# Theory of nematic and polar active fluid surfaces

Guillaume Salbreux<EMAIL_ADDRESS>University of Geneva, Quai
Ernest Ansermet 30, 1205 Genève, Switzerland  The Francis Crick Institute, 1
Midland Road, NW1 1AT, United Kingdom  Frank Jülicher  Max Planck Institute
for the Physics of Complex Systems, Nöthnitzer Str. 38, 01187 Dresden, Germany
Jacques Prost  Institut Curie, PSL University 26 rue d’Ulm F-75248 Paris Cedex
05, France  Mechanobiology Institute, NUS, 5A Engineering Dr 1, 117411,
Singapore  Andrew Callan-Jones  Laboratoire de Matière et Systèmes Complexes,
Université de Paris, France

###### Abstract

We derive a fully covariant theory of the hydrodynamics of nematic and polar
active surfaces, subjected to internal and external forces and torques. We
study the symmetries of polar and nematic surfaces and find that in addition
to 5 different types of in-plane isotropic surfaces, polar and nematic
surfaces can be classified into 5 polar, 2 pseudopolar, 5 nematic and 2
pseudonematic types of surfaces. We give examples of physical realisations of
the different types of surfaces we have identified. We obtain expressions for
the equilibrium tensions, moments, and external forces and torques acting on a
passive polar or nematic surface. We calculate the entropy production rate
using the framework of thermodynamics close to equilibrium and find
constitutive equations for polar and nematic active surfaces with different
symmetries. We study the instabilities of a confined flat planar-chiral polar
active layer and of a confined deformable polar active surface with broken up-
down symmetry.

Figure 1:  A. In this manuscript, we consider active surfaces with a tangent
polar or nematic order. B. The distribution of stresses within a thin layer
gives rise to tensions and torques acting within the surface, when integrated
across the thickness of the layer. C. Internal and external forces and torques
act on a surface element with surface area $dS$. D. The order parameter
changes in time and is transported by the flow. Lagrangian time derivatives
are taken by comparing the order parameter at two times $t$ and $t+dt$,
following the flow of the surface.

##  I Introduction

Living systems contain a rich repertoire of dazzling and seemingly
choreographed surface movements. These underlie vital biological functions,
ranging from the subcellular up to the organ scale. Biological surfaces are
frequently active, being driven by chemical reactions at the microscopic
level. They usually also possess internal degrees of freedom corresponding to
in-plane order and that interplay with each other over large scales, giving
rise to self-organized behavior. For example, self-generated flow and nematic
re-ordering in the actomyosin cortex drive cell surface deformations at the
late stage of cell division Reymann:2016aa . At a larger scale, morphogenetic
movements involve coherent flows in epithelia, thin tissues that grow and
deform to give shape to organs. The constituent cells often carry an in-plane
polarity that dynamically couples to tissue flow to help establish patterns in
the developing organism Eaton:2011aa .

At the root of cell and tissue surface mechanics is the actomyosin
cytoskeleton, a type of filamentous, soft active material Marchetti_RMP .
Theoretical descriptions of these active gels have been successfully applied
to a number of problems involving three dimensional active flows and nematic
or polar re-ordering prost2015active . Yet, surface movements pose special
theoretical difficulties owing to the geometric nonlinearities inherent in
them. Thus, transposing active gel physics from three to two dimensions is a
challenging problem.

A step in this direction was made in recent work establishing a theoretical
framework for active isotropic surfaces salbreux2009hydrodynamics . From the
different types of broken symmetries compatible with isotropic surfaces,
generic relations were derived linking the tensions and moments across a
surface cut to variables such as the local curvature, velocity field, and
chemical activity. This theory has been successfully used to describe mechano-
chemical instabilities of isotropic active fluid surfaces mietke2019minimal ;
mietke2019self . This and other continuum theories have also been used to
address the deformations of epithelia, which have distinct apical and basal
interfaces and can therefore be seen as surfaces with broken up-down symmetry
messal2019tissue ; haas2019nonlinear ; morris2019active ; al2021active .
Despite these advances, understanding and systematically exploring how in-
plane order couples to surface deformation and activity is beyond the reach of
current theories.

Numerous experimental observations have shown that broken symmetry variables
play a key role in biological systems. Nematic order has been observed to
emerge in the cell cortex Reymann:2016aa ; spira2017cytokinesis , but polar
order could also exist. Long-range coherent patterns of cell planar polarity
emerge during development goodrich2011principles ; sagner2012establishment .
Epithelia cultured in vitro exhibit patterns of nematic order when the
individual cells are elongated saw2018biological . Recent experiments on
reconstituted and in vivo systems have revealed how activity and topological
defects in the surface nematic field can lead to dramatic surface shape
changes keber2014topology . Coupling between nematic ordering and flows and
deformations have been shown to play a key role in epithelial tissues
blanch2021integer ; blanch2021quantifying ; doostmohammadi2021physics ;
guillamat2022integer  and can even explain regeneration in Hydra Maroudas-
Sacks:2021aa . Along these lines, coarse-grained simulations have
recapitulated many of these findings and have shed some light on the basic
physics Alaimo:2017aa ; Metselaar:2019aa . Still, there is no comprehensive
continuum framework yet that can be used to predict how in-plane polar or
nematic order interacts with active surface deformation and flow. We expect
that an understanding of these effects will have broad relevance, including in
living systems and in designing engineered materials manna2021harnessing . We
propose here to develop such a framework.

In this work we derive a covariant theory for polar and nematic surfaces which
are driven out of equilibrium by chemical reactions. We classify these
surfaces according to their symmetries, and find 7 polar surfaces and 7
nematic surfaces (Fig. 3, section III). We calculate equilibrium tensions and
torques of a generic polar or nematic fluid surface, and obtain the entropy
production rate close to equilibrium. We then obtain linear constitutive
equations, out of equilibrium, for the different classes of polar and nematic
surfaces (sections V and VI). We then discuss flows and deformation of a
confined active polar film with broken planar-chiral or up-down symmetry
(section VII).

##  II Tensor variations

We start by introducing notations and differential operators for vectors and
tensors which will be convenient in the next sections. Notations for
differential geometry are as in salbreux2017activesurfaces  and are summarized
in Appendix A. Briefly, we denote by latin indices $i,j$ the surface
coordinates $s^{1}$, $s^{2}$. The position of the surface in three-dimensions
is denoted by a vector $\mathbf{X}(s^{1},s^{2})$. The tangent vectors to the
surface are denoted $\mathbf{e}_{i}=\partial_{i}\mathbf{X}$ and the unit
normal vector
$\mathbf{n}=\mathbf{e}_{1}\times\mathbf{e}_{2}/|\mathbf{e}_{1}\times\mathbf{e}_{2}|$.
The velocity field on the surface is denoted $\mathbf{v}$. In the following we
consider polar and nematic surfaces, whose local order is described
respectively by a tangent polar vector $\mathbf{p}$ and a tangent symmetric
traceless tensor $\mathbf{Q}$. To describe variations of vectors and tensors
on the surface, we introduce a corotational operator for vectors and second-
rank tensors varying together with the shape $\mathbf{X}$. For a given
infinitesimal variation of the shape $d\mathbf{X}$, we denote an infinitesimal
rotation by $d\boldsymbol{\vartheta}=\frac{1}{2}\nabla\times d\mathbf{X}$ (Eq.
211). Similarly, we introduce the vorticity
$\boldsymbol{\omega}=\frac{1}{2}\nabla\times\mathbf{v}$ associated to the
surface velocity field $\mathbf{v}$. The corotational variation of a vector
$\mathbf{a}$ or second-rank tensor $\mathbf{B}$, not necessarily tangent to
the surface, is then denoted with a capital $D$, and the corotational time
derivative with a capital $D_{t}$:

$\displaystyle D\mathbf{a}$
$\displaystyle=d\mathbf{a}-d\boldsymbol{\vartheta}\times\mathbf{a}~{},$ (1)
$\displaystyle D_{t}\mathbf{a}$ $\displaystyle=\frac{{\rm d}\mathbf{a}}{{\rm
d}t}-\boldsymbol{\omega}\times\mathbf{a}~{},$ (2) $\displaystyle D\mathbf{B}$
$\displaystyle=d\mathbf{B}-d\boldsymbol{\vartheta}\times_{1}\mathbf{B}-d\boldsymbol{\vartheta}\times_{2}\mathbf{B}~{},$
(3) $\displaystyle D_{t}\mathbf{B}$ $\displaystyle=\frac{{\rm
d}\mathbf{B}}{{\rm
d}t}-\boldsymbol{\omega}\times_{1}\mathbf{B}-\boldsymbol{\omega}\times_{2}\mathbf{B}~{},$
(4)

where the notation $\times_{1}$ and $\times_{2}$ indicates vector products
with the first and second members, respectively, of the dyad forming
$\mathbf{B}$ (Eq. A). The corotational operator $D$ applied to a vector/tensor
describes the vector/tensor variation that does not arise from rotation of the
surface in which the vector/tensor is embedded. In Eqs. 2 and 4, the time
derivatives of vectors $\mathbf{a}$ and second-rank tensors $\mathbf{B}$,
$d\mathbf{a}/dt$ and $d\mathbf{B}/dt$, are Lagrangian time derivatives and are
taken following the flow of material points on the surface (Fig. 1D). In the
following, when considering time-dependent surface fields and surface
deformations, we adopt an Eulerian perspective where the coordinates of the
surface move along the normal to the surface, but do not vary with the
tangential flow. In that case, for a vector field $\mathbf{a}(s^{1},s^{2},t)$
or a tensor field $\mathbf{B}(s^{1},s^{2},t)$ (Appendix B.4),

$\displaystyle\frac{{\rm d}\mathbf{a}}{{\rm d}t}$
$\displaystyle=\partial_{t}\mathbf{a}+v^{i}\partial_{i}\mathbf{a}~{},$ (5)
$\displaystyle\frac{{\rm d}\mathbf{B}}{{\rm d}t}$
$\displaystyle=\partial_{t}\mathbf{B}+v^{i}\partial_{i}\mathbf{B}~{}.$ (6)

To keep notations compact, it is useful to introduce a covariant derivative
operator which returns the components of the infinitesimal variations of a
tangent vector or tensor. We therefore introduce the following notation for
covariant variations of a vector $\mathbf{a}$ and a second-rank tensor
$\mathbf{B}$:

$\displaystyle\nabla a^{i}$ $\displaystyle=(d\mathbf{a})\cdot\mathbf{e}^{i}$
(7) $\displaystyle\nabla_{j}a^{i}$
$\displaystyle=\left(\partial_{j}\mathbf{a}\right)\cdot\mathbf{e}^{i}$ (8)
$\displaystyle\nabla_{t}a^{i}$ $\displaystyle=\left(\frac{{\rm
d}\mathbf{a}}{{\rm d}t}\right)\cdot\mathbf{e}^{i}$ (9) $\displaystyle\nabla
B^{ij}$ $\displaystyle=(d\mathbf{B}):(\mathbf{e}^{i}\otimes\mathbf{e}^{j})$
(10) $\displaystyle\nabla_{k}B^{ij}$
$\displaystyle=(\partial_{k}\mathbf{B}):(\mathbf{e}^{i}\otimes\mathbf{e}^{j})$
(11) $\displaystyle\nabla_{t}B^{ij}$ $\displaystyle=\left(\frac{{\rm
d}\mathbf{B}}{{\rm d}t}\right):(\mathbf{e}^{i}\otimes\mathbf{e}^{j})~{},$ (12)

which can be distinguished, e.g., from component variations such as
$da^{i}=d(\mathbf{a}\cdot\mathbf{e}^{i})$. The operator $\nabla_{j}$ is the
standard covariant derivative with respect to the surface coordinates. The
operator $\nabla_{t}$ takes the Lagrangian time derivative of a vector or
tensor and projects it on the surface tangent directions.

The components of the corotational operator read for a vector $\mathbf{a}$ or
a second-rank tensor $\mathbf{B}$:

$\displaystyle Da^{i}$ $\displaystyle=(D\mathbf{a})\cdot\mathbf{e}^{i}$ (13)
$\displaystyle D_{t}a^{i}$
$\displaystyle=(D_{t}\mathbf{a})\cdot\mathbf{e}^{i}$ (14) $\displaystyle
DB^{ij}$ $\displaystyle=(D\mathbf{B}):(\mathbf{e}^{i}\otimes\mathbf{e}^{j})$
(15) $\displaystyle D_{t}B^{ij}$
$\displaystyle=(D_{t}\mathbf{B}):(\mathbf{e}^{i}\otimes\mathbf{e}^{j})~{}.$
(16)

The operators $\nabla$ and $D$ obey the product rule for derivatives; e.g.
$D(a^{i}b^{j})=(Da^{i})b^{j}+a^{i}(Db^{j})$, and indices of components of the
operators $\nabla$ and $D$ can be lowered or raised with the metric; for
instance $Da_{i}=(D\mathbf{a})\cdot\mathbf{e}_{i}=g_{ij}Da^{j}$ (Appendix A).
These definitions can be generalized to higher-order tensors, using
generalized cross-products between a vector and a tensor defined in Eq. A.

##  III Symmetries of polar and nematic surfaces

In this section we discuss symmetries of possible phases for nematic and polar
surfaces. We present a classification of surfaces according to how a surface
element transforms under mirror operations and rotations.

###  III.1 Symmetries, order parameters, and phases

To discuss surface symmetries, we first introduce the signatures of scalars,
vectors and tensors under a change of coordinates. We define the signature
$\epsilon_{A}^{T}$ of a tensorial quantity $A_{i_{1}...i_{n}}$ on the surface
under a 3D linear transformation $\mathbf{T}$, such that

$A^{\prime}_{i_{1}...i_{n}}=\epsilon_{A}^{T}T_{i_{1}}{}^{k_{1}}...T_{i_{n}}{}^{k_{n}}A_{k_{1}...k_{n}}\,,$
(17)

i.e., $\epsilon_{A}^{T}=\pm 1$ is a factor that distinguishes a tensor from a
pseudo-tensor. A true tensor has $\epsilon_{A}^{T}=1$ for all transformations
$\\{T\\}$; a pseudo-tensor has $\epsilon_{A}^{T}=-1$ for a subset of
transformations $\\{T\\}$. For a surface, the effect of a local change of
coordinates can be decomposed into a tangential and a normal part:

$\displaystyle\mathbf{T}^{\rm 3D}=\mathbf{T}^{\rm 2D}\mathbf{T}^{n}\,,$ (18)

such that $\mathbf{T}^{\rm 2D}$ corresponds to a change of coordinates
$(s^{1},s^{2})$ on the surface and $\mathbf{T}^{n}$ is a change of coordinates
perpendicular to the surface. Vectorial and tensorial quantities on the
surface have their components transformed by $\mathbf{T}^{\rm 2D}$ only.

Figure 2:  Examples of signatures of scalar and vectorial order parameters
under linear transformations (Eq. 17). As an example we consider two
transformations, a mirror operation with respect to the tangent plane $M_{n}$
and a rotation by $\pi$ around the $x$ axis, $R_{x}$ (Fig. 3A). For each
transformation, the top schematic indicates how the surface as a whole is
modified by the transformation, and the order parameter is redefined
accordingly, while the lower schematic shows the order parameter modified by
the tangential part of the transformation. The signature ($+1$ or $-1$ in the
right-hand side) then indicates whether the two redefined order parameters are
the same (signature $+1$) or not (signature $-1$). A. A true tangent polar
vector has signature $+1$ under all transformations. B. The up-down
pseudoscalar $\epsilon_{\rm UD}$ has signature $-1$ under $M_{n}$ and $R_{x}$.
C. $\epsilon_{\rm PC}$ corresponds to a direction of rotation on the surface,
and is invariant under $M_{n}$ but ahs signature $-1$ under $R_{x}$. D. The
pseudopolar order parameter $\mathbf{p}_{\rm C}$ has signature $-1$ under a
mirror symmetry, but signature $+1$ under $R_{x}$.

The introduction of signatures of order parameters under linear
transformations is helpful for the following reason. Scalar, tangent vectorial
and tangent tensorial order parameters can be introduced to characterise the
statistical distribution and symmetry properties of molecules within the
surface. Under 3D spatial linear transformations, molecules within a surface
element are modified and change their position and orientation. Order
parameters can then be recalculated to characterize this transformed state,
according to Eq. 17. The recalculated order parameters may not simply
correspond to a transformation of the original order parameters by the
tangential part of the linear transformation (Fig. 2); i.e. order parameters
do not necessarily behave as simple scalar, vectorial and tensorial quantities
on the surface. Instead, they may acquire an additional sign change under a
subset of transformations. The corresponding set of signatures allow to
classify order parameters, as further described below.

###  III.2 Isotropic surfaces

We first briefly discuss isotropic surfaces, which were considered in Ref.
salbreux2017activesurfaces . For such a surface broken symmetries are
described by pseudoscalar fields. To classify possible pseudoscalar fields, we
consider their signatures under the following set of transformations (Fig. 3):
$M_{n}$ (reflection operation with respect to a plane tangent to the surface),
$M_{t}$ (reflection operation with respect to a plane normal to the surface
and containing the tangent vector $\mathbf{t}$), $R_{t}$ (rotation by an angle
$\pi$ around a tangent vector $\mathbf{t}$), $R_{n}$ (rotation by an angle
$\pi$ around the normal $\mathbf{n}$), and $I$ (full inversion of space).
These transformations verify the composition relations $M_{n}R_{n}=I$,
$M_{t}R_{t}=M_{n}$ and $M_{n}R_{t}=M_{t}$. As a result, a pseudoscalar field
must break none, or at least two, of the symmetries within each set
$\\{M_{n},R_{n},I\\}$ and $\\{M_{t},R_{t},M_{n}\\}$. For instance, a field
that breaks the $M_{t}$ symmetry, but is invariant under $R_{t}$, also breaks
the mirror symmetry $M_{n}$. Furthermore, a surface scalar field must be
invariant under $R_{n}$. Taking these rules into account, one can define three
distinct types of scalar fields for a surface that breaks various symmetries
salbreux2017activesurfaces . The signatures of these three pseudoscalars,
denoted $\epsilon_{\rm UD}$ (for up-down broken symmetry), $\epsilon_{\rm C}$
(for chiral broken symmetry) and $\epsilon_{\rm PC}$ (for planar-chiral broken
symmetry) are given in Table 1.

| $M_{n}$ | $M_{t}$ | $R_{t}$ | $I$ | $R_{n}$
---|---|---|---|---|---
$\epsilon_{\rm UD}$ | -1 | 1 | -1 | -1 | 1
$\epsilon_{\rm C}$ | -1 | \- 1 | 1 | -1 | 1
$\epsilon_{\rm PC}$ | 1 | -1 | -1 | 1 | 1
Table 1:  Table of signatures of surface pseudoscalars.

These pseudoscalars are order parameters that distinguish different types, or
phases, of isotropic surface. We note that the 2D Levi-Civita tensor has the
same signature as $\epsilon_{\rm PC}$, corresponding to a planar-chiral
surface, while the 3D Levi-Civita tensor has the same signature as
$\epsilon_{\rm C}$, corresponding to a chiral surface.

By considering how pseudoscalar order parameters can combine, we find 5 phases
of isotropic surfaces which are listed in Table 2. There, we also give the
corresponding Schoenflies notation ashcroft1976solid  which describes the
symmetry properties of the corresponding phase, as well as the orientation of
the rotation axis relative to the tangent plane of the surface.

Schoenflies notation |  Rotation axis |  Order parameters | $M_{n}$ | $M_{t}$ | $R_{t}$ | $I$ | $R_{n}$
---|---|---|---|---|---|---|---
$D_{\infty\rm h}$ | $\perp$ | $\emptyset$ | 1 | 1 | 1 | 1 | 1
$C_{\infty\rm v}$ | $\perp$ | $\epsilon_{\rm UD}$ | X | 1 | X | X | 1
$D_{\infty}$ | $\perp$ | $\epsilon_{\rm C}$ | X | X | 1 | X | 1
$C_{\infty\rm h}$ | $\perp$ | $\epsilon_{\rm PC}$ | 1 | X | X | 1 | 1
$C_{\infty}$ | $\perp$ |  $\\{\epsilon_{\rm UD}$, $\epsilon_{\rm C}$, $\epsilon_{\rm PC}\\}$ | X | X | X | X | 1
Table 2: Schoenflies notation, order parameters, and conserved and broken
symmetries for the 5 types of isotropic surfaces. An “X” in the Table stands
for a broken symmetry and a “1” for a conserved symmetry. Here, a broken
symmetry is a transformation which changes the order parameter, which does not
imply a signature $-1$ of the order parameter for the transformation. The
second column refers to the orientation, with respect to the surface tangent
plane, of the main rotation axis (here leaving the system invariant by any
angle) associated to the symmetry group described by the Schoenflies notation.

More complex surface phases can be characterized by (pseudo-) vectorial and
(pseudo-) tensorial order parameters, as described further below.

Figure 3:  Classification of polar and nematic surfaces. A. We consider
changes in the surface state under up-down mirror symmetry $M_{n}$, mirror
symmetry with respect to a plane perpendicular to the surface $M_{t}$, up-down
rotation symmetry of angle $\pi$ $R_{t}$, rotation by $\pi$ around the normal
$R_{n}$, and rotation by $\pi/2$ around the normal $R_{n}^{\frac{1}{2}}$.
Combinations of $R_{n}^{\frac{1}{2}}$ with other symmetries are considered but
not represented. B. Order parameters. We consider combinations of 3
pseudoscalar order parameters with polar or nematic order parameters tangent
to the surface. C, D. Polar surfaces can be classified in 7 types according to
their symmetries : 5 polar surfaces and 2 pseudopolar surfaces. E, F. Nematic
surfaces can be categorised in 7 types according to their symmetries: 5
nematic surfaces and 2 pseudonematic surfaces. In panels C-F, schematics give
examples of actual surfaces belonging to each category. Red and green letters
indicate respectively broken and preserved symmetries. The Schoenflies
notation for the surface symmetry is given for each surface type. When
relevant, we add a $\parallel$ or $\perp$ superscript to the standard
Schoenflies notation to indicate if the rotation axis, (or, for $C_{\rm
s}=C_{\rm 1h}$, the normal to the mirror plane of symmetry, or for $D_{\rm
2d}$ and $S_{4}$, the axis of rotation-reflection), is tangent or
perpendicular to the surface.

###  III.3 Polar and pseudopolar surfaces

We now consider polar surfaces, that is, surfaces with a broken rotational
symmetry characterized by a tangent vector, $\mathbf{p}$. Such a vector
transforms under a change of coordinates $\mathbf{T}^{\rm 3D}=\mathbf{T}^{\rm
2D}\otimes\mathbf{T}^{n}$ as:

$\displaystyle p^{\prime}_{i}=\epsilon_{\mathbf{p}}^{T}{T^{\rm
2D}}_{i}{}^{j}p_{j}~{},$ (19)

where $\epsilon_{\mathbf{p}}^{T}$ is the vector signature. If $\mathbf{p}$ is
a true tangent vector, it has signature $\epsilon_{\mathbf{p}}^{T}=1$ for all
types of transformations. In addition, one can define three pseudovectors
$\mathbf{p}_{\rm UD}$, $\mathbf{p}_{\rm C}$, $\mathbf{p}_{\rm PC}$, which have
a $-1$ signature under at least one of the transformations $M_{n}$, $M_{t}$,
$R_{t}$, $R_{n}$, $I$. The corresponding signatures of vectors and pseudo
vectors are listed in Table 3. Subscripts of the pseudovectors
$\mathbf{p}_{\rm UD}$, $\mathbf{p}_{\rm C}$, $\mathbf{p}_{\rm PC}$ are given
in analogy between their signatures in Table 3 and signature of the
pseudoscalars $\epsilon_{\rm UD}$, $\epsilon_{\rm C}$ and $\epsilon_{\rm PC}$
in Table 1.

| $M_{n}$ | $M_{t}$ | $R_{t}$ | $I$ | $R_{n}$
---|---|---|---|---|---
$\mathbf{p}$ | 1 | 1 | 1 | 1 | 1
$\mathbf{p}_{\rm UD}$ | -1 | 1 | -1 | -1 | 1
$\mathbf{p}_{\rm C}$ | -1 | \- 1 | 1 | -1 | 1
$\mathbf{p}_{\rm PC}$ | 1 | -1 | -1 | 1 | 1
Table 3: Signatures of polar and pseudopolar order parameters.

We note that a rotation of $\mathbf{p}$ by $\pm\pi/2$ about the unit normal
vector to the surface $\mathbf{n}$, that is,
$\hat{\mathbf{p}}=\mathbf{n}\times\mathbf{p}$ or
$-\hat{\mathbf{p}}=-\mathbf{n}\times\mathbf{p}$, defines an alternative order
parameter. This corresponds to a different convention to describe the ordering
state of the surface, which does not change the physics. In component notation
this is equivalent to defining $\hat{p}^{i}=-\epsilon^{i}{}_{j}p^{j}$ or
$-\hat{p}^{i}=\epsilon^{i}{}_{j}p^{j}$. Here, $\epsilon_{ij}$ is the Levi-
Civita tensor (Eq. 173) and has the same signatures under the transformations
$M_{n}$, $M_{t}$, $R_{t}$, $R_{n}$ as $\epsilon_{\rm PC}$. For this reason
$\mathbf{p}_{\rm PC}$ does not describe a different phase than a vector
$\mathbf{p}$, as $\hat{\mathbf{p}}_{\rm PC}=\mathbf{n}\times\mathbf{p}_{\rm
PC}$ has the same signatures as a true vector $\mathbf{p}$. Similarly, the
pseudovectors $\mathbf{p}_{\rm UD}$ and $\mathbf{p}_{\rm C}$ describe the same
phase, as $\hat{\mathbf{p}}_{\rm UD}=\mathbf{n}\times\mathbf{p}_{\rm UD}$ has
the same signatures as the pseudovector $\mathbf{p}_{\rm C}$. Overall, there
are therefore only two distinct phases characterized only by a vectorial order
parameter: a simple polar surface (e.g. with polar order $\mathbf{p}$) or a
simple pseudopolar surface (e.g. with pseudopolar order $\mathbf{p}_{\rm C}$).

Schoenflies notation |  Rotation axis or mirror plane normal orientation |  Order parameters | $M_{n}$ | $M_{\mathbf{p}}$ | $R_{\mathbf{p}}$ | $M_{\hat{\mathbf{p}}}$ | $R_{\hat{\mathbf{p}}}$ | $I$ | $R_{n}$ |  number in Fig. 3C-D
---|---|---|---|---|---|---|---|---|---|---
$C_{2\rm v}$ | $\parallel$ | $\mathbf{p}$ | 1 | 1 | 1 | X | X | X | X | (i)
$C_{\rm s}=C_{\rm 1h}$ | $\parallel$ |  $\\{\epsilon_{\rm UD}$, $\mathbf{p}\\}$ | X | 1 | X | X | X | X | X | (ii)
$C_{2\rm}$ | $\parallel$ |  $\\{\epsilon_{\rm C}$, $\mathbf{p}\\}$ | X | X | 1 | X | X | X | X | (iii)
$C_{\rm s}=C_{\rm 1h}$ | $\perp$ |  $\\{\epsilon_{\rm PC}$, $\mathbf{p}\\}$ | 1 | X | X | X | X | X | X | (iv)
$C_{\rm 1}$ |  |  $\\{\epsilon_{\rm UD}$, $\epsilon_{\rm C}$, $\epsilon_{\rm PC}$, $\mathbf{p}\\}$ | X | X | X | X | X | X | X | (v)
$C_{2\rm h}$ | $\parallel$ | $\mathbf{p}_{\rm C}$ | X | X | 1 | 1 | X | 1 | X | (vi)
$C_{\rm i}=S_{2}$ |  |  $\\{\epsilon_{\rm PC}$, $\mathbf{p}_{\rm C}\\}$ | X | X | X | X | X | 1 | X | (vii)
Table 4: Schoenflies notation, order parameters, and conserved and broken
symmetries for the 7 types of polar surfaces. When relevant, the second column
indicates whether the axis of rotation in the Schoenflies notation (or, for
$C_{\rm s}=C_{\rm 1h}$, the normal to the mirror plane of symmetry) is
parallel ($\parallel$) or orthogonal ($\perp$) to the tangent plane of the
surface. The first 5 types are polar surfaces, while the 2 last types are
pseudopolar surfaces. In the table, a “X” stands for a broken symmetry and a
“1” for a conserved symmetry, and
$\hat{\mathbf{p}}=\mathbf{n}\times\mathbf{p}$ is a tangent vector orthogonal
to $\mathbf{p}$. See Fig. 3C-D for examples of surfaces of different types.

Scalar and vectorial order parameters can also exist simultaneously in a
surface, resulting in more symmetries being broken. To obtain a complete list
of surface polar phases, we consider any combination of the vectorial order
parameters $\mathbf{p}$, $\mathbf{p}_{\rm UD}$, $\mathbf{p}_{\rm C}$,
$\mathbf{p}_{\rm PC}$ and the scalar order parameters $\epsilon_{\rm C}$,
$\epsilon_{\rm PC}$, and $\epsilon_{\rm UD}$, but eliminate combinations which
are redundant as they describe the same phase. For instance, the combination
$\\{\epsilon_{\rm C},\mathbf{p}_{\rm C}\\}$ is redundant with the combination
$\\{\epsilon_{\rm C},\mathbf{p}\\}$, as $\epsilon_{\rm C}\mathbf{p}_{\rm C}$
has the same signature under symmetry operations as a true vector
$\mathbf{p}$. Furthermore, those vectorial order parameters, which when
rotated by $\pm\pi/2$ have the same symmetry properties as other vectorial
order parameters, are redundant. For instance the combination of order
parameters $\\{\epsilon_{\rm C},\mathbf{p}_{\rm UD}\\}$ can be redefined as
$\\{\epsilon_{\rm C},\hat{\mathbf{p}}_{\rm UD}\\}$ which has the same
signatures as $\\{\epsilon_{\rm C},\mathbf{p}_{\rm C}\\}$. The set
$\\{\epsilon_{\rm C},\epsilon_{\rm C}\mathbf{p}_{\rm C}\\}$ can be obtained
from the order parameters $\\{\epsilon_{\rm C},\mathbf{p}_{\rm C}\\}$, and has
the same symmetry properties as the polar chiral surface $\\{\epsilon_{\rm
C},\mathbf{p}\\}$. Therefore surfaces described by $\\{\epsilon_{\rm
C},\mathbf{p}_{\rm UD}\\}$ and $\\{\epsilon_{\rm C},\mathbf{p}\\}$ are
actually the same phase.

Listing all resultant independent combinations, we find 7 distinct surface
phases which are listed in Table 4, together with the Schoenflies notation
ashcroft1976solid  which characterize the symmetry properties of the
corresponding phases. There, we also indicate whether the order parameter is
invariant under a subset of orthogonal transformations. In addition to the
transformations $M_{n}$, $R_{n}$, $I$ introduced earlier, we consider the
transformations $M_{\mathbf{p}}$, $M_{\hat{\mathbf{p}}}$ (mirror operations in
a plane containing $\mathbf{n}$ and $\mathbf{p}$ and $\mathbf{n}$ and
$\hat{\mathbf{p}}$, respectively) and $R_{\mathbf{p}}$, $R_{\hat{\mathbf{p}}}$
(rotation of $\pi$ around the axis $\mathbf{p}$ or $\hat{\mathbf{p}}$). These
transformations satisfy the composition relations
$M_{\mathbf{p}}R_{\mathbf{p}}=M_{\hat{\mathbf{p}}}R_{\hat{\mathbf{p}}}=M_{n}$,
$M_{\mathbf{p}}M_{\hat{\mathbf{p}}}=R_{\mathbf{p}}R_{\hat{\mathbf{p}}}=R_{n}$,
$M_{\mathbf{p}}R_{\hat{\mathbf{p}}}=R_{\mathbf{p}}M_{\hat{\mathbf{p}}}=I$.
These relations constrain the set of possible conserved and broken symmetries
for each surface phase, since invariance with respect to two transformations
implies invariance with respect to their composition. For instance, all polar
phases break the symmetry $R_{n}$; therefore none of the phases can be
invariant under both $M_{\mathbf{p}}$ and $M_{\hat{\mathbf{p}}}$, or under
both $R_{\mathbf{p}}$ and $R_{\hat{\mathbf{p}}}$. It is possible to find
additional combinations of conserved and broken symmetries compatible with the
multiplicative table of transformations discussed above, which are not listed
in Table 4: indeed we find that for these cases, the redefinition of the polar
order parameter $\hat{\mathbf{p}}=\mathbf{n}\times\mathbf{p}$ allows to
recover one of the categories already listed in Table 4 (Appendix C).

Examples of the polar surface phases listed in Table 4 are given in Fig. 3C-D.
There, we denote a surface to be polar if its set of order parameters includes
a true vector $\mathbf{p}$, and pseudopolar otherwise.

###  III.4 Nematic and pseudonematic surfaces

We now discuss the symmetries of nematic surfaces, with tangent nematic order.
Following the same reasoning as for polar surfaces, we find that there are 5
nematic surfaces and 2 pseudonematic surfaces. Nematic order is characterized
by a traceless symmetric second-rank tangent tensor $\mathbf{Q}$
($Q_{k}{}^{k}=0$ and $Q_{ij}=Q_{ji}$). Such a tensor transforms under a change
of coordinates $\mathbf{T}^{\rm 3D}=\mathbf{T}^{\rm 2D}\otimes\mathbf{T}^{n}$
as:

$\displaystyle Q^{\prime}_{ij}=\epsilon_{\mathbf{Q}}^{T}{T^{\rm
2D}}_{i}{}^{k}{T^{\rm 2D}}_{j}{}^{l}Q_{kl}~{},$ (20)

where $\epsilon_{\mathbf{Q}}^{T}$ is the tensor signature. A true tangent
tensor has signature $\epsilon_{\mathbf{Q}}^{T}=1$ under all transformations
$\mathbf{T}^{\rm 3D}$. As for polar surfaces, three different types of
pseudonematic tensors $\mathbf{Q}_{\rm UD}$, $\mathbf{Q}_{\rm C}$,
$\mathbf{Q}_{\rm PC}$ can be defined, according to their signatures under
reflections and rotations. The signatures of nematic and pseudonematic order
parameters are given in Table 5.

| $M_{n}$ | $M_{t}$ | $R_{t}$ | $I$ | $R_{n}$
---|---|---|---|---|---
$\mathbf{Q}$ | 1 | 1 | 1 | 1 | 1
$\mathbf{Q}_{\rm UD}$ | -1 | 1 | -1 | -1 | 1
$\mathbf{Q}_{\rm C}$ | -1 | \- 1 | 1 | -1 | 1
$\mathbf{Q}_{\rm PC}$ | 1 | -1 | -1 | 1 | 1
Table 5: Signatures of nematic and pseudonematic order parameters.

The transformation of a nematic tensor
$\mathbf{Q}\rightarrow\hat{\vphantom{\rule{1.0pt}{5.71527pt}}\smash{\hat{\mathbf{Q}}}}$,
with
$\hat{\vphantom{\rule{1.0pt}{5.71527pt}}\smash{\hat{Q}}}_{ij}=-\epsilon_{i}{}^{k}Q_{kj}$
or
$-\hat{\vphantom{\rule{1.0pt}{5.71527pt}}\smash{\hat{Q}}}_{ij}=\epsilon_{i}{}^{k}Q_{kj}$,
corresponds to a rotation by $\pm\pi/4$ around the normal to the surface of
the eigenvectors of the order parameter. The tensors $\mathbf{Q}$ and
$\hat{\vphantom{\rule{1.0pt}{5.71527pt}}\smash{\hat{\mathbf{Q}}}}$ correspond
to two different conventions to quantify nematic order, but describe the same
nematic phase. As the 2D antisymmetric tensor $\epsilon_{ij}$ has the same
signatures under transformations as $\epsilon_{\rm PC}$, not all pseudonematic
tensors describe physically distinct phases. Specifically, $\mathbf{Q}_{\rm
C}$ and $\mathbf{Q}_{\rm UD}$ describe surfaces with the same symmetry.
Similarly, $\mathbf{Q}_{\rm PC}$ and $\mathbf{Q}$ also describe surfaces with
the same symmetry. This can be seen from the fact, for instance, that the
order parameter
$\hat{\vphantom{\rule{1.0pt}{5.71527pt}}\smash{\hat{\mathbf{Q}}}}_{\rm PC}$ is
a true tensor with the same signatures under transformations as $\mathbf{Q}$
in Table 5. Overall, there are only two distinct phases characterized only by
a nematic order parameter: a simple nematic surface (e.g. with nematic order
$\mathbf{Q}$) or a simple pseudonematic surface (e.g. with nematic order
$\mathbf{Q}_{\rm UD}$).

As for polar surfaces, to obtain the full set of nematic surfaces, we consider
the combination of a nematic or pseudonematic order parameter $\mathbf{Q}$,
$\mathbf{Q}_{\rm UD}$, $\mathbf{Q}_{\rm C}$, $\mathbf{Q}_{\rm PC}$ with the
pseudoscalar order parameters $\epsilon_{\rm UD}$, $\epsilon_{\rm PC}$,
$\epsilon_{\rm C}$, and eliminate redundant combinations. The combination of
the nematic order parameter $\mathbf{Q}$ with the pseudoscalar order
parameters $\epsilon_{\rm UD}$, $\epsilon_{\rm PC}$, $\epsilon_{\rm C}$, gives
5 different types of nematic surfaces. Further combinations of pseudoscalar
and pseudonematic order parameters can also correspond to surfaces with the
same symmetry using the transformation
$\mathbf{Q}\rightarrow\hat{\vphantom{\rule{1.0pt}{5.71527pt}}\smash{\hat{\mathbf{Q}}}}$:
for instance the set $\\{\epsilon_{\rm PC}$, $\mathbf{Q}_{\rm UD}\\}$
describes a surface with the same symmetries as $\\{\epsilon_{\rm PC}$,
$\mathbf{Q}_{\rm C}\\}$.

Listing all combinations which are not redundant, we obtain a list given in
Table 6. For each order parameter, we indicate whether the order parameter is
conserved under a subset of orthogonal transformations. Nematic order
parameters are all invariant under the rotation by $\pi$ around the normal,
$R_{n}$, and are all modified by the rotation of $\pm\pi/2$ around the normal,
$R_{n}^{\pm\frac{1}{2}}$. In Table 6, pseudonematic surfaces are distinguished
from nematic surfaces by their behaviour under the transformations
$M_{\hat{\vphantom{\rule{1.0pt}{6.16736pt}}\smash{\hat{\mathbf{q}}}}}$ (a
mirror operation with respect to a plane perpendicular to the surface and
containing
$\hat{\vphantom{\rule{1.0pt}{5.71527pt}}\smash{\hat{\mathbf{q}}}}$),
$R_{\hat{\vphantom{\rule{1.0pt}{6.16736pt}}\smash{\hat{\mathbf{q}}}}}$ (a
rotation of $\pi$ around the axis
$\hat{\vphantom{\rule{1.0pt}{5.71527pt}}\smash{\hat{\mathbf{q}}}}$) and the
composition $R_{n}^{\pm\frac{1}{2}}M_{n}$. Here,
$\hat{\vphantom{\rule{1.0pt}{5.71527pt}}\smash{\hat{\mathbf{q}}}}=\mathbf{R}_{n}^{\frac{1}{4}}\mathbf{q}$
is the vector obtained by rotation of $\pi/4$ around the normal to the surface
$\mathbf{n}$, of the eigenvector of $\mathbf{Q}$ with positive eigenvalue,
$\mathbf{q}$. These transformations satisfy the composition relations
$M_{\hat{\vphantom{\rule{1.0pt}{6.16736pt}}\smash{\hat{\mathbf{q}}}}}R_{\hat{\vphantom{\rule{1.0pt}{6.16736pt}}\smash{\hat{\mathbf{q}}}}}=R_{\hat{\vphantom{\rule{1.0pt}{6.16736pt}}\smash{\hat{\mathbf{q}}}}}M_{\hat{\vphantom{\rule{1.0pt}{6.16736pt}}\smash{\hat{\mathbf{q}}}}}=M_{n}$,
$M_{\hat{\vphantom{\rule{1.0pt}{6.16736pt}}\smash{\hat{\mathbf{q}}}}}M_{\mathbf{q}}=R_{\hat{\vphantom{\rule{1.0pt}{6.16736pt}}\smash{\hat{\mathbf{q}}}}}R_{\mathbf{q}}=R_{n}^{\frac{1}{2}}$,
$M_{\mathbf{q}}M_{\hat{\vphantom{\rule{1.0pt}{6.16736pt}}\smash{\hat{\mathbf{q}}}}}=R_{\mathbf{q}}R_{\hat{\vphantom{\rule{1.0pt}{6.16736pt}}\smash{\hat{\mathbf{q}}}}}=R_{n}^{-\frac{1}{2}}$,
$M_{\hat{\vphantom{\rule{1.0pt}{6.16736pt}}\smash{\hat{\mathbf{q}}}}}R_{\mathbf{q}}=R_{\hat{\vphantom{\rule{1.0pt}{6.16736pt}}\smash{\hat{\mathbf{q}}}}}M_{\mathbf{q}}=R_{n}^{\frac{1}{2}}M_{n}$
and
$M_{\mathbf{q}}R_{\hat{\vphantom{\rule{1.0pt}{6.16736pt}}\smash{\hat{\mathbf{q}}}}}=R_{\mathbf{q}}M_{\hat{\vphantom{\rule{1.0pt}{6.16736pt}}\smash{\hat{\mathbf{q}}}}}=R_{n}^{-\frac{1}{2}}M_{n}$.
As for polar surfaces, these multiplication relations restrict the set of
possible conserved and broken symmetries for each surface phase, as invariance
with respect to two transformations implies invariance with respect to their
composition. As for polar surfaces, additional combinations of broken and
conserved symmetries not listed in Table 6 are possible; we find that these
combinations arise from the redefinition
$\hat{\vphantom{\rule{1.0pt}{5.71527pt}}\smash{\hat{Q}}}_{ij}=-\epsilon_{i}{}^{k}Q_{kj}$
of one of the already listed surface phases (Appendix C).

Examples of the nematic surface phases listed in Table 6 are given in Fig.
3E-F. There, we denote a surface to be nematic if its set of order parameter
includes a true tensor $\mathbf{Q}$, and pseudonematic otherwise.

Schoenflies notation |  Rotation or rotation-reflection axis |  Order parameters | $M_{n}$ |  $M_{\mathbf{q}}$ or $M_{\hat{\mathbf{q}}}$ |  $R_{\mathbf{q}}$ or$R_{\hat{\mathbf{q}}}$ | $I$ | $R_{n}$ | $R_{n}^{\pm\frac{1}{2}}$ | $R_{n}^{\pm\frac{1}{2}}M_{n}$ | $M_{\hat{\vphantom{\rule{1.0pt}{6.16736pt}}\smash{\hat{\mathbf{q}}}}}$ | $R_{\hat{\vphantom{\rule{1.0pt}{6.16736pt}}\smash{\hat{\mathbf{q}}}}}$ |  number in Fig. 3E-F
---|---|---|---|---|---|---|---|---|---|---|---|---
$D_{\rm 2h}$ $D_{\rm 2h}$ |  $\perp$ $\parallel$ | $\mathbf{Q}$ | 1 | 1 | 1 | 1 | 1 | X | X | X | X | (i)
$C_{\rm 2v}$ | $\perp$ |  $\\{\epsilon_{\rm UD}$, $\mathbf{Q}\\}$ | X | 1 | X | X | 1 | X | X | X | X | (ii)
$D_{\rm 2}$ $D_{2}$ |  $\perp$ $\parallel$ |  $\\{\epsilon_{\rm C}$, $\mathbf{Q}\\}$ | X | X | 1 | X | 1 | X | X | X | X | (iii)
$C_{\rm 2h}$ | $\perp$ |  $\\{\epsilon_{\rm PC}$, $\mathbf{Q}\\}$ | 1 | X | X | 1 | 1 | X | X | X | X | (iv)
$C_{\rm 2}$ | $\perp$ |  $\\{\epsilon_{\rm UD}$, $\epsilon_{\rm C}$, $\epsilon_{\rm PC}$, $\mathbf{Q}\\}$ | X | X | X | X | 1 | X | X | X | X | (v)
$D_{\rm 2d}$ | $\perp$ | $\mathbf{Q}_{\rm UD}$ | X | 1 | X | X | 1 | X | 1 | X | 1 | (vi)
$S_{4}$ | $\perp$ |  $\\{\epsilon_{\rm PC}$, $\mathbf{Q}_{\rm UD}\\}$ | X | X | X | X | 1 | X | 1 | X | X | (vii)
Table 6:  Schoenflies notation, order parameters, and conserved and broken
symmetries for the 7 types of nematic surfaces. An “X” stands for a broken
symmetry and a 1 for a conserved symmetry. $\mathbf{q}$ denotes a vector going
along the eigenvector of $\mathbf{Q}$ with the largest eigenvalue,
$\hat{\mathbf{q}}$ its rotation by $\pi/2$ around the normal,
$\hat{\hat{\mathbf{q}}}$ its rotation by $\pi/4$ around the normal. When
relevant, the second column indicates whether the axis of rotation in the
Schoenflies notation (or, for $D_{\rm 2d}$ and $S_{4}$, the rotation-
reflection axis) is parallel ($\parallel$) or orthogonal ($\perp$) to the
tangent plane of the surface. See Fig. 3E-F for examples of surfaces of
different types.

We note that the Schoenflies notation we use here refers to the symmetry
properties of the phase, rather than of individual molecules. Due to the
statistical arrangements of molecules within a surface element, these symmetry
properties differ in general. For instance, molecules belonging to the
symmetry group $C_{2\rm v}$ in the Schoenflies notation, with a uniform
orientation and a mirror plane of symmetry constrained to be tangent to the
surface, give rise to a $C_{\rm 2v}^{\parallel}$ polar surface. Letting the
orientation axis of the molecules tilt towards one side of the surface gives
rise instead to a polar up-down surface $C_{\rm s}^{\|}$.

##  IV Force and torque balance and virtual work

The surface mechanics is described by a tension tensor $\mathbf{t}^{i}$ and a
moment tensor $\mathbf{m}^{i}$, which, respectively, characterize momentum
flux and angular momentum flux in the surface. The force $\mathbf{f}$ and
torque $\boldsymbol{\Gamma}$ acting on an infinitesimal surface line element
with length $dl$ and unit vector $\boldsymbol{\nu}$, oriented perpendicular to
the line element and tangent to the surface, are then given by
$\mathbf{f}=dl\nu_{i}\mathbf{t}^{i}$ and
$\boldsymbol{\Gamma}=dl\nu_{i}\mathbf{m}^{i}$. Ignoring the moment of inertia
tensor for simplicity, conservation of momentum and of angular momentum result
in the force and torque balance expressions (Appendix D):

$\displaystyle\nabla_{i}\mathbf{t}^{i}$ $\displaystyle=$
$\displaystyle-\mathbf{f}^{\rm{ext}}+\rho\mathbf{a},$ (21)
$\displaystyle\nabla_{i}\mathbf{m}^{i}$ $\displaystyle=$
$\displaystyle\mathbf{t}^{i}\times\mathbf{e}_{i}-\boldsymbol{\Gamma}^{\rm{\rm{ext}}},$
(22)

where $\mathbf{a}={\rm d}\mathbf{v}/{{\rm d}t}$ is the local center-of-mass
acceleration, $\mathbf{f}^{\rm ext}$ is the external force density and
$\boldsymbol{\Gamma}^{\rm ext}$ is the external torque density acting on the
surface (Fig. 1C).

The force and torque balance equations can be expressed in terms of the
components of the tension and moment tensors (Appendix D):

$\displaystyle\nabla_{i}t^{ij}+C_{i}{}^{j}t_{n}^{i}$ $\displaystyle=$
$\displaystyle-f^{{\rm ext},j}+\rho a^{j},$ (23)
$\displaystyle\nabla_{i}t_{n}^{i}-C_{ij}t^{ij}$ $\displaystyle=$
$\displaystyle-f^{\rm{ext}}_{n}+\rho a_{n},$ (24)
$\displaystyle\nabla_{i}m^{ij}+C_{i}{}^{j}m_{n}^{i}$ $\displaystyle=$
$\displaystyle\epsilon_{i}{}^{j}t_{n}^{i}-\Gamma^{{\rm ext},j},$ (25)
$\displaystyle\nabla_{i}m_{n}^{i}-C_{ij}m^{ij}$ $\displaystyle=$
$\displaystyle-\epsilon_{ij}t^{ij}-\Gamma^{\rm{ext}}_{n}~{},$ (26)

where $C_{ij}$ is the surface curvature tensor.

We now write the infinitesimal virtual work for an element of surface
$\mathcal{S}$ with closed contour $\mathcal{C}$, associated to an
infinitesimal virtual surface displacement $d\mathbf{X}$, as follows:

$\displaystyle dW$ $\displaystyle=dW_{\rm ext}+dW_{\rm contour}$ (27)
$\displaystyle dW_{\rm ext}$
$\displaystyle=\int_{\mathcal{S}}dS\,\left[\left(\mathbf{f}^{\rm
ext}-\rho\mathbf{a}\right)\cdot d\mathbf{X}+\boldsymbol{\Gamma}^{\rm ext}\cdot
d\boldsymbol{\vartheta}\right]$ (28) $\displaystyle dW_{\rm contour}$
$\displaystyle=\oint_{\mathcal{C}}dl\nu_{i}\left(\mathbf{t}^{i}\cdot
d\mathbf{X}+\mathbf{m}^{i}\cdot d\boldsymbol{\vartheta}\right)\,.$ (29)

where we follow the form given in Ref. salbreux2017activesurfaces . The
infinitesimal rotation associated with the infinitesimal shape change
$d\mathbf{X}$ is $d\boldsymbol{\vartheta}=\frac{1}{2}\nabla\times d\mathbf{X}$
(Eq. 211). The first contribution $dW_{\rm ext}$ corresponds to the effect of
external forces and torques acting on the surface, while the second
contribution $dW_{\rm contour}$ corresponds to the work associated to forces
and torques internal to the surface, acting on the contour of the element of
surface considered here. Here the vector $\boldsymbol{\nu}$ is a unit vector
tangent to the surface and normal to the contour $\mathcal{C}$.

Using the force and torque balance equations 21 and 22, the infinitesimal
virtual work can be re-written (see Appendix E):

$\displaystyle dW$
$\displaystyle=\int_{\mathcal{S}}dS\left[\overline{t}^{ij}\frac{dg_{ij}}{2}+\overline{m}^{ij}DC_{ij}+m^{i}_{n}(\partial_{i}d\boldsymbol{\vartheta})\cdot\mathbf{n}\right]~{},$
(30)

where expressions for the infinitesimal metric variation $dg_{ij}$ and the
corotational curvature infinitesimal variation $DC_{ij}$ are given in Eqs. 207
and 222. In Eq. 30, we have introduced the modified tension and moment
tensors, which are coupled in the virtual work respectively to the change of
metric and the corotational curvature change salbreux2017activesurfaces :

$\displaystyle\overline{t}^{ij}$
$\displaystyle=t^{ij}_{s}+\frac{1}{2}\left(\overline{m}^{ik}C_{k}{}^{j}+\overline{m}^{jk}C_{k}{}^{i}\right)$
(31) $\displaystyle\overline{m}^{ij}$
$\displaystyle=-m^{ik}\epsilon_{k}{}^{j}~{},$ (32)

where the $s$ subscript denotes the symmetric part of the tensor (Eq. 177). We
note that with the definitions introduced above for the tension and moment
tensors, the modified moment tensor $\bar{m}^{ij}$ can always be symmetrized
at the cost of a redefinition of the tension tensor $\mathbf{t}^{i}$ (Appendix
E).

##  V Polar active fluid surface

We now turn to a description of equilibrium and non-equilibrium thermodynamics
of polar active fluid surfaces. Here we use concepts from irreversible
thermodynamics to derive constitutive equations for a curved polar fluid. We
restrict ourselves to the case of a polar vector tangent to the surface,
$\mathbf{p}$. We consider a fluid surface, consisting of several species
$\alpha=1...N$ with concentrations $c^{\alpha}$. The local mass density is
given by $\rho=\sum_{\alpha}m^{\alpha}c^{\alpha}$ with $m^{\alpha}$ the
molecular mass of species $\alpha$. The corresponding mass conservation
equations are given in Appendix D.2.

In the following, we derive equilibrium tensions and torques from a generic
expression for a polar surface virtual work. We then obtain the entropy
production rate of the surface close to equilibrium, and obtain linear
constitutive equations for the non-equilibrium generalized fluxes.

###  V.1 Free energy and external potential

We consider here a region of surface $\mathcal{S}$ of a polar fluid membrane,
with free energy given by

$\displaystyle
F=\int_{\mathcal{S}}dSf(\mathbf{v},c^{\alpha},\mathbf{C},\mathbf{p},\boldsymbol{\nabla}\otimes\mathbf{p})~{},$
(33)

with $f$ the total free energy density, which has two contributions:

$f(\mathbf{v},c^{\alpha},\mathbf{C},\mathbf{p},\boldsymbol{\nabla}\otimes\mathbf{p})=\frac{1}{2}\rho
v^{2}+f_{0}(c^{\alpha},\mathbf{C},\mathbf{p},\boldsymbol{\nabla}\otimes\mathbf{p})~{},$
(34)

where the kinetic energy is given by $\frac{1}{2}\rho
v^{2}=\frac{1}{2}\rho\left[v_{i}v^{i}+(v_{n})^{2}\right]$ and $f_{0}$ is the
free energy density in the rest frame. The tensor
$\boldsymbol{\nabla}\otimes\mathbf{p}=\mathbf{e}^{k}\otimes\partial_{k}\mathbf{p}$
is the gradient of polarity (Eq. 186). Here we consider for simplicity a
situation at fixed temperature and where the polar vector $\mathbf{p}$ is
tangent to the surface. The differential of $f_{0}$ can be written:

$df_{0}=\mu^{\alpha}dc^{\alpha}+\mathbf{K}:d\mathbf{C}-\mathbf{h}_{0}\cdot
d\mathbf{p}+\boldsymbol{\pi}:d(\boldsymbol{\nabla}\otimes\mathbf{p}),\\\ $
(35)

where $\mu^{\alpha}$ is the chemical potential of component $\alpha$,
$\mathbf{K}$ is the passive bending moment, $\mathbf{h}_{0}$ and
$\boldsymbol{\pi}$ are the forces conjugate to the polarity field and to its
gradient on the surface. These conjugate forces are obtained by taking partial
derivatives of the free energy density $f_{0}$:

$\displaystyle\mu^{\alpha}=$ $\displaystyle\frac{\partial f_{0}}{\partial
c^{\alpha}},~{}\boldsymbol{K}=\frac{\partial f_{0}}{\partial\boldsymbol{C}},$
$\displaystyle\mathbf{h}_{0}=$ $\displaystyle-\frac{\partial
f_{0}}{\partial\mathbf{p}},~{}\boldsymbol{\pi}=\frac{\partial
f_{0}}{\partial(\boldsymbol{\nabla}\otimes\mathbf{p})}~{}.$ (36)

We assume that $f_{0}$ is such that $\mathbf{K}$ and $\mathbf{h}_{0}$ are
tangent to the surface, that $\mathbf{K}$ is symmetric, and that
$\boldsymbol{\pi}$ can be written
$\boldsymbol{\pi}=\mathbf{e}_{i}\otimes\boldsymbol{\pi}^{i}$. Because of
invariance by rotation of the free energy (Appendix F and Eq. 41), the
differential of the free energy density can also be written:

$df_{0}=\mu^{\alpha}dc^{\alpha}+\mathbf{K}:D\mathbf{C}-\mathbf{h}_{0}\cdot
D\mathbf{p}+\boldsymbol{\pi}:D(\boldsymbol{\nabla}\otimes\mathbf{p}).\\\ $
(37)

As $\mathbf{C}$ and $\mathbf{p}$ are tangent to the surface, $D\mathbf{C}$ and
$D\mathbf{p}$ are also tangent to the surface (Eq. B.3). We also introduce the
total molecular field $\mathbf{h}$, which is decomposed in two contributions:

$\displaystyle\mathbf{h}$ $\displaystyle=-\frac{\delta
F}{\delta\mathbf{p}}=\mathbf{h}_{0}+\nabla_{j}\boldsymbol{\pi}^{j}~{},$ (38)

and is not necessarily tangent to the surface. Note that we define here the
functional derivative of the functional $F[\phi]$ through
$F[\phi+d\phi]-F[\phi]=\int_{\mathcal{S}}dS\frac{\delta F}{\delta\phi}d\phi$
to first order in $d\phi$.

In the following we also assume that the surface is subjected to an external
potential:

$\displaystyle
U=\int_{\mathcal{S}}dSc^{\alpha}u^{\alpha}(\mathbf{X},\mathbf{n},\mathbf{p}),$
(39)

where $u^{\alpha}$ is an external potential density acting on species
$\alpha$. Such a potential could be for instance generated by an external
magnetic field.

###  V.2 Invariance of free energy by translation and rotation

Using the fact that the free energy density
$f_{0}(c^{\alpha},\mathbf{C},\mathbf{p},\boldsymbol{\nabla}\otimes\mathbf{p})$
is only a function of the surface variables and does not explicitly depend on
space, one obtains the Gibbs-Duhem relation (Appendix F)

$\displaystyle\nabla_{j}\left[(f_{0}-\mu^{\alpha}c^{\alpha})g_{i}{}^{j}-K^{jk}C_{ik}-\boldsymbol{\pi}^{j}\cdot\partial_{i}\mathbf{p}\right]$
$\displaystyle+C_{ik}\nabla_{j}K^{jk}=-(\partial_{i}\mu^{\alpha})c^{\alpha}-\mathbf{h}\cdot\partial_{i}\mathbf{p}~{}.$
(40)

A similar calculation using that the free energy density is invariant by local
solid rotations, leads to the generalized Gibbs-Duhem relation (see Appendix
F):

$\displaystyle\boldsymbol{\pi}^{i}\times\partial_{i}\mathbf{p}+(\boldsymbol{\pi}^{i}\cdot\partial_{j}\mathbf{p})\epsilon_{i}{}^{j}\mathbf{n}=\mathbf{h}_{0}\times\mathbf{p}+\mathbf{C}\times_{1}\mathbf{K}+\mathbf{C}\times_{2}\mathbf{K}~{}.$
(41)

where the notations for cross-product of tensors is introduced in Appendix A.
These relations are valid at and out of equilibrium, and reflect a
mathematical relation between the surface variables and their conjugated
forces in the free energy. These arise from the form of the free energy, which
does not depend on the position or orientation of the surface in space.

###  V.3 Equilibrium relations

To obtain equilibrium relations and expressions for the equilibrium tensions
and torques, we consider changes of free energy associated with an
infinitesimal virtual surface displacement $d\mathbf{X}$, virtual change of
polarity $d\mathbf{p}$ and change of concentration $dc^{\alpha}$ of the
component $\alpha$. Therefore, we write that for a surface patch $\mathcal{S}$
enclosed by the contour $\mathcal{C}$, the variation of
$F_{0}=\int_{\mathcal{S}}dSf_{0}$ is:

$\displaystyle dF_{0}=dW+\int_{\mathcal{S}}dS[-\mathbf{h}\cdot
D\mathbf{p}+\mu^{\alpha}{\rm d}\bar{}\hskip
1.00006ptc^{\alpha}]+\oint_{\mathcal{C}}dl\nu_{i}\boldsymbol{\pi}^{i}\cdot
D\mathbf{p}~{},$ (42)

where $dW$ is the mechanical differential work associated with the surface
deformation (Eqs. 27 and 30). The corotational differential $D\mathbf{p}$ has
been defined in Eq. 1. In this equation, the molecular field $\mathbf{h}$
couples to the corotational variation of the tangent polarity vector
$\mathbf{p}$, while $\boldsymbol{\pi}^{i}$ is the internal molecular field
coupled to corotational variation of $\mathbf{p}$ at the boundary of the patch
$\mathcal{S}$. The chemical potential $\mu^{\alpha}$ couples to the variation
of concentration of molecular species $\alpha$ which does not arise from
dilution, ${\rm d}\bar{}\hskip 1.00006ptc^{\alpha}$ defined in Eq. 210. For
surfaces perturbations where the polarity rotates with the surface,
$D\mathbf{p}=0$, and the concentration fields change only by geometric
dilution, ${\rm d}\bar{}\hskip 1.00006ptc^{\alpha}=0$, the variation of free
energy is equal to the work $dW$ associated with the surface deformation.

Similarly, the variation of the external potential $U$ has a mechanical
contribution, corresponding to the work $dW_{\rm ext}$ (Eq. 27), and
contributions from the variations in the polar field $D\mathbf{p}$ and the
concentration fields ${\rm d}\bar{}\hskip 1.00006ptc^{\alpha}$:

$\displaystyle dU=-dW_{\rm ext}+\int_{\mathcal{S}}dS[\mathbf{h}^{\rm ext}\cdot
D\mathbf{p}-\mu^{{\rm ext},\alpha}{\rm d}\bar{}\hskip
1.00006ptc^{\alpha}]~{},$ (43)

where external molecular field $\mathbf{h}^{\rm ext}$ and external chemical
potential $\mu^{{\rm ext},\alpha}$ are defined by:

$\displaystyle\mathbf{h}^{\rm ext}=$ $\displaystyle\frac{\delta
U}{\delta\mathbf{p}}~{},$ (44) $\displaystyle\mu^{{\rm ext},\alpha}=$
$\displaystyle-\frac{\delta U}{\delta c^{\alpha}}~{}.$ (45)

We assume that $U$ is such that $\mathbf{h}^{\rm ext}$ is tangent to the
surface. Eqs 42 and 43 can be combined such that:

$\displaystyle dF_{0}+dU=$
$\displaystyle\int_{\mathcal{S}}dS\left[(\mathbf{h}^{\rm ext}-\mathbf{h})\cdot
D\mathbf{p}+(\mu^{\alpha}-\mu^{{\rm ext},\alpha}){\rm d}\bar{}\hskip
1.00006ptc^{\alpha}\right]$ $\displaystyle+dW_{\rm
contour}+\oint_{\mathcal{C}}dl\nu_{i}\boldsymbol{\pi}^{i}\cdot
D\mathbf{p}~{}.$ (46)

The equilibrium tension and moment tensors can be obtained by calculating the
infinitesimal change of surface free energy 35 under a surface deformation,
and using the expression of the virtual work, Eq. 30 together with the
equilibrium condition 42. This procedure then gives (Appendix G.1.1):

$\displaystyle t_{e}^{ij}=$
$\displaystyle(f_{0}-\mu^{\alpha}c^{\alpha})g^{ij}-K^{ik}C_{k}{}^{j}-\boldsymbol{\pi}^{i}\cdot\partial^{j}\mathbf{p}~{},$
(47) $\displaystyle t_{e,n}^{i}=$
$\displaystyle\nabla_{j}K^{ji}+\epsilon^{i}{}_{j}\left[\Gamma_{e}^{{\rm
ext},j}-(\mathbf{h}\times\mathbf{p})\cdot\mathbf{e}^{j}\right]~{},$ (48)
$\displaystyle\mathbf{m}_{e}^{i}=$ $\displaystyle
K^{ik}\epsilon_{k}{}^{j}\mathbf{e}_{j}+\mathbf{p}\times\boldsymbol{\pi}^{i}~{}.$
(49)

The last term in Eq. 47 is the surface tension tensor arising from distortion
of the polar field; $\gamma
g^{ij}-\boldsymbol{\pi}^{i}\cdot\partial^{j}\mathbf{p}$ with
$\gamma=f_{0}-\mu^{\alpha}c^{\alpha}$ is the equivalent, for a surface, of the
Ericksen stress tensor in a liquid crystal degennesprost ;
napoli2010equilibrium . The expression for the normal component of the
equilibrium tension tensor $t^{i}_{e,n}$ arises from the tangential torque
balance Eq. 25, and involves the external tangential torque density acting on
the surface.

To obtain external forces, external torques and molecular field deriving from
an external potential $U$, we obtain the differential $dU$ from the potential
$U$ defined in Eq. 39, and use the equilibrium condition 43 and external
virtual work 28 to identify $\mathbf{f}^{\rm ext}_{e}$,
$\boldsymbol{\Gamma}^{\rm ext}_{e}$, $\mathbf{h}^{\rm ext}$, and $\mu^{{\rm
ext},\alpha}$. One then obtains (Appendix G.1.2):

$\displaystyle\mathbf{f}^{\rm ext}_{e}$
$\displaystyle=-c^{\alpha}\frac{\partial u^{\alpha}}{\partial\mathbf{X}}~{},$
(50) $\displaystyle\boldsymbol{\Gamma}^{\rm ext}_{e}$
$\displaystyle=c^{\alpha}\left(\frac{\partial
u^{\alpha}}{\partial\mathbf{n}}\times\mathbf{n}+\frac{\partial
u^{\alpha}}{\partial\mathbf{p}}\times\mathbf{p}\right)~{},$ (51)
$\displaystyle\mathbf{h}^{\rm ext}$ $\displaystyle=c^{\alpha}\frac{\partial
u^{\alpha}}{\partial\mathbf{p}}~{},$ (52) $\displaystyle\mu^{{\rm
ext},\alpha}$ $\displaystyle=-u^{\alpha}~{}.$ (53)

Next, minimising the total free energy $F_{\rm tot}=F_{0}+U$ with respect to
the polarity field $\mathbf{p}$ and the concentration fields $c^{\alpha}$ at
fixed surface shape, results in the equilibrium relations for the tangential
part of the molecular field and for the chemical potential, using Eq. 46:

$\displaystyle h^{i}=$ $\displaystyle h^{{\rm ext},i}~{},$ (54)
$\displaystyle\mu^{\alpha}=$ $\displaystyle\mu^{{\rm ext},\alpha}~{}.$ (55)

We now discuss relations between equilibrium tensions and torques and
invariance of the surface properties under a rigid translation or rotation,
Eqs. V.2 and 41. Using the expressions for the equilibrium tensors 47-49 and
the relations 298 and G.1.2, the Gibbs-Duhem relations V.2 and 41, obtained by
invariance of the free energy under solid translations and rotations, can be
re-written in the simpler form

$\displaystyle\nabla_{i}t_{e}^{ij}+C_{i}{}^{j}t_{e,n}^{i}+f_{e}^{{\rm
ext},j}=$ $\displaystyle(h_{i}^{{\rm ext}}-h_{i})\nabla^{j}p^{i}$
$\displaystyle+c^{\alpha}\partial^{j}(\mu^{{\rm
ext},\alpha}-\mu^{\alpha})~{},$ (56)
$\displaystyle\nabla_{i}m_{n,e}^{i}-C_{ij}m_{e}^{ij}+\epsilon_{ij}t_{e}^{ij}+\Gamma_{e,n}^{\rm
ext}=$ $\displaystyle\epsilon^{ij}(h_{i}^{{\rm ext}}-h_{i})p_{j}~{}.$ (57)

The right-hand sides of these two equations vanish when the equilibrium
equations for the concentration field (Eq. 55) and for the polar field (Eq.
54) are satisfied. In that case the left-hand sides of these two equations
also vanish, corresponding to the tangential force balance Eq. 23 and normal
torque balance Eq. 26, applied to equilibrium tension, moments, external force
and torque densities.

Overall, at equilibrium the tangential force balance equation and normal
torque balance equations are satisfied by expressing the chemical potential
balance $\mu^{\alpha}=\mu^{\rm ext,\alpha}$ and that the polar field is at
equilibrium, Eq. 54. The tangential torque balance expression gives the
expression of the normal tension $t_{n}^{i}$, Eq. 48. The remaining normal
force balance equation following from Eq. 24:

$\displaystyle\nabla_{i}t_{e,n}^{i}-C_{ij}t_{e}^{ij}$
$\displaystyle=-f^{\rm{ext}}_{e,n}~{},$ (58)

together with the equilibrium relation for the tension tensor
$\mathbf{t}_{e}^{i}$, Eqs. 47 and 48, and for the external force density Eq.
50, is an equation for the equilibrium shape of the surface.

###  V.4 Entropy production rate

We can now calculate the entropy production, by calculating the time
derivative of the surface free energy. We consider a region of surface
$\mathcal{S}$ enclosed by a contour $\mathcal{C}$, which can deform in three
dimensions. We consider a situation where the contour $\mathcal{C}$ is
deforming with the surface. We also consider that the total external force
density $\mathbf{f}^{\rm ext}$ and external torque density
$\boldsymbol{\Gamma}^{\rm ext}$ have a conservative part arising from an
external potential $U$, which are written $\mathbf{f}^{\rm ext}_{e}$,
$\boldsymbol{\Gamma}^{\rm ext}_{e}$, with corresponding definitions given in
Eqs. 50 and 51. We consider that species $\alpha$ can change due to the
surface tangent flux $\mathbf{J}^{\alpha}$ and due to a source term
$r^{\alpha}$ arising from chemical reactions. Corresponding balance equations
for the concentration of species $\alpha$ in the surface are discussed in
Appendix D.2. The total flux $\mathbf{J}^{\alpha}$ can be separated into a
part that depends on the center-of-mass velocity,
$c^{\alpha}\mathbf{v}^{\alpha}$, and a relative flux $\mathbf{j}^{\alpha}$.
The rate of the total free energy change $F_{\rm tot}=F+U$ is calculated in
Appendix H.1, and we obtain:

$\displaystyle\dot{F}_{\rm tot}=$
$\displaystyle\int_{\mathcal{S}}dS\,\Big{\\{}-\left(\overline{t}^{ij}-\overline{t}_{e}^{ij}\right)v_{ij}-(\overline{m}^{ij}-\overline{m}_{e}^{ij})D_{t}C_{ij}$
$\displaystyle-(m_{n}^{i}-m_{n,e}^{i})\omega_{in}-(\mathbf{h}-\mathbf{h}^{\rm
ext})\cdot D_{t}\mathbf{p}$ $\displaystyle+(\mu^{\alpha}-\mu^{{\rm
ext},\alpha})r^{\alpha}+j^{\alpha,i}\partial_{i}(\overline{\mu}^{\alpha}-\overline{\mu}^{{\rm
ext},\alpha})$ $\displaystyle+(\mathbf{f}^{\mathrm{ext}}-\mathbf{f}^{\rm
ext}_{e})\cdot\mathbf{v}+(\boldsymbol{\Gamma}^{\mathrm{ext}}-\boldsymbol{\Gamma}^{\mathrm{ext}}_{e})\cdot\boldsymbol{\omega}\Big{\\}}$
$\displaystyle+\oint_{\mathcal{C}}dl\nu_{i}\left[\mathbf{t}^{i}\cdot\mathbf{v}+\mathbf{m}^{i}\cdot\boldsymbol{\omega}+\boldsymbol{\pi}^{i}\cdot
D_{t}\mathbf{p}\right.$ $\displaystyle\left.\hskip
42.67912pt-(\overline{\mu}^{\alpha}-\overline{\mu}^{\rm
ext,\alpha})j^{\alpha,i}\right]~{},$ (59)

where we have introduced the gradient of flow $v_{ij}$, the vorticity
$\boldsymbol{\omega}$, the normal vorticity gradient $\omega_{in}$ and the
corotational derivative of the curvature tensor $D_{t}C_{ij}$, which are given
by:

$\displaystyle v_{ij}=$
$\displaystyle\frac{1}{2}(\nabla_{i}v_{j}+\nabla_{j}v_{i})+C_{ij}v_{n}~{},$
(60) $\displaystyle\boldsymbol{\omega}=$
$\displaystyle\epsilon^{ij}(\partial_{j}v_{n}-C_{jk}v^{k})\mathbf{e}_{i}+\frac{1}{2}\epsilon^{ij}(\nabla_{i}v_{j})\mathbf{n}~{},$
(61) $\displaystyle\omega_{in}=$
$\displaystyle(\partial_{i}\boldsymbol{\omega})\cdot\mathbf{n}=\partial_{i}\omega_{n}-C_{i}{}^{j}\omega_{j}~{},$
(62) $\displaystyle D_{t}C_{ij}=$
$\displaystyle-\nabla_{i}(\partial_{j}v_{n})-v_{n}C_{ik}C^{k}{}_{j}+v^{k}\nabla_{k}C_{ij}$
$\displaystyle+\omega_{n}(\epsilon_{i}{}^{k}C_{kj}+\epsilon_{j}{}^{k}C_{ki})\,,$
(63)

and

$\overline{\mu}^{\alpha}=\mu^{\alpha}-\frac{m^{\alpha}}{m^{1}}\mu^{1},\overline{\mu}^{{\rm
ext},\alpha}=\mu^{{\rm ext},\alpha}-\frac{m^{\alpha}}{m^{1}}\mu^{{\rm
ext},1}~{},$ (64)

are the relative chemical potential of species $\alpha$. Here we have made the
arbitrary choice of taking chemical potentials relative to the chemical
potential of species $\alpha=1$, which in practice can be chosen to be the
solvent. As a result, the sum over species $\alpha$ in Eq. 59 can be taken for
$\alpha=2...N$. If one considers $I=1..M$ chemical reactions, the contribution
$(\mu^{\alpha}-\mu^{{\rm ext},\alpha})r^{\alpha}$ can be rewritten as a sum
over chemical reactions (Appendix D.2):

$\displaystyle(\mu^{\alpha}-\mu^{{\rm
ext},\alpha})r^{\alpha}=-\sum_{I}\Delta\mu^{I}r^{I}~{},$ (65)

where $\Delta\mu^{I}=\sum_{I}a^{\alpha,I}(\mu^{\alpha}-\mu^{{\rm
ext},\alpha})$ with $a^{\alpha,I}$ the stoichiometric coefficients for
$\alpha$ in reaction $I$, and $r^{I}$ the rate of reaction $I$ as defined in
Eq. 260. In the following we consider a single chemical reaction between a
fuel and its product; such that $(\mu^{\alpha}-\mu^{{\rm
ext},\alpha})r^{\alpha}=-r\Delta\mu$ with $r$ the rate of fuel consumption and
$\Delta\mu$ the chemical potential of conversion of fuel to the product.

For an isothermal surface, the free energy density evolves according to the
balance equation (Eq. 264):

$\displaystyle\partial_{t}f+\nabla_{i}(fv^{i}+j^{f,i})+v_{n}C_{i}{}^{i}f$
$\displaystyle=J^{f}_{n}-T\theta~{},$ (66)

where $J_{n}^{f}$ and $j^{f,i}$ are respectively the normal flux of free
energy and tangential flux relative to the center of mass of free energy, and
the entropy production rate within the surface is denoted $\theta$. Using Eq.
265, identification with the rate of change of free energy then gives the rate
of entropy production within the surface:

$\displaystyle T\theta=$
$\displaystyle\left(\overline{t}^{ij}-\overline{t}_{e}^{ij}\right)v_{ij}+(\overline{m}^{ij}-\overline{m}_{e}^{ij})D_{t}C_{ij}$
$\displaystyle+(m_{n}^{i}-m_{n,e}^{i})\omega_{in}+(\mathbf{h}-\mathbf{h}^{\rm
ext})\cdot D_{t}\mathbf{p}$
$\displaystyle+r\Delta\mu-j^{\alpha,i}\partial_{i}(\overline{\mu}^{\alpha}-\overline{\mu}^{{\rm
ext},\alpha})~{},$ (67)

and the flux of free energy with normal and tangential components:

$\displaystyle J_{n}^{f}$
$\displaystyle=(\mathbf{f}^{\mathrm{ext}}-\mathbf{f}^{\rm
ext}_{e})\cdot\mathbf{v}+(\boldsymbol{\Gamma}^{\mathrm{ext}}-\boldsymbol{\Gamma}^{\mathrm{ext}}_{e})\cdot\boldsymbol{\omega}~{},$
(68) $\displaystyle j^{f,i}$
$\displaystyle=-\mathbf{t}^{i}\cdot\mathbf{v}-\mathbf{m}^{i}\cdot\boldsymbol{\omega}-\boldsymbol{\pi}^{i}\cdot
D_{t}\mathbf{p}+(\overline{\mu}^{\alpha}-\overline{\mu}^{\rm
ext,\alpha})j^{\alpha,i}.$ (69)

Here we have assumed that external forces and torques do not contribute to the
surface entropy production. The conjugate fluxes and forces can be read from
the entropy production rate Eq. V.4 and are listed in Table 7.

Flux | Force
---|---
In-plane deviatoric tension tensor $\overline{t}^{ij}_{d}=\overline{t}^{ij}-\overline{t}^{ij}_{e}$ | In-plane shear tensor $v_{ij}$
In-plane deviatoric bending moment tensor $\overline{m}^{ij}_{d}=\overline{m}^{ij}-\overline{m}^{ij}_{e}$ | Bending rate tensor $D_{t}C_{ij}$
Normal deviatoric moment $m_{n,\,d}^{i}=m_{n}^{i}-m_{n,e}^{i}$ | Vorticity gradient $\omega_{in}=(\partial_{i}\boldsymbol{\omega})\cdot\mathbf{n}$
Convected, corotational derivative of polarity $D_{t}p^{i}$ |  Deviatoric molecular field $h^{i}_{d}=h^{i}-h^{\rm ext,i}$
Rate of fuel consumption $r$ | Fuel hydrolysis chemical potential $\Delta\mu$
Flux of species $\alpha=2.,.N$, $j^{\alpha,i}$ |  Deviatoric gradient of relative chemical potential of species $\alpha=2...N$, $-\partial_{i}\bar{\mu}_{d}^{\alpha}=-\partial_{i}(\bar{\mu}^{\alpha}-\bar{\mu}^{{\rm ext},\alpha})$
Table 7:  List of conjugate forces and fluxes for an active fluid polar
surface.

###  V.5 Constitutive equations

To obtain constitutive equations, we perform a linear expansion of
thermodynamic fluxes into thermodynamic forces listed in Table 7 de2013non .
We discuss coupling terms that are allowed for different types of polar
surfaces classified in section III.

####  V.5.1 Polar surfaces

We first consider true polar surfaces, associated with a vectorial order
parameter $\mathbf{p}$. The deviatoric parts of the stress and moment tensors,
the convected time derivative of the polarity field, the flux of species
$\alpha$, relative to the centre of mass and the rate of fuel consumption can
be decomposed as

$\displaystyle\overline{t}^{ij}_{d}$
$\displaystyle=\overline{t}^{ij}_{0}+\epsilon_{\rm
UD}\overline{t}^{ij}_{\rm{UD}}+\epsilon_{\rm
C}\overline{t}^{ij}_{\rm{C}}+\epsilon_{\rm PC}\overline{t}^{ij}_{\rm{PC}},$
$\displaystyle\overline{m}^{ij}_{d}$
$\displaystyle=\overline{m}^{ij}_{0}+\epsilon_{\rm
UD}\overline{m}^{ij}_{\rm{UD}}+\epsilon_{\rm
C}\overline{m}^{ij}_{\rm{C}}+\epsilon_{\rm PC}\overline{m}^{ij}_{\rm{PC}},$
$\displaystyle m^{i}_{n,d}$ $\displaystyle=m^{i}_{n0}+\epsilon_{\rm
UD}m^{i}_{n\rm{UD}}+\epsilon_{\rm C}m^{i}_{n\rm{C}}+\epsilon_{\rm
PC}m^{i}_{n\rm{PC}}$ $\displaystyle D_{t}p^{i}$
$\displaystyle=P_{0}^{i}+\epsilon_{\rm UD}P_{\rm UD}^{i}+\epsilon_{\rm
C}P_{\rm C}^{i}+\epsilon_{\rm PC}P_{\rm PC}$ $\displaystyle j^{\alpha,i}$
$\displaystyle=j_{0}^{\alpha,i}+\epsilon_{\rm UD}j_{\rm
UD}^{\alpha,i}+\epsilon_{\rm C}j_{\rm C}^{\alpha,i}+\epsilon_{\rm PC}j_{\rm
PC}^{\alpha,i}$ $\displaystyle r$ $\displaystyle=r_{0}+\epsilon_{\rm UD}r_{\rm
UD}+\epsilon_{\rm C}r_{\rm C}+\epsilon_{\rm PC}r_{\rm PC}~{},$ (70)

where $\overline{t}^{ij}_{0}$ is the part of the stress tensor that exists for
any surface, $\overline{t}^{ij}_{\rm{UD}}$ correspond to terms present when
the surface breaks up-down symmetry, $\overline{t}^{ij}_{\rm{C}}$ exist for
chiral surfaces, and $\overline{t}^{ij}_{\rm{PC}}$ for planar-chiral surfaces.
Similar rules apply for the decomposition of other thermodynamic fluxes.

To express constitutive equations for each of the components, one can then
write possible terms of the expansion at linear thermodynamic fluxes into
generalized thermodynamic forces, and ask whether the corresponding terms
break the symmetries $M_{n}$, $M_{t}$, $R_{t}$, $I$, according to the
signatures given in section III. To check which symmetries are broken or
satisfied by couplings in the constitutive equations, we note that
$\overline{t}^{ij}$ is a true tensor, whereas $\overline{m}^{ij}$ is a
pseudotensor with signature $-1$ under $M_{n},R_{t},I$ and signature $+1$
under $M_{t},R_{n}$. Indeed, $\overline{m}^{ij}$ is the product of $m^{ij}$
and the tensor $\epsilon^{ij}$ (Eq. 32). $m^{ij}$ is proportional to a torque
(Eq. 253) and has therefore signature $-1$ under the mirror and inversion
symmetries $M_{n}$, $M_{t}$, $I$. The tensor $\epsilon^{ij}$ has signature
$-1$ under $M_{t},R_{t}$ and $+1$ under $M_{n},R_{n},I$. Also,
$\mathbf{m}_{n}=m_{n}^{i}\mathbf{e}_{i}$ is a pseudo-vector with signature
$-1$ under $M_{t},R_{t}$ and $+1$ under $M_{n},R_{n},I$, since $\mathbf{n}$ is
a pseudo-vector with signature $-1$ under $M_{n},R_{t},I$ and $+1$ under
$M_{t},R_{n}$. These observations can be summarized in the Table of
transformation 8.

| $M_{n}$ | $M_{t}$ | $R_{t}$ | $I$ | $R_{n}$
---|---|---|---|---|---
$\overline{t}^{ij}$ | 1 | 1 | 1 | 1 | 1
$m^{ij}$ | -1 | -1 | 1 | -1 | 1
$\overline{m}^{ij}$ | -1 | 1 | \- 1 | -1 | 1
$m_{n}^{i}$ | 1 | -1 | -1 | 1 | 1
Table 8:  Signatures of tension and moment tensors under a set of linear
transformations.

The corresponding constitutive equations for polar surfaces are given in
Appendix I. Because many coupling terms are possible, for simplicity we have
imposed some restrictions on the coupling terms considered. We limit ourselves
to an expansion of phenomenological coefficients to first order in the
curvature tensor, the polarity vector and its associated nematic tensor
$q_{ij}=p_{i}p_{j}-\frac{1}{2}p^{2}g_{ij}$. Viscous coupling terms between
$\bar{t}^{ij}_{d}$, $\bar{m}^{ij}$, $m_{n,d}^{i}$ and $v_{ij}$, $D_{t}C_{ij}$,
$\omega_{in}$ and diagonal coupling terms between a flux and its conjugated
force that are dependent on the polarity vector and on the curvature tensor
are not listed. We do not include cross-couplings between the various
mechanical tensors and the molecular field $\mathbf{h}$ that depend on the
curvature tensor. Also, we do not include cross-coupling terms involving the
gradient of the chemical potential, $\partial_{i}\bar{\mu}_{d}^{\alpha}$,
except for active couplings of the relative flux $\mathbf{j}^{\alpha}$ with
the fuel hydrolysis chemical potential, $\Delta\mu$. Finally, we do not write
active terms which are proportional to the gradient of the polarity,
$\partial_{i}\mathbf{p}$.

We give below the contributions to constitutive equations which depend on the
polarity vector. Viscous terms and curvature-coupling terms for in-plane
isotropic, chiral and planar-chiral surfaces, which do not depend on the
polarity vector, were derived in Ref. salbreux2017activesurfaces . The
polarity-dependent contributions to the deviatoric contributions to the
tension tensor then read

$\displaystyle\overline{t}^{ij}_{0}=$
$\displaystyle\frac{\nu}{2}(p^{i}h_{d}^{j}+p^{j}h_{d}^{i}-h_{d}^{k}p_{k}g^{ij})+\nu^{\prime}h_{d}^{k}p_{k}g^{ij}+\zeta_{\rm
n}\Delta\mu q^{ij}$ $\displaystyle\overline{t}^{ij}_{\rm{UD}}=$
$\displaystyle\tilde{\zeta}_{\rm n}\Delta\mu
q^{kl}C_{kl}g^{ij}+\tilde{\zeta}^{\prime}_{\rm n}q^{ij}C_{k}{}^{k}\Delta\mu$
$\displaystyle\overline{t}^{ij}_{\rm{C}}=$
$\displaystyle\zeta_{\rm{Cn}}\Delta\mu(\epsilon^{ik}q_{k}{}^{l}C_{l}{}^{j}+\epsilon^{jk}q_{k}{}^{l}C_{l}{}^{i})$
$\displaystyle+\tilde{\zeta}_{\rm{Cn}}\Delta\mu(\epsilon^{ik}C_{k}{}^{l}q_{l}{}^{j}+\epsilon^{jk}C_{k}{}^{l}q_{l}{}^{i})$
$\displaystyle\overline{t}^{ij}_{\rm{PC}}=$ $\displaystyle\frac{\nu_{{\rm
PC}}}{2}(\epsilon^{i}{}_{k}h_{d}^{k}p^{j}+\epsilon^{j}{}_{k}h_{d}^{k}p^{i}-\epsilon_{lk}h_{d}^{k}p^{l}g^{ij})$
$\displaystyle+\nu_{\rm
PC}^{\prime}\epsilon_{kl}p^{k}h_{d}^{l}g^{ij}+\zeta_{\rm PC{\rm
n}}\Delta\mu\epsilon^{ik}q_{k}{}^{j},$ (71)

where we use the notation $h_{d}^{i}=h^{i}-h^{{\rm ext},i}$ for the deviatoric
molecular field, and we have introduced the nematic tensor that can be
constructed from the polar order parameter $\mathbf{p}$:

$\displaystyle q^{ij}=p^{i}p^{j}-\frac{1}{2}p^{2}g^{ij}~{}.$ (72)

We do not restrict ourselves here to $|\mathbf{p}|=1$; although we note that
the modulus of $\mathbf{p}$ is not in general a hydrodynamic variable, except
close to a polar/non-polar transition. In the linear expansion of
thermodynamic fluxes into thermodynamic forces, we have used that the tensor
$\epsilon^{ik}q_{k}{}^{j}$ is symmetric (Eq. 200) and that
$q_{i}{}^{k}\epsilon_{kj}=-\epsilon_{ik}q^{k}{}_{j}$ (Eq. 201). To avoid
introducing redundant couplings, we have also used that
$[\epsilon^{ik}p_{k}h_{d}^{j}]_{s}=[\epsilon^{i}{}_{k}h_{d}^{k}p^{j}]_{s}-\epsilon_{kl}p^{k}h_{d}^{l}g^{ij}$
as well as the identities which follow from Eqs 197-198,
$[C_{ik}q^{k}{}_{j}]_{s}=[q_{ik}C^{k}{}_{j}]_{s}=\frac{1}{2}C_{k}{}^{k}q_{ij}+\frac{1}{2}C_{kl}q^{kl}g_{ij}$.
Here we use the notation $[\cdot]_{s}$ for the symmetric part of a tensor, as
defined in Eq. 177. We find only two independent active terms in the
contribution $\bar{t}_{\rm C}^{ij}$ proportional to
$[\boldsymbol{\epsilon}\mathbf{q}\mathbf{C}]_{s}$ and
$[\boldsymbol{\epsilon}\mathbf{C}\mathbf{q}]_{s}$, due to the tensor product
identities
$\mathbf{q}\boldsymbol{\epsilon}\mathbf{C}=-\boldsymbol{\epsilon}\mathbf{q}\mathbf{C}$,
$\mathbf{C}\boldsymbol{\epsilon}\mathbf{q}=-\mathbf{C}\mathbf{q}\boldsymbol{\epsilon}$,
$[\mathbf{C}\mathbf{q}\boldsymbol{\epsilon}]_{s}=-[\boldsymbol{\epsilon}\mathbf{q}\mathbf{C}]_{s}$
and
$[\mathbf{q}\mathbf{C}\boldsymbol{\epsilon}]_{s}=-[\boldsymbol{\epsilon}\mathbf{C}\mathbf{q}]_{s}$,
as well as the two relations which follow from Eqs. 197 and 198:

$\displaystyle[\epsilon_{ik}C^{kl}q_{lj}]_{s}=\frac{1}{2}C_{k}{}^{k}\epsilon_{ik}q^{k}{}_{j}-\frac{1}{2}\epsilon_{kl}q^{lm}C_{m}{}^{k}g_{ij}~{},$
(73)
$\displaystyle[\epsilon_{ik}q^{kl}C_{lj}]_{s}=\frac{1}{2}C_{k}{}^{k}\epsilon_{ik}q^{k}{}_{j}+\frac{1}{2}\epsilon_{kl}q^{lm}C_{m}{}^{k}g_{ij}~{}.$
(74)

The deviatoric contributions to the moment tensor $\bar{m}_{d}^{ij}$ which
depend on the polarity vector $\mathbf{p}$ read:

$\displaystyle\overline{m}^{ij}_{0}=$ $\displaystyle\tilde{\zeta}_{c{\rm
n}}\Delta\mu q^{kl}C_{kl}g^{ij}+\tilde{\zeta}^{\prime}_{c{\rm n}}\Delta\mu
q^{ij}C_{k}{}^{k}$ $\displaystyle\overline{m}^{ij}_{\rm{UD}}=$
$\displaystyle\frac{\beta}{2}(p^{i}h_{d}^{j}+p^{j}h_{d}^{i}-p_{k}h_{d}^{k}g^{ij})+\beta^{\prime}p_{k}h_{d}^{k}g^{ij}+\zeta_{c{\rm
n}}\Delta\mu q^{ij}$ $\displaystyle\overline{m}^{ij}_{\rm{C}}=$
$\displaystyle\frac{\beta_{\rm{C}}}{2}(\epsilon^{i}{}_{k}h_{d}^{k}p^{j}+\epsilon^{j}{}_{k}h_{d}^{k}p^{i}-\epsilon_{lk}h_{d}^{k}p^{l}g^{ij})+\beta_{\rm{C}}^{\prime}\epsilon^{k}{}_{l}p_{k}h_{d}^{l}g^{ij}$
$\displaystyle+\zeta_{c{\rm C}{\rm n}}\Delta\mu\epsilon^{ik}q_{k}{}^{j}$
$\displaystyle\overline{m}^{ij}_{\rm{PC}}=$
$\displaystyle\zeta_{c\rm{PCn}}\Delta\mu(\epsilon^{ik}q_{k}{}^{l}C_{l}{}^{j}+\epsilon^{jk}q_{k}{}^{l}C_{l}{}^{i})$
$\displaystyle+\tilde{\zeta}_{c\rm{PCn}}\Delta\mu(\epsilon^{ik}C_{k}{}^{l}q_{l}{}^{j}+\epsilon^{jk}C_{k}{}^{l}q_{l}{}^{i})~{},$
(75)

where to list independent terms, we have used the same relations as for the
tension tensor $\bar{t}_{ij}$. In Eq. 75, we also have only introduced
symmetric contributions to the bending moment tensor $\bar{m}_{ij}$. We find
the following contributions to the tensor $m_{n}$:

$\displaystyle m^{i}_{n0}$ $\displaystyle=\chi_{\rm
p}\Delta\mu\epsilon^{ij}p_{j}+\psi\,\epsilon^{i}{}_{j}h_{d}^{j}$
$\displaystyle m^{i}_{n\rm{UD}}$ $\displaystyle=(\overline{\chi}_{\rm
p}\epsilon^{i}{}_{k}C^{kj}+\overline{\chi}^{\prime}_{\rm
p}\epsilon^{j}{}_{k}C^{ki})\Delta\mu p_{j}$ $\displaystyle m^{i}_{n\rm{C}}$
$\displaystyle=(\chi_{\rm{C}p}C^{ij}+\chi_{\rm{C}p}^{\prime}C_{k}{}^{k}g^{ij})\Delta\mu
p_{j}$ $\displaystyle m^{i}_{n\rm{PC}}$
$\displaystyle=\chi_{\rm{PC}\rm{p}}\Delta\mu p^{i}+\psi_{\rm PC}h_{d}^{i}~{}.$
(76)

We have used the relation
$\epsilon^{i}{}_{k}C^{kj}=\epsilon^{j}{}_{k}C^{ki}+C_{k}{}^{k}\epsilon^{ij}$
(Eq. 200) to avoid redundant couplings in the equation for the normal moment
tensor $\mathbf{m}_{n\rm{UD}}$.

In the equations above, active coupling coefficients with the nematic tensor
$q_{ij}$ are denoted with a subscript “${\rm n}$”, while active polar coupling
coefficients with the polar vector $\mathbf{p}$ are denoted with a subscript
“${\rm p}$”. Terms contributing to the dynamics of the polarity field reads:

$\displaystyle P_{0}^{i}=$
$\displaystyle\frac{1}{\gamma}h_{d}^{i}-\nu\tilde{v}^{ij}p_{j}-\nu^{\prime}v_{k}{}^{k}p^{i}+\lambda\Delta\mu
p^{i}+\psi\,\epsilon^{ij}\omega_{jn}$ $\displaystyle P_{\rm UD}^{i}=$
$\displaystyle-\beta
p_{j}D_{t}\tilde{C}^{ij}-\beta^{\prime}p^{i}D_{t}C_{k}{}^{k}$
$\displaystyle+\lambda_{\rm UD}\Delta\mu\,C^{ij}p_{j}+\lambda^{\prime}_{\rm
UD}\Delta\mu\,C^{j}{}_{j}p^{i}$ $\displaystyle P_{\rm C}^{i}=$
$\displaystyle\beta_{\rm{C}}\epsilon^{i}{}_{k}D_{t}\tilde{C}^{kj}p_{j}+\beta_{\rm{C}}^{\prime}\epsilon^{ij}p_{j}D_{t}C_{k}{}^{k}$
$\displaystyle+(\lambda_{\rm C}\epsilon^{i}{}_{k}C^{kj}+\lambda_{\rm
C}^{\prime}\epsilon^{j}{}_{k}C^{ki})p_{j}\Delta\mu$ $\displaystyle P_{\rm
PC}^{i}=$ $\displaystyle\frac{1}{\gamma_{\rm
PC}}\epsilon^{i}{}_{j}h_{d}^{j}+\nu_{\rm{PC}}\epsilon^{i}{}_{k}\tilde{v}^{kj}p_{j}+\nu_{\rm{PC}}^{\prime}v_{k}{}^{k}\epsilon^{ij}p_{j}$
$\displaystyle+\lambda_{\rm{PC}}\Delta\mu\epsilon^{ij}p_{j}-\psi_{\rm
PC}\omega^{i}_{n}\,,$ (77)

where we have introduced the notation
$\tilde{A}_{ij}=A_{ij}-\frac{1}{2}A_{k}{}^{k}g_{ij}$ for the traceless part of
a tensor $A_{ij}$. Here $1/\gamma_{\rm PC}$ is an odd inverse rotational
viscosity, analogous to odd or Hall viscosities that can arise in surfaces
with broken planar-chiral symmetry (Appendix I). Note that $1/\gamma_{\rm PC}$
can be nonzero in systems close to equilibrium only if time-reversal symmetry
is broken, for example in the presence of a magnetic field. If time-reversal
symmetry holds, Onsager symmetry require this odd inverse rotational viscosity
to vanish.

The flux of species $\alpha=2...N$, relative to the centre of mass has
different contributions which are dependent on the polarity vector and are
given by:

$\displaystyle j^{\alpha,i}_{0}$ $\displaystyle=\kappa_{\rm
p}^{\alpha}p^{i}\Delta\mu$ $\displaystyle j^{\alpha,i}_{\rm{UD}}$
$\displaystyle=(\kappa_{\rm{UD}p}^{\alpha}C^{ij}+\kappa_{\rm{UD}p}^{\prime\alpha}C_{k}{}^{k}g^{ij})p_{j}\Delta\mu$
$\displaystyle j^{\alpha,i}_{\rm{C}}$ $\displaystyle=(\kappa_{\rm
Cp}^{\alpha}\epsilon^{i}{}_{k}C^{kj}+\kappa_{\rm
Cp}^{\prime\alpha}\epsilon^{j}{}_{k}C^{ki})p_{j}\Delta\mu$ $\displaystyle
j^{\alpha,i}_{\rm PC}$ $\displaystyle=\kappa_{\rm
PCp}^{\alpha}\epsilon^{ij}p_{j}\Delta\mu~{},$ (78)

where we have used the relation
$\epsilon^{i}{}_{k}C^{kj}=\epsilon^{j}{}_{k}C^{ki}+C_{k}{}^{k}\epsilon^{ij}$
to avoid redundant couplings in the equation for $\mathbf{j}_{\rm C}$.
Finally, the rate of fuel consumption has the following contributions which
depend on the polarity vector:

$\displaystyle r_{0}=$ $\displaystyle-\zeta_{\rm
n}v^{ij}q_{ij}-(\tilde{\zeta}_{c\rm
n}q^{kl}C_{kl}g^{ij}+\tilde{\zeta}^{\prime}_{c\rm
n}C_{k}{}^{k}q^{ij})D_{t}C_{ij}$ $\displaystyle-\chi_{\rm
p}\epsilon^{ij}p_{j}\omega_{in}+\lambda p_{i}h_{d}^{i}-\kappa_{\rm
p}^{\alpha}p^{i}\partial_{i}\bar{\mu}_{d}^{\alpha}$ (79) $\displaystyle r_{\rm
UD}=$ $\displaystyle-(\tilde{\zeta}_{\rm
n}q^{kl}C_{kl}g^{ij}+\tilde{\zeta}_{\rm
n}^{\prime}q^{ij}C_{k}{}^{k})v_{ij}-\zeta_{c\rm n}q^{ij}D_{t}C_{ij}$
$\displaystyle-(\overline{\chi}_{\rm
p}\epsilon^{i}{}_{k}C^{kj}+\overline{\chi}^{\prime}_{\rm
p}\epsilon^{j}{}_{k}C^{ki})p_{j}\omega_{in}$ $\displaystyle+(\lambda_{\rm
UD}C^{ij}+\lambda^{\prime}_{\rm UD}C_{k}{}^{k}g^{ij})p_{j}h_{d,i}$
$\displaystyle-(\kappa_{\rm UDp}^{\alpha}C^{ij}+\kappa_{\rm
UDp}^{\prime\alpha}C_{k}{}^{k}g^{ij})p_{j}\partial_{i}\bar{\mu}_{d}^{\alpha}$
(80) $\displaystyle r_{\rm C}=$ $\displaystyle-(2\zeta_{\rm
Cn}\epsilon^{ik}q_{kl}C^{lj}+2\tilde{\zeta}_{\rm
Cn}\epsilon^{ik}C_{kl}q^{lj})v_{ij}$ $\displaystyle-\zeta_{c\rm
Cn}\epsilon^{ik}q_{k}{}^{j}D_{t}C_{ij}$ $\displaystyle-(\chi_{\rm
Cp}C^{ij}+\chi_{\rm Cp}^{\prime}C_{k}{}^{k}g^{ij})p_{j}\omega_{in}$
$\displaystyle+(\lambda_{\rm C}\epsilon^{i}{}_{k}C^{kj}+\lambda_{\rm
C}^{\prime}\epsilon^{j}{}_{k}C^{ki})p_{j}h_{d,i}$ $\displaystyle-(\kappa_{\rm
Cp}^{\alpha}\epsilon^{i}{}_{k}C^{kj}+{\kappa^{\prime}}_{\rm
Cp}^{\alpha}\epsilon^{j}{}_{k}C^{ki})p_{j}\partial_{i}\bar{\mu}_{d}^{\alpha}$
(81) $\displaystyle r_{\rm PC}=$ $\displaystyle-\zeta_{\rm
PCn}\epsilon^{ik}q_{k}{}^{j}v_{ij}$ $\displaystyle-(2\zeta_{c\rm
PCn}\epsilon^{ik}q_{k}{}^{l}C_{l}{}^{j}+2\tilde{\zeta}_{c\rm
PCn}\epsilon^{ik}C_{k}{}^{l}q_{l}{}^{j})D_{t}C_{ij}$ $\displaystyle-\chi_{\rm
PCp}p^{i}\omega_{in}+\lambda_{\rm PC}\epsilon^{ij}p_{j}h_{d,i}-\kappa_{\rm
PCp}^{\alpha}\epsilon^{ij}p_{j}\partial_{i}\bar{\mu}_{d}^{\alpha}.$ (82)

In the equations above, coefficients containing the letters $\nu$, $\beta$,
$\zeta$, $\chi$ and $\psi$ are all reactive. Coefficients containing the
letters $\gamma$, $\lambda$ and $\kappa$ are dissipative. The Onsager symmetry
relations impose symmetry relations between couplings which are reactive or
dissipative. Couplings which relate flux and forces with the same signature
under time reversal are reactive and are antisymmetric. Conversely, couplings
which relate flux and forces with opposite signatures under time reversal are
dissipative and are symmetric.

####  V.5.2 Pseudopolar surfaces

We now discuss pseudopolar surfaces. As discussed in section III, several
combinations of pseudoscalar and pseudovector order parameters can be used to
describe the order of a surface, such that only two physically distinct
pseudopolar surfaces exist. Here we consider surfaces described by the order
parameter $\mathbf{p}_{\rm C}$ and by the combination $\\{\epsilon_{\rm
PC},\mathbf{p}_{\rm C}\\}$. We note that for surfaces whose order is described
by a pseudovector $\mathbf{p}_{\rm C}$, a true nematic tensor $q^{ij}=p_{\rm
C}^{i}p_{\rm C}^{j}-\frac{1}{2}p_{{\rm C},k}~{}p_{\rm C}^{k}g^{ij}$ can be
defined. We then obtain the following constitutive equations, writing here
only terms that depend on the pseudopolar vector (other terms are given in
Appendix I):

$\displaystyle\overline{t}_{d}^{ij}=$ $\displaystyle\zeta_{\rm n}\Delta\mu
q^{ij}+\frac{\nu}{2}(p_{\rm C}^{i}h_{{\rm C},d}^{j}+p_{\rm C}^{j}h_{{\rm
C},d}^{i}-h_{{\rm C},d}^{k}p_{{\rm C},k}g^{ij})$
$\displaystyle+\nu^{\prime}h_{{\rm C},d}^{k}p_{{\rm C},k}g^{ij}+\epsilon_{\rm
PC}\left[\zeta_{\rm PC{\rm n}}\Delta\mu\epsilon^{ik}q_{k}{}^{j}\right.$
$\displaystyle\left.+\frac{\nu_{{\rm PC}}}{2}(\epsilon^{i}{}_{k}h^{k}_{{\rm
C},d}p_{\rm C}^{j}+\epsilon^{j}{}_{k}h_{{\rm C},d}^{k}p_{\rm
C}^{i}-\epsilon_{lk}h_{{\rm C},d}^{k}p_{\rm C}^{l}g^{ij})\right.$
$\displaystyle\left.+\nu_{\rm PC}^{\prime}\epsilon_{kl}p_{\rm C}^{k}h_{{\rm
C},d}^{l}g^{ij}\right]$ (83) $\displaystyle\overline{m}_{d}^{ij}=$
$\displaystyle\tilde{\zeta}_{c{\rm
n}}q^{kl}C_{kl}g^{ij}\Delta\mu+\tilde{\zeta^{\prime}_{c{\rm
n}}}q^{ij}C_{k}{}^{k}\Delta\mu$ $\displaystyle+\epsilon_{\rm
PC}\left[\zeta_{\rm{cPCn}}\Delta\mu(\epsilon^{ik}q_{k}{}^{l}C_{l}{}^{j}+\epsilon^{jk}q_{k}{}^{l}C_{l}{}^{i})\right.$
$\displaystyle\left.+\tilde{\zeta}_{\rm{cPCn}}\Delta\mu(\epsilon^{ik}C_{k}{}^{l}q_{l}{}^{j}+\epsilon^{jk}C_{k}{}^{l}q_{l}{}^{i})\right]$
(84) $\displaystyle m^{i}_{n,d}$
$\displaystyle=(\chi_{\rm{C}p}C^{i}{}_{j}+\chi_{\rm{C}p}^{\prime}C_{k}{}^{k}g^{i}{}_{j})p_{\rm
C}^{j}\Delta\mu$ $\displaystyle+\epsilon_{\rm PC}(\overline{\chi}_{\rm
p}\epsilon^{ik}C_{kj}+\overline{\chi}^{\prime}_{\rm
p}\epsilon_{jk}C^{ki})p_{\rm C}^{j}\Delta\mu$ (85) $\displaystyle D_{t}p_{\rm
C}^{i}=$ $\displaystyle\frac{1}{\gamma}h_{{\rm
C},d}^{i}-\nu\tilde{v}^{ij}p_{{\rm C},j}-\nu^{\prime}v_{k}{}^{k}p_{\rm
C}^{i}+\lambda\Delta\mu p_{\rm C}^{i}$ $\displaystyle+\epsilon_{\rm
PC}\left[\frac{1}{\gamma_{\rm PC}}\epsilon^{i}{}_{j}h_{{\rm
C},d}^{j}+\nu_{\rm{PC}}\epsilon^{ik}\tilde{v}_{kj}p_{\rm C}^{j}\right.$
$\displaystyle\left.+\nu_{\rm{PC}}^{\prime}v_{k}{}^{k}\epsilon^{ij}p_{{\rm
C},j}+\lambda_{\rm{PC}}\Delta\mu\epsilon^{i}{}_{j}p_{\rm C}^{j}\right]$ (86)
$\displaystyle j^{\alpha,i}=$ $\displaystyle(\kappa^{\alpha}_{\rm
Cp}\epsilon^{ik}C_{kj}+\kappa_{\rm
Cp}^{\prime\alpha}\epsilon_{jk}C^{ki})p_{\rm C}^{j}\Delta\mu$
$\displaystyle+\epsilon_{\rm
PC}(\kappa^{\alpha}_{\rm{UD}p}C^{i}{}_{j}+\kappa_{\rm{UD}p}^{\prime\alpha}C_{k}{}^{k}g^{i}{}_{j})p_{\rm
C}^{j}\Delta\mu$ (87) $\displaystyle r=$ $\displaystyle-\zeta_{\rm
n}v^{ij}q_{ij}-(\tilde{\zeta}_{c{\rm
n}}q^{kl}C_{kl}g^{ij}+\tilde{\zeta}^{\prime}_{c\rm
n}C_{k}{}^{k}q^{ij})D_{t}C_{ij}$ $\displaystyle-(\chi_{\rm Cp}C^{ij}+\chi_{\rm
Cp}^{\prime}C_{k}{}^{k}g^{i}{}_{j})p_{\rm C}^{j}\omega_{in}$
$\displaystyle+\lambda p_{{\rm C},i}h_{{\rm C}d}^{i}-(\kappa_{\rm
Cp}^{\alpha}\epsilon^{ik}C_{kj}+\kappa_{\rm
Cp}^{\prime\alpha}\epsilon_{jk}C^{ki})p_{\rm
C}^{j}\partial_{i}\bar{\mu}_{d}^{\alpha}$ $\displaystyle-\epsilon_{\rm
PC}\left[-\lambda_{\rm PC}\epsilon_{ij}p_{\rm C}^{j}h_{{\rm
C},d}^{i}+\zeta_{\rm PCn}\epsilon^{ik}q_{k}{}^{j}v_{ij}\right.$
$\displaystyle\left.+(\overline{\chi}_{\rm
p}\epsilon^{i}{}_{k}C^{kj}+\overline{\chi}^{\prime}_{\rm
p}\epsilon^{j}{}_{k}C^{ki})p_{j}\omega_{in}\right.$
$\displaystyle\left.+(\kappa^{\alpha}_{\rm UDp}C^{ij}+\kappa_{\rm
UDp}^{\prime\alpha}C_{k}{}^{k}g^{ij})p_{{\rm
C},j}\partial_{i}\bar{\mu}_{d}^{\alpha}\right.$
$\displaystyle\left.+(2\zeta_{c\rm
PCn}\epsilon^{ik}q_{k}{}^{l}C_{l}{}^{j}+2\tilde{\zeta}_{c\rm
PCn}\epsilon^{ik}C_{k}{}^{l}q_{l}{}^{j})D_{t}C_{ij}\right]~{}.$ (88)

In the above, we have used the notation for the molecular field associated to
the pseudovector, $\mathbf{h}_{\rm C}=-\frac{\delta F}{\delta\mathbf{p}_{\rm
C}}$. All the couplings in these equations were already introduced for polar
surfaces, and only a subset of couplings for polar surfaces are permitted for
pseudopolar surfaces.

##  VI Nematic active fluid surfaces

We discuss here nematic surfaces by introducing a traceless nematic order
parameter $\mathbf{Q}$ on the surface, tangent to the surface. Our analysis
follows the same presentation as the previous section for polar surfaces. The
force and torque balance equations 21 and 22 also apply here for a nematic
surface.

###  VI.1 Free energy and external potential

We consider here a region of surface $\mathcal{S}$ of a multicomponent nematic
fluid membrane, with free energy given by

$\displaystyle
F=\int_{\mathcal{S}}dSf(\mathbf{v},c^{\alpha},\mathbf{C},\mathbf{Q},\boldsymbol{\nabla}\otimes\mathbf{Q})~{},$
(89)

where the total free energy density $f$ has two contributions:

$f(\mathbf{v},c^{\alpha},\mathbf{C},\mathbf{Q},\boldsymbol{\nabla}\otimes\mathbf{Q})=\frac{1}{2}\rho
v^{2}+f_{0}(c^{\alpha},\mathbf{C},\mathbf{Q},\boldsymbol{\nabla}\otimes\mathbf{Q})~{},$
(90)

with $f_{0}$ is the free energy density in the rest frame. The tensor
$\boldsymbol{\nabla}\otimes\mathbf{Q}=\mathbf{e}^{k}\otimes\partial_{k}\mathbf{Q}$
is the gradient of nematic tensor (Eq. 188). We introduce conjugated fields to
the thermodynamic variables, such that the differential of the free energy
density $f_{0}$ reads:

$\displaystyle
df_{0}=\mu^{\alpha}dc^{\alpha}+\mathbf{K}:d\mathbf{C}-\mathbf{H}_{0}:d\mathbf{Q}+\boldsymbol{\Pi}\mathrel{\vbox{\hbox{.}\hbox{.}\hbox{.}}}d(\boldsymbol{\nabla}\otimes\mathbf{Q})~{},$
(91)

where in addition to the chemical potential $\mu^{\alpha}$ and the passive
bending moment tensor $\mathbf{K}$, we have introduced the force conjugate to
the nematic order parameter, $\mathbf{H}_{0}$, and the force conjugate to the
gradient of the nematic order parameter, $\boldsymbol{\Pi}$. These conjugate
forces are obtained by taking partial derivatives of the free energy density
$f_{0}$:

$\displaystyle\mu^{\alpha}=$ $\displaystyle\frac{\partial f_{0}}{\partial
c^{\alpha}},~{}\mathbf{K}=\frac{\partial f_{0}}{\partial\mathbf{C}},$
$\displaystyle\mathbf{H}_{0}=$ $\displaystyle-\frac{\partial
f_{0}}{\partial\mathbf{Q}},~{}\boldsymbol{\Pi}=\frac{\partial
f_{0}}{\partial(\boldsymbol{\nabla}\otimes\mathbf{Q})}~{}.$ (92)

We assume that $f_{0}$ is such that $\mathbf{K}$ and $\mathbf{H}_{0}$ are
tangent to the surface, that $\mathbf{H}_{0}$ is symmetric traceless, and that
one can write $\boldsymbol{\Pi}=\mathbf{e}_{i}\otimes\boldsymbol{\Pi}^{i}$,
with the tensor $\boldsymbol{\Pi}^{k}$ taken to satisfy
$\Pi^{k}{}_{ij}=\Pi^{k}{}_{ji}$, $\Pi^{k}{}_{ni}=\Pi^{k}{}_{in}$ and
$\Pi^{k}{}_{nn}=0$.

Because of rotation invariance of the free energy (Appendix F and Eq. VI.2),
the differential of the free energy density can also be written:

$df_{0}=\mu^{\alpha}dc^{\alpha}+\mathbf{K}:D\mathbf{C}-\mathbf{H}_{0}:D\mathbf{Q}+\boldsymbol{\Pi}\mathrel{\vbox{\hbox{.}\hbox{.}\hbox{.}}}D(\boldsymbol{\nabla}\otimes\mathbf{Q}).\\\
$ (93)

As $\mathbf{C}$ and $\mathbf{Q}$ are tangent to the surface, $D\mathbf{C}$ and
$D\mathbf{Q}$ are also tangent to the surface (Appendix B.3). We note that as
$\mathbf{Q}$ is tangent, symmetric and traceless, $D\mathbf{Q}$ is also
tangent, symmetric and traceless (Appendix B.3).

The total force conjugate to the nematic field,
$\mathbf{H}=-\tilde{\frac{\delta F}{\delta\mathbf{Q}}}$, reads

$\displaystyle\mathbf{H}=\mathbf{H}_{0}+\nabla_{i}\boldsymbol{\Pi}^{i}-\frac{1}{2}(\nabla_{i}\Pi^{i}{}^{j}{}_{j})\mathbf{e}_{k}\otimes\mathbf{e}^{k}~{}.$
(94)

where we have imposed that the tangential part $H^{ij}$ is symmetric
traceless, and we have used that $\mathbf{H}_{0}$ is tangent, symmetric and
traceless, and that the tangent part of the tensor
$\nabla_{k}\boldsymbol{\Pi}^{k}$ is symmetric since
$(\nabla_{k}\boldsymbol{\Pi}^{k})\cdot(\mathbf{e}_{i}\otimes\mathbf{e}_{j})=\nabla_{k}\Pi^{k}{}_{ij}+\Pi^{k}{}_{nj}C_{ki}+\Pi^{k}{}_{in}C_{kj}$.

As for polar surfaces, we assume that the surface is subjected to an external
potential:

$\displaystyle
U=\int_{\mathcal{S}}dSc^{\alpha}u^{\alpha}(\mathbf{X},\mathbf{n},\mathbf{Q}),$
(95)

where $u^{\alpha}$ is an external potential density acting on species
$\alpha$. Here again, the external potential could be provided by an external
magnetic field.

###  VI.2 Invariance of free energy by rotation and translation

Using the fact that the free energy density
$f_{0}(c^{\alpha},\mathbf{C},\mathbf{Q},\boldsymbol{\nabla}\otimes\mathbf{Q})$
is only a function of the surface variables and does not explicitly depend on
space, one obtains the Gibbs-Duhem relation (Appendix F)

$\displaystyle\nabla_{j}\left[(f_{0}-\mu^{\alpha}c^{\alpha})g_{i}{}^{j}-K^{jk}C_{ik}-\boldsymbol{\Pi}^{j}:\partial_{i}\mathbf{Q}\right]+C_{ik}\nabla_{j}K^{jk}$
$\displaystyle=-(\partial_{i}\mu^{\alpha})c^{\alpha}-\mathbf{H}:\partial_{i}\mathbf{Q}~{}.$
(96)

Invariance by rotation of the free energy also leads to the following
identity, which can be seen as a generalized Gibbs-Duhem relation:

$\displaystyle\boldsymbol{\Pi}^{k}\times_{1}\partial_{k}\mathbf{Q}+\boldsymbol{\Pi}^{k}\times_{2}\partial_{k}\mathbf{Q}+\epsilon_{i}{}^{j}(\boldsymbol{\Pi}^{i}:\partial_{j}\mathbf{Q})\mathbf{n}=$
$\displaystyle\mathbf{H}_{0}\times_{1}\mathbf{Q}+\mathbf{H}_{0}\times_{2}\mathbf{Q}+\mathbf{C}\times_{1}\mathbf{K}+\mathbf{C}\times_{2}\mathbf{K}~{},$
(97)

where we use a notation for the cross-product of tensors introduced in
Appendix A.

###  VI.3 Equilibrium relations

To obtain equilibrium relations and expressions for the equilibrium tensions
and torques, we consider changes of free energy associated to an infinitesimal
virtual surface displacement $d\mathbf{X}$, virtual change of nematic tensor
$d\mathbf{Q}$ and change of concentration $dc^{\alpha}$ of the component
$\alpha$. Therefore, we write that for a patch of surface $\mathcal{S}$
enclosed by the contour $\mathcal{C}$, the variation of
$F_{0}=\int_{\mathcal{S}}dSf_{0}$ is:

$\displaystyle
dF_{0}=dW+\int_{\mathcal{S}}dS[-\mathbf{H}:D\mathbf{Q}+\mu^{\alpha}{\rm
d}\bar{}\hskip
1.00006ptc^{\alpha}]+\oint_{\mathcal{C}}dl\nu_{i}\boldsymbol{\Pi}^{i}:D\mathbf{Q}~{}.$
(98)

The corotational differential $D\mathbf{Q}$ has been defined in Eq. 3. In this
equation, the molecular field $\mathbf{H}$ couples to the tangent nematic
tensor $\mathbf{Q}$, while $\boldsymbol{\Pi}$ is the internal molecular field
coupled to $\mathbf{Q}$ on the contour of the surface patch, $\mathcal{C}$. As
for polar surfaces, the chemical potential $\mu^{\alpha}$ couples to the
variation of concentration of molecular species $\alpha$ which does not arise
from geometric dilution, ${\rm d}\bar{}\hskip 1.00006ptc^{\alpha}$ defined in
Eq. 210. For surfaces perturbations where the nematic tensor rotates with the
surface, $D\mathbf{Q}=0$, and where the concentration fields change only by
geometric dilution, ${\rm d}\bar{}\hskip 1.00006ptc^{\alpha}=0$, the variation
of free energy is equal to the work associated with the surface deformation,
$dW$.

Similarly, the variation of the external potential $U$ has a mechanical
contribution, corresponding to the work $dW_{\rm ext}$ (Eq. 27), and
contributions from the variations in the nematic field $D\mathbf{Q}$ and the
concentration field ${\rm d}\bar{}\hskip 1.00006ptc$:

$\displaystyle dU=-dW_{\rm ext}+\int_{\mathcal{S}}dS[\mathbf{H}^{\rm
ext}:D\mathbf{Q}-\mu^{{\rm ext},\alpha}{\rm d}\bar{}\hskip
1.00006ptc^{\alpha}]~{},$ (99)

where external molecular field $\mathbf{H}^{\rm ext}$ and external chemical
potential $\mu^{\alpha,{\rm ext}}_{e}$ are defined by:

$\displaystyle\mathbf{H}^{\rm ext}=$ $\displaystyle\frac{\delta
U}{\delta\mathbf{Q}}~{},$ (100) $\displaystyle\mu^{\alpha,\rm ext}=$
$\displaystyle-\frac{\delta U}{\delta c^{\alpha}}~{}.$ (101)

We assume that $U$ is such that that $\mathbf{H}^{\rm ext}$ is traceless,
symmetric and tangent to the surface. As a result the variation of the total
free energy $F_{0}+U$ reads:

$\displaystyle dF_{0}+dU=$ $\displaystyle\int_{\mathcal{S}}dS[(\mathbf{H}^{\rm
ext}-\mathbf{H}):D\mathbf{Q}+(\mu^{\alpha}-\mu^{{\rm ext},\alpha}){\rm
d}\bar{}\hskip 1.00006ptc^{\alpha}]$ $\displaystyle+dW_{\rm
contour}+\oint_{\mathcal{C}}dl\nu_{i}\boldsymbol{\Pi}^{i}:D\mathbf{Q}~{}.$
(102)

For a nematic surface with the free energy differential introduced in Eq. 91,
we then obtain (Appendix G.2):

$\displaystyle t_{e}^{ij}=$
$\displaystyle(f_{0}-\mu^{\alpha}c^{\alpha})g^{ij}-K^{ik}C_{k}{}^{j}-\boldsymbol{\Pi}^{i}:\partial^{j}\mathbf{Q}$
(103) $\displaystyle t_{e,n}^{i}=$
$\displaystyle\nabla_{j}K^{ji}+\epsilon^{i}{}_{j}\left[\boldsymbol{\Gamma}_{e}^{{\rm
ext}}-(\mathbf{H}\times_{1}\mathbf{Q}+\mathbf{H}\times_{2}\mathbf{Q})\right]\cdot\mathbf{e}^{j}$
(104) $\displaystyle\mathbf{m}_{e}^{i}=$ $\displaystyle
K^{ik}\epsilon_{k}{}^{j}\mathbf{e}_{j}-\boldsymbol{\Pi}^{i}\times_{1}\mathbf{Q}-\boldsymbol{\Pi}^{i}\times_{2}\mathbf{Q}~{}.$
(105)

The expression for the normal component of the equilibrium tension tensor
$t^{i}_{e,n}$ arises from the tangential torque balance Eq. 25, and involves
the external tangential torque density acting on the surface.

The external forces, external torques, molecular field $\mathbf{H}$ and
external molecular potential $\mu^{\rm ext,\alpha}$ deriving from the external
potential $U$ read (Appendix G.2.2):

$\displaystyle\mathbf{f}^{\rm ext}_{e}$
$\displaystyle=-c^{\alpha}\frac{\partial u^{\alpha}}{\partial\mathbf{X}}$
(106) $\displaystyle\boldsymbol{\Gamma}^{\rm ext}_{e}$
$\displaystyle=c^{\alpha}\left(\frac{\partial
u^{\alpha}}{\partial\mathbf{n}}\times\mathbf{n}+\frac{\partial
u^{\alpha}}{\partial\mathbf{Q}}\times_{1}\mathbf{Q}+\frac{\partial
u^{\alpha}}{\partial\mathbf{Q}}\times_{2}\mathbf{Q}\right)$ (107)
$\displaystyle\mathbf{H}^{\rm ext}$ $\displaystyle=c^{\alpha}\frac{\partial
u^{\alpha}}{\partial\mathbf{Q}}$ (108) $\displaystyle\mu^{{\rm ext},\alpha}$
$\displaystyle=-u^{\alpha}~{}.$ (109)

Imposing that the total free energy $F_{0}+U$ is minimised at equilibrium also
results in the equilibrium relations for the tangential part of the molecular
field and for the chemical potential:

$\displaystyle H^{ij}=$ $\displaystyle H^{{\rm ext},ij}~{},$ (110)
$\displaystyle\mu^{\alpha}=$ $\displaystyle\mu^{{\rm ext},\alpha}~{}.$ (111)

Using these expressions for the equilibrium tensors, the Gibbs-Duhem relations
VI.2 and VI.2, obtained by invariance of translations and rotations, can be
rewritten in the simpler form, using Eqs. G.2.2 and G.2.2:

$\displaystyle\nabla_{i}t_{e}^{ij}+C_{i}{}^{j}t_{e,n}^{i}+f_{e}^{{\rm
ext},j}=$ $\displaystyle(H^{{\rm ext}}_{ik}-H_{ik})\nabla^{j}Q^{ik}$
$\displaystyle+c^{\alpha}\partial^{j}(\mu^{{\rm
ext},\alpha}-\mu^{\alpha})~{},$ (112)
$\displaystyle\nabla_{i}m_{n,e}^{i}-C_{ij}m_{e}^{ij}+\epsilon_{ij}t_{e}^{ij}+\Gamma_{e,n}^{\rm
ext}=$ $\displaystyle 2(H^{{\rm ext},ij}-H^{ij})Q_{ik}\epsilon_{j}{}^{k}~{}.$
(113)

At equilibrium, the conditions 111 and 110 imply that the right-hand side of
these two equations cancel; such that Eqs. VI.3 and 113 are then equivalent to
the tangential force balance and normal torque balance.

###  VI.4 Entropy production

For a nematic surface, the rate of change of the total free energy $F_{\rm
tot}=F+U$ reads:

$\displaystyle\dot{F}_{\rm tot}=$
$\displaystyle\int_{\mathcal{S}}dS\,\Big{\\{}-\left(\overline{t}^{ij}-\overline{t}^{ij}_{e}\right)v_{ij}-(\overline{m}^{ij}-\overline{m}^{ij}_{e})D_{t}C_{ij}$
$\displaystyle-(m_{n}^{i}-m_{n,e}^{i})\omega_{in}-(\mathbf{H}-\mathbf{H}^{{\rm
ext}}):D_{t}\mathbf{Q}$ $\displaystyle+(\mu^{\alpha}-\mu^{{\rm
ext},\alpha})r^{\alpha}+j^{\alpha,i}\partial_{i}(\overline{\mu}^{\alpha}-\overline{\mu}^{{\rm
ext},\alpha})$
$\displaystyle+(\mathbf{f}^{\mathrm{ext}}-\mathbf{f}_{e}^{\mathrm{ext}})\cdot\mathbf{v}+(\boldsymbol{\Gamma}^{\mathrm{ext}}-\boldsymbol{\Gamma}_{e}^{\mathrm{ext}})\cdot\boldsymbol{\omega}\Big{\\}}$
$\displaystyle+\oint_{\mathcal{C}}dl\nu_{i}\left[\mathbf{t}^{i}\cdot\mathbf{v}+\mathbf{m}^{i}\cdot\boldsymbol{\omega}+\boldsymbol{\Pi}^{i}:D_{t}\mathbf{Q}\right.$
$\displaystyle\left.\hskip
42.67912pt-(\overline{\mu}^{\alpha}-\overline{\mu}^{\rm
ext,\alpha})j^{\alpha,i}\right]\,.$ (114)

As for polar surface, we consider a single chemical reaction between a fuel
and its product, such that we rewrite the term $(\mu^{\alpha}-\mu^{{\rm
ext},\alpha})r^{\alpha}=-r\Delta\mu$ with $r$ the rate of fuel consumption and
$\Delta\mu$ the chemical potential of conversion of fuel to product. Using Eq.
265, we then identify the entropy production rate introduced in Eq. 66:

$\displaystyle T\theta=$
$\displaystyle\left(\overline{t}^{ij}-\overline{t}^{ij}_{e}\right)v_{ij}+(\overline{m}^{ij}-\overline{m}_{e}^{ij})D_{t}C_{ij}$
$\displaystyle+(m_{n}^{i}-m_{n,e}^{i})\omega_{in}+(H^{ij}-H^{{\rm
ext},ij})D_{t}Q_{ij}$
$\displaystyle+r\Delta\mu-j^{\alpha,i}\partial_{i}(\overline{\mu}^{\alpha}-\overline{\mu}^{{\rm
ext},\alpha})~{},$ (115)

and the normal and tangential fluxes of free energy:

$\displaystyle J_{n}^{f}=$
$\displaystyle(\mathbf{f}^{\mathrm{ext}}-\mathbf{f}^{\rm
ext}_{e})\cdot\mathbf{v}+(\boldsymbol{\Gamma}^{\mathrm{ext}}-\boldsymbol{\Gamma}_{e}^{\mathrm{ext}})\cdot\boldsymbol{\omega}$
(116) $\displaystyle j^{f,i}=$
$\displaystyle-\mathbf{t}^{i}\cdot\mathbf{v}-\mathbf{m}^{i}\cdot\boldsymbol{\omega}-\Pi^{ijk}D_{t}Q_{jk}$
$\displaystyle+(\overline{\mu}^{\alpha}-\overline{\mu}^{\rm
ext,\alpha})j^{\alpha,i}~{}.$ (117)

Here we have assumed that external forces and torques do not contribute to the
surface entropy production. In Table 9, we list the corresponding conjugate
fluxes and forces that can be read from the expression for the entropy
production rate, Eq. VI.4. There we introduce the deviatoric molecular field,
the tangent tensor $H_{d}^{ij}=H^{ij}-H^{{\rm ext},ij}$, and the deviatoric
chemical potential, $\mu_{d}^{\alpha}=\mu^{\alpha}-\mu^{{\rm ext},\alpha}$.

Flux | Force
---|---
In-plane deviatoric tension tensor $\overline{t}^{ij}_{d}$ | In-plane shear tensor $v_{ij}$
In-plane deviatoric bending moment tensor $\overline{m}^{ij}_{d}$ | Bending rate tensor $D_{t}C_{ij}$
Normal deviatoric moment $m_{n,\,d}^{i}$ | Vorticity gradient $\omega_{in}=(\partial_{i}\boldsymbol{\omega})\cdot\mathbf{n}$
Convected, co-rotational derivative of nematic tensor $D_{t}Q_{ij}$ | Deviatoric molecular field $H_{d,ij}=H_{ij}-H^{\rm ext}_{ij}$
Rate of fuel consumption $r$ | Fuel hydrolysis chemical potential $\Delta\mu$
Flux of species $\alpha=2...N$, $j^{\alpha,i}$ |  Deviatoric gradient of relative chemical potential of species $\alpha=2...N$, $-\partial_{i}\overline{\mu}_{d}^{\alpha}=-\partial_{i}(\overline{\mu}^{\alpha}-\overline{\mu}^{{\rm ext},\alpha})$
Table 9:  List of conjugate forces and fluxes for an active fluid nematic
surface.

###  VI.5 Constitutive equations

We now obtain constitutive equations for different types of nematic surfaces,
as illustrated in Fig. 3.

####  VI.5.1 Nematic surfaces

The deviatoric part of the stress and moment tensor can be decomposed as in
Eq. 70. In writing down these relations, the nematic couplings already present
in Eqs. 71-75 carry over here, with $Q_{ij}$ replacing $q_{ij}$. Whenever
possible we give the same name to phenomenological coefficients which resemble
those already introduced for a polar surface. As for polar surfaces, for the
sake of simplicity we make a number of restrictions on couplings: we do not
include couplings between the various mechanical tensors and the deviatoric
molecular field $H^{ij}_{d}$ that depend on the curvature tensor. Couplings
are included to linear order in the curvature tensor $\mathbf{C}$ and in the
nematic tensor $\mathbf{Q}$, except for diagonal coupling terms whose
dependency in $\mathbf{C}$ and $\mathbf{Q}$ is not written. Viscous coupling
terms between $\bar{t}^{ij}_{d}$, $\bar{m}^{ij}$, $m_{n,d}^{i}$ and $v_{ij}$,
$D_{t}C_{ij}$, $\omega_{in}$ that are dependent on the nematic and curvature
tensors are not listed. We do not include cross-coupling terms in the gradient
of the chemical potential, $\partial_{i}\bar{\mu}_{d}^{\alpha}$ or the flux
$j^{\alpha,i}$, except for active couplings with the fuel hydrolysis chemical
potential, $\Delta\mu$.

Below we only list terms which involve the nematic tensor $Q^{ij}$ and the
deviatoric molecular field $H^{ij}_{d}$; the complete constitutive equations
are given in Appendix I. The contributions to the tension tensor which depend
on the nematic tensor $\mathbf{Q}$ read:

$\displaystyle\overline{t}^{ij}_{0}=$ $\displaystyle\nu
H_{d}^{ij}+\nu^{\prime}Q_{kl}H_{d}^{kl}g^{ij}+\zeta_{\rm n}\Delta\mu Q^{ij}$
$\displaystyle\overline{t}^{ij}_{\rm{UD}}=$ $\displaystyle\tilde{\zeta}_{\rm
n}\Delta\mu Q^{kl}C_{kl}g^{ij}+\tilde{\zeta}^{\prime}_{\rm n}\Delta\mu
Q^{ij}C_{k}{}^{k}$ $\displaystyle\overline{t}^{ij}_{\rm{C}}=$
$\displaystyle\zeta_{\rm{Cn}}\Delta\mu(\epsilon^{ik}Q_{k}{}^{l}C_{l}{}^{j}+\epsilon^{jk}Q_{k}{}^{l}C_{l}{}^{i})$
$\displaystyle+\tilde{\zeta}_{\rm{Cn}}\Delta\mu(\epsilon^{ik}C_{k}{}^{l}Q_{l}{}^{j}+\epsilon^{jk}C_{k}{}^{l}Q_{l}{}^{i})$
$\displaystyle\overline{t}^{ij}_{\rm{PC}}=$ $\displaystyle\nu_{\rm
PC}\epsilon^{i}{}_{k}H_{d}^{kj}+\nu_{\rm
PC}^{\prime}\epsilon^{kl}Q_{km}H_{d}^{ml}g^{ij}$ $\displaystyle+\zeta_{\rm
PC{\rm n}}\Delta\mu\epsilon^{ik}Q_{k}{}^{j}~{},$ (118)

where we have used that
$[Q_{ik}H_{d}^{k}{}_{j}]_{s}=\frac{1}{2}Q_{kl}H_{d}^{kl}g_{ij}$, that
$[\boldsymbol{\epsilon}\mathbf{Q}\mathbf{H}_{d}]_{s}=[\mathbf{Q}\mathbf{H}_{d}\boldsymbol{\epsilon}]_{s}=[\mathbf{H}_{d}\boldsymbol{\epsilon}\mathbf{Q}]_{s}=-[\boldsymbol{\epsilon}\mathbf{H}_{d}\mathbf{Q}]_{s}=-[\mathbf{H}_{d}\mathbf{Q}\boldsymbol{\epsilon}]_{s}=-[\mathbf{Q}\boldsymbol{\epsilon}\mathbf{H}_{d}]_{s}$
and that
$[\epsilon^{ik}Q_{d,kl}H_{d}^{lj}]_{s}=\frac{1}{2}\epsilon^{kl}Q_{d}^{lm}H_{d,mk}g^{ij}$,
which follows from $\mathbf{H}_{d}$ and $\mathbf{Q}$ being tangent, symmetric
and traceless (Eqs. 202-A), to avoid redundant couplings. We also note that
the tensor $\epsilon^{ik}Q_{k}{}^{j}$ is symmetric. We find only two
independent active terms in the contribution $\bar{t}_{\rm C}^{ij}$
proportional to $[\boldsymbol{\epsilon}\mathbf{Q}\mathbf{C}]_{s}$ and
$[\boldsymbol{\epsilon}\mathbf{C}\mathbf{Q}]_{s}$; following the same
arguments as given for the polar case after Eq. 71.

The contributions to the moment tensor, in the decomposition indicated in Eq.
70, which depend on the nematic tensor, read

$\displaystyle\overline{m}^{ij}_{0}=$ $\displaystyle\tilde{\zeta}_{c{\rm
n}}\Delta\mu Q^{kl}C_{kl}g^{ij}+\tilde{\zeta}^{\prime}_{c\rm n}\Delta\mu
Q^{ij}C_{k}{}^{k}$ $\displaystyle\overline{m}^{ij}_{\rm{UD}}=$
$\displaystyle\beta
H_{d}^{ij}+\beta^{\prime}Q_{kl}H_{d}^{kl}g^{ij}+\zeta_{c{\rm n}}\Delta\mu
Q^{ij}$ $\displaystyle\overline{m}^{ij}_{\rm{C}}=$ $\displaystyle\beta_{\rm
C}\epsilon^{i}{}_{k}H_{d}^{kj}+\beta_{\rm
C}^{\prime}\epsilon^{kl}Q_{km}H_{d}^{ml}g^{ij}+\zeta_{c{\rm C}{\rm
n}}\Delta\mu\epsilon^{ik}Q_{k}{}^{j}$
$\displaystyle\overline{m}^{ij}_{\rm{PC}}=$
$\displaystyle\zeta_{c\rm{PCn}}\Delta\mu(\epsilon^{ik}Q_{k}{}^{l}C_{l}{}^{j}+\epsilon^{jk}Q_{k}{}^{l}C_{l}{}^{i})$
$\displaystyle+\tilde{\zeta}_{c\rm{PCn}}\Delta\mu(\epsilon^{ik}C_{k}{}^{l}Q_{l}{}^{j}+\epsilon^{jk}C_{k}{}^{l}Q_{l}{}^{i})~{}.$
(119)

In Eq. 119, we have only introduced symmetric contributions to the bending
moment tensor. With our simplifying assumptions, there is no contribution to
the normal moment $\mathbf{m}_{n}$, and to the flux of component $\alpha$,
$\mathbf{j}^{\alpha}$, involving the nematic order parameter $\mathbf{Q}$ or
the molecular field $\mathbf{H}$.

The co-moving, co-rotating derivative of the nematic order parameter field
reads:

$\displaystyle D_{t}Q^{ij}=\mathcal{Q}^{ij}_{0}+\epsilon_{\rm
UD}\mathcal{Q}^{ij}_{\rm UD}+\epsilon_{\rm C}\mathcal{Q}^{ij}_{\rm
C}+\epsilon_{\rm PC}\mathcal{Q}^{ij}_{\rm PC}$ (120)

with

$\displaystyle\mathcal{Q}_{0}^{ij}=$
$\displaystyle\frac{1}{\gamma}H_{d}^{ij}-\nu\tilde{v}^{ij}-\nu^{\prime}v_{k}{}^{k}Q^{ij}+\lambda\Delta\mu\,Q^{ij}$
$\displaystyle\mathcal{Q}_{\rm UD}^{ij}=$ $\displaystyle-\beta
D_{t}\tilde{C}^{ij}-\beta^{\prime}D_{t}C_{k}{}^{k}Q^{ij}$
$\displaystyle+\lambda_{\rm UD}\Delta\mu\,\tilde{C}^{ij}+\lambda_{\rm
UD}^{\prime}\Delta\mu C_{k}{}^{k}Q^{ij}$ $\displaystyle\mathcal{Q}_{\rm
C}^{ij}=$ $\displaystyle\beta_{\rm
C}\epsilon^{i}{}_{k}D_{t}\tilde{C}^{kj}+\beta_{\rm
C}^{\prime}\epsilon^{i}{}_{k}Q^{kj}D_{t}C_{l}{}^{l}$
$\displaystyle+\lambda_{\rm
C}\Delta\mu(\epsilon^{i}{}_{k}C^{kj}+\epsilon^{j}{}_{k}C^{ki})+\lambda_{\rm{Cn}}\Delta\mu
C_{k}{}^{k}\epsilon^{ik}Q_{k}{}^{j}$ $\displaystyle\mathcal{Q}_{\rm PC}^{ij}=$
$\displaystyle\frac{1}{\gamma_{\rm PC}}\epsilon^{i}{}_{k}H_{d}^{kj}+\nu_{\rm
PC}\epsilon^{ik}\tilde{v}_{k}{}^{j}$ $\displaystyle+\nu^{\prime}_{{\rm
PC}}\epsilon^{i}{}_{k}Q^{kj}v_{l}{}^{l}+\lambda_{\rm
PC}\Delta\mu\epsilon^{ik}Q_{k}{}^{j}~{}.$ (121)

where we have used that
$[C^{i}{}_{k}Q^{kj}]_{s}-\frac{1}{2}C_{kl}Q^{kl}g^{ij}=\frac{1}{2}C_{k}{}^{k}Q^{ij}$,
following Eq. 197, to avoid redundant terms. We have also used the identities
73-74 as well as the tensor identities
$\mathbf{Q}\boldsymbol{\epsilon}\mathbf{C}=-\boldsymbol{\epsilon}\mathbf{Q}\mathbf{C}$,
$\mathbf{C}\boldsymbol{\epsilon}\mathbf{Q}=-\mathbf{C}\mathbf{Q}\boldsymbol{\epsilon}$,
$[\mathbf{C}\mathbf{Q}\boldsymbol{\epsilon}]_{s}=-[\boldsymbol{\epsilon}\mathbf{Q}\mathbf{C}]_{s}$
and
$[\mathbf{Q}\mathbf{C}\boldsymbol{\epsilon}]_{s}=-[\boldsymbol{\epsilon}\mathbf{C}\mathbf{Q}]_{s}$,
and that $D_{t}Q^{ij}$ is traceless, to obtain the only term proportional to
$\boldsymbol{\epsilon}$, $\mathbf{C}$ and $\mathbf{Q}$. With a similar
reasoning we obtained one term proportional to $\boldsymbol{\epsilon}$,
$v^{ij}$ and $\mathbf{Q}$ and one term proportional to
$\boldsymbol{\epsilon}$, $D_{t}C^{ij}$ and $\mathbf{Q}$. As for the polar
case, the odd inverse rotational viscosity $1/\gamma_{\rm PC}$ can be nonzero
in systems close to equilibrium only if time-reversal symmetry is broken, for
example in the presence of a magnetic field. If time-reversal symmetry holds,
Onsager symmetry require this odd inverse rotational viscosity to vanish.

Finally, the rate of fuel consumption has the following contributions which
depend on the nematic tensor:

$\displaystyle r_{0}=$ $\displaystyle-\zeta_{\rm
n}v^{ij}Q_{ij}-(\tilde{\zeta}_{c\rm
n}Q^{kl}C_{kl}g^{ij}+\tilde{\zeta}^{\prime}_{c\rm
n}C_{k}{}^{k}Q^{ij})D_{t}C_{ij}$ $\displaystyle+\lambda Q_{ij}H_{d}^{ij}$
(122) $\displaystyle r_{\rm UD}=$
$\displaystyle-(\tilde{\zeta}_{n}Q^{kl}C_{kl}g^{ij}+\tilde{\zeta}_{n}^{\prime}Q^{ij}C_{k}{}^{k})v_{ij}-\zeta_{c\rm
n}Q^{ij}D_{t}C_{ij}$ $\displaystyle+(\lambda_{\rm
UD}\tilde{C}^{ij}+\lambda^{\prime}_{\rm UD}C_{k}{}^{k}Q^{ij}){H_{d}}_{ij}$
(123) $\displaystyle r_{\rm C}=$ $\displaystyle-(2\zeta_{\rm
Cn}\epsilon^{ik}Q_{kl}C_{l}{}^{j}+2\tilde{\zeta}_{\rm
Cn}\epsilon^{ik}C_{k}{}^{l}Q_{l}{}^{j})v_{ij}$ $\displaystyle-\zeta_{c\rm
Cn}\epsilon^{ik}Q_{k}{}^{j}D_{t}C_{ij}+2\lambda_{\rm
C}\epsilon_{jk}C^{k}{}_{i}{H_{d}}^{ij}$
$\displaystyle+\lambda_{\rm{Cn}}C_{k}{}^{k}\epsilon^{ik}Q_{k}{}^{j}{H_{d}}_{ij}$
$\displaystyle r_{\rm PC}=$ $\displaystyle-\zeta_{\rm
PCn}\epsilon^{ik}Q_{k}{}^{j}v_{ij}+\lambda_{\rm
PC}\epsilon_{ik}Q^{k}{}_{j}H_{d}^{ij}$ $\displaystyle-(2\zeta_{\rm
cPCn}\epsilon^{ik}Q_{k}{}^{l}C_{l}{}^{j}+2\tilde{\zeta}_{\rm
cPCn}\epsilon^{ik}C_{k}{}^{l}Q_{l}{}^{j})D_{t}C_{ij}~{}.$ (124)

In the equations above, coefficients which contain the letters $\nu$, $\beta$
or $\zeta$ are all reactive. Coefficients containing the letters $\gamma$,
$\lambda$ are dissipative.

####  VI.5.2 Pseudonematic surfaces

Here we consider the two types of pseudonematic surfaces and assume that their
order is described by, respectively, the pseudotensor $\mathbf{Q}_{\rm UD}$
and the combination of the pseudoscalar and pseudovector $\\{\epsilon_{\rm
PC},\mathbf{Q}_{\rm UD}\\}$. With this restriction, the contributions to the
constitutive equations for the pseudonematic surfaces which depend on the
pseudonematic tensor read:

$\displaystyle\overline{t}^{ij}_{d}=$ $\displaystyle\nu^{\prime}Q_{{\rm
UD},kl}H_{{\rm UD}d}^{kl}g^{ij}+\tilde{\zeta}_{\rm n}\Delta\mu Q_{\rm
UD}^{kl}C_{kl}g^{ij}$ $\displaystyle+\tilde{\zeta}^{\prime}_{\rm n}\Delta\mu
Q_{\rm UD}^{ij}C_{k}{}^{k}+\epsilon_{\rm PC}\left[\nu_{\rm
PC}^{\prime}\epsilon^{kl}Q_{{\rm UD},km}H_{{\rm UD}d}^{ml}g^{ij}\right.$
$\displaystyle\left.+\zeta_{\rm{Cn}}\Delta\mu(\epsilon^{i}{}_{k}Q_{\rm
UD}^{kl}C_{l}{}^{j}+\epsilon^{j}{}_{k}Q_{\rm UD}^{kl}C_{l}{}^{i})\right.$
$\displaystyle\left.+\tilde{\zeta}_{\rm{Cn}}\Delta\mu(\epsilon^{ik}C_{kl}Q_{\rm
UD}^{lj}+\epsilon^{jk}C_{kl}Q_{\rm UD}^{li})\right]$ (125)
$\displaystyle\overline{m}^{ij}_{d}=$ $\displaystyle\beta H_{{\rm
UD}d}^{ij}+\zeta_{c{\rm n}}\Delta\mu Q_{\rm UD}^{ij}$
$\displaystyle+\epsilon_{\rm PC}\left[\beta_{\rm C}\epsilon^{i}{}_{k}H_{{\rm
UD}d}^{kj}+\zeta_{c{\rm C}{\rm n}}\Delta\mu\epsilon^{i}{}_{k}Q_{\rm
UD}^{kj}\right]$ (126) $\displaystyle D_{t}Q^{ij}_{\rm UD}=$
$\displaystyle\frac{1}{\gamma}H_{{\rm UD}d}^{ij}+\lambda\Delta\mu Q_{\rm
UD}^{ij}+\lambda_{\rm UD}\Delta\mu\,\tilde{C}^{ij}$
$\displaystyle-\nu^{\prime}v_{k}{}^{k}Q_{\rm UD}^{ij}-\beta
D_{t}\tilde{C}^{ij}$ $\displaystyle+\epsilon_{\rm
PC}\left[\frac{1}{\gamma_{\rm PC}}\epsilon^{i}{}_{k}H_{{\rm
UD}d}^{kj}+\nu^{\prime}_{{\rm PC}}\epsilon^{i}{}_{k}Q_{\rm
UD}^{kj}v_{l}{}^{l}\right.$ $\displaystyle\left.+\frac{1}{2}\beta_{\rm
C}\,\left(\epsilon^{i}{}_{k}D_{t}C^{kj}+\epsilon^{j}{}_{k}D_{t}C^{ki}\right)\right.$
$\displaystyle\left.+\lambda_{\rm
C}\Delta\mu(\epsilon^{i}{}_{k}C^{kj}+\epsilon^{j}{}_{k}C^{ki})+\lambda_{\rm
PC}\Delta\mu\epsilon^{i}{}_{k}Q_{\rm UD}^{kj}\right]$ (127) $\displaystyle r=$
$\displaystyle\lambda Q_{\rm UD}^{ij}H_{{\rm UD}d,ij}+\lambda_{\rm
UD}\tilde{C}_{ij}H_{{\rm UD}d}^{ij}-\zeta_{c\rm n}Q_{\rm UD}^{ij}D_{t}C_{ij}$
$\displaystyle-(\tilde{\zeta}_{\rm n}Q_{\rm
UD}^{kl}C_{kl}g^{ij}+\tilde{\zeta}_{\rm n}^{\prime}Q_{\rm
UD}^{ij}C_{k}{}^{k})v_{ij}$ $\displaystyle+\epsilon_{\rm
PC}\left[-(2\zeta_{\rm Cn}\epsilon^{i}{}_{k}Q_{{\rm
UD}}^{kl}C_{l}{}^{j}+2\tilde{\zeta}_{\rm Cn}\epsilon^{ik}C_{kl}Q_{\rm
UD}^{lj})v_{ij}\right.$ $\displaystyle\left.-\zeta_{\rm
cCn}\epsilon^{i}{}_{k}Q_{\rm UD}^{kj}D_{t}C_{ij}+2\lambda_{\rm
C}\epsilon^{j}{}_{k}C^{ki}H_{{\rm UD}d,ij}\right.$
$\displaystyle\left.+\lambda_{\rm PC}\epsilon_{ik}{Q_{\rm
UD}}^{k}{}_{j}H_{{\rm UD}d}^{ij}\right]~{}.$ (128)

##  VII Active polar film in a confined geometry

Figure 4: Schematic for a confined polar active film, with planar anchoring at
boundaries. A. Flat planar-chiral membrane. B. Deformed active polar surface
with broken up-down symmetry.

We now discuss a one-component polar active film confined along its $x$ axis
in between $x=0$ and $x=L$, invariant by translation in the $y$ direction
(Fig. 4). We consider the limit of low Reynolds number where inertial terms
can be neglected. For a flat, polar, achiral surface, and planar anchoring of
the polarity at the wall, an instability arises at a threshold value of
nematic active stress, leading to a distorted polarity profile and spontaneous
flow in the active film voituriez2005spontaneous . Here we consider a similar
geometry and explore a more general situation. We first investigate the case
of a planar-chiral flat surface, and then consider an active film which can
deform weakly in the third dimension and has broken up-down symmetry (Fig. 4).

We describe a polar active surface with a tangent polar vector $\mathbf{p}$.
We assume that the equilibrium behaviour of the surface is described by the
free energy

$\displaystyle
F=\int_{\mathcal{S}}dS\left[\gamma+\frac{\kappa}{2}(C_{i}{}^{i})^{2}+\frac{K}{2}(\partial_{i}\mathbf{p})^{2}+\frac{\lambda_{p}}{2}(\mathbf{p}^{2}-1)\right]~{},$
(129)

where $\lambda_{p}$ is a Lagrange multiplier enforcing the norm of
$\mathbf{p}$, $\gamma$ is the passive surface tension, $\kappa>0$ the surface
bending modulus. $K>0$ is the polarity distortion modulus in the single-
constant approximation, and we denote
$(\partial_{i}\mathbf{p})^{2}=(\partial_{i}\mathbf{p})\cdot(\partial^{i}\mathbf{p})$.
Here we assume that the passive surface has no spontaneous curvature. We note
that the polarity distortion term in
$(\partial_{i}\mathbf{p})^{2}=(\nabla_{i}p_{j})(\nabla^{i}p^{j})+p_{i}C^{ij}p^{k}C_{kj}$
introduces a coupling between polarity and curvature. For simplicity we did
not introduce additional coupling terms between the curvature tensor and
polarity in Eq. 129. The surface is assumed to be free from external force and
is not subjected to an external potential.

Differentation of the free energy density
$f_{0}=\gamma+\frac{\kappa}{2}(C_{i}{}^{i})^{2}+K(\partial_{i}\mathbf{p})^{2}/2+\frac{\lambda_{p}}{2}(\mathbf{p}^{2}-1)$
with respect to $\mathbf{p}$ and the curvature tensor results in the following
expressions for thermodynamic fields conjugate to $\mathbf{C}$ and
$\mathbf{p}$, as defined in Eq. 35:

$\displaystyle K^{ij}=$ $\displaystyle\kappa C_{k}{}^{k}g^{ij}~{},$ (130)
$\displaystyle\mathbf{h}_{0}=$ $\displaystyle-\lambda_{p}\mathbf{p}~{},$ (131)
$\displaystyle\boldsymbol{\pi}=$ $\displaystyle
K\mathbf{e}^{i}\otimes\partial_{i}\mathbf{p}~{},$ (132)

and following Eq. 38, the total molecular field is then given by

$\displaystyle\mathbf{h}=-\lambda_{p}\mathbf{p}+K\nabla_{i}(\partial^{i}\mathbf{p})~{}.$
(133)

The corresponding equilibrium tensors are (Eqs. 47, 49):

$\displaystyle t_{e}^{ij}=$
$\displaystyle\left[\gamma+\frac{\kappa}{2}(C_{k}{}^{k})^{2}+\frac{K}{2}(\partial_{i}\mathbf{p})^{2}\right]g^{ij}-\kappa
C_{k}{}^{k}C^{ij}$
$\displaystyle-K(\partial^{i}\mathbf{p})\cdot(\partial^{j}\mathbf{p})~{},$
(134) $\displaystyle\mathbf{m}_{e}^{i}=$ $\displaystyle\kappa
C_{k}{}^{k}\epsilon^{ij}\mathbf{e}_{j}+K\mathbf{p}\times\partial^{i}\mathbf{p}~{}.$
(135)

Deviatoric, non equilibrium contributions to the tension and moment tensors,
and the polarity dynamic equation, are determined from the general
constitutive equations 332-335 with the following simplifications: we set
$\eta_{\rm PC}=\eta_{c{\rm PC}}=1/\gamma_{\rm PC}=0$ and we do not include
cross-coupling viscosities; we also do not include non-equilibrium
contributions to $m_{n}^{i}$ nor coupling to the vorticity gradient
$\omega_{in}$. We also consider the surface to be incompressible,
corresponding to the limit $\eta_{b}\rightarrow\infty$ and $v_{k}{}^{k}=0$. In
the discussion below, we assume that $\Delta\mu>0$.

###  VII.1 Flat planar-chiral membrane

Figure 5:  Phase diagram for a quasi one-dimensional, confined, flat polar
active surface with broken planar-chiral symmetry. A. Depending on the active
tension parameters $\zeta_{{\rm PCn}}$ and $\zeta_{\rm n}$, a principal axis
of the active tension tensor deviates by an angle $-\theta_{p}/2$ from the
polarity axis, for a surface with broken planar-chiral symmetry. Here the
active tension is represented in the contractile case, $\zeta_{\rm n}>0$. B.
The parameter $\lambda_{\rm PC}$ triggers an active rotation of the polar
vector. C, D: Phase diagram of homogeneous polarity orientation as a function
of the two planar-chirality coefficients, for the contractile and extensile
case, for $\nu=\nu_{\rm{PC}}=0$. Outside of the “hourglass” shape, the
polarity is rotating with a time-varying angular velocity. E. Bifurcation plot
showing the angle at the center of the confined active film as a function of
system size $L$ ($\lambda_{\rm PC}=0$, $\zeta_{\rm n}>0$, $\nu=\nu_{\rm
PC}=0$, $\ell=\sqrt{K(1+4\eta/\gamma)/(\zeta_{\rm n}\Delta\mu)}$). For small
values of $L$ and $\zeta_{{\rm PCn}}=0$, the polar angle is homogeneous and
equal to $\frac{\pi}{2}$; for larger values of $L$ a pitchfork bifurcation
appears. The pitchfork bifurcation becomes imperfect for $\zeta_{{\rm
PCn}}\neq 0$. F. Bifurcation plot as a function of system size $L$, for
different values of $\lambda_{\rm PC}$ ($\zeta_{{\rm PCn}}=0$, $\zeta_{\rm
n}>0$, $\nu=\nu_{\rm PC}=0$, $\ell=\sqrt{K(1+4\eta/\gamma)/(\zeta_{\rm
n}\Delta\mu)}$). For large enough values of $\lambda_{\rm PC}$, the magnitude
of $\theta(x=\frac{L}{2})$ keeps increasing as $L$ increases, corresponding to
the formation of repeated polarity turns in the system. (G, H). Steady-state
polarity profiles for $\nu=\nu_{\rm PC}=\zeta_{{\rm PCn}}=0$, $\zeta_{\rm
n}>0$, $L/\ell=5$, $4\eta\lambda_{\rm PC}/\zeta_{\rm n}=3$ or as indicated.

We first consider a flat surface, such that $C_{ij}=0$, confined at its
boundaries, such that $v_{x}(x=0)=v_{x}(x=L)=0$, and we also assume a
vanishing shear stress at the boundaries, resulting in $t_{xy}=0$ (Appendix
K.1). The constitutive equations for the tension and the polarity dynamics
are:

$\displaystyle t^{ij}=$ $\displaystyle-
Pg^{ij}+2\eta\tilde{v}^{ij}+\frac{\nu}{2}(p^{i}h^{j}+p^{j}h^{i}-p^{k}h_{k}g^{ij})$
$\displaystyle+\frac{\nu_{{\rm
PC}}}{2}(\epsilon^{i}{}_{k}h^{k}p^{j}+\epsilon^{j}{}_{k}h^{k}p^{i}-\epsilon_{kl}h^{k}p^{l}g^{ij})$
$\displaystyle-\frac{1}{2}\epsilon_{kl}p^{k}h^{l}\epsilon^{ij}+\zeta_{\rm
n}\Delta\mu q^{ij}+\zeta_{\rm PC{\rm n}}\Delta\mu\epsilon^{ik}q_{k}{}^{j}$
$\displaystyle-K((\nabla^{i}p_{k})(\nabla^{j}p^{k})-\frac{1}{2}(\nabla_{l}p_{k})(\nabla^{l}p^{k})g^{ij})~{},$
(136) $\displaystyle D_{t}p^{i}$
$\displaystyle=\frac{1}{\gamma}h^{i}-\nu\tilde{v}^{ij}p_{j}+\nu_{\rm{PC}}\epsilon^{i}{}_{k}\tilde{v}^{kj}p_{j}+\lambda_{\rm{PC}}\Delta\mu\epsilon^{ij}p_{j}\,,$
(137)

where we have introduced a two dimensional pressure $P$ which enforces the
incompressibility condition. We take here $\lambda=0$, as the corresponding
term acts in the polarity equation to change the norm of the polarity, which
is constrained here. In general this term also renormalizes tensions by
modifying the molecular field $\mathbf{h}$. In the equations above, $\gamma$
is an inverse rotational viscosity, $\nu$ is a flow-coupling alignment whose
role have been discussed extensively degennesprost ; voituriez2005spontaneous
. Constitutive equations related to Eqs. 136 and 137 have been considered in
Refs. duclos2018spontaneous ; maitra2019spontaneous ; li2020pattern . Ref.
maitra2019spontaneous  pointed out that solutions with uniformly rotating
polarity can emerge due to the coupling term in $\lambda_{\rm{PC}}$.

Since we consider $|\mathbf{p}|=1$, the polarity vector can be denoted
$\mathbf{p}=\cos\theta\mathbf{e}_{x}+\sin\theta\mathbf{e}_{y}$. Starting from
Eqs. 136\- 137, we obtain a dynamic equation for the angle $\theta$ (Appendix
K.1):

$\displaystyle\partial_{t}\theta=$ $\displaystyle
K\left(\frac{1}{\gamma}+\frac{(1-\nu\cos 2\theta-\nu_{\rm PC}\sin
2\theta)^{2}}{4\eta+\gamma(\nu\sin 2\theta-\nu_{\rm{PC}}\cos
2\theta)^{2}}\right)\partial_{x}^{2}\theta$
$\displaystyle-\lambda_{\rm{PC}}\Delta\mu$ $\displaystyle-\frac{(1-\nu\cos
2\theta-\nu_{\rm PC}\sin 2\theta)}{4\eta+\gamma(\nu\sin
2\theta-\nu_{\rm{PC}}\cos 2\theta)^{2}}\times$ $\displaystyle(\zeta_{\rm
n}\Delta\mu\sin 2\theta-\zeta_{{\rm PCn}}\Delta\mu\cos 2\theta)~{}.$ (138)

In this expression the angle $\theta_{p}$, which depends on active tension
parameters, and is defined by

$\displaystyle\tan\theta_{p}=\frac{\zeta_{{\rm PCn}}}{\zeta_{\rm
n}},~{}-\frac{\pi}{2}<\theta_{p}<\frac{\pi}{2}~{},$ (139)

plays a special role. Indeed the active tension contribution to Eq. 136
(contribution proportional to $\Delta\mu$) can be rewritten:

$\displaystyle t^{ij}_{a}=\frac{\zeta_{\rm
n}\Delta\mu}{2\cos\theta_{p}}\left(\begin{array}[]{cc}\cos(2\theta-\theta_{p})&\sin(2\theta-\theta_{p})\\\
\sin(2\theta-\theta_{p})&-\cos(2\theta-\theta_{p})\end{array}\right)~{},$
(142)

such that $-\theta_{p}/2$ corresponds to the angle, introduced by the planar-
chiral order parameter, between one of the principal directions of the active
tension tensor and the polarity vector direction (Fig. 5A). In addition, in
Eq. VII.1, the active coupling $\lambda_{\rm PC}$ induces a rotation of the
polar order parameter on the surface (Fig. 5B).

We now assume that the flow-aligning parameters $\nu$ and $\nu_{\rm PC}$ can
be neglected. We first consider the situation where the polarity field can be
considered uniform (limit of $L\rightarrow\infty$). Steady-states solution for
the polarity angle $\theta$ then exists for

$\displaystyle\left|\frac{\zeta_{{\rm PCn}}}{\zeta_{\rm
n}}\right|>\sqrt{\frac{16\eta^{2}\lambda_{\rm{PC}}^{2}}{\zeta_{\rm
n}^{2}}-1}~{},$ (143)

and the corresponding steady-state solutions are

$\displaystyle\theta_{s}^{1}=$
$\displaystyle\frac{1}{2}\left(\pi+\arcsin\left(\frac{4\eta\lambda_{\rm
PC}}{\zeta_{\rm n}}\cos\theta_{p}\right)+\theta_{p}\right)+k\pi~{},$ (144)
$\displaystyle\theta_{s}^{2}=$
$\displaystyle\frac{1}{2}\left(-\arcsin\left(\frac{4\eta\lambda_{\rm
PC}}{\zeta_{\rm n}}\cos\theta_{p}\right)+\theta_{p}\right)+k\pi~{},$ (145)

with $k$ an integer. A stability analysis (Appendix K.1) shows that
$\theta_{s}^{1}$ is stable for $\zeta_{\rm n}<0$ (extensile active stress),
while $\theta_{s}^{2}$ is stable for $\zeta_{\rm n}>0$ (contractile active
stress). For a simple surface, not planar-chiral, the coefficients
$\zeta_{{\rm PC}{\rm n}}=\lambda_{\rm PC}=0$ and the angle $\theta_{p}=0$: in
that case $\theta=0$, $\pi$ are the stable solutions for $\zeta_{\rm n}>0$ and
$\theta=\pi/2$, $3\pi/2$ are stable solutions for $\zeta_{\rm n}<0$. When
$\theta_{p}\neq 0$ or $\lambda_{\rm PC}\neq 0$, the steady-state angles
deviate from this solution as $|\zeta_{\rm PCn}|$ or $|\lambda_{\rm PC}|$ are
increased (Fig. 5C-D). For sufficiently large $|\lambda_{\rm PC}|$, above the
threshold defined by Eq. 143, the steady-state solutions are lost and the
polarity rotates with the average angular velocity (Fig. 5C):

$\displaystyle\omega=|\lambda_{\rm PC}|\Delta\mu\sqrt{1-\left(\frac{\zeta_{\rm
n}}{4\eta\lambda_{\rm PC}\cos\theta_{p}}\right)^{2}}~{},$ (146)

which approaches a constant velocity, $|\lambda_{\rm PC}|\Delta\mu$ for
$|\lambda_{\rm PC}|\rightarrow\infty$, and vanishes at the transition line
given by Eq. 143. The direction of rotation is determined by the sign of
$\lambda_{\rm PC}$.

For finite values of $L$ and planar anchoring
($\theta(x=0)=\theta(x=L)=\frac{\pi}{2}$), a competition arises between the
distortion term proportional to $K$ in Eq. VII.1, favoring uniform orientation
$\theta=\frac{\pi}{2}$, and other physical effects promoting a different
polarity orientation. For a simple surface, $\zeta_{{\rm PCn}}=\lambda_{\rm
PC}=0$, and for contractile stress $\zeta_{\rm n}>0$, this competition gives
rise to a critical length, below which the polarity orientation stays uniform
and oriented along the $y$ axis, and above which a spontaneous flow and
polarity distortion emerges voituriez2005spontaneous . The transition is a
pitchfork bifurcation which gives rise to two symmetric configurations, with
the polarity tilted to the left or the right. For small, but non-zero values
of $\zeta_{{\rm PCn}}$ and $\lambda_{\rm PC}$, the pitchfork bifurcation
becomes imperfect (Fig. 5E-F); as the planar-chiral terms break the symmetry
between the two polarity orientations away from $\theta=\frac{\pi}{2}$. For
sufficiently large $\lambda_{\rm PC}$, a new regime emerges where the polarity
transiently rotates and sets up a number of spatial polarity turns in the
system, until the distortion energy prevents the polarity from rotating
further and a high-distortion steady-state is reached (Fig. 5F-G). At steady-
state and for large $\lambda_{\rm PC}$, $n\sim L^{2}|\lambda_{\rm
PC}|\Delta\mu\eta\gamma/(2\pi K(\gamma+4\eta))$ spatial turns can form in the
confined system: as $\lambda_{\rm PC}$ or $L$ are increased, more and more
bands of full $2\pi$ polarity rotations emerge (Fig. 5G-H).

###  VII.2 Weakly deformed active polar surface with broken up-down symmetry

Figure 6:  Phase diagram for a quasi one-dimensional, confined, polar active
surface with broken up-down symmetry. A. The coefficient
$\zeta_{c\rm{n}}\Delta\mu$, an active bending moment, results in internal
torques and a preferred curvature direction, depending on the polarity axis
orientation. We assume here $\Delta\mu>0$. The schematic on the right
indicates the curvature induced in a free surface by the active bending moment
$\sim\zeta_{c{\rm n}}$, assuming that the surface deforms only along the $x$
direction. B. The coefficient $\lambda_{\rm UD}\Delta\mu$, with dimension of a
velocity, is an active response of the polarity field to curvature. The
schematic on the right indicates the stationary orientations induced by the
coupling $\sim\lambda_{\rm UD}$. C, D. Phase diagrams of possible stable
steady homogeneous states for $\zeta_{c}>0$ and $\zeta_{\rm n}=0$ (C), or
$\zeta_{\rm n}\Delta\mu\kappa/(\zeta_{c}\Delta\mu)^{2}=1$ (D). Green:
$\theta=\theta_{0}=\frac{1}{2}\arccos\left(-\frac{2\zeta_{c}}{\zeta_{c\rm
n}}+\frac{\zeta_{n}\kappa}{\eta\bar{\lambda}_{\rm UD}\Delta\mu\zeta_{c{\rm
n}}}\right)$ or $\theta=\pi-\theta_{0},\pi+\theta_{0},2\pi-\theta_{0}$; Brown
region: $\theta=0,\pi$; Blue region: $\theta=\pi/2,3\pi/2$; orange region:
$\theta=0,\pi/2,\pi,3\pi/2$. The dashed line indicates a change of curvature
sign of the solutions $\theta=0,\pi$, which does not correspond to a
bifurcation. One assumes here $\beta=\nu=0$.

We now consider an active polar surface which does not have a chiral or
planar-chiral broken symmetry but has broken up-down symmetry. We consider a
situation when the boundaries of the active surface at $x=0$ and $x=L$ are
forced to be straight, but the surface can otherwise move without external
forces at its boundaries (no tension, no torque). In that case $t_{xx}=0$,
$\bar{m}_{xx}=0$ and $t_{xy}=0$. We assume that active terms contributing
isotropic and anisotropic bending moduli ($\zeta^{\prime}_{c}$,
$\tilde{\zeta}_{c}$, $\tilde{\zeta}_{c\rm n}$, $\tilde{\zeta}^{\prime}_{c\rm
n}$) are vanishing. We also take $\beta^{\prime}=\lambda=\lambda_{\rm
UD}^{\prime}=0$, as these terms act in the polarity equation to change the
norm of the polarity which is constrained here, so that they only renormalize
couplings in the constitutive equations for tensions and torques by modifying
the molecular field $\mathbf{h}$. With these simplifications, the constitutive
equations for the tangential tension tensor, the tangential bending moment and
the polarity dynamics are (Eqs. 332-335, K-363, 134-135):

$\displaystyle t^{ij}=$ $\displaystyle-Pg^{ij}+2\eta\tilde{v}^{ij}+(\zeta_{\rm
n}+\tilde{\zeta}^{\prime}_{\rm n}C_{k}{}^{k})\Delta\mu
q^{ij}+2\tilde{\zeta}\Delta\mu\tilde{C}^{ij}$
$\displaystyle+\frac{\nu}{2}(p^{i}h^{j}+p^{j}h^{i}-p_{k}h^{k}g_{ij})$
$\displaystyle-\frac{1}{2}\left[\epsilon_{kl}p^{k}h^{l}-K\epsilon^{km}p_{k}p_{l}C^{nl}C_{nm}\right]\epsilon^{ij}$
$\displaystyle-K((\nabla^{i}p^{k})(\nabla^{j}p_{k})-\frac{1}{2}(\nabla^{l}p^{k})(\nabla_{l}p_{k})g^{ij})$
$\displaystyle-\frac{1}{2}(\bar{m}^{ik}C_{k}{}^{j}+\bar{m}^{jk}C_{k}{}^{i}-\bar{m}^{lk}C_{kl}g^{ij}$
$\displaystyle-\bar{m}^{lk}C_{lm}\epsilon_{k}{}^{m}\epsilon^{ij})$ (147)
$\displaystyle\overline{m}^{ij}=$ $\displaystyle(\kappa
C_{k}{}^{k}+\eta_{cb}D_{t}C_{k}{}^{k})g^{ij}+KC^{i}{}_{k}p^{k}p^{j}$
$\displaystyle+2\eta_{c}D_{t}\tilde{C}^{ij}+\zeta_{c}\Delta\mu
g^{ij}+\zeta_{c\rm n}\Delta\mu q^{ij}$
$\displaystyle+\frac{\beta}{2}(p^{i}h^{j}+p^{j}h^{i}-p^{k}h_{k}g^{ij})$ (148)
$\displaystyle D_{t}p^{i}=$
$\displaystyle\frac{1}{\gamma}h^{i}-\nu\tilde{v}^{ij}p_{j}+\lambda_{\rm
UD}\Delta\mu\,C^{ij}p_{j}-\beta p^{j}D_{t}\tilde{C}^{i}{}_{j}~{}.$ (149)

In the equations above, $\eta_{c}$ and $\eta_{cb}$ are shear and bulk bending
viscosities. The term in $\zeta_{c}$ corresponds to an active bending moment
which results in a spontaneous curvature of the film
$C_{k}{}^{k}=-\zeta_{c}\Delta\mu/\kappa$ in the absence of other effects.
Without loss of generality, we assume that the surface is oriented such that
$\zeta_{c}>0$. The term in $\zeta_{c{\rm n}}$ corresponds to an active