Shear dynamics of an inverted nematic emulsion
SShear dynamics of an inverted nematic emulsion
A. Tiribocchi ∗ a , M. Da Re ∗ a , D. Marenduzzo b , E. Orlandini a Received Xth XXXXXXXXXX 20XX, Accepted Xth XXXXXXXXX 20XXFirst published on the web Xth XXXXXXXXXX 200X
DOI: 10.1039/b000000x
Here we study theoretically the dynamics of a 2D and a 3D isotropic droplet in a nematic liquid crystal under a shear flow.We find a large repertoire of possible nonequilibrium steady states as a function of the shear rate and of the anchoring of thenematic director field at the droplet surface. We first discuss homeotropic anchoring. For weak anchoring, we recover the typicalbehaviour of a sheared isotropic droplet in a binary fluid, which rotates, stretches and can be broken by the applied flow. Forintermediate anchoring, new possibilities arise due to elastic effects in the nematic fluid. We find that in this regime the 2Ddroplet can tilt and move in the flow, or tumble incessantly at the centre of the channel. For sufficiently strong anchoring, finally,one or both of the topological defects which form close to the surface of the isotropic droplet in equilibrium detach from it andget dragged deep into the nematic state by the flow. In 3D, instead, the Saturn ring associated with normal anchoring disclinationline can be deformed and shifted downstream by the flow, but remains always localized in proximity of the droplet, at leastfor the parameter range we explored. Tangential anchoring in 2D leads to a different dynamic response, as the boojum defectscharacteristic of this situation can unbind from the droplet under a weaker shear with respect to the normal anchoring case. Ourresults should stimulate further experiments with inverted liquid crystal emulsions under shear, as most of the predictions can betestable in principle by monitoring the evolution of liquid crystalline orientation patterns or by tracking the position and shape ofthe droplet over time.
Dispersions of particles in a host fluid are important exam-ples of soft matter with many potential applications in foodindustry, drugs, paints and design of new composite materi-als . While in colloidal suspensions the hosted particles aresolid, emulsions are dispersions of liquid droplets coated witha surfactant. When the particles are dispersed in a nematicliquid crystal, i.e., an anisotropic fluid, where elongated or-ganic molecules are, on average, aligned along a common di-rection (director), additional long-range forces, due to elasticdeformations of the director field in proximity of the dropletsurface, are induced and topological defects are observed. De-fects mediate anisotropic droplet-droplet interactions of dipo-lar or quadrupolar type that typically lead to the formation ofordered droplet structures, such as chains and hexagonal lat-tices .Due to a large body of theoretical and experimental works,the equilibrium properties of isotropic droplets in a nematichost (inverted nematic emulsions) are now well understood:depending on the strength and direction of the anchoring ofthe nematogens at the droplet surface, many equilibrium struc-tures occur, each with a well defined defect and droplet shapeconformation . For example, isotropic droplets with normal a Dipartimento di Fisica e Astronomia and Sezione INFN di Padova, Univer-sit´a di Padova, Via Marzolo 8, 35131 Padova, Italy. b SUPA and The School of Physics and Astronomy, University of Edinburgh,Edinburgh EH9 3FD, United Kingdom anchoring of the director field at their surface are often accom-panied by a hyperbolic hedgehog defect while, for weakeranchoring a Saturn ring defect (or a pair of antipodal defectsof topological charge -1/2 in 2D) surrounding the droplet isobserved . The processes of defects formation and direc-tor field orientation in proximity of the droplet surface havebeen also investigated numerically by different means suchas molecular dynamics simulations , Monte Carlo simu-lations and through minimisation of a Landau-de Gennesfree energy functional , although in many cases the ap-proximation of undeformable droplet has been considered.Much less is known, however, on the rheological and re-sponse properties of isotropic-nematic mixtures when theyare, for example, subject to external perturbations such aselectric, magnetic and flow fields. In fact, although molec-ular dynamics approaches are in principle able to describethe time evolution of these systems, they are limited to verysmall length and short time scales where hydrodynamic modesare still irrelevant. Continuum models, based on a Landau-de Gennes free energy description of the nematogens, havebeen used in the past to study either the phase separation dy-namics of symmetric nematic-isotropic mixtures or the ef-fect of hydrodynamic flow on the orientational order of thenematic liquid crystal in presence of spherical, rigid, inclu-sions . More recently, other theoretical studies, based on afree-energy description of the system and Lattice Boltzmannapproaches, have been carried out with the aim of characteris-ing either the equilibrium properties of the interface between a r X i v : . [ c ond - m a t . s o f t ] S e p ematic and isotropic fluids or the shape of a droplet of anisotropic fluid immersed in a nematic liquid crystal in pres-ence of a surfactant . In particular Sulaiman et al. haverecently introduced and tested a Lattice Boltzmann algorithmwhich solves numerically the hydrodynamic equations of mo-tion of a nematic coexisting with an isotropic phase, either inabsence or in presence of an electric field.Here we adapt this algorithm to study the effect that an ex-ternally imposed shear flow can have on the dynamical proper-ties of inverted nematic emulsions, described as a single two-or three-dimensional droplet of isotropic fluid surrounded bya nematic liquid crystal. By varying the shear rate and theratio between the elastic energy scales of the nematic in thebulk and at the droplet surface, we observe a rich dynamicalresponse. For weak anchoring, the behaviour of the systemresembles that of a sheared isotropic droplet in a binary fluid,whereas for intermediate and strong anchoring many othersteady states are observed. Of particular interest is the oc-curence of oscillatory steady states in which the droplet tum-bles and deforms in the flow (for intermediate anchoring), orin which the topological defects created by the anchoring de-tach from the droplet surface and move around in the bulk.Such oscillatory steady states are possible since the system isdriven far from equilibrium by the applied shear. In our work,we characterise the steady states in terms both of the dropletshape (aspect ratio and tilt) and of the defect textures withinthe nematic host.The paper is structured as follows. In Section 2, we in-troduce the model which describes the equilibrium phase be-haviour and the hydrodynamics of 2D droplets of isotropicfluid suspended in a lyotropic nematic liquid crystal. In par-ticular we write down the Landau-de Gennes free energy ofthe system and the Beris-Edwards equations of motion for ne-matic liquid crystals coupled to Cahn-Hilliard dynamics forthe diffusion of the two species. In Section 3 we partitionour results as follows. First we discuss the equilibrium prop-erties of a single droplet of Newtonian fluid dispersed into anematic host fluid, for different anchoring conditions (eitherhomeotropic, i.e., normal, or homogeneous, i.e., tangential).We next present the main findings of the paper referring to theeffect that linear shear flow has on the shape and dynamicalproperties of the emulsion. The end of the section is devoted toextend the study to 3D systems where the defect configurationat equilibrium is given by a Saturn ring hugging the droplet.The effect of the shear on the dynamics of the droplet-Saturnring pair is presented and compared with the 2D counterparts.Finally Section 4 is devoted to a discussion of the results andto conclusions. We consider an inverted nematic emulsion in an extremely di-luted regime in which a single isotropic droplet is dispersedin a nematic liquid crystal. In this two-component system thedroplet is made up of anisotropic-shaped molecules (such asrods) randomly distributed and oriented in the space. The re-sulting phase is that of a liquid crystal in the isotropic phasein which, unlike in the nematic phase, there is neither ori-entational nor positional order. The physics of the systemcan be described in terms of a set of coarse-grained variables φ ( (cid:126) r , t ) , ρ ( (cid:126) r , t ) , (cid:126) v ( (cid:126) r , t ) and Q ( (cid:126) r , t ) which are respectively: anorder parameter related to the relative concentration of the ne-matic phase ( φ ( (cid:126) r , t ) = , describes the ne-matic phase. More specifically, in the uniaxial approximation Q αβ = q ( ˆ n α ˆ n β − δ αβ ) , where ˆ n is the director field (Greeksubscripts denote Cartesian coordinates) and q is the localdegree of nematic order related to the largest eigenvalue of Q (0 ≤ q ≤ ).The equilibrium properties of the system are encoded in aLandau-de Gennes free energy F = (cid:82) V f dV , where f = f b f ( φ ) + f lc ( φ , Q ) + f int ( φ , Q ) + f W ( Q ) . (1)The term f b f ( φ ) = a φ ( φ − φ ) + κ | ∇φ | (2)stems from a typical binary fluid formalism and is made bytwo contributions: the first one is a double well potentialallowing bulk phase separation into a nematic (outside with φ (cid:39) φ ) and isotropic (inside with φ =
0) phase in the dropletgeometry, whereas the second one creates an interfacial ten-sion between these phases whose strength depends on κ . Theterm f lc ( φ , Q ) = A (cid:20) (cid:18) − ζ ( φ ) (cid:19) Q αβ − ζ ( φ ) Q αβ Q βγ Q γα + ζ ( φ ) ( Q αβ ) (cid:21) + K ( ∂ α Q βγ ) , (3)is the free energy density of the liquid crystal phase. It is madeby four contributions (summation over repeated indexes is as-sumed). The first three, multiplied by the positive constant A ,are the bulk free energy density for an uniaxial liquid crystalsystem with an isotropic-nematic transiton at ζ ( φ ) = .
7. Theparameter ζ ( φ ) , which determines which phase (isotropic ornematic) is the stable one, is assumed to be linearly dependenton the concentration φ , namely ζ = ζ + ζ s φ , (4)here ζ and ζ s are constants controlling the boundary of thecoexistence region. The fourth term creates an elastic penaltyfor local distortion of the nematic order, within the (standard)“one elastic constant” approximation , with K being the re-sulting single elastic constant. The term f int ( φ , Q ) = L ( ∂ α φ ) Q αβ ( ∂ β φ ) (5)takes into accout the anchoring of the nematic liquid crystal onthe surface of the droplet. The constant L controls the anchor-ing strength: if negative the director is aligned perpendicularly(homeotropic anchoring) to the surface, whereas if positive thedirector is aligned tangentially to the surface (planar anchor-ing). Finally, in presence of confining walls, one has to takeinto account the anchoring of the nematic with these bound-aries. This is described by the last term of Eq. (1) f W ( Q ) = W ( Q αβ − Q αβ ) , (6)where the constant W controls the strength of the nematic an-choring at the walls and Q αβ = S (cid:16) n α n β − δ αβ / (cid:17) with n α and S being respectively the direction and the magnitude ofthe nematic ordering at the walls.The dynamical equations governing the evolution of thesystem are ∂ t φ + ∂ α ( φ u α ) = ∇ (cid:18) M ∇ δ F δφ (cid:19) , (7) ( ∂ t + (cid:126) u · ∇ ) Q − S ( W , Q ) = Γ H , (8) ∇ · (cid:126) u = , (9) ρ ( ∂ t + u β ∂ β ) u α = ∂ β σ total αβ . (10)The first equation, that governs the time evolution of theconcentration φ ( (cid:126) r , t ) , is a convection-diffusion equation for amodel B where M is a thermodynamic mobility parameterand δ F / δφ is the chemical potential.The dynamics of the liquid crystal order parameter, the ten-sor Q , is described by Eq. (8) which is a convection relaxationequation. The first two terms on the left hand side are thematerial derivative. Moreover, since for rod-like moleculesthe order parameter distribution can be rotated and stretchedby flow gradients, a further contribution S ( W , Q ) is needed .Its explicit expression is S ( W , Q ) = ( ξ D + ω )( Q + I / ) + ( Q + I / )( ξ D − ω ) − ξ ( Q + I / ) Tr ( QW ) , (11)where D = ( W + W T ) / ω = ( W − W T ) / W αβ = ∂ β u α and I is the unit matrix. The con-stant ξ takes into account the aspect ratio of the molecules of a given liquid crystal and determines the dynamical behaviourof the director field under shear, in particular whether it is flowaligning ( ξ ≥ . ξ < . Γ is the collective rotational diffusion constant and H = − δ F δ Q + I Tr δ F δ Q . (12)is the molecular field (this is the analogue for Q of the chemi-cal potential δ F / δφ ).Finally, the last two equations are respectively the conti-nuity and the Navier-Stokes equations for an incompressiblefluid. Here σ total is the total hydrodynamic stress which is thesum of three contributions. The first term is the viscous stress σ visc αβ = η ( ∂ α u β + ∂ β u α ) , (13)where η is an isotropic shear viscosity . The second one isthe contribution to the stress due to the liquid crystalline orderand is given by σ lc αβ = − P δ αβ − ξ H αγ ( Q γβ + δ γβ ) − ξ ( Q αγ + δ αγ ) H γβ + ξ ( Q αβ − δ αβ ) Q γ µ H γ µ + Q αν H νβ − H αν Q νβ , (14)where the pressure P is given by P = ρ T − K ( ∇ Q ) (where T is the temperature), and, except in proximity of the dropletsurface and of the defect cores, is costant in our simulationsto a very good approximation. The last term is the sum of theinterfacial stress between the isotropic and the liquid crystalphases with the elastic stress due to the distortions within theliquid crystal phase σ s αβ = − (cid:18) δ F δφ φ − F (cid:19) δ αβ − δ F δ ( ∂ β φ ) ∂ α φ − δ F δ ( ∂ β Q γ µ ) ∂ α Q γ µ . (15) Eqs. (7,8,9,10) are solved numerically by using a hybrid lat-tice Boltzmann method previously used in similar systemssuch as binary fluids , liquid crystals and active mat-ter . Unless explicitely stated otherwise, most of the simu-lations are performed on a two-dimensional rectangular lattice( Lx = Ly = φ is initially set to zero inside thedroplet and to a constant value φ in the bulk nematic phase.Similarly, the order parameter Q is initially set to zero insidethe droplet and different from zero elsewhere. In particularthe initial direction of the director in the nematic phase equalsthe one imposed at the walls. This means that the director inthe bulk is along the y -direction for homeotropic (or perpen-dicular) anchoring at walls and along the x -direction if homo-geneous (or tangential) anchoring is instead considered. Weincidentally note that these initial conditions lead to a dropletwhose final size, after equilibration, depends on the anchoringconditions of the director on its surface. A similar effect hasbeen observed in Ref. in which an isotropic droplet is em-bedded in a polar liquid crystal, and has been ascribed to thedeviation of the equilibrium values of φ of the two phases fromthe ones at the minima of the free energy (which are φ = , <
1% and does not affect in an appreciable way the dynamicsof the droplet under shear. Finally at the walls we impose no-slip boundary conditions for the velocity field, neutral-wetting(meaning that there are no flows of matter across the walls )for the concentration field and strong anchoring of the nematic(this is achieved by setting the anchoring strength W = . a = × − , D = × − , φ = Γ = A = η (cid:39) .
67. Note that the parameter ξ , appearing in Eq. (8),tunes the dynamical response of the director field under shear.We have set ξ = . . For completness we have alsoperfomed simulations with ξ = .
5. This corresponds to theflow tumbling regime where a steady solution under a con-stant shear rate no longer exists and the director rotates in adirection consistent with the vorticity of the flow. Althoughthe overall dynamics of the director field in both regimes is di-verse, we have not found appreciable differences in the defectdynamics under a shear flow and here we only report the flowaligning case.The key parameters for determining the shape of the dropletare the tension coefficient, which we have kept fixed to κ = .
14 (for stability reasons), the interface cross-gradient coef-ficient L , which ranges from ± − to ± × − (negativevalues for strong homeotropic anchoring and positive valuesfor strong tangential anchoring), and the elastic constant K whose values varies between 8 × − and 8 × − . A suit-able dimensionless quantity which characterises the shape ofthe droplet is the ratio λ = F int R / ( Σ K ) , where F int = (cid:82) V f int dV (measured in units of N = Jm − ), Σ is the perimeter of the droplet (see Appendix 1 for the calculation of Σ when thedroplet shape is deformed under shear) and R its radius. Thisquantity measures the strength of the surface anchoring rela-tive to the bulk elastic deformations . Typical values for theanchoring strength F int / Σ found in the literature rangefrom 10 − J m − to 10 − J m − , for an elastic constant K of 10 − N and for a droplet size of 1-10 µ m. This leads toa corresponding value of λ varying within the interval 10 − -10 . If λ < λ (cid:29) σ = F κ φ / Σ , where F κ φ = (cid:82) V dV κ / ( ∇φ ) . If κ = . σ (cid:39) − in simulation units. This corresponds to a physi-cal value of 10 − J m − , whereas experimental values varyin the range (cid:39) ( − − − ) J m − . Unless stated oth-erwise, we have kept this quantity constant. In the next sec-tion we show that, depending on the values of these param-eters, several possible equilibrium shapes, whose structure isaffected by the position of topological defects, can be identi-fied. These equilibrium states will in turn respond differentlywhen sheared. Similar shapes have been found and discussedin Ref. .Note that all the aforementioned values are in simulationunits. In order to map them into physical units we follow theapproach given in Ref. . In particular, assuming to modela flow-aligning regime, ∼ µ m thick device with a rotationalviscosity of roughly 1 poise (a typical value of 5CB), one getsthe length-scale and time-scale to be respectively ∆ x = − mand ∆ t = − s. Furthermore, an elastic constant K rangingbetween 8 × − − × − in simulation units, correspondsto (cid:39) −
100 pN, within the typical values of nematic liquidcrystals. To compare the effect of anchoring in simulationsand experiments, we can use the dimensionless parameter λ defined previously. We first characterize the equilibrium properties of an isotropicdroplet in nematic when either homogeneous or homeotropicanchoring of the director at the surface of the droplet is con-sidered. For each case we also look at the effect that differentdirections of the anchoring at the walls (i.e. either perpendic-ular or parallel) may have on the equilibrium configurations.Later on the dynamical response of the equilibrated config-urations when subject to a shear flow (flow aligning regime) isstudied for different shear rates.
We first consider an isotropic droplet of radius R =
15 locatedat the centre of a rectangular box of size L x = L y = (a) (b)(d)(c) Fig. 1
Equilibrium director profile of an isotropic droplet in a nematic host when homeotropic anchoring is set on its surface, while on bothwalls there is strong homeotropic anchoring. In the background is shown the corresponding profile of the largest eigenvalue of the tensor orderparameter Q : this goes from zero, inside the droplet (black region), to (cid:39) .
33 in the nematic phase (yellow region). Red circles indicate theposition of the defects. Parameters are: (a) K = × − , L = − × − , λ (cid:39) . × − ; (b) K = × − , L = − × − , λ (cid:39) .
1; (c) K = . × − , L = − × − , λ (cid:39) .
1; (d) K = × − , L = − × − , λ (cid:39)
13. As mentioned in the text the simulation box isrectangular but here only its central part is shown. he anchoring on the surface of the droplet is homeotropicand the director is perpendicularly anchored on both walls.The corresponding simulation is run for 5 × time-stepsuntil the droplet and the surrounding medium are completelyequilibrated, a state achieved when the total free energy is atits minimum. In Fig. 1 we show four equilibrium configura-tions obtained by appropriately changing the elastic constant K and the surface anchoring strength L . In terms of the num-ber λ , these correspond to λ (cid:39) . × − (Fig. 1a), λ (cid:39) . λ (cid:39) . λ (cid:39)
13 (Fig. 1d) ∗ . When stronghomeotropic anchoring is set on the surface of the droplet(see Fig. 1), an imaginary defect of topological charge 1 isenucleated at its centre. The conservation of the topologicalcharge requires though the formation of two defects of topo-logical charge − /
2, which are located on opposite parts ofthe droplet (see, for instance, Fig. 1c or d). This is the 2Dversion of the well-known Saturn ring . The position ofthe defects pair can be controlled by properly balancing thestrength of the surface anchoring and the bulk elastic distor-tions (namely the parameter λ ). For λ (cid:28)
1, for instance, sur-face anchoring is very weak and defects disappear, leaving theshape of the droplet unaltered (Fig. 1a). Notice that this isin agreement with the topological charge conservation, as inabsence of anchoring the imaginary defect (of charge 1) doesnot form inside the droplet, maintaining the total charge zero.When the anchoring strength becomes comparable with theelastic nematic energy ( λ (cid:39) λ , both de-fects emerge on opposite sides of the droplet along the equa-tor (Fig. 1c). Lastly, for λ (cid:29)
1, since the elastic liquid crystalenergy becomes very small compared to the surface energy,the defects pair are clearly far apart from the droplet and wellinside the bulk nematic phase. Similar droplet configurationshave been also discussed in Ref. in which their dynamics inpresence of an applied electric field is studied, and in Ref. where the role of a surfactant has also been taken into account.For homogeneous anchoring at the droplet surface, differ-ent steady states are expected. In this case we have consid-ered an isotropic droplet of radius R =
15 embedded in thenematic phase with homogeneous anchoring on both walls.This choice determines a final configuration in which two de-fects of topological charge − / K we have identified two equilibrium shapes (see Fig. 2).When the distortion energy overcomes the surface one, defects ∗ Since F int is negative we will implicitly consider its absolute value.† Other configurations are possible, such as a hyperbolic hedgehog (a defect ofcharge − are completely absorbed within the droplet and the directorprofile smoothly surrounds its surface (Fig. 2a). When botheffects become comparable, two defects form on both sideof the droplet, keeping unaltered the total topological charge(Fig. 2b). The position of defects changes if homeotropic an-choring is set on both walls. In this case they will form alongthe north-south direction of the system, on opposite sides ofthe droplet (not shown). However this final state is unstableto small perturbations and both defects eventually shift andmove along the droplet surface until they find a more stableconformation.We finally mention that, in line with previous nematody-namics studies , we found a small degree of biaxiality es-pecially close to defects. By following the approach usedin Ref. , regions of biaxial order can be found by calcu-lating the values of three parameters, namely e l = θ − θ , e p = ( θ − θ ) and e s = θ , where θ , θ and θ (with θ ≥ θ ≥ θ ) are the eigenvalues of the diagonalised ma-trix D αβ = Q αβ + δ αβ /
3. These parameters have the follow-ing properties: 0 ≤ e l , e p , e s ≤ e l + e p + e s =
1. Awell-ordered uniaxial nematic arrangement will give e l (cid:39) e s (cid:39) e p (cid:39)
1, respectively. Inparticular we found e p (cid:39) .
008 near the defects and e p (cid:39) By starting from the equilibrated droplets previously de-scribed, we now impose a shear flow on the system by mov-ing the top wall along the y -axis with velocity u and the bot-tom wall in the opposite direction with velocity − u . Thissets a shear rate ˙ γ = u / L y measured in ∆ t − in simulationunits. In all the cases studied, for low shear rates (typically for˙ γ (cid:39) − ) the initial droplet configuration is the same as foran unsheared system, independently on the values of the elas-tic constant K and on the surface anchoring constant L . Wewill therefore discuss only results whose shear rate is strongenough to induce deformations and/or droplet motion.Besides the parameter λ , the shape of the droplet and therole played by the forces in the system can be characterizedby introducing the capillary number Ca = R ˙ γηΣ F κφ . This adimen-sional quantity, often used in rheological experiments, mea-sures the strength of viscous forces relative to the surface ten-sion acting at the interface between two immiscible fluids. Inthese simulations Ca is expected to range from ∼ . ∼ D = a − ba + b ( a and b represent the majorand the minor axis respectively), which measures the dropletdeformation under shear and goes from 0 (no deformation) to1 (full deformation). A further parameter worth considering isthe Reynolds number Re = ρ ˙ γ R / η , which measures the im- (a) (b) Fig. 2
Equilibrium director profile of an isotropic droplet in a nematic host when tangential anchoring is set on its surface. The anchoring isstrong and homogeneous on both walls. On the background the corresponding profile of the largest eigenvalue of the tensor order paramter Q ,which goes from zero inside the droplet (black region) to (cid:39) .
33 in the nematic liquid crystal phase (yellow region). Red circles indicate theposition of the defects. Paramters are: (a) K = × − , L = × − , λ (cid:39) .
56 and (b) K = . × − , L = × − , λ (cid:39) .
92. Only thecentral part of the rectangular box is shown. portance of inertial forces relative to the viscous ones: for lowshear Re (cid:28) Re increases upto ∼ L =
0, or λ =
0) is embedded into a nematic liquid crystal having a mod-est or low elastic energy. This is the closest approximation ofour system to the simple binary fluid case. In Fig. 3a we showthe steady state attained by the droplet after imposing a shear(with ˙ γ = . × − ) and the director profile of the nematicphase. Similarly to what observed in immiscible mixtures ofNewtonian fluids, the droplet elongates and afterwards alignsalong the direction of the shear flow, with a deformation ratethat, at steady state, is D (cid:39) . θ formed by the major axis of the dropletwith the shear direction is (cid:39) . the authors found an angle of 25 degrees for Re = Re (cid:39) .
31 and Ca (cid:39) . λ >
0) and higher shear rates are considered.Finally, we mention a couple of subtle points which needto be kept in mind in setting and interpreting our simulations.First, it is well known that, by increasing the shear-rate, thetemperature at which the isotropic-nematic transition occursdecreases . This would correspond to a conversion of theisotropic phase of the droplet into a nematic one. In our sim-ulations we avoid this effect by keeping the shear rate smallenough. Second, one needs to avoid interface-interface in-teractions that can occur in strongly deformed droplets understrong shear. This is achieved by considering a droplet witha sufficiently large radius (provided that its interaction withwalls at equilibrium remains negligible) embedded in a rela-tively long rectangular lattice (necessary to diminish the effectof the periodic image of the droplet). In our simulations, as thetypical interface thickness separating the droplet and the liquidcrystal is around 8 lattice sites (which corresponds to roughly µ m), a radius of at least 15 lattice sites is needed. In thenext Sections we will show the results obtained for dropletswith a larger radius ( R =
22) and compare these with the onesobserved for a smaller radius ( R = λ . We initially consider the case in which homeotropic anchor-ing is set on the surface of a droplet. This anchoring canbe experimentally achieved by means of chemical treatments,such as coating the droplet with a surfactant which favours aperpendicular alignment. As previously mentioned, we havesimulated a droplet of radius R =
22 located in a rectangularbox of size Lx = Ly =
120 (area fraction at equilibrium A eqiso / A eqlc (cid:39) . × − ) and a droplet of radius R =
15 in abox of size Lx = Ly =
80 (area fraction at equilibrium A eqiso / A eqlc (cid:39) . × − ).When λ (cid:28) − < ˙ γ < × − ) is expectedto be very similar to the case discussed in Fig. 3, in which adroplet with no surface anchoring ( L =
0) has been studied.Indeed for R =
22, besides the different elastic constant (now K = × − ) and the weak contribution of the surface an-choring ( L = − − ), the shear stretches and elongates thedroplet along the shear flow without generating any net mo-tion ‡ . In particular the droplet reaches a steady state with themajor axis forming an angle θ (cid:39)
43 degrees with the sheardirection. The steady state values of the deformation and cap-illary number are respectively D (cid:39) .
095 and Ca (cid:39) . L = R =
15. In thiscase an angle θ (cid:39) . γ = . × − , with a deformation rate D (cid:39) .
082 anda capillary number Ca (cid:39) . γ . However larger droplets permit the studyof the physics in a wider dynamical range and unveil unex-pected properties. For instance, a well-know effect observedon a droplet in a binary fluid mixture is its break-up as theshear is increased . This phenomenon is still observed whenisotropic droplets are surrounded by a nematic liquid crystal,regardless of the presence of topological defects. In particular,we have identified two possible rupture regimes related to twodifferent values of ˙ γ . For ˙ γ = . × − the droplet breaksin two smaller droplets whereas for higher values, for instance ‡ We set ˙ γ = . × − necessary to avoid an excessive shrinkage of theisotropic phase (droplet), hence Re (cid:39) . ˙ γ = . × − , three smaller droplets result. In Fig. 4 weshow the dynamics of the former case. The droplet, initiallymoderately stretched along the shear direction, afterwards un-dergoes a deeper deformation (see Fig. 4a-b) which furtherelongates it. Due to the high shear rate, the rod-like state ex-hibited in Fig. 4b is however unstable, and the droplet breaks(Fig. 4c) and divides in two separate smaller droplets (Fig. 4d)which move along opposite directions. Before the rupture therate of deformation increases up to D (cid:39) .
9. Unlike the singledroplet case in which the velocity field displays one vortex in-side the droplet (as in Fig. 3b), now two vortices are formed inthe two smaller droplets, with a magnitude much lower thanthe one near the walls. At late times both resulting dropletsgradually shrink and completely disappear.
Due to the low value of the anchoring strength L , the dynam-ics discussed so far shows several similarities with the oneobserved in a single isotropic droplet (binary fluid-like) un-der shear. On the other hand, when λ (cid:39) γ = . × − , the droplet, similarly to the previous case,stretches and elongates along the shear direction (Fig. 5a-b).The tilt of the major axis of the droplet can be measured bylooking at the time evolution of the angle θ it forms with theshear direction, as reported in Fig. 6b (left scale): the dropletorientation angle initially achieves a maximum at (cid:39)
45 de-grees and afterwards relaxes to a constant value of (cid:39)
41 de-grees. Notice that the value of the angle at the steady stateis very similar to the one measured for the λ (cid:28) x -axis with an almost constant speed (seeFig. 5c-d) as the position of the y component of the centre ofmass is slightly shifted upwards (see Fig. 6a in which the x andthe y components of the centre of mass are reported), and at- (a) (b) θ Fig. 3 (a) Steady state director profile of an isotropic droplet in a nematic host with no surface anchoring ( L = θ indicates the direction of the major axis of the droplet with the shear direction. The elastic constant is K = × − . (b) Steady state velocity field when a shear is applied. A vortex forms inside the droplet whereas intense flow fields, inopposite directions, appear near the walls. tains a final steady state whose shape is only weakly deformedby the shear (Fig. 6b, green crosses, plots the deformation pa-rameter D ). We call it bound state (BS), as both defects remainon the droplet surface.A persistent rotation of the defects can be achieved if theshear rate is further increased. Movie S2 shows the dynam-ics with ˙ γ = . × − . Similarly to the previous case, thedroplet quickly deforms and aligns along the shear directionwhile both defects rotate clockwise. However, due to highvalue of ˙ γ , the velocity field (very intense near the walls butweak in the centre, where the typical vortex pattern inside thedroplet forms) pushes the defects over the position attainedfor lower ˙ γ and drives them around the entire surface of thedroplet. In particular, during an entire cycle, they speed uptheir motion near the walls of the cell and then, at the end ofeach cycle, slow down as their respective distance from thewalls augments. This dynamics also leaves temporary wake-like signatures on the director field departing from the defects(red wakes in Movie S2) and short-living opposite charge to-pogical defects which annihilate quickly (red spots appearingin the nematic phase, see Movie S2). A measure of the angle θ that the major axis forms with the shear direction is reportedin Fig. 7 with a plot of the elastic free-energy F el = K ( ∂ α Q βγ ) + L ( ∂ α φ ) Q αβ ( ∂ β φ ) . (16)At long times the angle θ displays two close local maxima,stabilized around 38 degrees, spaced out by two local minima, one short and one large, both around 30 degrees. Althoughdefects continuously rotate on the surface of the droplet themajor axis exhibits a characteristic oscillatory behaviour, inwhich the two minima are achieved when defects slow downtheir motion (far from the walls) and the two close maximaduring the successive cycle (near the walls) (see Fig. 7, redplusses, left scale). The elastic free energy of the nematic os-cillates as well at long times (see Fig. 7, green crosses, rightscale); in particular its local maximum corresponds roughly tothe large minimum of θ and its minimum to the local maxi-mum (on its left) of θ observed during a cycle. A crude ex-planation of this can be arguably related to the backflow: thevelocity field, much higher near the walls than in the middleof the cell, aligns the director field more strongly when defectsare closer to the walls (hence diminishing the elastic free en-ergy) then far from them. Indeed, the director profile observedin these states supports this interpretation (see the insets (c)and (d) of Fig. 7). On the other hand, the director field at theother extremes of the angle θ is significatively different (seethe insets (a) and (b) of Fig. 7). In particular the state (b) has alower elastic free energy than that in (d), meaning that elasticdeformations are weaker in (b) than in (d) (notice that thesestates correspond to the two minima of θ ) whereas the elasticfree energy of the state (a) has a similar value of that in (b).Interestingly, both the director and the corresponding velocityfield (in the state (a)) acquire an out of plane component (alongthe z -direction) which accompanies the escape of the cores of (b)(c)(a)(d) Fig. 4
Dynamic evolution of a droplet of radius R =
22 without defects under shear with ˙ γ = . × − , K = × − and L = − − (atequilibrium without shear λ (cid:39) . he defect pair into the third dimension. The out of plane com-ponent of the velocity field in particular is roughly one orderof magnitude lower than the other two components. Unlikea droplet-free flow aligning nematic liquid crystal, the solepresence of topological defects on the droplet surface unveilsan unexpected flow-tumbling-like dynamics which is overallakin to that observed in a strictly two-dimensional system butfor a slightly higher value of the shear rate. In the latter inparticular both the director and the velocity field are not signi-ficatively different from those observed in the quasi-2d case,and the final steady state (the oscillatory bound state) is pre-served. In addition, at the steady state the rate of deformation D of the droplet weakly oscillates around 0 . (cid:39) .
28. The steady state described above, in whichthe shear stress induces an oscillatory-rotational motion of theantipodal defects pair close to the droplet, is, to our knowl-edge, a new result for an inverted nematic emulsion and wecall it the oscillatory bound state (OBS).When λ (cid:29) γ < − ) bothdefects remain close to the droplet surface, as in the oscilla-tory bound state; for intermediate shear rate only one defectgoes away from the droplet ( single bound (SB) state) and fi-nally, for sufficiently high shear rates, both defects move intothe bulk § ( unbound (U) state). In Fig. 8 and in Movie S3we report the dynamics of the SB state. For ˙ γ = . × − the droplet initially behaves similarly to what seen for theBS case where both defects, located on opposite sides alongthe equator, rotate simultaneously in the clockwise direction(Fig. 8a-b). Later on the rotation is arrested and the droplet-defect system, dragged by the fluid flow, slowly moves unidi-rectionally along the x -axis (Fig. 8c), whereas the defect nearthe bottom wall gradually leaves the droplet and moves intothe bulk along the opposite direction (from right to left in thebottom half part of the system) (Fig. 8d). This defect mo-tion leaves a characteristic comet-like signature of the directorfield in the bulk. Note that the SB state of Fig. 8d occurs be-cause, at equilibrium (i.e. before the shear is switched on), thedroplet centre of mass is not equidistant from the two walls( y cm (cid:39) . > L / x -direction and, more importantly, reduces the dis-tance between its surface and the top defect, increasing at thesame time the distance from the bottom defect, which is thenpushed backwards by the fluid. It is interesting to note that,after the bottom defect moves into the bulk, the droplet ac- § For extremely high shear rates the isotropic phase disappears as usual. celerates attaining a novel steady state with a higher velocity.During this process the total and the elastic free energies showa very similar behaviour: at early times (when defects just ro-tate around the droplet) they increase rapidly. At intermediatetimes instead they increase very slowly (this regime refers tothe situation in which the defects move slowly along oppositedirection parallel to the walls). Finally, at longer times, whenthe droplet and the bottom defect move apart, they increasequite rapidly again (see Fig. 9b).A state in which both defects migrate into the bulk (U,unbound, state) is achieved by further increasing the shearrate. In Fig. 10 and in Movie S4 we show this situation(˙ γ = . × − ). The early times dynamics is similar to thatobserved for the SB state (see Fig. 10a-b), except for a largerdeformation of the droplet ( D (cid:39) . D (cid:39) .
08 inthe SB state of Fig. 8). However, the shear rate is now intenseenough along the x -direction to detach both defects from thedroplet. Interestingly, there is not an appreciable motion of thedroplet along the shear direction until the defects, reappear-ing at the periodic side of the lattice, reapproach the dropletagain. It is worth noticing that an unbound state could bealso achieved by suitably controlling the elasticity of the liq-uid crystal. Indeed, we have found a very similar dynamics fora droplet with a smaller radius ( R =
15) with the parameters ofFig. 1d (hence with K = × − ) but with ˙ γ = . × − . The results just presented can be summarised and rationalisedinto a phase diagram in which the observed steady states arereported in the λ − ˙ γ plane. The diagram displays differentdynamical responses of the droplet under a linear shear. For λ (cid:28)
1, when defects are not present, a dynamical responsetypical of a Netwonian emulsion dominates for all values ofshear rates ˙ γ . In particular if ˙ γ ≤ − the droplet deforms andelongates along the shear flow whereas, for higher values of˙ γ , it breaks into smaller droplets and eventually evaporates ¶ .If, on the other hand, λ (cid:39)
1, defects pair forms very closeto the droplet surface and, due to the interplay between theirdynamics and the deformation-rotation dynamics of droplet,three new interesting steady states can be identified. For smallshear rates defects are bound in proximity of the surface ofthe droplet and no appreciable difference with the standard bi-nary fluid behaviour is observed. Increasing ˙ γ unveils a regionin which the motion of the droplet along the shear directioncombines with a partial rotation of the defects along its sur-face. If ˙ γ is further increased, a second type of steady stateoccurs, in which defects are still bound to the droplet surfacebut now persistently rotate around it. Clearly, for sufficiently ¶ We borrow this term from the liquid-gas thermodynamics as the shrinkageof the isotropic phase is due to the shift of the temperature of the isotropic-nematic transition which favours the liquid-crystal phase. (b)(c)(a)(d) Fig. 5
Dynamic evolution of the droplet of radius R =
22 with defects located on its surface (see Fig. 1b for the corresponding equilibriumstate) with ˙ γ = . × − . While they rotate in the clockwise direction ((a)-(b)), the droplet deforms and elongates along the shear flow((a)-(b)) and, at the same time, acquires unidirectional motion along the x -axis dragged by the fluid ((c)-(d)). The capillary number is (cid:39) . A iso / A lc (cid:39) . × − . See Movie S1 for the complete dynamics.
200 220 240 260 280 300 320 340 0 200000 400000 600000 60.4 60.45 60.5 60.55 60.6 60.65 60.7 x c m y c m tx cm y cm θ D t θ D (a) (b) ab c d a b c d Fig. 6
In panel (a) a plot of the x and y components of the centre of mass (red plusses-left scale and green crosses-right scale, respectively) ofthe droplet of Fig. 5 are reported. The droplet acquires motion with an almost constant speed since the shear is switched on. In panel (b) theangle the major axis forms with the shear direction (red plusses, left scale) and the deformation parameter D (green crosses, right scale) arereported. Logarithmic time scale is set on the x-axis. The droplet attains a weakly deformed steady state with the major axis forming an angleof (cid:39)
43 degrees with the shear direction. The black lines in both panels indicate the times at which snapshots (a)-(d) of Fig. 5 are taken. high values of ˙ γ , the droplet breaks up and eventually evapo-rates. For λ (cid:29) γ , only one defect remains close to the droplet whilethe other one starts to move freely with the shear flow (singlebound state). When, on the other hand, ˙ γ is sufficiently high,the second defect leaves the droplet as well and moves in thebulk (unbound state). When homogeneous (or tangential) anchoring is set on thesurface of the droplet a different dynamical response of thesystem is observed. For these cases, our simulations are per-formed by using a rectangular box of size Lx = Ly = R =
15. When defects are imaginary(see Fig. 2a), the dynamical behaviour under shear is similarto the one observed with homeotropic anchoring at low shearrate (˙ γ = . × − ): the droplet aligns with shear flow with θ (cid:39)
40. In the presence of defects at the droplet surface (asin Fig. 2b), however, the situation is different from that withhomeotropic anchoring. In this case, we observe a slow driftof the droplet along the x -direction induced by the defect atits right. Accordingly, the droplet increases its distance fromthe defect located on its left, which gradually leaves the sur-face of the droplet (see Movie S5). This establishes a transientsingle-bound state which lasts until the “free” defect wrapsthe periodic boundaries: when this happens, the mobile de- fect rotates clockwise around the surface and this time remainsbound, trapped by the elastic interactions. A similar dynam-ics has also been observed for higher shear rates, before thedroplet evaporates. Steady unbound states with defects in thebulk of the nematic host could instead be found either by de-creasing the nematic elastic constant K or by increasing thesurface anchoring L . Indeed, when imposing L = . × − the equilibrated droplet looks very similar to the one of Fig. 2;however, under a shear rate of ˙ γ = . × − we now observedefect detachment. This is possible because a large value of L augments the distance between each defect, which favoursdetachment. The rich phenomenology found in the 2D system may be ob-served in thin film of inverted nematic emulsions under shear;however it is of interest to ask to which extent this also occursin a fully 3D system. For simplicity, we only consider herethe case where without shear the droplet is accompanied by aSaturn ring defect (the case of a hyperbolic hedgehog defectmay also be of interest). Previous experimental and numeri-cal studies on micron-size droplets hosted in a liquid crytstaland advected by a flow indicate that the Saturn ring defectsformed around them are visibly displaced in the downstreamdirection and eventually collapse into a hyperbolic point de-fect . Here we look at how a linear shear may impacton the Saturn ring-droplet system by restricting ourselves to3D isotropic droplets hosted in a liquid crystal fluid that is θ F e l t θ F el a b c d Fig. 7
A plot of the angle θ the major axis of the droplet forms with the shear direction (red plusses, left scale) and a plot of the elastic liquidcrystal free-energy (green crosses, right scale) versus simulation time are reported for the dynamics shown in Movie S2. The angle θ displaysa periodic behaviour in which two close maxima around 38 degrees correspond to the dynamic state of both defect close to the walls, whereasat the local minima defects are far from walls. The elastic free-energy (the sum of the elastic term in the bulk free energy multiplied by theconstant K and the surface anchoring contribution multiplied by the constant L ) oscillates too. The four insets show the droplet and the nearbydirector profile. Red and blue circles indicate the position of the defects. Here A iso / A lc (cid:39) . × − at steady state. (b)(c)(a)(d) Fig. 8
Dynamic evolution of a droplet of radius R =
22 with defects located out of its surface with ˙ γ = . × − (see Fig. 1c or d for thecorresponding equilibrium state). The droplet is stretched along the shear flow and the two defects, initially on opposite side of the droplet (a),move in clockwise direction along its surface (b). Afterwards the one near the bottom wall detaches from the surface (c) and migrate awayfrom it (d), leaving a characteristic comet-like signature on the director field, whose bulk structure is now almost completely aligned along theshear flow. At steady state the capillary number is (cid:39) .
095 and A iso / A lc (cid:39) . × − . See also Movie S3 for the complete dynamics.
200 220 240 260 280 300 320 340 360 380 0 150000 300000 450000 60 60.5 61 61.5 62 62.5 63 63.5 x c m y c m tx cm y cm -310.4-310.2-310-309.8-309.6-309.4-309.2-309-308.8 0 150000 300000 450000 -0.8-0.6-0.4-0.2 0 0.2 0.4 0.6 F F e l tFF el (a) a b c d (b) a b c d Fig. 9 (a) x and y components of the centre of mass (red plusses, left scale and green crosses, right scale respectively) relative to the dynamicsshown in Fig. 8. After the shear is switched on the droplet acquires unidirectional motion along the x -axis, with an almost constant velocity atearly times, up to t (cid:39) × . Afterwards, when the lower defect detaches from the droplet, this moves faster attaining a second steadystate with a higher velocity. On the other hand the y -component remains almost constant although initially, when the droplet is at rest, slightlyshifted upwards. This breaks the symmetry and allows the droplet to move along a preferential direction. (b) A plot of the total free energy(red plusses, left scale) and the elastic one (green crosses, right scale) for the dynamics shown in Fig. 8 is reported. Black lines indicate thetimes at which the corresponding snapshots of Fig. 8 are taken. homeotropically anchored to the droplet surface. The systemis made by a single droplet of radius R =
14 sandiwched be-tween two walls at L z = L z =
40. Along the x and y directions, periodic boundary conditions are considered with L x =
60 and L y =
40. Since, as a first approximation, thedroplet is expected to assume a generic ellipsoidal shape, wecompute a set of three dimensionless numbers, c l , c s and c p ,measuring respectively the degree of prolateness, sphericityand oblateness of the ellipsoid.These quantities are defined as c l = a − ba + b + c , (17) c s = ca + b + c , (18) c p = ( b − c ) a + b + c , (19)where a , b and c are the principal axes of the ellipsoid, suchthat a ≥ b ≥ c ≥
0. For example, to establish whether the el-lipsoid is more prolate than oblate one can compare c l with c p :if c l > c p , the ellipsoid tends to be prolate whereas if c l < c p the droplet is more oblate. Two extreme cases can be identi-fied: for c l = c p = a = b > c =
0) the ellipsoid degeneratesinto a two-dimensional circle. Finally if c s = a = b = c ). Since, similarly to the 2D case, we expect that the equilib-rium location of the defect and its dynamics under shear flowdepends on the adimensional ratio λ , we consider the two ex-treme cases of λ (cid:39) λ >>
1. In both cases the systemis first let to equilibrate in absence of shear; the resulting con-figuration is then used as initial condition of the shear experi-ment.The equilibrium state for λ (cid:39) c l (cid:39) . c s (cid:39) .
89 and c p (cid:39) . z -axis, except in proximity of the droplet surface where,due to the strong homeotropic anchoring, a large splay-benddistortion emerges (Fig. 12a). When a moderate shear flow isimposed on the system (˙ γ = . × − ), the droplet does notmove with the flow but simply stretches and elongates alongthe shear flow (resembling the 2D case of Fig. 5), and achievesa final steady state whose shape is that of a prolate ellipsoid( c l (cid:39) . (b)(c)(a)(d) Fig. 10
Dynamic evolution of a droplet of radius R =
22 with defects located out of its surface with ˙ γ = . × − (see Fig. 1c or d for thecorresponding equilibrium state). The dynamics at early times ((a) and (b)) is analogous to that of the SB state, except for a larger dropletdeformation D = .
2. Indeed, the droplet stretches along the shear flow and defects simultaneously rotate clockwise. Afterwards, due to thehigh shear rate, both defects are dragged away from the surface of the droplet, whose position, unlike the SB case, is not appreciably changed.At steady state the capillary number is (cid:39) .
28 and A iso / A lc (cid:39) . × − . ˙γ λ Single boundstate Unbound stateBound state -3 -4 -4 Newtonianlike Oscillatorybound state -3 Breakup Fig. 11
A qualitative phase diagram of the steady states observed under shear in the λ − ˙ γ plane (both in simulation units). For λ (cid:28) λ (cid:39) γ the droplet is either quiescent or acquires motion along the shear direction combined with a partial rotation of the defects. This isthe bound state (BS). If ˙ γ is further augmented, a persistent rotation of both defects is found. This is the oscillatory bound state (OBS). For λ (cid:29) γ one of the defects leaves the droplet while the other one remainsconnected (single bound state (SB)). For higher values of ˙ γ both defects are dragged by the shear flow and disconnect from the droplet(unbound state (US)). In all cases, for very high ˙ γ , the droplet breaks into smaller droplets which eventually evaporate. The circles indicate theposition in the phase diagram of the systems described in the figures of the manuscript. More specifically we have: λ =
0, ˙ γ = . × − (redcircle); λ (cid:39) .
49, ˙ γ = . × − (green circle); λ (cid:39) .
58, ˙ γ = . × − (blue circle); λ (cid:39) .
58, ˙ γ = . × − (orange circle); λ (cid:39) γ = . × − (yellow circle); λ (cid:39)
3, ˙ γ = . × − (cyan circle). alls. The Saturn ring itself does not experience an apprecia-ble deformation but rotates in the y − z plane with its symmetryaxis (passing through the centre of the ring) remaining almostparallel to the major axis of the ellipsoid.For higher values of the shear rate the dynamics of thedroplet and accompanying Saturn ring is significatively differ-ent. In Fig. 13 we report a simulation in which ˙ γ = . × − (see also Movie S7). After the shear is switched on, thedroplet initially aligns and elongates along the flow direction(Fig. 13b) achieving a highly prolate ( c l ∼ .
18) intermedi-ate (non-steady) state (Fig. 13c). In this state the Saturn ringremains firmly anchored at the droplet surface. Interestingly,though, instead of surrounding the equator of the droplet (asfor the case when ˙ γ = . × − ), the disclination ring fol-lows the droplet deformation and stretches along the entiresurface. Later on a more complex rearrangement is observed:the droplet moves slightly upwards (along the z -axis), in re-gions of the system in which the flow (direct along the x -axis)is more intense, and rotates around its major axis (Fig. 13d).Finally it is pushed forwards and the Saturn ring slips down-stream, opposite to the direction of motion (Fig. 13e-f). Thedownstream motion of the Saturn ring agrees with the previ-ous studies on similar systems and can be considered asthe 3D concounterpart of the 2D bound state observed before(see Fig. 5).For λ (cid:29) λ (cid:39) γ = − ). Initially the droplet stronglydeforms ( c l (cid:39) . c s (cid:39) . c p (cid:39) .
08) with its major axisaligning almost parallel to the shear flow, while the Saturnring, although located almost at the centre of the droplet, un-dergoes an S -like deformation (Fig. 14b-c), less pronouncedthough than that observed for λ (cid:39) γ = . × − . Af-terwards, the droplet is advected by the flow along the x -axiswhile the disclination ring slips towards the rear part of theemulsion where eventually it gets pinned (Fig. 14d-e-f). In-terestingly the shift of the Saturn ring occurs whenever thedroplet acquires motion, namely when, for instance, its centreof mass shifts (either upwards or downwards). Thus, unlike in2d, where for λ (cid:29) In conclusion, we have studied the dynamics of a 2D isotropicdroplet immersed in a nematic host under an imposed sym-metric shear flow. The physics of the steady states is mostlydetermined by the shear rate and the strength of the anchor-ing of the director field at the droplet surface. For weak orno anchoring, the system behaves similarly to a droplet in abinary mixture of isotropic, Newtonian, fluids. On the otherhand, for intermediate or strong anchoring a variety of non-trivial non-equilibrium steady states are possible. One op-tion is to create an oscillatory steady state, where the droplet,for instance, tumbles in the flow. Another option is to createhighly dynamic states where one or both the defects, whichaccompany the droplet when quiescent, detach from the sur-face of the droplet and become mobile due to the imposedshear. For 3D systems and homeotropic anchoring the defectpair becomes a Saturn ring surrounding the droplet. We haveshown that, regardless of the shear rate and of the surface an-choring strength, the disclination ring always remains local-ized around the droplet although it can be highly deformedand shifted downstream (similarly to the moving bound stateobserved in 2D). In this respect the unbound and single de-fect states observed in 2D can be seen as a limiting case ofan inverted emulsion confined within a very thin film of liquidcrystal.All these findings suggest that the rheological response ofan isotropic droplet in a nematic host is much richer than pre-viously probed, and can further be tuned by varying strengthand nature of the anchoring, together with the magnitude ofthe imposed flow. A viable route to test our results experimen-tally is, for instance, in microfluidic and rheology experimentson water-liquid crystal emulsions (or other mixtures contaninga liquid crystal and a Newtonian fluid).
Here we briefly describe how we calculate the perimeter ofthe droplet when its shape is deformed under shear. Morespecifically, our simulations show that the droplet aligns alongthe direction of the shear flow, elongates and often acquiresmotion, with a shape reminding that of an ellipse. In order tocalculate the ellipse which best reproduces the boundary of b)(a) xxy yz
Fig. 12 (a) Equilibrium configuration of an isotropic 3D droplet embedded in a nematic liquid crystal homeotropically anchored both at thedroplet surface and at the walls. The equilibrium value of the parameter λ is roughly equal to 1. (b) Steady state of the system described in (a)under a moderate shear (˙ γ = . × − ). The droplet (namely its major axis) orients parallel to the shear flow while the Saturn ring (almostundeformed) rotates in the y − z plane following the major axis of the droplet. The director field is almost everywhere aligned along the sheardirection except in proximity of both the droplet surface and the walls, where strong hometropic anchoring is set. Here K = × − and L = − × − . (b)(a) y yz (c)(d) (e) (f) x x z xyxy z z xx yy Fig. 13 (a) Equilibrium configuration of a 3D isotropic droplet embedded in a nematic liquid crystal homeotropically anchored both at thedroplet surface and at the confining walls. At equilibrium λ (cid:39)
1. Panels (b-d) represent intermediate states of the system when it is subject to asufficiently strong shear rate (˙ γ = . × − ). When the shear is turned on, the droplet is initially stretched by the fluid (b) and afterwards italigns along the direction of the shear flow (c). Later on it slightly rotates around its major axis and moves upwards (along the z -axis), towardsregions of higher shear flow (d). The droplet then acquires unidirectional motion along the x -axis and the Saturn ring moves downstream, i.e.,in the opposite direction of the motion (e-f). Here K = × − and L = − × − . b)(a) y yz (c)(d) (e) (f) x x z xyxy z z xx yy Fig. 14 (a) Equilibrium configuration of a 3D isotropic droplet embedded in a nematic liquid crystal homeotropically anchored both at thesurface droplet and at the walls. Here λ (cid:29)
1. When a sufficiently high value of shear is established (˙ γ = − ), the droplet aligns along theshear flow and elongates into an elliptic prolate shape. The Saturn ring, initially surrounding the equator of the droplet, rotates (with the axisalmost parallel to the major axis of the emulsion) and slightly deforms ((b)-(c)). Later on, when the droplet is advected by the fluid along the x -axis, the Saturng ring moves downstream, and eventually gets pinned at the downstream extremity of the droplet ((d)-(e)-(f)). Here K = × − , L = − × − . the droplet, we compute the inertia tensor of a flat ellipticaldisk, which is given by I = (cid:18) ∑ i m i ( y i − y cm ) − ∑ i m i ( x i − x cm )( y i − y cm ) − ∑ i m i ( x i − x cm )( y i − y cm ) ∑ i m i ( x i − x cm ) (cid:19) , where x cm and y cm are the coordinates of the centre of massof the droplet. The mass m i of the i -th lattice site is related tothe local concentration φ i as follows: m i = − ( φ i − φ min ) φ max − φ min , (20)where φ min and φ max are the minimum and the maximum val-ues of φ in the system. In particular φ max = φ min = m i (cid:39) m i (cid:39) m i = m i < .
04. By diagonalizing the matrix I , one gets the prin-cipal axes of inertia, which coincide with the x and y axes inan unsheared system. The principal moments of inertia are1 / √ I x and 1 / (cid:112) I y , where I x = Mb / I y = Ma / M = ∑ i m i , and a and b are the semiaxes. Anestimation of the perimeter can be thus obtained through the Ramanujan formula Σ = π (cid:20) ( a + b ) + ( a − b ) ( a + b ) + √ a + ab + b (cid:21) . (21)Lastly, the angle that the major axis forms with the directionof the shear flow can be calculated from the eigenvectors of thematrix. References
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