The microscopic origin for the existence of the D-state is attributed to the cyclic dissociation and association of condensed ions. Figure 2, which is taken from Ref. [9], illustrates the mechanism through which such a cyclic dissociation and association is kinetically induced. First consider a nematic domain with its director aligned along the direction of the electric field (stage (I) in Fig. 2). In Fig. 2, only two rods out of the entire domain are depicted for clarity. The domain orientation along the field direction is a consequence of single-particle torques resulting from field-induced polarization of the double-layer and the layer of condensed ions. In such an orientation, their is a relatively large amount of excess condensed (positive) ions at the top side of the rod (indicated in red), while there is a depleted region at the bottom part (indicated in blue). The excess of condensed ions at the top creates an electric field that pushes the ions out-of the condensed layer into bulk solution (indicated by the arrows in stage (I)). The opposite happens at the bottom part of the rod. As the concentration of ions in the condensed layer is much larger than in the diffuse double layer, the net result is a dissociation of condensed ions. After a time comparable to the time needed for the released ions to diffuse into the bulk solvent, the ionic strength will increase. This will decrease the Debye length (the extent of the diffuse double layer is indicated in Fig. 2 by the dotted blue lines around the core of the rods). Hence, in stage (II), the rods carry less condensed ions, while the Debye length is smaller. A reduction of the Debye length corresponds to a reduction of the effective concentration (the effective concentration will be quantified in the subsequent paragraph). When the effective concentration becomes less than the lower isotropic-nematic binodal, the nematic domain becomes unstable and melts, which is accompanied by a de-alignment, as depicted in Fig. 2 in stage (III). Association of condensed ions occurs as the rods take orientations towards directions perpendicular to the electric field. This leads in turn to a decreased ambient ionic strength, and thereby to an increase of the Debye length (as depicted in stage (IV)). The accompanied increase of the effective concentration renders the nematic phase stable again. The resulting nematic domain aligns along the electric field direction due to single-particle torques (see stage (V)), during which polarization takes place leading to stage (I), after which the cycle subsequently repeats itself.
The above-mentioned time-dependent “effective concentration” is to be understood as follows. For suspensions of very long and thin colloidal rods with hard-core interactions, Onsager showed that the location of isotropic-nematic binodal- and spinodal concentrations depends on the dimensionless concentration (L/d) φ, with L the length of the rod, d the core diameter, and φ=(π/4) d
2
L
ρ the volume fraction (the fraction of the total volume occupied by the cores of the rods), with ρ the number concentration of rods [26, 27]. In case the rods are charged, the same Onsager theory can be employed, except that the core thickness is larger due to the additional repulsive electrostatic interactions. This defines an effective, time-dependent diameter d
e
f
f
, and thereby an effective dimensionless concentration (L/d
e
f
f
) φ
e
f
f
. The following expression for the effective diameter can be derived [9, 10],
$$\begin{array}{@{}rcl@{}} d_{ef\!f}\;=\;\kappa^{-1}\,\left[\,\ln K_{Q}+\gamma_{E}\,\right]\;, \end{array} $$
(1)
where κ
−1 is the Debye length and γ
E
=0.5772⋯ is Euler’s constant, and where,
$$\begin{array}{@{}rcl@{}} K_{Q}\;=\;\frac{2\,\pi\,\exp\{\kappa\,d\}}{\left( \,1+\frac{1}{2}\kappa\,d\,\right)^{2}}\, \frac{l_{B}}{\kappa\,L^{2}}\,\left( \,N_{0}-N_{c,0}\,\right)^{2}\;, \end{array} $$
with d, as before, the core diameter, l
B
is the Bjerrum length, N
0 is the number of immobile charges chemically attached to the surface of a rod, and N
c,0 the number of condensed ions of a rod in the absence of an electric field. Considerations concerning the quantification of an effective diameter can also be found in Refs. [26–30]. Note that the total rod-surface charge that is relevant for the ion-concentrations within the diffuse double layer is equal to −e (N
0−N
c,0 ) (with e the elementary charge), which the total immobile surface-charge plus the total charge within the layer of condensed ions.
In order to describe the dynamics of melting and forming of nematic domains, an equation of motion for the orientational order parameter tensor S should be derived. This tensor is defined as the ensemble average of the dyadic product of the unit vector \(\hat {\mathbf {u}}\) that specifies the orientation of a rod,
$$\begin{array}{@{}rcl@{}} \mathbf{S}(t)\;\equiv\;<\hat{\mathbf{u}}\,\hat{\mathbf{u}}>(t)\;. \end{array} $$
(2)
The largest eigenvalue λ of the orientational order parameter tensor quantifies the degree of alignment of the rods. In the isotropic phase λ=1/3 while in a perfectly aligned state λ=1. The concentration dependence of the order parameter (without the electric field) is most conveniently understood on the basis of the bifurcation diagram given in Fig. 3. The values of the effective concentration (L/d
e
f
f
) φ
e
f
f
where the isotropic-nematic binodals and spinodals are located, according to Onsager [26, 27], are given on the lower axis of the bifurcation diagram. Here, \(C_{bin}^{(-)}\) and \(C_{bin}^{(+)}\) are the lower and upper binodal concentrations. For initial overall concentrations in between the two binodal concentrations, the equilibrium state is a coexistence between an isotropic and nematic phase. For concentrations below \(C_{bin}^{(-)}\), the isotropic phase is stable, and above \(C_{bin}^{(+)}\) the nematic phase is stable. The spinodal concentrations \(C_{spin}^{(\pm )}\) relate to the stability of the uniform isotropic and nematic state. The spinodal concentration \(C_{spin}^{(+)}\) marks the concentration where a uniform isotropic state becomes unstable against the uniform nematic state upon increasing the concentration. The vertical dashed arrow in Fig. 3 depicts the temporal increase of the orientational order parameter towards the nematic branch (the solid line in blue). On lowering the concentration of a uniform nematic below the spinodal concentration \(C^{(-)}_{spin}\), the nematic becomes unstable against the isotropic state. The order parameter of the uniform nematic state now decreases towards 1/3. We thus find that the uniform nematic is unstable for effective concentrations \((L/d_{ef\!f})\,\varphi _{ef\!f}<C_{spin}^{(-)}\), while the uniform isotropic state is meta-stable for concentrations \((L/d_{ef\!f})\,\varphi _{ef\!f}<C_{spin}^{(+)}\), as indicated in Fig. 3. The red, closed curve depicts the limit cycle corresponding to the alternating crossing of the lower binodal in the dynamical state under the action of an electric field. Hence, melting of the nematic state occurs from the unstable state, through spinodal decomposition. The isotropic state grows from the meta-stable state, through nucleation and growth. Nucleation times in the present case are probably small, as there is some reminiscent alignment after melting.
In order to quantify the dynamics of melting and forming of nematic domains in the dynamical state, according to the above discussion, we need two equations of motion for the orientational order parameter tensor (2): one equation of motion for spinodal melting of the nematic state when \((L/d_{ef\!f})\,\varphi _{ef\!f}<C_{bin}^{(-)}\), and one for nucleation and growth of the nematic from an (near-) isotropic state when \((L/d_{ef\!f})\,\varphi _{ef\!f}>C_{bin}^{(-)}\).
An equation of motion for spinodal decomposition of a nematic state can be derived from the Smoluchowski equation, through a Ginzburg-Landau expansion upto fourth order in the orientational order parameter. Such a Ginzburg-Landau expansion can only be employed to describe the kinetics of an initially unstable state, and therefore describes melting of the nematic, as discussed above. For frequencies of the external field that are sufficiently large that during a cycle of the field the configuration of the rods is essentially unchanged (for the fd-suspensions under consideration, this frequency is about 50–100 Hz), an equation for the orientational order parameter tensor can be derived from the Smoluchowski equation [9, 10]. This equation of motion can be written as a sum of various contributions,
$$\begin{array}{@{}rcl@{}} \frac{\partial\,\mathbf{S}}{\partial\,\tau}\,=\,\mathbf{\Delta}_{id}+\mathbf{\Delta}_{Q,hc}+\mathbf{\Delta}_{twist} +\mathbf{\Delta}_{pol}+\mathbf{\Delta}_{torque}\,, \end{array} $$
(3)
with the dimensionless time variable,
$$\begin{array}{@{}rcl@{}} \tau\;=\;D_{r}\,t\;, \end{array} $$
where D
r
is the free rotational diffusion coefficient. The various contributions are as follows. First of all, Δ
i
d
is the contribution from free diffusion,
$$\begin{array}{@{}rcl@{}} \mathbf{\Delta}_{id}\;=\;6\,\left[\,\frac{1}{3}\,\hat{\mathbf{I}}\,-\mathbf{S}\,\right]\;. \end{array} $$
The second contribution Δ
Q,h
c
stems from interactions, unperturbed by the external field, with an effective hard-core diameter that accounts for the above discussed electrostatic interactions, as indicated by the subscript “Q”,
$$\begin{array}{@{}rcl@{}} \mathbf{\Delta}_{Q,hc}=\frac{9}{2}\,\frac{L}{d_{ef\!f}}\,\varphi_{ef\!f}\, \left\{\,\mathbf{S}\cdot\mathbf{S}-\mathbf{S}\,\mathbf{S}:\mathbf{S}\,\right\}\;. \end{array} $$
The third contribution Δ
t
w
i
s
t
is the twist contribution,
$$\begin{array}{@{}rcl@{}} \mathbf{\Delta}_{twist}\;=\;-\,\frac{9}{2}\,\left[\frac{5}{4}-\ln 2\right]\,\frac{1}{\kappa\,d_{ef\!f}}\,\frac{L}{d_{ef\!f}}\,\varphi_{ef\!f}\, \left\{\,\mathbf{S}\cdot\mathbf{S}-\mathbf{S}\,\mathbf{S}:\mathbf{S}\,\right\}\;. \end{array} $$
This contribution describes the effect, referred to as “the twist effect” [28, 29], that there is a preference for a non-parallel, twisted orientation of two rods due to the energetically unfavorable overlap of diffuse double layers in parallel orientation. The contribution Δ
p
o
l
is the contribution due to interactions from polarization charges,
$$\begin{array}{@{}rcl@{}} \mathbf{\Delta}_{pol}\;=\; \frac{7}{60}\,\left[\frac{K_{E}}{K_{Q}}\right]^{2}\,\frac{1}{\kappa\,d_{ef\!f}} \,\frac{L}{d_{ef\!f}}\,\varphi_{ef\!f}\,h({\Omega})\,\mathcal{E}_{0}^{4}\,\left( \mathbf{S}: \hat{\mathbf{E}}_{0}\hat{\mathbf{E}}_{0}\right)\,\mathbf{F}(\mathbf{S},\hat{\mathbf{E}}_{0})\;, \end{array} $$
while Δ
t
o
r
q
u
e
accounts for single-particle torques with which the external field acts on polarization charges,
$$\begin{array}{@{}rcl@{}} \mathbf{\Delta}_{torque}\;=\;\frac{1}{80}\,\frac{L}{l_{B}}\,\tilde{F}\,I({\Omega})\,\mathcal{E}_{0}^{2}\, \mathbf{F}(\mathbf{S},\hat{\mathbf{E}}_{0})\;, \end{array} $$
where \(\mathcal {E}_{0}\) is the dimensionless external field strength (with β=1/k
B
T, and E
0 the external field strength),
$$\begin{array}{@{}rcl@{}} \mathcal{E}_{0}\;=\; \beta\,e\,L\,E_{0}\;, \end{array} $$
(4)
and \(\mathbf {F}(\mathbf {S},\hat {\mathbf {E}}_{0})\) is an abbreviation for,
$$\begin{array}{@{}rcl@{}} \mathbf{F}(\mathbf{S},\hat{\mathbf{E}}_{0})&\equiv&\,\frac{3}{2}\mathbf{S}\cdot\hat{\mathbf{E}}_{0}\hat{\mathbf{E}}_{0} +\frac{3}{2}\,\hat{\mathbf{E}}_{0}\hat{\mathbf{E}}_{0}\cdot\mathbf{S}+\mathbf{S}\cdot\mathbf{S} \cdot\hat{\mathbf{E}}_{0}\hat{\mathbf{E}}_{0} \\ &&+\hat{\mathbf{E}}_{0}\hat{\mathbf{E}}_{0}\cdot\mathbf{S}\cdot\mathbf{S}-2\,\mathbf{S}\cdot \hat{\mathbf{E}}_{0}\hat{\mathbf{E}}_{0}\cdot\mathbf{S}-3\,\mathbf{S}\,\mathbf{S}:\hat{\mathbf{E}}_{0} \hat{\mathbf{E}}_{0}, \end{array} $$
with \(\hat {\mathbf {E}}_{0}\) is the unit vector in the direction of the external field. The frequency-dependent functions appearing in the above equations are,
$$\begin{array}{@{}rcl@{}} h(\Omega)&=&\left[\!\frac{1}{\Omega}\frac{\sin\{\Omega\}\!+\!\sinh\{\Omega\}}{\left[\cos\{2\,\Omega\}\!+\! \cosh\{2\,\Omega\}\right]^{2}}\!\right]^{2}\left[1\!+\!\frac{4}{3}\Omega^{4}\!+\! \frac{2}{5}\Omega^{8}\right]\;,\\ I(\Omega)&=&\frac{1}{2\,\Omega^{3}}\;\frac{\sinh\{2\,\Omega\}-\sin\{2\,\Omega\}}{\cosh\{2\,\Omega\}+ \cos\{2\,\Omega\}}\;, \end{array} $$
where [31],
$$\begin{array}{@{}rcl@{}} {\Omega}\;=\;\sqrt{\frac{\omega\,L^{2}}{8\,D_{ef\!f}}}\;, \end{array} $$
is a dimensionless frequency, with ω the frequency of the external field, and with D
e
f
f
the effective translational diffusion coefficient of the condensed ions, which is equal to,
$$\begin{array}{@{}rcl@{}} D_{ef\!f}\;=\;D\,\left[\,1+2\,\kappa_{c}\,a\,\mathcal{K}(\kappa\,a)\,\right]\;, \end{array} $$
(5)
where D is the free translational diffusion coefficient of condensed ions. The second term within the square brackets accounts for the repulsive interactions between the condensed ions, with,
$$\begin{array}{@{}rcl@{}} \kappa_{c}\;=\;\frac{2\,l_{B}}{d\,L}\,N_{c}\;, \end{array} $$
the inverse “condensate length.” The frequency-dependent functions h(Ω) and I(Ω) are essentially zero for Ω>3, for which the polarization of the layer of condensed ions ceases to occur. At that frequency, the H-phase becomes the stable phase, as discussed in “The electric phase/state diagram” section. The constants K
E
and \(\tilde {F}\) are equal to [9, 10, 31],
$$\begin{array}{@{}rcl@{}} K_{E}&=&\frac{\pi\,\exp\{\kappa\,d\}}{2\,\left( 1+\kappa\,a\right)^{2}\,\left( 1+2\,k_{c}\,a\,\mathcal{K} (\kappa\,a)\right)^{2}}\,\frac{l_{B}}{\kappa\,L^{2}}\,{N_{c}^{2}}\;, \\ \tilde{F}&=& V(\kappa_{c}a)\,\left[\,W(\kappa_{c}a,\kappa a)+1\,\right]\\&&\times \left\{2\left[1+\kappa_{c}\,a\, \mathcal{B}(\kappa\,a)\right]^{2}-\kappa_{c}\,a\left[\,1+\kappa_{c}\,a\,\mathcal{B}(\kappa\,a)\,\right]\,\right\} , \end{array} $$
with a=d/2 the hard-core radius, and where V and W stand for,
$$\begin{array}{@{}rcl@{}} V(\kappa_{c}a)&=&\frac{\kappa_{c}\,a}{\left( \,1+\kappa_{c}\,a\,\mathcal{B}(\kappa\,a)\right)^{2}}\;,\\ W(\kappa_{c}a,\kappa a)&=&-\,\frac{2\,\kappa_{c}\,a\,\mathcal{K}(\kappa\,a)}{1+2\,\kappa_{c}\,a\,\mathcal{K}(\kappa\,a)}\;, \end{array} $$
with (K
0 is the modified Bessel function of the second kind of zeroth order),
$$\begin{array}{@{}rcl@{}} \mathcal{K}(\kappa\,a)&\equiv&\frac{1}{2\,\pi}\;{\int}_{\!\!\!\!0}^{2\pi}d\varphi\;K_{0}\!\left( \kappa\,a\sqrt{2\, (1-\cos\varphi)}\,\right)\;,\\ \mathcal{B}(\kappa\,a)&\equiv&\frac{1}{\pi}{\int}_{\!\!\!\!0}^{2\,\pi}\!\!\!d\varphi\,\cos\{\varphi\}\,K_{0}\! \left( \kappa\,a\,\sqrt{2(1\,-\,\cos\varphi)}\,\right)\;. \end{array} $$
An equation of motion for the nucleation and growth of a nematic domain from the isotropic state requires the solution of the Smoluchowski equation including all orders of the orientational order parameter, as well as the spatial dependence of the orientational order parameter tensor. This is a problem that is probably too complicated to allow for an analytical treatment. We therefore adopt the exponential growth that is found in simulations [32],
$$\begin{array}{@{}rcl@{}} \frac{\partial\,\mathbf{S}}{\partial\,\tau}\;=\;\frac{\bar{\mathbf{S}}-\mathbf{S}}{\mathcal{T}}+ \mathbf{\Delta}_{pol}+\mathbf{\Delta}_{torque}\;, \end{array} $$
(6)
where \(\bar {\mathbf {S}}\) is the order parameter tensor of the nematic phase in equilibrium, without the electric field, and where \(\mathcal {T}\) is the time scale on which the internal orientational order of domains increases. The last term is responsible for the orientation of the nematic director towards the electric-field direction, which is an essential ingredient for the existence of the dynamical state.
We note that both equations of motion (3, 6) neglect the finite size of nematic domains. Spatial variation of the orientational order parameter is not considered here. This is probably a reasonable approximation in view of the quite fuzzy interface between the nematic and isotropic regions, as evidenced from microscopy images. On the other hand, the dynamics of a given domain might be affected by adjacent domains.
An oscillatory state is only found when dissociation/association of condensed ions is included. In Ref. [9], the following semi-empirical equation of motion for the number N
c
of condensed ions is proposed,
$$\begin{array}{@{}rcl@{}} \frac{d\,N_{c}}{d\tau}&=&\pm\,\mathcal{C}_{d}\,\left\{\,{N_{c}^{2}}\,-\,N_{lim}^{2}\,\right\} \left( \frac{z^{2}l_{B}}{L\left[1\,+\,2\,\kappa_{c}\,a\,\mathcal{K}(\kappa\,a)\right]}\right)^{2} \\ &&\times\,\mathcal{E}_{0}^{2}\,\left( \,\hat{\mathbf{E}}_{0}\hat{\mathbf{E}}_{0}:\,\left[\,\mathbf{S}(t)-\alpha_{thr}\,\hat{\mathbf{I}}\,\right] \,\right)\,I({\Omega})\;, \end{array} $$
(7)
where C
d
is the “effective dissociation constant.” Dissociation occurs only when there is sufficient polarization of the layer of condensed ions along the long axis of the rod, which requires a minimum component of the orientation of a rod along the external field. The number α
t
h
r
thus specifies the minimum value of the orientation along the field direction upon which dissociation can occur. When \((\mathbf {S}:\hat {\mathbf {E}}_{0}\hat {\mathbf {E}}_{0})>\alpha _{thr}\) dissociation occurs (and the “ − ” in Eq. 7 applies), whenever the actual number of condensed ions is larger than the limiting number of condensed ions N
l
i
m
, which is given by,
$$\begin{array}{@{}rcl@{}} N_{lim}\;=\;\frac{\alpha_{lim}\,N_{c,\,0}}{\alpha_{lim}+\mathcal{E}_{0}^{2}\,\left( \hat{\mathbf{E}}_{0}\hat{\mathbf{E}}_{0}:\mathbf{S}\right)\,I({\Omega})}\\ \text{when},\;\;\;(\mathbf{S}:\hat{\mathbf{E}}_{0}\hat{\mathbf{E}}_{0})>\alpha_{thr}\;.\qquad\qquad\, \end{array} $$
(8)
This is the limiting, time averaged number of condensed ions in the stationary state when a rod with a fixed orientation is subjected to the external field for a long time. When, on the other hand, \((\mathbf {S}: \hat {\mathbf {E}}_{0}\hat {\mathbf {E}}_{0})<\alpha _{thr}\) (and the “ + ” applies), association of condensed ions occurs, and,
$$\begin{array}{@{}rcl@{}} N_{lim}\;=\;N_{c,0}\;\;\;,\;\;\;\text{when},\;\;\;(\mathbf{S}:\hat{\mathbf{E}}_{0}\hat{\mathbf{E}}_{0})<\alpha_{thr}\;, \end{array} $$
where, as before, N
c,0 is the number of condensed ions in the absence of the external field (note that N
l
i
m
≤N
c,0).
It takes some time before the ion-concentration within the bulk of the solvent is affected by the dissociation or association of condensed ions. Ions that dissociate from the condensed layer must diffuse over distances of the order of a rod length, in order to change the bulk ionic strength. Similarly, it takes some time for ions to diffuse from the bulk to the condensed layer as association occurs. The change of the bulk concentration of ions at time t is thus approximately proportional to the number ΔN
c
=N
c, 0−N
c
of released ions at an earlier time t−τ
d
i
f
, where τ
d
i
f
is the time required for ions to diffuse over distances of the order of a rod length. The time-dependent (inverse) Debye length at time t is therefore taken equal to,Footnote 1
$$\begin{array}{@{}rcl@{}} \kappa(t)\;=\;\sqrt{\frac{\beta\,e^{2}\,\left[\,2\,c_{0}+\bar{\rho}\,{\Delta} N_{c}(t-\tau_{dif})\,\right]}{\epsilon}}\;, \end{array} $$
(9)
where \(\bar {\rho }\) is the number density of rods, and c
0 is the ambient ionic strength,
$$\begin{array}{@{}rcl@{}} c_{0}\;=\;\frac{1}{2}{\sum}_{\alpha}\,c_{\alpha}\,z_{\alpha}^{2}\;, \end{array} $$
where the summation ranges over all species of ions in bulk solution, and c
α
is the number concentration of species α (with valency z
α
), in the absence of the electric field. Therefore, the effective diameter in Eq. 1 becomes time-dependent,
$$\begin{array}{@{}rcl@{}} \frac{d_{ef\!f}(t)}{d}\;=\;\frac{1}{\kappa (t)\,d}\,\left[\,\ln\{K_{Q}(\kappa\equiv\kappa(t))\}+\gamma_{E}\,\right]\;, \end{array} $$
where the interaction strength K
Q
is evaluated with an inverse Debye length equal to κ(t). This time dependence quantifies the variation of the effective concentration upon dissociation/association of condensed ions, which is at the origin of the dynamical state.