Telephone-cord instabilities in thin smectic capillaries

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1 Telephone-cord instabilities in thin smectic capillaries Paolo Biscari, Politecnico di Milano, Italy Maria Carme Calderer, University of Minnesota, USA P.Biscari, M.C.Calderer Tel-cord instabilities Cortona 05 1/19

2 Overview Free energy functional and geometry Chirality-induced SmA-SmC transition Bent SmA* Helicoidal SmC* P.Biscari, M.C.Calderer Tel-cord instabilities Cortona 05 2/19

3 Nematic liquid crystals - Frank energy n Order parameter: n, director σ nem = K 1 ( div n ) 2+ K2 ( n curln + qch ) 2+ K3 n curln + v0 2 + ( K 2 + K 4 ) ( tr( n) 2 (div n) 2) q ch : cholesteric pitch v 0 : spontaneous bend (below) P.Biscari, M.C.Calderer Tel-cord instabilities Cortona 05 3/19

4 Smectic liquid crystals - elastic energy Order parameter: ψ = ρ e iω, σ sm = C n ( ψ i qsm ψn) 2 + C ψ i qsm ψn 2 + ζ(ρ) q sm : smectic pitch n ( ψ i qsm ψn) 2 = ( n ρ ) 2 + ρ 2 ( n ω q sm ) 2 ψ i qsm ψn 2 = ρ 2 + ρ2 ω 2 C 0; σ sm positively defined provided C cos 2 α + C sin 2 α > 0 ( α, angle (n, ω) ) P.Biscari, M.C.Calderer Tel-cord instabilities Cortona 05 4/19

5 Spontaneous polarization Let P = P p be the (spontaneous) polarization vector. [ ] ( ) 2 σ pol P = G1 div P + G P 2 ( + G P). G: scalar potential depending only on P. Induced charges on the boundary: σ anch [P] = ω P P ( 1 p ν ). The spontaneous bend depends on P: λ P if P 0 (spontaneous-polarization induced bend) v 0 = b 0 ν n if P = 0 (flexoelectric ; ν, layer normal) P.Biscari, M.C.Calderer Tel-cord instabilities Cortona 05 5/19

6 s Ω Geometry N T (B := T N) curvature κ torsion τ Ω = { P R 3 : P = c(s) + ξ e, for s [0, l], ξ [0, r], e = cos ϑ N + sinϑb }. Free-boundary conditions on nematic and smectic fields ( ) Thin capillary: all fields depend only on s ω(s) ν = T n = cos α T + sinαcos ϕ N + sinαsinϕb. P.Biscari, M.C.Calderer Tel-cord instabilities Cortona 05 6/19

7 Linear shapes 1/3 Let κ = τ 0 and P = 0. A non-zero cholesteric pitch may induce a SmA-SmC transition even if C > 0. σ = σ nem + σ sm = ( K 1 sin 2 α + K 3 cos 2 α ) α 2 ( + K 2 qch ϕ sin 2 α ) 2 [ + K 3 sin 2 α (b 0 ϕ cos α) 2 + C ρ 2 cos 2 α + ρ 2 (ω cos α q sm ) 2] + C sin 2 α ( ρ 2 + ρ 2 ω 2) + ζ(ρ) Integrate E.L. eqn s for ϕ,ω... Look for solutions with α α 0 and ρ ρ 0... P.Biscari, M.C.Calderer Tel-cord instabilities Cortona 05 7/19

8 Linear shapes 2/3 σ(α 0, ρ 0 ) = K 2K 3 (q ch cos α 0 b 0 sin 2 α 0 ) 2 K 2 sin 2 α 0 + K 3 cos 2 α 0 + C C ρ 2 0q 2 sm sin 2 α 0 C cos 2 α 0 + C sin 2 α 0 + ζ(ρ 0 ). SmA phase (α 0 = 0) always a stationarity point. SmA unstable even when C > 0, provided that since σ(α 0, ρ 0 ) = σ(0, ρ 0 ) + C q 2 smρ 2 0 < K 2 K 3 q ch ( K2 q ch + 2b 0 K 3 ), + O(α 4 0) as α 0 0. (C q 2smρ 20 K ( )) 2q ch K2 q ch + 2b 0 K 3 K 3 α 2 0 P.Biscari, M.C.Calderer Tel-cord instabilities Cortona 05 8/19

9 Linear shapes 3/3 π 2 π 4 π 2 π q ch /q sm q ch /q sm q ch /q sm α opt 0 as a function of q ch /q sm when K 2 = K 3 = C ρ 2 0; C = 1 3 C (left), C = 2 3 C (center) or C = C (right), and b 0 /q sm = 0 (bold), 1 4, 1 2, 3 4, 1 (dotted). P.Biscari, M.C.Calderer Tel-cord instabilities Cortona 05 9/19

10 Flexoelectric bending? No spontaneous bending if v 0 is of flexoelectric origin. σ = f(κr)k 13 A 2 + f(κr) 1 K 23 Φ 2 f(κr) ξ,ϑ f(κr) + K 23 (Φ q 1 ) 2 + K 2K 3 K 23 q ch cos α b 0 sin 2 α 2 + C C ρ 2 0q 2 sm sin 2 α C cos 2 α + C sin 2 α with A := α + κ cos ϕ Φ := (ϕ τ) sin α κ cos α sin ϕ K 13 := K 1 sin 2 α + K 3 cos 2 α K 23 := K 2 sin 2 α + K 3 cos 2 α q 1 := (K 2q ch + K 3 b 0 cos α)sin α K 23 2 := C 2 ρ 4 0q 2 sm cos 2 α C cos 2 α + C sin 2 α f(x) := if x = 0, 1 x 2 ºx 2 if x (0, 1] P.Biscari, M.C.Calderer Tel-cord instabilities Cortona 05 10/19

11 Bent SmA* 1/4 Let α 0. Assume ρ ρ 0, P P 0 =: λ 0 /λ, ([λ 0 ] = L 1 ). Let Γ := GP 2 0, Γ 1 := G 1 P 2 0. Let P = P 0 ( cos φ B + sinφn ). Bulk free-energy density σ b = σ nem + σ sm + σ pol [ ( ) 2 σ b = K 2 qch 2 κ + K 3 λ 0 + 2κλ ( )] 0 1 sinφ 1 κξ cos ϑ 1 κξ cos ϑ ( + C ρ 2 ω ) κξ cos ϑ q sm + Γ 1 κ 2 sin 2 φ (1 κξ cos ϑ) 2 + Γ κ2 sin 2 φ + (φ + τ) 2 (1 κξ cos ϑ) 2. Anchoring: σ anch = ω P P 0 ( 1 sin(ϑ + φ) ). P.Biscari, M.C.Calderer Tel-cord instabilities Cortona 05 11/19

12 Bent SmA* 2/4 Integrate over the transverse section, insert equilibrium value of ω... 1 πr 2 σ = K 2 qch 2 ( + K 3 λ 2 0 2κλ 0 sinφ + κ 2 f(κr) ) ξ,ϑ + C ρ 2 0q 2 sm f(κr) 1 f(κr) + (Γ + Γ 1 ) κ 2 sin 2 φ f(κr) + Γ (φ + τ) 2 f(κr) + ω Pλ 0 ( 2 + κr sinφ ) λr Optimal polarization direction for K 3 -term: P = P 0 N. P.Biscari, M.C.Calderer Tel-cord instabilities Cortona 05 12/19

13 Bent SmA* 3/4 With above choices F opt (κ, τ) πr 2 l = A 0 A 1 κr + A 2 (κr) 2 f(κr) A 3 f(κr) + B 1 f(κr) (τr) 2, where A s and B s are non-negative: A 0 = K 2 q 2 ch + K 3 λ C ρ 2 0q 2 sm + 2ω P P 0 /r A 1 = (2K 3 λ ω P ) P 0 r A 2 = ( ) K 3 + Γ + Γ 1 / r 2 A 3 = C ρ 2 0q 2 sm B 1 = Γ/r 2. Minimum attained at τ = 0 and positive κ: F opt (κ, 0) πr 2 l = (A 0 A 3 ) A 1 κr + A 2 + A 3 3 (κr) 2 + O(κr) 4 (κr 0) P.Biscari, M.C.Calderer Tel-cord instabilities Cortona 05 13/19

14 Bent SmA* 4/4 Preferred curvature of the axis of a SmA* capillary 1.0 κ opt r P 0 /P 0 P 0 := K 3 + Γ + Γ 1 (2K 3 λ ω P )r. Top to bottom, C ρ 2 0q 2 sm K 3 + Γ + Γ 1 = 0, 1, 10. P.Biscari, M.C.Calderer Tel-cord instabilities Cortona 05 14/19

15 Helicoidal SmC* 1/4 Assume α α 0, φ φ 0. Then: σ b = σ sm (α 0 ) + K 1 κ 2 cos 2 ϕ sin 2 α 0 (1 κξ cos ϑ) 2 + K 2 q ch sin α 0 1 κξ cos ϑ (ϕ τ) sin α 0 κ cos α 0 sin ϕ + K 3 λ 0 sin φ 0 κ cos ϕ cos α κξ cos ϑ 2 + λ 0 cos φ 0 (ϕ τ) sin α 0 κ cos α 0 sin ϕ 1 κξ cos ϑ 2 cos α Γ 1 (1 κξ cos ϑ) 2 (ϕ τ)cos α 0 + κsin α 0 sin ϕ cos φ 0 sin α 0 (ϕ τ)sin α 0 cos φ 0 + κ cos ϕ sin φ 0 κ cos α 0 sin ϕ cos φ 0 cos α0 2 Γ (ϕ 2 (1 κξ cos ϑ) 2 τ)cos α 0 + κ sin α 0 sin ϕ + (ϕ τ)sin α 0 cos φ 0 + κ cos ϕ sin φ 0 κ cos α 0 sin ϕ cos φ 0 2 P.Biscari, M.C.Calderer Tel-cord instabilities Cortona 05 15/19

16 Helicoidal SmC* 2/4 1-constant approximation + thin capillary regime: K 1 = K 2 = K 3 = Γ = ω P /λ =: K ; Γ 1 = 0 ; λ 0 r 1 = If q ch = 0, κ opt qch =0 = 3 cos 2α λ 0 and τ opt qch =0 = 3 8 sin2α 0 λ 0. In general, κ opt, τ opt are different from zero. P.Biscari, M.C.Calderer Tel-cord instabilities Cortona 05 16/19

17 Helicoidal SmC* 3/ κ opt τ opt q ch λ q ch λ π 4 α π π 4 α π 2 Curvature and torsion of an optimal-shaped SmC* capillary. The plots correspond to q ch /λ 0 = 0.1, 0.5, 1.0, 1.5, 2. Exp obs: r = 0.85µm, r hel = 2.25µm, p hel = 6.7µm; q ch = 0 = α 0 20 C. P.Biscari, M.C.Calderer Tel-cord instabilities Cortona 05 17/19

18 Helicoidal SmC* 4/4 Small-α 0 limit: κ opt = 2K 3λ ω P 2(K 3 + Γ + Γ 1 ) P 0 + O(α 0 ), τ opt = K ( ) 3 2(Γ + Γ1 )λ + ω P P 0 α 0 + O(α 2 2Γ(K 3 + Γ + Γ 1 ) 0) as α 0 0. Optimal free energy: [ F opt πr 2 l = σ sm(α 0 ) + K 3 λ 2 0 ( ) 2P 2 2K3 λ ω P 0 4(K 3 + Γ + Γ 1 ) + K 2qch 2 + 2ω PP 0 r ] 2K ( ) 2 2K3 λ ω P P0 q ch α 0 2(K 3 + Γ + Γ 1 ) + O(α 2 0)... SmA* phase unstable if q ch 0! P.Biscari, M.C.Calderer Tel-cord instabilities Cortona 05 18/19

19 Conclusions A non-null cholesteric pitch may induce a SmA-SmC transition, even when C > 0 Flexoelectric effects do not lead to a spontaneous capillary bending κ > 0 in a SmA* capillary Telephone-cord instabilities arise in SmC* Cholesteric pitch not a direct ingredient for the tel-cord instability, but it induces also a SmA*-SmC* transition. P.Biscari, M.C.Calderer Tel-cord instabilities Cortona 05 19/19

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