $n\_p$ involved can be very large and every chain has internal degrees of freedom, since they are assumed to be totally flexible. For this reason, it is convenient to introduce collective variables, which in this case is the polymer segment density:

\rho (\vec x)= \frac{1}{V} \sum_{\alpha=1}^{n_p} \int_0^N d \nu \delta (\vec x -\vec R_{\alpha}(\nu)). with $V$ the total solution volume.

$\backslash rho(\backslash vec\; x)$ can be viewed as a microscopic density operator whose value defines the density at an arbitrary point $\backslash vec\; x$.

The transformation $\backslash mathcal\; \{H\}([\backslash vec\; R\_\backslash alpha\; (\backslash nu)])\; \backslash rightarrow\; \backslash mathcal\; \{H\}([\backslash rho\; (\backslash vec\; x)])$ is less trivial than one might imagine and cannot be carried out exactly. The final result corresponds to the so-called random phase approximation (RPA) which has been frequently used in solid-state physics. To explicitly calculate the partition function using the segment density one has to switch to reciprocal space, change variables and only then execute the integration. For a detailed derivation see.{{cite journal | last1=Edwards | first1=S F | last2=Anderson | first2=P W | title=Theory of spin glasses | journal=Journal of Physics F: Metal Physics | publisher=IOP Publishing | volume=5 | issue=5 | year=1975 | issn=0305-4608 | doi=10.1088/0305-4608/5/5/017 | pages=965–974| bibcode=1975JPhF....5..965E }} With the partition function obtained, a variety of physical quantities can be extracted as previously described.

• File dynamics
• Kuhn length
• Important publications in polymer physics
• Persistence length
• Polymer characterization
• Random coil
• Worm-like chain

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Category:Condensed matter physics

Category:Polymers

Category:Polymer physics