Pankaj
Popli
^{a},
Saswati
Ganguly
^{b} and
Surajit
Sengupta
*^{a}
^{a}TIFR Centre for Interdisciplinary Sciences, 36/P Gopanapally, Hyderabad 500107, India. E-mail: pankajp@tifrh.res.in; surajit@tifrh.res.in
^{b}Institut für Theoretische Physik II: Weiche Materie, Heinrich Heine-Universität Düsseldorf, Universitätsstraße 1, 40225 Düsseldorf, Germany. E-mail: saswati@thphy.uni-duesseldorf.de

Received
18th September 2017
, Accepted 28th November 2017

First published on 29th November 2017

We show that dynamic, feed-back controlled optical traps, whose positions depend on the instantaneous local configuration of particles in a pre-determined way, can stabilise colloidal particles in finite lattices of any given symmetry. Unlike in a static template, the crystal so formed is invariant under uniform translations and retains all possible zero energy modes. We demonstrate this in silico by stabilising the unstable two-dimensional square lattice in a model soft solid with isotropic interactions.

In the first case, interactions between colloidal particles are tuned. This may be done either by controlling the shape,^{11} by alloying^{12,13} or by adding specially reactive patches or tethers.^{18,19} Apart from this, confinement, either between glass plates^{14,15} or at an interface,^{16,17} induces effective interactions between colloidal particles influencing their structure. While one obtains a degree of control over the symmetry and properties of the colloidal crystals so produced, producing complex structures is difficult and needs careful synthesis and/or cumbersome fine tuning of many parameters.

On the other hand, one may also produce ordered colloidal crystals using templates.^{20} These templates may be either permanent, such as etched onto a surface, or reconfigurable, if produced by optical means.^{21–23} The latter technique has the advantage that a large variety of crystals,^{21} quasi-crystals^{22} and even random structures^{23} may be produced. Nevertheless, all periodic crystals induced by static templates necessarily suffer from a fundamental flaw.

While crystallisation due to inter-particle interactions necessarily breaks the continuous translational and orientational symmetry of the liquid spontaneously, uniform translations of the crystal as a whole in any direction or global rotations about any axis still do not cost energy. This gives rise to Goldstone modes viz. acoustic phonons whose frequencies disappear with increasing wavelength (or decreasing wavenumber).^{24} On the other hand, templates break translational invariance explicitly by destroying spatial homogeneity.^{25} Uniform translations of the whole crystal in one or more directions now cost energy and some or all vibrational modes become massive, i.e. their frequencies do not disappear with decreasing wavenumber. In the extreme case of deep periodic traps, particles do not interact with each other at all and oscillate independently in isolation, making the spectrum of vibrational frequencies resemble that of a trivial Einstein crystal.^{26} This has a profound influence on scattering processes at small wavenumbers since fewer low energy vibrational modes remain. While for many applications, such as colloidal epitaxy,^{4,20} this may not be a problem, for many others it is an issue to be addressed. For example, constructing colloidal models of solid–solid transformations^{27} and interfaces,^{28} the mechanical behaviour of crystals,^{29} crystal–glass transition,^{30}etc. requires such spurious effects to be avoided.

In this paper, we propose a means to stabilise finite-sized colloidal crystals in any desired lattice symmetry. The particles may, in principle, have almost any kind of pairwise or many-body interactions. For example, one may now stabilise open lattices such as a square or honeycomb in two dimensions (2d) and a simple cube or diamond cube in three dimensions (3d), which are unstable for most colloids interacting with central pair potentials. As a specific example of the general principle, we stabilise a square lattice in such a system of colloidal particles using computer simulations. Our proposed method is able to generate additional interactions which stabilise such a lattice “on the fly” and requires tuning of only a single parameter. The lattices so formed are invariant under uniform translations and retain acoustic phonon modes.

The rest of the paper is organised in the following way. In the next section we introduce our proposal, which uses a feedback mechanism to stabilise target structures for any kind of colloidal particle. This is followed by our specific results for first a harmonic network of particles as a test case and then a colloidal solid modelled by an isotropic pair potential. This is followed by a three part discussion section where we first explain the mechanism behind our proposal, then indicate how our scheme may be generalised to more complex structures and to higher dimensions and end by discussing caveats for its experimental realisation, comparing our strategy with other existing methods. Finally we conclude with a summary and outlook for future work.

Additional forces are computed from the following extended Hamiltonian,^{32,34} = _{0} + _{X}. Here _{0} represents any Hamiltonian for the interacting particles and,

(1) |

_{X} involves the “projection operator” and the particle displacements u_{i} = r_{i} − R_{i}. The projection operator is a function only of the reference lattice and for any particle, i is given by and , where Greek indices go over spatial components and j ∈ Ω is a neighbour of i.

Note that (1) preserves translational invariance viz.u_{i} → u_{i} + constant. One can also show^{32} that where χ_{i} is the least square error made while replacing particle displacements in Ω by the “best fit” affine strains.^{35} The quantity χ_{i} = Δ^{T}PΔ where Δ is the column vector of displacement differences^{31} with components Δ_{j} = u_{j} − u_{i} between particles i and all its neighbours j within Ω. The projection operator therefore projects out of the non-affine part of Δ and χ_{i} is the local non-affine parameter. Finally, the forces ^{i}_{χ} = − ∂_{X}/∂u_{i}. We show below that suppressing non-affine fluctuations using negative values of h_{X} is sufficient to stabilise S_{T}. The Hamiltonian _{X} introduces new interactions among particles, which guarantee that the target lattice S_{T} is stabilised as long as |h_{X}| is larger than a system size dependent threshold.

(2) |

Here p_{i} is the momentum of the particle i, m is the mass, r_{i} are the instantaneous positions, and R_{i} is the reference position in the target lattice. In this system, the length and energy scales are set by the lattice parameter l and Kl^{2} respectively. The timescale is set by . We may choose l = m = K = 1 without the loss of generality. The dimensionless inverse temperature is given by β = Kl^{2}/k_{B}T, with k_{B} as the Boltzmann constant. The interactions have a range equal to the size of the coordination volume Ω. The nearest and next nearest neighbour vertices are connected by harmonic springs of strengths K_{1} = 1 and K_{2} respectively. The square lattice is mechanically unstable in the limit K_{2} → 0 due to softening of the transverse phonon modes.^{36} This is illustrated in Fig. 2a where we have plotted the phonon dispersion ω(q) for this harmonic square network for three different values of K_{2}. For K_{2} = 0 the transverse mode disappears. We now add the term proportional to h_{X} in (1) to the harmonic Hamiltonian _{0} using the interaction volume, namely, the first and second neighbour shells of the square lattice as our choice of Ω (see Fig. 2 inset). The dynamical matrix,^{24,26} corresponding to the full Hamiltonian, can be written as ^{μν} = ^{μν}_{0} + ^{μν}_{X}, where ^{μν}_{0} is the dynamical matrix from _{0} and

(3) |

In Fig. 2b we plot the resulting phonon dispersion curves all at K_{2} = 0 but for three different values of h_{X}. As h_{X} is made more negative, the transverse phonon mode appears to revive. One must note however that as q → 0 the nature of A_{X}(q) dictates that the speed of transverse sound vanishes in the hydrodynamic limit. Nevertheless, for finite lattices, this limit is never reached since wavenumbers are cut off at q = 2π/L where L ∝ N^{1/d} is the linear system size.

ϕ_{ij} = εexp(−r_{ij}^{2}/σ^{2}). |

When particles interacting with the GCM potential are arranged in a square lattice, one obtains a mechanically unstable solid. Small displacement fluctuations from the ideal square lattice positions (where forces still disappear due to symmetry) make the solid deform into a stable triangular structure via soft transverse modes.^{36} This is clear from the calculated dispersion curve shown in Fig. 3a for h_{X} = 0.

Fig. 3 (a) Phonon dispersion curve ω^{2}(q) plotted along high symmetry directions for a square lattice of particles interacting with the GCM potential at ρ = 0.5 and T = 1 × 10^{−3}. The small q region where ω^{2} < 0 is shown in the inset. We follow the same convention as in Fig. 2. There are two sets of curves for h_{X} = 0 (red curves) and h_{X} = 0.5 (blue curves). (b) A stability diagram for finite size square lattices constructed from the dispersion curve in (a). The green shaded region represents −h_{X} values below which the square lattice becomes unstable (ω^{2}(q) < 0). We have also marked six points on the graph such that black (white) circles denote stable (unstable) square lattices. |

We now turn on h_{X} defined exactly as in the case of the network. As h_{X} is decreased below zero, again, we see a revival of the transverse phonon mode. Unlike the network however, for all h_{X}, the transverse mode now has ω^{2} < 0 in a small region of q → 0. Making h_{X} more negative can, nevertheless, restrict this region to extremely small q values which are not accessed by a solid of finite size due to the infrared cut-off discussed earlier. Since within harmonic approximation temperature enters only as a pre-factor, this leads to a T independent “stability diagram” as shown in Fig. 3b.

To verify the stability diagram presented in Fig. 3b we perform Monte Carlo simulations with standard Metropolis updates^{42} of a GCM solid with various N and h_{X} values keeping ρ and T the same as before, equilibrating the system for a minimum of 10^{6} Monte Carlo steps (MCS) starting from the initial square lattice. To check our results we obtained the probability distribution of the non-affine parameter P(χ) and compared it with the predictions from the harmonic theory using the calculated dynamical matrix ^{μν} as the input.^{31,43}

Equilibrated configurations from our simulations are presented in Fig. 4a. The h_{X} and N values for these configurations are marked in Fig. 3b. It is clear that these results follow the expectations from our stability criterion. Once the stability threshold is breached, the square lattices destabilise due to q → 0 modulations. The local non-affine parameter χ rapidly rises as the crystal becomes unstable and is shown as a colour map. For all the stable square solids studied, our results were indistinguishable from the theoretical prediction though they deviate, as they should, if instability sets in (Fig. 4b). If ρ is increased, harmonic approximation becomes more accurate and the magnitude of h_{X} needed to stabilise the square lattices becomes even smaller.

Fig. 4 (a) Configurations obtained from Monte Carlo simulations after 6 × 10^{6} MCS arranged according to the corresponding points marked in Fig. 3b verifying the stability condition. Colours correspond to the local value of χ. (b) P(χ) obtained from the equilibrated configurations where the square lattice is stable (top) and unstable (bottom). Data points are computed from the results of our simulations and lines are predictions of the harmonic theory.^{43} The configurations and curves are labelled by the corresponding h_{X} and N values. |

For example, both square and triangular crystals may be considered as special cases of a general oblique lattice and one can be obtained from the other by a purely affine transformation. Along this transformation path, which involves a bulk homogeneous strain, h_{X} will not contribute. However, such an event is statistically unlikely except for extremely small systems. What is more likely is that a small patch of particles locally transforms to the S_{E} structure, creating χ at the interface. Within the classical nucleation theory^{24} the free energy cost of such a patch of size L_{p} is _{p} = AΔL_{p}^{2} + Bh_{X}L_{p}, where we ignore interfacial terms independent of h_{X}. Here Δ is the bulk free energy difference per unit area between the two lattices and A and B are constants. This patch is stable only if it is larger than a critical size L_{p}* ∼ −h_{X}/Δ and costs interfacial energy ∼h_{X}^{2}/Δ. L_{p}* can be large if Δ is small and can be made even larger by tuning h_{X}. If L_{p}* > L, again, a finite S_{T} crystal will be stable due to the inability of the system to create a sufficiently low energy S_{E}|S_{T} interface.

It is possible to demonstrate both these mechanisms for the square to triangular transition. It is known that a mechanically unstable square lattice may decay at nonzero temperatures into a stable triangular structure in many different ways.^{36} For example, (a) alternate rows of particles in the square lattice may shift by half-a-lattice spacing, hence producing a distorted triangular lattice that subsequently equilibrates. On the other hand, a patch of particles inside the square matrix may undergo distortion to a triangular structure (b). Our stabilisation strategy should be able to recover the target square lattice S_{T} from both (a & b) of these distortions.

In Fig. 5a and b we demonstrate this explicitly by starting from initial (square) configurations of an N = 2500 GCM solid incorporating these two kinds of distortions (a) and (b) and equilibrating with h_{X} = −2.0, where the square lattice is stable. In the first case (Fig. 5a), the local shuffles of particles cost energy and are quickly removed from the solid. In the second case (Fig. 5b(i)), we first create a large patch of particles with a local triangular order using an inhomogeneous affine strain. Next, we equilibrate the surrounding matrix keeping the particles within the patch immobile. This produces a mechanically relaxed (but high energy with large local χ) interface between a square solid and a triangular inclusion. The constraint is then removed and the whole system is equilibrated. Fig. 5b(iii) shows that the sub-critical patch thus created disappears. As a check (Fig. 6) we ensure that P(χ) obtained from the equilibrated configurations again matches theoretical predictions.^{31,43}

Fig. 6 The resulting P(χ) at the end of the runs in Fig. 5a (red circles) and Fig. 5b (blue squares); the solid line is the harmonic theory prediction.^{43} |

In 2d, long wavelength displacement fluctuations are known to destroy the crystalline order^{24} and therefore a discussion of this effect is germane for the 2d square solid discussed here. Crystals in 2d posses only “quasi”, rather than true, long-ranged order since the mean squared displacement 〈u^{2}〉 diverges as logL. This is an effect of dimensionality and appears in the thermodynamic limit for any 2d crystal, even those with non-pathological phonon spectra. These fluctuations have also recently been observed even for amorphous solids,^{45–47} considerably increasing the scope of its general applicability. How do these fluctuations affect our results?

Our proposal is designed to stabilise finite-sized crystals. Indeed, for any |h_{X}|, as the system size L is increased, one reaches a threshold above which the solid is no longer stable (see Fig. 3b). In order to stabilise larger systems, |h_{X}| has to be increased as ∼L^{2} = N. To obtain stable solids with large L requires large and probably experimentally un-realisable optical field strengths. The weak logarithmic divergence of displacements discussed above is therefore unobservable at the system sizes obtainable with any realistic |h_{X}| and does not alter our results.

Open lattices may be of two kinds. Firstly they may be sparsely coordinated primitive lattices. Maxwell's condition for stability^{48,49} requires that the coordination number, z, must be strictly >2 × d, which amounts to z > 4 in 2d and z > 6 in 3d. The square lattice in 2d and the simple cubic lattice in 3d are marginal having 4 and 6 nearest neighbours respectively. Including the second coordination shell while constructing the coarse grain volume Ω is essential in these cases. This ensures that phonon modes which are soft, such as the shear modes encountered in the 2d square lattice, involve particles within Ω. These modes need to be suppressed in order to ensure stability of the lattice.

Open lattices may also result from decorating a primitive lattice with a basis i.e. a motif consisting of more than one inequivalent particle. For example, in 2d one has a planar honeycomb lattice where a dimer replaces every site of the triangular lattice.^{50} Similarly, in the Kagomé lattice, the equivalent motif contains five particles arranged as a pair of equilateral triangles, which share a vertex.^{49} In 3d, the diamond cubic structure contains repeated vertex sharing tetrahedra.

Such lattices also contain “floppy” modes which involve relative twists of the basis with respect to the rest of the particles in the structure.^{43,49} In the planar honeycomb structure localised modes representing a twist of the dimer results in the creation of a Stone–Wales defect.^{43} The coarse grain volume needs to be large enough to be able to describe such modes and Ω needs to contain all symmetry dictated particles belonging to copies of the basis, centred on each site of the primitive lattice. The single parameter h_{X} may then be used to suppress these modes. In this case, however, our formulae for relative displacement Δ and projection matrices need to be generalised as follows.

To obtain Δ we arrange the entries Δ_{Jmn} = u_{Jm} − u_{In} as a column vector. The first (capitalized) index numbers the basis and the second index numbers the _{Ω} particles within the basis. The elements of the matrix of reference coordinates are similarly given by . These elements are arranged so that the matrix has the dimensions of d_{Ω} × d^{2}, where _{Ω} is the total number of distinct lattice vectors, {R}, in Ω. In terms of the d_{Ω} × d_{Ω} dimensional projection matrix remains, . Once a suitable Ω is chosen, _{Ω}, _{Ω}, and are known and the feedback loop can be implemented exactly as discussed earlier. As the complexity of the lattices increases, the calculation of the forces becomes more involved but always scales linearly with _{Ω}. Real time feedback is therefore still limited by the response time of the optical system and not computation.

Particle tracking at a frame rate of 10 kHz is possible using the current technology^{51} so this is not a rate limiting step. Since the computations needed to calculate forces even for the most complex lattices are simple, these are also quite fast. The crucial experimental step therefore is the deployment of traps to generate forces.

For this purpose one may use a spatial light modulator and holographic tweezer technology^{33} which produce a reconfigurable optical surface based on inputs. Typical spatial resolutions of these devices are about 1–3 μm which is appropriate for our purpose. The input frame rate can theoretically go from 500 up to 1 kHz, although the actual frame rates can be much slower. In any case, considering that large colloids are slow, one has a reasonable margin of exploration.

Are the values of h_{X} for reasonably sized lattices too high to be realised in the lab? The quantitative estimation of energy due to the h_{X} term shows that it is comparable to the energy of interactions which is of the order of a few k_{B}T. This is the regime where all previous optical manipulations of colloidal crystals have been traditionally performed.^{21–23} However, spatial light modulators also decrease the light intensity which depends on the make of the device; so higher laser powers are needed limiting the number of particles which may be trapped without causing excessive heating. An alternative procedure is to use time shared traps^{52} in a dense enough array fixed in space with their intensities modified intermittently to approximate the optical potential surface.

Feedback controlled traps of somewhat different nature have already been employed to study the crystallisation of colloids.^{53} The principle behind this implementation is quite different from ours. Instead of coupling to a local configuration-dependent parameter such as χ, the trap responds to a global order parameter for crystallisation. The symmetry of the crystal is still determined by inter-particle interactions and very limited control over the crystal structure is possible.

One may view the stabilising forces that we use as originating from an effective three-body potential viz., eqn (1). The form of this potential depends on the reference configuration, guaranteeing that these forces necessarily stabilise the target structure. Our method therefore automatically determines the potential which stabilises S_{T}. The advantage of our method is that no further assumption about particle interactions is necessary. While it may be possible, for the case of simple target structures, to guess interactions that give rise to them and perhaps design colloids which produce these structures,^{18} this becomes progressively difficult as the complexity of S_{T} increases. This is especially important when we realise that using a similar strategy even inhomogeneous structures such as surfaces and interfaces of any specified orientation as well as random glassy configurations may be stabilised.

Finally, we emphasise again that fast particle tracking and the realisation of fast response times of light fields required for our method to work is feasible for colloidal solids because of their slow timescales. Similarly heating problems associated with these light fields should also be minimal especially since the magnitude of the forces needed is small (≪k_{B}T). Of course, only actual experiments can finally decide upon the feasibility of our proposal. We hope that our work motivates experimental work in this direction in the near future. Extension of our method to stabilise complex open lattices in 2d and 3d, in collaboration with experiments, is an attractive future goal.

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