Spontaneous liquid crystal and ferromagnetic ordering of colloidal magnetic nanoplates

Ferrofluids are familiar as colloidal suspensions of ferromagnetic nanoparticles in aqueous or organic solvents. The dispersed particles are randomly oriented but their moments become aligned if a magnetic field is applied, producing a variety of exotic and useful magnetomechanical effects. A longstanding interest and challenge has been to make such suspensions macroscopically ferromagnetic, that is having uniform magnetic alignment in the absence of a field. Here we report a fluid suspension of magnetic nanoplates that spontaneously aligns into an equilibrium nematic liquid crystal phase that is also macroscopically ferromagnetic. Its zero-field magnetization produces distinctive magnetic self-interaction effects, including liquid crystal textures of fluid block domains arranged in closed flux loops, and makes this phase highly sensitive, with it dramatically changing shape even in the Earth's magnetic field.


Supplementary Note 1: Field-induced magnetization and birefringence in the Iso phase
In order to assess the nature of the magnetic ordering in the Iso phase, where Δn sat is the limiting Δn at high fields, is measured for several BF/BuOH concentrations, using 630 nm monochromatic light. The results are shown in Supplementary Fig. 5a. To analyze the data, we first tried to apply the standard Langevin-Weiss (LW) mean-field model 6 , accounting for the polydispersity of the particle sizes, which enables calculation of Q 1 and Q 2 for a system of magnetic dipoles in thermal equilibrium. For a given diameter distribution function, the free parameters in the model are the temperature T (here always 298 K), the total particle number density , and the mean nanoplate magnetic moment m o . For the purpose of this calculation, the nanoplates are divided into i = 23 species according to their diameters with a 5 nm bin size and low and high end cut-offs at 10 nm and 120 nm, respectively, following the , where κ = 5.2 and τ = 9.2. In the absence of an applied magnetic field, the nanoplate moments in the Iso suspensions of orient randomly. When the Iso suspensions are subjected to an applied field, the magnetic torques interaction tend to align the magnetic moments of individual nanoplates with the field. If the field is strong enough, the magnetic moments of all the nanoplates will be completely aligned and the magnetization of the suspension will reach the saturation value M sat where the subscribed i represents the ith species.
When the magnetic field is not strong enough to completely align the magnetic moments of the nanoplates, thermal fluctuations tend to randomize the orientation of the nanoplates. The effective magnetic moment of a nanoplate is its component along the field direction, m i cos ϑ i , where ϑ i is the angle between m i and the applied magnetic field B ext . The field-induced magnetization of the suspension is the total effective magnetic moment per unit volume. It can be described by the first-rank order parameter Q 1i (B ext ) with g(ϑ i ) being the orientational distribution function for the ith species. g(ϑ i ) can be expressed as where k B is the Boltzmann constant and B is the total magnetic field. For convenience, we define After integration, we obtain the so-called Langevin function The total magnetic field is a sum of the applied magnetic field B ext and the internal magnetic field from the magnetized suspension where the second term describes the local magnetic field on a nanoplate in a spherical cavity of surrounding nanoplates. Taking all the species in a suspension into consideration, the overall first-rank order parameter is then The field-induced birefringence, Δn(B ext ), can be expressed in terms of the second-rank order parameter Q 2 as where  is the optical polarizability anisotropy per unit area of the nanoplates, n m is the mean refractive index of the solution, and 2 1 cos The mean value of cos 2 ϑ i is obtained by integrating over the entire distribution which yields When the field is strong enough that all the nanoplates are perfectly aligned, Q 2i = 1 and Δn = Δn sat . Taking all the species in the suspension into consideration, the overall second-rank order parameter is For low concentration suspensions, the magnetic coupling between nanoplates should be weak and the LW theory a reasonable approximation. For a ϕ = 0.005 suspension, we find that m o = 2×10 -18 A·m 2 . This is very close to the value of 3×10 -18 A·m 2 measured 7 for aligned and dried colloidal nanoplates with a mean diameter of 70 nm.

Supplementary Note 2: Low-field susceptibility of the Iso phase
To further understand the magnetization process in the nanoplate suspension system, we measured the field-induced birefringence Δn(B ext )/Δn sat versus B ext in Iso suspensions of different concentration ϕ, shown in Supplementary Fig. 5a. From these data we extracted the initial (lowfield) magnetic susceptibility,  = dM/dH, using the response at low fields along with Supplementary Eqs. 11, 5 and 4, under the assumption of monodisperse particles. Supplementary Fig. 5b shows the resulting  (symbols) and the prediction of the LW mean field model. Both Supplementary Figs. 5a and b indicate that if the LW model is adjusted to fit  in the limit of low ϕ, then the predicted Iso/N F transition occurs at a much lower concentration than observed experimentally.
Since here the nanoplates are in the Iso phase, we may consider them to be discs with orientational diffusion, sweeping out volumes obtained by spinning the average size disc about its diameter and in doing so behaving like hard spheres, as in an Onsager isotropic phase. Pursuing this notion, we indicate in Supplementary Fig. 5b the range of susceptibilities obtained from several theoretical models for the susceptibility of monodisperse dipolar hard spheres 1 , which account for the dipolar coupling and correlation in different ways. The equivalent spherical volume fraction ϕ * is related to the nanoplate volume fraction ϕ by ϕ * = D 3 /6 = 2Dϕ/3t. These models all behave at low ϕ as the same distribution of independent plates, giving the same (ϕ)  ϕ, and are therefore all scaled in the same way to match the experiment at low ϕ by setting m o to 2×10 -18 A·m 2 . The dipolar coupling constant 1 used for the dipolar hard spheres curves in Supplementary Fig. 5b is The range of susceptibilities obtained from the dipolar hard sphere models in Fig. 4 of Ref. 1 are indicated by the cyan-shaded region in Supplementary Fig. 5b. At the low limit of the cyan susceptibility range is the Onsager approximation for dipolar spheres 8 , and at the high limit the Reference Limited Hypernetted Chain model by Patey 2 , the latter providing a reasonable qualitative description of (ϕ). The concentration variable ϕ * is that typically used in models of liquid crystal ordering by the steric interaction of discs, being defined such that ϕ * ~ 1 at the concentration where the effective spherical volumes occupied by nanoplates start to overlap each other, which is where the Onsager LC ordering of discs occurs. Thus, as ϕ * approaches 1, (ϕ) increases faster than the dipolar hard sphere models, which we interpret to be due to enhanced correlations between the discs coming from their flat shape, beyond that arising from the magnetic interaction. Note that for monodisperse hard spheres, the value of ϕ * can never exceed ~0.74, which is the upper limit of the packing fraction of spheres. However, for the case of our nanoplates, ϕ * = 1 corresponds to a nanoplate volume fraction of only ϕ ~ 0.22, and above this concentration is where the Onsager excluded volume theory predicts that an Iso to nematic phase transition occurs 9 . Numerical simulations of polydisperse and charged disc suspensions [10][11][12] show that an Iso/Nematic transition starts to take place at <D 3 > = 3.2 ~ 4.0. Considering that the volume of a nanoplate is π<D 2 t>/4, and assuming the size distribution shown in Supplementary  Fig. 1b, we can calculate the equivalent Iso/N F phase transition concentration ϕ = 0.27 ~ 0.34, which corresponds very well with our experimental observations. Thus, at ϕ ~ 0.28 the BF/BuOH system is near an Onsager-type nematic ordering transition that enhances translational entropy by orienting hard discs to reduce their mutual excluded volume. This nematic ordering transition leads to the dramatic increase of  and to the formation of the N F phase.

Supplementary Note 3: Orientational order parameters of N F BF/BuOH suspensions from X-ray diffraction
The birefringence and dichroism of the domains in the N F phase indicate orientational ordering of the nanoplate planes. The order parameter of this uniaxial N F phase can be calculated from these intensity distribution functions as follows. The orientational order of plates in the N F phase is described by an orientational distribution function f(β), in which β is the angle between the axis of the plates and the macroscopic symmetry axis of the N F phase, similar to studies on thermotropic liquid crystals with molecules that possess the cylindrical symmetry. Following the method first established by Leadbetter and Norris 13 , the scattered intensities from a uniaxially aligned liquid crystal depends on the orientational distribution as follows: , (13) where I(θ) is the azimuthal profile, as illustrated in Fig. 2d (insert). To find f(β) from the above function, we follow the method developed by Davidson et al 14 . First, f(β) is expanded as a Fourier series as follows . (14) By inserting this in Supplementary Eq. 13, we obtain, after integration, .
The nematic order can be described by the second-rank order parameter , where .
Applying Supplementary Eq. 14, we obtain , from which we may obtain the order parameter. Q 2 = 1 represents a perfectly oriented state, while Q 2 = 0 describes an isotropic state. It is found that Q 2 = 0.8 at Location N F and Q 2 = 0.4 at Location Iso/N F . The form factor of the nanoplates, which broadens the distribution by the ratio (2/D)/(2π/d) ~ 7º, was ignored in this estimate.

Supplementary Note 4: Ferromagnetic nematic magneto-elastic deformation energy
We analyze the textural features of the ferromagnetic nematic phase as phenomena resulting from the combined effects of magnetostatic and Frank nematic elastic energies, described by where we have taken K S = K B = K, H is generated by the magnetic charge, and  is in the range  o <  <  o + π. The soliton-like analytic solution for the wall structure is sketched in Supplementary Fig. 7c, a result initially obtained for the analogous electric case 3 . Assuming K the value of the splay constant K S = 6k B T/D = 5×10 -13 N, obtained from Monte Carlo simulation of cut-spheres 15 with thickness-over-diameter ratio of 1/10 and Q 2 ~ 0.8, we find  M ~ 0.1 µm.
Nematic regions substantially larger than this, such as those shown in Fig. 3, are thus expected to have uniform orientation of M(r) and n(r) 16 .
In the other limiting case, of a thin sample that is uniform along c, with spatial variation  in the a,b plane, but where  M is larger than with the sample thickness L, the normal mode energy/area becomes: The crossover length in this case is  ML = K/(µ o M 2 L), analogous to the result obtained for freely suspended ferroelectric LC films [16][17][18] . In the BF/5CB case, the length  M is much larger than the cell thickness, meaning that the M 2 self-interaction is irrelevant and the field response is determined by the balance of Frank elasticity and the interaction with B ext , yielding, for example, a Fréedericksz transition if B ext is applied antiparallel to M, with a threshold where B ext is large enough to reduce the magnetic coherence length  B = / to the cell thickness.

Supplementary Note 6: Analogy to high-polarization ferroelectric liquid crystals
The scenario of uniform blocks of magnetization separated by sharp domain walls is an example of "orientational fracture", similar to that found 17 in -1 topological defects in the director field structure and textures of thermotropic ferroelectric smectic C liquid crystals with large permanent polarization P. In this case, the polarization charge  p (r) = -·P(r) produced by splay deformation of the polarization density P(r) field is sufficiently costly in energy that splay of P(r) is expelled from the bulk of a texture, rather being confined to narrow, P-stabilized 1/2 rotation walls of width w ~  E = / separating "block" domains of uniform P(r).
Elastic energy functions of the form of Supplementary Eq. 19 have been employed extensively to describe such behavior in the absence of bulk free-charge screening of the polarization charge 19 .
In the electric case, observation of the self-interaction effects of P require the polarization to be large enough to overcome the screening effects of free charge, from ions in solution, for example.
Free charge on surfaces can also control the orientation of P(r) 18 . Since there are no magnetic free charges, the bulk and surface magnetic charges are only dipolar in nature, their selfinteractions are always present, and there are always equal amounts of positive and negative magnetic charges.