SUPPLEMENTARY INFORMATION

Transcription

1 SUPPLEMENTARY INFORMATION SUPPLEMENTARY INFORMATION for Memory and topological frustration in nematic liquid crystals confined in porous materials Takeaki Araki 1,2, Marco Buscaglia 3, Tommaso Bellini 3, and Hajime Tanaka 1 1 Institute of Industrial Science, University of Tokyo, Komaba, Meguro-ku, Tokyo , Japan 2 Department of Physics, Kyoto University, Kitashirakawa-Oiwakecho, Sakyo-ku, Kyoto , Japan 3 Dipartimento di Chimica, Biochimica e Biotecnologie per la Medicina, Università di Milano, Via F.lli Cervi 93, Segrate (MI), Italy Memory effects in NLC confined in Random Porous Media We have studied NLC incorporated in a random porous structure (RPM) as a function of the strength E of an external field. Among the bicontinuous geometries explored in this work, RPM is the only one lacking geometrical regularity, thus more suitable to be compared to most past experimental realization of NLC confined in porous matrices. In comparison to the BC structure, RPM provides smaller memory effects characterized by long non-exponential relaxations. The memory effects in RPM have been studied repeating the simulation procedure described for the BC structure. Similarly, to quantify the memory effect we use the nematic order parameter Q = Q/Q B, where Q B is the order parameter of the bulk material at the same temperature. Figure S1a reports Q of a NLC incorporated into a RPM as the field is first applied and later removed for various choices of field strengths, ranging from weak to strong. When the electric field is applied, the director is forced to be uniform. For the strongest fields, Q Q B. Upon removing the field, the system relaxes to a metastable state having intermediate order, characterized by a remnant order Q M. The field strength dependence of Q E - the order parameter in the presence of the field - and Q M is shown in Fig. S1b, whereas the trajectories of the DLs inside the structure before, during and after the application of a strong field are shown as red lines in Figs. S1c, d and e, respectively. nature materials 1

2 supplementary information a 1.0 E=0.1 E=0.2 E=0.3 E=0.4 Q * 0.5 E=0.5 Q * 0.5 E=1.0 b 1.0 Q * E c t(10 MCC) d e Q * M E/E c 3 z t=1x10 5 t=2x t=3 x10 Figure S1: Memory effects in NLC confined in RPM. a, Computer simulated evolutions of the nematic order parameter Q of a NLC incorporated into a RPM as the field is first applied (at t = 10 5 MCC) and later removed (at t = MCC) for various choices of field strengths, ranging from weak (E =0.1, black) to strong (E = 1, grey). b, Normalized nematic order parameters Q E (full symbols) and Q M (empty symbols), extracted from the simulations in panel a, as a function of the field strength E normalized by the threshold value E C. c-e, Snapshots of one realization of the simulated random porous medium as obtained by spinodal decomposition (green body; blue inside). l = 43.9 while the simulation cube is The nematic fraction is not shown. Disclination lines (DLs) (red lines) are displayed for before (c), during (d) and after (e) the application of a strong field (E = 1). The end points of disclination lines on the surface of a simulation box are marked by black circles in c-e. Interactions between defect lines and the surface of BC and SC c porous matrices After the removal of the field, the trajectory of DLs is determined by free energy minimization, which is achieved by shortening DLs and by positioning them relative to the surface so to minimize the elastic energy. BC surfaces are such to favour well defined trajectories of the defect loops. This is shown in Fig. S2, where we can see that half of the DL loops (those having topological charge s =1/2) encircle solid necks of BC, whereas the 2 nature MATERIALS

3 supplementary information other half (s = 1/2) are localized within the narrowest sections of the channels running along z. As described in the main text, this is due to the specific location of the surface position having the largest negative Gaussian curvature. Here we show that, by contrast, for a SC c case, there are no specific stable configurations (see Fig. S2d). This reflects the fact that the porous surface in SC c is characterized just by a one positive curvature and thus there are no preferred orientations for disclination loops. The difference between BC and SC c clearly indicates that the spatial distribution of curvatures plays a crucial role in the stabilization of the configuration of dislocation loops. a E b E c E d E Figure S2: Defect configuration memorised in BC and SC c porous structure. a, A view from x direction in BC. b, A view from the direction of the field (z) in BC. c, An overall view of the defect configuration in BC. d, An overall view of the defect configuration in SC c. The end points of disclination lines on the surface of a simulation box are marked by black circles in d. The system size are 64 3 in a and b, and 32 3 in c and d. The porous matrices are shrunken to be seen clearly. nature materials 3

4 supplementary information A large number of possible topological states for NLC confined in a bicontinuous porous media In all the explored cases (see, e.g., Fig. 1), DLs form loops within the N domain, the apparent end points of DLs in the figures being artefacts of the periodic boundary condition. A dense network of DL loops is generated when the system is first cooled into the nematic phase. Upon equilibration, the vast majority of them vanish by continuously contracting to a point. A fraction of DL loops are, however, permanently stabilized by the very geometry of the porous media. The multiply connected topology of the N domains in bicontinuous porous matrices, makes the system rich in irreducible loops, i.e., loops that encircle continuous portions of the solid fraction and thus cannot vanish by contraction. Moreover, the multiple connectedness of N provides the conditions for the existence of a host of topologically distinct configurations of irreducible loops that cannot be continuously transformed into one another. Although the relation between the topology of a porous structure and the number of DL paths is complex (see below), we can estimate the variety of DL paths by schematizing the porous structure as composed by nodes connected by branches. Accordingly, the total number of possible defect configurations is of the order (p 1) g, where p is the average number of branches at a node, typically, p 3 6, and g l 3 is number of nodes, l being the characteristic length of the porous structure (see Methods). Hence, in the macroscopic limit, the number of the possible topological states of the confined NLC can be enormously large. Relationship between the topology of a porous structure and disclination lines formed The sum of the topological charges of disclination lines is determined by the topology of the porous medium. This fact is demonstrated, in the case of homeotropic anchoring, by the Stein-Gauss theorem, proposed by Stein in 1979 [1]. The theorem relates the total topological charge of the defects in a NLC enclosed in a cavity, with the Euler characteristic EC of the enclosing surface. EC, in turn, can be explicitly computed when the detailed morphology of the surface is known [2]. For example, the surface of the section of BC matrix shown in Fig. 1 has EC = 256. According to the Stein-Gauss theorem we hence expect the total topological charge of the incorporated NLC to be 128. Despite the power of this theorem, its extensions to generic porous geometries aimed to 4 nature MATERIALS

5 supplementary information evaluate the number of defects and the potential of each porous matrix for metastability and memory effect are very difficult for a combination of factors. First, determining the total topological charge does not imply the number of defect lines. This can be easily seen by considering as an example two spherical particles with homeotropic surface anchoring immersed in NLC. Each particle behaves as a point defect of the topological charge +1 and a Saturn ring defect is formed around it. When the particles are placed nearby, however, there are two possible defect structures. One is composed of the two independent Saturn ring defects and the other has a single disclination adopting the figure of eight structure [3]. The same situation is found when more spheres are considered. Even in this case, the system can adjust its topology through a single disclination line having a path increasingly complex as the number of spheres increases. Analogously, if we count the disclination loops in the BC structure after cooling from the nematic without applied field and after long annealing, we obtain a number in the range 25-30, much lower, but necessarily with the same topological charge than the number of defect loops (128) found when the field is on and even after its removal. Second, determining the topological charge of disclination loops is not straightforward, as the examples above clearly show. Specifically, the topological charge of a disclination defect loop is +1 or 1 only in the simplest geometries, while more generally it depends on its topology and interconnectivity with other loops [1]. Finally, a general prescription to calculate the Euler characteristic for a generic bicontinuous structure based on topology and connectivity is still missing. Such a prescription would enable evaluating the total topological charge in a bicontinuous structure without relying on the detailed knowledge of its surface morphology. At the same time, also missing is a general criterion to evaluate the number of distinct paths that defect loops can adopt within bicontinuous structures. Such a tool would enable predicting the multiplicity and nature of metastable states. These various issues will be addressed in a forthcoming publication. Stability of defect loops under weak perturbation and its relation to the bicontinuous nature of the porous structure We have studied the stability of the paths of defect loops under the weak perturbation provided by weak external fields in NLC incorporated in bicontinuous porous media and nature materials 5

6 supplementary information hosting disconnected solid spheres. This investigation was aimed to clarify the importance of the bicontinuous nature of a porous structure on the metastability and memory effects. First we studied the stability of defect structures in BC by applying a field along the x axis on a sample previously ordered by a field along z. Figure S3a shows the evolution of the remnant order along z when a field E x is constantly applied to the system. The defect structure shows long term stability when the electric field is weaker than E c =0.28. Above this threshold electric field, the defect structure switches from configuration (A) to (C) via (B) in Fig. S3c. From Fig. S3b, we can see that the energy barrier required for the reconnection of disclination lines is more than 200 k B T. Next we describe the stability of defect structures in SC n. In this case, the remnant order along z decreases even by a very weak external field (less than 10% of the E c for BC) (see Fig. S4b) and the behaviour is strongly dependent on the temperature T (see Fig. S4a). This indicates that the barrier for the change in the defect configuration is comparable to the thermal energy. This is a direct consequence of the fact that the change of the orientation of defect lines does not require topological changes in the defect lines and can thus be achieved by simple rotation under quadrupolar interactions between nearby loops. The large difference in the behaviour of the two systems can be understood as a consequence of the different transformation required to rotate the direction of mean orientation. In the bicontinuous system rotation of defect loops is prevented by their being chained to close loops of the solid fraction of the porous matrix. This is not the case when the NLC hosts disconnected solid bodies. In this case, defect loops can be made circumnavigate the solid hosts in any direction. Overall, this comparison clearly indicates that the bicontinuous nature of a porous structure is a crucial factor for the stability of defect patterns against perturbation. 6 nature MATERIALS

7 a E/T Q zz b c E x =0.2 Ex=0.28 Ex= Ex=0.3 E x = (C) t (MCC) E x =0.2 Ex=0.28 Ex=0.29 Ex=0.3 Ex= t (MCC) (A) (B) (C) supplementary information (A) (B) z x Figure S3: Switching behaviour of the topological defects in BC. a, Decay of Q zz in BC (l =6.75) under various external fields along x direction. b, Temporal variation of the energy per unit cell in BC. The energy is given by the difference from that of the annealed structure in the rest (E = 0). c, Defect structures subject to external fields: (A) E =0.2, (B) E =0.28, (C) E =0.3 (see the labels in a). In (B), only defects free from the topological locking rotate towards the new field. In all cases, the annealing temperature is T =0.1. nature materials 7

8 supplementary information a Q T b Q zz t (MCC) 1 (A) (B) 0 (C) E x t (MCC) 10 6 c (A) (B) (C) (D) (E) (D) (E) z y x Figure S4: Behaviour of defects in a porous medium without topological constraint. a, Thermal decay of Q in SC n of a = 3 for various temperatures in the rest (E = 0). b, Temporal change of Q under various external fields along x direction in SC n of a =5 at fixed T =0.1. c, Defect structures observed around t = 10 6 after the application of the external field. (A)-(E) correspond to E x =0.02, 0.04, 0.06, 0.08, 0.1, in its order (see the labels in b). 8 nature MATERIALS

9 supplementary information Fitting of the relaxation of the orientational order after the field removal. The relaxation of Q(t) after the field removal has been quantitatively analysed by fitting the decays in Fig. 3 with the functions given in Eqs. (1) and (2). While the fitting of the relaxation of NLC in BC is rather straightforward, being it characterized by a single decay, the study of the RPM relaxations requires a more elaborate analysis. Having identified the slower decay as a manifestation of a glassy behaviour, there are two natural choices for its functional dependence. Relaxations in a system exhibiting the glass behaviour are in fact typically described either by a stretched exponential with a small stretching exponent, or by a logarithmic decay. Hence we tried them both. Figure S5 shows Q(t) for NLC in a RPM having l = 18.8 and T = 0.1. Simulation data (lower panel) are shown as red crosses. In Fig. S5a, the fitting with Eq. (2) is shown as a continuous line, hard to spot underneath the data because of the quality of the fitting. The resulting residual errors R are shown in the upper panel (R is on the same scale as Q). Besides the obvious quality of the fitting, a further confirmation of the appropriateness of the functional dependence in Eq. (2) is confirmed as follows. We have fitted the decays of NLC in RPM with a stretched exponential as in Eq. (1), considering only the short time data with Q>0.75. As visible, the resulting function, shown in the figure as a dotted line, closely matches the stretched exponential component of the composite fitting in all parameters. a R b R Q Q t t Figure S5: Fittings of the decay function of Q(t) for NLC in a RPM. a, Q(t) is for NLC in a RPM having l = 18.8 and T = 0.1. Simulation data (lower panel) are shown as red crosses and the fitting with Eq. (2) as a continuous line. The short dashed line indicates the remnant order Q M. Long dashed and dot-dashed lines represent, respectively, the fast 9 nature materials 9

10 supplementary information (stretched exponential) and slow (logarithmic) relaxations that combine in the fitting of the data, their amplitudes ( Q S and Q L ) indicated by vertical arrows close to the Q axis. The upper panel shows the residual errors in the fitting. b, The same fitting as a but with the sum of two stretched exponentials (a continuous line). The residual errors shown in the upper panel are larger than those in a. The same decay was also fitted by the sum of two stretched exponentials, as shown in Fig. S5b. The result is fairly good, although the residual errors (see the top panel) are larger than those in Fig. S5a. For this reason we adopted the logarithmic fitting for the long time decay. It is however interesting to mention that the value of the stretched exponent for the long time behaviour, about , is comparable to that obtained from the experimental finding in Ref. [4]. An even more sophisticated fitting was necessary to account for the relaxations in RPM in the intermediate T range where the fast decay appears to be the sum of two contributions: a first decay well fitted by a stretched exponential with exponent α>1 (i.e., a compressed exponential), reminiscent of the behaviour found at high T, and a second one also fitted by a stretched exponential whose parameters are well connected with those found in the low T regime. This behaviour indicates that in this structure the range of phase coexistence is rather large. T dependence of the characteristic time of the viscoelastic decay. The analysis of Q(t) at different T s for BC (l(bc)=6.75) and RPM (l(rpm)=12.7) enables us to characterize the T dependence of the viscoelastic decay. Here we report these results that were not included in the main text. We find, for both systems, that the stretched exponent is roughly T -independent. In the range explored, we found α to take values within the intervals for RPM and for BC. These ranges match with the l dependence reported in Fig. 4a. The decay time τ S instead depends on T, as expected. Data for τ S vs. T are shown in Fig. S6 for RPM and for BC. The former are systematically larger because of the l dependence of τ S (see Fig. 4a). τ S is generally expected to be determined by the ratio between the viscosity and the elastic coefficient (see below), both of them depending on T. The resulting τ S (T ) is rather complex, even in structures simpler than ours. Ref. [5] reports the viscoelastic relaxation in the nearly spherical pores 10 nature MATERIALS

11 supplementary information of Polymer-Dispersed LC devices. In such a case, τ S (T ) is non-monotonic. At low T, τ S (T ) is, in principle, dominated by the Arrhenius behaviour, although such a behaviour is not experimentally confirmed. Ref. [6] reports (the inset of Fig. 5 there) a T dependence of the viscoelastic dynamics that bears strong resemblances with our observations: as T is lowered, τ S increases but less steeply than expected by an Arrhenius dependence. This is attributed to an anomalous T dependence of the viscosity due to the interactions of LC with the surfaces. Additionally, the T dependence of the memory effect reported in Fig. 4d, indicates that the energy landscape itself is T -dependent. Hence, it is not surprising that, even at low T, the T dependence of the viscoelastic relaxation time is not fully explained by a simple activation process a b τ S (MCC) T /T Figure S6: T dependence of τ S for RPM and BC. RPM (filled blue diamonds) and BC (filled red circles). The marked increment of τ S at low T apparent in panel a cannot be simply interpreted through an Arrhenius behaviour, as evident in the 1/T plot in panel b. Decay of the orientational order in RPM. Here we discuss the mechanisms responsible for peculiar decay processes of the orientational order in RPM after the field removal (see above, Fig. 3, and the related discussion). The process can be divided into two regimes. The fast process represents the relaxation of the elastic deformation of the director field while keeping the defect topology, whereas the second slow mode stems from the reorganization of the topological structure of the defects. Fast decay process: In a coarse-grained theory, the free energy of a nematic liquid crystal nature materials 11

12 supplementary information confined in a porous medium without external fields is given by F = dv K 2 ( n)2 ds W 2 (n s)2, (1) where n is the director field, s is the normal vector of the surface, K is the typical elastic modulus, and W is the anchoring strength. A simple scaling argument predicts that a pattern of the director field, or a defect structure, is characterized by a non-dimensional parameter W l/k in an equilibrium state. It is known that this scaling behaviour is realized for a nematic liquid crystal containing a spherical particle of the radius l. Thus, we expect that the relaxing process of the oriented nematic order in a porous medium is also determined by W l/k. The dynamics of the director fields in the bulk and on the surface, respectively, obey t t n s n = γ (I nn) δf δn = γ(i nn) K 2 n, (2) = γ/a (I nn) δf δn s = γ/a(i nn) {K n s W (n s)n s }, (3) where n s is the director field on the surface. γ is a kinetic coefficient and a is the characteristic length of surface anchoring, which is the lattice size in our simulation. However, it is difficult to solve the set of Eqs. (2) and (3) analytically. Under an external field, the elastic deformation is localized near the surface because of the constraint from surface anchoring. Soon after the external field is removed, thus, the relaxation first takes place near the surfaces to release the stored elastic energy and then in the bulk later. Assuming that the surface anchoring dominates the bulk elastic contribution, or W K/l, in Eq. (3), the characteristic relaxation time of n s can be approximated as τ a = a/w γ. On the other hand, the subsequent relaxation of the elastic deformation in the bulk is characterized by τ e = l 2 /Kγ from Eq. (2). Under the condition of W K/l and l>a, we can assume τ e τ a. Thus, the decay rate of the director field in a channel of size l is approximately estimated as Γ s (τ a + τ e ) 1 = τ 1 e = γk/l 2. (4) We note that for W =1.0 we employed, the anchoring is so strong that the relation W K/l is almost satisfied. As shown in Fig. 3, the fast mode does not decay exponentially and it is more stretched for larger l. This is because the relaxation of n occurs in a spatially 12 nature MATERIALS

13 supplementary information inhomogeneous manner. The inhomogeneity increases with an increase in l, since Eq. (2) is parabolic as the diffusion equation. Slow logarithmic decay process: In the late stage after the initial decay, the director field is subjected to an effective field, which stems from the topological locking of the defect structures. This means that the remnant order cannot fully relax during the fast process. As mentioned above, the second mode comes from an activation process, which accompanies defect breakages. We consider that a defect reorganization occurs locally in a subsystem of the volume l 3. Its energy barrier is related to the defect length as U = ckl, where c is a numerical constant. Now we come to the origin of the logarithmic decay of this last relaxation process observed in RPM. Such a logarithmic decay is known for flux creep in hard superconductors as the Anderson-Kim theory [7, 8]. In the light of this theory, we can explain this peculiar decay by assuming that the activation energy is a simple (linear) decreasing function of the (remnant) orientational order. This is quite natural at least on a qualitative level since the difference between the remaining orientational order along the field applied and the orientational state favoured under the surface field without an external field should be the driving force of defect reorganization. The rate of the change in the remnant order is assumed to be ( ) Q M U(QM ) Aexp, (5) t k B T where k B is the Boltzmann constant, T the temperature, and A a positive constant. Then assuming that the change in Q M is through the change in U or vice versa, we obtain ( )( ) 1 ( ) ( dq M du du du = = A exp U(Q ) M). (6) dt dt dq M dq M k B T The asymptotic solution of this equation is U(Q M ) = k B T ln(t/t 0 ) with t 0 = k B T/(A(dU/dQ M )). t 0 is an effective attempt time, which is different from a usual microscopic one. The defect reorganization is assisted by the driving force, which should be a decreasing function of Q M. The simplest expression for the barrier is a linearly decreasing function of Q M : [ U = U 0 1 Q ] M, (7) Q 0 where U 0 is the barrier height in the absence of a driving force, and Q 0 corresponds to the critical orientational order required to make the barrier zero. Combining the above two nature materials 13

14 supplementary information equations, we obtain Q M (t) =Q 0 [ 1 k BT U 0 ( )] t ln. (8) t 0 The key is that the barrier is a function of the remnant order and the temporal change of the remnant order is controlled by this barrier. This bi-directional relation may be the underlying physics behind the logarithmic decay observed in our system. Comparison of our simulation results with experimental results The simplest experimental access to the memory effect of NLC in porous media is the measurement of the intensity I of the light transmitted through a thin sample before, during and after the application of electric-field (E) pulses. The corresponding values of optical turbidity Θ 0,Θ E and Θ M can be extracted from I. Θ 0 is the optical turbidity of a sample that has been filled in the isotropic phase and cooled into the NLC state with no field applied ( zero-field cooling ). Θ E and Θ M are, respectively, the turbidity in the presence of the field and that after the removal of the field. In many NLC-filled porous materials Θ 0 is remarkably large, yielding a scattering mean free path as short as 2 µm [9]. Θ 0 is principally due to the scattering from the distortions of the NLC optical axis produced by the anchoring at the pore surfaces and depends on the length scale of such distortions, typically set by the pore size l of the NLC-imbibed media. In isotropic porous media, zerofield cooled samples are globally isotropic, and hence the nematic order parameter averaged on the whole sample is Q = 0. At the same time, the systems are locally ordered, their local nematic order parameter being approximately equal to Q B, the order parameter of the bulk material at the same temperature. If an electric field is applied, the optical axis is forced to be uniform. For the strongest fields Q Q B. The corresponding turbidity Θ E Θ 0, is determined essentially by the mismatch in refractive indices between the well ordered NLC and the material forming the porous medium. We found that upon removing the field, the system relaxes to a state having intermediate order Θ E < Θ < Θ 0 which is held indefinitely [10]. From Θ measured in this state, in which order has been memorized, it is possible to estimate Q by using a model suitable to account for the scattering of light in such continuously distorted media [9, 10]. Figure S7a shows Q (t) for a 16 µm thick samples of the thermotropic NLC 5CB incorporated in a microporous Millipore membrane filter (MF) with average pore size l MF 3 14 nature MATERIALS

15 supplementary information µm. Samples were prepared as described in [10] and cooled to 25 C with no field applied. E was turned on in the interval t 1 <t<t 2 shadowed in light blue, during which Q reaches Q E, its equilibrium value in the presence of E. After the field is removed, Q relaxes to the new stationary value Q M. In Fig. S7b we plot Q E and Q M as described in [9]. remnant order Q M for the MF sample prepared Upon increasing E, here normalized by the threshold value E c, the grows, indicating that the removal of the field leaves the system in a metastable state bearing the memory of its previous thermo-electric history. The sample is representative of NLC incorporated into random porous medium with locally smooth surfaces, as shown by the electron microscope image reported in Fig. S7c. The analogy with the simulated RPM structure is evident also from the comparison of the behaviours of Q E and Q M reported in Fig. S7b and d, although the alignment of the NLC on the surface of the MF matrix prepared as described [10] is not clearly determined. a 1.0 Q * E b 1.0 Q * E Q * Q * Q * M 0.0 E=0 E=0 Q * M 0.0 c t (s) d 1.0 Q * E E/E c Q * 0.5 Q * M E/E c 3 Figure S7: Experimental memory effects in NLC confined in porous membranes. a, Measured time dependence of Q, the nematic order parameter normalized to the bulk nematic order, in a 5CB-MF sample (l MF =3µm). E = 0 for t<t 1 and t>t 2, while E = 1.25 V/µm for t 1 <t<t 2 (light blue region). t 1 = 150 s, t 2 = 530 s. Values of Q E and Q M are also indicated. b, Measured normalized nematic order parameters Q E and Q M as a function of the field strength E normalized by the threshold value E C. Lines are as nature materials 15

16 supplementary information follows. Continuous line is Q E, whereas dashed blue line is Q M for MF. E C =0.54 V/µm. c, Scanning electron microscope picture of a MF membrane with nominal pore size of 3 µm. The length bar is 10 µm. Courtesy of Marina Carpineti (Universitá degli Studi di Milano). d, Normalized nematic order parameters Q E (full symbols) and Q M (empty symbols) in the simulated RPM structure as a function of the field strength E normalized by the threshold value E C. The behaviour found for NLC in RPM by simulations well reproduces the kinetics experimentally observed in 5CB-MF (see also [4]). In the experiments, as the field is switched off, the system viscoelastically relaxes to a first plateau. On a longer time scale, a second decay is observed, whose characteristic time markedly grows as T decreases, a behaviour consistent with our findings for RPM at low T. This example for RPM clearly indicates the predictability of our simulations. Effects of the external field on surface anchoring The orientation of molecules (spins in the simulation) at the solid-lc interface is affected by all three terms in the Hamiltonian (Eq. (3) of the main text). For weak fields, the dominant term is the anchoring coupling W. However, as the applied field is increased, the coupling with the field may become comparable or even stronger than the anchoring term, possibly producing breaking of the surface anchoring. In this condition the Stain- Gauss theorem does not apply any more and the overall topology of the NLC may be modified. However, for typical values of parameters (K = 10 pn, W = J/m 2, and ϵ =1.8pF/m), Ẽ = 1 corresponds to E = 2.3V/µm with d =1µm, we confirm that even this field strength leads to little breaking of the surface anchoring in our simulations (see below). It is expected that anchoring breaking will take place when the electric coherence length ξ E, which is given by ξ E = K/( ϵe 2 ), exceeds the surface extrapolation length ξ s [11]. This criterion ξ E >ξ s can be expressed as E> W 2 /(K ϵ), i.e., Ẽ>1in our scaled units. Since electric-field-induced defect reconfigurations already take place for Ẽ<0.5 (see Fig. 6), we can assume that there is no anchoring breaking (see also below). We also stress that our main conclusion is not affected by whether the electric field breaks anchoring or not. To check whether in our study the application of a field modifies the topological charge 16 nature MATERIALS

17 supplementary information through field-induced anchoring breaking, we have studied in better detail the switching behaviour of NLC confined in BC (L = 32). Figure S8a shows that the switching of the whole system is completed slightly below E =0.3. This evaluation was performed using as the initial state a random configuration prepared by zero-field cooling at T =0.1 (see Fig. 1). Figure S8b, on the other hand, quantifies the behaviour of the spins at the solid-lc interface as a function of the field strength. This is done by studying two parameters, both computed on the restricted ensemble of spins in the surface sites S, the number of which is denoted as N S : (i) the degree of anchoring Q anc, defined as Q anc = 3 { (n i s) 2 1 }, (9) 2N S 3 i S where s is the normal vector of the surface of a porous medium, and (ii) the nematic order along the field Q sur, defined as Q sur = 3 2N S i S { (n i z) 2 1 }, (10) 3 where z is the unit vector towards the external field. The subscripts E and M represent the values of the parameter during and after the application of the field, respectively. In this case, the initial state is the ordered state depicted in Fig. S2, obtained by having previously applied a strong enough field. Because of this preordering, Q sur is not zero even without the field. Q anc = 1 means that the director field is normal, i.e., perfectly anchored, to the porous surface. Here the deviation of Q anc from 1 is mainly due to frustration effects coming from the local curvature of the surface. For E < 0.4, the degree of anchoring Q anc E is almost the same as that for E = 0. Above E = 0.4, it starts to decrease with an increase in E, which may be regarded as a crossover from a strong to a weak anchoring regime. The anchoring breaking takes place around E =1.1. This can be clearly seen by comparing Figs. S8c and d: While for E =1.0 disclination lines can be clearly identified within the nematic matrix (red curves), as the field is increased to E = 1.1 they collapse on the surfaces, so that their identification becomes doubtful. nature materials 17

18 supplementary information a 1.0 b 1.0 anc Q M Q 0.5 Q E Q M sur anc Q,Q 0.5 anc Q E sur Q E c E d 0.0 sur Q M E z e y t=0 t=50000 t= t= Figure S8: Field-induced switching of topological defects and anchoring breaking for NLC in BC. a, Induced and remnant nematic order parameters, Q E and Q M, for NLC confined in BC (L = 32), as a function of the field strength E. This plot is the same as in Fig. 2b (main text) but with no normalization for E c. The blue coloured region is a where surface anchoring breaks (the same for panel b). b, Induced and remnant order parameters Q sur M and Qsur E computed for the spins on the surface, as a function of E. In this plot, E is applied on a system with large remnant order previously recorded by a field pulse. The order parameters of surface spins with respect to the surface normal are also plotted: Q anc E and Qanc M express the degree of anchoring during and after the field application. c, Director field in a y-z plane under the field E =1.0. Here, the cross section cutting the thinnest portion of the arms of BC structure is shown. The black segments represent the local director orientation. Red curves mark disclination lines. d, The same as panel c with a field of E =1.1. In this condition the anchoring is broken and there are no defects. e, The process of the topological change of the disclination lines under E =0.19, which is strong 18 nature MATERIALS

19 supplementary information enough to align the orientation, are shown. We can see the topological change proceeds without anchoring breaking. Having identified the onset of surface anchoring breaking, we can go back to the data in Fig. S8a, to conclude that below E =0.3 the surface anchoring is affected only weakly. We can thus conclude that at these field strengths, the topological transformations of disclinations yielding to the remnant order are not influenced by anchoring breaking. Another important element indicating that anchoring breaking has no bearing in the memory effects here described is given by the behaviour of Q anc and Q sur. Both quantities are basically unaffected by E up to values even above E =1.1. Hence, even if anchoring breaking effects were at play and even when using strong fields, they would not modify our conclusions, based on the variety of topological states compatible with the porous structure in the absence of external fields. Fields are here used only to switch between such metastable states. Concepts for exploiting the memorization capability of confined NLC in memory devices Technologies for microfabrication are improving fast. Among them, microconstructions based on two-photon polymerization appear the most versatile and, with some additional technology development, we expect that they will soon enable polymerize structures with accurate control down to 1 µm, or even below [12]. Hence, the practical realization of porous matrices such as the BC here simulated should be achievable in not too long a time. On this basis, we can envisage two basic concepts for the realization of image-retaining displays based on the memory effects we found for NLC in BC. (i) A device controlling the transmitted intensity via the control of the birefringence of the confined NLC. The porous matrix should in this case be enclosed between crossed polarizers and the electrodes should be designed so to enable the application of the field both perpendicularly to the polarizers (z direction) and along the polarizers. Given the cubic symmetry of the BC structure, the application of an electric field either along z or x yields the same degree of remnant order. When the field is applied along z, the director is oriented along z. Therefore, if light also propagates along z, the material will appear isotropic with a refractive index approximately equal to the ordinary refractive index of the NLC material. Since this state is meant to perturb as little as possible the incoming polarization, the renature materials 19

20 supplementary information fractive index of the porous material needs to be chosen so to match the ordinary refractive index of the NLC. As the field is applied along x, the material will appear birefringent to light propagating along z. Hence, by suitably choosing orientation of polarizer and analyzer with respect to x, and by selecting the thickness of the porous matrix, a large transmitted intensity can easily be achieved. Metastability ensures the permanence of on or off states without the need of electric field permanently applied. (ii) A device based on the modulation of the back-scattering of light, to be used as electronic paper. In this case the illumination is provided by ambient light and no polarizers are used. The aim is to obtain a highly scattering white state and a transparent black state in which light penetrates the material to be absorbed by a black layer on its rear plane. A transparent state is obtained when the field is applied in the direction of the propagation of light. The large value of the remnant order and the matching between refractive indices of porous matrix and NLC ensures that this state is weakly scattering, enabling a large fraction of light to be transmitted by the material and absorbed on the back of the display. The production of the white state exploits the high optical turbidity that can be obtained in disordered nematics [9, 10]. Turbidity of disordered NLC can be particularly high because of the intrinsic property of light scattered by a collection of randomly oriented birefringent domains. Specifically, the polarization, amplitude and sign of the field scattered by each domain all depend on the director orientation. Because of this reason, the total scattered intensity is an incoherent summation of the field scattered by each domain. In this way, the overall scattering crosssection does not suffer from the cancellations due to the structure factor, which typically limit the turbidity of closely packed structures. In other words, the birefringence of NLC, and the consequent depolarized scattering are the crucial elements making disordered NLC very turbid. A highly scattering state can be obtained in NLC confined in BC by applying fields in the z directions if materials with negative dielectric anisotropy are used. In this case, the coupling with the electric field forces the director in the x-y plane. The 2D degeneracy of this coupling makes the system become very inhomogeneous, with the nematic director pointing in various direction in the x-y plane. This creates a highly scattering state for light propagating along the z direction. This state has been explored in Fig. 6b of the main text. Both the transparent and the turbid state can be produced when the system is realized by making use of a dual-frequency NLC compound in which dielectric anisotropy inverts sign with frequency. One example is the compound MLC2048 produced by Merk [13]. In such 20 nature MATERIALS

21 supplementary information a material a low frequency pulse produces alignment with the field and hence a transparent state, while a high frequency pulse favours the molecular alignment perpendicular to the field, and hence a highly turbid state. Light is scattered from NLC because of the fluctuations in the director direction. In the Rayleigh-Gans-Debye approximation, the intensity of the scattered light is proportional to the mean squared fluctuations of the tensorial order parameter Q. The practical realization of structures here discussed will involve characteristic sizes of one or few microns and thus larger than 1/q max, where q max is the largest scattering vector involved in the scattering of light q max =2π/λ, λ being the wavelength of light. In this regime, the total scattered intensity S is proportional to the volume averaged mean squared fluctuations of the nematic tensorial order parameter, as Eq. (4) in the main text. The behaviour of S is shown in Fig. 6b. The most noticeable fact is the large difference in S between the two states obtained by inverting the electric coupling (the ratio of the two values of S being approximately equal to 7). Again, a device would require optimizing thickness so to yield a large contrast between the amount of light back-scattered by the two states. Current displays used as book readers have a white state that back-scatters light 6-10 times more efficiently than the black state. Given the large difference in scattering cross section, this should be within reach of devices based on confined NLC: The combination of increased availability of microfabrication tools with the concept for NLC functionalization through geometrical confinement open various scenarios for the realization of a new generation of image-retaining NLC displays and optical memories. Movies S1 and 2: Reconfigurations of defects in a random porous medium shown in Fig. 5. Movie S1 demonstrates the reconfiguration process of the disclination lines at T =0.1 after a strong field E = 1 is removed. l = 43.9 while the simulation cube is The white lines represent the disclination lines. Each frame corresponds to MCC before t = MCC, and then after that time the movie is speeded up by 5. Movie S2 is the same as movie S1 except that the porous domain is also shown there. nature materials 21

22 supplementary information [1] Stein, D. L. Topological theorem and its applications to condensed matter systems. Phys. Rev. A 19, (1979). [2] Góźdź, W. T. & Ho lyst, R. Triply periodic surfaces and multiply continuous structures from the Landau model of microemulstions. Phys. Rev. E 54, (1996). [3] Araki, T. & Tanaka, H. Colloidal aggregation in a nematic liquid crystal : Topological arrest of particles by a single stroke disclination line. Phys. Rev. Lett. 97, (2006). [4] Bellini, T. et al. Nematics with quenched disorder: How long will it take to heal? Phys. Rev. Lett. 88, (2002). [5] Amundson, K. Electro-optical properties of a polymer-dispersed liquid-crystal film: Temperature dependence and phase behavior. Phys. Rev. E 53, (1996). [6] Mertelj, A., Spindler, L. & Copic, M. Dynamic light scattering in polymer-dispersed liquid crystals. Phys. Rev. E 56, (1997). [7] Anderson, P. W. Theory of flux creep in hard superconductors. Phys. Rev. Lett. 9, (1962). [8] Kim, Y. B., Hempstead, C. F. & Strnad, A. R. Critical persistent currents in hard superconductors. Phys. Rev. Lett. 9, (1962). [9] Bellini, T., Clark, N. A., Degiorgio, V., Mantegazza, F. & Natale, G. Light-scattering measurement of the nematic correlation length in a liquid crystal with quenched disorder. Phys. Rev. E 57, (1998). [10] Buscaglia, M. et al. Memory effects in nematics with quenched disorder. Phys. Rev. E 74, (2006). [11] Stark, H. Physics of colloidal dispersions in nematic liquid crystals. Phys. Rep. 351, (2001). [12] Maruo, S. & Fourkas, J. T. Recent progress in multiphoton microfabrication. Laser & Photon. Rev. 108, (2008). [13] Yin, Y., Shiyanovskii, S. V., Golovin, A. B. & Lavrentovich, O. D. Dielectric torque and orientation dynamics of liquid crystals with dielectric dispersion. Phys. Rev. Lett. 95, (2005). 22 nature MATERIALS