\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
Ninth MSU-UAB Conference on Differential Equations and Computational 
Simulations.
\emph{Electronic Journal of Differential Equations},
Conference 20 (2013),  pp. 165--174.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document} \setcounter{page}{165}
\title[\hfilneg EJDE-2013/Conf/20/ \hfil Extended constitutive laws]
{Extended constitutive laws for lamellar phases}

\author[C.-D. Yoo,  J. Vi\~nals\hfil EJDE-2013/conf/20 \hfilneg]
{Chi-Deuk Yoo, Jorge Vi\~nals}  % in alphabetical order

\address{Chi-Deuk Yoo \newline
School of Physics and Astronomy,
and Minnesota Supercomputing Institute,
University of Minnesota, 116 Church Street SE, Minneapolis,
 MN 55455, USA}
\email{yoo@physics.umn.edu}

\address{Jorge Vi\~nals \newline
School of Physics and Astronomy,
and Minnesota Supercomputing Institute,
University of Minnesota, 116 Church Street SE, Minneapolis,
 MN 55455, USA}
\email{vinals@umn.edu}

\thanks{Published October 31, 2013.}
\subjclass[2000]{74Q15, 76A10, 82C05, 82D60, 82D80}
\keywords{Block copolymer lamellae; rheology;
 effective constitutive relations of structured fluids}

\begin{abstract}
 Classically, stress and strain rate in linear viscoelastic materials
 are related by a constitutive relationship involving the viscoelastic
 modulus $G(t)$. The same constitutive law, within Linear Response
 Theory, relates currents of conserved quantities and gradients of
 existing conjugate variables, and it involves the autocorrelation
 functions of the currents in equilibrium. We explore the consequences
 of the latter relationship in the case of a mesoscale model of a
 block copolymer, and derive the resulting relationship between viscous
 friction and order parameter diffusion that would result in a lamellar
 phase. We also explicitly consider in our derivation the fact that the
 dissipative part of the stress tensor must be consistent with the uniaxial
 symmetry of the phase. We then obtain a relationship between the stress
 and order parameter autocorrelation functions that can be interpreted as
 an extended constitutive law, one that offers a way to determine them
 from microscopic experiment or numerical simulation.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks


\section{Introduction}

Modulated phases are equilibrium phases which in terms of symmetry are
intermediate between disordered liquids and fully ordered crystalline
solids. They are ubiquitous in soft-matter systems \cite{re:seul95},
and are currently being investigated for potential applications at the
nanoscale, as well as for their role in biological materials.
Their equilibrium state is characterized by a spatially periodic order
parameter, with the modulation generally resulting from the
competition between effective attractive interactions at short distances, and
repulsive at long distances. The details of the competition determine both the
symmetry of the phase and the characteristic length scale of its periodic
structure. Dynamically, the response arising from the system's structural
irreducible units introduces characteristic relaxation times that may span
several disparate scales. Transport at any given scale is often
viscoelastic as shorter scales do not completely decouple.

The main applications motivating our research concern the
dynamical response of block copolymer phases to shears (rheology), and
the nonequilibrium evolution of macroscopicallty disordered samples in
the extended system limit. Applications of block copolymers in
several areas of nanotechnology
that leverage their self assembly has spurred a large body of
experimental work \cite{re:bates12}.
Much of the equilibrium phenomenology is well
understood theoretically and through simulation. The equilibrium free
energy of the phases is determined through
Self Consistent Field Theory, a treatment that can include a fair
amount of microscopic and architectural detail of the polymer chains
\cite{re:hong81,re:fredrickson06,re:guo08}. In fact,
block copolymer phases may well be a first example of a complex
system in which a mesoscopic model allows a quantitative description
of the equilibrium phase diagram. On the other hand, key aspects of
their nonequilibrium behavior that are needed to optimize their
processing, or to describe their rheology are not well understood.

A structured fluid in equilibrium possesses some degree of broken
symmetry which is described by an appropriate mesoscopic order parameter set
$\psi(\mathbf{x},t)$. The equilibrium free energy $F$ is then a functional of
$\psi$ and its gradients. Nonequilibrium evolution is often modeled as
purely dissipative or relaxational, and assumed to be driven by free
energy reduction,
\begin{equation}
\frac{\partial \psi}{\partial t}
= -\Lambda \;\frac{\delta F}{\delta \psi}, \quad 
F = \int d \mathbf{x} f(\psi,\partial_{i} \psi),
\label{eq:TDGL}
\end{equation}
where $\Lambda$ is an Onsager kinetic coefficient of microscopic
origin, constant in some
cases, or $\Lambda = - M \nabla^{2}$ if the order parameter satisfies
a conservation law, with $M$ a constant mobility.
A number of well established methods exist that allow a quantitative
determination of
$F$ at or near equilibrium. In the case of block copolymers, we
mention Self Consistent Field Theory
\cite{re:fredrickson02,re:shi04,re:fredrickson06}, and extensions of
classical density functional theory
\cite{re:fraaije97,re:maurits98}. A self
consistent calculation of the generalized chemical potential $\mu =
\delta F/\delta \psi$ is obtained by computing the partition
function of a single polymer chain in a self-consistently calculated
chemical potential field (in the commonly used mean field version of
the theories).

Recent extensions of \eqref{eq:TDGL} explicitly introduce the two
point direct correlation function of the order parameter
\cite{re:elder07,re:archer12}
\begin{equation}
 C_2(\mathbf{x} - \mathbf{x'}) = - \frac{1}{k_{B}T}
 \Big(\frac{\delta^2 F}{\delta \psi(\mathbf{x}) \delta
\psi(\mathbf{x'})}\Big)_{\psi_0},
\label{eq:c2}
\end{equation}
where the derivative is computed around a reference state $\psi_0$,
usually the equilibrium state. Gradient models of $F$ in
\eqref{eq:TDGL} are recovered by expanding the correlation
function in gradients
\begin{equation}
C_2(\mathbf{x} - \mathbf{x'}) = - \frac{1}{k_{B}T} \left( \hat{C}_0 +
\hat{C}_2 \nabla^{2} + \hat{C}_{4} \nabla^{4} + \ldots \right) \delta(
\mathbf{x} - \mathbf{x'})
\label{eq:c2_gradient}
\end{equation}
The correlation function determines the linear part of
\eqref{eq:TDGL}, to which non linearities are added
phenomenologically. Different choices are now possible depending on
whether $\psi$ is a broken
symmetry variable ($\hat{C_0} = 0$ according to Goldstone's Theorem),
the equilibrium phase is uniform in equilibrium (the
Ginzburg-Landau free energy), or is modulated
(the Brazovskii free energy, for example,\cite{re:ohta86,re:fredrickson94})
\begin{equation}
F = \int d^3 x \Big\{
- \frac{r}{2} \psi^2 + \frac{u}{4} \psi^4 + \frac{\xi}{2}
\big[ (\nabla^2 + q_0^2) \psi \big]^2
\Big\},
\label{eq:brazovskii}
\end{equation}
where $q_0$ is the characteristic wavenumber of the modulation, and
$r$, $u$, and $\xi$
are coefficients that depend on the system in question. These
extensions have allowed quantitative descriptions of non
uniform systems at the mesoscale.

In a formal expansion in frequency ($\omega = 0$ corresponds
to equilibrium), the lowest order transport model must correspond to
reversible (non dissipative or quasi static) motion. Incorporation of
fluid flow requires consideration of an energy density that depends on
kinetic energy $\frac{1}{2} \rho v^{2}$. The momentum density
$\mathbf{g} = \rho \mathbf{v}$ and the velocity field $\mathbf{v}$ are
then conjugate variables. Momentum  density is a
conserved variable, with stress its associated current, $\partial_{t}
g_{i} + \partial_{j} \pi_{ij} = 0$. We will decompose the total stress
as $\pi_{ij} = P \delta_{ij} - \sigma_{ij}^{R} - \sigma_{ij}^{D}$,
explicitly separating the hydrostatic pressure $P$, and the reversible
and dissipative contributions to $\pi_{ij}$. Advection of order
parameter, $v_{i} \partial_{i} \psi$, can be added to Eq.~\eqref{eq:TDGL}
and carries with it the requirement
to introduce a reversible stress in the momentum conservation equation
through a Maxwell relation \cite{re:martin72}
\begin{equation}
\frac{\partial \dot{\psi}}{\partial v_{i}} = \frac{\partial \dot{g}_{i}}
{\partial(\partial_{j} \alpha_{j})} \quad \text{with }  \alpha_{j} =
\frac{\partial f}{\partial (\partial_{j}\psi)},
\label{eq:maxwell}
\end{equation}
where the dot signifies time derivative. For example if, as is often
the case, the energy density $f(\psi, \partial_{i} \psi)$ is quadratic
in the order parameter gradient $f = \frac{K}{2} \| \nabla \psi \|^{2}
+ g (\psi)$, a so called non standard reversible stress arises which
is given
by $\sigma_{ij}^{R} =K \partial_{i} \psi \; \partial_{j} \psi$.
\cite{re:gurtin96,re:anderson98}

Dissipative motion then appears formally at the next order in
$\omega$. In existing models of transport in modulated phases it
arises from diffusive relaxation of $\psi(\mathbf{x},t)$ in
\eqref{eq:TDGL}. In addition, viscous forces in the
equation of conservation of momentum are introduced explicitly or
implicitly through the dependence of the reversible body force $\partial_{j}
\sigma_{ij}^{R}$ on $\psi$. Such a description is incomplete for
partially fluid (modulated) phases. Consider, for example, the
simplest modulated phase: a lamellar phase. Let
$\mathbf{\hat{q}}$ be the normal to the lamellar planes, and
assume that the system is subjected to a shear strain of amplitude
$\gamma$. There are three possible orientations of
the lamellar planes relative to the shear velocity $\mathbf{v}_0$
(Figure \ref{fi:orientations}):
transverse ($\mathbf{\hat{q}} \parallel \mathbf{v}_0$), parallel  ($\hat{\bf
 q} \parallel \nabla \mathbf{v}_0$), and
perpendicular ($\hat{\mathbf{q}} \parallel \nabla \times
\mathbf{v}_0$). As expected, shear along the transverse
direction leads to an elastic stress linear in $\gamma$ 
(solid-like response).

\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.75\textwidth]{fig1}
\end{center}
\caption{Schematic representation of the three possible lamellar
  orientations relative to a shear}
\label{fi:orientations}
\end{figure}

 Dissipation, however, can be shown to appear only nonlinearly
(proportional to $\gamma^{3}$ for small $\gamma$). Furthermore, both parallel
and perpendicular orientations are completely decoupled from the flow
since $\mathbf{\hat{q}}\cdot\mathbf{v}_0 = 0$, and there is no effect
from the shear. To remedy this lack of linear dissipation, a Newtonian
constitutive law for a dissipative stress $\sigma_{ij}^{D}$ is introduced
\cite{re:fredrickson94,re:hall06}. However, while  linear in
$\gamma$, this relation is not consistent with the symmetry of the
phase, so that, for example, it does not distinguish,
between locally parallel and perpendicular orientations.

We explore here a direct connection between order parameter relaxation and
friction forces, and do so while allowing for a dissipative stress that is
consistent with the uniaxial symmetry of the lamellar phase.

\section{Theoretical framework}

\subsection{Linear Viscoelastic Modulus}

For an incompressible fluid of density $\rho$ the equation of conservation of
momentum reads
\begin{equation}
\rho \partial_t v_i + \partial_j \pi_{ij} = 0.
\label{eq:mom.cons.eqn}
\end{equation}
As indicated above, the reversible stress $\sigma_{ij}^{R}$ is determined by the
choice of gradient terms in the free energy,
\eqref{eq:maxwell}.
However, one still needs to introduce a constitutive law for the 
dissipative stress $\sigma_{ij}^{D}$.
In general, $\sigma_{ij}^{D}$ is assumed to be linear in the small strain rate
$v_{ij} = (\partial_i v_j + \partial_j v_i)/2$,
\begin{equation}
\sigma_{ij}^{D} = \eta_{ijkl} \; v_{kl},
\label{eq:stress_strain}
\end{equation}
where $\eta_{ijkl}$ is the viscosity tensor. The symmetry of the viscosity
tensor needs to be compatible with the broken symmetry of the modulated phase.
In the simple case of an isotropic incompressible Newtonian fluid, the coefficient
of proportionality is constant, so that $\partial_j
\sigma_{ij}^\text{D} = \eta \partial^2 v_i$ where  $\eta$ is the shear
viscosity.

More generally, a constitutive relation for viscoelastic fluids can be
written as a generalization of \eqref{eq:stress_strain} including
non locality in both space and time \cite{re:doi.polymer.86}
\begin{equation}
\sigma_{ij}({\bf x},t) =
\int_{-\infty}^{t} dt' \int d^3 x' \;
G_{ijkl}({\bf x}-{\bf x}', t- t')
\Big[ \partial_k' v_l({\bf x}',t')
+ \partial_l' v_k({\bf x}', t') \Big],
\label{eq:gen.constitutive.relation}
\end{equation}
where $\partial_i' = \partial / \partial x_i'$,
$\sigma_{ij}=\sigma_{ij}^\text{R}+\sigma_{ij}^\text{D}$ and
$G_{ijkl}({\bf x},t)$ is the viscoelastic modulus.
Because of causality the viscoelastic modulus is nonzero only for
$t' < t$.  The so called complex modulus is then
defined as \cite{re:doi.polymer.86}
\begin{equation}
G_{ijkl}^{*}(\omega) = - i\omega \int_0^{\infty} d t \; e^{i\omega t} G_{ijkl}(t)
 = G_{ijkl}'(\omega) + i G_{ijkl}''(\omega),
\label{eq:complex.modulus}
\end{equation}
where $G_{ijkl}'(\omega)$ is the storage modulus and
$G_{ijkl}''(\omega)$ is the loss modulus. The number of
independent elements of $G_{ijkl}(t)$ (or $G_{ijkl}^{*}(\omega)$)
depends on the underlying broken symmetry of the modulated phase.

\subsection{Response Functions}

In Linear Response Theory, changes in a field in response to its conjugated
external field are described through a response function. Strictly speaking,
the relaxation modulus in \eqref{eq:gen.constitutive.relation} is not
a response function because the strain rate is not the conjugate field to
the shear stress, rather the strain tensor $u_{ij} = (\partial_i u_j +
\partial_j u_i)/2$ is the appropriate variable.
The response function relating stress to strain $\chi_{\sigma_{ij}
\sigma_{kl}}({\bf x},t)$ is \cite{re:forster.hydro.75}
\begin{equation}
\sigma_{ij}({\bf x},t) =
\int_{-\infty}^{t} dt' \int d^3 x' 
\chi_{\sigma_{ij}\sigma_{kl}}({\bf x}-{\bf x}', t- t')
[ \partial_k' u_l({\bf x}',t')
  + \partial_l' u_k({\bf x}',t')],
\label{eq:rel.sigma.strain}
\end{equation}
where both temporal and spatial translation invariances are assumed.
The response function 
$\chi_{\sigma_{ij}\sigma_{kl}}({\bf x}-{\bf x}', t- t')$ is nonzero 
only for $t > t'$ because of causality. The
Laplace transform is then
\begin{equation}
\chi_{\sigma_{ij}\sigma_{kl}}({\bf x}-{\bf x}', z) =
\int_0^\infty dt \; e^{izt} \chi_{\sigma_{ij}\sigma_{kl}}({\bf x}-{\bf x}', t),
\end{equation}
which is analytic in the upper half $z$-plane.
Also, its half time Fourier transform is
\begin{equation}
\chi_{\sigma_{ij}\sigma_{kl}}({\bf x}-{\bf x}', \omega) =
\lim_{\epsilon \to 0} \chi_{\sigma_{ij}\sigma_{kl}}({\bf x}-{\bf x}',
z=\omega+ i\epsilon),
\end{equation}
which can be decomposed into real and imaginary parts
\begin{equation}
\chi_{\sigma_{ij}\sigma_{kl}}({\bf x}-{\bf x}', \omega) =
\chi_{\sigma_{ij}\sigma_{kl}}'({\bf x}-{\bf x}', \omega)
+ i \chi_{\sigma_{ij}\sigma_{kl}}''({\bf x}-{\bf x}',
\omega).
\end{equation}
Then, the fluctuation-dissipation theorem states that
the imaginary part of the response function is related to the auto
correlation function of the stress
\begin{equation}
\chi_{\sigma_{ij}\sigma_{kl}}^{\prime \prime}({\bf k}, \omega)
= \frac{\beta\omega}{2 V} \left< \sigma_{ij}({\bf k}, \omega)
  \sigma_{kl}(-{\bf k}, t=0) \right>,
\label{eq:fluctuation.dissipation}
\end{equation}
where $V$ is the volume of the system.

Of course, the viscoelastic relaxation modulus $G_{ijkl}(t)$ is related to
the correlation function as well. First, noting that $v_i = \partial_t
u_i$, we integrate \eqref{eq:gen.constitutive.relation} by parts to obtain
\begin{align*}
\sigma_{ij}({\bf x},t)
&=- \int_{-\infty}^{t} dt' \int d^3 x' \;
\partial_{t'} G_{ijkl}({\bf x}-{\bf x}', t- t')
\Big[ \partial_k' u_l({\bf x}',t')
+ \partial_l' u_k({\bf x}',t') \Big]\\
&\quad + \int d^3 x' \; G_{ijkl}({\bf x}-{\bf x}', t = t')
\Big[ \partial_k' u_l({\bf x}',t) + \partial_l'
u_k({\bf x}',t) \Big].
\end{align*}
By comparing this equation with \eqref{eq:rel.sigma.strain} we find
\begin{equation}
\chi_{\sigma_{ij}\sigma_{kl}}({\bf x} - {\bf x}', t-t')
= - \partial_{t'} G_{ijkl}({\bf x} - {\bf x}',t-t')
+ G_{ijkl}({\bf x} - {\bf x}', t-t') \delta(t'- t).
\end{equation}

Since both $\chi_{\sigma_{ij}\sigma_{kl}}$ and $G_{ijkl}$ are nonzero for
$t>t'$
we take the Laplace transform of the above equation. Since boundary
terms at $t=t'$ cancel out, and by using the fact that
$\chi_{\sigma_{ij}\sigma_{kl}}({\bf k}, \omega) =
\lim_{\epsilon \to 0} \chi_{\sigma_{ij}\sigma_{kl}}({\bf k}, z=\omega+
i\epsilon)$ and the definition of the complex modulus
\eqref{eq:complex.modulus} we find
\begin{equation}
\chi_{\sigma_{ij}\sigma_{kl}}({\bf k}, \omega) = G_{ijkl}^{*}({\bf k},\omega).
\label{eq:rel.chi.and.complex.modulus}
\end{equation}
In other words, the complex modulus is the momentum current autocorrelation
function. In terms of this function, the storage and loss moduli are given by
\begin{gather}
G_{ijkl}''({\bf k},\omega)
= \frac{\beta\omega}{2 V} \left< \sigma_{ij}({\bf k}, \omega)
  \sigma_{kl}(-{\bf k}, t=0) \right>,
\label{eq:loss.modulus} \\
G_{ijkl}'({\bf k},\omega)
= \frac{ \beta }{2V} \mathcal{P} \int_{-\infty}^\infty \frac{d \omega_1}{\pi} \;
\frac{\omega_1  \left< \sigma_{ij}({\bf k}, \omega_1)
\sigma_{kl}(-{\bf k}, t=0) \right>}{\omega_1 - \omega},
\label{eq:storage.modulus}
\end{gather}
where we have used the fluctuation-dissipation theorem
\eqref{eq:fluctuation.dissipation},
and the Kramers-Kronig relation between $\chi'$ and
$\chi''$. In \eqref{eq:storage.modulus}
$\mathcal{P}$ stands for the principal value of the integral.
In short, Equations \eqref{eq:loss.modulus} and \eqref{eq:storage.modulus} 
show that a general, but linear, viscoelastic constitutive relation follows from
the stress autocorrelation function. We also mention that the momentum
conservation equation allows one to express the stress autocorrelation
function in terms of the velocity autocorrelation function.
In general, if the fluid is incompressible, one can use the longitudinal
part (parallel to the velocity gradient) of the momentum conservation
equation to eliminate the pressure, and from the resulting
transverse part of the momentum conservation equation one finds
\begin{equation}
\left< v_i({\bf k}, \omega) v_k(-{\bf k},t=0) \right> =
\frac{k_j k_l}{\rho^2\omega^2} \left< \sigma_{ij}({\bf k}, \omega)
  \sigma_{kl}(-{\bf k},t=0) \right>.
\label{eq:rel.velvel.to.sigmasigma}
\end{equation}
By inserting \eqref{eq:rel.velvel.to.sigmasigma} into
\eqref{eq:rel.chi.and.complex.modulus}, it is possible to relate
the complex modulus to the velocity autocorrelation function.

The Green-Kubo relation follows immediately from \eqref{eq:loss.modulus}.
In the limit of small frequency and wavenumber, one has
\begin{equation}
\eta_{ijkl} = \lim_{\omega \to 0} \lim_{k \to 0}
\frac{G_{ijkl}''({\bf k}, \omega)}{\omega},
\label{eq:Green.Kubo}
\end{equation}
which is the viscosity tensor defined in \eqref{eq:stress_strain}.

\subsection{Extended Constitutive Relations}

In the spirit of Eqs.~\eqref{eq:c2} and \eqref{eq:c2_gradient}, one
can ask whether it is possible to use microscopic information contained
in the stress autocorrelation function (for example, from experiments
or molecular level simulation) to formulate extended constitutive
equations for the stresses that will appear as nonlocal contributions in
the momentum conservation equation. This is analogous to nonlocal contributions
from the pair correlation function $C_2$ in the equation governing the
evolution of the order parameter. Such a representation could be
particularly useful in modulated phases in which structural correlations
will affect molecular friction.


If there exists a hydrodynamic variable associated with a broken
symmetry, flow advects the new variable, and the reversible momentum
current $\sigma_{ij}^\text{R}$ contains an additional contribution
associated with the restoring force to an equilibrium ordered state
when the system is driven out of equilibrium, as shown in
\eqref{eq:maxwell} \cite{re:martin72}. Assume now that
\eqref{eq:gen.constitutive.relation} remains valid. In general,
the equation of conservation of momentum can be formally linearized as
\begin{equation}
\rho\partial_t v_i - \partial_j \sigma_{ij}^\text{D}
= - \partial_i P + \partial_j \sigma_{ij}^\text{R}.
\label{eq:mom.cons.eqn.1}
\end{equation}
We rewrite this equation, in Fourier space,
\begin{equation}
\hat{\mathcal{L}}_{ij}({\bf k},\omega) v_j({\bf k},\omega) =
\hat{\mathcal{F}}_i (\psi),
\end{equation}
where the linear operator $\hat{\mathcal{L}}_{ij}({\bf k},\omega)$ is defined from
the LHS of \eqref{eq:mom.cons.eqn.1}, and $\hat{\mathcal{F}}_i
(\psi)$ follows from the dependence of the reversible shear stress on
the additional order parameter fields. The velocity auto correlation
function can be formally written as
\begin{equation}
\left< v_i({\bf k}, \omega) v_k(-{\bf k},t=0) \right> =
\hat{\mathcal{L}}_{ij}^{-1}({\bf k},\omega) \hat{\mathcal{L}}_{kl}^{-1} (-{\bf k}, -\omega)
\big<\hat{\mathcal{F}}_j (\psi)({\bf k},\omega) \hat{\mathcal{F}}_l
(\psi)(-{\bf k},t=0) \big>,
\label{eq:model.constitutive.rel}
\end{equation}
where $\hat{\mathcal{L}}^{-1}({\bf k},\omega)$ is the inverse of
$\hat{\mathcal{L}}({\bf k},\omega)$
such as $\hat{\mathcal{L}}_{ij}^{-1}({\bf
  k},\omega)\hat{\mathcal{L}}_{jk}({\bf k},\omega) = \delta_{ij}$.
Note now that, in the linear regime, the RHS of
Eq.~\eqref{eq:model.constitutive.rel} is proportional
to the autocorrelation function of the order parameter.
Equation~\eqref{eq:model.constitutive.rel} is interesting as it shows
the formal relationship between correlations of the viscous degrees of
freedom (and hence momentum response), and
the structural order parameters that characterize the mesophase. As
such, it can be thought of as a generalized constitutive equation.
We next illustrate the relationship in the specific case of a block
copolymers in the lamellar phase.

\section{Viscoelastic response of a Lamellar phase}

Diblock copolymers are macromolecules comprising two chemically distinct
and mutually incompatible segments (monomers) that are covalently
bonded. Their equilibrium properties are
dictated by $N$, the degree or polymerization (i.e., the length of the chain),
$f$, the volume fraction of one of the monomers, and $\chi$, the Flory-Huggins
interaction parameter between the distinct segments
\cite{re:fredrickson06}. The first two parameters can be
controlled through processing, whereas the third is determined by the
choice of monomers involved. Above an order-disorder transition
temperature $T_{ODT}$, the equilibrium phase is fluid like (disordered).
Below $T_{ODT}$, equilibrium structures of a wide
variety of symmetries have been predicted and experimentally observed.
Around $f=0.5$ (symmetric mixture), a so called lamellar phase is
observed, in which nanometer sized
layers of A and B rich regions alternate in space. Phases of complex
symmetries, including bi-continuous structures, have been predicted and
observed in higher order multiblocks \cite{re:zheng95,re:lee10}.

At frequencies low compared with the smallest inverse relaxation time of the
polymer, chain conformation fluctuations can be
adiabatically eliminated, and a mesoscopic theory can be
developed. The evolution of a melt is
described by an order parameter field $\psi(\mathbf{x},t)$ which
represents the local density difference of the constituent monomers.
The coarse grained free energy functional $F$ was first
given by \cite{re:ohta86}. In the weak segregation limit near the order-disorder
transition point, the simpler free energy in \eqref{eq:brazovskii}
has been introduced \cite{re:fredrickson94,re:leibler80}.

The linearized momentum conservation equation describing the dynamics of block copolymer
in lamellar phases is given by \eqref{eq:mom.cons.eqn.1}, with the
reversible stress tensor given by\begin{equation}
\partial_j \sigma_{ij}^\text{R} =
- \psi \partial_i \big( \frac{\delta F}{\delta \psi} \big).
\end{equation}

We now introduce the following constitutive law for the dissipative 
stress tensor $\sigma_{ij}^\text{D}$ that is compatible with the uniaxial 
symmetry of a lamellar phase. For perfectly ordered lamellae
with $\hat{q}$ being the unit normal to the layers, we
have\cite{re:ericksen59}
\begin{equation}
\sigma_{ij}^\text{D} = \alpha_1 \hat{q}_i \hat{q}_j \hat{q}_k \hat{q}_l v_{kl} + \alpha_4 v_{ij}
+ \alpha_{56} \hat{q}_k ( \hat{q}_i v_{kj} + \hat{q}_j v_{ki} ),
\end{equation}
where $\alpha_1$, $\alpha_4$ and $\alpha_{56}$ are viscosity coefficients.
Now we can use this momentum conservation equation to derive a
relation in terms of the autocorrelation function of the
order parameter.

As a reference state we consider one in which flow is absent, and planar lamellae with
periodicity ${\bf q}$ are stationary. The function
$\psi_0 = \psi_1 \cos({\bf q} \cdot {\bf x}) + \ldots $
minimizes the free energy
Eq.~\eqref{eq:brazovskii} to lowest order in $r/\xi q_0^4$ with
\begin{equation}
\psi_1^2 = \frac{4}{3} \big[ \frac{r}{u} - \frac{\xi}{u}(q^2 - q_0^2)^2 \big].
\end{equation}
Next, we consider disturbances from the reference state as $\delta
\psi = \psi - \psi_0$ and $v_i$,
and linearize \eqref{eq:mom.cons.eqn.1}, resulting in
\begin{equation}
\rho \partial_t v_i = \partial_j \delta \sigma_{ij}^\text{R} 
+ \partial_j \sigma_{ij}^\text{D},
\label{eq:linearized.mom.eqn}
\end{equation}
where
\begin{equation}
\partial_j \delta \sigma_{ij}^\text{R} = - \psi_0 \partial_i 
\big[ \bar{\mu}({\bf r}) \delta \psi \big].
\end{equation}
with $\bar{\mu}({\bf r}) = - r + 3 u \psi_0^2 + \xi (\nabla^2 + q_0^2)^2$.
We can then identify the linear operator $\hat{\mathcal{L}}_{ij}$ from the terms
proportional to the flow velocity in \eqref{eq:linearized.mom.eqn},
\begin{equation}
\hat{\mathcal{L}}_{ij}({\bf k},\omega) v_j({\bf k},\omega)
= -i\omega \rho v_i({\bf k},\omega) - \partial_j \sigma_{ij}^\text{D}({\bf k},\omega),
\end{equation}
and
\begin{equation}
\hat{\mathcal{F}}_i(\psi) = \partial_j \delta \sigma_{ij}^\text{R}.
\end{equation}

The resulting relation is a function of the orientation of the lamellae 
relative to the local velocity gradient. To be specific, we consider 
two examples: Perpendicularly and transverse oriented
lamellae (Figure \ref{fi:orientations}).

\subsection{Perpendicular Orientation}

In this case ${\bf v} = v \hat{x}$, ${\bf q} = q \hat{y}$,
${\bf k} = k \hat{z}$, and there is no reversible contribution 
to the stress tensor because
$\partial_j \delta \sigma_{ij}^\text{R} \propto k_i - q_i$ 
is perpendicular to $v_i$. Then \eqref{eq:linearized.mom.eqn} 
when linearized reduces to
\begin{equation}
\rho \partial_t v({\bf k},t) = - \frac{\alpha_4}{2} k^2 v({\bf k},t).
\label{eq:mom.cons.eqn.perp}
\end{equation}
Now it is straightforward to calculate the velocity-velocity 
correlation function.
First, we Laplace transform \eqref{eq:mom.cons.eqn.perp} to get
\begin{equation}
v({\bf k},z) = \frac{1}{-iz + \alpha_4 k^2/2\rho} v({\bf k},t=0).
\end{equation}
This equation and Equation (3.14) onf \cite{re:forster.hydro.75} imply that
\begin{equation}
\chi_{vv}({\bf k},z) 
= \frac{\alpha_4 k^2/2\rho}{-iz + \alpha_4 k^2/2\rho}\chi_{vv}({\bf k}).
\end{equation}
Since $\chi_{vv}({\bf k},\omega) = \lim_{\epsilon \to 0} \chi_{vv}({\bf k},
z=\omega + i \epsilon)
= \chi'_{vv}({\bf k},\omega) + i \chi''_{vv}({\bf k},\omega)$ and
$\chi_{vv}({\bf k}) = \rho^{-1}$ we find from equations 
\eqref{eq:loss.modulus} and
\eqref{eq:storage.modulus} that
\begin{equation}
G_{xzxz}'({\bf k},\omega) =
\frac{\alpha_{4}}{2} \omega^2 \frac{\alpha_4 k^2 / 2 \rho}{\omega^2 +
  (\alpha_4 k^2 / 2 \rho)^2},
\end{equation}
and
\begin{equation}
G_{xzxz}'' ({\bf k},\omega) =
\frac{\alpha_{4}}{2} \frac{\omega^3}{\omega^2 + (\alpha_4 k^2 / 2 \rho)^2}.
\end{equation}
The viscosity coefficient can be obtained by taking small $k$ and $\omega$ limits
\begin{equation}
\eta_\perp = \lim_{\omega \to 0} \lim_{k \to 0} \frac{G_{xzxz}'' ({\bf k},\omega)}{\omega}
= \frac{\alpha_4}{2}.
\end{equation}
The lamellar phase in this orientation responds as a terminal fluid of 
shear viscosity $\alpha_{4}$ (terminal behavior has at low frequencies
$G_{xzxz}'' \propto \omega$ and $G_{xzxz}' \propto \omega^2$). 
This is an obvious result and illustrates that there is no  coupling
 between the order parameter and the fluid velocity. The same is the 
case for the parallel orientation. Interestingly, however, the resulting
 viscosity is $\alpha_4+\alpha_{56}$. Even though the lamellar phase 
responds as a terminal fluid for both orientation, the resulting 
viscosity is nevertheless different.

\subsection{Transverse Orientation}

Consider now a transverse orientation
in which ${\bf v} = v \hat{x}$, ${\bf q} = q \hat{x}$ and ${\bf k} = k \hat{z}$.
In contrast to the perpendicular orientation,
since ${\bf v} \parallel {\bf q}$, $\hat{F}_x(\psi) \propto q_x$ does not vanish, and there exists coupling between $\psi$ and ${\bf v}$. Elastic and viscous response are coupled in this case.
If we use the incompressibility condition (${\bf v} \cdot {\bf k} = 0$), we find
\begin{equation}
\hat{\mathcal{L}}_{xx} = -i \omega \rho + \frac{\alpha_4 + \alpha_{56}}{2} k^2,
\end{equation}
and
\begin{equation}
\hat{\mathcal{F}}_x(\psi) =
i q \frac{\psi_1}{2} \xi \big[ k^4 + 2 k^2 (q^2 - q_0^2) \big]
\big[ \delta\psi({\bf k}-{\bf q},t) - \delta\psi({\bf k}+{\bf q},t) \big].
\end{equation}
Defining the phase of the perturbation $\phi_2({\bf k},t) = [\delta\psi({\bf k}+{\bf q},t)
-\delta\psi({\bf k}-{\bf q},t)]/\sqrt{2}$,
we find from equations \eqref{eq:loss.modulus}, 
\eqref{eq:rel.velvel.to.sigmasigma}
and \eqref{eq:model.constitutive.rel},
\begin{equation}
\begin{split}
G_{xzxz}' ({\bf k},\omega) =&
- \frac{\beta}{V} \frac{\psi_1^2}{4} q^2 \xi^2
k^2 \Big[ k^2 + (q^2 - q_0^2) \Big]^2
\\
& \times
\mathcal{P} \int_{-\infty}^{\infty} \frac{d \omega_1}{\pi}
\frac{\omega_1^3\big< \phi_2({\bf k}, \omega_1) \phi_2 (-{\bf k},t=0) \big>}
{(\omega_1-\omega)[\omega_1^2 + (\alpha_4+\alpha_{56})^2 k^4 / 4 \rho^2]},
\label{eq:trans.storage.modulus}
\end{split}
\end{equation}
and
\begin{equation}
\frac{G_{xzxz}'' ({\bf k},\omega)}{\omega} =
- \frac{\beta}{V} \frac{\psi_1^2}{4} q^2 \xi^2
k^2 \Big[ k^2 + 2 (q^2 - q_0^2) \Big]^2
\frac{\omega^2\big< \phi_2({\bf k}, \omega) \phi_2 (-{\bf k},t=0) \big>}
{\omega^2 + (\alpha_4+\alpha_{56})^2 k^4 / 4 \rho^2}.
\label{eq:trans.loss.modulus}
\end{equation}

Equations \eqref{eq:trans.storage.modulus} and 
\eqref{eq:trans.loss.modulus} directly relate the viscoelastic 
modulus and the order parameter correlation function. Of course, 
their specific form here depend on the functional form of 
the reversible stress and of the dissipative stress tensor chosen 
(the prefactor and denominator in both equations respectively). 
The prefactor is related to the equilibrium inverse susceptibility 
of the order parameter (not any kinetics), and the denominator 
indicates viscous diffusion with viscosity $\alpha_{4} + \alpha_{6}$. 
It is quite straightforward to conduct the calculation for other 
free energies or forms of the dissipative stress. Assuming, however, 
that both functions can be experimentally determined (e.g., micro 
rheology and X-Ray scattering) or by numerical simulation of a copolymer
 melt, including dynamical density functional theory, their relationship 
would give information about the microscopic mechanisms responsible for 
viscous friction. These mechanisms are, at present, unknown for block 
copolymers, fact that hinders a solid theoretical understanding of 
their nonequilibrium evolution, and, as indicated in the Introduction, 
underpins the need for a great deal of empiricism in their processing.

\subsection*{Acknowledgments}

We thank the Minnesota Supercomputing Institute for supporting this
research.

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\end{document}