1− (χD2M0)2γ1
bD⊥
. (2.18)
At a critical field µ0H = µ0Hc = −KAα1qMz2+A0Kα1Mq2z02, where qz =π/d, the dynamics of the acoustic modes slows down. If the field is more negative than this critical field, the magnetization and with it the director starts to reverse.
mn
δmδn
Hc< H <0
n m
δn, δm H = 0
mn
δmδn
H >0
mn
δmδn
H → ∞
n m
δn δm
Hc< H <0
n m
δn δm
H = 0
n m
δn δm
H >0
n m
δn δm
H → ∞
Figure 2.5: A schematic of the magnetic field dependence of the slower (top) and the faster (bottom) mode. The vectors δn and δm represent the deviations of the director and the magnetization orientation from the equilibrium.
2.2 Effects of flow on the dynamics in a ferromagnetic nematic liquid crystal
In order to understand the dynamic behavior of a system better one needs to study its rheological properties. A common technique used in rheology is to study the effects of an imposed shear rate. In a ferromagnetic nematic, the orientations of the molecules and the platelets are influenced by a shear flow, which in turn influences the measured viscosity. In Ref. [45]
measurements have been done on a sample of ferromagnetic nematic using a cone-plate rheometer.
It was found for example that an imposed magnetic field of order of 20 mT enhances the viscosity by two times.
Effects of flow on the dynamics in a ferromagnetic nematic liquid crystal 21
Γx= -1.0 s-1 Γx=1.0 s-1
-40 -20 0 20 40
0.00 0.02 0.04 0.06 0.08 0.10
μ0H[mT] νeff[Pas]
Γx= -1.0 s-1 Γx=1.0 s-1
-2000 -1000 0 1000 2000
0.00 0.05 0.10 0.15
μ0H[mT] νeff[Pas]
(a)
(b)
Figure 2.6: The behavior of the effective viscosity for (a) small and (b) large values of the applied magnetic field at oppositely equal shear rates. The dashed lines represent two of the Miesowicz viscosities (ηxx and ηzz). Figure taken from Ref. [138].
We study the Couette flow of a ferromagnetic nematic between two parallel plates. Several theoretical predictions on the effects of flow were made by studying the dynamic interplay of the director field, the magnetization and the velocity field [53],
(∂
∂t +vj∇j)Mi+ijkMjωk+XiR+XiD = 0, (2.19) (∂
∂t+vj∇j)ni+ijknjωk+YiR+YiD = 0, (2.20) ρ(∂
∂t +vj∇j)vi+∇j(σijR+σDij +σthij)− ∇ip = 0, (2.21) where YiR,D,XiR,D and σR,Dij are the reversible and the dissipative currents for the director, the magnetization and the velocity field. We shall not write down all the currents explicitly, but we mention the analogue of the flow alignment tensor for the magnetization, described by the tensor cRijk [53],
XiR =. . .−cRijkAjk, (2.22) σRij =. . .−cRkijhMk , (2.23)
22 Effects of flow on the dynamics in a ferromagnetic nematic liquid crystal
where . . . contains all the other parts of the currents, 2Aij =∇ivj +∇jvi is the symmetrized velocity gradient and hMi is the thermodynamic force, defined by hMi = δMδε
i. The tensor cRijk contains 6 coefficients
cRijk = cR1Minjnk+cR2(δijMk+δikMj) +cR3Miδjk
+ cR4niMpnpδjk+cR5(niMjnk+niMknj) +cR6niMpnpnjnk. (2.24) Our numerical results on the viscosity as a function of the applied magnetic field agree qualitatively well with experiments. One observes a quick increase of the viscosity for low magnetic fields, Fig. 2.6. A further increase is observed at fields of order 1 T, when the diamagnetic energy becomes important.
We also calculate analytically the distortion of the director and the magnetization as a function of the imposed shear rate and the magnetic field. The angles are defined by n = cosθˆex + sinθˆez and m = cosψeˆx + sinψˆez. The solution at large values of the applied magnetic field is,
θ±(z) = ±π
2 −CΓx∓π
2 ∓CΓx
cosh [q(z−d/2)]
cosh (qd/2) , (2.25)
where
q2 =q20 µ0|H|M0
A1M02+µ0|H|M0
, (2.26)
withq0 =p
A1M02/K, andCis a constant determined by the dynamic and the static parameters [138]. We point out that Eq. (2.25) also predicts the fact that the director can rotate by more than π/2 in the middle of the cell. These results could be used in future experiments on shear flow, while simultaneously measuring the magneto-optic response.
In a complex fluid, such as a nematic liquid crystal, there are more dynamical coefficients compared to ordinary fluids. A simple way to determine some of the additional viscosities in a nematic is to measure the so called Miesowicz viscosities [139,140], where one fixes the director by external fields and imposes a simple shear flow. Depending on the direction of the shear plane, the velocity and the director, there are three different limiting cases for the Miesowicz viscosities in a nematic. In a ferromagnetic we find 9 different viscosities, which is due to the additional variable of spontaneous magnetization, see Fig. 2.7. If the director is fixed along the z axis, for example, the Miesowicz viscosities take the following form
ηzx=ν3+ γ1
4 (1 +λ)2− (cD)2
bD⊥ + M02(1−2cR2 −2cR5)2
4bDk , (2.27)
ηzy =ν3+ γ1
4 (1 +λ)2− (cD)2
bD⊥ , (2.28)
ηzz =ν3+ 1 4
1− (χD2)2M02γ1
bD⊥
−1
(2.29)
×
γ1(1 +λ)2−4(cD)2+ M02
bD⊥ (1 + 2cR2) 1 + 2cR2 −2χD2 γ1(1 +λ) .
We furthermore study the flow alignment, which is a phenomenon well known in usual nematic liquid crystals, where the director is tilted by a finite angle with respect to the velocity
Effects of flow on the dynamics in a ferromagnetic nematic liquid crystal 23
xy z
η
xxη
xyη
xzη
yxη
yyη
yzη
zxη
zyη
zzFigure 2.7: Miesowicz viscosities in a ferromagnetic nematic where the director is indicated by yellow double-headed arrows. Figure taken from Ref. [138].
field. In ferromagnetic nematics we have an additional dynamic equation for the magnetization and a different dynamic coupling to the velocity field. This not only leads to different stationary directions of the magnetization and the director, but both of them can move out of the shear (xz) plane, see Fig. 2.8. We studied the case, where the magnetic field is pointing along the x axis. The solution for the angle ψ is
sinψ± =−ΓHx Γx ±
sΓHx Γx
2
+2cR2 −1
4cR2 , (2.30)
where ΓHx = 4cRbD⊥
2M02µ0HM0 is the characteristic shear rate determined by the magnetic field. In the absence of the magnetic field and in the large shear rate limit, the stationary solution exists if |λ| ≥1 and |cR2| ≥ 12. In usual nematic liquid crystals, if|λ|<1, the stationary solution does not exist and the system shows a tumbling behavior. In such a system flow alignment can be recovered if a sufficiently large electric field is applied, see Refs. [141, 142]. In Ref. [138], we show that in ferromagnetic nematics this can be achieved by using low magnetic fields.
Lastly the effects of flow on the switch-ON dynamics are investigated. In this case flow is generated by the reorientation of the director field and the magnetization field. An inclusion of only the reversible couplings of n and Mhas shown a very small influence on the reorientation
24
Dynamic interplay of nematic, magnetic, and tetrahedral order in ferromagnetic nematics
y x z
M n
Initial state: n= ˆex andM=M0ˆex
M n
Final state: BothnandM can move out of the shear (xz) plane.
System is exposed to a shear flowv= Γxzˆex.
Figure 2.8: A schematic of the director field and the magnetization under imposed shear flow.
dynamics, which is typically observed in usual nematics. On the other hand the dissipative cross-coupling of n to the velocity field can have a strong effect. Such a coupling induces a nonzero rotation of the director out of the shear plane, which could potentially be detected using polarizing microscopy techniques.