A polymer is a macromolecule, composed of many similar or identical repeated subunits. Polymers are common in, but not limited to, organic media. They range from familiar synthetic plastics to natural biopolymers such as DNA and proteins. Their unique elongated molecular structure produces unique physical properties, including toughness, viscoelasticity, and a tendency to form glasses and semicrystalline structures. The modern concept of polymers as covalently bonded macromolecular structures was proposed in 1920 by Hermann Staudinger. One sub-field in the study of polymers is polymer physics. As a part of soft matter studies, Polymer physics concerns itself with the study of mechanical properties and focuses on the perspective of condensed matter physics. Because polymers are such large molecules, bordering on the macroscopic scale, their physical properties are usually too complicated for solving using deterministic methods. Therefore, statistical approaches are often implemented to yield pertinent results. The main reason for this relative success is that polymers constructed from a large number of monomers are efficiently described in the thermodynamic limit of infinitely many monomers, although in actuality they are obviously finite in size. Thermal fluctuations continuously affect the shape of polymers in liquid solutions, and modeling their effect requires using principles from statistical mechanics and dynamics. The path integral approach falls in line with this basic premise and its afforded results are unvaryingly statistical averages. The path integral, when applied to the study of polymers, is essentially a mathematical mechanism to describe, count and statistically weigh all possible spatial configuration a polymer can conform to under well defined potential and temperature circumstances. Employing path integrals, problems hitherto unsolved were successfully worked out: Excluded volume, entanglement, links and knots to name a few. Prominent contributors to the development of the theory include Nobel laureate P.G. de Gennes, Sir Sam Edwards, M.Doi, F.W. Wiegel and H. Kleinert.
Path integral formulation Early attempts at path integrals can be traced back to 1918. A sound mathematical formalism wasn't established until 1921. This eventually lead Richard Feynman to construct a formulation for quantum mechanics, 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, denoting a path integral as:
∫ G [ f ( x ) ] D f ( x ) {\displaystyle \int G[f(x)]{\mathcal {D}}f(x)}
with G [ f ( x ) ] {\displaystyle G[f(x)]} as the functional and D f ( x ) {\displaystyle {\mathcal {D}}f(x)} the functional differential.
Ideal polymers
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:
ψ ( r → ) = 1 4 π l 2 δ ( | r → | − l ) {\displaystyle \psi ({\vec {r}})={\frac {1}{4\pi l^{2}}}\delta (\left|{\vec {r}}\right\vert -l)}
With δ ( ) {\displaystyle \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 {\displaystyle l} , our constant bond length. A second crucial feature of the ideal model is that the bond vectors r → n {\displaystyle {\vec {r}}_{n}} are independent of each other, meaning we can write the distribution function for the complete polymer conformation as:
Ψ ( { r → n } ) = ∏ n = 1 N ψ ( r → n ) {\displaystyle \Psi (\left\{{\vec {r}}_{n}\right\})=\prod _{n=1}^{N}\psi ({\vec {r}}_{n})}
Where we assumed N {\displaystyle \textstyle N} monomers and n {\displaystyle \textstyle n} acts as a dummy index. The curly brackets { } mean that Ψ {\displaystyle \Psi } is a function of the set of vectors r → n {\displaystyle {\vec {r}}_{n}}
Salient results of this model include:
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: ⟨ R → 2 ⟩ = N l 2 {\displaystyle \left\langle {\vec {R}}^{2}\right\rangle =Nl^{2}} with the end to end vector defined as: R → ≡ ∑ n = 1 N r → n {\displaystyle \textstyle {\vec {R}}\equiv \sum _{n=1}^{N}{\vec {r}}_{n}} . Thus, a first crude approximation for the polymer size is simply R 0 ≡ ⟨ R → 2 ⟩ = N l {\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:
Φ ( R → , N ) = ( 3 2 π N l 2 ) 3 2 exp ( − 3 R → 2 2 N l 2 ) {\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 N l {\displaystyle Nl} , clearly an unreasonable result which stems from the limit taken N → ∞ {\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 N → ∞ {\displaystyle N\rightarrow \infty } and l → 0 , {\displaystyle l\rightarrow 0,} under the constraint N l = c o n s t {\displaystyle Nl=const} one comes to a differential equation for the probability distribution:
∂ Φ ∂ N = l 2 6 ∇ 2 Φ {\displaystyle {\frac {\partial \Phi }{\partial N}}={\frac {l^{2}}{6}}\nabla ^{2}\Phi }
With the laplacian ∇ 2 {\displaystyle \textstyle \nabla ^{2}} taken in respect to actual space. One way to derive this equation is via Taylor expansion to Φ ( R → , N {\displaystyle \Phi ({\vec {R}},N} ) and Φ ( R → , N + Δ N ) . {\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:
Φ ( R → , N ) = ∫ 0 , 0 R → , N exp { − ∫ 0 N L 0 d ν } D R → ( ν ) {\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 L 0 = 3 2 l 2 ( d R → d ν ) 2 . {\displaystyle \textstyle L_{0}={\frac {3}{2l^{2}}}\left({\frac {d{\vec {R}}}{d\nu }}\right)^{2}.}
Here ν {\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.
Spatial obstructions 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.
Dust Consider a space filled with small impenetrable particles, or "dust". Denote the fraction of space excluding a monomer end point by f ( R → ) {\displaystyle f({\vec {R}})} so its values range: 0 ≤ f ( R → ) ≤ 1 {\displaystyle 0\leq f({\vec {R}})\leq 1} . Constructing a Taylor expansion for Φ ( R → , N + Δ N ) . {\displaystyle \Phi ({\vec {R}},N+\Delta N).} , one can arrive at the new governing differential equation:
∂ Φ ∂ N = l 2 6 ∇ 2 − f Φ {\displaystyle {\frac {\partial \Phi }{\partial N}}={\frac {l^{2}}{6}}\nabla ^{2}-f\Phi }
For which the corresponding path integral is:
Φ ( R → , N ) = ∫ 0 , 0 R → , N exp { − ∫ 0 N [ L 0 + f ( R → ) ] d ν } D R → ( ν ) {\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 f ( R → ) l 2 → + ∞ {\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: V ( R → ) {\displaystyle \textstyle V({\vec {R}})} . The potential will be incorporated through a Boltzmann factor. Taken for the entire polymer this takes the form:
exp { − β ∑ j = 0 N V ( R → j ) } ≅ exp { − β ∫ 0 N V ( R → ( ν ) ) } {\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 β = ( k b T ) − 1 {\displaystyle \beta =(k_{b}T)^{-}1} with T {\displaystyle T} as Temperature and k b {\displaystyle k_{b}} the Boltzmann constant. In the right hand side, our usual limits N → ∞ & L → 0 {\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:
Q V ( R → N , N | R → 0 , 0 ) = ∫ R → 0 , 0 R → N , N exp { − ∫ 0 N [ L 0 ] d ν } D R → ( ν ) {\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:
∂ f ∂ N = l 2 6 ∇ 2 f − β V ( R → ) f {\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 Q V ( R → N , N | R → 0 , 0 ) = ∑ n f n ( R → N ) f n ∗ ( R → 0 ) exp ( − E N N ) {\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:
[ l 2 6 ∇ 2 f − β V ( R → ) ] f n ( R → n ) = E + n f n ( R → n ) {\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 T c {\displaystyle T_{c}} determined by the specific problem parameters l , V ( R → ) {\displaystyle l,V({\vec {R}})} : In high temperatures T > T c {\displaystyle T>T_{c}} , the potential well has no bound states, meaning all eigenvalues are positive and the corresponding eigenfunction takes the asymptotic form < ( x → ∞ ) {\displaystyle <(x\rightarrow \infty )} :
f n ≅ A n sin ( 6 λ n / L 2 x ) + B m cos ( 6 λ m / L 2 x ) {\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 λ n {\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 x = 0 {\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 T < T c {\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 ( x → ∞ ) {\displaystyle (x\rightarrow \infty )} takes the form:
f ( x 0 ) ≅ A 0 exp ( − 6 | λ 0 | / l 2 x ) {\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 l 6 | λ 0 | {\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.
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. 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 r < l / 2 {\displaystyle r<l/2} , otherwise successive spheres would overlap. A path integral approach affords a relatively simple method to derive an approximated solution: The results presented are for three dimensional space, but can be easily generalized to any dimensionality. The calculation is based on two reasonable assumptions:
Statistical characteristics for the volume excluded case resemble that of a polymer without excluded volume but with a fraction f ( R → ) {\displaystyle f({\vec {R}})} occupied by small spheres of an identical volume to the hypothesized monomer sphere. These aforementioned characteristics can be approximated by a calculation of the most probable chain configuration. In accordance with the path integral expression for Q V ( R → N , N | R → 0 .0 ) {\displaystyle \textstyle Q_{V}({\vec {R}}_{N},N|{\vec {R}}_{0}.0)} previously presented, the most probable configuration will be the curve R → ∗ ( ν ) {\displaystyle {\vec {R}}^{*}(\nu )} that minimizes the exponent of the original path integral:
S [ R → ( ν ) ] ≡ ∫ 0 N { 3 2 l 2 ( d R → d ν ) 2 + f ( R → ) } d ν {\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:
3 l 2 d 2 R → ∗ d ν 2 = ∇ f ( R → ∗ ) {\displaystyle {\frac {3}{l^{2}}}{\frac {d^{2}{\vec {R}}^{*}}{d\nu ^{2}}}=\nabla f({\vec {R}}^{*})}
We set R ≡ R ∗ {\displaystyle R\equiv R^{*}} . To determine the appropriate function f ( R → ) {\displaystyle f({\vec {R}})} , consider a sphere of radius R {\displaystyle R} , thickness d R {\displaystyle dR} and profile 4 π R 2 {\displaystyle 4\pi R^{2}} centered around the origin of the polymer. The average number of monomers in this shell should equal