The Physics of Liquid Crystals P. G. de Gennes and J. Prost (Oxford University Press, 1993) Liquid crystals are beautiful and mysterious; I am fond of them for both reasons. My hope is that some readers of this book will feel the same attraction, help to solve the mysteries, and raise new questions.
SmA SmC
N a = 1/ N a = 0 z i= 1 iz N z = 1/ iz i= 1 a N a
a = a = a, a + a + a = 1 x y z x y z a = a = a = 1/3 x y z a = 1 z S 0 S = 3 1 S = a z S =1 3 1 0< S < 1 = 3 a z 1 ( )
r' r r ' = Ur r = ( xyz,, ) = ( x1, x, x3) r ' = ( x', y', z') = ( x ', x ', x ') 1 3 U11 U1 U13 3 U1 U U x U = α ' = Uαβ xβ = U αβ x β 3 β = 1 U31 U3 U 33 r a θ b a ' a b= a' b' θ b' a b= ab α α
P = χε P E χ P1 χ11 χ1 χ13 E1 P = χ χ χ E 1 3 P 3 χ31 χ3 χ 33 E 3 P ' = χε ' ' UP = χ ' UE P' = UP ' = Ε UE χ' χ χ' = U U t = U U χ αβ αγ βδ γδ ( 1 χ ' ) P = U U E t U χ a aa χ αα α β
S 1 ( S = 3 a 1 ) z 1 = ( 3 a a δ ) αβ α β αβ S αβ n n S S = (3 n n δ ) αβ α β αβ
- F S αβ A B C ( ) C 3 4 4 1 F = F0 + Sαβ Sβα + Sαβ Sβγ Sγα + Sαβ Sβα + Sαβ Sβγ Sγδ Sδα 3A B 9C F F S S S 4 4 16 S S = (3 n n δ ) αβ α β αβ A= A ( T T ) 3 4 = 0 + + + 0 0 C = C + C / 1 T c
S n / x = 0 n / x 0 α β ( αβ=, 1,,3) α β f d n / x F = f dv d α β nr ()
f d f d f d f d n n 1 1 fd = K1( n) + K( n ( n) ) 1 + K3( n ( n) ) + K' n ( n) (.) ( / x, / y, / z) n= div n, n= rotn K ( i = 1,,3) i
1 1 fd = K1( n) + K( n ( n) ) 1 + K3( n ( n) ) + K' n ( n) = ( / x, / y, / z) n = n / x+ n / y+ n / z x y z ( n / y n / z, n / z n / x, n / x n / y) n = z y x z y x n ( n ) K '
x-z n = (sin θ,0,cos θ) ( θ,0,1) 1 ( ) 1 K n n θ / x = θ = cx c θ K, n 1 K, ( n ) n K, 3 ( n ) n
1 1 fd = K1( n) + K( n ( n) ) 1 + K3( n ( n) ) + K' n ( n) = 1 K ( n) + 1 K ( n ( n) + q ) + 1 K ( n ( n) ) 1 K q 1 0 3 0 q = K '/ K 0 ( ) 0 0 n n + q = nr ( ) = (cos qz,sin qz,0) 0 0 z
x y nr () = (cos φ(),sin r φ(),0) r 1 1 1 fd = K1( n) + K( n ( n) ) + K3( n ( n) ) K1 = K = K3 = K f d φ 1, φ K φ φ = + x y x y (3.47)
F = f dv d φ() r δφ() r φ() r φ() r + δφ() r F δf δf = f ( ( φ+ δφ)/ x, ( φ+ δφ)/ y)dv d δ F fd ( φ/ x, φ/ y)dv = δφdv + L > 0 δφ δ F δφ fd fd fd φ x ( φ/ x) y ( φ/ y) = 0 δ F δφ : φ x φ y + = 0 f d 1 K φ φ = + x y
45 φ = sα + c (3.49) α = 1 tan ( y/ x) s =± 1, ±, ± 3, L ± 1/, ± 3/, ± 5 /, L c :
f ( ) el = 1 D E = 1 εε 1 1 0 E E = εε 0 E ε0 ε n E (3.3) ε = ε δ + ( ε ε )nn (3.) αβ αβ α β ε : ε ε : 0 ε = ε ε : ε > 0 n E ε < 0 n E 1 1 1 1 f ( ) mag = µ 0 χ B µ 0 χ n B (3.4) µ : µ µ : 0 µ = µ µ :
n = (cos θ( z),sin θ( z),0) 1 dθ 1 f fd + fmag = K sin dz ξ θ (3.6) K ξ = µ χ 1 0 1 B
dθ F = f θ( z), dz θ( z) dz f d f = θ d z (d θ /d z) 0 d θ 1 + sin θ cos θ =0 dz ξ (3.6) (3.14) θ x, z t ξ d x 1 cos = x dt x
θ (b) z = 0 θ (0) = 0, z θ = ± π/, = 0 z θ π ± z / ξ tan + = 4 e ξ K = µ χ 1 0 1 B
θ(0) = θ( d) = 0 3. (a) (3.14)
1 3 θ < θm << 1 sin θ θ θ 6 d 1 d θ 1 ( 4 F ( f ) = 0 d + fmag)d z K θ θ /3 dz 0 z ξ θ() z = θ sin( qz) q= π / d m F d 1 1 F = K q θ θ 4 ξ + 4ξ 4 m m θ m B = π K µ χ c 1 d 0 :
F / θ = 0 m F d 1 1 = K θ q + θ = m m θm ξ ξ 0 B< B q ξ θ = c ( > m 0 B > B q ξ θ ( < c c m =± 1 ξ q =± B B ( B+ B )( B B ) B ( B B ) B B =± ± =± B c c c c c B Bc Bc
x = ρcos α, y = ρsinα (3.47) 1 φ 1 φ (3.49) 1 s f = K K ρ + = ρ α ρ d ρ 1 s F = fd x y = K πρ ρ = π Ks ρ a max a ρ max dd d log( / ) ρmax a: (3.55)
σ line σ line F l = l T F : l l Fl = Fl σ = line F
dr dt = Γ R R = Γ t0 t ( ) R Γ Formation, Dynamics and Statistics of Patterns (World Scientific, 1993)
K = K = K = K φ = sα + s α + c 1 1 1 3 = s tan ( y / x) + s tan ( y /( x d)) + c 1 1 1 (3.59) z K φ φ F = + dd x y x y (3.60) φ φ ε = K, Ex =, Ey = (3.61) y x ε { (3.60) F = E } x + Ey dd x y
(3.59) (3.61) x x d y y ( Ex, Ey) = s1 + s, s1 + s x + y ( x d) + y x + y ( x d) + ε σ1 x σ x d ( Ex, Ey) = +, πε x + y πε ( x d) + y σ1 y σ y + πε x + y πε ( x d) + y σ i i s = σ σ, s πε = πε (3.65) 1 1 y
ε f int = σσ 1 πε d (3.65) f int = π Ks s d 1
G dl = d t l l = G( t0 t) l G
εε, ε : ε: 0 0 z E ( z, t) = acos( kz ωt+ δ ), E ( z, t) = 0 (1.a) x E ( z, t) = 0, E ( z, t) = bcos( kz ωt+ δ ) (1.b) x y 1 y δ = δ 1 k = πn/ λ n= ε, λ : (1.a) (1.b)
(1.a) (1.b) E ( z, t) = acos( kz ωt+ δ ), E ( z, t) = asin( kz ωt+ δ ) x 1 y 1 E ( z, t) = bcos( kz ωt+ δ ), E ( z, t) = bsin( kz ωt+ δ ) x y E ( z, t) = acos( kz ωt+ δ ), E ( z, t) = bcos( kz ωt+ δ ) x 1 y
( E, E ) = ( E cos( kz ωt), 0) x 0 0 y 0 ( ω ω ) ( ω ω ) = E / cos( kz t), sin( kz t) + E / cos( kz t), -sin( kz t) E ( E ) 0 ( ) 0 x, Ey = cos( krz ωt), sin( krz ωt) + ( cos( klz ωt), sin( klz ωt) ) kr kl kr kl kr + kl = E0 cos z,sin z cos z ωt ψ = ( k k )/ d = ( v v ) ωd / = π( n n ) d / λ -1-1 R L L R L R E (5.39) nl, nr :
ε x 0 0 0 ε y 0 0 0 ε z z E ( z, t) = acos( k z ωt+ δ ), E ( z, t) = 0 x x 1 y E ( z, t) = 0, E ( z, t) = bcos( k z ωt+ δ ) x y y k = πn / λ n = ε x x x x k = πn / λ n = ε y y y y
d
E ( z, t) = E e i kz t x E ( z, t) = E sinα e E ( z, t) = E cosα e 0 0 0 ( ω ) i( πn / λ z ωt) i( πn / λ z ωt) x E ( d, t) = E ( d, t)sin α + E ( d, t)cosα y 1 = E0 α n, n : ( / / ) i πn d λ i πn d λ iωt sin e e e π nd a I = Ey = I0sin α sin, na = n n : λ
z Er (,) t = ( E(,), zt E(,),0), zt Hr (,) t = ( H(,), zt H (,),0) zt x y x y H = D = εε E 0, E = B = µ H 0 t t t t ε E E = ( E) = E ( E) = (5.45) c t z (3.) ( ε + ε )/ 0 0 cosqz 0 sinqz 0 0 ε a ε ( z) = 0 ( ε + ε )/ 0 + sinqz 0 cosq0z 0 0 0 ε 0 0 0 q : 0
1 iωt 1 E(,) zt = E()e z + E()*e z iωt = Re[ E( z)e ] ± E ( z) E ( z) ± ie ( z) x y iωt E + ( z) E ( z) z ( + ) ( ikz ) E ( z), E ( z) = e,0 ( k > 0) ikz ikz ( E ( z), E ( z)) = (e /, ie /) x y E ( z, t), E ( z, t) = cos( kz ωt)/,sin( kz ωt)/ ( ) ( ) x y
± E ( z) E ( z) ± ie ( z) x y + iq0z + d E ( z) k0 k1 e E ( z) = iq0z dz E ( z) k1 e k0 E ( z) k ω ω ε a 0 = ε, k1 = c c ε = ( ε + ε )/, ε = ε ε a + E z a il ( + q0 ) z ( ) = e, E ( z) = be il ( q ) z 0 (5.60)
( l+ q0) k0 k a 1 = k1 ( l q0) k b 0 0 (5.61) ( l+ q ) k k 0 0 1 k1 ( l q0) k0 4 = ( k0 l q0) 4q0l k1 = 0 (5.6) ω ω ± () l = c ( l + q ) ± 4 q l + ( ε / ε) ( l q ) ε ε ε 0 0 a 0 1 ( a / )
L R +R + L L +R R + L + R: z L: z
l ± iκ ( κ > 0) cq ω (0) < ω < ω (0) ω (0) =, ω (0) = cq 0 0 + + n n (5.60) + E z a a il ( + q0) z mκ iqz 0 ( ) = e = e z e, E z = b = b il ( q0) z mκ z iqz 0 ( ) e e e
(5.6) 4 k1 kl k0 + 8 kq( q + k) k k + R 0 0 0 0 0 4 k1 8 kq( q k) 0 0 0 0 (5.39) 4 k1 ( kl kr) /= 8 q ( k q ) 0 0 0 q n 0 n 1 = n + n λ λ 0 0 3 ' (1 ' ) λ' = λ/ P= q / k, λ:, P:
ω l ( λ << ( n n ) P) (5.6) ω ω l n ( ω ( l) ) l n ( ω+ ( l) ) c c a = b (5.61) a = b Ex (,) z t cos qz 0 acos( lz t) E ( zt, ) ω = sin qz y 0 Ex (,) z t sin qz 0 = asin( lz ω t) Ey ( z, t) cos qz 0 cos qz 0 n( z) = sin qz 0
TN(Twisted Nematic)
n = (cos( φ ( z/ d 1)),sin( φ ( z/ d 1)),0) 0 0
n() z = (sin θ()cos z φ(),sin z θ()sin z φ(),cos z θ()) z 1 θ 1 φ fd = g( θ) + h( θ) z z g( θ) = K sin θ + K cos θ, h( θ) = ( K sin θ + K cos θ)sin θ 1 3 3 θ = π / ψ ( ψ << 1) 1 ψ 1 φ 1 d = 1 + + 3 ψ ε0 εψ { ( ) } f K z K K K z E (5.86) (3.3)
m 0 ( z d ) ψ = ψ sin( πz/ d), φ = φ / 1 1 1 F / d = K + ( K K ) E + K 4 d d d π φ0 φ0 1 3 ε0 ε ψm ψ m V K K K π c = 1/ 1 + ( 3 ) φ0 ( ε0 ε) 1/
vr (): ρ div( ρ ) t = v ρ divv = 0 ρ d v vr ( + vdt, t + d t) vr (, t) dt dt v = + ( v grad) v t
ρ dv v v σ ρ + vβ = dt t dxβ dx α α α βα β σ αβ v / x v x α β A W = A + W αβ αβ αβ αβ 1 v v α = + xβ x 1 v v α = xβ x ω = ( W, W, W ) β α β α β α (4.16a) (4.16b) yz zx xy
v y x = A + W xy xy 0
σ α αβ β σ βα n β (s) f = σ / x (s) α βα β f α
σ αβ = (visc) σ α β pδ αβ A v η : (visc) α σαβ = η αβ = η + xβ p : v x β α vα ρ + ρv v = p+ η v t xβ xα vα = 0 x α β α α
edθ= Ω dt = n dn Ω Ω d = n n dt I dω dt = Γ I Γ
Γ hr () nr () nr () +δ nr () h α δf = hr () δn()d r V (4.6) () r f f δ F nα xβ ( nα / xβ) δ nα d d = = (4.7) ( nr hr ) ( nr () δ nr ()) δ F = hr () δ nr ()dv = () () dv ( F ) eδθ = n δ n: Γ = n h : (4.56)
Γ, Γ (visc) (visc) 1 N dn dt = Ω n d N ( Ω ω) n= n ω n(4.55) dt γ 1 Ω? ω ( Ω ω) = n N + ( n Ω ) n (4.55) 0 dn = dt (visc) Γ 1 = γ1n N γ1n ω n 1 γ
Γ = γ n (visc) An I dω = Γ = Γ + Γ + Γ dt dn = n h n ω n dt ( F ) (visc) (visc) 1 γ1 γ n Γ = Γ + Γ (visc) (visc) (visc) An 1 dn = γ n γ A 1 ω n n n dt (4.63) d γ n n h= 1n ω n + γn An dt
σ = σ α + σ (e) (visc) αβ β αβ σ σ = f n (e) d γ βα ( nγ / nβ) xα pδ = α A + α n n n n A + α n n A + α n n A (visc) αβ 4 αβ 1 α β µ ρ µρ 5 α µ µβ 6 β µ µα + α nn + α nn α β 3 β α αβ α i dn N = ( Ω ω) n= ω n dt
α nnnna 1 α β µ ρ µρ A = 1, A = 1 n = ( n, n,0) x y xx yy nnnna α β µ ρ µρ n ( ) x nxny nx ny nn x y ny
γ = α α 1 3 γ = α α 6 5 α α = α + α 6 5 3
dvα ( (e) (visc) ρ = σ ) βα + σβα (4.76a) dt xβ Ω dn I ( =0 ) = n h γ1n ω n γn An t dt (4.76b) vα = 0 (4.76c) x α σ = α A + α n n n n A + α n n A + α n n A (visc) αβ 4 αβ 1 α β µ ρ µρ 5 α µ µβ 6 β µ µα + αnn α β + α3nn β α (4.77) (e) f nγ σ βα = pδαβ (4.78a) ( nγ / nβ) xα f fd hα = (4.78b) xβ ( nα / xβ) nα dnα dnα Nα = ( ω n) α = Wαβnβ (4.78c) dt dt vn,, p
v = (0,0, γ& x) γ& n n = (sinθ cos φ,sinθsin φ,cos θ) (4.80) (4.16) 1 Axz = Wzx = ωy = γ& (4.81) (4.78c) 1 1 Nz = ωynx = γ& nx, Nx = ωynz = γ& nz (4.77) γ& = u/ h, σ = K / S ( visc) σ xz = ηθφγ (, )& η( θφ, ) 1 η( θφ, ) (α cos θ α + α)sin θcos φ+ ( α + α)cos θ+ α { } 1 5 3 6 4 xz
( θ = 90, φ = 0 ) (a) 1 η1 = ( α + α4 + α5) ( ) (b) θ= 0 1 η = ( α3 + α4 + α6) (c) ( θ = 90, φ = 90 ) 1 η3 = α 4
4.1(c) (4.63),(4.80),(4.81) Γ (visc) = 0 4.1(a) (b) ( φ= 0) n = (sin θ,0,cos θ),(4.63),(4.81) Γ = γ ( nn nn) γ ( nn A nn A ) (visc) y 1 z x x z z µ µ x x µ µ z 1 = γ& γ1+ γ θ Γ = (visc) y 0 { cos } cos θ = γ / γ ( γ / γ < 1 ) θ 0 1 1 0
3.3 vr (, t = 0) = 0 (4.78 c) (4.77) vr (, t) = 0 ω = 0, A= 0 (4.76b) z d γ n 1 n = ( n h) z dt z (4.56) ( n h) e d ( n h) δf = δθ V = δθdv z δ F ( nr () hr ()) z = δθ n = (cos θ,sin θ,0) dn dθ n = dt dt z
dθ δf (3.6) θ 1 1 = = K + 0 B γ µ χ sinθcosθ dt δθ z θ(0) = θ( d) = 0 B= 0 θ( z, t) = θm ( t)sin( qz) ( q= π / d) dθm π K = θ m dt γ d 1 θ ( t) = θ (0)exp( t/ τ), τ = γ d / π K m m 1