Department of Physics Seminar - 4th year Nematodynamics of a Colloidal Particle Author: David Seˇ c Adviser: doc. dr. Daniel Svenšek Co-Adviser: dr. Miha Ravnik Ljubljana, January 2009 Abstract Dynamic flow phenomena in complex fluids are presented, with the particular focus on the flow around colloidal particles. Phenomenological Ericksen-Leslie theory of nemato- dynamics is presented. Possible simplifications of the complex nematodynamic formalism are discussed. Relevant numerical techniques are presented that are commonly used to solve fluid dynamic equations. To demonstrate the differences between fluid dynamics in isotropic and anisotropic liquids (nematic), the Stokes drag of a moving sphere in isotropic liquid and nematic liquid crystal is calculated.
Transcript
Department of Physics
Seminar - 4th year
Nematodynamics of a Colloidal Particle
Author: David Sec
Adviser: doc. dr. Daniel Svenšek
Co-Adviser: dr. Miha Ravnik
Ljubljana, January 2009
Abstract
Dynamic flow phenomena in complex fluids are presented, with the particular focus onthe flow around colloidal particles. Phenomenological Ericksen-Leslie theory of nemato-dynamics is presented. Possible simplifications of the complex nematodynamic formalismare discussed. Relevant numerical techniques are presented that are commonly used tosolve fluid dynamic equations. To demonstrate the differences between fluid dynamics inisotropic and anisotropic liquids (nematic), the Stokes drag of a moving sphere in isotropicliquid and nematic liquid crystal is calculated.
Various types of flows occur in natural and technical environment. Without fluid flows lifewould probably be impossible or would be at least extremely different. The important role offlows can be seen in various transport processes in human beings, which supply our body withthe oxygen and other essential nutrients. Flows are vital in rivers, lakes, and seas, as well as inall atmosphere, where they influence the weather and thus climate greatly. Finally, flows are oflarge importance in technical environment, where multiple processes depend on fluid flows. Forexample, fluid flow coupled with chemical reactions provide the combustion in piston engines,whereas flow around airplane wings generate required lift forces. That is why, the study of fluiddynamics is important. One can therefore, predict the flow development (e.g. forecast weather),improve technological equipment (minimize energy loss resulting from the flow resistance incars, airplanes, etc.), gain knowledge that would result in novel applications [1].
Interesting field of fluid dynamics is microfluidics, the science and technology of systemsthat process or manipulate small (10−9 to 10−18 liters) amounts of fluids, using channels withdimensions of tens to hundreds of micrometers [2]. There exist important advantages in usingsuch microscopic analysis: relatively small quantities of samples can be used, high resolution,good sensitivity and low costs. Analysis could be made by Lab-on-a-chip, which is a small de-vice with size to few square centimeters that has one or several laboratory functions integrated
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[3]. One of the elements in a way to the Lab-on-a-chip is fabrication of micro pumps. An inter-esting solution was just recently provided by our faculty research group under the supervisionof Professor Igor Poberaj [4]. A microscale rotor pump (Fig. 1) was made from self-assembledsuperparamagnetic colloidal spheres and driven by an external magnetic field. Magnetic fieldwas induced with two orthogonal coil pairs, driven by sinusoidal current. Electric field, whichgoverns the cluster position and thus the direction of flow, was produced with separate mi-croelectrodes at each pump. This allows to control each pump’s speed and direction of flowseparately. Such pumps can be also fabricated with existing technologies and easily integratedinto microfluidic devices.
Figure 1: (a) An ilustration of microscale pump made from a cluster of superparamagnetic colloids thatrotate in the same direction as the magnetic field B. Microelectrodes are presented with black color. (b)A pump in a microfluidic channel, the position of the cluster determines the direction of the flow [4].
The fluid dynamics is governed by the Navier-Stokes equation. It arises from the New-ton’s second law applied to fluid motion. Most often the incompressibility of liquid can beassumed, because the changes in pressure and temperature due to the flow are sufficiently smallthat changes in density are negligible. For example, compressibility of water, which describesthe relative change of volume as a response to change of pressure, is roughly 5 × 10−10Pa−1
[5]. That means that even in oceans at the depth of 4 km there is only 1.8% increase in density.When the fluid is not isotropic the equations describing flow complicate even more.
The seminar is organized as follows. In Chapter 2 liquid crystal physics is introduced. Then,in Chapter 3 the Ericksen-Leslie equations governing the nematodynamics are described. InChapter 4, some numerical methods common for solving these equations are described. Finally,the differences in fluid flow for isotropic and nematic liquid around the spherical particle arepresented.
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2 Liquid crystals
Liquid crystals are materials that are more ordered than ordinary liquids but less ordered thansolids. They are made of rigid organic molecules that self-organize at a certain temperatureor molecular concentration. Nematic mesophase is the least ordered among liquid crystals,since only long-range orientational order of the molecules is present, but the centers of mass ofthe molecules are disordered like in a liquid phase. Nematics are formed by achiral elongatedmolecules, which can be approximated by rotational ellipsoids [6].
The study of liquid crystal hydrodynamics has recently turned out to be useful also fordescribing some biological systems. Certain types of bacteria in suspensions swim in swirlsthat suggest long-range orientational order [7]. Another examples of self-organizing systemare cytoskeletal filaments and motor-proteins with intrinsically non-equilibrium ordering. Thecollective behavior in these systems can be described in the continuum limit. Such systemare analogous to liquid crystals and can be thus represented by the same phenomenologicalhydrodynamic model, here referred to as active-liquid-crystal model.
2.1 Orientational order
To characterize the orientational order of molecules in liquid crystals, nematic order parametersare introduced [8]. The vectorial order parameter, named director n(r, t), describes the averageorientation of nematic molecules at a given position r at a time t. The ordering is such that thestates n and −n are equivalent (no dipolar ordering).
The nematic degree of order S quantifies orientational fluctuations of uniaxial moleculesaround the director [8]
S =12
⟨(3 cos2 θ − 1
)⟩=
12
∫f (θ)
(3 cos2 θ − 1
)dΩ, (1)
where θ denotes azimuthal angle in spherical coordinates, f (θ) the orientational distributionfunction of molecules, dΩ solid angle and < . > denotes ensemble average. The values ofS lie in the interval [−1
2 , 1]. For a perfectly ordered nematic where all the molecules pointexactly along the director S = 1 and for a perfectly disordered state where molecules have nopreferential direction S = 0. The value S = −1
2 represents the state where all molecules arealigned in the plane perpendicular to n. Typical values for liquid crystals are S ∼ 0.3 − 0.8 [6].
The nematic degree of order can be spatially dependent. When the molecules are not uni-axial or molecular fluctuations about the director are not rotationally symmetric to director, fullorder parameter tensor must be introduced [9]. The full order parameter tensor is usually neededwhen describing defects or when external constraints (surfaces, magnetic or electric fields) breakthe uniaxial symmetry.
2.2 Frank elastic theory
Nematic liquid crystal with uniform director field has the lowest free energy [10]. Any defor-mations of the orientational ordering increase energy, and as a result nematic acts as an effectiveelastic medium. When director is deformed, nematic tends to align to a spatially uniform di-rector field. For example, supposing that one could grab and bend the director field, it wouldeffectively act as a bent stick and try to relax to a uniform configuration. Any elastic defor-mation can be decomposed in three basic deformation modes (splay, twist and bend) that areschematically presented in Fig. 2.
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Figure 2: Liquid crystals act as effective elastic materials and any elastic deformation can be decom-posed in three basic deformation modes: (a) splay, (b) twist, and (c) bend. The arrows show the directionof the director [8].
Slowly varying spatial deformations, i.e. director gradients are small (no defects), are de-scribed phenomenologically by expanding free energy volume density into invariants of n thatare allowed by symmetry. The result is Frank-Oseen free energy volume density [8, 11]
fE =12
K1 (∇ · n)2 +12
K2 [n · (∇ × n)]2 +12
K3 [n × (∇ × n)]2 , (2)
where K1, K2, K3 are elastic constants that characterize splay, twist, and bend deformations,respectively. Besides the volume density terms, there also exist surface contributions (anchoring,surface elastic terms). The constants Ki have the nematic degree of order incorporated and areproportional to S 2. One can note that the free energy density (Eq. (2)) is a quadratic form forthree basic deformation modes and obeys the n→ −n inversion symmetry.
Many materials have similar elastic constants. Therefore, a common approximation for freeenergy volume density is to use only one average elastic constant that sets K1 = K2 = K3 = K.This simplifies the complex spatially dependent Frank-Oseen free energy functional which canbe now written as
f oneE =
12
K[(∇ · n)2 + (∇ × n)2
], (3)
which is without surface terms the same as
f ′Eone=
12
K(∂in j
)2. (4)
Equilibrium director field is obtained by the variation of the free energy F
δF =∫δ fE dV =
∫ ∂ fE
∂niδni +
∂ fE
∂(∂ jni
)δ (∂ jni) dV = 0. (5)
One obtains Euler-Lagrange equations for the director
∂ f∂ni− ∂ j
∂ f
∂(∂ jni
) = 0 with the constraint n2 = 1, (6)
where summation over repeated indices is assumed. For the Frank-Oseen free energy densitywith one elastic constant (Eq. (3)) this yields Laplace equation for components of the director
∇2ni = 0 with the constraint n2 = 1. (7)
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When not describing the equilibrium, minimizing the free energy yields the molecular fieldh [12]
hi = −∂ f∂ni+ ∂ j
∂ f
∂(∂ jni
) . (8)
Since h has no physical meaning when pointing along the director (the length of the director isfixed, n · n = 1), only the normal component of h represents a thermodynamic force acting onthe director and thus driving it towards equilibrium. Therefore, the normal molecular field H isintroduced
Hi = hi − (h j n j) ni. (9)
However, besides the elastic force that drives director field towards equilibrium, there alsoexist an opposing force. Compared to molecular time scales, the reorientation of the directorfield is a slow process. That is why a molecule that receives angular momentum conveys it toother molecules by collisions or interactions. Effectively, one can think of director being over-damped. This results in no inertial term (n) in the equation of motion of the director. Balanceof forces becomes
Hi = γ1ni, (10)
where elastic forces (Hi) are set equal to rate dependent friction force (γ1ni) and material con-stant γ1 denotes the rotational viscosity. Director reorientation can be viewed as a viscousprocess, where during reorientation the elastic energy stored in the director field is dissipated.
2.3 Defects in nematic
When liquid crystals are confined by surfaces or external fields, regions where director field isnot defined are created [8]. This regions are called defects. At the molecular level defects canbe viewed as regions where strong fluctuations of the molecular orientations are present, i.e.molecules have no preferential orientation, therefore, S = 0 and n is not defined. Defects can beviewed upon as topological objects as it is presented in [9]. Defects in a nematic liquid crystalcan be either lines or points. In point defects, the region where strong fluctuations are present isof spherical shape, typically, with diameters of ∼ 10 nm. In line defects, the regions with strongfluctuations have a cylindrical shape with similar dimensions [10].
Liquid crystal defects are especially important when colloidal particles are added to theliquid crystal. The symmetry and spatial configuration of the defects namely crucially determinethe ordering of the particles. Liquid crystal molecules can orient differently at surfaces. If thesurface forces are strong enough to impose a well-defined direction of the director, then this iscalled strong anchoring [11]. There are two main types of anchoring, homeotropic anchoringwhen the molecules are aligned perpendicularly to the surface and planar anchoring when themolecules are aligned tangentially to the surface. The direction of the director can be alsoimposed by external electric or magnetic fields due to anisotropy of dielectric constant anddiamagnetism of most organic molecules.
Imagine a uniformly ordered nematic liquid crystal, with director pointing in the z directionas presented in Fig. 3a. Then a spherical particle, which prefers strong homeotropic anchoring,is added to a nematic. This is opposed to the preferential uniform director ordering and conse-quently a defect in a proximity of the particle is formed [10]. One possibility is a hyperbolicpoint defect that is tightly bound with the particle, the structure is then called a dipole (Fig. 3b).Another possibility is a disclination ring that encircles the spherical particle at its equator anda Saturn ring configuration (Fig. 3c) is formed. Of course, the disclination ring can be moved
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upward or downward, and by shrinking it into a hyperbolic point defect the Saturn ring configu-ration can be transformed into a dipole. It was shown that a dipole configuration is energeticallyfavored with larger particles, while, for smaller particles Saturn ring configuration is preferred.
Figure 3: Schematically presented director fields for a colloidal particle with homeotropic anchoring in auniform nematic. (a) A particle is added to uniformly ordered nematic. The particle prefers homeotropicanchoring, which is opposed to the nematic ordering and thus a defect in a proximity of the particle isformed. (b) When a hyperbolic point defect occurs, the structure is called a dipole. (c) If a line defect thatencircles the particle at its equator is formed, then the configuration is called a Saturn-ring. A dipolestructure is favored with larger particles and Saturn-ring configuration with smaller particles [10].
3 Nematodynamics
3.1 Navier-Stokes equation
The Navier-Stokes equation is basically the second Newton’s law applied to a little volumeelement of fluid
ρ a = f, (11)
where ρ denotes fluid density, a = dv(r,t)dt acceleration of the fluid element, v velocity, t time and
f volume density of forces. If we assume that there are no external forces, the only forces actingon the fluid element are the forces from pressure and viscous forces
f = −∇ · p + η∇2v, (12)
where p denotes pressure and η fluid viscosity. The incompressibility (∇ · v = 0) was also as-sumed and thus neglecting the additional term in viscous forces that is proportional to ∇ (∇ · v).The second term in Eq. (12) can be formally written as a divergence of stress tensor σ
η∇2v = ∇ · σ, (13)
where σ is proportional to symmetrized velocity gradient Ai j [24]
σi j = η Ai j =12η(∂iv j + ∂ jvi
), (14)
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where η denotes shear viscosity.Formally, the stress tensor is obtained via the dissipative function D that determines the rate
of dissipation, i.e. the rate of decrease of energy E [13, 14]
dEdt= −2D. (15)
It is assumed that friction forces depend linearly on velocity gradients ∂iv j. Dissipative functionis a scalar invariant (i.e. D is invariant to rotations and coordinate inversion) composed ofvelocity gradient terms ∂ jvi that are allowed by symmetry. Besides it has to be invariant to timeinversion, as it has to be positive disregarding direction of fluid flow. Furthermore, D and thestress tensor have to be zero for a velocity field corresponding to a rigid rotation (v = ω × r).The stress tensor is then obtained by differentiation
σi j =∂D
∂(∂iv j
) . (16)
For incompressible isotropic fluid the dissipative function is constructed only from the scalarinvariants of the tensor ∂iv j
Diso =12η(∂iv j
)2+
12η′ δik
(∂iv j
) (∂ jvk
). (17)
The request that Diso has to be zero for the velocity field corresponding for rigid rotations resultsin the relation η = η′, leaving us with only one independent viscosity coefficient.
We arrived at the Navier-Stokes (NS) equation describing fluid dynamics of an incompress-ible fluid [15]
ρdvi
dt= ρ
(∂vi
∂t+ (v j ∂ j)vi
)= −∂ j p + ∂ j σi j. (18)
On the lhs of the Eq. (18) the total derivative of velocity is decomposed into ordinary Eulerianderivative ( ∂∂t ) and into advective term (v · ∇). The changes of velocity at a given position(∂v∂t =
dvdt − (v · ∇)v) are due to changes in velocity with respect to time when we follow the fluid
element ( dvdt ) and due to changes that are a consequence of fluid flow ((v · ∇)v).
3.2 Stress tensor
For isotropic liquid, the stress tensor contains terms due to viscosity and is proportional to thesymmetrized velocity gradient (Eq. (14)).
In a nematic liquid the viscous part of the stress tensor is complicated due to the viscosityanisotropy. For a time-independent director field, there exist three elementary shear flow ge-ometries, which differ in relative directions of director n, velocity v and shear velocity ∇v andare presented in the Fig. 4. This gives three distinct viscosity coefficients ηa, ηb, and ηc, namedMiesowicz viscosities that are usually measured [11].
The viscous part of the stress tensor is derived from the dissipative function D. In nematicthe dissipative function is composed of director ni, total time derivative of director ni, and gradi-ents of velocity ∂ jvi that are allowed by symmetry. Due to the fact that D and the viscous stresstensor have to be zero for a velocity field corresponding to a rigid rotation ni can only be in formof Ni
Ni = ni −12εi jkω jnk. (19)
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Figure 4: Three simple shear flow geometries that differ in relative directions of director n, velocity vand shear velocity ∇v [11].
Ni actually represents a relative rotation of the director with respect to local rotation of the fluid(which is given by the curl of the velocity field, ωi =
12εi jk∂ jvk).
For uniaxial nematic the dissipative function contains additional invariants
D = Diso +
+12ξ1
(∂iv jni
)2+
12ξ2
(∂iv jn j
)2+
12ξ3
(∂iv jni
) (∂iv jn j
)+
+12ξ4
(∂iv jnin j
)2+
12ξ5 ∂iv jnin j +
12ξ6 ∂iv jnin j +
12ξ7 nini. (20)
Only four coefficients ξi are independent, due to the requirement that dissipative function andthe stress tensor have to be zero for a rigid rotation. More often a different set of six viscositycoefficients is used, named Leslie viscosity coefficients αi, where only five of them are indepen-dent.
Finally the viscous stress tensor for a nematic reads
where the ni denotes the director, Ni rate of change of the director relative to the local rotationof the fluid, Ai j symmetrized velocity gradient and coefficients αi are Leslie viscosities. TheLeslie viscosity coefficients are connected to rotational viscosity γ1 = α3 −α2 and are linked byone relation α2 + α3 = α6 − α5. The values are of comparable magnitude and typically in therange 10−3 to 10−2 Pa s [11]. The term with α4 remains also in isotropic liquid. The second andthird term could be called “active” terms, since they are responsible for flow when director fieldis rotating. The other terms (with α1, α5 and α6) are “passive” terms as making the viscosityanisotropic.
However, in the case of nematic liquid crystal, which is anisotropic, the stress tensor involvesalso an elastic part σe
i jσi j = σ
vi j + σ
ei j. (22)
The cause for the existence of the elastic part of the stress tensor is the nematic elasticity and isillustrated in the following example. Imagine a confined nematic with director configuration asin Fig. 5. We assume that the walls prefer homeotropic anchoring. Such configuration wouldrelax to a spatially uniform director field, perpendicular to the confining walls. On the otherhand, if the director is kept fixed, elastic force would be exerted on the walls pushing them
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apart. Because by moving walls apart, the director distortion energy would be decreased. Theorigin of these forces is the elastic part of the stress tensor. Thus, hydrodynamic flow can begenerated by the director distortion, which is a signature of the elastic stress tensor.
Figure 5: Imagine director configuration as in the figure with walls preferring homeotropic anchoring.Such configuration would relax to a spatially uniform director. If the orientation of the director is keptfixed, an elastic stress results pushing the walls apart.
The elastic stress tensor has the form [12]
σei j = −
∂ fE
∂(∂ink)∂ jnk, (23)
where fE denotes Frank-Oseen free energy volume density (Eq. (2)). As can be seen from theEq. (18), in equilibrium the elastic force (the divergence of elastic stress tensor) is balanced bythe pressure gradient.
The Eq. (10) accounts only for elastic forces due to the elastic stress tensor. However,also the viscous stress tensor effects the motion of director. Thermodynamic viscous forces areobtained by differentiating the dissipative function D with respect to ni
hvi =∂D∂ni. (24)
Elastic and viscous forces govern motion of the director
Hi = γ1Ni + (α3 + α2)(Ai jn j
)⊥n, (25)
where γ1 = α3 − α2 is the rotational viscosity, which was already introduced in Eq. (10). In thelast term only the component perpendicular to n has to be taken, which results in the change ofdirector being perpendicular to director, i.e. the director only rotates, but the magnitude is keptfixed.
3.3 Dimensionless quantities
Typical dimensionless number that measures the ratio of inertial forces (ρv2) to viscous forces(ηv/l) is the Reynolds number [15]
Re =ρvlη. (26)
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For nematic the typical intrinsic velocity is the quotient between typical length l and typicaltime τ = γ1l2/K in which the director reorientates. This gives the typical Reynolds number forintrinsic flow in nematic [12]
Re ∼ ρKγ2
1
≈ 10−6. (27)
Because it is very low (Re ≪ 1) advective inertial forces (ρ(v · ∇)v) are small compared toviscous forces and can be thus neglected.
Due to director elastic degree of freedom, another dimensionless number named Ericksennumber can be introduced. It is defined as the ratio of viscous and elastic forces acting on thedirector [24]. The viscous forces are f v ∝ η∇v ∼ ηv/l and the elastic forces read f e ∝ K∇2n ∼K/l2. This gives the Ericksen number
Er =ηvlK, (28)
where l denotes the typical length in our system (e.g. radius of the sphere). For small Ericksennumbers (Er ≪ 1), the viscous forces are too weak to distort the director field and thus thedirector field is such as it would be in the equilibrium in case of v = 0.
3.4 Ericksen-Leslie equations
The hydrodynamic flow has to be calculated from the Navier-Stokes equation, however, thecoupling between director field and hydrodynamic flow that is inherited in the Eq. (25) has tobe taken into account. This is the full description of a nematodynamic problem based on theFrank elastic theory, which results in the so-called Ericksen-Leslie equations. Equations writtenin vectorial form are
ρ∂v∂t+ ρ(v · ∇)v = −∇p + ∇ · (σe + σv), (29)
γ1∂n∂t+ γ1(v · ∇)n = H − (α3 − α2)(A · n)⊥n +
12γ1(∇ × v) × n (30)
∇ · v = 0. (31)
The first equation is the NS equation (Eq. (18)), the second describes elastic and viscous torqueson the director (Eq. (25)) and the last is the equation of continuity for incompressible liquid.
For low Reynolds numbers (Re ≪ 1) the second term on lhs in NS can be neglected. Insuch conditions the flow is called the Stokes or creeping flow. Another simplification can bemade, since the velocity field relaxes almost instantaneously compared to director field and canbe then taken as stationary
(∂v∂t = 0
). This is called the adiabatic approximation. Therefore, the
Due to the complexity, the Ericksen-Leslie equations (Eqs. 29 - 31) typically can not be solvedanalytically. Therefore, the equations have to be approached numerically. There exist differentnumerical procedures to solve the nematodynamic problem, such as finite differences, finiteelements, finite volumes and Lattice-Boltzmann method [16].
The finite difference method (FDM) is the most simple and intuitive. The solutions of dif-ferential equations are approximated by replacing derivatives with approximately equivalentdifference quotients [17]. For a first derivative of the function f (x) at a point a it reads
f ′(a) ≈ f (a + h) − f (a)h
, (35)
where h is a small value. One then gets a difference equation, which gives an approximatesolution. Accuracy of a numerical method is defined as a difference between the exact analyticalsolution and approximation. Typically, the errors are proportional to the step size h. This methodis used especially for simple differential equations and rarely for hydrodynamics.
The finite element method (FEM) consists of two basic steps: firstly, one chooses a gridof the region of interest (i.e. discretizes whole space to smaller subspaces), and secondly, onechooses basic functions [18]. Often the grid consists of triangles as can be seen in the Fig.6a, however, it can consist of any shapes. For a basis, frequently piecewise linear functionsare used. The solution is then written as a linear combination of basis functions with unknowncoefficients that are calculated in a way to give the best approximation. This can be done viaGalerkin or variational method [19]. As an example a 1D model will be described. As can beseen in the Fig. 6b, the mesh consists of nodes xi and for each interval [xk−1, xk+1] a piecewiselinear function is used. The result is the sum of all functions, which are weighed by coefficientscalculated in a way to give the best approximation (by minimizing the residual). In comparisonwith finite difference method, FEM can relatively easily handle complicated geometries and thequality of approximation is often higher.
Figure 6: (a) An example of FEM mesh of a 2D problem that uses triangles. Colored is the approximationfor a resulting function [18]. (b) An example of FEM on a 1D model: note mesh points xi and piecewiselinear functions (blue). Red line represents the resulting function [18]. (c) In LBM a particle (blue circle)has a certain probability pi that will propagate with velocity vi.
The finite volume method (FVM) is one of the most commonly used methods in compu-tational fluid dynamics. This method is similar to FDM, due to the fact that the values arecalculated at discrete places on a meshed geometry [20]. The volume integrals that contain di-vergence terms can be then converted to surface integrals of each small volume surrounding the
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mesh point. The main advantage of this method is that it is flux-conservative. Surface integralscan be understood as fluxes and the flux leaving a given volume is entering the neighboring oneand is thus conserved.
In Lattice-Boltzmann method (LBM) rather than solving Navier-Stokes equation, the dis-crete Boltzmann equation is solved to simulate the flow of a fluid [21, 22]. The LBM modelsfluid as fictive particles that perform steps over the discrete mesh. Also the velocity directionsare discretized and the propagation of the particle is defined by distribution functions, i.e. a fluidparticle has a certain probability pi to propagate in a certain direction with velocity vi. Particlethen propagates with velocity v =
∑i pi vi to another mesh point, where it “collides” with an-
other particle. There the collision rules are applied that govern the particle movement. The mainadvantages of the LBM are easy implementing (complex) boundary conditions, incorporatingthe microscopic interactions and fully parallel algorithms.
In large-level fluid simulations, the FVM and LBM are the most widely used methods.
5 Stokes drag of a moving spherical particle
Stokes drag force on a spherical particle with radius a is presented as an illustration of theapplication of the nematodynamics. In this way the differences between particle in isotropicliquid and for different configurations in a nematic are highlighted.
5.1 Isotropic fluid
A stationary velocity field and low Reynolds number regime (Re ≪ 1) in an incompressibleisotropic fluid are assumed. Instead of calculating the velocity field of a moving sphere, theequivalent problem of the flow around the sphere is solved. The equations read
−∇p + ∇ · σ = 0 and ∇ · v = 0, (36)
where p denotes pressure, v velocity and σ stress tensor (Eq. (14)). For a nonslip conditionat the surface of the particle (v|r=a = 0) and a uniform velocity v∞ at infinity, the solution canbe obtained analytically. The nonslip condition indeed well describes the interaction betweenthe wall and the liquid. It is generally valid in bulk systems, whereas in confined systems isrealistic to dimensions of the confining channels of few 10 molecular layers [23]. One gets thewell-known Stokes formula for drag force
FisoS = γv∞ with γ = 6πηa, (37)
where η denotes fluid viscosity and γ is friction coefficient. One should note that the Stokesforce is always parallel to v∞.
5.2 Nematic fluid
In nematic environment similar assumptions are made: incompressibility of a nematic, station-ary director and velocity field and Re ≪ 1. The coupling between director field and hydrody-namic flow has to be considered, therefore Ericksen-Leslie equations (Eqs. 32, 33) have to besolved. The solutions cannot be obtained analytically, thus a numerical procedure has to be ap-plied. We will follow the approach as described in [24]. The director configuration is calculatedfrom the balance of thermodynamic forces (Eq. (33)), whereas the velocity field is obtainedfrom NS equation (Eq. (32)).
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The main difference from the isotropic fluid is that the friction coefficient now becomes atensor γ and, consequently, the Stokes drag can point in a direction different from the velocityfield
FnemS = γv∞ with γ =
γ⊥ 0 00 γ⊥ 00 0 γ∥
. (38)
There exist two independent components γ⊥ and γ∥ of the friction tensor, since the nematic isuniaxial. When the flow is parallel or perpendicular to the symmetry axis of the system, alsothe Stokes force is parallel to v∞. Otherwise, a component perpendicular to v∞, called lift-forceappears.
In analogy with the isotropic fluid (Eq. (37)), effective viscosities η⊥e f f and η∥e f f for a nematicare introduced as
γ⊥ = 6π η⊥e f f a and γ∥ = 6π η∥e f f a. (39)
Of course, uniformly ordered nematic has well defined viscosity coefficients η⊥,∥ for flow per-pendicular or parallel to the director, respectively. However, the viscosities in Eq. (39) areeffective, as the specific director configurations and the shape of the particle affect the flow.
5.3 Streamline patterns
Streamline is a family of curves that are tangential to the velocity vector and show the directiona fluid element will follow at any point in time. In stationary flow they do not vary with time andcoincide with the paths of the fluid particles, whereas in non-stationary flow the coincidence nolonger occurs [15].
Figure 7: Streamline pattern around sphericalparticle for isotropic liquid (right) and uniformdirector field, which is everywhere parallel tov∞ (left). No anchoring is present. Note thatin a nematic the bent streamlines seem to fol-low the director configuration and occupy lessspace than in isotropic liquid [24].
Figure 8: Streamline pattern around sphericalparticle for isotropic liquid (right) and directorfield for dipole configuration that is parallel tov∞ (left). Note the broken mirror symmetry forthe dipole streamlines [24].
In the case of a uniform director field with no anchoring present, which points everywherein the same direction as v∞, the streamline pattern is very similar to the one for isotropic liquidas can be seen in Fig. 7. However, in isotropic liquid the bent streamlines occupy more space,
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whereas in a nematic liquid crystal they seem to follow the director field lines [24]. This isnot a surprise, since the shear flow along the director possesses the smallest Miesowicz shearviscosity (called ηb in the Fig. 4).
Figure 9: The distance rd between the point defect and the center of the sphere (left scale) and theeffective viscosity (right scale) for different Ericksen numbers that actually represent velocity v∞ (Er > 0represents flow from below and Er < 0 flow from above, relative to the configuration). The defect willnot detach from the sphere as that would cost too much elastic energy [25].
In the case of topological dipole parallel to v∞ in the low Ericksen limit (i.e. the directorconfiguration is as it would be without flow) an asymmetry in the streamlines can be observed.The mirror symmetry of the streamline pattern is broken as can be seen in Fig. 8. At higherEricksen numbers there exist a difference when the fluid flows from above or belove as presentedin the Fig. 9 [25]. When the fluid flows from above, the defect is pulled towards the particleand also the effective viscosity (and Stokes force) decreases, as the director field is then moreuniform. On the other hand, for the flow from below the defect is pulled away from the particle,which results in an increase of effective viscosity. As can be seen in Fig. 9, the effect is alsohighly non-linear. It should be also mentioned that the point defect will not detach from theparticle as might be suggested in the figure, since this costs too much elastic energy, but forlarge Er numbers the defect becomes unstable and transforms into a Saturn-ring configurationand possibly moves to a position above the particle (which would correspond to flow fromabove).
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Figure 10: Streamline pattern around spheri-cal particle for a Saturn-ring (right) and dipoledirector field configuration parallel to velocityfar away (left). Note the mirror symmetry forthe Saturn ring configuration. Defect ring pro-duces the dip in the nearest streamline [24].
Figure 11: Streamline pattern around spheri-cal particle for dipole configuration perpendic-ular to v∞. The similarity to the Magnus effectcan be noticed, yet the particle experiences nolift force. Note the non-zero torque that inducesparticle rotation [25].
In the case of Saturn ring configuration the streamline pattern would have a mirror symme-try, present in the isotropic fluid. On the other hand, the position of the line defect is clearlyobservable by the dip in the streamlines. The comparison of the streamlines to the dipole can beseen in Fig. 10.
When a dipole is oriented perpendicular to v∞ the streamline pattern (Fig. 11) resemblesthe one of the Magnus effect due to the density of the streamlines [25]. However, no lift forceis present, since the symmetry dictates only the force parallel to velocity at infinity. What ismore, a non zero torque appears that induces particle rotation. This should be also observablein experiments when a particle falling under the influence of gravity in a nematic would start torotate.
6 Conclusion
The flow phenomena in a nematic liquid crystal was described. The basic difference betweenisotropic fluid and nematic is the existence of the dynamic director field and the material anisotropy.As nematic acts as effective elastic medium, stress induced by spatially non-uniform directorfield results in hydrodynamic flow, which decreases free energy. The complexity of nematody-namic equations results in need for strong numerical methods that are required to solve evenbasic fluid geometries.
The difference between nematic and isotropic fluid was presented via the Stokes force andstreamline patterns. Reasonable assumptions were made, such as low Reynolds and low Erick-sen regime (due to small velocities) and static velocity field (due to very short dynamic timecompared to director field). In the uniform director field, the streamlines try to follow the direc-tor field lines. The dipole lacks a mirror plane symmetry and when perpendicular to the velocityfield a torque appears that induces the rotation of the particle. In Saturn-ring configuration, thestreamlines are symmetrical and the presence of the disclination line exhibits in a small dip inthe nearest streamlines.
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In this seminar only the “stationary” solutions of nematodynamics where the flow was ex-ternally imposed were presented. In general, however, despite stationary fluid flow the directorfield can be time dependent. Nematics that respond with time-dependent orientational behaviourare called tumbling nematics [26].
Nematodynamic coupling is of great importance in understanding the behavior of colloidalparticles in liquid crystals and controlling liquid crystal flows in microfluidic cells. For example,the physics of the microscale rotor pump (from [4]) would be greatly complicated when insteadof isotropic liquid nematic is present. Further complications arise from the higher Ericksennumber and complex director field. But assuming uniform director field as in the Fig. 12and Miesowicz viscosities for liquid crystal 5CB [24], one can approximately calculate thedissipated power due to viscous flow. In the first case (Fig. 12a), the velocity gradient is parallel
Figure 12: A microscale rotor pump from [4] with nematic instead of isotropic fluid. When director isparallel to velocity gradient (a), the dissipated energy per unit time is 5.6 times greater than in the caseof director perpendicular to velocity gradient (b).
to director, where flow is determined by Miesowicz viscosity ηc, but in the second configuration,the velocity gradient is perpendicular to director, which gives viscosity ηb. The dissipated poweris roughly P ∼ f orce ·velocity ∼ η∇v ·v ∝ η. Therefore, the power needed in the case of directorfield perpendicular to the fluid flow is approximately 5.6 times greater than in the case of seconddirector field
Pa
Pb∼ ηc
ηb=
0.1296 Pas0.0229 Pas
≈ 5.6 .
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References[1] R. N. Ibragimov, Fluid Mechanics, Lecture notes. Available at http://infohost.nmt.edu/∼iranis/
[19] G. P. Nikishkov, Introduction to the Finite Element Method, Lecture notes (2007). Available athttp://web-ext.u-aizu.ac.jp/∼niki/feminstr/introfem/introfem.html (23.12.2009).
