Continuum Theory of Liquid Crystals continued...

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1 Continuum Theory of Liquid Crystals continued... Iain W. Stewart 25 July 2013 Department of Mathematics and Statistics, University of Strathclyde, Glasgow, UK Raffaella De Vita, Don Leo at Virginia Tech, USA Fiona Stewart, Strathclyde NSF, EPSRC funding Iain W. Stewart Continuum Theory of Liquid Crystals continued...

2 Ericksen Leslie theory for nematic liquid crystals 2. Continuum description Energy density for nematics 5 / 42

3 Ericksen Leslie theory for nematic liquid crystals 2. Continuum description Energy density for nematics Nematic energy density The energy density, W F, for nematic liquid crystals goes back to Oseen and Zocher in the 1920s and was essentially established by Frank in 1958, now commonly called the Frank Oseen energy. This energy density for nematics, when it is assumed that the energy is invariant to a change in the sign of n, may be expressed as w F (n, n) = 1 2 K 1( n) K 2(n n) K 3(n n) (K 2 + K 4 ) [(n )n ( n)n] (1) where K 1 to K 4 are elastic constants. The total elastic energy is, of course, w F integrated over the sample volume V: W = w F (n, n) dv V 6 / 42

4 Ericksen Leslie theory for nematic liquid crystals 2. Continuum description Constitutive equations The experiments of Miesowicz (1936) and Zwetkoff (1939) suggested that t ij has a linear dependence upon N i and A ij so that a constitutive assumption takes the form t ij = A ij + B ijk N k + C ijkp A kp (19) where the coefficients A ij, B ijk and C ijkp are functions of n i. From the work of Smith and Rivlin (1957) tensors of such forms can be expressed explicitly. After invoking nematic symmetries, and the dissipation inequality, the final result was given by Leslie (1966) t ij = α 1 n k A kp n p n i n j + α 2 N i n j + α 3 n i N j + α 4 A ij + α 5 n j A ik n k + α 6 n i A jk n k (20) The coefficients α 1, α 2,..., α 6, are called the Leslie viscosity coefficients, or simply the Leslie viscosities. 15 / 42

5 Ericksen Leslie theory for nematic liquid crystals 2. Continuum description Ericksen Leslie equations Ericksen Leslie equations Director constraint: Incompressibility: Balance of linear momentum: n i n i = 1 v i,i = 0 ρ v i = ρf i (p + w F ),i + g j n j,i + G j n j,i + t ij,j Balance of angular momentum: ( ) wf w F + g i + G i = λn i n i,j n i,j G i is the generalised body force, related to the external body moment K i per unit mass through relation ρk i = ɛ ipq n p G q. λ is a Lagrange multiplier due the director constraint. g i is defined by g i = γ 1 N i γ 2 A ip n p 18 / 42

6 Continuum Model for Smectic A Liquid Crystals 3. Applications Twisted nematic device The twisted nematic device polariser d glass H z #! n "! o!"! o y x glass polariser (a) off state H < H c (b) on state H > H c This was studied theoretically by Leslie in 1970 and was applied by Schadt and Helfrich in the early 1970s to twisted nematic liquid crystal devices using an electric rather than a magnetic field (Figure from IWS The Static and Dynamic Continuum Theory of Liquid Crystals Taylor & Francis, 2004).

7 Smectic A liquid crystals Introduction and Motivation Mathematical description Continuum equations Connections between SmA and lipid bilayers Shear flow instability Conclusions and Future Plans Iain W. Stewart Continuum Model for Smectic A Liquid Crystals

8 Ericksen Leslie theory for nematic liquid crystals 2. Continuum description Energy density for nematics Nematic liquid crystals n anisotropic axis nematic liquid crystal director Ericksen and Leslie based their theories of dynamics on extensions to the static equilibrium theory of nematic liquid crystals. These equations require some knowledge of the energy density related to the distortions of the director. 4 / 42

9 Smectic Liquid Crystals (a) n (b) smectic A!! n smectic C 3 / 28

10 Simple Shear Problem 5 / 28

11 Motivation R. Ribotta and G. Durand, J. de Physique, 38, (1977) S.J. Elston, Liq. Cryst., 16, (1994); see also IWS J. Phys. A: Math. Theor., 40, (2007). G.K. Auernhammer, H. Brand and H. Pleiner Phys. Rev. E, 66, , (2002) An important reference since Iain W. Stewart n and a have been decoupled (no longer the same) a is no longer forced to be zero The linearised equations of the theory presented below coincide with those of Auernhammer et al. for SmA liquid crystals; however, their theory has been developed from a different technique and differs in certain nonlinear aspects. Continuum Model for Smectic A Liquid Crystals

12 Balance Laws Summary of the theory in Continuum Mech. Thermodyn., 18, (2007). The three conservation laws for mass, linear momentum and angular momentum.are, respectively, D ρ dv = 0, Dt D Dt V ρ v dv = V D ρ(x v) dv Dt V = V V ρf dv + t ds, S ρ(x F + K) dv + (x t + l) ds, where ρ denotes the density, x is the position vector, v is the velocity, F is the external body force per unit mass, t is the surface force per unit area, K is the external body moment per unit mass, l is the surface moment per unit area and D/Dt represents the material time derivative. S 8 / 28

13 A rate of work postulate is taken to be ρ(f v+k w) dv + Φ τ ν) ds = V S(t v+l w+ D Dt V ( 1 2 ρ v v+w A) dv + where w is the local angular velocity of the director, so that ṅ = w n and D 0 is the rate of viscous dissipation per unit volume. Equation (1) is similar to that used elsewhere by Leslie (1992) and Stewart (2004) except for the contribution due to Φ τ ν ; this, as discussed by E (1997), is the rate of work done by the layers at the boundary S. The term Φ is the rate of displacement of the layers and τ represents the permeation force at the boundary S applied to the layers in the volume V. This postulate is based on that for polar materials: see for example, Malvern (1969) and Atkin and Fox (1973). V D dv 9 / 28

14 The compressible smectic layers are described by Φ, a scalar function that depends on the spatial coordinates and time. The layer normal a is defined by Φ/ Φ and so it satisfies The director n must fulfil the constraint a i = Φ,i Φ, a ia i =1. (1) The incompressibility condition is given by where v is the velocity. n i n i =1. (2) v i,i =0, (3) 10 / 28

15 The equations that arise from the balance law for linear momentum are ρ v i = ρf i p,i + g j n j,i + G j n j,i + Φ a i J j,j + t ij,j, (4) where ρ is the density, F i is the external body force per unit mass, G i is the generalised external body torque that is related to the external body moment K i per unit mass, p = p + w A where p is the pressure and w A is the energy density, and J is defined by J i = w A Φ,i + 1 Φ [ ( ) wa a p,k,k w A a p ] (δ pi a p a i ). (5) This phase flux term J is also related to that introduced by Shalaginov, Hazelwood and Sluckin (1998 & 1999). 11 / 28

16 The balance of angular momentum leads to the equations ( ) wa w A + g i + G i = λn i, (6) n i n i,j,j where the scalar function λ is a Lagrange multiplier that arises from the constraint (2) and can usually be eliminated from these equations or evaluated by taking the scalar product of (6) with n. The permeation equation is Φ = λ p J i,i, (7) where λ p 0 is the permeation coefficient. The stress tensor and couple stress tensor are given by, respectively, t ij = p δ ij + Φ a i J j w A n p,j n p,i w A l ij = ɛ ipq ( n p w A n q,j + a p w A a q,j a p,i + t ij, (8) a p,j ). (9) 12 / 28

17 Equations (2) to (4), (6) and (7) provide nine equations in the nine unknowns Φ, n i, v i, p and λ ; the smectic layer normal a is, of course, determined by (1) from the solution for Φ. 13 / 28

18 The rate of strain tensor A and vorticity tensor W are defined in the usual way: A ij = 1 2 (v i,j + v j,i ), W ij = 1 2 (v i,j v j,i ), (10) and, following the standard procedure for nematics, the co-rotational time flux N of the director n is introduced as It is also known that N = ṅ Wn. (11) N = ω n, (12) where, in the terminology used by Cowin (1974), ω is the relative angular velocity, i.e., the difference between the usual regional angular velocity ŵ of a fluid, defined by ŵ = 1 2 v, and the local angular velocity w of the director, so that ω = w ŵ = w 1 2 v. (13) 14 / 28

19 The constitutive equations for the viscous stress t ij and g i are given by t ij = α 1 (n k A kp n p )n i n j + α 2 N i n j + α 3 n i N j + α 4 A ij +α 5 (n j A ip n p + n i A jp n p )+(α 2 + α 3 )n i A jp n p +τ 1 (a k A kp a p )a i a j + τ 2 (a i A jp a p + a j A ip a p ) +κ 1 (a i N j + a j N i + n i A jp a p n j A ip a p ) +κ 2 (n k A kp a p )(n i a j + a i n j ) +κ 3 [(n k A kp n p )a i a j +(a k A kp a p )n i n j ] +κ 4 [2(n k A kp a p )n i n j +(n k A kp n p )(a i n j + n i a j )] +κ 5 [2(n k A kp a p )a i a j +(a k A kp a p )(n i a j + a i n j )] +κ 6 (n j A ip a p + n i A jp a p + a i A jp n p + a j A ip n p ). (14) g i = (α 3 α 2 )N i (α 2 + α 3 )A ip n p 2κ 1 A ip a p. (15) 15 / 28

20 Remarks If all terms containing the director n are neglected then t ij = α 4 A ij + τ 1 (a k A kp a p )a i a j + τ 2 (a i A jp a p + a j A ip a p ). (16) This viscous stress is precisely that which is known from previous descriptions of incompressible SmA liquid crystals by Martin it et al. (1972), E (1997)and Forster et al. (1971). It also coincides with the incompressible case highlighted by de Gennes and Prost (1993) for a linearised description of planar samples of SmA. The viscosities α 1 to α 5 are nematic-like while τ 1 and τ 2 are SmA-like. The viscosity coefficients κ 1 to κ 6 occur in contributions that depend upon both a and n these terms are coupling terms that reflect the links between the traditional nematic and SmA modes of behaviour. 16 / 28

21 Summary of key equations Incompressibility: Linear momentum: v i,i =0. (17) ρ v i = ρf i p,i + g j n j,i + G j n j,i + Φ a i J j,j + t ij,j. (18) Angular momentum: ( ) wa n i,j,j w A n i + g i + G i = λn i. (19) Permeation equation: Φ = λ p J i,i. (20) 17 / 28

22 An energy density has been proposed for lipid bilayers by May (2000): w A = K n 1 2 ( ) θ(r) 2 + θ θ(r)θ (r) (r)+s 0 K 4 r r ( ) b(r) 2 + B b 2 B 1(θ(r) b (r)) 2 0 ( )( ) θ(r) b(r) + B 2 + θ (r) 1 r b 0 (21) where θ(r) and b(r) are defined in previous figure; the constant coefficients K n 1, K 4, B 0, B 1 and B 2 have been introduced here rather than May s original notation so that a direct comparison can be made with liquid crystal theory. 19 / 28

23 A general energy per unit volume, w A, has been derived from geometrical considerations by De Vita and Stewart (to!2013"#$ appear): w A = 1 2 K n 1 ( n s 0 ) K 4 [(n )n ( n)n] B 0 Φ 2 (1 Φ ) B 1 ( 1 (n a) 2 ) + B 2 ( n)(1 Φ 1 )). (22) The first term is the usual elastic splay energy with s 0 denoting the spontaneous splay, the second is the saddle-splay energy, the third is the compression-expansion energy, the fourth accounts for the energy cost of decoupling n and a, and the fifth measures the coupling between the elastic splay and compression of the layers. Except for B 2, versions of these terms have been discussed by Ribotta and Durand (1977), Kleman and Parodi (1977), E (1997), Capriz and Napoli (2001) and Stewart (2007). The quadratic order approximation of (22) is identical to that proposed by May, who derived his terms in an ad hoc manner.!"# 20 / 28

24 Simple Shear Problem Iain W. Stewart Continuum Model for Smectic A Liquid Crystals

25 Φ = z u(x, y, z, t), a = Φ Φ n = (sin θ cos φ, sin θ sin φ, cos θ), a =( u x, u y, 1) v = ( γz, 0, 0) γ : shear rate u : displacement of the layers Iain W. Stewart Continuum Model for Smectic A Liquid Crystals

26 Governing dynamic equations Continuum Mech. Thermodyn., 18, (2007) n i n i = 1, a i a i = 1, v i,i = 0 ρ v i = p,i + g n k n k,i + t ij,j + Φ a i J j,j ( w n i,j ),j w n i + g n i + µn i =0 Φ = λ p ( J) g n i = 2(λ 1 D a i + λ 4 A i ) Iain W. Stewart Continuum Model for Smectic A Liquid Crystals

27 Dynamic Equations v = ( γz, 0, 0) satisfies linear momentum equations and enables us to determine the pressure, p. The remaining equations for angular momentum and permeation reduce to the following: ( λ +1 2 ) λ sin 2 (θ 0 ) γ = B 1 sin(θ 0 ) cos(θ 0 )+ B 0 sin(θ 0 )[1 cos(θ 0 )], γ 1 γ 1 Iain W. Stewart Continuum Model for Smectic A Liquid Crystals

28 Dynamic Equations cont. [ A θ 2 γλsin(θ 0 ) cos(θ 0 )+ B 0 [ sin 2 (θ 0 ) cos 2 (θ 0 ) + cos(θ 0 ) ] γ 1 B 1 γ 1 [sin 2 (θ 0 ) cos 2 (θ 0 )] A φ 1 2 γ(λ + 1) A u [ ] A u B 0 γ 1 sin(θ 0 )q z =0, ] B 0 1 cos(θ 0 ) q x + B 1 q x γ 1 cos(θ 0 ) γ 1 =0, A θ λ p B 0 sin(θ 0 )q z + A φ λ p [B 1 q x sin(θ 0 ) cos(θ 0 )+B 0 q x sin(θ 0 )(1 cos(θ 0 ))] + A u λ p [B 0 q 2 x(1 cos(θ 0 )) 2 B 1 q 2 xcos 2 (θ 0 ) K 1 q 4 x B 0 q 2 z ]=0. Iain W. Stewart Continuum Model for Smectic A Liquid Crystals

29 Example: shear flow instability See figure on slide 5. An instability can be detected by finding the critical shear rate k c for the onset of an instability to a steady state solution. Consider small disturbances to a steady planar state of the form: {ˆθ, ˆφ, û, v x, v y, v z, p} = { } A θ, A φ, A u, A vx, A vy, A vz, A p e i(q y y+q z z), (23) where A θ, A φ, A u, A vx, A vy, A vz and A p are small amplitudes. Inserting (23) into the relevant seven dynamic continuum equations and linearising results in a matrix system of the form Ax = 0, (24) where A is the appropriate constant coefficient matrix and x =[A θ, A φ, A u, A vx, A vy, A vz, A p ] T. A non-zero solution requires det(a) = / 28

30 Shear Rate tilt angle 0 (rad) (k) nonlinear 0 (k) linear approximation shear rate k ( s -1 ) Iain W. Stewart Continuum Model for Smectic A Liquid Crystals

31 Criticality (a) c (rad) minimal set coupling to velocity B 0 (N m -2 ) (b) q c (m -1 ) x10 7 6x10 7 4x10 7 minimal set coupling to velocity 2x B 0 (N m -2 ) Iain W. Stewart Continuum Model for Smectic A Liquid Crystals

32 Criticality (a) c (rad) (b) 3.645x minimal set coupling to velocity B 1 (N m -2 ) q c (m -1 ) 3.636x x10 7 minimal set coupling to velocity B 1 (N m -2 ) Iain W. Stewart Continuum Model for Smectic A Liquid Crystals

33 Dependence on the permeation coefficient λ p : 9.5x10 6 k c (s -1 ) 9.0x x x x q c (m -1 ) 3.6x x p (m 2 Pa -1 s -1 ) Extensive details of these and related results can be found in Stewart & Stewart, J. Phys.: Condens. Matter, Vol. 21, (2009). 25 / 28

34 Conclusions and future work The experimental work of Elston (1994) can be modelled successfully by the equilibrium version of the above theory of sma features could be modelled that were missed by the classical Helfrich model for SmA. Iain W. Stewart Continuum Model for Smectic A Liquid Crystals

35 Experimental result S.J. Elston, Liq. Cryst., 16, 151 (1994); IWS J. Phys. A: Math. Theor., 40, 5297 (2007). Iain W. Stewart Continuum Model for Smectic A Liquid Crystals

36 Conclusions and future work The experimental work of Elston (1994) can be modelled successfully by the equilibrium version of the above theory of sma features could be modelled that were missed by the classical Helfrich model for SmA. Effects of weak anchoring in finite domains: De Vita & IWS, J. Phys.: Condens. Matter (2008). Nonlinearities in tilt snd layer displacements of planar lipid bilayers: De Vita & IWS, Eur. Phys. J. E (2010). Comparison of results with a lipid bilayer model: De Vita & IWS, Soft Matter (2013). Iain W. Stewart Continuum Model for Smectic A Liquid Crystals

37 Acknowledgements EPSRC for two PhD studentships (Fiona Stewart, Jamie Cowley) NSF and Virginia Tech for funding my visits to Leo, Sarles and De Vita in the USA, and for funding De Vita to visit the UK To the Isaac Newton Institute for the invitation to participate in this programme and for the opportunity to present this work Thank you for your attention Iain W. Stewart Continuum Model for Smectic A Liquid Crystals

38 Material parameters Material parameter values (cf. de Gennes and Prost (1993), Auernhammer et al. (2002), Kleman and Lavrentovich (2003), Stewart (2004): parameter d q z K B 0 B 1 λ p α 1 α 2 α 3 α 4, τ 1, τ 2 α 5 γ 1 = α 3 α 2 γ 2 = α 2 + α 3 ρ typical value 10 5 m π/d N Nm Nm m 2 Pa 1 s Pa s Pa s Pa s Pa s Pa s Pa s Pa s 1000 kg m 3 28 / 28