colloidal suspensions in modulated light fields

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Colloidal suspensions in modulated light fields

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2008 J Phys Condens Matter 20 404220

(httpiopscienceioporg0953-89842040404220)

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IOP PUBLISHING JOURNAL OF PHYSICS CONDENSED MATTER

J Phys Condens Matter 20 (2008) 404220 (12pp) doi1010880953-89842040404220

Colloidal suspensions in modulated lightfieldsM C Jenkins and S U Egelhaaf

Condensed Matter Physics Laboratory Lehrstuhl fur Physik der weichen MaterieHeinrich-Heine-Universitat Dusseldorf Universitatstraszlige 1 D-40225 Dusseldorf Germany

E-mail matthewjenkinsuni-duesseldorfde

Received 13 May 2008 in final form 17 June 2008Published 10 September 2008Online at stacksioporgJPhysCM20404220

AbstractPeriodically modulated potentials in the form of light fields have previously been applied toinduce reversible phase transitions in dilute colloidal systems with long-range interactionsHere we investigate whether similar transitions can be induced in very dense systems whereinter-particle contacts are important Using microscopy we show that particles in such systemsare indeed strongly affected by modulated potentials We discuss technical aspects relevant togenerating the light-induced potentials and to simultaneously imaging the particles We alsoconsider what happens when the particle size is comparable to the modulation wavelength Theeffects of selected modulation wavelengths as well as pure radiation pressure are illustrated

(Some figures in this article are in colour only in the electronic version)

1 Introduction

One of the defining features of soft matter systems theirintermediate or mesoscopic lengthscale places them at aremarkable confluence of physical properties where they aresusceptible to influence by lsquoeverydayrsquo external forces Notonly are they lsquosoftrsquo in their response to applied shear andtheir constituents sufficiently buoyant to display an interestinginterplay between gravity and thermal (Brownian) motion [1]colloids are also affected by the radiation pressuremdashordinarilythought of as negligible in laboratory experimentsmdasharising asthey scatter light which impinges on them

The coincidence of the magnitude of these forces(Brownian gravitational and that arising from scattered EMradiation) in these systems set fundamentally by the relativemagnitudes of Boltzmannrsquos constant kb the gravitationalconstant G the speed of light c and Planckrsquos constant h iscompounded by a coincidence of the particle size with thewavelength of visible light The remarkable consequence isthat we are able to study in real space Brownian particlesunder the influence of light-induced potentials Ashkin iscredited with first realizing this possibility [2ndash7] leadingultimately to his development of optical tweezers [8] Theseare now a widely-used tool in physics [9] and biology [10ndash12]Studies using optical tweezers generally consider very deeppotential minima where objects are tightly trapped the presentwork considers in addition the case where potentials do

not necessarily irreversibly capture particles but do bias themtowards certain locations

Colloids have found favour as models of atomic systemsamong their advantages being the possibility of imagingdirectly large colloids with light microscopy the relativelyslow speed at which they diffuse and the relative ease withwhich interactions can be tailored [13 14] In the apparentlysimple case of hard-sphere interactions colloidal systemsshow the predicted entropy-driven fluidndashsolid transition athigh density [15] and at higher densities still a glasstransition instead of the expected crystalline equilibrium stateagain in analogy with atomic and molecular glass-formingsystems [15ndash19] Similar behaviour has been seen in othersystems for example charged particles interacting via long-ranged electrostatic repulsion [20]

Fluidndashsolid transitions are ubiquitous in nature and arefrequently the expected equilibrium state Often solid phasesoccur upon cooling whereupon distinct density modulationsare enhanced resulting in translational and orientationalorder Periodically modulated potentials can have a similareffect giving rise to a controllable phase transition effectedat will by the application and removal of an externalpotential Chowdhury was the first to demonstrate this intwo dimensions in so-called laser-induced freezing (LIF)experiments [21 22] These were followed by furtherexperimental [23ndash26] and theoretical [27ndash30] confirmationIn LIF a one-dimensionally modulated potential gives rise to

0953-898408404220+12$3000 copy 2008 IOP Publishing Ltd Printed in the UK1

J Phys Condens Matter 20 (2008) 404220 M C Jenkins and S U Egelhaaf

a two-dimensionally modulated crystal The registration ofadjacent rows is mediated by Brownian motion perpendicularto the direction of the potential which the particles continueto undergo although they sit on average at the minima ofthe potential For deeper potentials the lateral motion ofthe particles becomes sufficiently restricted that the systemis only modulated in one dimension but remains liquid-likealong the direction of the fringes laser-induced melting (LIM)occurs [31ndash39] There are many open questions relatingto light-induced phase transitions ranging from the kineticsof LIF and LIM to as yet untested predictions for systemswith more complex inter-particle potentials [40] time-varyingpotentials [41] and binary mixtures [42]

LIF and LIM both represent transitions between equilib-rium states brought about by the application of external laser-induced potentials In each of these cases removal of the po-tential leads to a re-establishment of the initial state In con-trast to reversible light-induced transitions there is the possi-bility that externally-applied modulated potentials may induceexpected but unobserved transitions to ordered states A primeexample is the hard-sphere glass transition described abovethough other glass and gel transitions may be relevant Dy-namical arrest due to co-operative effects such as caging is fre-quently offered as a possible explanation for the glass transi-tion eg [18 43 44] It is known that shear can lead to order-ing in glasses and gels [45ndash49] though the underlying mech-anism is still far from understood [50 51] (Note that shearcan also lead to melting of crystals [52 53]) If the emergingpicture of co-operative arrest is accurate and given the aboveevidence of external field-induced ordering it seems plausi-ble that periodic light fields might also induce ordering in verydense systems Biroli et al have developed an inhomogeneousmode-coupling theory (MCT) which predicts the response of asupercooled liquidrsquos dynamical structure factor when exposedto a static inhomogeneous potential [54] Agreement of thesecalculations with experimental results would represent a pow-erful test for MCT

This paper describes a first investigation into whetherperiodically modulated light fields can indeed induce an effectin very dense colloidal systems In a few initial steps towardsthe grand goals mentioned above we present some technicalaspects relating to the experimental realization of an apparatuswhich permits exposure of a sample to the light fields and itssimultaneous observation Our first results with this apparatusshow that even the most dense samples do show a clearrearrangement under the influence of an applied light field

2 Light as an external potential for colloidalparticles

Light forces acting on small particles have been describedin considerable detail both from a ray-optics and aRayleigh perspective The necessary basic physics is wellestablished [55ndash58] We outline the two features mostimportant to the present work the so-called scattering forceFscat which acts in the direction of beam propagation and thegradient force Fgrad which depends on the shape of the lightintensity distribution

21 Scattering force Fscat

Historically the scattering force preceded full optical trappingin the form of Ashkinrsquos levitation experiments [2ndash4] It arisesas a result of momentum transfer to a particle from incidentphotons as they are scattered from it

A photon of light with frequency ν wavelength λ andphase velocity vP carries energy Ephot = hν = hvPλ andmomentum p = hλ = EphotvP In absorbing a photonan object therefore experiences a force Fscat = partppart t =part(EphotvP)part t = PvP = n Pc where P is the powerof the photon source The last equality follows since thephase velocity of light in a material of refractive index nvP = cn where c is the speed of light in vacuo [8 9]1 In ourcase photons are scattered rather than absorbed and thereforetransfer only a portion of their momentum to the particle in thecase of dielectric spheres a prefactor q sim 01 is customarilyassumed [2 9]

In our experiments we have calculated Fscat to be on theorder of a tenth of the particlesrsquo own weight for each mW ofapplied laser power (as measured at the laser as always in thisarticle) in a typical experiment this corresponds to around atenfold increase in the particlesrsquo effective buoyant mass

22 Gradient force Fgrad

The gradient force Fgrad allows a single beam suitably focusedto act as an optical trap Its origin is intuitively clear in theray-optics formulation [9 58] although since we ultimatelyconsider the sum of forces over all infinitesimal volumeelements of finite-sized spheres we discuss the origin of thegradient force as it applies to Rayleigh particles In this regimethe electric field over a particle is approximately uniformand induces a dipole moment p = αE where α is thepolarizability of the (dielectric) particle given by the ClausiusndashMosotti relation [60]

α = n2s

(n2 minus 1

n2 + 2

)a3

with n = ncns the ratio of the refractive index of the colloidalparticle nc to that of the surrounding medium (solvent) nsand a the particlersquos radius Having acquired a dipole momentthe particle experiences a Lorentz force fL = (p middot nabla)E +(1c)dpdt times B This can be re-written [55]

fL = α

(nabla

(1

2E2

)+ 1

c

part

part t(E times B)

)

The second term is the scattering force Fscat discussed aboveand is directed along the direction of propagation The firstterm Fgrad = (α2)nablaE2 is the gradient force and expressesthat in a non-uniform electric field a particle will movetowards regions of higher electric field if nc gt ns and viceversa This indicates that the modulated potentials we seek canbe realized by spatially modulated electric fields

1 We have assumed the Minkowski form which differs from the Abrahamform by a factor n2 Although an important unresolved theoretical issue [59]this subtlety does not concern us unduly since optical forces can be calibratedexperimentally (and n2 sim 1)

2


Figure 1 Fringe pattern calculated for typical experimental parameters which result in a fringe spacing d = 929 μm (a) an observed fringepattern (b) and the magnitude of the gradient force calculated based on the fringe pattern shown on the left (c) Parameters are (section 31)wavelength λ = 532 nm crossing angle θ = 328 beam radius at the sample R = 122λ fD = 232 μm [21] for a laser beam of diameterD = 280 mm focused by a lens of focal length f = 200 mm

Figure 2 A vector plot showing the position-dependent forceexperienced by a particle found by numerical differentiation (see theappendix) of the calculated pattern in figure 1(a)

23 Modulated potentials from modulated light fields

One conceptually simple means of generating modulated lightfields is using a crossed beam experiment as employed inprevious studies similar to ours [22 33] and in thermaldiffusion forced Rayleigh scattering studies [61 62] Theexperimental arrangement is also similar to the so-called lsquodual-scatterrsquo or lsquodual-beamrsquo configurations used in laser Doppleranemometry [63 64] For two crossed coherent TEM00 mode(Gaussian profile) laser beams the resulting intensity profileis [24]

I (x) = 2 I01 + cos

[2kx sin(θ2)

]eminus2x2 cos2(θ2)R2

(1)

where θ is the beam crossing angle I0 the intensity ofeach beam R the laser beam radius (the eminus2 point) andk = 2πλ the incident beam wavevector The term inbraces 1 + cos[2kx sin(θ2)] = 1 + cos[qx] is thesinusoidally-varying interference pattern of fringe spacing d =λ(2 sin(θ2)) and fringe wavevector q = 2πd The lastterm represents the overlying Gaussian envelope due to thefinite beam size Figure 1 compares a calculated fringepattern (a) with an experimentally observed pattern (b) Fromthe gradient the magnitude of the force can be calculated

5 10 15 20

00

02

04

06

08

10

3j1(q

a)q

a

0 25qa

Figure 3 The factor 3 j1(qa)(qa) arising from the finite particleradius a (equation (2))

ie Fgrad = (α2)nablaE2 (c) These images illustrate thatnot only are particles drawn to the fringes but they are alsoconfined by the Gaussian envelope Figure 2 illustrates thisin the form of a vector field plot which was obtained bynumerical differentiation (see the appendix)

From the first term of the modulated light fieldequation (1) we obtain the oscillatory part of the potentialV (x) = V01 + cos[qx] where V0 absorbs the particlesrsquopolarizability and the laser beam intensity Loudiyi et aladditionally normalize this quantity by the thermal energy [24]

24 Effect of finite particle size

Implicit in all of the previous section was that the potentialacts only at the centre of the particles or alternatively that theparticle radius a d This is not satisfied in the present workFor finite particle size integration over infinitesimal volumeelements of the particle results in a modified potential [24]

V (x) = V0

1 + 3

j1(qa)

qacos

[qx

] (2)

where x is the position of the particle centre and j1 the firstorder spherical Bessel function

The finite size of the particles leads to the additionalfactor 3 j1(qa)(qa) (figure 3) This factor resembles the form

3


Figure 4 Schematic diagram of the experimental arrangement used to generate a modulated light field and simultaneously image the sampleTwo coherent beams are created by beamsplitter BS and subsequently brought parallel by two mirrors (M1 M2) and a pair of moveablemirrors The position of the moveable mirrors determines the beam separation s and therefore the beam crossing angle θ following focusinglens L1 For observation white illuminating light is combined from above using dichroic mirror D1 and the sample imaged by an objectiveThe majority of the intense laser light is deflected to beam dump BD using a second dichroic mirror D2 A series of three optical filters somerotatable in the objective infinity space allows for variable attenuation of the remaining laser light and thus for an adjustment of the brightnessof the fringes to be viewed Images are recorded after suitable magnification using a digital camera

factor of a sphere as it occurs in scattering experiments [65]In a scattering experiment the sample is illuminated by abeam and the intensity scattered under a scattering angle θ orequivalently a scattering vector q is determined This impliesan identical geometry with the incident and scattered beamhere represented by the two crossing beams

The additional factor demonstrates several interestingfeatures [66] First for large fringe spacing (qa 0) theparticles behave like point particles Conversely for very smallfringe spacing (qararrinfin) the effective potential is averaged tozero Interestingly there are fringe spacings where the signof the potential is reversed indicating that spherical particlescan be either drawn into the fringes or repelled from themdepending on qa As long as the fringe spacing is greater thand = 0699 times 2a (corresponding to qa = 4493 the first rootof j1(qa)) the factor 3 j1(qa)(qa) gt 0 This means that forfringe spacings larger than the particle diameter but still finite(and indeed slightly smaller too) the behaviour is qualitativelysimilar to that for point particles albeit with reduced potency

3 Experiment

31 Apparatus

Figure 4 shows the experimental arrangement used to createthe modulated potential and at the same time observe theresponse of the sample The modulated potential is createdby splitting a linearly-polarized laser beam (Coherent Verdi V5with P = 5 W λ = 532 nm) and subsequently crossing the twobeams The beam is split using a 5050 beamsplitter (BS) witha preceding half-wave plate (λ2) to adjust the polarizationfor optimum performance of the beamsplitter The two beamsare brought parallel to one another by means of two mirrors

(M1 M2) and a moveable pair of mirrors Translation of themoveable mirrors adjusts the beam separation s and after thefocusing lens (L1) the crossing angle θ of the two beamsA half-wave plate (λ2) in one of the beams allows rotationof the polarization of one beam with respect to the otherthereby controlling the amplitude of the interference fringeswhilst maintaining a constant mean intensity and thus radiationpressure

The introduction of the sample changes the crossing angleFor a typical sample cell (depth 170 μm) and fringe spacing(7 μm) the angle of incidence is reduced from θi2 =sinminus1(λ2d) = 22 to θr2 = sinminus1(sin(22)133) = 16corresponding to change in the focal position of 170 μm times(tan(θi2)tan(θr2) minus 1) 64 μm This is corrected bya linear translation of the lens L1 (figure 4) Note thatdespite the change in the crossing angle the fringe spacingremains unchanged (since λ also changes upon entering thenew medium)

Concurrently the sample is imaged with a home-builtinverted bright-field microscope Kohler illumination isprovided from above the sample The extremely long workingdistance condensing lens provides sufficient space for a lsquonotchrsquodichroic mirror (D1) reflective in a narrow range aroundλ = 532 nm but otherwise transmitting in the visible Thisdichroic mirror combines the imaging and modulated lightat the sample After the sample a standard high numericalaperture microscope objective (Nikon times100 PA VC NA =14) forms an image at infinity before a tube lens (TL)and subsequent telescope (L2 CO) adjust the magnificationas appropriate for the camera Additional optics can beintroduced straightforwardly into the (relatively long) so-calledinfinity space behind the objective We use this to separatethe intense laser light damaging to the camera from the

4


white imaging light The bulk (98) is deflected to a beamdump (BD) using a second dichroic mirror (D2) As wellas eliminating safely the majority of the laser light this lightcan be re-used for example by retro-reflection to achieve acounter-propagating arrangement whereby Fscat can be reducedindependently of the laser intensity Even after the seconddichroic mirror the intensity of the laser light is far too high forthe camera Three additional optical filters are used to adjustthe level of the modulated light field while retaining most ofthe imaging light This permits simultaneous imaging of thesample and an appropriate fraction of the light field

The intense laser light has to be reduced to about 10minus9ndash10minus8 W at the camera to obtain satisfactory images of thefringes The fringes become essentially invisible upon a furtherreduction by a factor of about 100 Since the laser powervaries depending on the experiment for optimum simultaneousimaging filters with a variable optical density at 532 nmOD532 are desirable This can be achieved for interferencefilters by changing the angle of incidence γ (figure 4) thefilters have a sharp transmittance edge slightly above 532 nmwhich shifts to lower wavelength as γ is increased (similarto notch filters [67]) Since the slope of the edge is finitethis provides control over OD532 (figure 5) When placedin the infinity space of the microscope these filters can bestraightforwardly rotated to allow the fringes to be imagedor not as desired The weak dependence of OD532 withinminus5 γ 5 (figure 5) is important for imaging sincethis range is slightly larger than the divergence in the infinityspace of the microscope2 Nevertheless for different parts ofthe field of view the effective γ and thus OD532 is differentand hence an image of the fringes is no longer quantitativelycorrect A correct image can however be obtained withγ = 0 or by using neutral density filters (in which case thewhite light is attenuated beyond usefulness) The bright-fieldimages remain good since the transmittance of each filter atwavelengths λ = 532 nm is T 09

32 Analysis

Having determined the particle coordinates [68 69] a range ofparameters can be calculated for example the particle densityφ the pair correlation function g(r) the mean coordinationnumber 〈z〉 the distribution of coordination numbers p(z) andbond-orientational order parameters eg ψ6 [70] Calibrationof distances necessary for determining the fringe spacing aswell as for structural analyses is performed using a high-resolution microscope test slide (Richardson Test Slide Model80303) [69]

33 Samples

We have used polystyrene sulfate spheres of radius a =2 μm (Interfacial Dynamics Corporation) suspended in water

2 Light from the focal plane is focused at infinity but except for lightoriginating from the point on the optical axis is nonetheless divergent Thefocal length of an objective is the microscope tube lens focal length (here about200 mm) divided by its magnification (here 100) Together with the radius ofthe field of view (here about 125μm) this results in a divergence in the infinityspace of the microscope of around tanminus1(125 μm2 mm) 36

-30 -20 -10 0 10 20 30

γ o-50

-40

-30

-20

-10

00

log 10

(PP

0) =

-O

D53

2

Figure 5 Variation of the filter optical density at 532 nm OD532with the angle of incidence γ P0 and P are the measured powervalues before and after the filters respectively

The large refractive index difference between particles (nc =159) and water (ns = 133) results in large optical gradientforces but the concomitant multiple scattering limits their useeffectively to a single layer ie two dimensions These spherescarry negative charges which in the present study are screenedby high salt concentrations We regard them as (almost)hard spheres which is supported by the observed distance ofclosest approach and the shape of the pair correlation function(section 52) Though the salt concentration is high it is stilllow enough to avoid problems with coagulation

Samples are prepared by pipetting a suitably dilutedhomogenized stock solution directly into the sample cellwhich fills largely by capillary action The particles quicklysediment onto the coverglass For dilute samples this resultsin two-dimensional samples while at higher concentrationsa few layers form (which can be reduced to a singlelayer by application of radiation pressure section 4) Theconcentrations we refer to in the following are the volumefractions φ of the initial homogenized bulk solutions Thisis a nominal value in the final sample inhomogeneities in thedensity may occur depending on the settling process

The sample cells consist of coverslips glued togethergiving a sample volume of about 20 mmtimes3 mmtimes170μm [69]The sample only comes into contact with glass and possiblythe UV-cure glue used to seal the cells whose effect isassumed negligible Since glass becomes negatively chargedin the presence of water the particles are repelled fromthe surfaces of the cell and become attached only veryoccasionally [71 72]

4 Radiation pressure results

We first investigate the effect of radiation pressure only as afunction of laser intensity and particle concentration (figure 6)The radiation pressure is applied by turning the second half-wave plate (λ2) until minimum contrast is achieved as judgedfrom images of the interference patterns formed using neutraldensity filters The concentrations are chosen such that a firstlayer of particles next to the coverslip (slightly out of focus in

5


Figure 6 The effect of pure radiation pressure as a function of sample concentration (from top to bottom initial homogenized volumefractions φ = 0015 0020 and 0030) and laser intensity (left to right P = 000 010 020 and 050 W) Images are taken following20 min of irradiation except for the top right image which shows the sample (φ = 0015 P = 010 W) 30 min after the laser is turned offNote that only the particles in the second layer are in focus they show a bright spot at their centre Particles in the first layer (most prominentin the image second from left top row) are out of focus but still clearly identifiable

figure 6) as well as an incomplete second layer (in focus) areformed

As the sample concentration increases (downwards infigure 6) the density of the second layer increases Thepresence of a second layer does not imply that the maximumpossible density has been achieved in the first layer Indeedthis is observed not to be the case upon increasing theradiation pressure particles are pushed from the second layerinto the first layer (left to right) In each of these cases theradiation pressure was applied for 20 min In the least densesample (top row) a laser intensity P = 010 W is alreadysufficient to insert all of the particles into the first layer (Thetwo highest laser intensity results for the lowest concentrationare omitted in figure 6) This forms a dense hexagonally-closepacked (HCP) layer With increasing density a greater laserintensity is required to insert all of the particles within thelaser beam into the first layer Beyond a certain density it isno longer possible to insert all of the particles into the firstlayer even for very large radiation pressures With increasingconcentration the area with only a single layer gets smallercorresponding to the Gaussian profile of the laser beam andthus the applied radiation pressure This can also be seen withincreasing laser intensity

The top rightmost image shows the least concentratedsample 30 min after the field is removed (Similar relaxationbehaviour is observed in all samples) The second layerhas become repopulated This indicates that the osmoticpressure experienced within the highly concentrated first layeris sufficient to cause particles to lsquopop uprsquo into the second layerThis also implies that the inability of particles to enter thefirst layer under their own weight cannot be explained by puregeometrical frustration We have observed that lsquopopping uprsquooccurs with a characteristic time of about 10 s If this upward

movement into the second layer is thermally driven ie isa chance Brownian excursion (and a return to the first layeris hindered by particle rearrangements within the first layer)its timescale should be given by Kramerrsquos escape time witha ramp potential of depth U0 and extent 2a representing thegravitational potential [49 73]

τ = 1

Ds

int 2a

0dx prime eβU(xprime )

int xprime

minusinfindx eminusβU(x)

= (2a)2

Ds

eminusβU0 minus (1 minus βU0)

(βU0)2

where β = 1kBT and Ds is the self-diffusion constant for afree particle Using appropriate parameters gives τ sim 106 s 1310 s indicating the presence of a driving force namely theosmotic pressure

This seems plausible in light of the established connectionbetween the statistical geometry of hard spheres and theirthermodynamic properties [74ndash76] These references suggestthat insertion of particles into a disordered layer by theapplication of radiation pressure should as in the Widominsertion method permit study of the thermodynamicproperties of particles in the first layer

5 Modulated potential results

Having established a two-dimensional sample we nowintroduce a modulated potential In the present study themodulation is always as great as possible at the specifiedlaser intensity ie both beams have the same directionof polarization In each experiment we prepare a densehexagonally-close packed (HCP) layer using radiation pressure(at the indicated power) before the half-wave plate is rotatedto lsquoturn onrsquo the modulation The parameters we vary are the

6


Figure 7 Sample (initial concentration φ = 0020) after 30 min of pure radiation pressure (a) followed by a further 30 min with a modulatedpotential of wavelength d = radic

3a (b) The laser intensity in both cases is P = 05 W and the superimposed line indicates the approximatefringe direction

0 30 60 90 120 150 180

Angle o

0

20

40

60

80

100

120

Occ

urre

nces

Figure 8 Distribution of the nearest-neighbour bond direction for asample (initial concentration φ = 0020) after 30 min of pureradiation pressure (solid (black) line) followed by a further 30 minwith a modulated potential of wavelength d = radic

3a (dashed (red)line) The laser intensity is P = 05 W

fringe separation and the amplitude of the modulation TheHCP symmetry suggests a few fringe spacings d (figure 12(a))here we investigate d = radic

3a (section 51) and d = 2radic

3a(section 52)

51 Natural fringe spacing

For a modulated potential of fringe spacing d = radic3a it is

possible for all particles forming an HCP layer to lie at thepotential minimum We thus consider this a natural fringespacing

Pure radiation pressure (a single beam of P = 050 Wfor 30 min) leads to randomly-oriented crystallites (figure 7left) After exposure to the modulated potential (P = 050 Wfor 30 min) the crystallites have rotated and consolidated toa near-perfect crystal with a clear direction aligned with thefringes (right) This is also reflected in the distribution of thenearest-neighbour bond direction which shows three strongpeaks separated by 60 (figure 8)

The crystallites thus seem to be able to rearrange despitethe high density It is interesting to investigate exactly howthis process occurs One observation is that as part of a

Figure 9 A dilute sample exposed to a modulated light field(P = 050 W) with spacing d = 2

radic3a

crystallite rotates the total energy in the light field does notdecrease monotonically until the particles are aligned withthe field At some angles ψ between the light field and thecrystal orientation relatively many particles are near to thepotential minima When the particles are aligned with the field(ψ = 0) all of the particle centres occupy a minimum Forangles ψ 18 31 and 42 there are only around 4060 and 40 of the particles in the minimum respectivelywhereas for in-between angles there are far fewer Supposinga large crystallite were to rotate towards the global minimumtherefore it may do so at varying speed perhaps even pausingat these intermediate metastable orientations depending on theamplitude of the field

52 Twice natural fringe spacing

We now consider a fringe spacing d = 2radic

3a correspondingto twice the spacing between two rows of an HCP layer Forsufficiently dilute samples the particles align along the fringes(figure 9)

In dense samples more complex structures develop(figure 10) The initially disordered sample (top left) develops

7


Figure 10 Micrographs of a sample (initial concentration φ = 0020) before irradiation (a) following 1 h of radiation pressure at a laserintensity of P = 040 W (b) and 100 s (c) and about 5 h (d) after the introduction of fringes with spacing d = 2

radic3a

randomly-oriented crystallites following the application ofradiation pressure (1 h of P = 040 W top right) as describedpreviously (section 4) Relatively soon after a modulation ofwavelength d = 2

radic3a is introduced (100 s still with P =

040 W bottom left) the sample is altered with the emergenceof voids which run broadly in the direction of the fringesAfter substantially more time (about 5 h bottom right) thefield has caused significant structural rearrangement In time-lapse movies of images groups of clusters can be seen movingco-operatively leading to arrangements along the potentialminima In particular the motif highlighted in figure 10(bottom right) occurs frequently with an orientation relativeto the fringes as indicated in figure 12(d) This rotation isunderstandable on energetic grounds which we discuss furtherbelow Other samples show similar behaviour

The structural evolution of the sample has been investi-gated more quantitatively by following the rearrangements in-duced by a modulated potential (figure 11) We determinedthe positions of particles which were located in a rectangu-lar region within the single-layer region and thus under theinfluence of the modulated potential Over the course of thewhole period the number of particles N(t) within the obser-vation region and thus the particle density steadily decreases(figure 11(a)) In addition a particlersquos average number ofneighbours 〈z(t)〉 drops from around 44 to 39 after 35 h (fig-ure 11(b)) This is also reflected in the distribution of the num-ber of neighbours p(z t) (figure 11(c)) which indicates an in-creasing probability of weakly connected particles consistentwith the appearance of voids along the fringes Although thenumber of neighbours decreases the bond-orientational orderparameter ψ6 (section 32) does not change significantly over

the course of the experiment (figure 11(d)) This indicates thatthose particles which remain bonded do so in a morpholog-ically similar way This is supported by the fact that the paircorrelation function g(r) is also essentially unaffected through-out the experiment (figure 11(e))

How can we understand these observations In dilutesamples all of the particles can be arranged in the potentialminima For the dense samples half of the particles can still liealong the minima (figure 12(a)) (section 51) but the remainingparticles are forced to lie between the fringes and thus at themaximum of the potential While the intensity gradient andhence the force is zero at the maximum this arrangement ismetastable with very small fluctuations inevitably resultingin large gradient forces These forces attempt to insertparticles into the minima ie the fringes (two such particlesare indicated as blue rings in figure 12(b)) and in so doingpush other particles along the fringes (as indicated by thearrows in the right-hand image in figure 12(b)) This isachieved without penalty provided the density at the end ofthe fringe is suitably low When the density of the sample islarge there is a significant osmotic penalty associated withpushing particles along the fringes and into the bulk Abalance must be struck between the optical gradient forceand the osmotic force which are opposed in their preferencefor density modulations This explains why the expectedmodulations in density are observed at low concentrations(figure 9) but not at very high concentrations At highdensities the system aims to accommodate as many particles aspossible within the fringe without significant extension alongthe fringe We have observed structures which achieve thisone example is that highlighted in figure 10 (bottom right) and

8


Figure 11 Effect of a modulated potential on the evolution ofdifferent parameters Shown are the time dependence of (a) thenumber of particles N(t) within the observation region (b) theaverage number of neighbours 〈z(t)〉 (c) the distribution of thenumber of neighbours p(z t) with time (direction of increasing timeindicated by arrow) (d) the bond-orientational order parameterψ6(t) and (e) the pair correlation function g(r t)

explained in figure 12(d) This rhombic lsquomotifrsquo represents apart of the crystal which after rotation through 30 reachesan energetically advantageous state (which depends on theprecise details of the potential see below) without a largeextension along the fringe direction The rearrangement ofsmall crystalline parts leaves bond orientations unchangedconsistent with the observed essentially constant ψ6 as well asleaving inter-particle distances largely unchanged This latterobservation is consistent with our finding that g(r) does notchange substantially What modest extension along the fringesthere is expels some particles in agreement with the decreasein the particle number N(t) and in turn the mean number ofneighbours 〈z(t)〉

These observations might have interesting consequencesFirst if what we observe are equilibrium structures it isremarkable that they form via small crystalline parts which arebroken away and simply reoriented with respect to the appliedpotential It is however also conceivable that due to thegeometrical frustration in a dense system these co-operativemotions are the only means by which the system can rearrangeIn this case the observed structures would correspond tonon-equilibrium states liable to further evolution indeed theevolution of particle number N(t) and mean coordinationnumber 〈z(t)〉 suggests that the samples are still evolving(figures 11(a) (b)) Whether equilibrium or not it is clearthat the modulated potential has a profound effect even inthese dense samples Our experiments also suggest that atintermediate (in the present context though these are stillrelatively very dense samples) concentrations novel structuresmight form due to the competition between the imposedpotential which favours density modulations and the osmoticpressure of the system which opposes them

Which structure is energetically or kinetically preferabledepends on the shape of the potential For example for asquare-well potential the particles can to some extent movelaterally within the fringe without penalty Depending on thepotential width and separation a wealth of structures has beenpredicted for this case [77] Although in that study the colloidsremain near to one another due to mutual attraction ratherthan osmotic pressure (as in our case) the effect is seeminglysimilar For a potential with monotonically increasingcurvature eg a quadratic potential it is advantageous todisplace particles from the minimum as little as possiblelsquozig-zagrsquo lines are expected In the present case howeverthe curvature of the potential is non-monotonic and it seemsreasonable that some particles maintain their position whileothers are significantly displaced from the potential minimumTogether with the influence of the osmotic pressure due to thebulk sample this energy-minimization argument justifies theexistence of the observed motifs

6 Conclusion

We have described an apparatus used to expose a sample tosinusoidally-varying light fields and simultaneously image thesample To demonstrate its capabilities we have investigatedthe response of colloidal particles to the modulated potentialswhich arise from the light field We have shown that

9


Figure 12 (a) Hexagonally-close packed (HCP) layer of particles with radius a and inter-layer spacingradic

3a (b) Particles located along thefringes (minima) are stable while those at the maxima (two of which are shown in the figure as blue rings) are metastable and as a result offluctuations experience a force toward the fringes They can join a minimum if particles which are already present in the minimum canadvance along it (arrows in the right-hand image) (c) When this is hindered the particles can locally rearrange (eg rotate by 30) to adoptmore favourable structures

these potentials influence even samples dense enough that thedynamics of their constituent particles are severely restrictedCurrently we are further improving the apparatus by includinga counter-propagating beam which will allow us to controlthe modulated potential and radiation pressure independentlyThis will be achieved by replacing the beam dump by a retro-reflector

Densely packed effectively two-dimensional samples havebeen generated using radiation pressures of different intensityThe behaviour of these samples upon exposure to modulatedpotentials has been investigated for two different modulationwavelengths This has revealed co-operative structuralrearrangements and final structures which seem to result from acompetition between the optical gradient force and the osmoticpressure of the bulk sample While theoretical predictions for asinusoidal potential are lacking similar theoretical calculationssuggest structures comparable to those we have observed

With this apparatus we can now investigate differentsituations first for disorder-to-order transitions specificpredictions exist for binary hard disc mixtures under similarconditions to those described here [42] Second disorder-to-disorder transitions are expected for systems with attractiveinteractions exposed to modulated potentials [40] Boththese transitions represent reversible transitions starting fromequilibrium states in which the initial states are recoveredon removal of the modulated potential In contrast ina third situation high-density non-equilibrium systems inparticular repulsive and attractive glasses might undergoirreversible transitions from their non-equilibrium state toan ordered equilibrium state upon exposure to a modulatedpotential In this case structural rearrangements lead to stableconfigurations that persist even after removal of the externalpotential In addition to revealing new physics this might alsohave implications for material sciences

Acknowledgments

We thank Hartmut Lowen Wilson Poon and Richard Hanesfor helpful discussions We also thank Jurgen Liebetrau fortechnical assistance and Beate Moser for help in preparing thediagrams This work was funded by the Deutsche Forschungs-gemeinshaft (DFG) within the GermanndashDutch Collaborative

Research Centre Sonderforschungsbereich-Transregio 6 (SFB-TR6) Project Section C7

Appendix Numerical differentiation of patterns

The calculated intensity profile was differentiated numericallyusing the Sobel method [78 section 713] to obtain anapproximation to the force field experienced by the particlesIf f is the image then the gradient of the image

nablaf =[

Gx

G y

]=

[part fpartxpart fparty

]

with magnitude |nablaf| = (G2x + G2

y)12 and direction ϕ(x y) =

tanminus1(G yGx) is formed by convolution of the image with thefollowing kernels

References

[1] Haw M D 2002 Colloidal suspensions Brownian motionmolecular reality a short history J Phys Condens Matter14 7769ndash79

[2] Ashkin A 1970 Acceleration and trapping of particles byradiation pressure Phys Rev Lett 24 156ndash9

[3] Ashkin A and Dziedzic J M 1971 Optical levitation byradiation pressure Appl Phys Lett 19 283ndash5

[4] Ashkin A and Dziedzic J M 1974 Stability of optical levitationby radiation pressure Appl Phys Lett 24 586ndash8

[5] Ashkin A 1980 Applications of laser radiation pressureScience 210 1081ndash8

[6] Smith P W Ashkin A and Tomlinson W J 1981 Four-wavemixing in an artificial Kerr medium Opt Lett 6 284ndash6

[7] Ashkin A Dziedzic J M and Smith P W 1982 Continuous-waveself-focusing and self-trapping of light in artificialKerr media Opt Lett 7 276ndash8

[8] Ashkin A Dziedzic J M Bjorkholm J E and Chu S 1986Observation of a single-beam gradient force optical trap fordielectric particles Opt Lett 11 288ndash90

[9] Molloy J E and Padgett M J 2002 Lights action opticaltweezers Contemp Phys 43 241ndash58

10


[10] Svoboda K and Block S M 1994 Biological applications ofoptical forces Annu Rev Biophys Biomol Struct 23 247ndash85

[11] Sheetz M P 1998 Laser Tweezers in Cell Biology (Methods inCell Biology) (New York Academic)

[12] Greulich K O 1999 Micromanipulation by Light in Biology andMedicine (Berlin Springer)

[13] Pusey P N 1991 Liquids Freezing and Glass Transition(Amsterdam Elsevier) chapter 10 (Colloidal Suspensions)pp 763ndash942

[14] Poon W C-K 2002 The physics of a model colloidndashpolymermixture J Phys Condens Matter 14 R859ndash80

[15] Pusey P N and van Megan W 1986 Phase behaviour ofconcentrated suspensions of nearly hard colloidal spheresNature 320 340ndash2

[16] Pusey P N and van Megan W 1987 Observation of a glasstransition in suspensions of spherical colloidal particlesPhys Rev Lett 59 2083ndash6

[17] van Megen W and Underwood S M 1993 Dynamic-light-scattering study of glasses of hard colloidal spheresPhys Rev E 47 248ndash61

[18] Weeks E R Crocker J C Levitt A C Schofield A B andWeitz D A 2000 Three-dimensional direct imaging ofstructural relaxation near the colloidal glass transitionScience 287 627ndash31

[19] Ferrer M L Lawrence C Demirjian B G Kivelson DAlba-Simionesco C and Tarjus G 1998 Supercooled liquidsand the glass transition temperature as the control variableJ Chem Phys 109 8010ndash5

[20] Hartl W Versmold H and Zhang-Heider X 1995 The glasstransition of charged polymer colloids J Chem Phys102 6613ndash8

[21] Chowdhury A H 1986 Laser induced freezing PhD ThesisOklahoma State University

[22] Chowdhury A and Ackerson B J 1985 Laser-induced freezingPhys Rev Lett 55 833ndash7

[23] Ackerson B J and Chowdhury A H 1987 Radiation pressure asa technique for manipulating the particle order in colloidalsuspensions Faraday Discuss Chem Soc 83 309ndash16

[24] Loudiyi K and Ackerson B J 1992 Direct observation of laserinduced freezing Physica A 184 1ndash25

[25] Loudiyi K and Ackerson B J 1992 Monte Carlo simulation oflaser induced freezing Physica A 184 26ndash41

[26] Wei Q-H Bechinger C Rudhardt D and Leiderer P 1998Structure of two-dimensional colloidal systems under theinfluence of an external modulated light field Prog ColloidPolym Sci 110 46ndash9

[27] Xu H and Baus M 1986 Freezing in the presence of a periodicexternal potential Phys Lett A 117 127ndash31

[28] Barrat J L and Xu H 1990 The phase diagram of hard spheres ina periodic external potential J Phys Condens Matter2 9445ndash50

[29] Chakrabarti J Krishnamurthy H R and Sood A K 1994 Densityfunctional theory of laser-induced freezing in colloidalsuspensions Phys Rev Lett 73 2923ndash6

[30] Sood A K 1996 Some novel states of colloidal mattermodulated liquid modulated crystal and glass Physica A224 34ndash47

[31] Chakrabarti J Krishnamurthy H R Sood A K andSengupta S 1995 Reentrant melting in laser field modulatedcolloidal suspensions Phys Rev Lett 75 2232ndash5

[32] Wei Q-H Bechinger C Rudhardt D and Leiderer P 1998Experimental study of laser-induced melting intwo-dimensional colloids Phys Rev Lett 81 2606ndash9

[33] Bechinger C Wei Q H and Leiderer P 2000 Reentrant meltingof two-dimensional colloidal systems J Phys CondensMatter 12 A425ndash30

[34] Bechinger C Brunner M and Leiderer P 2001 Phase behaviorof two-dimensional colloidal systems in the presence ofperiodic light fields Phys Rev Lett 86 930ndash3

[35] Bechinger C and Frey E 2001 Phase behaviour of colloids inconfining geometry J Phys Condens Matter 13 R321ndash36

[36] Bechinger C 2002 Colloidal suspensions in confinedgeometries Curr Opin Colloid Interface Sci 7 204ndash9

[37] Strepp W Sengupta S and Nielaba P 2001 Phase transitions ofhard disks in external potentials a Monte Carlo study PhysRev E 63 046106

[38] Strepp W Sengupta S and Nielaba P 2002 Phase transitions ofsoft disks in external potentials a Monte Carlo study PhysRev E 66 056109

[39] Strepp W Sengupta S Lohrer M and Nielaba P 2002 Phasetransitions of hard and soft disks in external periodicpotentials a Monte Carlo study Comput Phys Commun147 370ndash3

[40] Gotze I O Brader J M Schmidt M and Lowen H 2003Laser-induced condensation in colloidndashpolymer mixturesMol Phys 101 1651ndash8

[41] Rex M Lowen H and Likos C N 2005 Soft colloids driven andsheared by traveling wave fields Phys Rev E 72 021404

[42] Franzrahe K and Nielaba P 2007 Entropy versus energy thephase behavior of a hard-disk mixture in a periodic externalpotential Phys Rev E 76 061503

[43] Pham K N Puertas A M Bergenholtz J Egelhaaf S UMoussaıd A Pusey P N Schofield A B Cates M EFuchs M and Poon W C K 2002 Multiple glassy states in asimple model system Science 296 104ndash6

[44] Gotze W and Sjogren L 1992 Relaxation processes insupercooled liquids Rep Prog Phys 55 241ndash376

[45] Ackerson B J and Pusey P N 1988 Shear-induced order insuspensions of hard spheres Phys Rev Lett 61 1033ndash6

[46] Haw M D Poon W C K Pusey P N Hebraud P andLequeux F 1998 Colloidal glasses under shear strainPhys Rev E 58 4673ndash82

[47] Haw M D Poon W C K and Pusey P N 1998 Direct observationof oscillatory-shear-induced order in colloidal suspensionsPhys Rev E 57 6859ndash64

[48] Vermant J and Solomon M J 2005 Flow-induced structure incolloidal suspensions J Phys Condens Matter17 R187ndash216

[49] Smith P A Petekidis G Egelhaaf S U and Poon W C K 2007Yielding and crystallization of colloidal gels underoscillatory shear Phys Rev E 76 041402

[50] Besseling R Weeks E R Schofield A B and Poon W C K 2007Three-dimensional imaging of colloidal glasses under steadyshear Phys Rev Lett 99 028301

[51] Koumakis N Schofield A B and Petekidis G 2008 Effects ofshear-induced crystallization on the rheology and ageing ofhard sphere glasses Preprint 08041218

[52] Ackerson B J and Clark N A 1981 Shear-induced meltingPhys Rev Lett 46 123ndash7

[53] Stevens M J Robbins M O and Belak J F 1991 Shear meltingof colloids a nonequilibrium phase diagram Phys Rev Lett66 3004ndash7

[54] Biroli G Bouchaud J-P Miyazaki K and Reichman D R 2006Inhomogeneous mode-coupling theory and growing dynamiclength in supercooled liquids Phys Rev Lett 97 195701

[55] Gordon J P 1973 Radiation forces and momenta in dielectricmedia Phys Rev A 8 14ndash21

[56] Harada Y and Asakura T 1996 Radiation forces on a dielectricsphere in the Rayleigh scattering regime Opt Commun124 529ndash41

[57] Tlusty T Meller A and Bar-Ziv R 1998 Optical gradient forcesof strongly localized fields Phys Rev Lett 81 1738ndash41

[58] Ashkin A 1992 Forces of a single-beam gradient laser trap on adielectric sphere in the ray optics regime Biophys J61 569ndash82

[59] Leonhardt U 2006 Momentum in uncertain light Nature444 823ndash4

[60] Jackson J D 1975 Classical Electrodynamics 2nd edn(New York Wiley)

11


[61] Wiegand S 2004 Thermal diffusion in liquid mixtures andpolymer solutions J Phys Condens Matter 16 R357ndash79

[62] Kohler W and Schafer R 2000 Polymer analysis bythermal-diffusion forced Rayleigh scattering Adv PolymSci 151 1ndash59

[63] Brayton D B and Goethert W H 1971 A new dual-scatter laserDoppler-shift velocity measuring technique ISA Trans10 40ndash50

[64] Durst F Melling A and Whitelaw J H 1976 Principles andPractice of Laser-Doppler anemometry (New YorkAcademic)

[65] Lindner P and Zemb T (ed) 2002 Neutrons X-rays and LightScattering Methods Applied to Soft Condensed Matter(Amsterdam Elsevier)

[66] Chowdhury A H Wood F K and Ackerson B J 1991 Transverseradiation pressure forces for finite sized colloidal particlesOpt Commun 86 547ndash54

[67] Semrock Inc Notch filter spectra versus angle of incidencehttpwwwsemrockcomCatalogNotch SpectrumvsAOIhtm (obtained March 2008)

[68] Crocker J C and Grier D G 1996 Methods of digital videomicroscopy for colloidal studies J Colloid Interface Sci179 298ndash310

[69] Jenkins M C and Egelhaaf S U 2008 Confocal microscopy ofcolloidal particles towards reliable optimum coordinatesAdv Colloid Interface Sci 136 65ndash92

[70] de Villeneuve V W A Dullens R P A Aarts D G A LGroeneveld E Scherff J H Kegel W K andLekkerkerker H N W 2005 Colloidal hard-sphere crystalgrowth frustrated by large spherical impurities Science309 1231ndash3

[71] Prieve D C and Loo F 1987 Brownian motion of a hydrosolparticle in a colloidal force field Faraday Discuss ChemSoc 83 297ndash307

[72] Prieve D C 1999 Measurement of colloidal forces with TIRMAdv Colloid Interface Sci 82 93ndash125

[73] Kramers H A 1940 Brownian motion in a field of force and thediffusion model of chemical reactions Physica 7 284ndash304

[74] Dullens R P A Aarts D G A L and Kegel W K 2006 Directmeasurement of the free energy by optical microscopyProc Natl Acad Sci USA 103 529ndash31

[75] Dullens R P A Aarts D G A L Kegel W K andLekkerkerker H N W 2005 The Widom insertion method andordering in small hard-sphere systems Mol Phys103 3195ndash200

[76] Widom B 1963 Some topics in the theory of fluids J ChemPhys 39 2808ndash12

[77] Harreis H M Schmidt M and Lowen H 2002 Decorationlattices of colloids adsorbed on stripe-patterned substratesPhys Rev E 65 041602

[78] Gonzalez R C and Woods R E 1992 Digital Image Processing(Reading MA Addison-Wesley)

12

1 Introduction

2 Light as an external potential for colloidal particles

21 Scattering force F_scat

22 Gradient force F_grad



3 Experiment

31 Apparatus

32 Analysis

33 Samples





6 Conclusion

Acknowledgments


References

IOP PUBLISHING JOURNAL OF PHYSICS CONDENSED MATTER

J Phys Condens Matter 20 (2008) 404220 (12pp) doi1010880953-89842040404220

Colloidal suspensions in modulated lightfieldsM C Jenkins and S U Egelhaaf

Condensed Matter Physics Laboratory Lehrstuhl fur Physik der weichen MaterieHeinrich-Heine-Universitat Dusseldorf Universitatstraszlige 1 D-40225 Dusseldorf Germany

E-mail matthewjenkinsuni-duesseldorfde

Received 13 May 2008 in final form 17 June 2008Published 10 September 2008Online at stacksioporgJPhysCM20404220

AbstractPeriodically modulated potentials in the form of light fields have previously been applied toinduce reversible phase transitions in dilute colloidal systems with long-range interactionsHere we investigate whether similar transitions can be induced in very dense systems whereinter-particle contacts are important Using microscopy we show that particles in such systemsare indeed strongly affected by modulated potentials We discuss technical aspects relevant togenerating the light-induced potentials and to simultaneously imaging the particles We alsoconsider what happens when the particle size is comparable to the modulation wavelength Theeffects of selected modulation wavelengths as well as pure radiation pressure are illustrated

(Some figures in this article are in colour only in the electronic version)

1 Introduction

One of the defining features of soft matter systems theirintermediate or mesoscopic lengthscale places them at aremarkable confluence of physical properties where they aresusceptible to influence by lsquoeverydayrsquo external forces Notonly are they lsquosoftrsquo in their response to applied shear andtheir constituents sufficiently buoyant to display an interestinginterplay between gravity and thermal (Brownian) motion [1]colloids are also affected by the radiation pressuremdashordinarilythought of as negligible in laboratory experimentsmdasharising asthey scatter light which impinges on them

The coincidence of the magnitude of these forces(Brownian gravitational and that arising from scattered EMradiation) in these systems set fundamentally by the relativemagnitudes of Boltzmannrsquos constant kb the gravitationalconstant G the speed of light c and Planckrsquos constant h iscompounded by a coincidence of the particle size with thewavelength of visible light The remarkable consequence isthat we are able to study in real space Brownian particlesunder the influence of light-induced potentials Ashkin iscredited with first realizing this possibility [2ndash7] leadingultimately to his development of optical tweezers [8] Theseare now a widely-used tool in physics [9] and biology [10ndash12]Studies using optical tweezers generally consider very deeppotential minima where objects are tightly trapped the presentwork considers in addition the case where potentials do

not necessarily irreversibly capture particles but do bias themtowards certain locations

Colloids have found favour as models of atomic systemsamong their advantages being the possibility of imagingdirectly large colloids with light microscopy the relativelyslow speed at which they diffuse and the relative ease withwhich interactions can be tailored [13 14] In the apparentlysimple case of hard-sphere interactions colloidal systemsshow the predicted entropy-driven fluidndashsolid transition athigh density [15] and at higher densities still a glasstransition instead of the expected crystalline equilibrium stateagain in analogy with atomic and molecular glass-formingsystems [15ndash19] Similar behaviour has been seen in othersystems for example charged particles interacting via long-ranged electrostatic repulsion [20]

Fluidndashsolid transitions are ubiquitous in nature and arefrequently the expected equilibrium state Often solid phasesoccur upon cooling whereupon distinct density modulationsare enhanced resulting in translational and orientationalorder Periodically modulated potentials can have a similareffect giving rise to a controllable phase transition effectedat will by the application and removal of an externalpotential Chowdhury was the first to demonstrate this intwo dimensions in so-called laser-induced freezing (LIF)experiments [21 22] These were followed by furtherexperimental [23ndash26] and theoretical [27ndash30] confirmationIn LIF a one-dimensionally modulated potential gives rise to

0953-898408404220+12$3000 copy 2008 IOP Publishing Ltd Printed in the UK1













α = n2s

(n2 minus 1

n2 + 2

)a3


fL = α

(nabla

(1

2E2

)+ 1

c

part

part t(E times B)

)



2






I (x) = 2 I01 + cos

[2kx sin(θ2)


(1)


5 10 15 20

00

02

04

06

08

10

3j1(q

a)q

a

0 25qa






V (x) = V0

1 + 3

j1(qa)

qacos

[qx

] (2)



3





3 Experiment

31 Apparatus





4




32 Analysis


33 Samples



-30 -20 -10 0 10 20 30

γ o-50

-40

-30

-20

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00

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(PP

0) =

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τ = 1

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int 2a


int xprime


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Ds


(βU0)2





6




0 30 60 90 120 150 180

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0

20

40

60

80

100

120

Occ

urre

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3a(section 52)







radic3a






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9







Acknowledgments





nablaf =[

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G y

]=


]




References










10





















































11




















12

1 Introduction






3 Experiment

31 Apparatus

32 Analysis

33 Samples





6 Conclusion

Acknowledgments


References













α = n2s

(n2 minus 1

n2 + 2

)a3


fL = α

(nabla

(1

2E2

)+ 1

c

part

part t(E times B)

)



2






I (x) = 2 I01 + cos

[2kx sin(θ2)


(1)


5 10 15 20

00

02

04

06

08

10

3j1(q

a)q

a

0 25qa






V (x) = V0

1 + 3

j1(qa)

qacos

[qx

] (2)



3





3 Experiment

31 Apparatus





4




32 Analysis


33 Samples



-30 -20 -10 0 10 20 30

γ o-50

-40

-30

-20

-10

00

log 10

(PP

0) =

-O

D53

2







5







τ = 1

Ds

int 2a


int xprime


= (2a)2

Ds


(βU0)2





6




0 30 60 90 120 150 180

Angle o

0

20

40

60

80

100

120

Occ

urre

nces





3a(section 52)







radic3a






7



radic3a







8






6 Conclusion


9







Acknowledgments





nablaf =[

Gx

G y

]=


]




References










10





















































11




















12

1 Introduction






3 Experiment

31 Apparatus

32 Analysis

33 Samples





6 Conclusion

Acknowledgments


References






I (x) = 2 I01 + cos

[2kx sin(θ2)


(1)


5 10 15 20

00

02

04

06

08

10

3j1(q

a)q

a

0 25qa






V (x) = V0

1 + 3

j1(qa)

qacos

[qx

] (2)



3





3 Experiment

31 Apparatus





4




32 Analysis


33 Samples



-30 -20 -10 0 10 20 30

γ o-50

-40

-30

-20

-10

00

log 10

(PP

0) =

-O

D53

2







5







τ = 1

Ds

int 2a


int xprime


= (2a)2

Ds


(βU0)2





6




0 30 60 90 120 150 180

Angle o

0

20

40

60

80

100

120

Occ

urre

nces





3a(section 52)







radic3a






7



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6 Conclusion


9







Acknowledgments





nablaf =[

Gx

G y

]=


]




References










10





















































11




















12

1 Introduction






3 Experiment

31 Apparatus

32 Analysis

33 Samples





6 Conclusion

Acknowledgments


References





3 Experiment

31 Apparatus





4




32 Analysis


33 Samples



-30 -20 -10 0 10 20 30

γ o-50

-40

-30

-20

-10

00

log 10

(PP

0) =

-O

D53

2







5







τ = 1

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int 2a


int xprime


= (2a)2

Ds


(βU0)2





6




0 30 60 90 120 150 180

Angle o

0

20

40

60

80

100

120

Occ

urre

nces





3a(section 52)







radic3a






7



radic3a







8






6 Conclusion


9







Acknowledgments





nablaf =[

Gx

G y

]=


]




References










10





















































11




















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1 Introduction






3 Experiment

31 Apparatus

32 Analysis

33 Samples





6 Conclusion

Acknowledgments


References




32 Analysis


33 Samples



-30 -20 -10 0 10 20 30

γ o-50

-40

-30

-20

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log 10

(PP

0) =

-O

D53

2







5







τ = 1

Ds

int 2a


int xprime


= (2a)2

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(βU0)2





6




0 30 60 90 120 150 180

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0

20

40

60

80

100

120

Occ

urre

nces





3a(section 52)







radic3a






7



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8






6 Conclusion


9







Acknowledgments





nablaf =[

Gx

G y

]=


]




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10





















































11




















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1 Introduction






3 Experiment

31 Apparatus

32 Analysis

33 Samples





6 Conclusion

Acknowledgments


References







τ = 1

Ds

int 2a


int xprime


= (2a)2

Ds


(βU0)2





6




0 30 60 90 120 150 180

Angle o

0

20

40

60

80

100

120

Occ

urre

nces





3a(section 52)







radic3a






7



radic3a







8






6 Conclusion


9







Acknowledgments





nablaf =[

Gx

G y

]=


]




References










10





















































11




















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1 Introduction






3 Experiment

31 Apparatus

32 Analysis

33 Samples





6 Conclusion

Acknowledgments


References




0 30 60 90 120 150 180

Angle o

0

20

40

60

80

100

120

Occ

urre

nces





3a(section 52)







radic3a






7



radic3a







8






6 Conclusion


9







Acknowledgments





nablaf =[

Gx

G y

]=


]




References










10





















































11




















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1 Introduction






3 Experiment

31 Apparatus

32 Analysis

33 Samples





6 Conclusion

Acknowledgments


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radic3a







8






6 Conclusion


9







Acknowledgments





nablaf =[

Gx

G y

]=


]




References










10





















































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1 Introduction






3 Experiment

31 Apparatus

32 Analysis

33 Samples





6 Conclusion

Acknowledgments


References






6 Conclusion


9







Acknowledgments





nablaf =[

Gx

G y

]=


]




References










10





















































11




















12

1 Introduction






3 Experiment

31 Apparatus

32 Analysis

33 Samples





6 Conclusion

Acknowledgments


References







Acknowledgments





nablaf =[

Gx

G y

]=


]




References










10





















































11




















12

1 Introduction






3 Experiment

31 Apparatus

32 Analysis

33 Samples





6 Conclusion

Acknowledgments


References





















































11




















12

1 Introduction






3 Experiment

31 Apparatus

32 Analysis

33 Samples





6 Conclusion

Acknowledgments


References




















12

1 Introduction






3 Experiment

31 Apparatus

32 Analysis

33 Samples





6 Conclusion

Acknowledgments


References

colloidal suspensions in modulated light fields

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