document: colloidal suspensions present the possibility to develop novel materials via self - assembly . of particular interest are colloidal crystals , whose optical properties can generate iridescent colours , and provide a means by which photonic crystals may be produced@xcite , while further applications range from lasers @xcite to display devices @xcite , with recent advances demonstrating tunable colours through control of lattice spacing with an external field @xcite . 
   further to the practical importance of colloidal crystals , their well - defined thermodynamic temperature allows colloidal dispersions to be viewed as mesoscopic ` model atoms ' @xcite . 
   recently , the ability to tune the colloid - colloid interactions has led to the observation of a wide variety of structures @xcite . 
   of particular interest here , to first order ac electric fields can induce dipolar interactions between the colloidal particles , leading to anisotropic interparticle potentials and exotic crystal structures , some of which are not observed in atomic and molecular systems @xcite , while external control of the colloid - colloid interactions allows direct observation of phase transitions @xcite . 
   furthermore , direct microscopic observation at the single - particle level allows an unprecedented level of detail to be accessed @xcite , opening the possibility of tackling long - standing problems in condensed matter , such as freezing @xcite . 
   the introduction of a rotating ac field opens up even more possibilities . in this case , the dipolar interactions lead to an attraction in the plane of rotation and to repulsions above and below . 
   studies with a rotating magnetic field on granular matter indeed produced disc like patterns consistent with expectations @xcite . unlike granular matter , since colloidal dispersions exhibit brownian motion , thermodynamic equilibrium structures ( ie crystals ) , may be obtained @xcite . in previous work snoswell 
   _ @xcite showed that lattice spacing within quasi-2d colloidal crystals could be controlled _ in - situ _ 
   , by means of coplanar rotating electric field . 
   the interparticle dipolar interactions in the plane of the electric field may be treated to first order as a circularly symmetric attraction , due to the time averaging effect of a rapidly rotating field ( 1000hz ) on relatively large particles on the micron lengthscale , where the diffusive timescale is of the order of seconds @xcite .    in considering the interactions between the particles , 
   the asymmetry between the colloids ( 10nm-1@xmath0 m ) and smaller molecular and ionic species must be addressed . 
   a number of coarse - graining schemes have been developed where the smaller components are formally integrated out @xcite . 
   this generates a one - component picture , where only the effective colloid - colloid interactions are considered , and the complexity of the description is vastly reduced . 
   the equilibrium behaviour of the colloids in the original multi - component system may then be faithfully reproduced by appeal to liquid state theory @xcite and computer simulation @xcite . 
   central to the success of this one - component approach is the use of a suitable effective colloid - colloid interaction @xmath1 .    in this study 
   , we use a simple numerical treatement in which we can predict the lattice spacing in the quasi 2d crystal from the electric field strength . 
   we consider a model system of charged colloids , in a rotating electric field @xcite . by exploiting the knowledge both of the electrostatic repulsions and dipolar attractions , we present a direct , quantitative comparison of a tunable interaction and a material property of the crystalline ` rafts ' formed . 
   we combine experimental measurements of the crystal lattice constant @xmath2 as a function of field strength @xmath3 with monte - carlo simulations according to a screened coulomb repulsion plus dipolar attraction where the only fitting parameter is the debye screening length .    in the simulations , we use pairwise interactions , in other words 
   we assume that at the higher densities at which the crystalline rafts are formed , the system is still accurately described by interactions calculated for two particles in isolation . 
   we note that deviations from this assumption of pairwise additivity have been measured both in the case of strongly charged colloids @xcite and in the case of _ repulsive _ dipolar interactions @xcite . 
   we further compare simulation results with the minimum of the effective potential , which we take as a measure of the lattice constant of the crystalline rafts , which we also determine from experimental data . 
   this paper is organised into six sections . in section 
   [ theory ] we present expressions for the effective interactions between the colloids , summing the attractions and repulsions to provide an effective one - component description of the system . 
   section [ experimental ] describes our experimental metholodogy . 
   section [ simulation ] outlines the monte - carlo simulation technique employed . 
   the comparison of simulation and experimental results is presented in section [ results ] and in section [ tunability ] we extrapolate our findings to maximise the tunability of the crystal lattice constant , which may be useful for applications . 
   we conclude our findings in section [ conclusions ] . 
   in the following we will consider a system consisting of two particles in a surrounding medium . 
   we shall assume that these particles are charged , leading to a repulsive interaction , and that the rotating ac electric field induces a dipole moment in the two particles and thus induces an attractive interaction . to describe this system , we start from the derjaguin , landau , verwey and overbeek ( dlvo ) approach @xcite , which consists of attractive van der waals interactions at short range , and long - ranged repulsive electrostatic interactions . 
   the van der waals interactions are very short - ranged , and are neglected , as electrostatic repulsions inhibit the close approach at which van der waals interactions become important . 
   we shall therefore assume that the only relevant attractions result from the long - ranged dipolar interactions induced by the rotating electric field .    in the linear poisson - boltzmann regime 
   , the electrostatic repulsions may be expressed as a hard core yukawa , or screened coulomb interaction @xcite ,    @xmath4    where @xmath5 where @xmath6 is boltzmann s constant , @xmath7 is temperature and @xmath8 is the colloid diameter . 
   the potential at contact , @xmath9 is given by    @xmath10    where @xmath11 is the colloid charge , @xmath12 is the bjerrum length and @xmath13 is the inverse debye screening length where @xmath14 is the total number density of monovalent ions . 
   now the regime of linear poisson - boltzmann theory in which equation ( [ eqyuk ] ) holds corresponds to relatively weak charging . 
   although this is not the case here , a potential of the yukawa form is recovered at larger separations , if a smaller , renormalized charge is considered @xcite . 
   we tabulate measurements of the @xmath15-potential of dilute suspensions in table [ tablekappasigma ] . 
   these values suggest that we expect a renormalised charge , in the conditions under which the colloids form the crystalline rafts , rather high colloid concentration , hence high counter ion concentration @xcite . therefore , noting that the debye length is much smaller than the colloid radius , we follow bocquet _ et . 
   _ @xcite and take the following expression for the renormalised charge @xmath16 :    @xmath17    which we substitute for @xmath11 in equation ( [ eqepsilonyuk ] ) . 
   this expression gives good agreement with measurements of the effective colloid charge for particles with a comparable @xmath15-potential @xcite . 
   we recall that many - body effects can lead to a density - dependence in the effective colloid - colloid interactions @xcite . 
   however , we shall neglect these effects in the present work . 
   the attractive potential between the colloids resulting from the rotating ac electric field is treated to first order as a dipolar interaction : @xmath18    where @xmath19 is the induced dipole moment of each particle , @xmath20 is the dielectric constant and @xmath21 is the permittivity of free space . 
   the dipole moment may be calculated from the strength of the electric field    @xmath22    where @xmath23 is the clausius - mosotti factor and takes values between 1 and -1/2 depending upon the origin of the dipole moment . 
   we note that in this case of an alternating field , the root - mean - square of the time dependent field is taken . 
   substituting @xmath24 leads us to a potential at contact    @xmath25    which has a cubic dependence on the colloid diameter , in the case of all other contributions being unchanged .    in considering the value of the clausius - mosotti factor , 
   two regimes are relevant , corresponding to applied electric fields of high and low frequency , in which the dipolar interactions result from dielectric permittivity differences between the particles and the solvent and differences in conductivity respectively @xcite . 
   the crossover frequency @xmath26 between these regimes is given by    @xmath27    where @xmath28 are conductivities and the indices @xmath19 and @xmath29 refer to the colloidal particles and the medium respectively . 
   following @xcite the conductivity of the particles @xmath30 is taken to be the sum of the bulk conductivity @xmath31 and the conductivity on the surface @xmath32 . 
   @xmath33    in principle equation ( [ eqconductivity ] ) holds for particles larger than 1 @xmath0 m . in the case of much smaller particles 
   , one should take into account that the contribution of the diffuse layer to the conductivity of the particles . noting that the smallest particles used in the experiments are 757 nm in diameter and that the conductivity of the de - ionised water is relatively low , neglect the double layer contribution and 
   thus use equation ( [ eqconductivity ] ) . 
   since the largest particles studied are 2070 nm in diameter and the frequency is 1 khz and the bulk ionic strength is @xmath34 mmol as determined from conductivity measurements , we work in the low frequency regime where the contributions from the conductivities dominate @xcite . 
   the clausius - mosotti factor therefore takes the following form :    @xmath35    due to the surface conductivity , @xmath30 is much larger than the conductivity of the medium @xmath36 . 
   the value of @xmath37 is therefore close to one . 
   we now combine the contributions from the electrostatic repulsions and the dipolar attractions , yielding the expression . 
   @xmath38    @xmath39 has the form indicated in fig . 
   we see a minimum in the potential , which one might expect to provide a first approximation to the lattice constant in the 2d colloidal crystal . in sections [ results ] and [ tunability ] we shall compare this minimum to our simulation results . 
   colloidal crystals underwent self - assembly as a result of an applied electric field . 
   we used anionic , sulphate stabilised polystyrene latex particles , either synthesised using a standard technique , surfactant free emulsion polymerisation @xcite in the case of @xmath40757 and 945 nm , or particles produced by the same method but purchased from microparticles gmbh for @xmath401390 and 2070 nm . 
   particle electrophoretic mobility was measured using a brookhaven zetaplus light scattering instrument . 
   particle sizes were determined by scanning electron microscopy , either in - house using a jeol jsm-6330f , in the case of @xmath40757 and 945 nm or by microparticles gmbh for @xmath401390 and 2070 nm are are listed in table [ tablekappasigma ] . 
   experiments were performed with dilute aqueous suspensions ( 0.5 - 1.5 wt% ) . 
   schematics of the experimental set up are shown in fig . 
   [ figschematic ] all glass surfaces were chemically washed with 0.1 m koh and washed with copious quantities of milliq water . 
   particles were deionised by direct contact with ion exchange beads before being made up to the desired electrolyte concentration with kcl . 
   very low salt conditions are required as even moderate ( mmol ) salt concentrations lead to electrohydrodynamic pattern formation due to ion flow @xcite . 
   for further details of the experimental set up and procedure the reader is referred to snoswell _ et . 
   _ @xcite . 
   we consider our experiment as a 2d system . 
   however , occasionally we noticed some overlap of the crystalline rafts . 
   we do not include these results in our analysis . in order to treat the system in 2d 
   , one might expect the gravitational length @xmath41 , where @xmath29 is the bouyant mass of the particle and @xmath42 is the acceleration due to gravity , to be much less than some characteristic length such as the particle diameter . due to the fairly small density mismatch between polystyrene and water , in fact even for the largest particles @xmath43 nm , @xmath44 . 
   however , the dipolar interactions between the particles are attractive in - plane , but strongly repulsive in the vertical direction . 
   this promotes the formation of ` rafts ' and ` sheets ' , and these large assemblies of many particles have very small gravitational lengths . 
   thus we argue that the system behaves in a quasi 2d manner . 
   the lattice parameter @xmath2 was taken as the average of typically ten crystalline ` rafts ' , measured across the ` raft ' , of around ten lattice spacings . the response to changing 
   the electric field strength was determined to be less than 100 ms , we waited 5s after changing the field strength before acquiring data . 
   we use standard monte - carlo ( mc ) simulations in the nvt ensemble @xcite . 
   the particles interact via equation ( [ equ ] ) in two dimensions . to mimic the experiments , we initialise the system in a random configuration at a relatively low concentration , corresponding to an area fraction @xmath45 . 
   we confirmed that different area fractions gave indistinguishable results , however smaller area fractions often led to longer equilibration times . 
   the attractive interactions cause the particles to approach one another , and form crystallites which then coalesce to form a crystalline ` rafts ' which contain of order @xmath46 particles , in a qualitatively similar way to the experimental system . 
   each simulation was equilibrated for typically 30000 mc moves per particle , followed by 3000 production moves per particle . 
   we recall that a potential of the form @xmath47 converges in 2d . 
   the lattice constant for this 2d hexagonal crystal was taken as    @xmath48    where @xmath49 is the pair correlation function , and @xmath50 the minimum between the first and second peaks of @xmath49 . 
   we found that when we used @xmath51 , that typically one crystalline domain was formed , around the centre of the simulation box . 
   we therefore do not use periodic boundary conditions , and consider the lattice spacing of the single crystal formed in the simulation . 
   one parameter is the number of particles typically present in each crystalline domain . 
   this is estimated to be around @xmath52@xmath53 from experimental data ( see fig . 
   [ figpix ] ) . 
   this value is governed by the overall concentration of the colloidal suspension . 
   we have considered the effects of varying @xmath54 as shown in fig . 
   [ figfinitesize ] . 
   the results of a considerable range of @xmath54 resulted in a variation of a around one percent in @xmath2 for the @xmath55 nm system for a field @xmath3=20 @xmath56 . 
   each simulation was repeated four times except @xmath57 . 
   we found a slight trend to tighter binding for larger @xmath54 , however , as shown in fig . 
   [ figfinitesize ] , this effect is rather small . 
   we henceforth use one simulation per state point unless otherwise stated . the slight scatter in the simulation data 
   is perhaps indicative the existence of different magic numbers for these crystalline rafts @xcite . 
   these may be thought of as 2d hexagonal clusters , whose ` magic numbers ' , ie low energy states , are expected to include @xmath58 , @xmath59 , @xmath60 ... close to a ` magic number ' the binding may be expected to be relatively tighter . 
   a more detailed exploration of this phenomenon lies beyond the scope of this work . 
   overall , the variation in @xmath2 as a function of @xmath54 is smaller than the experimental scatter , nonetheless we take both @xmath61 and @xmath62 values for @xmath2 when comparing with experimental data . 
   the slightly larger value for @xmath63 in fig . 
   [ figfinitesize ] is attributed to the small size of the crystalline ` raft ' , such that the surface particles which are more widely spaced make a greater contribution . 
   the domain size is of order 100 particles , thus , for larger @xmath54 , defects and grain boundaries may lead to a smaller contribution to the measurement of @xmath2 . 
   this also applies to experimental data .    for fitting , 
   all parameters are known , except the debye length @xmath64 which we take as a free parameter for each particle size and salt concentration , although we note that the effective colloid charge is itself a function of the debye length , equation ( [ eqzeff ] ) . 
   it has previously been shown that comparison of simulation and experimental data can yield a reasonable _ local _ measure of the debye length @xcite . 
   furthermore , in the region of the sample in which the electric field is applied , the colloid volume fraction is much higher , and due to the colloidal counter - ions the ionic strength may increase , leading to a reduction in the debye length with respect to the bulk . 
   however , it is the debye length in this region which is relevant for the effective colloid - colloid interactions . the bulk debye length that may be determined , for example from conductivity measurements , 
   may therefore be taken only as an upper bound . 
   system parameters for different particle sizes are tabulated in table [ tablekappasigma ] , and the debye length is plotted in fig . [ figkappasigma](a ) and the contact potential of the yukawa interaction [ equation ( [ eqepsilonyuk ] ) ] in fig . [ 
   figkappasigma](b ) are plotted as a function of particle diameter . 
   clearly , our mc simulations may be expected to provide a reasonable treatment of the crystalline rafts in ( quasi ) equilibrium , rather than to describe the formation process . 
   we therefore restrict our analysis to a simple characterisation of the crystal rafts , and primarily consider the lattice spacing . 
   we furthermore note both here and in previous work @xcite that considerable variation in shapes of crystalline rafts was observed , but that this had little impact upon the measurement of the lattice spacing @xmath2 . likewise , at the level of this work , we neglect possible local variations in @xmath2 due to the proximity of an interface . in any case 
   we note that for display applications , all lattice spacings contribute to the diffraction . 
   a few words on equilibration are in order . 
   this applies both to the experimental system , and to the simulations . 
   neither system is strictly in equilibrium , in that case we might expect a rather regularly - shaped raft such as a hexagon . 
   however , the insensitivity of the lattice parameter either as a function of @xmath54 ( fig . 
   [ figfinitesize ] ) , and the very close agreement between statistically independent simulation runs lead us to conclude that our approach is sufficient to compare lattice parameters between experiment and simulation . 
   the effect of changing the electric field strength is readily demonstrated in fig . [ figpix ] , which shows optical microscope images of the region of the sample in which the field is applied . 
   two field strengths are shown , 29 kvm@xmath65 ( a ) and 80 kvm@xmath65 ( b ) for @xmath55 nm and we see a correspondingly tighter lattice in the case of the higher field strength . 
   similar behaviour was observed in @xcite . 
   figure [ figpix](c ) illustrates simulated coordinates for @xmath66@xmath67 @xmath68 , @xmath69 , @xmath70@xmath71 @xmath68 , which corresponds to @xmath3=29.0 vm@xmath65for the @xmath55 nm system ( see table [ tablekappasigma ] ) .    a more quantitative comparison between experiment and simulation is shown in fig . 
   [ figg ] , which plots the radial distribution function @xmath49 . 
   we see that the structure appears to be crystalline , with reasonably long - ranged correlations , although , as is clear from fig . 
   [ figpix ] , the system is too small to test for truly long - ranged order . 
   we thus note that it is difficult to distinguish the true crystal from a hexatic phase in this system . 
   however , our motivation is to model the experimental system , with @xmath72 and thus we argue that identification of the hexatic phase lies beyond the scope of this work . 
   having illustrated the general behaviour of the system , we now turn our attention to the fitting of the experimental data , with the model described in section ii . 
   the experimental data was fitted to the theory by taking the debye - length as the fitting parameter . 
   the main results are shown in figs . 
   [ figsmallexpmc ] and [ figbigexpmc ] for @xmath73 and 945 nm , and @xmath74 and 2070 nm systems respectively . 
   these concern the lattice parameter @xmath2 as a function of applied field strength @xmath3 . in general , 
   the simulation is able to capture the behaviour of the system in a reasonably quantitative manner . 
   furthermore , simply plotting the minimum in the potential @xmath75 given by equation ( [ equ ] ) provides a first approximation to the lattice constant @xmath2 . 
   figure @xmath76(a ) shows the results for @xmath73 nm particles . due to the small particle size , and the inverse cubic dependence of the strength of the dipolar attractions on the particle diameter , as characterised by @xmath77 [ equation ( [ eqepsilondip ] ) ] 
   the attraction for a given field strength is comparitively small , so we find relatively larger lattice parameters for this system . at low field strengths ( @xmath78 @xmath56 ) 
   we see an increase of the lattice constant for both the experimental data and the simulation , compared to the minimum in the potential . 
   apparently , as the crystal approaches melting , fluctuations become more pronounced , leading to an increase in @xmath2 , which is not captured in equation ( [ equ ] ) , or perhaps a molten surface layer increases the apparent value of @xmath2 . 
   we leave this intriguing question for further investigations . 
   meanwhile , at high field strengths , we find some deviation between simulation , experiment and equation ( [ equ ] ) . apparently , due to the long ranged nature of the interaction , second nearest neighbours experience an attraction of sufficient strength that the crystal is compressed by its own cohesion , leading to a smaller lattice constant . 
   in the case of the @xmath73 nm particles , we find that for small field strengths , ( @xmath79 @xmath56 ) , the system does not form crystallites , rather it remains as a colloidal liquid . 
   we determined this both by experimental observation and with simulations . in the latter case , we identified crystallisation with a splitting in the second peak of the pair correlation function @xmath49 . 
   the kink around @xmath80 @xmath56 in fig . 
   [ figsmallexpmc ] is likely related an artifact of measuring multiple rafts , so defects and grain boundaries contribute to the value of @xmath2 . at weak field strengths the rafts 
   tend to be smaller , thus increasing this effect .    at larger particle sizes , ( fig . 
   [ figbigexpmc ] ) , we again see reasonable agreement between experiment and simulation , albeit with some deviation between simulation and equation ( [ equ ] ) at higher field strengths , as the simulation predicts a smaller lattice constant than that observed in the experiments . 
   however , for the @xmath74 nm system in particular , in fact equation ( [ equ ] ) provides a more accurate description of the experimental data . apparently some of our assumptions in the simulations , perhaps that of pairwise additivity , begin to break down at high field strengths . according to equation ( [ equ ] ) , 
   the potential at the minimum of the attractive well is some 250 @xmath68 for an applied field of 50 kvm@xmath65 for @xmath74 nm . thus we conclude that for moderate interaction strengths , equation ( [ equ ] ) provides a reasonable description of the system . 
   we now consider the dependence of the system upon the colloid diameter @xmath8 . 
   firstly , we see from fig . 
   [ figkappasigma](a ) that the absolute value of the ( fitted ) debye length @xmath64 does not change significantly for all the four particle sizes studied . 
   thus , the reduced inverse debye length @xmath81 is linear in @xmath8 . 
   the contact potential @xmath82 has an approximately inverse cubic dependence on @xmath8 for @xmath81@xmath83 , equation ( [ eqepsilonyuk ] ) , but also depends upon the ( effective ) charge @xmath16 . 
   plotting @xmath82 as a function of @xmath8 in fig . [ 
   figkappasigma](b ) we find an approximate power law dependence with an exponent of @xmath84 . 
   as noted above , the prefactor of the dipolar attraction , @xmath77 [ equation ( [ eqepsilondip ] ) ] , has a @xmath85 dependence . 
   although the electrostatic interactions are non - negligible , we nonetheless expect from equation ( [ eqepsilondip ] ) that upon decreasing @xmath8 we find a relatively larger lattice constant for a given field strength , and that larger field strengths are required to provide sufficient interactions that the crystalline rafts form . 
   this we indeed find , as shown in figs . 
   [ figsmallexpmc ] and [ figbigexpmc ] , and also in the next section . 
   now we have noted that this system has optical display applications , due to the possibility of externally tuning the lattice parameter @xmath2 . 
   these applications may be most usefully realised when the possibility to tune the system is maximised , ie when the range of @xmath2 is maximised . 
   we are therefore motivated to consider a range of particle sizes , and calculate the range of tunability of @xmath2 . 
   now @xmath8 sets a lower bound to @xmath2 while the upper bound is set by the melting transition which can be determined by monte - carlo simulation . 
   hitherto , we have used the debye length @xmath64 as a fitting parameter . however , as fig . 
   [ figkappasigma](a ) shows , there is relatively little change in the absolute value of the debye length for the range of colloid diameters investigated . 
   we therefore fix @xmath86 nm and vary the colloid size , and apply the same methodology as that outlined in sections [ theory ] and [ simulation ] , in calculating the response for @xmath87 nm colloids to an external electric field . 
   while an accurate determination of the melting transition is complicated by the small system size , melting is approximately determined by the splitting of the second peak in the radial distribution function @xmath49 @xcite . 
   we decrease the electric field to identify the weakest field at which the second peak in @xmath49 exhibits clear splitting [ fig . 
   [ figtuning](a ) , inset ] , thus yielding @xmath88 , the lattice spacing just prior to melting . 
   the melting value of the lattice parameter @xmath89 is taken as    @xmath90    where @xmath91 is the amount by which the field strength is varied between simulations , typically @xmath92 @xmath56 . 
   the error in @xmath89 is then taken as @xmath2@xmath93 .    in terms of the particle diameter 
   , @xmath89 grows considerably at small sizes , as the relatively larger debye length leads to a minimum in the effective potential at relatively larger distances . 
   however , we recall that our analysis suggests a roughly @xmath94 dependence of @xmath77 the strength of the dipolar attraction . 
   this suggests that we might expect a stronger electric field to be required for relatively small colloids , and indeed we find that a higher field is required to provide sufficient cohesive energy to hold the colloids together in the crystal - like ` rafts ' , so at 100 @xmath56 , a value we take as a reasonably accessible maximum field strength , the lattice spacing is still around @xmath95@xmath8 for @xmath96 @xmath97 , where @xmath98 is the lattice spacing corresponding to a field strength of 100 @xmath56 . 
   conversely , at larger particle sizes , the debye length is relatively small , so we find , even at melting , that the lattice constant only approaches around @xmath99@xmath8 . 
   we note that , at high field strengths , the assumptions of section [ theory ] will ultimately break down . 
   the results presented in figs . [ figsmallexpmc ] and [ figbigexpmc ] show that @xmath100 @xmath56 is reasonable , at least for @xmath101 nm , supporting our conclusion of a maximum in tuneability around @xmath99@xmath8 . 
   we show the values of the well depth of equation ( [ equ ] ) at melting for the different experimental system in table [ tablekappasigma ] , which indicates a trend towards deeper wells for larger particles , i.e. shorter ranges of the repulsive interaction relative to the particle size . 
   larger field strengths may be expected to yield greater tunability for the small particle sizes . since the overall minimum lattice spacing is @xmath102 , in principle smaller particles have greater tunability 
   , however experimental observations reveal intense fluid turbulance disrupts crystal formation at higher field strengths . 
   this is caused by large field gradients at the electrode edges that induce strong dielectrophoretic forces . 
   in addition , higher field strengths can cause bubble formation by electrolysis and electrochemical degradation of the electrodes . 
   the result is thus that intermediate particle sizes have the highest degree of tuneability , for a given maximum field strength . 
   here we have considered 100 @xmath56 , which leads to a maximum tuneability of @xmath103 in the range 1000 nm @xmath1041500 nm . 
   2d colloidal crystalline ` rafts ' with externally controllable properties have been modeled with monte - carlo simulations , and the resulting lattice constant , both experimental and simulated , has been compared to the minimum in the effective interaction potential . in treating the effective colloidal interactions , we find that a straightforward , pairwise approach provides a reasonable description in describing the lattice parameter . 
   the minimum in the effective interaction formed from combining the electrostatic repulsions and dipolar attractions is a useful means to approximate the lattice parameter , although close to melting fluctuations lead to a larger value in the lattice parameter than this minimum . 
   conversely , at high field strengths , our treatment appears to be less accurate . 
   apparently , higher order terms which we have not accounted for may become important , such as non - linear poisson - boltzmann contributions to the electrostatic interactions , leading to a deviations from the yukawa form [ equation ( [ eqyuk ] ) ] @xcite or limitations in our pairwise treatment of the dipolar attractions @xcite . 
   other possibilities include inaccuracies in our assumptions for the effective colloid charge [ equation ( [ eqzeff ] ) ] , limitations of the 2d behaviour of the experimental system . perhaps due to a cancellation of errors , at high field strengths , 
   the minimum in the effective interaction can sometimes provide a more accurate value of the lattice parameter than the simulations . 
   another important assumption lies in our derivation on the electrostatic interactions . 
   while their yukawa - like form is well - accepted @xcite , the values the effective colloid charge comes from equation ( [ eqzeff ] ) . 
   we note that the debye length we determine seems rather constant across the four particle sizes considered , [ fig.[figkappasigma ] ( a ) ] , which suggests that out approach is at least consistent . 
   however challenging it may be , a more quantitative measurement of the local ion concentration in the vicinity of the crystalline rafts is desirable and will be considered in the future . 
   furthermore , there may be some variation in the value of @xmath16 . 
   in fact this should affect all state points in a similar manner ( for each particle size ) , and therefore we anticipate a similar outcome , but that a different value might be arrived at in the fitting of the debye length , however , owing to the relatively short - ranged nature of the interaction , the effect of a different value for the effective charge is unlikely to impinge significantly the debye length and should have a negligible effect on our main results . 
   we have also assumed that a given system is described by _ _ 
   one single__debye length . 
   in fact , in the counterion - dominated regime , the debye length is in fact a function of colloid concentration @xcite , an effect we have neglected . 
   this might lead to a tightening at higher colloid concentration ( i.e. high field strength ) , which may be expected to lead to an increase in deviation with the experimental results ( figs . 
   [ figsmallexpmc ] and [ figbigexpmc ] ) . 
   nevertheless , varying the debye length as a function of local colloid concentration would be worth considering in the future . 
   we have extended our approach a range of colloid size , to optimise the lattice tuneability for display applications . 
   tunable 2d crystal rafts can behave as tunable diffraction gratings capable of filtering white light into visible colours@xcite . 
   similar tunable diffraction has been proposed for display devices @xcite.the range of colours obtainable for a given geometry is governed by the lattice tunability . 
   fig 9b demonstrates that lattice tunability in our system is maximised for particles of approximately 1 to 1.5 microns . 
   only higher field strengths applied to smaller particles can increase the lattice tunability . 
   as already indicated , higher field strengths are not practical in the current experimental system , however in future experiments metal coated colloids would exhibit much stronger dipolar interactions , enabling a reduction in field strength for a given attraction . 
   regardless , our methodology illustrates using the well - developed machinery of effective colloidal interactions , as a means to model potentially complex interactions , useful for engineering purposes . 
   * acknowledgments * we thank jeremy baumberg , martin cryan , daan frenkel and junpei yamanaka for helpful discussions . 
   cpr acknowledges the royal society for funding and hajime tanaka for kind provision of lab space and computer time . 
   des thanks kodak european research centre , cambridge and epsrc ( ep / do33047/1 ) for funding . ) 
   .fitted parameters of the debye screening length used in the mc simulations for systems with different colloid diameters . 
   errors in the diameter are standard deviations in the data obtained found from electron microscopy measurements , while those in @xmath105 , @xmath81 , and the ionic strength are estimated from the fitting of @xmath2 from mc data.@xmath15 potential measurements were made in a 0.01 mmol kcl solution [ tablekappasigma ] [ cols="^,^,^,^,^,^ " , ]
summary: we compare the behaviour of a new 2d aqueous colloidal model system with a simple numerical treatment . to first order 
   the attractive interaction between the colloids induced by an in - plane rotating ac electric field is dipolar , while the charge stablisation leads to a shorter ranged , yukawa - like repulsion . in the crystal - like ` rafts ' formed at sufficient field strengths , we find quantitative agreement between experiment and monte carlo simulation , except in the case of strongly interacting systems , where the well depth of the effective potential exceeds 250 times the thermal energy . 
   the ` lattice constant ' of the crystal - like raft is located approximately at the minimum of the effective potential , resulting from the sum of the yukawa and dipolar interactions . 
   the experimental system has display applications , owing to the possibility of tuning the lattice spacing with the external electric field . 
   limitations in the applied field strength and relative range of the electrostatic interactions of the particles results in a reduction of tunable lattice spacing for small and large particles respectively . the optimal particle size for maximising the lattice spacing tunability 
   was found to be around 1000 nm .