remark ∎ example ∎

# Statistical Field Theory of Polarizable Polymer Chains with Nonlocal Dipolar
Interactions

Pratik Khandagale  0000-0002-3105-1439<EMAIL_ADDRESS>Department of
Mechanical Engineering, Carnegie Mellon University  Carlos Garcia-Cervera
Mathematics Department, University of California, Santa Barbara  BCAM, Basque
Center for Applied Mathematics, E48009 Bilbao, Basque Country, Spain  Gal
deBotton  Department of Mechanical Engineering, Ben Gurion University
Department of Biomedical Engineering, Ben Gurion University  Timothy Breitzman
Air Force Research Laboratory  Carmel Majidi  Department of Mechanical
Engineering, Carnegie Mellon University  Department of Civil and Environmental
Engineering, Carnegie Mellon University  Department of Materials Science and
Engineering, Carnegie Mellon University  Kaushik Dayal  0000-0002-0516-3066
Department of Civil and Environmental Engineering, Carnegie Mellon University
Department of Mechanical Engineering, Carnegie Mellon University  Center for
Nonlinear Analysis, Department of Mathematical Sciences, Carnegie Mellon
University

###### Abstract

The electromechanical response of polymeric soft matter to applied electric
fields is of fundamental scientific interest as well as relevant to
technologies for sensing and actuation. Several existing theoretical and
numerical approaches for polarizable polymers subject to a combined applied
electric field and stretch are based on discrete monomer models. In these
models, accounting for the interactions between the induced dipoles on
monomers is challenging due to the nonlocality of these interactions. On the
other hand, the framework of statistical field theory provides a continuous
description of polymer chains that potentially enables a tractable way to
account for these interactions. However, prior formulations using this
framework have been restricted to the case of weak anisotropy of the monomer
polarizability.

This paper formulates a general approach based in the framework of statistical
field theory to account for the nonlocal nature of the dipolar interactions
without any restrictions on the anisotropy or nonlinearity of the
polarizability of the monomer. The approach is based on 3 key elements: (1)
the statistical field theory framework, in which the discrete monomers are
regularized to a continuous dipole distribution; (2) a replacement of the
nonlocal dipole-dipole interactions by the local electrostatics PDE with the
continuous dipole distribution as the forcing; (3) the use of a completely
general relation between the polarization and the local electric field. Rather
than treat the dipole-dipole interactions directly, the continuous description
in the field theory enables the computationally-tractable nonlocal-to-local
transformation. Further, it enables the use of a realistic statistical-
mechanical ensemble wherein the average far-field applied electric field is
prescribed, rather than prescribing the applied field at every point in the
polymer domain.

The model is applied, using the finite element method (FEM), to study the
electromechanical response of a polymer chain in the ensemble with fixed far-
field applied electric field and fixed chain stretch. The nonlocal dipolar
interactions are found to increase, over the case where dipole-dipole
interactions are neglected, the magnitudes of the polarization and electric
field by orders of magnitude as well as significantly change their spatial
distributions. Next, the effect of the relative orientation between the
applied field and the chain on the local electric field and polarization is
studied. The model predicts that the elastic response of the polymer chain is
linear, consistent with the Gaussian approximation, and is largely unchanged
by the orientation of the applied electric field, though the polarization and
local electric field distributions are significantly impacted.

††preprint: Physical Review E 109:044501, 2024
(https://link.aps.org/doi/10.1103/PhysRevE.109.044501).

##  1 Introduction

Stimuli-responsive polymeric soft matter is central to actuators and sensors
in applications such as soft robotics [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11],
stretchable electronics [12, 13, 14, 15], energy harvesting [16, 17, 18, 19],
healthcare [20, 21, 22, 23, 24], and functional systems broadly [25, 26, 27,
28, 29, 30, 31, 32, 33, 34, 35]. Electro-responsive polarizable polymers such
as dielectric elastomers (DEs) are naturally soft, lightweight, compliant, and
can undergo large deformation under an applied electric field, making them
promising candidate materials. However, there are also shortcomings with
currently-available polymerics materials, e.g., they often need highly applied
electric fields to achieve a meaningful level of actuation [36]. A fundamental
understanding of the physics of polymers subjected to electric fields is
essential to improve existing, and discover new, polarizable polymeric
materials, e.g., in the case of statistical mechanics applied to soft matter
flexoelectricity [37, 38, 39, 40, 41, 42].

#### Prior Work.

The physics of polarizable polymeric soft matter is governed by the polymer
chain entropy, the interaction between the applied electric field and the
induced dipoles, and the nonlocal dipolar interactions between the polymer
segments (Fig. 1). Existing models for electro-responsive polymeric soft
matter can be broadly divided into two categories: continuum based-approaches,
e.g., [43, 44, 45, 46, 47, 48, 49, 50, 51], and statistical mechanics-based
approaches, e.g., [52, 53, 54, 55, 56, 37, 57, 58].

Continuum approaches typically formulate the free energy density by coupling
established rubber elasticity models to continuum electrostatics. These
approaches are useful in enabling the study of electro-responsive polymers in
complex and realistic geometries and boundary conditions. However, these
approaches cannot provide predictive insights that are based in the response
of the individual monomers.

The statistical mechanics based-approaches for polymers, on the other hand,
are capable of accounting for the molecular details of the polymer chain.
Statistical mechanics has been employed for several decades to study the
mechanical response of polymers and their networks, e.g., [59, 60, 61, 62,
63]. In the context of electro-responsive polymers, the first works that
applied statistical mechanics appear to be [52] and [53]. Broadly, their work
derives an approximate expression for the most probable density of monomer
orientations that is exact when the polymer chain is not stretched. Building
on this, a statistical mechanics-based discrete monomer model for an electro-
responsive polarizable polymer chain was presented in [54, 57]. Using the
maximum term approximation assumption, they evaluated the most probable
density of monomer orientation and the free energy of polymer chain applicable
at large stretches. Although these theoretical approaches for electro-
responsive polymers provide valuable insights, they all ignore the nonlocal
dipolar interactions; they only model the interaction between the applied
electric field and the induced dipoles. A key reason for this limitation is
that these approaches all use a discrete description of the polymer chain, and
it is computationally very expensive to account for all pairs of interactions.

In contrast, a statistical field theoretic formulation [64, 65, 66] for
polarizable polymer chains in an external electric field was presented in [67]
that accounts for the dipolar interactions. A key feature of the field
theoretic approach is that the polymer is described as a continuous — rather
than discrete — object, enabling the replacement of dipole-dipole interaction
sums by integrals [68, 69, 70]. However, the formulation in [67] is limited to
the setting of weak anisotropy of the polymer polarizability. Also, in the
field theoretic formulation, it is challenging to account for realistic
electrical boundary conditions, i.e., typically the statistical mechanics
ensemble assumes that the applied field is given at every point, rather than
only the far-field or average value which is more realistic.

Figure 1:  A polarizable polymer chain in an externally-applied far-field
electric field $\displaystyle{\mathbold E}_{0}$ induces dipoles in the polymer
segments. The dipole-dipole interactions are long range — they decay as the
inverse third power of the distance — and lead to nonlocal effects that cannot
be truncated without very large error [71]. These, in turn, affect the
configuration of the chain and the mechanical response. We show in this paper
that neglecting the dipole-dipole interactions and only accounting for the
applied field-dipole interactions, as in prior work, leads to large errors.

#### Contributions of This Work.

To overcome the limitations of the existing models for polarizable polymeric
soft matter, we have developed a statistical field theoretic framework for
polarizable polymer chains that enables us to account for the nonlocal dipole-
dipole interactions among the polymer segments. The approach is based on three
elements.

First, the statistical field theory framework, in which the polymer chain is
regularized to a continuous description. We model the polarizable flexible
polymer chain using a worm-like chain (WLC) model with field-induced dipoles
along the length of the chain. The continuous description enables us to avoid
treating individual interactions between discrete dipoles, but instead as a
tractable continuous polarization distribution.

Second, the continuous polarization description enables us to replace the
nonlocal dipole-dipole interactions — an integral operation — by the local
electrostatics partial differential equation (PDE) which uses instead the
interaction with the local electric field set up the dipoles. To compute the
effective bound charge, we introduce a polarization operator for a polymer
chain, and derive an expression for the thermodynamically-averaged induced
polarization.

Third, we use a completely general dielectric response function that relates
the induced polarization to the local electric field. While we use a linear
anisotropic relation between the polarization and the electric field for the
numerical calculations, the method is directly applicable to general nonlinear
response functions. Further, the dielectric response is necessarily nonlinear
in the orientation to satisfy frame invariance. These elements provide a self-
consistent field theoretic formulation to obtain the properties of the polymer
chain — such as segment density, polarization distribution, and local electric
field distribution around the polymer chain — under an externally applied
electric field.

A significant aspect of our formulation is that we apply the external electric
field only on the boundary of the spatial domain that includes the polymer
chain as well as free space. This corresponds to a thermodynamic ensemble with
specified far-field or average applied electric field [54], which is realistic
in terms of experimental configurations. The local electric field is obtained
self-consistently through solving for the electrostatic equation that accounts
for the polarization distribution. By using the finite element method (FEM)
with an unstructured discretization, we are able to efficiently solve by
refining the mesh around the polymer chain where variations are large and
keeping it coarse in the free space away from the chain.

The FEM implementation is applied to study the electromechanical response of a
polymer chain in the ensemble with fixed far-field applied electric field and
fixed chain stretch. We find that the nonlocal dipolar interactions are found
to increase, over the case where dipole-dipole interactions are neglected, the
magnitudes of the polarization and electric field by orders of magnitude as
well as change significantly their spatial distributions. Next, we study the
effect of the relative orientation between the applied field and the chain on
the local electric field and polarization. When the applied electric field is
aligned with the chain end-to-end vector, the larger values of the
polarization and electric field are primarily concentrated near the
constrained chain ends. In contrast, when the applied field is orthogonal to
the chain end-to-end vector, the larger values of the polarization and
electric field are distributed along the chain. However, we observe that
despite these differences in the polarization and local field, the elastic
response of the chain is linear and largely unchanged by the different
orientations of the applied electric field.

#### Organization.

Section 2 presents our approach; Section 3 presents the numerical method; and
Section 4 presents results from our calculations.

##  2 Formulation

This section presents the formulation of the framework for a polarizable
polymer chain under a far-field applied electric field. First, we summarize
the standard self-consistent statistical field theoretic description of the
polymer chain, following [66]. Next, we introduce the polarization operator
and its thermodynamic average for the polarizable polymer chain, and describe
the coupling between electrostatics and polymer chain description.

###  2.A Self-Consistent Statistical Field Theory Description of a Polymer
Chain

Figure 2: Kinematic quantities used to describe the polymer chain.

We use a worm-like chain (WLC) model in this paper with a small but nonzero
value for the persistence length. The chain has $\displaystyle N$ coarse-
grained polarizable polymer segments, each with length $\displaystyle a$;
$\displaystyle L_{c}=aN$ is the total contour length of the chain. The
persistence length is denoted by $\displaystyle\lambda$, i.e., the distance
along the polymer chain contour over which the orientational correlations
decay. The ratio $\displaystyle\lambda/L_{c}$ determines the flexibility of
the chain: $\displaystyle\lambda/L_{c}\ll 1$ gives a very flexible polymer
chain, whereas $\displaystyle\lambda/L_{c}\gg 1$ gives a rigid rod-like
polymer chain. To model a flexible polymer chain in this work, we use
$\displaystyle\lambda/L_{c}=10^{-3}$.

A coarse-grained configuration of the polymer chain is represented as a
continuous 3-d space curve $\displaystyle{\mathbold r}(s)$ in Figure 2, where
$\displaystyle s$ is the chain contour coordinate that is non-dimensionalized
with chain contour length $\displaystyle L_{c}$ such that $\displaystyle 0\leq
s\leq 1$. The position and orientation of the chain segment with contour
coordinate $\displaystyle s$ are given by $\displaystyle{\mathbold
x}={\mathbold r}(s)$ and $\displaystyle{\mathbold u}=\frac{1}{L_{c}}\frac{\
\mathrm{d}{\mathbold r}}{\ \mathrm{d}s}$, respectively. The chain is assumed
to be inextensible, hence $\displaystyle{\mathbold u}$ is a unit vector. The
position and orientation at the ends of the chain are denoted
$\displaystyle{\mathbold x}^{0}$ and $\displaystyle{\mathbold u}^{0}$ at
$\displaystyle s=0$ and $\displaystyle{\mathbold x}^{1}$ and
$\displaystyle{\mathbold u}^{1}$ at $\displaystyle s=1$.

The fundamental statistical mechanics quantity is the partition function,
$\displaystyle Q[w]$, which has the expression:

$Q[w]=\frac{1}{4\pi V}\int\ \mathrm{d}{\mathbold x}\int\ \mathrm{d}{\mathbold
u}\ q({\mathbold x},{\mathbold u},{\mathbold x}^{0},{\mathbold
u}^{0},s)q^{*}({\mathbold x},-{\mathbold u},{\mathbold x}^{1},{\mathbold
u}^{1},1-s),$ (1)

where $\displaystyle w({\mathbold x},{\mathbold u})$ is the potential of the
external field, originating in the interaction between the induced dipole on
the polymer segment and the local electric field and whose form is given in
(13); $\displaystyle q({\mathbold x},{\mathbold u},{\mathbold
x}^{0},{\mathbold u}^{0},s)$ and $\displaystyle q^{*}({\mathbold
x},-{\mathbold u},{\mathbold x}^{1},{\mathbold u}^{1},1-s)$ are the partial
partition functions of the chain for the two chain fragments, one from
$\displaystyle 0$ to $\displaystyle s$ and the other from $\displaystyle 1$ to
$\displaystyle s$, respectively (Fig. 2); and $\displaystyle V=Na^{2}$ is the
volume of the polymer chain in 2-d (we would use $\displaystyle V=Na^{3}$ for
3-d). The domain of integration is over the spatial domain in
$\displaystyle{\mathbold x}$ and over the unit sphere in
$\displaystyle{\mathbold u}$.

The partial partition functions $\displaystyle q$ and $\displaystyle q^{*}$
are obtained by solving the PDEs below [66]:

$\displaystyle\displaystyle\frac{\partial q}{\partial s}$
$\displaystyle\displaystyle=-w({\mathbold x},{\mathbold u})q-L_{c}{\mathbold
u}\cdot\nabla_{{\mathbold x}}q+\frac{L_{c}}{2\lambda}\nabla^{2}_{{\mathbold
u}}q$ (2) $\displaystyle\displaystyle\frac{\partial q^{*}}{\partial
s^{\prime}}$ $\displaystyle\displaystyle=-w({\mathbold x},{\mathbold
u})q^{*}-L_{c}{\mathbold u}\cdot\nabla_{{\mathbold
x}}q^{*}+\frac{L_{c}}{2\lambda}\nabla^{2}_{{\mathbold u}}q^{*}$ (3)

where $\displaystyle s^{\prime}=1-s$ varies along the chain contour in the
opposite sense as $\displaystyle s$ (Fig. 2).

The corresponding initial conditions are:

$\displaystyle\displaystyle q({\mathbold x},{\mathbold u},{\mathbold
x}^{0},{\mathbold u}^{0},s)\Big{|}_{s=0}$ $\displaystyle\displaystyle=V\
\delta({\mathbold x}-{\mathbold x}^{0}).$ (4) $\displaystyle\displaystyle
q^{*}({\mathbold x},-{\mathbold u},{\mathbold x}^{1},{\mathbold
u}^{1},s^{\prime})\Big{|}_{s^{\prime}=0}$ $\displaystyle\displaystyle=V\
\delta({\mathbold x}-{\mathbold x}^{1})$ (5)

The initial conditions in (4) and (5) specify the physical constraint that the
ends of the chain are fixed at $\displaystyle{\mathbold x}^{0}$ and
$\displaystyle{\mathbold x}^{1}$. We do not constrain the chain orientations
at the ends.

The linear PDEs in (2) and (3) are Fokker–Planck equations that govern the
propagation of correlations in segment position and orientation for a worm-
like polymer chain under an external field $\displaystyle w({\mathbold
x},{\mathbold u})$. This system of PDEs is derived using a recursive relation
based on the Markov property of the polymer chain partition function.
Physically, these PDE imply that the partition function for a chain fragment
from $\displaystyle 0$ to $\displaystyle s+\Delta s$ can be composed of two
contributions: first, the partition function for the chain from $\displaystyle
0$ to $\displaystyle s$, and second, the partition function for a small
additional chain segment between $\displaystyle s$ and $\displaystyle s+\Delta
s$. The partition function for this additional segment is then written, using
a Boltzmann weight, in terms of the energy that consists of a quadratic
contribution from chain bending and the energy due to the external field
$\displaystyle w$. Taking the limit $\displaystyle\Delta s\to 0$ leads to the
system of PDE; for the details of the derivation, we refer to Section 2.5 in
[66].

The first term on the right sides of (2) and (3) relate to the external field
acting on the polymer chain. The operator $\displaystyle\nabla_{{\mathbold
u}}^{2}$ in the third term on the right side is the rotational diffusion
operator that generates diffusive motion on the unit sphere.

In general, by using the appropriate form for the functional dependence of
$\displaystyle w({\mathbold x},{\mathbold u})$ on $\displaystyle{\mathbold
u}$, these equations can be used to describe the interaction between worm-like
polymer chains with a broad class of external potentials, including electric
and magnetic fields [66]. Examples include polymer chains with induced or
permanent electric or magnetic dipoles along the polymer backbone, and
anisotropic potential fields that can model liquid crystalline behavior. In
this work, we use $\displaystyle w({\mathbold x},{\mathbold u})$ to model the
electrostatic energy of the dielectric polymer chain due to induced electric
dipoles in the polymer segments in an externally applied electric field.

###  2.B Spatial Dipole Distribution

The density operator $\displaystyle\hat{\rho}({\mathbold x},{\mathbold u})$
for the WLC is defined as [66]:

$\hat{\rho}({\mathbold x},{\mathbold u}):=\int\limits_{0}^{1}\ \mathrm{d}s\
\delta({\mathbold x}-{\mathbold r}(s))\delta\left({\mathbold
u}-\frac{1}{L_{c}}{\mathbold r}^{\prime}(s)\right)\delta(|{\mathbold u}|-1),$
(6)

The Dirac measures build in the kinematic definitions of
$\displaystyle{\mathbold x}$ and $\displaystyle{\mathbold u}$ as constraints.
The thermodynamically-averaged segment density,
$\displaystyle\langle\hat{\rho}({\mathbold x},{\mathbold u})\rangle$, is then
obtained as [66]:

$\langle\hat{\rho}({\mathbold x},{\mathbold u})\rangle=\frac{1}{4\pi
VQ[w]}\int\limits_{0}^{1}\ \mathrm{d}s\ q({\mathbold x},{\mathbold
u},{\mathbold x}^{0},{\mathbold u}^{0},s)q^{*}({\mathbold x},-{\mathbold
u},{\mathbold x}^{1},{\mathbold u}^{1},1-s).$ (7)

where $\displaystyle\langle\cdot\rangle$ corresponds to the statistical
average performed over all possible conformations of the polymer chain.

Next, we introduce the polarization operator $\displaystyle\hat{{\mathbold
p}}({\mathbold x},{\mathbold u})$ as:

$\hat{{\mathbold p}}({\mathbold x},{\mathbold u}):=4\pi V\int\limits_{0}^{1}\
\mathrm{d}s\ {\mathbold p}_{seg}({\mathbold x},{\mathbold u})\
\delta({\mathbold x}-{\mathbold r}(s))\delta\left({\mathbold
u}-\frac{1}{L_{c}}{\mathbold r}^{\prime}(s)\right)\delta(|{\mathbold u}|-1),$
(8)

where $\displaystyle{\mathbold p}_{seg}({\mathbold x},{\mathbold u})$ is the
polarization response function, i.e., the induced polarization at the point
$\displaystyle({\mathbold x},{\mathbold u})$ in configuration space.

Defining $\displaystyle\langle\hat{{\mathbold p}}({\mathbold x},{\mathbold
u})\rangle$ as the thermodynamically-averaged polarization of the polymer
segment at $\displaystyle({\mathbold x},{\mathbold u})$ in configuration
space, we write:

$\begin{split}\langle\hat{{\mathbold p}}({\mathbold x},{\mathbold
u})\rangle&=\left\langle 4\pi V\int\limits_{0}^{1}\ \mathrm{d}s\ {\mathbold
p}_{seg}({\mathbold x},{\mathbold u})\ \delta({\mathbold x}-{\mathbold
r}(s))\delta\left({\mathbold u}-\frac{1}{L_{c}}{\mathbold
r}^{\prime}(s)\right)\delta(|{\mathbold u}|-1)\right\rangle\\\ &=4\pi
V{\mathbold p}_{seg}({\mathbold x},{\mathbold
u})\left\langle\int\limits_{0}^{1}\ \mathrm{d}s\ \ \delta({\mathbold
x}-{\mathbold r}(s))\delta\left({\mathbold u}-\frac{1}{L_{c}}{\mathbold
r}^{\prime}(s)\right)\delta(|{\mathbold u}|-1)\right\rangle\\\ &=4\pi
V{\mathbold p}_{seg}({\mathbold x},{\mathbold
u})\left\langle\hat{\rho}({\mathbold x},{\mathbold
u})\right\rangle,\quad\text{using \eqref{eq:rho_def} and
\eqref{eq:rho_avg}.}\end{split}$ (9)

We define $\displaystyle{\mathbold p}({\mathbold x})$, the polarization at the
spatial location $\displaystyle{\mathbold x}$, as the average over
$\displaystyle{\mathbold u}$ at the location $\displaystyle{\mathbold x}$:

${\mathbold p}({\mathbold x}):=\frac{1}{4\pi}\int\ \mathrm{d}{\mathbold u}\
\langle\hat{{\mathbold p}}({\mathbold x},{\mathbold u})\rangle=V\int\
\mathrm{d}{\mathbold u}\ {\mathbold p}_{seg}({\mathbold x},{\mathbold
u})\langle\hat{\rho}({\mathbold x},{\mathbold u})\rangle$ (10)

This final quantity $\displaystyle{\mathbold p}({\mathbold x})$ will appear in
the electrostatic equation in the bound charge density.

###  2.C Electrostatics

We next obtain the electric field through the local electrostatics PDE, which
is tractable numerically since the charge distribution described through
$\displaystyle{\mathbold p}({\mathbold x})$ does not involve singular dipole
distributions. This also lets us directly apply realistic boundary conditions
— i.e., specified potential on the boundary of the domain, corresponding to a
given far-field applied electric field — without having to compute the Greens
function for a given geometry. As shown in the Appendix of [54], this ensemble
is equivalent to prescribing the average field over the domain. The interior
field within the domain is a superposition of the electrostatic interaction
between the applied field and the induced dipole response of the polymer
segments as well as the nonlocal dipole-dipole interactions among the induced
dipoles. The usual electrostatics PDE accounts automatically for all of these
interactions.

We start from the electrostatic equation for the electrostatic potential
$\displaystyle\phi({\mathbold x})$:

$-\epsilon_{0}\nabla^{2}\phi({\mathbold x})=-\operatorname{div}{\mathbold
p}\text{ on }\Omega,\quad\text{ given }\phi({\mathbold x})=-{\mathbold
E}_{0}\cdot{\mathbold x}\text{ on }\partial\Omega$ (11)

where $\displaystyle{\mathbold p}$ is obtained from (10);
$\displaystyle-\operatorname{div}{\mathbold p}$ is the bound charge density;
$\displaystyle{\mathbold E}_{0}$ is the given average electric field; and
$\displaystyle\Omega$ is the region of space with boundary
$\displaystyle\partial\Omega$ over which we solve the electrostatic problem.

The electric field $\displaystyle{\mathbold E}({\mathbold x})$ is related to
$\displaystyle\phi({\mathbold x})$ through the classical relation:

${\mathbold E}({\mathbold x})=-\nabla\phi({\mathbold x}).$ (12)

In turn, the field $\displaystyle w({\mathbold x},{\mathbold u})$ is related
to $\displaystyle{\mathbold E}({\mathbold x})$ as [66]:

$w({\mathbold x},{\mathbold u})=-\frac{4\pi V}{2k_{B}T}{\mathbold
p}_{seg}({\mathbold x},{\mathbold u})\cdot{\mathbold E}({\mathbold x})$ (13)

###  2.D Monomer Dipole Response

We assume that the monomers that compose the chain have a dielectric response
that is linear in the electric field111 While we use a dielectric response
that is linear in the electric field for this paper to perform explicit
numerical calculations, it is equally easy to incorporate more general
nonlinear responses. . However, the response is necessarily nonlinear in the
orientation $\displaystyle{\mathbold u}$ to satisfy rotation invariance of the
monomer.

We define the polarization response of a segment of the polymer chain as:

${\mathbold p}_{seg}({\mathbold x},{\mathbold
u}):=\epsilon_{0}\mathbold{\beta}({\mathbold u}){\mathbold E}({\mathbold x}).$
(14)

where $\displaystyle\mathbold{\beta}$ is related to the molecular
polarizability tensor of the chain segment, and depends on the orientation of
the chain segment $\displaystyle{\mathbold u}$. We model the polarizability
tensor $\displaystyle\mathbold{\beta}$ as transversely isotropic [53, 52] with
the expression:

$\mathbold{\beta}({\mathbold u})=\beta_{\parallel}{\mathbold
u}\otimes{\mathbold u}+\beta_{\perp}\left({\mathbold I}-{\mathbold
u}\otimes{\mathbold u}\right).$ (15)

where $\displaystyle\beta_{\parallel}$ and $\displaystyle\beta_{\perp}$ are
the polarizabilities of the segment along the segment orientation and
transverse to the segment orientation, respectively.

###  2.E Model Summary

The external field $\displaystyle w({\mathbold x},{\mathbold u})$ connects to
the PDEs for the partial partition functions $\displaystyle q$ and
$\displaystyle q^{*}$ in (2) and (3). The solutions for $\displaystyle q$ and
$\displaystyle q^{*}$ in turn relate to the partition function $\displaystyle
Q[w]$ and average segment density $\displaystyle\langle\hat{\rho}({\mathbold
x},{\mathbold u})\rangle$ using (1) and (7), respectively. Finally, to close
the loop, the polarization, $\displaystyle{\mathbold p}({\mathbold x})$, is
related by (10) to $\displaystyle Q[w]$,
$\displaystyle\langle\hat{\rho}({\mathbold x},{\mathbold u})\rangle$, and
$\displaystyle{\mathbold p}_{seg}$.

##  3 Numerical Method

###  3.A Self-consistent Iteration

Algorithm 1 Self-consistent iterative algorithm to compute the equilibrium
properties of a polymer chain

while $\displaystyle\Delta Q>\epsilon=10^{-3}$ do

Compute $\displaystyle{\mathbold p}({\mathbold x})=V\int\ \mathrm{d}{\mathbold
u}\ {\mathbold p}_{seg}({\mathbold x},{\mathbold
u})\langle\hat{\rho}({\mathbold x},{\mathbold u})\rangle$
$\displaystyle\triangleright$ $\displaystyle{\mathbold p}_{seg}({\mathbold
x},{\mathbold u})=\epsilon_{0}\mathbold{\beta}({\mathbold u}){\mathbold
E}({\mathbold x})$

Solve for $\displaystyle\phi({\mathbold x})$:
$\displaystyle\nabla^{2}\phi({\mathbold
x})=\frac{1}{\epsilon_{0}}\operatorname{div}{\mathbold p}({\mathbold x})$,
given $\displaystyle\phi({\mathbold x})=-{\mathbold E}_{0}\cdot{\mathbold x}$
on $\displaystyle{\partial\Omega}$

Compute $\displaystyle{\mathbold E}({\mathbold x})=-\nabla\phi({\mathbold x})$

Compute $\displaystyle w({\mathbold x},{\mathbold u})=-\frac{4\pi
V}{2k_{B}T}\Big{[}\epsilon_{0}\mathbold{\beta}({\mathbold u}){\mathbold
E}({\mathbold x})\Big{]}\cdot{\mathbold E}({\mathbold x})$

Compute $\displaystyle q$ and $\displaystyle q^{*}$, by solving (2) and (3),
respectively

Compute $\displaystyle Q[w]$ and $\displaystyle\langle\hat{\rho}({\mathbold
x},{\mathbold u})\rangle$, using (1) and (7), respectively

end while

Outputs: $\displaystyle{\mathbold E}^{eq}({\mathbold
x}),Q^{eq},\langle\hat{\rho}({\mathbold x},{\mathbold
u})\rangle^{eq},{\mathbold p}^{eq}({\mathbold x})$

Algorithm 1 shows the iterative procedure in the proposed self-consistent
field theory formulation to obtain the equilibrium properties of a polarizable
polymer chain.

To initialize, i.e., guess the electric field for the initial step, we simply
use $\displaystyle\phi({\mathbold x})=-{\mathbold E}_{0}\cdot{\mathbold x}$,
and use this to compute $\displaystyle{\mathbold p}$ from (14). To continue
the numerical iteration from step $\displaystyle n$ to step $\displaystyle
n+1$, we use $\displaystyle{\mathbold p}^{n}({\mathbold x})$, the polarization
at iteration step $\displaystyle n$, to obtain the electric potential at the
next iteration step, $\displaystyle\phi^{n+1}({\mathbold x})$, using:

$\nabla^{2}\phi^{n+1}({\mathbold
x})=\frac{1}{\epsilon_{0}}\operatorname{div}{\mathbold p}^{n}({\mathbold
x}),\quad\text{ given }\phi^{n+1}({\mathbold x})=-{\mathbold
E}_{0}\cdot{\mathbold x}\text{ on }\partial\Omega$ (16)

$\displaystyle\phi^{n+1}({\mathbold x})$ is used to obtain $\displaystyle
w^{n+1}({\mathbold x},{\mathbold u})$ using (12) and (13). Using
$\displaystyle w^{n+1}({\mathbold x},{\mathbold u})$, we again solve (2) and
(3) to compute $\displaystyle Q[w^{n+1}]$ and
$\displaystyle\langle\hat{\rho}({\mathbold x},{\mathbold u})\rangle^{n+1}$
using (1) and (7), respectively. This in turn lets us compute
$\displaystyle{\mathbold p}^{n+1}({\mathbold x})$ using (9) and (10), which is
used in the equation above to continue the iteration. The self-consistent
iteration procedure is continued until the energy term $\displaystyle-
k_{B}T\log Q$ has converged, which we check by using the change in
$\displaystyle Q$ across successive iterations.

###  3.B Finite Element Formulation

We use FEM to solve the PDEs (2) and (3) for the partial partition functions
as in [72]. We use FEniCS, an open-source FEM framework, for the numerical
implementation [73, 74]. We work in 2 spatial dimensions (i.e.,
$\displaystyle{\mathbold x}=(x_{1},x_{2})\in\Omega\subset\mathbb{R}^{2}$) and
restrict the unit orientation vector to the unit circle (i.e., it can be
represented as $\displaystyle{\mathbold u}=(\cos\phi,\sin\phi)$, where
$\displaystyle\phi\in[0,2\pi)$). The configuration space in
$\displaystyle({\mathbold x},{\mathbold u})$ is 3-dimensional, enabling us to
use standard FEM meshing and shape functions.

In terms of $\displaystyle q(x_{1},x_{2},\phi,s)$, we can rewrite (2) as:

$\frac{\partial q}{\partial s}=-wq-L_{c}\left(\cos{\phi}\frac{\partial
q}{\partial x_{1}}+\sin{\phi}\frac{\partial q}{\partial
x_{2}}\right)+\frac{L_{c}}{2\lambda}\left(\frac{\partial^{2}q}{\partial\phi^{2}}\right),$
(17)

The contour coordinate $\displaystyle s$ is treated as a time-like variable.
Derivatives with respect to $\displaystyle s$ in (2) and (3) are approximated
using a Crank-Nicolson finite difference method. We discretize in
$\displaystyle s$ using a uniform discretization with $\displaystyle 100$
steps along the chain contour. We can then write:

$\frac{q^{i+1}-q^{i}}{\Delta s}=\frac{f^{i+1}+f^{i}}{2},\quad\text{ with
}f^{i}=-wq^{i}-L_{c}\left(\cos{\phi}\frac{\partial q^{i}}{\partial
x_{1}}+\sin{\phi}\frac{\partial q^{i}}{\partial
x_{2}}\right)+\frac{L_{c}}{2\lambda}\left(\frac{\partial^{2}q^{i}}{\partial\phi^{2}}\right)$
(18)

where the superscripts $\displaystyle i$ and $\displaystyle i+1$ represent the
discretized quantities along $\displaystyle s$.

The domain in configuration space is discretized using first-order Lagrange
family finite elements. We use a mesh with $\displaystyle 20\times 40$ finite
elements to discretize in $\displaystyle{\mathbold x}$ and $\displaystyle 30$
finite elements to discretize in $\displaystyle\phi$, which is sufficiently
refined that the quantities of interest are independent of the mesh. The
spatial mesh is finer around the chain ends, and the Dirac delta functions in
(4) and (5) are approximated as peaked Gaussians. The mesh is uniform in the
$\displaystyle{\mathbold u}$ discretization.

Following the usual FEM procedure, we first multiply (18) by a test function
$\displaystyle v(x_{1},x_{2},\phi)$; second, integrate over
$\displaystyle{\mathbold x}$ and $\displaystyle{\mathbold u}$; third, use
integration-by-parts and the divergence theorem to convert the second
derivatives $\displaystyle\frac{\partial^{2}q}{\partial\phi^{2}}$ to a product
of first derivatives; and, fourth, eliminate the boundary terms using the
assumed Neumann boundary condition in $\displaystyle(x_{1},x_{2},\phi)$ to get
the FEM weak form:

$\begin{split}\int\limits_{\tilde{{\mathbold
x}},\phi}\left(q^{i+1}v+\frac{\Delta s}{2}wq^{i+1}v+\frac{\Delta
s}{2}\cos\phi\frac{\partial q^{i+1}}{\partial\tilde{x}_{1}}v+\frac{\Delta
s}{2}\sin\phi\frac{\partial q^{i+1}}{\partial\tilde{x}_{2}}v+\frac{L_{c}\Delta
s}{4\lambda}\frac{\partial q^{i+1}}{\partial\phi}\frac{\partial
v}{\partial\phi}\right)\\\ =\int\limits_{\tilde{{\mathbold
x}},\phi}\left(q^{i}v-\frac{\Delta s}{2}wq^{i}v-\frac{\Delta
s}{2}\cos\phi\frac{\partial q^{i}}{\partial\tilde{x}_{1}}v-\frac{\Delta
s}{2}\sin\phi\frac{\partial q^{i}}{\partial\tilde{x}_{2}}v-\frac{L_{c}\Delta
s}{4\lambda}\frac{\partial q^{i}}{\partial\phi}\frac{\partial
v}{\partial\phi}\right),\end{split}$ (19)

where $\displaystyle\tilde{{\mathbold
x}}=(\tilde{x}_{1},\tilde{x}_{2})=\left(\frac{x_{1}}{L_{c}},\frac{x_{2}}{L_{c}}\right)$
is the nondimensional spatial coordinate.

##  4 Results and Discussion

In this section, we apply the model to examine the effect of dipole-dipole
interactions; specifically, we compare the electric field and dipole
distributions with and without accounting for dipole interactions. Then, we
examine the effect of the orientation of the applied electric field, relative
to the chain orientation, on the elastic response, the electric field, and the
dipole distribution.

Various quantities are either nondimensionalized or rescaled, and will then be
denoted by an overhead tilde $\displaystyle\tilde{\cdot}$. The chain is
assumed to have $\displaystyle N=100$ polymer segments, each having length of
$\displaystyle a=L_{c}/N$. The length scales in the problem are
nondimensionalized by chain contour length $\displaystyle L_{c}$. The
computational domain is chosen to be $\displaystyle-0.1\leq\tilde{x}_{1}\leq
0.1,-0.2\leq\tilde{x}_{2}\leq 0.2$. The electric field and polarization are
both rescaled by dividing by
$\displaystyle\left(\sqrt{\frac{2k_{B}T}{L_{c}^{3}}}\right)$, i.e.,
$\displaystyle\tilde{{\mathbold E}}=\frac{{\mathbold
E}}{\sqrt{\frac{2k_{B}T}{L_{c}^{3}}}}$ and $\displaystyle\tilde{{\mathbold
p}}=\frac{{\mathbold p}}{\sqrt{\frac{2k_{B}T}{L_{c}^{3}}}}$. For the
polarizability tensor, we use $\displaystyle\beta_{\parallel}=1$ and
$\displaystyle\beta_{\perp}=0.5$ following [52]. The angle between the applied
electric field $\displaystyle{\mathbold E}_{0}$ with
$\displaystyle\hat{{\mathbold e}}_{1}$ is denoted by
$\displaystyle\theta_{{\mathbold E}_{0}}$.

###  4.A Field and Dipole Distributions: Comparing with and without Dipole-
dipole Interaction

(a) Electric field distribution $\displaystyle\tilde{{\mathbold E}}$

(b) Chain segment density $\displaystyle\tilde{\rho}$

(c) Polarization distribution $\displaystyle\tilde{{\mathbold p}}$

Figure 3:  The baseline case with dipole-dipole interactions neglected, with
externally applied electric field $\displaystyle\tilde{{\mathbold
E}}_{0}=\hat{{\mathbold e}}_{2}$. The arrows show the direction of the vector
field, and the background color shows the magnitude. (a) The electric field
distribution $\displaystyle\tilde{{\mathbold E}}$, which is constant in space
and equal to $\displaystyle\tilde{{\mathbold E}}_{0}$, as expected. (b) The
chain segment density $\displaystyle\tilde{\rho}$. (c) The polarization
distribution $\displaystyle\tilde{{\mathbold p}}$. Both the electric field and
polarization distributions are close to zero on the scale of the plot, which
is chosen to allow comparison with the results when dipole-dipole interactions
are included.

(a) Polarization distribution $\displaystyle\tilde{{\mathbold p}}$ for
$\displaystyle\theta_{{\mathbold E}_{0}}=0$

(b) Polarization distribution $\displaystyle\tilde{{\mathbold p}}$ for
$\displaystyle\theta_{{\mathbold E}_{0}}=\pi/2$

Figure 4:  Comparison of the polarization distribution
$\displaystyle\tilde{{\mathbold p}}$, comparing without and with accounting
for dipole-dipole interactions for different orientations of the applied
electric field (a) $\displaystyle\theta_{{\mathbold E}_{0}}=0$, and (b)
$\displaystyle\theta_{{\mathbold E}_{0}}=\pi/2$, both with
$\displaystyle|\tilde{{\mathbold E}}_{0}|=1$. In both (a) and (b), the left
panel is without accounting for the interactions and the right panel is with
accounting for the interactions. The polarization is at least one order of
magnitude larger when dipole-dipole interactions are accounted for. The arrows
show the direction of the vector field, and the background color shows the
magnitude.

We fix the chain end-to-end vector length to $\displaystyle 1.5aN^{1/2}$ by
fixing the chain ends at $\displaystyle\pm 0.075\ {\mathbold e}_{2}$ as shown
in Figure 3(b). We apply a far-field electric field
$\displaystyle\tilde{{\mathbold E}}_{0}=\hat{{\mathbold e}}_{2}$ by using the
appropriate electric potential on the boundaries of the computational domain.

Figure 3 shows the electric field, chain segment density, and polarization for
the baseline case when the dipole-dipole interactions are neglected. We
observe that the average segment density of the chain is highest at the
constrained ends, and largely concentrated along the chain end-to-end vector.
We also observe that the induced dipoles are essentially all oriented along
$\displaystyle\tilde{{\mathbold E}}_{0}$.

Figure 4 shows a direct comparison of the polarization,
$\displaystyle\tilde{{\mathbold p}}$, obtained without and with the dipole-
dipole interactions for the extreme cases of the relative orientation between
the applied field and the chain orientation. We observe that the polarization
with the interactions considered is at least an order of magnitude higher than
the case that neglects the interactions. Further, when the interactions are
not considered, essentially all of the dipoles are oriented along
$\displaystyle\tilde{{\mathbold E}}_{0}$, whereas there are much more complex
dipole distributions when the interactions are considered. When
$\displaystyle\tilde{{\mathbold E}}_{0}$ is aligned with the chain end-to-end
vector, the polarization is largely concentrated around the chain ends.
However, when $\displaystyle\tilde{{\mathbold E}}_{0}$ is orthogonal to the
chain end-to-end vector, the polarization is relatively uniformly distributed
along the chain.

We conclude that the dipole-dipole interactions not only increase the
polarization by orders of magnitude, but also changes the distribution, and
depends strongly on the relative orientation of the applied electric field.

###  4.B Effect of the Orientation of the Electric Field Relative to the Chain

(a) Electric potential $\displaystyle\tilde{\phi}$ for
$\displaystyle\theta_{{\mathbold E}_{0}}=0$

(b) Electric field $\displaystyle\tilde{{\mathbold E}}$ for
$\displaystyle\theta_{{\mathbold E}_{0}}=0$

(c) Polarization $\displaystyle\tilde{{\mathbold p}}$ for
$\displaystyle\theta_{{\mathbold E}_{0}}=0$

(d) Electric potential $\displaystyle\tilde{\phi}$ for
$\displaystyle\theta_{{\mathbold E}_{0}}=\pi/2$

(e) Electric field $\displaystyle\tilde{{\mathbold E}}$ for
$\displaystyle\theta_{{\mathbold E}_{0}}=\pi/2$

(f) Polarization $\displaystyle\tilde{{\mathbold p}}$ for
$\displaystyle\theta_{{\mathbold E}_{0}}=\pi/2$

Figure 5:  Effect of orientation $\displaystyle\theta_{{\mathbold E}_{0}}$ of
the applied electric field $\displaystyle\tilde{{\mathbold E}}_{0}$, with full
accounting of the dipole-dipole interactions. We use
$\displaystyle|\tilde{{\mathbold E}}_{0}({\mathbold x})|=1$. (a), (d) plot the
electric potential; (b), (e) plot the electric field; and (c), (f) plot the
polarization, for $\displaystyle\theta_{{\mathbold E}_{0}}=0$ and
$\displaystyle\theta_{{\mathbold E}_{0}}=\pi/2$ respectively. The arrows show
the direction of the vector field, and the background color shows the
magnitude.

Next, we study the effect of the orientation of the applied electric field on
the properties of the polymer chain. As in the earlier section, we fix the
chain ends and fix the magnitude of the applied electric field
$\displaystyle|\tilde{{\mathbold E}}_{0}|=1$, and study the effect of the
field being orthogonal to the chain orientation compared to being aligned
parallel to the chain orientation.

Figure 5 shows the electric potential, the electric field, and the
polarization for $\displaystyle\theta_{{\mathbold E}_{0}}=0$ and
$\displaystyle\theta_{{\mathbold E}_{0}}=0\pi/2$. When the applied electric
field is orthogonal to the chain end-to-end vector
($\displaystyle\theta_{{\mathbold E}_{0}}=0$), we observe that the dipole
distribution is approximately uniformly distributed along the chain. However,
when the applied electric field is aligned with the chain end-to-end vector
($\displaystyle\theta_{{\mathbold E}_{0}}=\pi/2$), the dipole distribution is
primarily concentrated near the chain ends. The electric field distribution
follows from the electric potential. Further, we observe that when the applied
electric field is orthogonal to the chain end-to-end vector, the electric
field is roughly uniformly distributed along the chain. However, when the
applied electric field is aligned with the chain end-to-end vector, the
electric field is largely concentrated near the chain ends. We also observe
that the local strength of the electric potential, the electric field, and the
polarization is higher when the applied electric field is orthogonal to the
chain end-to-end vector as compared to when it is aligned.

###  4.C Elastic Response

Figure 6:  Elastic response (force against stretch) of the polymer chain for
$\displaystyle\theta_{{\mathbold E}_{0}}=0$. The insets show the electric
field $\displaystyle\tilde{{\mathbold E}}$ and polarization
$\displaystyle\tilde{{\mathbold p}}$ for $\displaystyle
L=aN^{1/2},2aN^{1/2},3aN^{1/2}$. The arrows show the direction of the vector
field, and the background color shows the magnitude.  Figure 7:  Elastic
response (force against stretch) of the polymer chain for
$\displaystyle\theta_{{\mathbold E}_{0}}=\pi/2$. The insets show the electric
field $\displaystyle\tilde{{\mathbold E}}$ and polarization
$\displaystyle\tilde{{\mathbold p}}$ for $\displaystyle
L=aN^{1/2},2aN^{1/2},3aN^{1/2}$. The arrows show the direction of the vector
field, and the background color shows the magnitude.

To study the effect of chain stretch, we vary the length of the chain end-to-
end vector, denoted $\displaystyle L$. The total free energy of the polymer
chain at equilibrium, $\displaystyle F$, is obtained as [75]:

$F=-k_{B}T\log Q[w]-\frac{k_{B}T}{2}\int\ \mathrm{d}{\mathbold x}\
\mathrm{d}{\mathbold u}\ w({\mathbold x},{\mathbold
u})\langle\hat{\rho}({\mathbold x},{\mathbold u})\rangle,$ (20)

The first term is the free energy of the non-interacting polymer chain in the
external field $\displaystyle w({\mathbold x},{\mathbold u})$, which double
counts the electrical energy due to dipole-dipole interactions. The second
term in (20) corrects for this double counting. We define the elastic force in
the polymer chains by $\displaystyle f:=\frac{\partial F}{\partial L}$.

Figures 6 and 7 show the elastic force as a function of $\displaystyle L$ for
the applied electric field orthogonal and parallel to the chain orientation
respectively. The insets in the figures show the electric field
$\displaystyle\tilde{{\mathbold E}}$ and polarization
$\displaystyle\tilde{{\mathbold p}}$. We observe that the elastic response is
linear, which is consistent with the Gaussian nature of the polymer chain,
even with the applied electric field at different orientations. Further, the
elastic force response is essentially unchanged even when we change the
orientation of the applied electric field. However, the distributions of the
electric field and polarization change very significantly with the direction
of the applied electric field. When the applied electric field is orthogonal
to the chain end-to-end vector ($\displaystyle\theta_{{\mathbold E}_{0}}=0$,
Fig. 6), we observe that as the chain is stretched, the distributions of the
electric field and polarization remain roughly uniformly distributed along the
chain, but are increasingly concentrated along the chain end-to-end vector
with increasing stretch. However, when the applied electric field is aligned
with the chain end-to-end vector ($\displaystyle\theta_{{\mathbold
E}_{0}}=\pi/2$, Fig. 7), the electric field and the polarization are primarily
concentrated near the chain ends, with the concentration changing with
increasing stretch.

##  5 Concluding Remarks

We have developed a statistical field theory framework for polarizable polymer
chains that overcomes key limitations of existing approaches in accounting for
the nonlocal dipole-dipole interactions between polymer segments. Our approach
is applicable to general nonlinear polarization-electric field responses, by
reformulating the nonlocal dipole-dipole interactions through the local PDE of
electrostatics. Regardless of the nonlinearity of the polarization-field
response, the PDE constraint is linear and directly amenable to efficient
numerical methods, such as boundary element methods that can account for
unbounded domains [76] or very efficient Fourier methods [77]. Our approach
also enables the use of a more realistic ensemble that corresponds to a far-
field applied electric field.

There are several possibilities for further development. An immediate
direction of study is to incorporate general nonlinear polarization-electric
field responses that go beyond the linear anisotropic model studied here.
Another interesting direction would be to apply the framework developed here
to study cross-linked polymer networks, e.g. following [78]. This can provide
key physical insights into the effects of inter-chain dipole-dipole
interactions on the response of electromechanical polymer networks such as
dielectric elastomers under external fields. Yet another interesting
possibility is to study polymers in confined settings with polarizable ambient
media, which have been shown in other contexts to lead to unusual effects,
e.g. [79, 80]. Further, by including excluded volume interactions, it is
possible to study the interplay between repulsive excluded volume interactions
and attractive dipole-dipole interactions to tailor the functional properties
of polymeric soft matter. Furthermore, an important aspect that we have not
accounted for in our formulation is the role of fluctuations; recent
developments in the statistical field theory provide tools to make progress in
this direction [81]. Finally, the dipole response can be generalized to
account for screening, rapid variations of the electric field, and so on [67].
In principle, these can be accounted for in our model — screening by using the
screened Poisson equation, for instance, and using gradients of the electric
field in the monomer dipole response to account for quadrupole and higher
moments — but it is important to demonstrate these in practice.

##

#### Software and Data Availability.

The code developed for this work and the associated data are available at
https://github.com/pkhandag/polarizable-polymer.git

#### Acknowledgments.

We thank AFRL for hosting visits by Kaushik Dayal; NSF (DMS 2108784, DMREF
1921857), BSF (2018183), and AFOSR (MURI FA9550-18-1-0095) for financial
support; and NSF for XSEDE computing resources provided by Pittsburgh
Supercomputing Center. This article draws from the doctoral dissertation of
Pratik Khandagale at Carnegie Mellon University [82].

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