Nematodynamics of a Colloidal Particle

Transcription

1 Department of Physics Seminar - 4th year Nematodynamics of a Colloidal Particle Author: David Seč 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 nematodynamics 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.

2 Contents 1 Introduction 2 2 Liquid crystals Orientational order Frank elastic theory Defects in nematic Nematodynamics Navier-Stokes equation Stress tensor Dimensionless quantities Ericksen-Leslie equations Solving the equations 12 5 Stokes drag of a moving spherical particle Isotropic fluid Nematic fluid Streamline patterns Conclusion 16 References 18 1 Introduction Various types of flows occur in natural and technical environment. Without fluid flows life would probably be impossible or would be at least extremely different. The important role of flows can be seen in various transport processes in human beings, which supply our body with the oxygen and other essential nutrients. Flows are vital in rivers, lakes, and seas, as well as in all atmosphere, where they influence the weather and thus climate greatly. Finally, flows are of large importance in technical environment, where multiple processes depend on fluid flows. For example, 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 fluid dynamics is important. One can therefore, predict the flow development (e.g. forecast weather), improve technological equipment (minimize energy loss resulting from the flow resistance in cars, airplanes, etc.), gain knowledge that would result in novel applications [1]. Interesting field of fluid dynamics is microfluidics, the science and technology of systems that process or manipulate small (10 9 to liters) amounts of fluids, using channels with dimensions of tens to hundreds of micrometers [2]. There exist important advantages in using such 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 device with size to few square centimeters that has one or several laboratory functions integrated 2

3 [3]. One of the elements in a way to the Lab-on-a-chip is fabrication of micro pumps. An interesting solution was just recently provided by our faculty research group under the supervision of Professor Igor Poberaj [4]. A microscale rotor pump (Fig. 1) was made from self-assembled superparamagnetic colloidal spheres and driven by an external magnetic field. Magnetic field was induced with two orthogonal coil pairs, driven by sinusoidal current. Electric field, which governs the cluster position and thus the direction of flow, was produced with separate microelectrodes at each pump. This allows to control each pump s speed and direction of flow separately. Such pumps can be also fabricated with existing technologies and easily integrated into microfluidic devices. Figure 1: (a) An ilustration of microscale pump made from a cluster of superparamagnetic colloids that rotate 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 Newton s second law applied to fluid motion. Most often the incompressibility of liquid can be assumed, because the changes in pressure and temperature due to the flow are sufficiently small that changes in density are negligible. For example, compressibility of water, which describes the relative change of volume as a response to change of pressure, is roughly Pa 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. In Chapter 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 are presented. 3

4 2 Liquid crystals Liquid crystals are materials that are more ordered than ordinary liquids but less ordered than solids. They are made of rigid organic molecules that self-organize at a certain temperature or 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 of the molecules are disordered like in a liquid phase. Nematics are formed by achiral elongated molecules, which can be approximated by rotational ellipsoids [6]. The study of liquid crystal hydrodynamics has recently turned out to be useful also for describing some biological systems. Certain types of bacteria in suspensions swim in swirls that suggest long-range orientational order [7]. Another examples of self-organizing system are cytoskeletal filaments and motor-proteins with intrinsically non-equilibrium ordering. The collective behavior in these systems can be described in the continuum limit. Such system are analogous to liquid crystals and can be thus represented by the same phenomenological hydrodynamic 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 parameters are introduced [8]. The vectorial order parameter, named director n(r, t), describes the average orientation of nematic molecules at a given position r at a time t. The ordering is such that the states n and n are equivalent (no dipolar ordering). The nematic degree of order S quantifies orientational fluctuations of uniaxial molecules around the director [8] S = 1 ( 3 cos 2 θ 1 ) = 1 f (θ) ( 3 cos 2 θ 1 ) dω, (1) 2 2 where θ denotes azimuthal angle in spherical coordinates, f (θ) the orientational distribution function of molecules, dω solid angle and <. > denotes ensemble average. The values of S lie in the interval [ 1 2, 1]. For a perfectly ordered nematic where all the molecules point exactly along the director S = 1 and for a perfectly disordered state where molecules have no preferential direction S = 0. The value S = 1 2 represents the state where all molecules are aligned in the plane perpendicular to n. Typical values for liquid crystals are S [6]. The nematic degree of order can be spatially dependent. When the molecules are not uniaxial or molecular fluctuations about the director are not rotationally symmetric to director, full order parameter tensor must be introduced [9]. The full order parameter tensor is usually needed when describing defects or when external constraints (surfaces, magnetic or electric fields) break the uniaxial symmetry. 2.2 Frank elastic theory Nematic liquid crystal with uniform director field has the lowest free energy [10]. Any deformations of the orientational ordering increase energy, and as a result nematic acts as an effective elastic medium. When director is deformed, nematic tends to align to a spatially uniform director field. For example, supposing that one could grab and bend the director field, it would effectively act as a bent stick and try to relax to a uniform configuration. Any elastic deformation can be decomposed in three basic deformation modes (splay, twist and bend) that are schematically presented in Fig. 2. 4

5 Figure 2: Liquid crystals act as effective elastic materials and any elastic deformation can be decomposed in three basic deformation modes: (a) splay, (b) twist, and (c) bend. The arrows show the direction of the director [8]. Slowly varying spatial deformations, i.e. director gradients are small (no defects), are described phenomenologically by expanding free energy volume density into invariants of n that are allowed by symmetry. The result is Frank-Oseen free energy volume density [8, 11] f E = 1 2 K 1 ( n) K 2 [n ( n)] K 3 [n ( n)] 2, (2) where K 1, K 2, K 3 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 K i have the nematic degree of order incorporated and are proportional to S 2. One can note that the free energy density (Eq. (2)) is a quadratic form for three basic deformation modes and obeys the n n inversion symmetry. Many materials have similar elastic constants. Therefore, a common approximation for free energy volume density is to use only one average elastic constant that sets K 1 = K 2 = K 3 = K. This simplifies the complex spatially dependent Frank-Oseen free energy functional which can be now written as f one E = 1 2 K [ ( n) 2 + ( n) 2], (3) which is without surface terms the same as f Eone = 1 2 K ( i n j ) 2. (4) Equilibrium director field is obtained by the variation of the free energy F f E δf = δ f E dv = δn i + f E n i ( )δ ( ) j n i dv = 0. (5) j n i One obtains Euler-Lagrange equations for the director f f j n i ( ) j n = 0 with the constraint n2 = 1, (6) i where summation over repeated indices is assumed. For the Frank-Oseen free energy density with one elastic constant (Eq. (3)) this yields Laplace equation for components of the director 2 n i = 0 with the constraint n 2 = 1. (7) 5

6 When not describing the equilibrium, minimizing the free energy yields the molecular field h [12] h i = f f + j n i ( ) j n. (8) i Since h has no physical meaning when pointing along the director (the length of the director is fixed, n n = 1), only the normal component of h represents a thermodynamic force acting on the director and thus driving it towards equilibrium. Therefore, the normal molecular field H is introduced H i = h i (h j n j ) n i. (9) However, besides the elastic force that drives director field towards equilibrium, there also exist an opposing force. Compared to molecular time scales, the reorientation of the director field is a slow process. That is why a molecule that receives angular momentum conveys it to other molecules by collisions or interactions. Effectively, one can think of director being overdamped. This results in no inertial term ( n) in the equation of motion of the director. Balance of forces becomes H i = γ 1 ṅ i, (10) where elastic forces (H i ) are set equal to rate dependent friction force (γ 1 ṅ i ) and material constant γ 1 denotes the rotational viscosity. Director reorientation can be viewed as a viscous process, 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 is not defined are created [8]. This regions are called defects. At the molecular level defects can be 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 be viewed upon as topological objects as it is presented in [9]. Defects in a nematic liquid crystal can be either lines or points. In point defects, the region where strong fluctuations are present is of spherical shape, typically, with diameters of 10 nm. In line defects, the regions with strong fluctuations have a cylindrical shape with similar dimensions [10]. Liquid crystal defects are especially important when colloidal particles are added to the liquid crystal. The symmetry and spatial configuration of the defects namely crucially determine the ordering of the particles. Liquid crystal molecules can orient differently at surfaces. If the surface forces are strong enough to impose a well-defined direction of the director, then this is called strong anchoring [11]. There are two main types of anchoring, homeotropic anchoring when the molecules are aligned perpendicularly to the surface and planar anchoring when the molecules are aligned tangentially to the surface. The direction of the director can be also imposed by external electric or magnetic fields due to anisotropy of dielectric constant and diamagnetism of most organic molecules. Imagine a uniformly ordered nematic liquid crystal, with director pointing in the z direction as 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 consequently a defect in a proximity of the particle is formed [10]. One possibility is a hyperbolic point 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 and a Saturn ring configuration (Fig. 3c) is formed. Of course, the disclination ring can be moved 6

7 upward or downward, and by shrinking it into a hyperbolic point defect the Saturn ring configuration can be transformed into a dipole. It was shown that a dipole configuration is energetically favored 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 a uniform nematic. (a) A particle is added to uniformly ordered nematic. The particle prefers homeotropic anchoring, which is opposed to the nematic ordering and thus a defect in a proximity of the particle is formed. (b) When a hyperbolic point defect occurs, the structure is called a dipole. (c) If a line defect that encircles the particle at its equator is formed, then the configuration is called a Saturn-ring. A dipole structure 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 volume element 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 acting on the fluid element are the forces from pressure and viscous forces f = p + η 2 v, (12) where p denotes pressure and η fluid viscosity. The incompressibility ( v = 0) was also assumed 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 where σ is proportional to symmetrized velocity gradient A i j [24] η 2 v = σ, (13) σ i j = η A i j = 1 2 η ( i v j + j v i ), (14) 7

8 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] de dt = 2D. (15) It is assumed that friction forces depend linearly on velocity gradients i v j. Dissipative function is a scalar invariant (i.e. D is invariant to rotations and coordinate inversion) composed of velocity gradient terms j v i that are allowed by symmetry. Besides it has to be invariant to time inversion, as it has to be positive disregarding direction of fluid flow. Furthermore, D and the stress 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 σ i j = D ( i v j ). (16) For incompressible isotropic fluid the dissipative function is constructed only from the scalar invariants of the tensor i v j D iso = 1 2 η ( i v j ) η δ ik ( i v j ) ( j v k ). (17) The request that D iso has to be zero for the velocity field corresponding for rigid rotations results in 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 incompressible fluid [15] ρ dv i dt ( ) vi = ρ t + (v j j )v i = j p + j σ i j. (18) On the lhs of the Eq. (18) the total derivative of velocity is decomposed into ordinary Eulerian derivative ( t ) and into advective term (v ). The changes of velocity at a given position ( v t = dv dt (v )v) are due to changes in velocity with respect to time when we follow the fluid element ( dv dt ) 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 the symmetrized velocity gradient (Eq. (14)). In a nematic liquid the viscous part of the stress tensor is complicated due to the viscosity anisotropy. For a time-independent director field, there exist three elementary shear flow geometries, which differ in relative directions of director n, velocity v and shear velocity v and are presented in the Fig. 4. This gives three distinct viscosity coefficients η a, η b, and η c, named Miesowicz viscosities that are usually measured [11]. The viscous part of the stress tensor is derived from the dissipative function D. In nematic the dissipative function is composed of director n i, total time derivative of director ṅ i, and gradients of velocity j v i that are allowed by symmetry. Due to the fact that D and the viscous stress tensor have to be zero for a velocity field corresponding to a rigid rotation ṅ i can only be in form of N i N i = ṅ i 1 2 ε i jkω j n k. (19) 8

9 Figure 4: Three simple shear flow geometries that differ in relative directions of director n, velocity v and shear velocity v [11]. N i 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, ω i = 1 2 ε i jk j v k ). For uniaxial nematic the dissipative function contains additional invariants D = D iso ξ ( ) i v j n i + 2 ξ ξ 4 ( ) 2 1 i v j n j + 2 ξ ( ) ( ) 3 i v j n i i v j n j + ( i v j n i n j ) ξ 5 i v j n i ṅ j ξ 6 i v j ṅ i n j ξ 7 ṅ i ṅ i. (20) Only four coefficients ξ i are independent, due to the requirement that dissipative function and the stress tensor have to be zero for a rigid rotation. More often a different set of six viscosity coefficients is used, named Leslie viscosity coefficients α i, where only five of them are independent. Finally the viscous stress tensor for a nematic reads σ v i j = α 1n i n j n k n l A kl + α 2 n j N i + α 3 n i N j + α 4 A i j + α 5 n j n k A ik + α 6 n i n k A jk, (21) where the n i denotes the director, N i rate of change of the director relative to the local rotation of the fluid, A i j symmetrized velocity gradient and coefficients α i are Leslie viscosities. The Leslie viscosity coefficients are connected to rotational viscosity γ 1 = α 3 α 2 and are linked by one relation α 2 + α 3 = α 6 α 5. The values are of comparable magnitude and typically in the range 10 3 to 10 2 Pa s [11]. The term with α 4 remains also in isotropic liquid. The second and third term could be called active terms, since they are responsible for flow when director field is rotating. The other terms (with α 1, α 5 and α 6 ) are passive terms as making the viscosity anisotropic. However, in the case of nematic liquid crystal, which is anisotropic, the stress tensor involves also an elastic part σ e i j σ i j = σ v i j + σe i j. (22) The cause for the existence of the elastic part of the stress tensor is the nematic elasticity and is illustrated in the following example. Imagine a confined nematic with director configuration as in Fig. 5. We assume that the walls prefer homeotropic anchoring. Such configuration would relax to a spatially uniform director field, perpendicular to the confining walls. On the other hand, if the director is kept fixed, elastic force would be exerted on the walls pushing them 9

10 apart. Because by moving walls apart, the director distortion energy would be decreased. The origin of these forces is the elastic part of the stress tensor. Thus, hydrodynamic flow can be generated 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 kept fixed, an elastic stress results pushing the walls apart. The elastic stress tensor has the form [12] σ e i j = f E ( i n k ) jn k, (23) where f E denotes Frank-Oseen free energy volume density (Eq. (2)). As can be seen from the Eq. (18), in equilibrium the elastic force (the divergence of elastic stress tensor) is balanced by the 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 are obtained by differentiating the dissipative function D with respect to ṅ i Elastic and viscous forces govern motion of the director h v i = D ṅ i. (24) H i = γ 1 N i + (α 3 + α 2 ) ( A i j n j ) n, (25) where γ 1 = α 3 α 2 is the rotational viscosity, which was already introduced in Eq. (10). In the last term only the component perpendicular to n has to be taken, which results in the change of director being perpendicular to director, i.e. the director only rotates, but the magnitude is kept fixed. 3.3 Dimensionless quantities Typical dimensionless number that measures the ratio of inertial forces (ρv 2 ) to viscous forces (ηv/l) is the Reynolds number [15] Re = ρvl η. (26) 10

11 For nematic the typical intrinsic velocity is the quotient between typical length l and typical time τ = γ 1 l 2 /K in which the director reorientates. This gives the typical Reynolds number for intrinsic flow in nematic [12] Re ρk γ (27) Because it is very low (Re 1) advective inertial forces (ρ(v )v) are small compared to viscous forces and can be thus neglected. Due to director elastic degree of freedom, another dimensionless number named Ericksen number can be introduced. It is defined as the ratio of viscous and elastic forces acting on the director [24]. The viscous forces are f v η v ηv/l and the elastic forces read f e K 2 n K/l 2. This gives the Ericksen number Er = ηvl K, (28) where l denotes the typical length in our system (e.g. radius of the sphere). For small Ericksen numbers (Er 1), the viscous forces are too weak to distort the director field and thus the director field is such as it would be in the equilibrium in case of v = Ericksen-Leslie equations The hydrodynamic flow has to be calculated from the Navier-Stokes equation, however, the coupling between director field and hydrodynamic flow that is inherited in the Eq. (25) has to be taken into account. This is the full description of a nematodynamic problem based on the Frank elastic theory, which results in the so-called Ericksen-Leslie equations. Equations written in vectorial form are ρ v t + ρ(v )v = p + (σe + σ v ), (29) n γ 1 t + γ 1(v )n = H (α 3 α 2 )(A n) n γ 1( v) n (30) v = 0. (31) The first equation is the NS equation (Eq. (18)), the second describes elastic and viscous torques on 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. In such conditions the flow is called the Stokes or creeping flow. Another simplification can be made, since the velocity field relaxes almost instantaneously compared to director field and can be then taken as stationary ( v t = 0 ). This is called the adiabatic approximation. Therefore, the equations governing the nematodynamics read 0 = p + (σ e + σ v ), (32) n t + (v )n = H (α 3 α 2 )(A n) n + 1 ( v) n 2 (33) v = 0. (34) 11

12 4 Solving the equations Due to the complexity, the Ericksen-Leslie equations (Eqs ) typically can not be solved analytically. Therefore, the equations have to be approached numerically. There exist different numerical procedures to solve the nematodynamic problem, such as finite differences, finite elements, finite volumes and Lattice-Boltzmann method [16]. The finite difference method (FDM) is the most simple and intuitive. The solutions of differential equations are approximated by replacing derivatives with approximately equivalent difference quotients [17]. For a first derivative of the function f (x) at a point a it reads f (a) f (a + h) f (a), (35) h where h is a small value. One then gets a difference equation, which gives an approximate solution. Accuracy of a numerical method is defined as a difference between the exact analytical solution and approximation. Typically, the errors are proportional to the step size h. This method is used especially for simple differential equations and rarely for hydrodynamics. The finite element method (FEM) consists of two basic steps: firstly, one chooses a grid of the region of interest (i.e. discretizes whole space to smaller subspaces), and secondly, one chooses 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 functions are used. The solution is then written as a linear combination of basis functions with unknown coefficients that are calculated in a way to give the best approximation. This can be done via Galerkin or variational method [19]. As an example a 1D model will be described. As can be seen in the Fig. 6b, the mesh consists of nodes x i and for each interval [x k 1, x k+1 ] a piecewise linear function is used. The result is the sum of all functions, which are weighed by coefficients calculated in a way to give the best approximation (by minimizing the residual). In comparison with finite difference method, FEM can relatively easily handle complicated geometries and the quality of approximation is often higher. Figure 6: (a) An example of FEM mesh of a 2D problem that uses triangles. Colored is the approximation for a resulting function [18]. (b) An example of FEM on a 1D model: note mesh points x i and piecewise linear functions (blue). Red line represents the resulting function [18]. (c) In LBM a particle (blue circle) has a certain probability p i that will propagate with velocity v i. The finite volume method (FVM) is one of the most commonly used methods in computational fluid dynamics. This method is similar to FDM, due to the fact that the values are calculated at discrete places on a meshed geometry [20]. The volume integrals that contain divergence terms can be then converted to surface integrals of each small volume surrounding the 12

13 mesh point. The main advantage of this method is that it is flux-conservative. Surface integrals can be understood as fluxes and the flux leaving a given volume is entering the neighboring one and is thus conserved. In Lattice-Boltzmann method (LBM) rather than solving Navier-Stokes equation, the discrete Boltzmann equation is solved to simulate the flow of a fluid [21, 22]. The LBM models fluid as fictive particles that perform steps over the discrete mesh. Also the velocity directions are discretized and the propagation of the particle is defined by distribution functions, i.e. a fluid particle has a certain probability p i to propagate in a certain direction with velocity v i. Particle then propagates with velocity v = i p i v i to another mesh point, where it collides with another particle. There the collision rules are applied that govern the particle movement. The main advantages of the LBM are easy implementing (complex) boundary conditions, incorporating the 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 the application of the nematodynamics. In this way the differences between particle in isotropic liquid 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 incompressible isotropic fluid are assumed. Instead of calculating the velocity field of a moving sphere, the equivalent 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 condition at the surface of the particle (v r=a = 0) and a uniform velocity v at infinity, the solution can be obtained analytically. The nonslip condition indeed well describes the interaction between the wall and the liquid. It is generally valid in bulk systems, whereas in confined systems is realistic to dimensions of the confining channels of few 10 molecular layers [23]. One gets the well-known Stokes formula for drag force F iso S = γv with γ = 6πηa, (37) where η denotes fluid viscosity and γ is friction coefficient. One should note that the Stokes force is always parallel to v. 5.2 Nematic fluid In nematic environment similar assumptions are made: incompressibility of a nematic, stationary director and velocity field and Re 1. The coupling between director field and hydrodynamic flow has to be considered, therefore Ericksen-Leslie equations (Eqs. 32, 33) have to be solved. The solutions cannot be obtained analytically, thus a numerical procedure has to be applied. We will follow the approach as described in [24]. The director configuration is calculated from the balance of thermodynamic forces (Eq. (33)), whereas the velocity field is obtained from NS equation (Eq. (32)). 13

14 The main difference from the isotropic fluid is that the friction coefficient now becomes a tensor γ and, consequently, the Stokes drag can point in a direction different from the velocity field F nem S = γv with γ = γ γ γ. (38) There exist two independent components γ and γ of the friction tensor, since the nematic is uniaxial. When the flow is parallel or perpendicular to the symmetry axis of the system, also the Stokes force is parallel to v. Otherwise, a component perpendicular to v, called lift-force appears. In analogy with the isotropic fluid (Eq. (37)), effective viscosities η e f f and η e f f for a nematic are 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 perpendicular or parallel to the director, respectively. However, the viscosities in Eq. (39) are effective, 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 direction a fluid element will follow at any point in time. In stationary flow they do not vary with time and coincide with the paths of the fluid particles, whereas in non-stationary flow the coincidence no longer occurs [15]. Figure 7: Streamline pattern around spherical particle for isotropic liquid (right) and uniform director field, which is everywhere parallel to v (left). No anchoring is present. Note that in a nematic the bent streamlines seem to follow the director configuration and occupy less space than in isotropic liquid [24]. Figure 8: Streamline pattern around spherical particle for isotropic liquid (right) and director field for dipole configuration that is parallel to v (left). Note the broken mirror symmetry for the dipole streamlines [24]. In the case of a uniform director field with no anchoring present, which points everywhere in the same direction as v, the streamline pattern is very similar to the one for isotropic liquid as can be seen in Fig. 7. However, in isotropic liquid the bent streamlines occupy more space, 14

15 whereas in a nematic liquid crystal they seem to follow the director field lines [24]. This is not a surprise, since the shear flow along the director possesses the smallest Miesowicz shear viscosity (called η b in the Fig. 4). Figure 9: The distance r d between the point defect and the center of the sphere (left scale) and the effective viscosity (right scale) for different Ericksen numbers that actually represent velocity v (Er > 0 represents flow from below and Er < 0 flow from above, relative to the configuration). The defect will not 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 director configuration 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 higher Ericksen numbers there exist a difference when the fluid flows from above or belove as presented in the Fig. 9 [25]. When the fluid flows from above, the defect is pulled towards the particle and also the effective viscosity (and Stokes force) decreases, as the director field is then more uniform. 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 also highly non-linear. It should be also mentioned that the point defect will not detach from the particle as might be suggested in the figure, since this costs too much elastic energy, but for large Er numbers the defect becomes unstable and transforms into a Saturn-ring configuration and possibly moves to a position above the particle (which would correspond to flow from above). 15

16 Figure 10: Streamline pattern around spherical particle for a Saturn-ring (right) and dipole director field configuration parallel to velocity far away (left). Note the mirror symmetry for the Saturn ring configuration. Defect ring produces the dip in the nearest streamline [24]. Figure 11: Streamline pattern around spherical particle for dipole configuration perpendicular to v. The similarity to the Magnus effect can be noticed, yet the particle experiences no lift force. Note the non-zero torque that induces particle rotation [25]. In the case of Saturn ring configuration the streamline pattern would have a mirror symmetry, present in the isotropic fluid. On the other hand, the position of the line defect is clearly observable by the dip in the streamlines. The comparison of the streamlines to the dipole can be seen in Fig. 10. When a dipole is oriented perpendicular to v the streamline pattern (Fig. 11) resembles the one of the Magnus effect due to the density of the streamlines [25]. However, no lift force is present, since the symmetry dictates only the force parallel to velocity at infinity. What is more, a non zero torque appears that induces particle rotation. This should be also observable in experiments when a particle falling under the influence of gravity in a nematic would start to rotate. 6 Conclusion The flow phenomena in a nematic liquid crystal was described. The basic difference between isotropic 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 director field results in hydrodynamic flow, which decreases free energy. The complexity of nematodynamic equations results in need for strong numerical methods that are required to solve even basic fluid geometries. The difference between nematic and isotropic fluid was presented via the Stokes force and streamline patterns. Reasonable assumptions were made, such as low Reynolds and low Ericksen regime (due to small velocities) and static velocity field (due to very short dynamic time compared to director field). In the uniform director field, the streamlines try to follow the director field lines. The dipole lacks a mirror plane symmetry and when perpendicular to the velocity field a torque appears that induces the rotation of the particle. In Saturn-ring configuration, the streamlines are symmetrical and the presence of the disclination line exhibits in a small dip in the nearest streamlines. 16

17 In this seminar only the stationary solutions of nematodynamics where the flow was externally imposed were presented. In general, however, despite stationary fluid flow the director field can be time dependent. Nematics that respond with time-dependent orientational behaviour are called tumbling nematics [26]. Nematodynamic coupling is of great importance in understanding the behavior of colloidal particles 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 instead of isotropic liquid nematic is present. Further complications arise from the higher Ericksen number and complex director field. But assuming uniform director field as in the Fig. 12 and Miesowicz viscosities for liquid crystal 5CB [24], one can approximately calculate the dissipated 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 is parallel to velocity gradient (a), the dissipated energy per unit time is 5.6 times greater than in the case of director perpendicular to velocity gradient (b). to director, where flow is determined by Miesowicz viscosity η c, but in the second configuration, the velocity gradient is perpendicular to director, which gives viscosity η b. The dissipated power is roughly P f orce velocity η v v η. Therefore, the power needed in the case of director field perpendicular to the fluid flow is approximately 5.6 times greater than in the case of second director field P a η c Pas = P b η b Pas

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