Generating an in situ tunable interaction potential for probing 2-D colloidal phase behavior

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Department of Chemical and Biomolecular E

MS 362, Houston, TX 77005, USA. E-mail: b

† Electronic supplementary informa10.1039/c3sm27620a

Cite this: DOI: 10.1039/c3sm27620a

Received 14th November 2012Accepted 7th February 2013

DOI: 10.1039/c3sm27620a

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Generating an in situ tunable interaction potential forprobing 2-D colloidal phase behavior†

Di Du, Dichuan Li, Madhuri Thakur and Sibani Lisa Biswal*

We present a novel method to tune the interaction potential from 5kBT to 40kBT in situ between micron-

sized superparamagnetic colloids. The potential is composed of a highly tunable long-range attraction

induced via a rotating magnetic field and a short-range electrostatic repulsion. Various 2-D

thermodynamic phases are observed in a colloidal suspension at different field strengths. Quantitative

agreement is found between theory and experiments for dipolar interactions. A theory to account for

the induced dipole due to neighboring particles is also presented. The effective three-body potential of

a dilute trimer system is measured to account for many-body effects in the system. These results

demonstrate an ideal model to study the phase behavior in 2-D systems.

Introduction

Controlling the interaction potential between colloids has led toordered systems with well-dened thermodynamics, allowingcolloids to be used as “model atoms”. Micron-sized colloids aresmall enough that thermal energy drives their dynamics. Thisallows colloids to be assembled into models for the classical gas,liquid, and solid phases.1 Direct microscopic visualization of thecolloids allows for their instantaneous locations to be trackedprecisely to provide “atomic” resolution, allowing colloidalsuspensions to be utilized to mimic both the equilibrium anddynamic behavior of their atomic counterparts. The ability todirectly measure the pair interaction potential has allowed for theunderstanding2,3 and prediction4,5 of bulk phase behavior, leadingto materials with specic optical or mechanical properties.

There continues to be growing interest in 2-D colloidal phasebehavior. The simplest example of a 2-D system is partiallywetting colloids trapped irreversibly at uid interfaces, whichhas been applied to the understanding of Pickering emulsions.6

In particular, since the thermodynamics of condensed mattersystems depends on their spatial dimension, 2-D systems haveled to new phenomena that are not observed in 3-D systems.Controlling these systems requires precise tuning of the effec-tive interactions. The majority of methods rely on repulsiveinteraction potentials, such as electrostatic dipole–dipolerepulsion,7,8 electrostatic ion–ion repulsion,9 magnetic dipole–dipole repulsion,10,11 and steric repulsion.12 Tuning attractiveinteractions has been more challenging and short-range,depletion-induced attractive interactions have been utilized in

ngineering, Rice University, 6100 Main St.

iswal@rice.edu; Tel: +1 713-348-6055

tion (ESI) available. See DOI:

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2-D systems.13 Recently, adjusting the in situ volume fraction ofa system has been achieved by using colloids consisting of athermally sensitive polymer microgel whose size can be tunedby temperature.14,15 However, tuning a long-range, isotropicattractive interaction potential over several kBT in 2-D systems islargely missing.

Here we present a method to generate an in situ isotropicpotential in a 2-D colloidal system using micron-sized super-paramagnetic colloidal particles dispersed in aqueous solution.In a uniform magnetic eld, superparamagnetic particlesacquire dipole moments, which interact to form chains ofcolloidal particles.16 This interaction potential is anisotropic:particles are attractive in the direction of the magnetic eld, butrepulsive perpendicular to the eld. A rotational magnetic eldcan be used to average this anisotropy. Once the eld isremoved, the chains redisperse into a suspension of Brownianparticles. A triaxial magnetic eld consisting of a rotating andnormal magnetic eld leads to a 3-D isotropic potential,resulting in 3-D colloidal networks or exible colloidalmembranes.17 Additionally, due to the close analogy tomagnetic elds, a rotational electric eld has also been intro-duced into the 2-D colloidal system to form colloidal crystals.18

The lattice constant of the crystal can be tuned by the magni-tude of the electric eld; however, the large elds required limitthe range of potentials that can be probed. Additionally, thecoupling between electrohydrodynamics and the dipolar inter-actions leads to difficulty in describing the exact mechanismgoverning charged colloids near an electrode in an AC electriceld,19 making the interaction potential of this system difficultto characterize. The method we employ uses a rotationalmagnetic eld. This potential between a pair of paramagneticparticles can be precisely tuned from 5kBT to 40kBT by varyingthe strength of the magnetic eld. Different phases formed in acolloidal suspension of paramagnetic particles show interesting

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thermodynamic behavior, conrming that this novel systemprovides an ideal platform to study the phase behavior anddynamics of 2-D suspensions.

Theory

Under a uniform magnetic eld, superparamagnetic particleshave an attractive dipolar interaction between particles i and j,calculated as:

Umag ¼ 1

4pm0r3

j � 3�m.

i$r��

j$r��

where r is the center-to-center distance between two particles, m0is the permeability of vacuum, and m

.i; j is the particle's

magnetic dipole. Similar to its electric analog,20 considering thelocal eld is the sum of external eld and the eld induced fromneighboring particles, the local magnetic dipole can bedecomposed into two components in polar coordinates:

m2;r ¼ m1;r ¼ 4pa3cm0H0

1� 2c

�3(2a)

m2;q ¼ m1;q ¼ 4pa3cm0H0

�sin q

�3(2b)

where c is the volumetric magnetic susceptibility of a sphere,H0

is the applied magnetic eld strength, a is the radius of a

Fig. 1 (a) Parameters of a dilute dimer system used in the model. (b) Thedifference between the simplified model and modified model for four fieldstrengths 4, 6, 8, and 10 Gauss. The blue lines correspond to the profile of thesimplified model and the red lines correspond to that of the modified model. Thearrow shows the direction of increasing field strength from 4 Gauss to 10 Gauss.

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sphere, and q is the angle between the connector vector and theapplied magnetic eld (Fig. 1(a)).

Typically a simplied dipolar model, in which the magneticinduction is based solely on the applied magnetic eld, is usedto describe the interaction potential between paramagnetic

particles:21,22 Uðr; qÞ ¼ m2

4pm0r3ð1� 3cos2 qÞ. This results in

particles that have identical dipole moments in the direction ofthe external magnetic eld. A more rigorous modied modelincludes mutual interaction from neighboring particles shownin eqn (2) so that the interaction energy between a pair ofparamagnetic particles can be expressed as:

Umagðr; qÞ ¼ m2

4pm0r3

1�1� 2c

�3�2

8><>:1� 3cos2 q

2640B@1� 2c

375sin2

9>=>;; (3)

where m ¼ 4pa3cm0H0

In a rotating magnetic eld at low frequencies, an aniso-tropic attractive potential between two particles leads to arotating dimer.23,24 As the frequency is increased above a criticalfrequency, the two q terms, cos2 q and sin2 q, in eqn (3) are time-averaged to 0.5 as the magnetic torque is balanced by frictionaltorques and the dimer no longer rotates, resulting in anisotropic interaction potential:

UmagðrÞ ¼ � am2

8pm0r3

a ¼ 1�1� 2c

�3�2

�1� 2c

�3�2

�1þ c

�3�2

1CCCA (4b)

Above this critical frequency, the interaction potential is nolonger a function of q, but is only effective in the direction of theconnector vector. Essentially the interaction potential isisotropic in the plane of the magnetic eld, similar to well-known, nondirectional, supramolecular interactions (e.g. vander Waals force or depletion interactions25) but the strength ofthis interaction is tunable in situ by changing the strength of themagnetic eld.

For charged colloids, the short-range electrostatic repulsioncould be calculated using Derjaguin–Landau–Verwey–Overbeek

(DLVO) theory:26 Uel ¼�64pkBTRrNg2

�expð�kDÞ, where R is

the radius of the spheres, rN is the number density of ions inthe bulk solution, k is the reciprocal of the Debye length, D is thesurface separation between particles, and g¼ tanh(zej0/4kBT) isthe reduced potential where j0 is the surface potential and e isthe unit charge. The total interaction potential including theattractive dipolar and repulsive electrostatic contributions forthe modied dipolar model is described by eqn (5),

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UmodðrÞ ¼ am2

8pm0r3þ �

64pkBTarNg2=k2�e�kðr�2aÞ (5)

Fig. 1(b) illustrates the difference between the simplied andmodied models. Note that neglecting the magnetic inductionfrom neighboring particles leads to an underestimation in thestrength of the interaction potential by 30% at the potentialwell.

Method and materialsExperimental magnetic and imaging system setup

Two pairs of air-core solenoids are placed perpendicular to eachother (Fig. 2) to generate an orthogonal magnetic eld. Amultifrequency power supply (Agilent N6784A) is programmedto provide sinusoidal currents through the coils to create amagnetic eld in the x-direction Hx ¼ H0sin(ut), and in the y-direction Hy ¼ H0sin(ut + p/2) between 5 and �5 V. This resultsin a homogeneous magnetic eld that rotates in the horizontalplane. The instantaneous direction of the eld is monitoredusing a digital oscilloscope embedded in the power supplymainframe (Agilent N6705A). A 100�/1.4 Olympus oil immer-sion objective is mounted at the center of the two orthogonalpairs of solenoids, beneath a piezo-controlled (Newport ESA-C)sample holder. A CCD camera (QImaging) captures the images.We monitor and record the colloid dynamics with the CCDcamera at a rate of 10–16 frames per second using Simple PCI(Hamamatsu). We batch-process the image les using MATLABto obtain x and y coordinates of spheres.27 A surface level tool isused to verify that the sample stage does not have any tilt from

Fig. 2 Coil setup schematic view: (a) top view and (b) expanded view of the corewith the objective, stage and sample clearly separated (the objective is moveddownward by 30 mm and the sample upward by 30 mm).

the center of the solenoid plane to ensure that themeasured 2-Dprojection of the trajectory gives the actual position.

Particle preparation

The particles used in this study are carboxyl-coated super-paramagnetic polystyrene M-270 Dynabeads (Dynal Biotech,Oslo, Norway). The mean particle diameter of the particles is2.8 � 0.08 mm, conrmed by dynamic light scattering andscanning electron microscopy (SEM) measurements. The volu-metric magnetic susceptibility of the beads is 0.96, as providedby the manufacturer. The particles are taken from a stocksolution, washed and vortexed in 0.1 mM NaCl solution, anddiluted to nal particle concentrations from 0.001 wt% to 0.01wt%. The zeta potential of the particles was measured to be�50.0 mV in 0.1 mM NaCl. The particles were injected into aow cell consisting of two glass coverslips (Thermo Scientic)sandwiched together with a paralm spacer. The spacingbetween the two coverslips is measured to be 50 mm, which issufficiently large to make the hydrodynamic backow negligibleand allows gravity to conne the particles to 2-D.9,28 Addition-ally, since the particles and coverslip are negatively charged, theparticles remain suspended at a xed distance from the bottomsubstrate.29 Both coverslips were rst cleaned in a plasmacleaner (Harrick Plasma PDC-32G) and rinsed with DI waterprior to experiments to prevent non-specic binding of theparticles to the glass surface. Aer particle injection, the cell isthen sealed with epoxy (Hardman Double/Bubble Extra FastDrying) to prevent evaporation of the solution. The cell was thenplaced in a sample holder at the center of two pairs of orthog-onal solenoids. One hour is allowed for the spheres to sedimentto the bottom of the ow cell and reach a quasi-equilibriumbefore any experimental operations or observations are made.

Simulation method

A Brownian dynamics (BD) simulation is used to model thissystem to study the frequency dependence and multi-bodyeffects on the interaction potential. The position of an ensembleof spheres in two dimensions is based on the algorithmproposed by Ermak and McCammon.30 In this algorithm, thecurrent position r(n) (2N vector) of the beads is moved to the nextposition r(n+1) according to

rðnþ1Þ ¼ rðnÞ þ Dt

kBTDðnÞFðnÞ þ xðnÞ (6)

where F(n) is the collective vector of internal and external (exceptBrownian and hydrodynamic) forces acting on a bead F(n) ¼Fmagi + Felei , x(n) is a random force generated at each time step

from a Gaussian distribution with zero mean and variancehx(n)x(n0)i ¼ 2D(n)Dtdnn0, where Dt is the time step and dnn0 is theKronecker delta, and D(n) is the 2 � 2 Rotne–Prager trans-lational diffusion matrix31 used to account for hydrodynamicinteractions between all beads.

The equation of motion that describes the dynamics of eachmagnetic bead is

d2ridt2

¼ Fmagi þ Fhydro

i þ FBrowni þ Fele

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where for the ith bead, mi is the mass of the ith bead (the le-hand side inertia term is negligible compared with right-handside terms in the case of low Reynolds number), ri is the posi-tion, Fhydroi is the hydrodynamic drag force, FBrowni is thestochastic Brownian force, Fmag

i is the magnetic dipole–dipoleforce that is dependent on time, magnetic eld strength andfrequency, and Felei is the electrostatic force between beads.32 Inthese BD simulations, time steps are between 0.01 ms and 0.05ms, and initial positions of particles are manually assigned to bewithin the vicinity of each other, typically 1.5 times the diameterof the bead away from each other. The transition between theanisotropic potential at low frequencies and the isotropicpotential above the critical frequency can be observed in ourBrownian dynamics simulations (movies provided in the ESI†).

Fig. 3 (a) The sweep angle of the dimer as a function of time at a field strengthof 6 Gauss at frequencies of 0.1, 0.5, 1, and 20 Hz; (b) a sequence of imagesshowing a particle dimer coming together and staying in this energeticallyfavorable configuration. The scale bar represents 5 mm.

ResultsColloidal dimers in a rotating magnetic eld

When a rotating magnetic eld is applied, the dipolar interac-tion between particles generates a magnetic torque, causing thedimer to follow the rotatingmagneticeld. Thismagnetic torqueis countered by the inertial, viscous, and Brownian torques.23 Atlow frequencies, the dimer will follow the externalmagnetic eldand rotate synchronously since the magnetic torque dominatesover the other frictional torques. For example, a dimer rotatingin a 6 Gauss magnetic eld at 0.1 Hz will make a 360� sweepangle in 10 seconds. As the frequency is increased, the frictionaltorques increase and cause the dimer to rotate slower than theapplied magnetic eld, leading to an increasing phase lagwhere the dimer slows and reverses directions to realign withthe applied eld. This results in the dimer covering a smallersweep angle at any given time compared to the synchronouslyrotating dimer. At a high enough frequency, the magnetictorque is counterbalanced by the frictional torques, so that thedimer simply moves stochastically due to Brownian motion.There is no longer an effective torque and the sweep anglechanges insignicantly over time. The pair interaction betweentwo particles is only effective in the direction of the connectorvector, as derived in eqn (5). Fig. 3(a) shows the measured sweepangle for a dimer in a 6 Gauss magnetic eld at differentfrequencies. This interaction potential is long-range, whicheasily brings together two particles that are over one particlediameter apart. Fig. 3(b) (movie shown in the ESI†) illustratesthat within 15 seconds, the particles are drawn together.

Fig. 4 Experimentally measured pair potentials compared with theoreticalresults: (a) 4 Gauss; (b) 6 Gauss; (c) 8 Gauss; and (d) 10 Gauss. By simply adjustingthe strength of the magnetic field, the interaction potential between a particlepair can be tuned from �5kBT to 40kBT.

Interaction pair potential

The pair potential between dimers at various magnetic eldstrengths H0 (4 Gauss, 6 Gauss, 8 Gauss and 10 Gauss) iscalculated by monitoring the position of a pair of particles andcalculating the interaction potential using the inverted Boltz-mann equation:33

UðrÞ �U�rref

�kBT

�n�rref

�nðrÞ

where n(r) is the number density extracted from the distancehistogram at a distance r. The most probable distance is chosen

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as the reference distance rref. From eqn (8), the pair potentialU(r) at a given magnetic eld strength is determined experi-mentally and is compared with theoretical proles computedusing corresponding magnetic eld strengths.

The experimentally measured pair potentials at fourdifferent magnetic eld strengths are shown and comparedwith theory in Fig. 4. The experimental pair potential prolesmatch very well with theoretical proles calculated from eqn (5).

Table 1 Parameters used in the theory

a/mm c T/K rN/(mmol L�1) J0/mV

1.4 0.96 298 0.1 50

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Note that no tting parameters are used in the theoreticalprole and all the input parameters used are listed in Table 1.The values for the particle size and susceptibility are obtainedfrom Invitrogen, the bead manufacturer. The temperature andparticle density are experimentally controlled, and the Debyelength is measured using a Zetasizer (Malvern instruments). Formoderate eld strengths (4 Gauss and 6 Gauss), the well depthis shallow enough to allow the particle pair to spend enoughtime in the potential well. Good statistics over sufficient inter-particle distances are sampled to provide the full shape of thepair potential prole. At higher eld strengths, the depth ofthe potential well is accurately captured, but the resolution ofthe full shape of the potential prole is difficult to obtainexperimentally due to the strong attractive potential.

Fig. 5 Comparison between the pair potential (red line) and the experimentalthree-body potential (blue dots) at various field strengths: (a) 4 Gauss; (b) 6 Gauss;(c) 8 Gauss; and (d) 10 Gauss. The experimental three-body potential is fit to eqn(5) (green line) with the dipole moment set as a fitting parameter.

Three-body effect

The pair potential oentimes is not sufficient to describe acollection of particles, therefore the many-body effect, of whichthe three-body effect is the leading term, needs to be consid-ered.34 The Born–Green equation relates the effective potentialto the pair potential35

vU ð2Þðr12Þvr1

� vUðr12Þvr1

ðvUðr13Þvr1

gð3Þðr1; r2; r3Þgðr12Þ dr3 (9)

with an integral of the force on particle 1 at r1 due to a particle atr3, where U(2)(r12) ¼ kBT ln(g

(2)) is the mean force, U(r12) is thepair potential, and g(3) (r1,r2,r3) is the triple correlation function.For an ideal uid, the triple correlation function of particledistance (r,s,t) could be simplied using Kirkwood Superposi-tion Approximation to g(3)KSA(r,s,t) ¼ g(r)g(s)g(t). For a non-idealuid, a modication factor G should be introduced35

g(3)(r,s,t) ¼ g(3)KSA(r,s,t)G(r,s,t) (10)

Thus the effective triple potential of mean force could beobtained by

U(3)(r,s,t) ¼ U(2)(r) + U(2)(s) + U(2)(t) + DU(3)(r,s,t), (11)

where DU(3)(r,s,t) ¼ �kT ln G(r,s,t).Since exact pair potential can be obtained experimentally

using isolated dimers, eqn (11) can be further simplied as

U(3)(r,s,t) ¼ U(r) + U(s) + U(t) + DU(3)(r,s,t) (12)

The information for the three-body correlation functioncould be extracted from the image sequence using the methodmentioned by Krumhansl andWang.36 The three-body potentialU(3)(r,s,t) can be calculated using eqn (13).

U(3)(r,s,t) ¼ �kT ln(g(3)(r,s,t)) (13)

Eqn (13) can be further simplied to illustrate the impor-tance of the three-body correlation,

U (3)(r,r,r) ¼ 3U(r) + DU(3)(r,r,r), (14)

where the difference between the three-body potential for anequilateral triangle, DU (3)(r,r,r), and superposition of three pairpotentials reveals the importance of the three-body effect. Fig. 5shows the comparison between these two potentials fordifferent eld strengths. The superimposed potential in thisthree body system is three times the pair potential found in thedimer system, shown by the red lines. The experimentallymeasured potential is determined using eqn (6) and plotted asblue dots.

For all eld strengths, the three-body potentials are similarto the pairwise superposition in shape but with shallower welldepths. The experimental three-body potential is t to eqn (5),shown by the green lines in Fig. 5, with the dipole moment setas a tting parameter. From this t, the difference DU(3)(r,r,r) ineqn (14) can be determined. The three-body effect is largelyattributed to mutual interaction among the dipoles. As shownin Fig. 6(a) when another free particle C is placed around thedimer, it will predominantly stay within the red area and form aquasi-equilateral triangle with the dimer AB (Fig. 6(b)). In orderto understand how the third particle affects the interaction ofdimer AB, we simplify the model by assuming that particle pairAB has no mutual interaction but AC and BC have. Thus themagnetic interaction potential for pair AB can be described by

the simplied model UmagðrAB; qÞ ¼ mA* $mB

4pm0rAB3ð1� 3cos2 qÞ,

while mA* and mB

* can be calculated using eqn (2). The resultingmagnetic potential at r ¼ 2.2a at a eld strength of 6 Gauss isshown in Fig. 6(c). Compared with the simplied model, thehypothetical model has a lower magnitude for both the

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Fig. 6 (a) Colored position probability map of a third particle relative to a dimerAB whose distance is fixed. The red region highlights the preferred position of thethird particle. (b) The equilateral configuration of trimer ABC. (c) The angledependent magnetic potential Umag at r ¼ 2.2a and 6 Gauss for simplified andhypothetical models. (d) The angle independent net potential U(r) at 6 Gauss forthe two models.

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attractive and repulsive component of the interaction potential.The time-averaged net potential U(r) for 6 Gauss in Fig. 6(d)conrms that the decrease in attractive interaction dominatesfor the hypothetical model and the resulting net potential has alower potential well depth. This illustrates how particle Cdecreases the pair potential of pair AB via mutual inductioninteractions.

Tunable 2-D phase behavior

For a suspension of particles, changing the magnitude of the20 Hz rotational magnetic eld affects the strength of theinteraction potential, leading to various 2-D thermodynamiccolloidal phases. The gas phase is found at a eld strength of 2Gauss (1.5kBT), the liquid phase at 2.2 Gauss (1.9kBT), the hex-atic phase at 2.6 Gauss (2.6kBT) and the crystal phase at 3.4Gauss (4.5kBT), shown in Fig. 7(a)–(d), respectively. No aggre-gation is observed when a control sample is placed in thesample chamber and leveled in the absence of a eld, con-rming that the applied rotational magnetic eld causes theobserved phases. Different phases are identied by density andtheir corresponding pair distribution function g(r), calculatedusing eqn (15):

gðrÞ ¼ A

d�r� rij

�(15)

where N is the total number of particles within the image, A isthe total area in the image and d(r � rij) is the delta function.Fig. 7(e) and (f) show that the g(r) for the gas phase has a nearlyconstant value at distances slightly larger than the particlediameter, while the liquid phase has a periodic response thatslowly decays with increased distance. The hexatic phase hascertain free topological defects, known as dislocations, whichdiffers from the crystal phase. The g(r) of the hexatic phase has a

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characteristic split of the second g(r) peak but has a quasi-long-range oscillation, as shown in Fig. 7(g). The crystal phase ischaracterized by the perfect regularity of a lattice arrangementand the complete splitting of the second g(r) peak, shown inFig. 7(h).

The structure factor (Fig. 7(i)–(l)) further conrms thehomogeneity of different phases. Both Fig. 7(i) and (j) haveuniform ring distribution indicating the homogeneity of theuid phases. Fig. 7(k) starts to show the enhanced intensity atthe edge of the ring implying the impending peak splitting. Thestructure factor for the crystal phase in Fig. 7(l) shows distinctpeaks, which constitute the reciprocal lattice arrangement.Another important function used to identify phases is thebond orientational correlation function g6(r), calculated usingeqn (16):

g6(r) ¼ hj*(0)j(r)i (16)

where jðrÞ ¼ jðrkÞ ¼ 1Nk

exp�6iq

�rkj*

��is the bond orien-

tational order parameter for particle k at rk distance away from

the reference particle, q�rkj*

�is the angle of the connector

vector between particle k and its neighboring particle j, andNk isthe number of nearest neighbors to particle k. Fig. 7(m) givesfurther conrmation of the hexatic phase from g6(r). Theguidelines for the crystal, hexatic phase and liquid phasecorrespond to a constant, algebraic decay with exponent �1/4and exponential decay with a correlation length of 2 respec-tively. The algebraic decay with exponent �1/4 for the hexaticphase also agrees very well with the KTHNY prediction.37–39

Two-phase equilibrium can also be readily observed asshown by the gas–liquid phase boundary and the solid–gasphase boundary in Fig. 8(a) and (c), respectively. Quanticationof the density change of the liquid–gas interface is provided inFig. 8(b), where the density of the liquid phase is shown to beapproximately 10 times as high as that of the gas phase at amagnetic eld strength of 2.2 Gauss. The quantication ofdensity change in Fig. 8(d) shows both the positional andorientational order for the solid phase, and nearly zero densityfor gas that is in equilibrium with the solid at a magnetic eldstrength of 3.4 Gauss.

Frequency effect

A Brownian dynamics (BD) simulation is used to predict thecritical frequency, at which there is no effective torque experi-enced by a dimer. Above this critical frequency, the averagedcenter-to-center distance and the center-to-center distanceuctuation do not change as the frequency increases. From oursimulations, the center-to-center distance and its uctuationversus frequency were obtained for a variety of eld strengths, asshown in Fig. 9(a) and (c). At larger eld strengths, such as 6, 8,and 10 Gauss, the large center-to-center distance uctuation atrelatively low frequencies is primarily due to the beating of thedimer resulted from the asymmetric eld (movie shown in theESI†). The transition between the anisotropic potential at lowfrequencies and the isotropic potential above the critical

Fig. 7 Characteristic images and pair distribution functions of various thermodynamic phases formed at different field strengths. (a)–(d) show images of the gas,liquid, hexatic and crystal phases respectively. (e)–(h) show the corresponding pair distribution function g(r), calculated from eqn (13). (i)–(l) show the correspondingstructure factor. (m) shows the bond-orientational correlation function g6(r) for different phases. The white scale bar represents 5 mm. Note that the images have beencropped to show detailed particle placement and that the g(r) and g6(r) are calculated using images that contain 600–1200 particles (typically two times larger than thecropped images).

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frequency can be observed from our simulations (moviesprovided in the ESI†). The center-to-center distance uctuationdecreases as the eld strength increases. At 4 Gauss, the center-to-center distance uctuation has a larger variation than that athigher eld strengths since the magnetic potential well depth issmall (6.4kBT) and thermal uctuations can noticeably inu-ence the center-to-center distance. Additionally, it should benoted that as the magnetic eld strength is increased, the

critical frequency increases, meaning that a higher frequency isrequired to eliminate the larger effective torque. The criticalfrequency can also be determined by the plateau in the averagedcenter-to-center distance plot as a function of frequency, asshown in Fig. 9(c). The critical frequencies for 6, 8, and 10 Gaussare determined to be 10, 15, and 25 Hz from Fig. 9(a) and (c).

There is also a threshold frequency to consider, which isassociated with the relaxation time of the magnetic

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Fig. 8 Characteristic images illustrating 2-D phase equilibrium. (a) Image of theliquid–gas interface; (b) density distribution of the liquid–gas interface; (c) imageof the solid–gas interface and (d) density distribution of the solid–gas interface.The color density scale depicts the number of particles per unit area. The scale barrepresents 10 mm.

Fig. 9 (a) The center-to-center distance fluctuations of a dimer system at variousfield frequencies for different field strengths simulated using BD simulation. (b)The center-to-center distance fluctuations at various field frequencies for differentfield strengths from experiment. (c) The averaged center-to-center distances atvarious field frequencies for different field strengths simulated using BD simula-tion. (d) The averaged center-to-center distances at various field frequencies fordifferent field strengths from experiment. (e) The potential landscape for 6 Gaussunder different frequencies. The blue dots correspond to the experimental dataunder 10, 15, 20, 15, 30, 40, 50, and 60 Hz respectively with the arrow indicatingthe direction of increasing frequency. The red lines are fit from eqn (5). (f) Thepotential well depth at different frequencies under different field strengths.

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nanoparticles dispersed within the particle matrix. At frequen-cies above the critical frequency, the particles do not experiencean effective torque. The individual magnetic nanoparticles

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embedded within the particles are able to rotate, resulting in anet particle dipole moment. However, if some of the magneticnanoparticles fail to relax, the magnetic susceptibility of thecolloidal particle would be affected. In a rotational magneticeld, the alignment between the magnetic moment of a nano-particle and the applied eld occurs through a combination oftwo mechanisms: the rotation of the magnetic moment withrespect to the crystal axis (Neel relaxation) or the rotation of theparticle in the surrounding material (Brownian relaxation).40

The nanoparticles embedded in the microspheres will relax viaNeel relaxation, while those that are not xed will relax via bothmechanisms. The characteristic relaxation frequency, which isthe inverse of the characteristic relaxation time, is the sum ofNeel relaxation frequency and Brownian relaxation frequency.Both frequencies are reported to be independent of eldstrengths smaller than 10 Gauss.41,42 The threshold frequency iswhere the real part of nanoparticle magnetic susceptibilitybegins to fall, and thus the colloidal particle susceptibilitydecreases. This threshold frequency is only a function of thecharacteristic relaxation frequency in an alternating magneticeld43 and is independent of the magnetic eld strength.

We determine this threshold frequency experimentally bymeasuring center-to-center distance uctuations of a dimersystem over various frequencies, which simulation is not able toillustrate due to the fact that ideal particles are used. Once thefrequency increases beyond the timescale required for relaxa-tion, the center-to-center distance uctuation increases.Fig. 9(b) illustrates the measured averaged distance uctuationsof three different dimers at various eld frequencies fordifferent eld strengths. For all dimers of interest, the distanceuctuations start to increase at the same threshold frequencyunder all eld strengths. Note that at eld strengths below 4Gauss, Brownian motion will affect the measurement of particledistance and the distance uctuations may vary for differentmeasurements. Likewise the critical frequency can also bedetermined by the increase of the averaged center-to-centerdistance. Fig. 9(d) shows for all four eld strengths, the aver-aged distance starts to decrease at 20 Hz, validating that theeld strength has a negligible effect on the relaxation of themagnetic nanoparticles in the colloid particle. From Fig. 9(d),notice that at 10 Gauss, the curve has a local maximum at 20 Hzinstead of a plateau below 20 Hz. Recall that in Fig. 9(a) thecritical frequency for 10 Gauss is 25 Hz and there is still a slighteffective torque experienced by the particle at 20 Hz. Howeverthis torque only minimally affects the pair potential as shown inboth Fig. 4(d) and 9(d). Nevertheless this effective torqueprevents us from further increasing the eld strength.

The change of the center-to-center distance and its uctua-tion is caused by the change of the interaction potential welldepth due to the change of nanoparticle susceptibility. Thepotential landscapes for 6 Gauss at different frequencies areshown in Fig. 9(e). When the frequency increases above 20 Hz inthe direction of the arrow, the potential well depth starts todecrease. When the frequency goes above 70 Hz, the dimer willnot be stable and will eventually separate due to the dominantthermal energy. Fig. 9(f) shows the relationship betweenpotential well depth and frequency under different eld

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strengths. This gure also conrms the information given byFig. 9(b) and (d) especially the threshold frequency of 20 Hz.

Conclusion

We demonstrate a 2-D colloidal system in which the pairpotential can be adjusted in situ using superparamagneticcolloidals and a rotating magnetic eld. We measure the exactpair potential as well as the three-body potential in a dilutesystem. A modied local magnetic eld theory for the pairpotential is developed to account for the induced dipole fromneighboring particles and is veried experimentally. Themeasured pair potential has quantitative agreement with ourmodied local magnetic eld theory. The measured effectivethree-body potential shows consistent deviation from thepotential between dimers at all eld strengths, which indicatesthat the many-body effect on the interaction potential is notnegligible. For a suspension of colloids, various thermodynamicphases are observed under different known eld strengthsconrming the virtuosity of this system. We believe this systemprovides an ideal platform to study the phase behavior anddynamics of colloidal systems in two dimensions.

Acknowledgements

Wegratefully thankDr JohnCrocker,DrWalterG.Chapman, JulieE. Byrom, Gautum Kini and Kai Gong for insightful discussionsand Douglas Chen for experimental help. We acknowledge theNational Science Foundation CAREER Award (CBET-0955003)and the Welch Foundation (C-1755) for nancial support.

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Soft Matter

Generating an in situ tunable interaction potential for probing 2-D colloidal phase behavior

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