Geometrical control of topological charge transfer in Shakti-Cairo colloidal ice

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Particle based ices recently emerge as an alternative model for geometric frustration, where repulsive colloids are confined within a lattice of double wells 26 .The mesoscopic length scale of colloidal particles simplifies their visualization and low intense external fields can be used to manipulate the particles or tune their interactions.In a particle ice, each double well is filled by one colloid, and an external field is used to induce tunable repulsive interactions 27 or to bias the particles along a crystallographic axis [28][29][30] .For an unconstrained system, the particles will form a triangular lattice, the ground state for colloids that repel isotropically in two dimensions.However, arranging the double wells in a lattice with an incompatible symmetry, generates competing interactions and geometric frustration sets in.Thus, the colloidal ice features a similar frustration-by-design as ASI.Such analogy can be extended further.If we associate an Ising-like spin to each double well, with the arrow pointing toward the well filled by the particle, it is possible to construct a set of vertex rules similar to the frustrated spin ice and the pyrochlore system, Fig. 1(a).
Spin ice is now a general concept for different systems spanning various geometries.Perhaps its best definition is in terms of the topological charge q associated to a vertex 31,32 , which depends on the connectivity, and it is invariant upon continuous deformations of the lattice.A vertex of coordination number z with n spins pointing towards it (in), has charge q = 2n − z corresponding to the number of spins pointing in minus those pointing out.Equivalently for a particle ice, it corresponds to the number of particles inside the vertex, minus the empty slots of the vertex.Thus, when z = 4 (square lattice), there are five possibilities (q = 0; ±2; ±4), and vertices that cancel the topological charge q = 0 obeying the ice rule.In contrast, when z = 3 as for the honeycomb lattice, there are no q = 0 vertices, but only positive or negative ones with q = ±1; ±3.In general, an ice rule manifold is a collection of spin configurations such that the absolute value of the topological charge is minimal: zero on even-coordinated vertices, or ±1 on odd-coordinated ones.
While the analogy between a magnetic spin ice and a colloidal ice was proved to be formally valid for a single coordinate lattice 26 , it breaks down for a mixed coordination system [33][34][35] .For example, in a decimated square lattice the ice-rule is spontaneously violated at the sublattice level, as negative topological charges q = −2 accumulate on the z = 4 vertices.Such topological charge transfer leads to the spontaneous screening of the q = −2 monopoles appearing on the z = 4 vertices, and was a clear manifestation of the fragile ice manifold of the particle ice system, in contrast to ASI which were proven to be robust 14,[36][37][38][39] .Here we show that in an artificial colloidal ice, topologically invariant deformations of the geometry can control the transfer of topological charges between different sublattices, because geometrical changes affect the relative energy of the vertices.This result highlights the general phenomenon of the topological charge transfer and, more importantly, demonstrates that geometric transformations can be used to control topological effects in the lattice.

RESULTS
Geometric transformation.Fig. 1(b) shows the transformation used, where we start from a Shakti lattice 38,40,41 , which can be considered as a decimated version of the square ice having a longer double well between the two z = 3 vertices.We convert this lattice in a Cairo using a continuous deformation which shortens the longer double wells and induces an alternating buckling of the two islands in each z = 3 vertex.In particular, we vary the distance a among the z = 3 vertices, the distance l between the z = 3 and z = 4 vertices, the plaquette side p and the coupling distance H between two potential minima in z = 3 and z = 4 vertices, Fig. 1(c).These quantities are related to the bond angle θ via the transformation: In this way we tessallate the 2D space by alternating horizontal and vertical plaquettes.In order to reduce as much as possible the number of coupling among vertices, we fix all the distances d in the z = 4 vertices and z = 3 vertices, Fig. 1(b).In the z = 3 vertices, d is kept constant due to the changes in the length of the trap in a and the only coupling that we vary is H = 2d cos (θ/2 + π/4).
Numerical simulations.The low energy states of the different lattices are investigated first by performing Brownian dynamics simulation, carefully parameterized to match the experimental system.We start from the simulation in order to span a wide range of geometries by varying θ without having to fabricate new structures each time.As shown in Fig. 1(e), we consider a two-dimensional system of paramagnetic colloidal particles with diameter δ and effective magnetic volume susceptibility κ, placed in a lattice of topographic double wells at a one to one filling.Under an external field B perpendicular to the plane of the particles, the colloids acquire a a dipole moment m = πδ 3 κB/(6µ 0 ), being µ 0 the permeability of the medium.Repulsive dipolar interactions between pair of particles (i, j) placed at distance r = |r i − r j | arise in form of an isotropic potential, U m = µ 0 m 2 /(4πr 3 ).For each colloidal particle i at position r i we integrate the overdamped equation of motion: The charge transfer in the Shakti lattice (q > 0 in z3, q < 0 in z4) inverts in the Cairo lattice (q < 0 in z3, q > 0 in z4).(c) Inversion of topological charge transfer by raising bond angle θ.Scattered data are simulation results at the maximum field B = 25mT while continuous lines are non-linear regression from the theoretical predictions (Method section).Green arrow in the plot denotes the "spin ice" cross-over at θ = 27.15• with no charge transfer between z = 3, 4 sublattices.
where γ is the friction coefficient, F T i is the force from the topographic double well which is considered as a bistable harmonic potential, F dd i the magnetic dipolar interaction that acts on the particle i due to its neighbors and η the random force due to thermal fluctuation, see Method section for more details.The external magnetic field B is used to raise the repulsive forces and it is applied via a slow ramp until a maximum value of B = 25mT.
The geometric transformation is obtained by using Eq. 1.Following this prescription, we vary θ ∈ [0, 30] • and determine for each bond angle the charge population q for all type of vertices by raising the field amplitude.Further, we extract the average topological charge per vertex type defined as q = q zi /N z for a number of vertices N z .
Topological charge transfer.In Fig. 2 we show our main findings in terms of q, first for the two extreme cases of the Shakti (θ = 0 • , Supplementary VideoS1) and the Cairo (θ = 30 • , Supplementary VideoS2) as a function of the applied field Fig. 2(a,b), and then as a function of θ for the maximum field value, Fig. 2(c).The first thing to notice is that at all angles but at exception of one point, the system is not in a spin ice manifold.Indeed, charges are present, and they are partitioned by coordination, so that we can talk about a "net charge transfer" among vertices of different coordination.The Shakti (θ = 0) shows an accumulation of negative monopoles q = −2 in the z = 4 vertices and their screening due to the positive monopoles q = +1 in the close z = 3 vertices, which induce a total charge difference of ∆q = |q z=3 − qz=4 | = 3.56 at B = 25mT.This situation is similar to the behaviour of a decimated square system 35 , and confirms the "fragile" nature of the colloidal ice manifold: the breakdown of the ice rule is observed for both cases.However, we find that in the Cairo lattice (θ = 30 • ) the charge transfer inverts, and a fraction of positive monopoles now accumulate in the z = 4 vertices being screened by negative ones in the z = 3, with a smaller charge difference than the Shakti, ∆q = 0.26.Such relatively small difference and the fluctuations in Figure 2(b) result from the stronger mobility of the topological charges in the Cairo system.Here we find that even at the largest field, topological charges are free to move as they appear and disappear spontaneously due to thermal fluctuations (Supplementary VideoS2).In contrast, in the Shakti system once they appear they become pinned at a lattice location (Supplementary VideoS1).The transition is continuous with the deformation, and it crosses a transition point (θ = 27.15• ) where all charges are equally balanced and the decimated system recover a spin ice behavior, with no charge transfer, Figure 2(c).
Charge transfer can be understood in terms of the fundamental differences between particle ice and spin ice 42 .By virtue of the dipole interaction, spin ice vertices are frustrated and thus their lowest energy configurations naturally obey the ice rule.Indeed, unlike the present system, magnetic spin ices in the Shakti 38,40,41 and Cairo 43 geometries both show no charges at low energy.Instead, colloids in particle ices repel each other, and the lowest energy configuration in each vertex corresponds to particles away from it, that give rise to a negative charge, as in Fig. 1(a), while ice rule vertices have higher energy.Particle-based ice is also frustrated, but its frustration is collective: clearly it is not possible to set all vertices in the lowest energy state, of negative topological charge.Therefore, in a geometry of single coordination, all the vertices are equivalent, and thus must be equally charge-balanced in terms of charges.There, the ice rule emerges as a collective compromise among vertices.But when there are multiple coordinations, it has been proposed 33 and then shown experimentally 35 , that higher coordination vertices break the ice rule, becoming negatively charged and transferring positive charge to lower coordination ones.This we also see in Fig. 2(c), for most angles, but, interestingly, not all.The same figure also shows that charge transfer can be inverted.This inversion is important, as it allows for a point of no-charge-transfer, θ = 27.15• , i.e. a spin ice point that corresponds to the intriguing topologically protected 41 Shakti ice.Theory.The charge inversion is due to the change in coupling of the vertices of coordination z = 3 due to the deformation.In general, they would receive a positive charge, but as the deformation increases, the energy cost of transferring the charge raises.Thus, we describe this phenomenon by considering the energy of a given vertex of coordination z with n colloids as, E = J z n(n − 1)/2, being J z an effective coupling constant for a vertex.Using this framework, one can arrive at the total charge as: Here ρ(q, n) is the probability that a vertex of coordination z has a charge q, Z z the partition function and the factor 2 before the second term is due to twice the abundance of z = 3 with respect of z = 4 vertices, see Method for more details.By minimizing Eq. 3, we determine the statistical frequencies for charges, and use it to compute the partial charges of the two set of vertices z = 3, 4. We find a good agreement with the data, as shown by the solid lines in Fig. 2(c), by choosing T = J 4 /7 and a linear dependence of x = J 3 /J 4 in the angle θ such that x = 0 when θ = 0 and x = 1.75 when θ = π/6.

Experimental realization.
We further strengthen our results by providing in Fig. 3 an experimental realization of the inverse charge transfer in the Cairo geometry, see Supplementary VideoS3.In contrast to previous works on different types of lattices 27,35 , we use an alternative lithographic protocol to produce a chemically stable, biocompatible and reusable micropatterned substrate, see Method section for more details.This structure, shown in Fig. 3(a) is composed of elliptical wells with a central, cylindrical post that give rises to an effective double-well confinement.These wells are filled with paramagnetic colloids of size d = 10µm and effective magnetic volume susceptibility κ = 0.0025 using optical tweezers.Once the particles are randomly located within the double wells one by one, we induce repulsive interactions by applying a magnetic field ramp from B = 0 to B = 10mT at a rate 0.03125mT s −1 .At the highest field the particles experience the strongest repulsion, so that for a distance r = 13µm (two particles in z = 3 vertex) the potential is U m = 524k B T .Using digital video microscopy, we extract the position of N = 180 particles, performing an ensemble average over 30 separate experiments.A typical field of view is shown in Fig. 3(b) where, in order to avoid boundary effects and loose of topological charges, we excluded in the statistics the outer region of vertices.
The experimental results confirms the topological charge inversion for the Cairo ice, which starts already at B ∼ 3 mT, Fig. 3(c).The difference with the previous simulation results arises from the finite size of the system limited by the optical field of view (Fig. 3(b)), and dispersion in the elevation of the central hill h.Moreover, we find that for B > 10mT the particles within the shorter double wells did not remain confined, but tend to pile up with their neighbors due to the strong dipolar forces.Such effect is difficult to prevent, as the particles are not strongly held by gravity within the double well, but float there due to electrostatic repulsion with the underlying lithographic structure.To adjust the simulation results to the experimental ones, we run further simulations with similar finite sizes and varying only two main parameters, particle susceptibility κ and hill elevation h.By adjusting both values, we obtain a good match with the experiments observing the same charge separation of ∆q = 0.88, as testified from the continuous lines in Fig. 3(e).

CONCLUSIONS
In this article, we have explored numerically the effect of topologically equivalent geometrical deformations on the "spin ice nature" of our system, quantitatively described in terms of transfer of topological charge.We have then chosen one configuration, the Cairo lattice, and probed it experimentally.Thus, we have designed a strategy to control and localize the topological charges in a two di-mensional lithographic network.From the application side, we have designed a strategy to control and localize the topological charges in a two dimensional lithographic network.Such ideas could be implemented in other artificially engineered nanoscale systems to trap, control and direct the motion of magnetic charges or other entities such as pinned vortices developed in superconducting systems 44 .The main technological drive of such research points toward the design of magnetic device such as memory or ports where logic information can be transported by dipolar switching, and thus being effectively dissipation free.Indeed one further avenue of this work could be to design a composite lattice made of different patches from the Shakti and the Cairo, where topological charges may sequentially be stored or released upon a simple external field command.

METHODS
Details of numerical simulations.We perform Brownian dynamics simulations to explore the transition from Shakti to Cairo colloidal ice by varying the bond angle θ and to obtain a large statistical data set.The numerical scheme consists of solving the overdamped Langevin equations given in the main text (Eqs.(1)) by Euler's method, where as friction coefficient we use the value of γ = 0.033pN s m −1 equal to previous work performed on similar colloidal particles.The latter was estimated by measuring experimentally the diffusion coefficient of the paramagnetic colloids in water 45 .We model the force from the double well on a particle i, F T i , as a piecewise harmonic bistable potential given by: where r and r ⊥ are components of the vector r parallel and perpendicular to the line which join the two stable positions of the trap.These positions are separated by a distance δ.
As stiffness we use k trap = 1 • 10 −4 pN/nm which keeps the particle confined to the elongated region around the centre of the trap, and k hill = 25 pN/nm that creates a potential hill able to pushes the particles away towards one of the bistable regions.The ratio k hill /k trap was chosen to match the experimental data.
The dipolar force on a particle i is given by, being m the magnetic moment induced by the external field B, and rij = (r i − r j )/|r i − r j |.Dipolar interactions are calculate in an iterative form such that the global field H includes that generated by all other dipoles, and we apply a large cut-off of 200µm to consider the effect of long range dipolar interactions.The last term in Eq.( 1) of the main text is a random force due to thermal fluctuation, which is given by η and characterized by a zero mean, η = 0 and delta correlated, η (t) η (t ′ ) = 2k B T γδ(t − t ′ ), being k B the Boltzmann constant and T = 300K the ambient temperature.
Our main simulations involve N = 2000 particles, namely 800 arranged along z = 3 vertices and 400 in z = 4 which is equivalent to a colloidal ice of 10 × 10 unit cells and open boundary conditions in order to mimic the experimental system.The statistics were taken neglecting the first shell of vertices, in order to minimize the effect of boundaries that may adsorb topological charges.To confirm that, for our system size the effect of boundaries did not influence the bulk behaviour, we have run simulation also with periodic boundary conditions and observe no significant differences in the measured parameters.Here we use as parameters κ = 0.005 and the hill spring constant k hill similar to a previous work on different lattices 46 .However to match the experimental results, we also run simulations on a much smaller system, made of N = 180 particles arranged along 3 × 3 unit cells, and open boundary conditions.Further, to match the experimental data on the Cairo system (different lithographic substrate), we have also adjusted the effective magnetic susceptibility and hill elevation.We find that the best matching was obtained for κ = 0.025 and k hill = 3 pN/nm.We use the same field ramp for all experiments and simulations and, in the latter case, numerically integrate the equation of motion using a time step of ∆t = 0.01s.
Details of the theoretical model.We explain here how we obtain the theoretical curves of Fig. 2c.It had been pointed out previously 33,34 that the similarities in behaviour between particle-based ices and magnetic spin ices depend on topology 33 and geometry 34 .In particular, ref 33 showed a way to compute topological charge transfers, which was later tested experimentally 35 .Presently, we use a modified version of the same method.The modification is needed to take into account that vertices of different coordination have different coupling constants due to the deformation, something not originally considered in ref. 33 .
In ref. 33 , there are strong approximations.Firstly, the energy for a given vertex of coordination z with n colloids in is given by only one coupling constant J, as Considering that the topological charge of a vertex of coordination z is q z,n = 2n − z and thus depends only on n, in our approach differences in energy in vertices of the same charge are neglected in our treatment.We know this is not true in general.However, id not too close to the ground state of the system, this methods has been proven to work.
Secondly, correlation among vertices is also neglected.That, and imposing the conservation of the number of colloids, return a Boltzmann distribution for the probability of finding a vertex of coordination z with n colloids (where Z z (T, φ) = n z n exp(−E φ z,n /T ) is the partition function), but in new effective energies E φ z,n which contain an "electrostatic" contribution from the emergent field φ coupled to the charge q z,n : and φ is a parameter to be determined by imposing that the total charge must be zero (that is, particle conservation).
One sees that the temperature independent choice is the solution for a lattice of single coordination z.It returns the effective energies (up to an irrelevant constant) which are quadratic in the topological charge, and are therefore spin-ice-like: the ensemble of lowest energy is the one where all vertices have minimal charge (±1 if z is odd, 0 if z is even).This would explain the similarities in behaviours between magnetic spin ices and particle ices, despite the fact that their vertex energetics differ: particle ices are controlled by an effective energetics that is spin-ice-like.Things become more interesting when there are multiple coordination.Then the same formalism applies to each subset of vertices, but φ (which is in fact an "entropic field") is the same.One sees that in such case φ depends on temperature.Ref. 33 shows how to deal with such situation at low temperature, and the result is "ice rule fragility": in the case of vertices of coordination z = 4, 3, vertices of coordination z = 4 become negatively charged, thus violating the ice rule, leaking charge to the subset of vertices of coordination z = 3.These latter do not, however, violate the ice rule.The reason is that z = 3 vertices already have a charge ±1 in the ice manifold, and they can adsorb the positive charge by reducing the number of vertices of charge 1 and replacing them with vertices of charge −1, so that overall charge neutrality is assured.This was demonstrated experimentally 35 by progressively decimating a square lattice to create z = 3 randomly dispersed vertices, and observing negative charges progressively appear on z = 4 vertices.The same behaviour is shown by the present system below the critical angle.
Our Shakti-Cairo system, however, also shows a change in the entity of the charge transfer and also an inversion of its sign.That comes not from topology (our argument until here has only invoked coordination) but from geometry.As the system deforms, the coupling constants in the two set of vertices differ.
To describe very approximatively this phenomenon in the context of this framework, we make the same approximations as above, because it has been proved sufficient by experiments.In particular, we use only a coupling constant per coordination: J 3 for vertices of coordination z = 3 and J 4 for coordination z = 4. Then in general Eq. ( 7) holds for each coordination number, and from that one obtains ρ ( q), the probability that a vertex of coordination z has a charge q.Then φ is found numerically by setting to zero the total charge given by Eq. 3 which of course depends on temperature and φ.Then, replacing φ in Eq. ( 7) one finds the statistical frequencies for charges, and uses it to compute the partial charges of the two set of vertices of coordination z = 3, 4 respectively.This formalism is at equilibrium and involves a temperature.The experimental system is essentially a-thermal.However, it had been previously shown theoretically 33,47 and demonstrated experimentally 27,29,35,42 that ramping up the perpendicular magnetic field leads to an equilibrium ensemble corresponding to low temperature, with respect of the coupling constants.

Experimental details
In contrast with previous works on colloidal ice 27,46 we change our fabrication process to realize the lattice of topographic double wells in Polydimethylsiloxane (PDMS) rather than in a UV curable resin.The PDMS is an optically clear and inert organosilicon compound that allow us to realize a reusable substrate.The hidrophobicity of PDMS helps prevent particles from being irreversible attached to the substrate, and it is easy to clean, reuse and replicate.
To start, we use a commercial software (AutoCad, Adobe) to design the Cairo lattice.The mask containing the desired structure is fabricated on a is fabricated on a 5inch soda-lime glass covered with a 500nm Cromium (Cr) layer.The designed motif are written via Direct Write Laser Lithography (DWL 66, Heidelberg Instruments Mikrotechnik GmbH) with a 405nm laser diode working at a speed of 5.7 mm 2 min −1 .
After being draw on the Cr mask, the double well lattice is then replicated in PDMS by first realizing an epoxy-based negative photoresist (SU-8) on the top of a silicon wafer.Then we cover the structure with liquid PDMS by spinning the sample at 4000 rpm for approximately during 1min with a angular speed of 2000rpm/s (Spinner Ws-650Sz, Laurell).With this process we obtain a layer of ≈ 20 µm thickness.The PDMS is solidified by baking for 4h at 65 • C in a levelled oven.After solidification of the PDMS, we peel off the structure with the help of a cover glass (MENZEL-GLASER, Deckglaser) of 130-170 µm height.The resulting sample is only around 200 µm thick, enabling optical access.

FIG. 1 .
FIG. 1. Lattice transformation from Shakti to Cairo.(a) Configurations of colloidal particles within z = 4, 3 vertices and associated topological charges q.In the first z = 4 vertex are shown the Ising-like spins corresponding to each double well.The ice-rule obeying vertices are highlighted in a green box with q = 0 (q = ±1 ) for the z = 4 (z = 3 resp.)vertices.(b) Geometric transformation from Shakti to Cairo by increasing the bond angle θ.(c) Definition of the parameters used in the transformation overlaid on a lithographic structure of double wells, scale bar is 10µm.One plaquette of side p is highlighted in blue.(d) Variation of the parameters l, p, a and H used in the simulation versus bond angle θ.(e) Topographic double-wells in a Cairo lattice filled with paramagnetic colloids.An external field B induces in the particles equal magnetic moments m and tunable repulsive interactions.

FIG. 2 .
FIG. 2.Inverse topological charge transfer.(a,b) Mean fraction of topological charges q versus applied field B in the z = 3 (blue circles) and z = 4 (red squares) vertices for the Shakti (a) and Cairo (b) lattices.The charge transfer in the Shakti lattice (q > 0 in z3, q < 0 in z4) inverts in the Cairo lattice (q < 0 in z3, q > 0 in z4).(c) Inversion of topological charge transfer by raising bond angle θ.Scattered data are simulation results at the maximum field B = 25mT while continuous lines are non-linear regression from the theoretical predictions (Method section).Green arrow in the plot denotes the "spin ice" cross-over at θ = 27.15• with no charge transfer between z = 3, 4 sublattices.

FIG. 3 .
FIG. 3. Experimental realization of the Cairo colloidal ice.(a) Scanning electron microscope image of the topographic double in the Cairo lattice, scale bar is 20µm.Inset shows enlargement of one double well.(b) A Cairo lattice of topographic double wells filled with paramagnetic colloids (black disks).Lattice parameters are a = 19.54µm, l = 26.7 µm and p = 46.24µm (Supplementary VideoS3).The shaded blue region is contains vertices excluded to avoid boundary effects.Scale bar is 20µm.(c) Fraction of charge q for all vertices, and (e) mean topological charge q for z = 3 (blue circles) and z = 4 (red squares) vertices versus applied field B. Symbols are experimental data, lines numerical simulations with adjusted parameters (κ = 0.025, h = 3pN • nm).