Instabilities and waves in thin films of living fluids

0.17 MB·English

by  Sumithra Sankararaman

Instabilities and waves in thin films of living fluids
Sumithra Sankararaman∗ and Sriram Ramaswamy†‡
Centre for Condensed Matter Theory, Department of Physics,
Indian Institute of Science, Bangalore 560012, India
Weformulatethethin-filmhydrodynamicsofasuspensionofpolarself-drivenparticlesandshow
thatitispronetoseveralinstabilitiesthroughtheinterplayofactivity,polarityandtheexistenceofa
freesurface. Ourapproachextends,toself-propellingsystems,theworkofBenAmarandCummings
[PhysFluids13(2001)1160]onthin-filmnematics. Basedonourestimatestheinstabilitiesshouldbe
seeninbacterialsuspensionsandthelamellipodium,andarepotentiallyrelevanttothemorphology
of biofilms. We suggest several experimental tests of our theory.
9
Activehydrodynamics,thecollectivebehaviourofself- v of the active particles, the mean film thickness h ,
0 0 0
0 driven, orientable particles in a fluid medium, is a topic the viscosity µ, an orientation mobility Γ, the coupling
2 of intense current research [1, 2, 3, 4, 5, 6, 7, 8]. In this C, present in equilibrium systems as well, of particle ori-
Letter we study the dynamics of a film of fluid, thin in entation to free-surface tilt, and the typical active con-
n
a the z direction and spread on a solid surface in the xy tractile or tensile stress σ0 ≡ Wc0. W is the strength
J plane which is also the easy plane for the orientational of the force-dipole associated with each active particle,

6 orderparameterpofthepolaractiveparticlessuspended and c the mean concentration of active particles. (iii)
0
2 in the fluid. For an active system, this polarity implies Low motility, i.e., small v , promotes the instability and
0

a current v cp with respect to the fluid, where v is a high motility suppresses it, a point we will return to at
] 0 0
t characteristic drift velocity and c the concentration of the end of this paper. The instability manifests itself in
f
o active particles. While our formulation is general, we moving (convective) and static (absolute) form. We also
s study mainly the properties of perturbations about an offerasimplephysicalpicture(Fig. 1)ofthemechanism
.
t ordered, uniform reference state (cid:104)p(cid:105)=xˆ. underlying this unique instability. (iv) We find several
a
m other possiblemodes of free-surface instability, which we
HIGH LOW discuss towards the end of the paper.
-
d Before presenting our model in detail, a few words are
n inorderaboutthesystemstowhichitmightapply. Bac-
o
terial colonies on a surface frequently take the form of
c
organised, rigid biofilms [9], e.g., those of Pseudomonas
[
aeruginosa [10]. An understanding of the formation of
4 LOW HIGH these structures should begin with the thin-film hydro-
v
dynamicsofactivefluids. Thelamellipodiumofcrawling
8
1 FIG. 1: Top view of film with contractile polar filaments cells is propelled by the ATP-driven force of a continu-
(solidarrows)withapreferenceforpointingdownhill. Active
9 ally polymerizing and depolymerizing, contractile actin
stresses cause fluid flow (dashed arrows) towards the open
4 network just below the cell membrane [11] and is thus a
endofasplayperturbation,leadingtogradientsintheheight
.
9 of the fluid film. Modes propagating forward experience a moving, thin, active fluid film.
0 height gradient that further torques (dash-dotted arrow) the We therefore consider a fluid film containing active
8
filaments in the direction in which they were already per- particles with concentration field c, and orientation de-
0
turbed. scribed by a vector field p = (p ,p ). Since we are
: ⊥ z
v interested here in perturbations about a macroscopically
Xi Our main results are as follows: (i) Active, ordered ordered state, we set |p|=1. We construct equations of
thin films, although dominated by viscosity, not inertia, motionforc(r ,t),p(r ,t),andtheheightfieldh(r ,t),
r ⊥ ⊥ ⊥
a show a wavelike response to external disturbances, as a i.e., the film thickness, as functions of in-plane position
result of the coupled dynamics of free-surface undula- r andtimet. Ourtreatmentgeneralizes[12]tothecase
⊥
tions the active stress field, and the concentration. (ii) of active systems.
In large regimes of parameter space this coupling pro- The kinematic boundary condition [13] h˙ = u −u ·
z ⊥
duces a novel instability whose growth rate, to leading ∇ h connects h to the hydrodynamic velocity field u =
⊥
order in small wavevector k = (kx,ky), varies as kykx1/2 (u⊥,uz) evaluated at the free surface. Incompressibility
for kx (cid:28) ξky2 and as ky2 for kx (cid:29) ξky2. The crossover ∇·u = 0 leads to the simplification
length ξ = Γ|Cσ |h2/v2µ depends on the drift velocity
0 0 0
(cid:90) h
∂ h+∇ · u dz =0. (1)
t ⊥ ⊥
0
†alsoatJNCASR,Bangalore560064India We eliminate u⊥ in favour of p, c and h through the
2
Stokes equation in the lubrication approximation [13] opposite happens. The behaviour of the active tension
u =0, |∇ u|(cid:28)|∂ u|: term is consistent with the idea that contractile stresses
z ⊥ z
pull in along the long axis of the particles, giving addi-
µ∂z2u⊥−∇⊥P −zˆ∂zP −∇·σa =0 (2) tional elastic resistance to stretching along that axis.
The dynamics of the polar orientation field p dif-
where µ and P are the viscosity and pressure field of the
fers from that of the conventional nematic director.
fluid, and σa(r) = Wc(r)p(r)p(r) is the intrinsic stress
First, symmetry cannot rule out a coupling of the form
field [3, 14] of the active particles. We discuss the role of −C(cid:82) d2xp ·∇ h which couples p to tilts of the free
gravitylaterinthepaper. W <0andW >0correspond ⊥ ⊥ ⊥
surface [17]. For C > 0 a tilt of the free surface tries to
respectively to contractile and tensile activity.
make p point uphill.
Theactive-particlecurrentisalongp,andtheparticles ⊥
We can think of two possible microscopic origins for
cannotleavethefilm. Thusp (z =0)=0, andp·Nˆ =0
at z = h, where Nˆ = (−∇zh,1)/(cid:112)1+(∇ h)2 is the such a term. Consider an imposed thickness variation
⊥ ⊥ tilt ∇ h of the free surface. One end of each polar par-
outward normal to the free surface. So p (cid:39) p ·∇ h ⊥
z ⊥ ⊥ ticle is in general expected to be fatter than the other,
at the free surface, interpolating linearly, through di-
and the particles will be best accommodated with the
rector elasticity [15], to p = 0 at the substrate, i.e.,
z fatendorientedtowardsthedirectionofincreasingh. In
p = (z/h)p · ∇ h. Therefore in a z-averaged de-
z ⊥ ⊥ addition, the spatial variation in the separation between
scription p = (1/2)∂ h and ∂ p (cid:39) h−1∂ h. Con-
z x z z x the free surface and the fixed substrate should give rise
sider small deviations [16] about a state aligned along
toanelectricfieldintheplane. Polarparticlesingeneral
xˆ: p = xˆ + θyˆ, θ (cid:28) 1. From the foregoing discus-
⊥ haveanelectricdipolemoment,andwillthusbeoriented
sion, the active force density has components ∇ σa =
i ix by this field. Secondly, consider the spontaneous-splay
σ0(∂yθ+∂xc/c0+h−1∂xh), ∇iσiay =σ0∂xθ and ∇iσiaz = term [18] H ≡ −(cid:82) d3xC˜∇·p in the effective Hamil-
σ ∂2h/2 to linear order. We now eliminate the pressure sp
0 x tonian for p, where C˜ is a phenomenological parameter
P in favour of h, p and c, through the z-component of
which can depend on the local concentration, say C˜ =
(2) and stress continuity at the free surface z = h, lead-
C˜(c )+C˜(cid:48)(c )δc+...≡C+C(cid:48)δc. Throughthe“equilib-
ingtoP(x,y,z,t)=P −γ∇2h+σ (h−z)∂2h/2, where 0 0
0 ⊥ 0 x rium” part of the dynamics of our system, the C and C(cid:48)
P is a reference pressure and γ the surface tension of
0 termswillcontribute−ΓδH/δp=ΓCδ(z)∇ h−ΓC(cid:48)∇δc
the fluid [13]. Since we are interested in the dynamics at ⊥
to the equation of motion for p, Γ being a kinetic coeffi-
large scales in the xy plane, we treat the active stresses
cient. On dimensional and symmetry grounds we expect
in a z−averaged approximation, allowing us to borrow
the methods of [13] for spreading under gravity, whose l
force density is z−independent.
l’ l
Using this in (2) and integrating twice over z with
∂ u (h)=0 and u (0)=0 we get R
z ⊥ ⊥
hz−z2/2(cid:18) 1 (cid:19)
u (z)= γ∇ ∇2h− σ h∂2∇ h−f , FIG. 2: An object with head and tail of sizes (cid:96) and (cid:96)(cid:48) re-
⊥ µ ⊥ ⊥ 2 0 x ⊥ ⊥
spectively, and length (cid:96), produces a spontaneous curvature
(3) 1/R(cid:39)((cid:96)−(cid:96)(cid:48))/(cid:96)2.
to linear order in perturbations of h, where f =
⊥
σ [(∂ θ+∂ c/c +h−1∂ h)xˆ+∂ θyˆ]. Using (3) in (1),
0 y x 0 x x C ∼F/R,whereF isatypicalforcescaleandR,thera-
linearizing h=h +δh, c=c +δc, the in-plane fourier-
0 0 dius of curvature associated with the particle shape (see
components δh (t), δc (t), θ (t) obey
k k k Fig. 2).,istheparticlethicknessdividedbythefractional
length difference between the two edges of the particle.
σ h2 δc
∂ δh = − 0 0[2h k k θ +h k2 k Combining the above with the results of [3], the dy-
t k 3µ 0 x y k 0 x c
0 namics of the the angle field θ becomes
1 γh3
+ (1− 2h20k2)kx2δhk]− 3µ0k4δhk. (4) ∂tθ =−a1v0∂xθ−ζ∂yδc+δ(z)ΓC∂yh+(λAyx−Ωyx)+D∇2θ.
(5)
The four effects of activity on the right-hand side of The first term on the right-hand side of (5) is advec-
(4) are, from left to right within the square bracket, tion of θ with a speed proportional to v [19]. The rest,
0
(i) curvature-induced flow; (ii) anisotropic osmotic flow; from left to right, are coupling to concentration gradi-
and, in parentheses, (iii) splay-induced flow from tilting ents, with ζ =ΓC(cid:48), coupling to the free surface through
the free surface and (iv) an active anisotropic contribu- the coefficient C, flow-orientation coupling as in nemat-
tion to the effective tension. The final term on the right ics [15], and gradient elasticity of p. The first three
of(4)isordinarysurfacetension. Forcontractile(σ <0) termsarepolar,andthefirstisaconsequenceofactivity.
0
stresses, term (iii) destabilises, and the active tension in A = 1(∇ u +∇ u ) and Ω = 1(∇ u −∇ u ). For
ij 2 i j j i ij 2 i j j i
(iv)stabilises,thesurface;fortensile(σ >0)stressesthe stable flow-alignment [15] |λ|>1.
0
3
Eliminating u in favour of θ and h through (3), and ω = a v k +iΓC 2σ0h20[k2−O( v0 )D k2](10)
averagingoverz, (5)becomes, toleadingordersink and 2 1 0 x v 3µa y ΓC ±
0 1
linear order in the fields,
Wenowincludetheconcentrationvia(7),butcontinue
∂tθ = +iΓh0Ckyδhk−(cid:0)D+kx2+D−ky2+ia1v0kx(cid:1)θ itnhethpeurseilmyprelilfayxinatgiolnimalitfoΓrmC/(v90),(cid:29)wit1h.cOoenffiecmienotdemroedtiafiiends
− (iζky−Φkxky)δc, (6) at O(ζ). The propagating mode in (10) becomes a pair
with speeds of order v ±O(ζ), with relaxation rates ∼
where D = D − (λ ± 1)h2σ /4µ, and Φ = (λ − 0
± 0 0 ±(h2σ CΓ/µv )[1+O(ζ)]k2. Theresultsfrom(8)tothis
1)h2σ /4c µ. 0 0 0
0 0 0 pointestablishthemainclaimsatthestartofthispaper.
The active contribution to particle currents with re-
Inparticular, itisreadilyseenthat(9), (10)crossoverto
spect to the fluid is v cp where v is a drift velocity.
0 0 (8) for k (cid:28)ξk2 with ξ =Γ|Cσ |h2/v2µ.
Thus in the lab frame of reference the continuity equa- x y 0 0 0
Somegeneralfeaturesareworthnoting: (a)Regardless
tionfortheconcentration,apartfromdiffusion,issimply
of the sign of σ C, there is always an instability; (b) in-
∂ δc=−∇·[(u+v p)c]. Linearizingaroundc andusing 0
t 0 0 creasingv weakenstheinstability,presumablybecausea
(5) and overall incompressibility, 0
collection of filaments drifting along x samples alternat-
∂ δc=−ic v k θ−iv k δc+O(k2δc ,k2δh ). (7) ing height gradients along y whose effect cancels; (c) for
t 0 0 y 0 x x k x k
σ C >0 the instability moves with a speed ∼v ±O(ζ),
0 0
TheO(k2)termsin(7)comefromincludingfluctuations combiningtheeffectsofthedriftspeedv andthe“pres-
0
inp aswellbyexplicitlyincludingdiffusionofparticles. sure” due to ζ, while for σ C < 0 there is an instabil-
x 0
Theinstabilitywearechieflyconcernedwithhere,aris- ity in mode ω , predominantly the height field h, that
1
ingfromthecombinationofactivityandthetiltcoupling grows without travelling, the remnant of the instability
C,isbestseenintheextremecaseofimmotile but active discussed for v =0.
0
polar particles, i.e. v =0 but σ (cid:54)=0. To leading order Itiscrucialtonoteherethatactivity andpolarity con-
0 0
ink theconcentrationplaysnorole, andthedynamicsis spiretoproducetheinstability,atleadingorderingradi-
determined by (4) and (5), yielding an unstable mode in ents, even at zero motility. Motility, also a consequence
the system which grows at a rate k1/2k . In detail, for of active polarity, is encoded in the active stress only at
x y
time-dependence exp(−iωt), to leading order in k, the next-to-leading order [14]. The stabilising effect of in-
mode frequency creasing v0 at constant σ0 is thus not paradoxical.
For ΓC/v (cid:28)1 all instabilities involving the interplay
1+isgn(k Cσ ) Γh2 0
ω =± √ x 0 ( 0)1/2|Cσ k |1/2|k | (8) of activity with the spontaneous-splay coupling C disap-
2 3µ 0 x y pear. Theremaininginstabilitiesinvolveneitherpolarity
nor a drift velocity. The most interesting of these, men-
has a real and an imaginary part; the mode displaying
tioned briefly below (4), arises as follows: Consider a
an instability travels in the +x direction (−x direction)
flat free surface with contractile filaments aligned along
if σ C >0 (σ C <0).
0 0
xˆ, and impose a small tilt δh ∝ x. The filaments at
Thisinstabilitycanbeunderstoodfromasimplephys-
the free surface are then tilted relative to those at the
ical picture, involving the interplay of active stresses σ
0
substrate. The resulting splay in the xz plane implies
andspontaneoussplayC. Tofixideas,considerasinFig.
thatactivestresseswillpumpfluidtowardstheopenend
1 a long-wavelength splay perturbation on a set of con-
of this splayed configuration, thus increasing the tilt. In
tractile filaments σ < 0 aligned with polarisation along
0
addition,theinstabilitiesofabulkactiveorderedsuspen-
+xˆ, with a preference for pointing downhill (C < 0), in
sion as originally discussed in [3, 5] can arise, modified
an initially flat film. The active stresses then generate
here by confinement to a thin film, and the anisotropic
a flow in the film that increases the height of the free
active tension [mentioned after (4)], if large enough, will
surface ahead of the outward-splayed filaments, and de-
destabilize tensile active films at order k4.
creasestheheightaheadoftheinward-splayedfilaments.
We now compare the relaxation rates in (8) with the
Themodepropagatingalong+xˆ willthencausethefila-
stabilizingcontributionsρgh3k2/µduetogravity,ρbeing
mentstosampleaheightgradientthattiltsthefilaments 0
themass-densityofthefilmandgtheaccelerationdueto
in the centre of the field further to the right. The same
gravity, γh3k4/µ due to surface tension, and Kk2/µ due
argument,mutatis mutandis,goesthroughforthetensile 0
toorientationalrelaxation, whereK isaFrankconstant.
and/or C >0 cases.
Ondimensionalgroundsandfromstandardliquid-crystal
For motile particles, v (cid:54)= 0, let us for the moment
0 physics[15]wetakeΓ≈1/µ,whereµ≈10−3 Pasisthe
continue to ignore the concentration field. In the limit
viscosity of the medium. With this estimate we find the
ΓC/v (cid:29)1, (4) and (5) then yield mode frequencies
0
instability survives these stabilizing agencies provided
ΓC 2σ h2 v kh <min[Cσ /ρ2g2h3,(Cσ h /γ2)1/5,Cσ h3/K2]. For
ω = −i 0 0[k2+O( 0 )k2] (9) 0 0 0 0 0 0 0
1 v0 3µa1 y ΓC x γ and ρ we use values for water. σ0 should be of order
4
fac =φf/a2 where f is the force exerted by the active 170 (2005); F. Ju¨licher et al., Phys. Rep. 449, 3 (2007).
0
particle on the fluid, a the particle size, and we take the [3] R.A. Simha and S. Ramaswamy, Phys. Rev. Lett 89,
particle volume fraction φ = c a3 of order unity since 058101 (2002).
0
[4] T.B.LiverpoolandM.C.Marchetti,Europhys.Lett.69,
we want an ordered phase. For a bacterium of size a ∼
846 (2005); Phys. Rev. Lett. 90, 138102 (2003)
a few µm moving at several µm/s through water f ∼ 1
[5] K. Kruse et al., Phys. Rev. Lett. 92, 078101 (2004).
pN. We argued that C ∼ F/R where F is a force scale
[6] A. Ahmadi et al., Phys. Rev. E 74, 061913 (2006); L.
and R the spontaneous curvature associated with a po- Giomi et al., arXiv:0805.1680; T. B. Liverpool, M. C.
lar object. For thermal hard-rod systems F ∼ k T/(cid:96) Marchetti, Phys. Rev. Lett. 97, 268101 (2006); J.-F.
B
where (cid:96) is a rod length. For µm-sized objects this would Joannyetal.,NewJ.Phys.9,422(2007);S.Ramaswamy
give C ∼10−6 dyn/cm. For a self-propelled object, it is and M. Rao, New J. Phys. 9, 423 (2007); R. Voituriez,
J-FJoannyandJ.Prost,Europhys.Lett.70,404(2005)
possiblymorereasonabletousetheforce-dipolestrength
D. Saintillan and M. J. Shelley, Phys. Rev. Lett. 100,
W =af asthenaturalenergyscale,inwhichcaseF ∼f.
178103(2008);V.MehandiaandP.Nott,J.FluidMech.
For an object of radius 2 µm moving through water at
595, 239 (2008); T. Ishikawa and T. J. Pedley, J. Fluid
20 µm/s this gives F in the range of a piconewton and Mech. 588, 399 (2007); C. Wolgemuth, Biophys. J. 95,
thus C ∼ 10−4 dyn/cm. For film thickness h (cid:39) 20 µm, 1564 (2008); M. E. Cates et al., Phys. Rev. Lett. 101
0
and C ∼ 10−6 dyn/cm, the instability should be seen (2008) 068102; I. Llopis, I. Pagonabarraga, Europhys.
if kh0 <∼10−3 . However, this is a pessimistic estimate: Lmeatrt,.,I.75P,ag9o9n9a(b2a0r0r6a)g;a,S.EuRra.mPahcsh.aJn.drEan2,0P,.15B1. (S2.00K6u)-;
First, activity rather than thermal energy is quite likely
J.P.Hernandez-Ortizet al.,Phys.Rev.Lett.95,204501
responsible for C. Secondly, the tension γ for biofilms is
(2005);P.Underhillet al.,Phys.Rev.Lett.100,248101
that between the film and the ambient aqueous medium (2008); G. Gr´egoire, H. Chat´e, and Y. Tu, Phys. Rev. E
andthereforemuchsmallerthantheair-watervalue. We 64, 011902 (2001).
expect therefore that the instability should be seen over [7] C.Dombrowskietal.,Phys.Rev.Lett.93,098103(2004);
a much wider wavenumber range, possibly kh0<∼1. I.Tuval,et. al.,Proc.Natl.Acad.Sci.102,2277(2005);
T. J. Pedley and J. O. Kessler, Annu. Rev. Fluid Mech.
The experiment of choice to test our ideas would be 24, 313 (1992); I. S. Aranson et al., Phys. Rev. E 75,
to compare the dynamics of two initially flat thin films 040901(R)(2007); X.-L. Wu and A. Libchaber, Phys.
of uniform concentration, one with highly motile bacte- Rev. Lett. 84, 3017 (2000).
ria, the other with a low-motility mutant, under condi- [8] O. Ignoshin, R. Welch, D. Kaiser and G. Oster, Proc.
Natl. Acad. Sci. 101, 4256 (2004).
tions of constant bacterial concentration. The mutant
[9] http://en.wikipedia.org/wiki/Biofilm
population would correspond to a system with v small,
0 [10] M. Klaussen et. al, Mol. Microbiol. 50, 61 (2003); ibid.
and should show our instabilities. The observations of
48, 1511 (2003).
[10] on Pseudomonas aeruginosa are of interest in this [11] K. Kruse et al., Phys. Biol. 3 (2006) 130137
regard, but are complicated by the fact that the bacte- [12] M.BenAmarandL.J.Cummings,Phys.Fluids.131160
ria are dividing. The relation of our mode structure to (2001)
the excitations seen in studies [8] of Myxococcus xanthus [13] H.A. Stone, in Nonlinear PDEs in Condensed Matter
and Reactive Flows,NATOScienceSeriesC:Mathemat-
and to the lamellipodium of crawling cells [11] remains
ical and Physical Sciences, 569, H. Berestycki and Y.
tobeexplored. Acompletetreatmentmustincludeactin
Pomeaueds.,KluwerAcademic,Dordrecht,TheNether-
treadmilling,aswellastheelasticityandanchoringofthe
lands(2002);A.Oron,S.H.DavisandS.G.Bankoff,Rev.
cell membrane. In all these examples, the dynamics of Mod. Phys. 69, 931 (1997).
thebulkfluidabovethefilmmustalsobeincluded. Since [14] Y. Hatwalne, S. Ramaswamy, M. Rao and R.A. Simha,
inhomogeneities in orientation give rise to mass flux, the Phys. Rev. Lett 92, 118101 (2004).
instabilities will produce large concentration variations [15] P.-G. de Gennes and J. Prost, The Physics of Liquid
Crystals, Clarendon, Oxford (1993).
which will be important especially when going beyond
[16] Even in a linearized description, p is not constant but
the linearised treatment. x
is slaved to the concentration and height fields, as seen
We thank S. Banerjee for pointing out an error in bygeneralizingthetreatmentof[1]toincludetheeffects
an earlier version of this paper, J. Prost, K. Vijay Ku- of the free surface. This results in subdominant contri-
butionstothedynamicsoftheheightfield(4),whichwe
mar and J. K¨as for valuable inputs, and CEFIPRA
ignore here.
project 3504-2 and the DST Math-Bio Centre grant
[17] We thank Shiladitya Banerjee for pointing out the error
SR/S4/MS:419/07 for support.
inourargumentsleadingtothisterminanearlierversion
of this paper.
[18] W.Kung,M.C.Marchetti,K.SaundersPhys.Rev.E73
(2006) 031708.
[19] The proportionality constant a is not necessarily unity.
1
∗ Electronic address: [email protected] Sincethevelocitiesaredefinedwithrespecttothefluid’s
‡ Electronic address: [email protected] frame of reference, shifting all velocities by a constant is
[1] J. Toner and Y. Tu, Phys. Rev. E 58, 4828 (1998). not an invariance [1].
[2] J. Toner, Y. Tu and S. Ramaswamy, Ann. Phys. 318,