Mechanics of active surfaces

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by  Guillaume Salbreux

Mechanics of active surfaces
Guillaume Salbreux1,2 and Frank Ju¨licher1
1Max Planck Institute for the Physics of Complex Systems, No¨thnitzer Str. 38, 01187 Dresden, Germany
2The Francis Crick Institute, 1 Midland Road, NW1 1AT, United Kingdom
(Dated: February 1, 2017)
We derive a fully covariant theory of the mechanics of active surfaces. This theory provides a
frameworkforthestudyofactivebiologicalorchemicalprocessesatsurfaces,suchasthecellcortex,
themechanicsofepithelialtissues, orreconstitutedactivesystemsonsurfaces. Weintroduceforces
and torques acting on a surface, and derive the associated force balance conditions. We show
7
that surfaces with in-plane rotational symmetry can have broken up-down, chiral or planar-chiral
1
symmetry. We discuss the rate of entropy production in the surface and write linear constitutive
0
relations that satisfy the Onsager relations. We show that the bending modulus, the spontaneous
2
curvatureandthesurface tension ofapassivesurfacearerenormalised byactiveterms. Finally,we

n identify novel active terms which are not found in a passive theory and discuss examples of shape
a instabilities that are related to active processes in the surface.
J

1
3 Biologicalsystemsexhibitastunningvarietyofcomplexmorphologiesandshapes. Organismsformfromafertilized
egg in a dynamic process called morphogenesis. Such shape forming processes in biology involve active mechanical

] events during which surfaces undergo shape changes that are driven by active stresses and torques generated in the
h
material. Important examples are two-dimensional tissues, so called epithelia. They represent surfaces that can
p
- deform their shape as a result of active cellular processes [1]. Cells also exhibit a variety of different shapes and can
o undergoactiveshapechanges. Forexampleduringcelldivision,cellsrounduptoasphericalshapedue toanincrease
i
b of active surface tension [2]. Cell shapes are governedby the cell cortex, a thin layer of an active contractile material
. atthe surfaceofthe cell[3]. Epithelialtissues andthe cellsurface areexamplesofactivesurfaces. Inaddition, recent
s
c experiments have reconstituted thin shells of active material in-vitro [4]. These are thin sheets of active matter that
i can deform due to the generation of internal forces and torques that are balanced by external forces (Figure 1A).
s
y The theory of active gels describe the large-scale properties of viscoelastic matter driven out-of-equilibrium due to
h a source of chemical free energy in the system [5]. A number of processes in living systems have been successfully
p described using this theoretical framework [6]. Living or artificial active systems often assemble into nearly two-
[
dimensional surfaces. To understand the physics of such active surfaces, requires a systematic analysis of force and

1 torque balances in curved two-dimensional geometries, taking into account active stresses and material properties.
v The shapes of passive fluid membranes has been described with considerable success by the Helfrich free energy, a
5 coarse-grained description of membranes with an expansion of the free energy in powers of the curvature tensor [7].
9
Expressions for the stress and torque tensors within a Helfrich membrane have been obtained. The associated force
0
and torque balance equations are equivalent to shape equations for miminal energy shapes [8, 9]. Active membranes
9
theories have expanded the description of passive membranes to include external forces induced by pumps contained
0
. in a membrane [10–12].
1
The morphogenesis of epithelial tissues is a highly complex problem involving forces generated actively within the
0
cells. Distribution of forces acting along the cross-sectionof a sheet-like tissue give rise to in-plane tensions, but also
7
to internal torques resulting from differential stresses acting along the cross-section of the tissue (Fig. 1B). These
1
: differential stresses are crucial to generate tissue shape changes [13]. However, no framework currently allows to
v
describe the mechanics of active thin surfaces with internal stresses and torque densities.
i
X In this work, we present such a general framework for the mechanics of active surfaces, driven internally out of
r equilibriumby molecularprocessessuchas achemicalreaction. We startby consideringforcesandtorquesgenerated
a in a surface of arbitrary shape. The corresponding expression for the virtual work shows that components of the
tension and torque tensors are coupled to the variation of the metric, of the curvature tensors and of the Christoffel
symbols defined for the surface (Eq. 15). Using these expressions, we then derive the entropy production for a fluid
surfaceundergoingchemicalreactions. We analysethe symmetries ofsurfaceswith rotationalsymmetry inthe plane,
and show that they can have up-down, chiral or planar-chiral broken symmetry. We write down the corresponding
constitutiveequationsforthecomponentsofthetensionandtorquetensorsandforthefluxes ofthe chemicalspecies.
Interestingly, the generic constitutive equations involve couplings of the curvature tensor with the chemical potential
of the surface chemical species. We then discuss the stability of a flat active fluid with broken up-down symmetry.
Finally, we show that generic equations for an active elastic thin shell can be obtained using the same framework.
2
A B
Apical
Tissue
Basal
Extracellular matrix
C OUT D
IN
FIG. 1: A.Filaments and motors near a surface and epithelial tissues are examples of active surfaces. B. The distribution of
stresses within a thin layer give rise to stresses and torques when integrated across the thickness of the layer. C. Local basis
of tangent vectors e1, e2 and normal vector n associated to the surface X(s1,s2). D. Internaland external forces and torques
acting on a surface element with surface area dS.
I. FORCE AND TORQUE BALANCE ON A CURVED SURFACE
We consider a curved surface X(s1,s2) parametrised by two generalised coordinates s1, s2 (Figure 1C). We use
latin indices to refer to surface coordinates and greek indices to refer to 3D euclidean coordinates. We introduce the
metric tensor g = e e where e = ∂ X with ∂ = ∂/∂si. The curvature tensor is defined as C = ∂ ∂ X n,
ij i j i i i ij i j
· − ·
wheren=e e /e e isthe unitnormalvector,whichweusuallyconsidertopointoutwardforaclosedsurface.
1 2 1 2
We denote×dl wi|th ×dl2 =| gijdsidsj a line element on the surface, and dS = √gds1ds2 a surface element, where
g =detg is the determinant of the metric tensor (Appendix A).
ij
The force f and torque Γ across a line of length dl with unit vector ν =νie , tangential to the surface and normal
i
to the line can be expressed as
f =dlνit =dlν ti, (1)
i i
Γ=dlνim =dl ν mi , (2)
i i
where we have introduced the tension t and moment m per unit length (Figure 1B,D). Decomposing ti and mi in
i i
tangential and normal components as
ti =tije +tin, (3)
j n
mi =mije +min , (4)
j n
defines the tension and moment per unit length tensors tij, ti, mij and mi.
n n
By expressing the total force acting on a region of surface with contour and using Newton’s law, one finds
S C
dSρa= dlν ti+ dSfext, (5)
i
S C S
Z I Z
where ρ is the surfacemass density, a is the localcenter-of-massacceleration,fext is anexternalforcesurfacedensity.
When the surface is embedded in a medium, the external force surface density is related to stresses exerted by the
mediumonthesurface,fext =σ n withσ the3-dimensionalstresstensorinthemedium. Thetotaltorqueobeys
α βα β αβ
dS[X ρa]= dlν mi+X ti + dS Γext+X fext . (6)
i
S × C × S ×
Z I Z
(cid:2) (cid:3) (cid:2) (cid:3)
3
where Γext is the external torque surface density, and where the left-hand side is the torque stemming from inertial
forces. Here, we ignore the moment of inertia tensor for simplicity. This results in the force balance expression
(Appendix C):
ti = fext+ρa, (7)
i
∇ −
mi =ti e Γext. (8)
i i
∇ × −
These equations can be expressed in terms of the components of the tension and torque tensors:
tij +C jti = fext,j +ρaj, (9)
∇i i n −
ti C tij = fext+ρa , (10)
∇i n− ij − n n
mij +C jmi =ǫ jti Γext,j, (11)
∇i i n i n−
mi C mij = ǫ tij Γext, (12)
∇i n− ij − ij − n
where the tangential and normal component of a vector v on the surface are written vi =v ei and v =v n.
n
· ·
II. VIRTUAL WORK
We introduce the virtual work δW, which is the mechanical work acting on a region of surface enclosed by a
contour , upon a small deformation δX of the surface, with X′(s1,s2) = X(s1,s2)+δX(s1,s2). SHere δX(s1,s2)
C
represents a displacement of a material point on the surface specified by (s1,s2). The virtual work can be defined as
1 1
δW = dlν (ti δX+ mi (∇ δX))+ dS((fext ρa) δX+ Γext (∇ δX)), (13)
i
C · 2 · × S − · 2 · ×
I Z
where is the surface region enclosed by , and we have introduced the rotational operator in euclidian space (Eq.
S C
A29):
(∇ δX) =ǫ ei(∂ δX )+ǫ n (∂ δX ). (14)
× α αβγ β i γ αβγ β n γ
In Eq.14, we have introduced the normal derivative of the surface deformation, ∂ δX. We consider here ∂ δX =
n n
(∂ δX n)ei (Appendix B).
i
− ·
The terms in the expression of the virtual work 13 describe the work due to forces and torques acting at the
boundary as well as external forces and torques acting on the surface . Using force balance and the divergence
C S
theorem, the virtual work can be re-expressedas (see Appendix D)
δW = dS tijδgij +mi δC j +mi ǫjkδΓkij . (15)
S 2 j i n 2
Z (cid:18) (cid:19)
Here,theexplicitexpressionofthemetricvariationδg ,curvaturevariationδC j andvariationofChristoffelsymbols
ij i
δΓk as a function of the surface variationδX are given in Appendix B. We have introduced the in-plane tension and
ij
bending moment tensors:
1
tij =tij + m¯kiC j +m¯kjC i , (16)
s 2 k k
mij = mikǫ (cid:0)j. (cid:1) (17)
k
−
where the s subscriptdenotes the symmetric partofthe tensor (Eq. A14). In Eq. 15, we haveuseda reference frame
that deforms with the material.
Thevirtualworkgivenbyequation15canbeinterpretedphysicallyasthemechanicalworkduetodifferenttypesof
deformations. The in-plane surface stress tensor t¯ij is conjugate to the variation of the metric tensor δg , describing
ij
internal shear and area compression. The in-plane tension tensor t introduced in Eq. 16 differs from the tension
ij
tensor t introduced in Eq. 3: this is because in a thin shell, a deformation leading to a change of metric of the
ij
surface mid-plane corresponds to a three-dimensional shear within the shell. As a result, the work to deform the
surface mid-plane depend on the in-plane bending moment tensor, which reflects the distribution of stresses across
the thickness of the shell. The in-plane tensor m¯ij of bending moments is conjugate to the variation of the curvature
tensor δC j due to bending of the surface. The normaltorque mi is conjugate to gradients of local rotationsǫi δΓk.
i n k ij
4
The expression of the virtual work 15 does not include shear perpendicular to the surface: this would require the
introduction of an additional variable.
The virtual work given in Eq. 15 is very general. In order to evaluate the virtual work for a given surface
ij
deformation, the values of the internal stresses characterised by the in-plane stress tensor t , the in-plane bending
moment tensor mij, and the normal torque m , have to be known. In general, they are provided by constitutive
n
relations describing the properties of the material associated with the surface.
We now discuss constitutive relations for active fluid and elastic curved surfaces. The case of a passive membrane
is discussed in Appendix H.
III. CURVED ACTIVE FILM
We now use concepts for irreversible thermodynamics to derive constitutive equations for a curved isotropic fluid.
We consider a fluid consisting of several species α = 1..N with concentrations cα. The local mass density is given
by ρ = mαcα with mα the molecular mass of species α. The free energy density in the rest frame is denoted
α
f (cα,C j,T) where C j is the curvature tensor of the film in mixed coordinates, and T the temperature. The
0 i i
P
differential of f is
0
df =µαdcα+Ki dC j sdT, (18)
0 j i
−
where µα is the chemical potential of component α, Ki is the passive bending moment and s the entropy density.
j
The total free energy density is
1
f = ρv2+f , (19)
0
2
where the kinetic energy is given by 1ρv2 = 1ρ v vi+(v )2 . We denote µα = df/dcα = µα +mαv2/2 the total
2 2 i n tot
chemical potential of the chemical species α.
(cid:2) (cid:3)
A. Conservation equations
FIG.2: Freeenergybalanceonasurfaceelementintheisothermalcase. Freeenergydensityisexchangedbetweenthesurface
element and thesurrounding surface with a flux Jf, with thesurrounding bulk with fluxJf, and is produced with rate −Tπ.
n
Westartbyderivingconservationequationsforthesurfacemass,concentrationofchemicalspecies,energy,entropy
and free energy. Using an Eulerian representation (Appendix E), mass balance reads
∂ ρ+ (ρvi)+v C iρ=Jρ, (20)
t ∇i n i n
where Jρ is a source term due to mass exchange with the environment and v = vie +v n is the center-of-mass
n i n
velocity.
The concentrations cα obey the balance equation
∂ cα+ Jαi+v C icα =Jα+rα, (21)
t ∇i n i n
where Jαi = cαvi +jαi is the tangential flux in the surface of molecule α, jα,i is the flux relative to the center of
mass, Jα describes exchanges between the surface and its surrounding environment, and rα denote source and sink
n
5
terms corresponding to chemical reactions in the surface. Mass conservation implies the following relation between
fluxes of molecules and chemical rates
mαJα =Jρ, (22)
n n
α
X
mαjα,i =0, (23)
α
X
mαrα =0. (24)
α
X
In the remaining of this work, summation over α is implicit. The conservation of energy and the balance of entropy
and free energy density have the form (Figure 2)
∂ e+ Je,i+v C ie = Je, (25)
t ∇i n i n
∂ s+ Js,i+v C is = Js+π, (26)
t ∇i n i n
∂ f + Jf,i+v C if = Jf Tπ Ji T ∂ Ts, (27)
t ∇i n i n − − s∇i − t
where e and s are the energy and entropy density respectively, Je and Js are energy and entropy fluxes entering
n n
the surface from the adjacent bulk, Je,i and Js,i are tangential energy and entropy fluxes within the surface, and
Jf = Je TJs and Jf,i =Je,i TJs,i are the normal and tangential fluxes of free energy. The entropy production
n n− n −
rate within the surface is denoted π. Eq. 27 is obtained from the relation f = e Ts and Eqs. 25 and 26. In the
−
following, we consider for simplicity the isothermal case.
B. Translation and rotation invariance
We now discuss relations between equilibrium tensions and torques implied by invariance of the surface properties
under a rigid translation or rotation.
1. Gibbs-Duhem relation
FIG. 3: Two Gibbs-Duhem relations for a fluid surface are obtained by considering a rigid translation of the surface by a
uniform infinitesimal vector δa or a rigid rotation with infinitesimal vector δθ. Coordinates on the new surface are obtained
by following thenormal nof theoriginal surface.
Using translation invariance of the free energy, we can derive a Gibbs-Duhem relation. We consider a infinitesimal
translation of the surface by a constant vector δa. The condition ∂ δa=0 implies using Eq. A20
i
δaj +C jδa =0, (28)
i i n
∇
∂ δa C δaj =0. (29)
i n ij
−
Duringtranslation,wereparametrizethenewsurfacesuchthateachpoint(s1,s2)movesnormaltotheoriginalsurface
on the new translated surface (Figure 3). Translation invariance then implies the relation (see Appendix F)
(f µαcα)g j KjkC +C Kjk = cα∂ µα. (30)
j 0 i ik ik j i
∇ − − ∇ −
Eq. 30isacovariantgeneralisatio(cid:2)nforsurfacesoftheGibb(cid:3)s-Duhemrelationforathree-dimensionalmulti-component
fluid [14, 15], with an additional term arising from the passive bending moment tensor.
6
2. Rotation invariance
WecanderiveageneralisedGibbs-Duhemrelationdescribingtorquebalancesusinginfinitesimalrotationdescribed
by the pseudo vector δθ, such that the surface is deformed as:
δX =ǫ δθ X . (31)
α αβγ β γ
The deformationdefines a new surface X′ =X+δX, which is reparametrizedsuch that (s1,s2) is constantalong the
normal to the original surface. Rotation invariance then implies (see Appendix F):
Kijǫ C k =0. (32)
jk i
implying that the tensor KijC k is symmetric.
i
3. Equilibrium tensions and torques
The equilibrium tension and bending moment tensors can be obtained by calculating the change of free energy
under a surface deformation and using the expression of the virtual work Eq. 15 (Appendix H). The equilibrium
tension and bending moments are given by
t¯ij = (f µαcα)gij, (33)
e 0−
m¯ij = Kij, (34)
e
mi = 0 . (35)
n,e
withγ =f µαcαthebaremembranesurfacetension. UsingEqs16and17,onealsoobtainsthesymmetricpartofthe
0
equilibrium−tensiontensortij =(f µαcα)gij (K iCkj+K jCki)/2andthebendingmomenttensormij =Kikǫ j.
e,s 0− − k k e k
Using the tangential torque balance equation 11 then yields the equilibrium tension ti = Kji ǫ iΓext,j. Using
e,n ∇j − j
Eq. 32 and 35, the normal torque balance equation 12 yields the equilibrium antisymmetric part of the stress,
ǫ tij = Γext.
ij − n
Combining the Gibbs-Duhem relation 30 and the tangential force balance given by Eq. 9, taking into account the
symmetry relation 32, leads to the equilibrium condition relating chemical equilibrium gradients to external forces:
1
cα∂ µα = fext ǫi (∂ Γext) C ǫ iΓext,k
j j − 2 j i n − ij k
= cα ∂ Uα+C (∂Uα/∂n) ei . (36)
j ij
− ·
In the second line, the external force and torque s(cid:2)urface densities derive fro(cid:3)m a potential Uα(s1,s2,n) acting on
component α (Eqs. H10 and H11). Eq. 36 shows that one can then introduce the effective chemical potential
µα (s1,s2)=µα(s1,s2)+Uα(s1,s2,n(s1,s2)), for which cα∂ µα =0.
eff i eff
The remaining normal force balance equation 10 then provides a shape equation for the equilibrium surface shape.
C. Entropy production rate
We can now calculate the entropy production rate using the variation of the free energy and the Gibbs-Duhem
relation derived above. We consider a region of surface enclosed by a fixed contour , which can deform in 3
S C
dimensions. The rate of change of the free energy F can be written as (see Appendix I):
dF DC
= dS tij tij v (m¯ij Kij) ij mi ∂ ω C ωj
dt S − − e ij − − Dt − n i n− ij
Z (cid:20) (cid:16) (cid:17) (cid:0) (cid:1)
+(∂ µα)jα,i+µαrα+µα Jα+fext v+Γext ω
i tot n · ·
(cid:21)
+ dlν fvi µαjα,i+ti v+mi ω , (37)
i
C − − · ·
Z (cid:20) (cid:21)
7
Flux Force
In-planeshear tensor vij In-planetension tensor t¯idj
Bending rate tensor DCij In-planebendingmoment tensor m¯ij
Dt d
Vorticity gradient (∂iω)·n Normal moment min
Diffusion fluxji,α Chemical potential gradient −∂iµα
Chemical reaction rate rα Chemical potential µα
TABLE I: List of pairs of conjugate thermodynamics fluxesand forces in a thin activesurface.
where we have introduced the symmetric in-plane shear tensor v , the rotational of the flow ω = ωie +ω n, and
ij i n
the corotationalderivative of the curvature tensor:
1
v = ( v + v )+C v , (38)
ij i j j i ij n
2 ∇ ∇
1
ω = ǫij(∂ v C vk)e + ǫij( v )n, (39)
j n jk i i j
− 2 ∇
DC
ij = (∂ v ) v C Ck +vk C +ω (ǫ kC +ǫ kC ) (40)
i j n n ik j k ij n i kj j ki
Dt −∇ − ∇
Note that the in-plane shear tensor v is the sum of a contribution from in-plane flows, equal to the symmetric part
ij
of the covariant gradient of flow v , and a contribution arising from normal flows vn, corresponding to in-plane
i j
shear induced by the deformation∇of the surface in three-dimensions. The vorticity ω of the flow has a normal part
arisingfromthetwo-dimensionalvorticityoftheflowǫij v /2,andatangentialpartspecifictocurvedsurfaces. The
i j
∇
bending rate tensor DC /Dt has the form of a corotational derivative, with the third term in Eq. 40 corresponding
ij
to advection of the curvature, and the last two terms to a corotational term. In Eq. 37, we have not included
contributions from the antisymmetric part of m¯ . Note that the bending moment tensor can always be chosen to be
ij
symmetric in the force balance equations, see Appendix I.
We can read off the entropy production rate in the surface per unit area from Eq. 37:
DC
Tπ=tijv +m¯ij ij +mi ∂ ω C ωj (∂ µα)jα,i µαrα, (41)
d ij d Dt n i n− ij − i −
(cid:0) (cid:1)
wheretij =t¯ij γgij andm¯ij =m¯ij Kij arethe dissipativepartofthe in-plane stressandbending momenttensor.
d − d −
The mechanicalcontributionto dissipationcan be alsounderstood starting fromEq. 15using Tπ =δW /δt, where
m d
δW is the workdoneby dissipativeforces,togetherwithEqs. E6andE8. Note thatthe entropyproductionrateis a
d
sum of products of conjugate thermodynamic fluxes and forces, which all vanish at thermodynamic equilibrium. The
pairs of conjugate fluxes and forces are listed in Table I.
ij
We now briefly discuss the conjugate fluxes and forces. The dissipative in-plane tension tensor t is conjugate to
d
the in-plane shear rate v , corresponding to the dissipative cost of introducing in-plane deformations in the surface.
ij
The couplingbetween the in-plane dissipativebending moment mij and the bending rate tensor DC /Dt arisesonly
d ij
for curved surfaces and is associated to the dissipative cost of changing the surface shape in three dimensions. The
couplingbetweenthenormalmomentmi andthevorticitygradientofflow(∂ ω) n=∂ ω C ωj isageneralisation
n i · i n− ij
tocurvedsurfaceofacouplingwhichalsoarisesforplanarsurfaces,andisassociatedtothedissipativecostofgradients
of rotations within the surface [16]. Finally, the two last terms in Eq. 41 correspond to couplings of the chemical
potential and its gradient to the rates of reactions and the flux of diffusion of species in the surface [15].
The flux of free energy entering the surface from the adjacent bulk reads
Jf =fext v+Γext ω+µα Jα, (42)
n · · tot n
which corresponds to the sum of the mechanical power acting on the surface and of the influx of chemical energy in
the surface. The flux of free energy tangential to the surface reads:
Jf,i =fvi ti v mi ω+µαjα,i (43)
− · − ·
where fvi is the advection of free energy, µαjα,i is the flux of chemical free energy,and the remaining terms describe
the mechanical power tangential to the surface at its boundaries.
8
A
Up-down mirror symmetry Normal plane mirror symmetry
Up-down rotation symmetry Symmetry with respect to
rotation of pi around the normal
B
(i) Up-down symmetric, (ii) Surfaces with broken
non chiral surfaces(O) up-down symmetry(UD)
(iii) Chiral surfaces preserving up-down rotation symmetry (C)
(iv) Planar-chiral surfaces with broken planar mirror symmetry ,
preserving up-down mirror symmetry(PC)
(v) Surfaces with broken chiral and up-down symmetry
FIG. 4: Classification of surfaces with in-plane rotation symmetry. A. The surface state can change under up-down mirror
symmetryMn,mirrorsymmetryMt,up-downrotationsymmetryRt,androtationbyπ aroundthenormalRn. Thesymmetry
Rnisnotbrokenforasurfacewithin-planerotationsymmetry. B.Surfaceswithin-planerotationsymmetrycanbecategorised
in 5 classes according to their symmetries. Schematics give examples of actual surfaces belonging to each category. Red and
green letters indicate respectively broken and preserved symmetries.
D. Mirror and rotation symmetries of surfaces
Constitutive relations describing the active surface must respect the symmetries satisfied by the surface [17]. We
therefore classify surfaces by asking whether the state of an element of surface is preserved under application of
symmetries (Fig. 4).
We restrict ourselves to surfaces with rotation symmetry in the plane. We then find that that 3 sets of discrete
symmetriescanbe associatedto thin shells: up-downmirrorsymmetryM , mirrorsymmetry withrespectto aplane
n
9
normal to the surface M , and up-down rotation symmetry R (Fig. 4A). M corresponds to a mirror symmetry by
t t t
a normal plane going along an arbitrary tangent vector t, R to a rotation of π around an arbitrary tangent vector
t
t. The corresponding transformations rules are given in Appendix G. Because inversion of space can be written
as the combination of M and the rotation of π around the normal R , inversion of space and M are broken or
n n n
preserved simultaneously for a surface with in-plane rotation symmetry. Furthermore, combination of two of the
symmetries M , M and R yield the third one, such that at least two of these symmetries must be broken. As a
n t t
result, surfaces can be classified into 5 different classes: (i) up-down symmetric, non-chiralsurfaces (type 0) preserve
all three symmetries, (ii) non-chiral surfaces with broken up-down symmetry (type UD) preserve M but break M
t n
and R , (iii) chiral surfaces with up-down rotation symmetry break all mirror symmetries M and M but preserve
t t n
R (type C) (iv) planar-chiral surfaces preserve up-down mirror symmetry M but break M and R (type PC) (v)
t n t t
up-down asymmetric and chiral surfaces break M , M and R (Fig. 4). Note that we choose to denote surfaces
n t t
breaking M and not M planar-chiralsurfaces because they break mirror-symmetry in the plane, but these surfaces
t n
are not necessarily made of chiral molecules (Fig. 4B).
E. Constitutive and hydrodynamic equations
UsingtheconjugatethermodynamicforcesandfluxesobtainedfromEq. 41andlistedinTableI,wewriteageneric
linear response theory taking into account the symmetries of an active fluid surface. For simplicity, we consider that
a single chemical reaction occurs in the surface converting a fuel species F into a product species P. The fuel and
product species have the same mass. We denote ∆µ=µF µP the difference of chemical potential between the field
and product species, r = rF =rP the rate of fuel consum−ption and j=jF = jP its flux. We also assume that no
chemical exchange exists b−etween the membrane and its surrounding, such that−the normal fluxes Jα and Jρ vanish.
n n
In the linear response theory, we expand the tensors t¯ij, m¯ij, mi, diffusion flux ji,α and chemical reaction rate rα
d d n
to linear order in the rates of deformation v , DC j/Dt, (∂ ω) n, chemical potential ∆µ and chemical potential
ij i i
·
gradient ∂ ∆µ.
i
The stress and moment tensor can then be decomposed as
ij ij ij ij ij
t = t +t +t +t ,
d 0 UD C PC
mij = mij +mij +mij +mij ,
d 0 UD C PC
mi = mi +mi +mi +mi , (44)
n n0 nUD nC nPC
ij ij
wheret is the partofthe stresstensorthatexists foranysurface, t correspondto terms presentwhenthe surface
0 UD
ij ij
breaks up-down symmetry, t exist for chiral surfaces, and t for planar-chiralsurfaces. Similar rules apply for the
C PC
decomposition of the bending moment tensor and normal moment tensor.
To express constitutive equations for each of the components, we then write all possible terms of the expansion of
the generalisedforcesinthe fluxesatlinearorder,andaskwhetherthe correspondingtermsbreakthe symmetryM ,
n
M , R according to the signatures given in Appendix G. The contributions to the stress tensor then read
t t
tij = 2ηv˜ij +η v kgij +ζgij∆µ
0 b k
DC˜ij DC k
tij = 2η¯ +η¯ k gij +2ζ˜C˜ij∆µ+ζ′C kgij∆µ
UD Dt b Dt k
DC j DC i
tij = η ǫik k +ǫjk k +ζ ǫi Ckj +ǫj Cki ∆µ
C C Dt Dt C k k
(cid:18) (cid:19)
tij = η ǫi vkj +ǫj vki , (cid:0) (cid:1) (45)
PC PC k k
where we have introduced the notation(cid:0)A˜ =A 1A(cid:1)kg for the traceless part of a tensor A. The moment tensor
ij ij− 2 k ij
reads
DC˜ij DC k
mij = 2η +η k gij +2ζ˜C˜ij∆µ+ζ′C kgij∆µ
0 c Dt cb Dt c c k
mij = 2η¯v˜ij +η¯ v kgij +ζ gij∆µ
UD b k c
mij = η ǫi vkj +ǫj vki
C − C k k
DC j DC i
mij = η (cid:0) ǫik k +ǫjk(cid:1) k +ζ ǫi Ckj +ǫj Cki ∆µ. (46)
PC cPC Dt Dt PC k k
(cid:18) (cid:19)
(cid:0) (cid:1)
10
In Eq. 46, we have only introduced symmetric contributions to the bending moment tensor. The normal moment
reads
mi = λ(∂iω Cijω )+χǫij∂ ∆µ
n0 n− j j
mi = 0
nUD
mi = χ Cij∂ ∆µ
nC C j
mi = λ ǫij(∂ ω C ωk)+χ ∂i∆µ. (47)
nPC PC j n− jk PC
The rate of fuel consumption then reads
r = (ζ+ζ′C k)v k 2ζ˜C˜ijv˜ 2ζ ǫi Ckjv
k k ij C k ij
− − −
DC k DC˜ DC
(ζ +ζ′C k) k 2ζ˜C˜ij ij 2ζ ǫi Ckj ij +Λ∆µ. (48)
− c c k Dt − c Dt − PC k Dt
and the fuel flux relative to the centre of mass is given by
ji = L∂i∆µ+ χǫij +χ Cij +χ gij (∂ ω C ωk). (49)
C PC j n jk
− −
η, η , η¯, η¯ , η , η , η , λ, Λ and L are d(cid:0)issipative couplings, ζ,(cid:1)ζ′, ζ˜, ζ , ζ˜, ζ′, χ, χ and χ are reactive
b b c cb C c c c C PC
couplings. The viscosities depend in general on the curvature tensor C ; here we have not taken this dependency
ij
into account for simplicity. We have introduced terms proportionals to η , η and λ corresponding to odd or
PC cPC PC
Hall viscosities which do not contribute to dissipation. These are reactive coefficients, and the time signatures of the
constitutive equationsimply thatthey changesignunder time reversal,whichcouldexistforexample inthe presence
of a magnetic field [18]. Active tensions and bending moments proportional to the difference of chemical potential
∆µ depend on the curvature tensor. In the constitutive equations 45-49, we have expanded these terms to first order
in the curvature tensor C . Although we have not written explicitly this dependency here, the phenomenological
ij
coefficients also depend in general in the concentration fields cα. Positivity of entropy productions implies that the
viscosities η, η , η , η , and λ are positive, however the up-down asymmetric viscosities η¯, η¯ and chiral viscosity η
b c cb b C
can be positive or negative.
In the equations above, the contribution to the two-dimensional stress t¯ij is the generalisation for curved surfaces
0
ofthegenerichydrodynamicequationsofathree-dimensionalactivegel[5]: η andη arerespectivelytheplanarshear
b
andbulkviscosityofthesurface,andζ∆µisanactivetensionarisinginthesurfacefromactiveprocesses. Additional
viscous tensions proportional to η¯, η¯ , and η arise for a curved surface due to the dissipative cost of changing the
b C
surface curvature. We also find new active terms for the tension tensor of a curved surface proportional to ζ˜, ζ′,
ζ , that depend on the curvature tensor C . In particular, anisotropic active stresses can arise in a curved surface
C ij
isotropic in the plane, due to the anisotropy of the curvature.
Active terms for the moment tensor introduced in Eq. 46 are specific to thin films and correspond to actively
induced torques in the film. The active torque ζ , arising in a surface with broken up-down symmetry, can induce
c
active bending of a flat surface.
Combining the constitutive equations45-49, the force andtorquebalanceequations7 and8,andthe concentration
balance equations 21 yield dynamic equations for the surface shape, the velocity field on the surface v and the
concentration fields on the surface ck. While the constitutive equations obtained here are linear, the dynamics
equations for the surface shape are non-linear due to geometric couplings.
F. Instabilities of a homogeneous active Helfrich membrane
Inthissection,werestrictourselvestonon-chiralsurfaceswithbrokenup-downsymmetryanddiscusslowReynolds
numbers where inertialterms canbe neglected. Starting froma descriptionof a passivesurface with the Helfrichfree
energy, we consider effects introduced by additional active terms.
1. General equations
Apassivefluidmembranedescribedbythe Helfrichenergywithmembranetensionγ,bending modulus κ,gaussian
bendingmodulusκ andspontaneouscurvatureC hastheequilibriumtensionandbendingmomenttensor(Appendix
g 0
H)
t¯ = γ+(κ+κ )(C k)2 4κC kC κ C kC l g , (50)
ij g k k 0 g l k ij
− −
m¯ij = (cid:0)2(κ+κg)Ckk 4κC0 gij 2κgCij. (cid:1) (51)
− −
(cid:0) (cid:1)