Communication: Radial distribution functions in a two-dimensional binary colloidalhard sphere systemAlice L. Thorneywork, Roland Roth, Dirk G. A. L. Aarts, and Roel P. A. Dullens

Citation: The Journal of Chemical Physics 140, 161106 (2014); doi: 10.1063/1.4872365 View online: http://dx.doi.org/10.1063/1.4872365 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/140/16?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Phase transitions in two-dimensional colloidal particles at oil/water interfaces J. Chem. Phys. 126, 034706 (2007); 10.1063/1.2409677 Theoretical direct correlation function for two-dimensional fluids of monodisperse hard spheres J. Chem. Phys. 125, 144504 (2006); 10.1063/1.2358133 On the radial distribution function of a hard-sphere fluid J. Chem. Phys. 124, 236102 (2006); 10.1063/1.2201699 Decay of correlation functions in hard-sphere mixtures: Structural crossover J. Chem. Phys. 121, 7869 (2004); 10.1063/1.1798057 The glass transition in binary mixtures of hard colloidal spheres AIP Conf. Proc. 519, 3 (2000); 10.1063/1.1291516

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THE JOURNAL OF CHEMICAL PHYSICS 140, 161106 (2014)

Communication: Radial distribution functions in a two-dimensionalbinary colloidal hard sphere system

Alice L. Thorneywork,1 Roland Roth,2 Dirk G. A. L. Aarts,1 and Roel P. A. Dullens1

1Department of Chemistry, Physical and Theoretical Chemistry Laboratory, University of Oxford,South Parks Road, Oxford OX1 3QZ, United Kingdom2Institut für Theoretische Physik, Eberhard Karls Universität Tübingen, Auf der Morgenstelle 14,72076 Tübingen, Germany

(Received 4 March 2014; accepted 11 April 2014; published online 29 April 2014)

Two-dimensional hard disks are a fundamentally important many-body model system in classicalstatistical mechanics. Despite their significance, a comprehensive experimental data set for two-dimensional single component and binary hard disks is lacking. Here, we present a direct comparisonbetween the full set of radial distribution functions and the contact values of a two-dimensional bi-nary colloidal hard sphere model system and those calculated using fundamental measure theory. Wefind excellent quantitative agreement between our experimental data and theoretical predictions forboth single component and binary hard disk systems. Our results provide a unique and fully quantita-tive mapping between experiments and theory, which is crucial in establishing the fundamental linkbetween structure and dynamics in simple liquids and glass forming systems. © 2014 AIP PublishingLLC. [http://dx.doi.org/10.1063/1.4872365]

The radial distribution function, g(r), is central to thestatistical mechanics of the liquid state and is proportional tothe probability of finding a particle in an infinitesimal shell ata distance r from another particle. The function quantifies theaverage structure of the system and, where pair interactionsmay be assumed, provides direct links to many importantquantities, such as the virial equation of state and the excessinternal energy.1 In addition, dynamic quantities like thediffusion coefficient and the intermediate scattering functionare often directly linked to structural measures such as thecontact value of g(r) or the structure factor.2–4 However, toexploit these fundamental relations, it is imperative to have afull quantitative understanding of the structure in the system.

In most simple fluids, the structure is largely determinedby the repulsive part of the potential5–8 and the simplest modelfluid with a purely repulsive potential is a fluid consisting ofthermal hard spheres. The hard sphere system is of paramountimportance because its free energy is governed by entropy,which allows for a detailed study of the behavior of entropicstatistical systems.9 In addition, hard spheres often serve asthe starting point for the typical theoretical treatment of a sim-ple fluid, where interactions are separated into a hard corereference system plus a perturbation.7, 10

Despite the fact that accounting for hard core repul-sion is difficult in theoretical approaches, the radial distri-bution functions of hard sphere and hard disk systems haveattracted a vast amount of attention from theory11–15 and incomputer simulations.16–20 Experimentally, colloids are oftenused as a hard sphere model system21, 22 and experimentaldetermination of the radial distribution function of colloidalsystems was achieved first in scattering experiments,23 andlater directly by confocal microscopy.24–26 Measurements ofthe radial distribution function in a variety of (quasi) two-dimensional (2D) colloidal systems have compared well to

computer simulations,27–38 but a satisfactory comparison totheory is not reported, not least due to the lack of an accu-rate theory. Recently, an accurate fundamental measure theory(FMT) for 2D hard disk mixtures has been developed and em-ployed for single component systems. Nevertheless, a com-prehensive quantitative mapping between experimental dataand theory for both single and binary hard disk systems hasbeen lacking, despite their importance as a reference systemfor simple liquids and glass forming systems.3, 29, 33

In this communication, we present a direct comparisonbetween the radial distribution functions and their contact val-ues obtained from experiments on a two-dimensional binarycolloidal hard sphere model system and those calculated usingfundamental measure theory for binary hard disks. We con-sider systems of a variety of compositions and over a largerange of total packing fractions, φt = (π/4)

∑i σ

2i ρi , with σ i

the diameter and ρ i the number density of component i.

(b)(a)

FIG. 1. (a) State diagram of the packing fractions of large (φl) versus small(φs) particles, showing the different compositions studied. The dashed linesindicate state points with a similar composition q = φl/φt. (b) A typical mi-croscopy image of the binary colloidal system at q = 0.71 and φt = 0.67.

0021-9606/2014/140(16)/161106/5/$30.00 © 2014 AIP Publishing LLC140, 161106-1

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161106-2 Thorneywork et al. J. Chem. Phys. 140, 161106 (2014)

(c) φt = 0.649

0 1 2 3 4 5 6 7

r / σ

0

1

2

3

4

5

g(r)

(a) φt = 0.346

0 1 2 3 4 5 6 7

r / σ

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2g(

r)(b) φt = 0.514

0 1 2 3 4 5 6 7

r / σ

0

0.5

1

1.5

2

2.5

3

g(r)

FIG. 2. The radial distribution function g(r) of a single component hard-disk system at three different packing fractions: (a) φt = 0.346, (b) φt = 0.514, and (c)φt = 0.649, as obtained from the experiments (symbols) and the theory (lines). Note that panels (a) and (b) show data for the small particles (q = 0) and panel(c) for the large particles (q = 1).

Our colloidal system consists of carboxylic acid func-tionalized melamine formaldehyde particles (MicroparticlesGmbH) with hard sphere diameters of σ s = 2.79 μm and σ l

= 4.04 μm dispersed in a 20/80 v/v% ethanol/water mixture.The polydispersities of the small and large particles are 2.1%and 1.2%, respectively, as determined by scanning electronmicroscopy. The particles are allowed to sediment onto thebase of a glass sample cell with a height of 200 μm, which iscleaned before use with a 2% solution of Hellmanex. The par-ticle mass density of 1.57 g/ml results in a gravitational lengthof 0.07 μm and 0.02 μm for the small and large particles, re-spectively. As this is a fraction of the size of the particles,the out of plane fluctuations are negligible and the system isstructurally two-dimensional.

The colloidal system is imaged using an OlympusCKX41 inverted bright-field microscope equipped with aPixeLink CMOS camera (1280 × 1080 pixels). Particlecoordinates are subsequently obtained from the microscopyimages with an error of 12 ± 10 nm using standard parti-cle tracking software.39 At the highest measured total pack-ing fraction φt for each of the different compositions, thereare typically between 3000 and 4000 particles in each frame.Large and small particles are readily distinguished based uponthe integrated brightness of the features found. The totalpacking fraction is varied over a range from approximatelyφt = 0.05 to 0.76 for six different systems with relative pack-ing fractions, q = φl/φt, of approximately 0, 0.17, 0.37, 0.50,0.71, and 1. Here, φl is the packing fraction of the large par-ticles. Figure 1(a) shows a state diagram with all the exper-imental compositions studied and Fig. 1(b) shows a typicalsnapshot of the binary system at q = 0.71 and φt = 0.67.

The partial radial distribution functions gij(r) are com-puted from the particle coordinates according to1

xixjgij (r) = 1

ρ

⟨1

N

Ni∑μ=1

Nj∑ν �=μ

δ(r + rμ − rν)

⟩, (1)

with N the total number of particles, ρ = N/A the total numberdensity of the system (A is the surface area), xi, j = Ni, j/N thefraction of species i or j, and rμ, ν the position of particle μ of

species i or ν of species j, respectively. In a single componentsystem particles μ and ν are by necessity of the same species,but in a mixture we can define radial distribution functionsfor i and j representing the same or different particle species.For the binary system considered here, this results in the fol-lowing three partial radial distribution functions, gss(r), gll(r),and gls(r) = gsl(r), which depend upon correlations betweenlike or unlike particles. The experimental radial distributionfunctions are averaged over at least 200 frames.

In order to compare the experimentally measured radialdistribution functions with theoretical predictions we calcu-late gij(r) within the framework of classical density functionaltheory40 (DFT) using the so-called test particle route. In thismethod one fluid particle of component i = s, l is fixed at theorigin of the frame of reference and thereby turned into anexternal field for the rest of the system. The inhomogeneousdensity distributions ρ

(i)j (r) of species j = s, l, of the fluid

mixture, subjected to an external field of a fixed particle ofspecies i, are calculated numerically by minimizing the den-sity functional. From the density distributions one can obtainthe radial distribution functions directly:

gij (r) = ρ(i)j (r)

ρj

. (2)

Since we are interested in the radial distribution functionsof fluid mixtures, we require a DFT for hard-disk mixtures.Fundamental measure theory41, 42 is a reliable and versatileDFT approach for hard-body mixtures. Despite this, an ac-curate FMT functional for hard-disk mixtures has only re-cently been constructed and, up to now, only been applied toone-component systems.43

To compare the experimental data to our FMT calcula-tions the size ratio, α = σ l/σ s, of the particles in the mixturemust be considered. For the experimental system in 3D thesize ratio is α = 4.04/2.79 ≈ 1.45, but at the base of the sam-ple cell the centers of the small and large particles are not inthe same plane. This makes the mixture slightly non-additive,i.e., σ ls = (1/2)(σ ll + σ ss)(1 + ) with �= 0. By project-ing the centers of a large and a small particle in contact ontothe base plane one finds an effective size ratio of αeff ≈ 1.41,

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161106-3 Thorneywork et al. J. Chem. Phys. 140, 161106 (2014)

FIG. 3. The partial radial distribution functions for the experiments (symbols) and theory (lines): gll(r) (squares and full lines), gls(r) = gsl(r) (circles anddashed-dotted lines), and gss(r) (diamonds and dashed lines) for various values of the packing fractions φl and φs as indicated in the panels. For clarity we haveshifted gls(r) and gll(r) vertically.

corresponding to a small negative value of ≈ −0.017. Inour FMT calculations for additive mixtures we assume a sizeratio in between these two values and set αDFT = 1.43. Wenote that αDFT = 1.45 (1.41) slightly improves the agree-ment between experiments and theory for systems rich inlarge (small) particles. The deviations from the results forαDFT = 1.43 are, however, small, indicating that the small de-gree of non-additivity has little effect.

First, we present radial distribution functions for a one-component fluid, which corresponds to either only small par-ticles (q = 0) or only large particles (q = 1). Figure 2 showsthe g(r) for various total packing fractions φt as a functionof the dimensionless radial distance r/σ . The experimentaldata (symbols) and theoretical predictions (lines) are in ex-cellent agreement for all details, i.e., the contact value, thewavelength of oscillation, the decay (correlation) length, forall values of φt without any adjustable parameters. The highlevel of agreement between our experimental data and FMTcalculations over the whole range of packing fractions con-firms the experimental values of particle diameters, the pack-ing fractions and shows that the colloidal system is an almostperfect model system for hard disks.

More challenging, both for experiment and theory, isthe determination of the partial radial distribution functions,gij(r), for a binary mixture. In Fig. 3 we show the three par-

tial radial distribution functions gll(r), gls(r) = gsl(r), andgss(r) for various combinations of the packing fractions ofthe large (φl) and small (φs) particles as obtained from theexperiments (symbols) and theory (lines). The agreement be-tween experiments and theory is excellent for all composi-tions, even at rather high values of the total packing fractionφt. Both the experimental data and theoretical results showthe same variation of the radial distribution functions arisingfrom complicated packing effects for different parts of theφl-φs plane (Fig. 1(a)). Note that for a binary mixture withtwo distinct diameters, a discontinuous change of the wave-length of oscillations in the asymptotic regime of the radialdistribution functions gij(r), the so-called structural crossover,was predicted46, 47 based on the theory of asymptotic decaysof correlation functions, and found experimentally.48 For thesize ratio used here, however, the diameters are too similar toexpect structural crossover.

Finally, we consider the contact values of the radial dis-tribution function, gij(σ ij), for both the single component andbinary systems. The experimental determination of the con-tact values, however, is rather subtle due to the fact that themeasured g(r) is the convolution of the “true” g(r) with thefinite bin size used in the measurement. As a result, the firstpeak is shifted to a slightly larger distance and the height ofthe first peak, g1, is smaller than the “true” contact value. The

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161106-4 Thorneywork et al. J. Chem. Phys. 140, 161106 (2014)

(a) (b)

FIG. 4. (a) Experimental contact values for the two single component systems (q = 0 and q = 1) with the prediction from 2D scaled particle theory (solidline).44 The inset shows the height of the first peak of g(r), g1, as a function of the bin size for three different centerings of the bins (4.06, 4.08, and 4.11 μm).(b) Contact values for gll (squares), gls (circles), and gss (triangles) for all studied compositions (q = 0.17, 0.37, 0.5, and 0.71). The coloured regions representthe predictions from Santos et al.45 for gij(σ ij) for compositions ranging from q = 0.17 to 0.71: gss (blue), gls (orange), and gll (green).

height of the first peak also depends on the bin size49 and thepositioning of the bins. This is shown in the inset in Fig. 4(a),where we plot g1 as a function of the bin size for three center-ing positions of the bins. As expected g1 decreases with thebin size, but is relatively independent of both the bin size andthe centering position of the bins for bin sizes in the range of0.02–0.05 μm. We therefore computed the distribution func-tions using a bin size of 0.0425 μm, which is chosen to be atthe upper end of this range to optimise statistics.

Next, the contact values gij(σ ij) are determined from ourexperimental radial distribution functions by fitting the de-cay of the first peak with an exponential function and ex-trapolating back to the hard disk diameter.49–53 The resultingcontact values for the single component fluid and the bi-nary systems are presented in Figs. 4(a) and 4(b), respec-tively. For the single component system our FMT gives anexpression for the contact values equivalent to that from 2Dscaled particle theory,44 and the agreement with the experi-mental data is excellent. In Fig. 4(b), we show the contactvalues for the three partial radial distribution functions ofthe binary system, gll, gls, and gss, for all studied compo-sitions (q = 0.17, 0.37, 0.5, and 0.71). Our FMT does notyield a closed expression for the contact values and scaledparticle theory54 does not agree well with contact valuesfrom simulations.55, 56 We therefore compare our data to theprediction of Santos et al.,45 which agrees well with sim-ulation data.56 As the contact values are weakly composi-tion dependent, we represent the prediction of Santos foreach gij(σ ij) as a coloured region, where the width accountsfor the composition dependence. Again excellent agreementwith our experimental data is observed, further corroboratingthe quantitative mapping between our colloidal system andtwo-dimensional hard disks.

In summary, we have presented a comprehensive compar-ison of the radial distribution functions of a two-dimensionalbinary colloidal hard sphere model system and fundamentalmeasure theory calculations. We find excellent agreement be-tween the experiments and fundamental measure theory forbinary hard disks. The contact values of the radial distribu-tion function also show excellent comparison to theoreticalexpressions, both for the single component and binary sys-

tems. Furthermore, our results confirm that the very smalldegree of non-additivity in the experimental system, whichcannot be treated in the present theory, has little influence onthe results. The unique and fully quantitative mapping be-tween our experimental colloidal system and theory is re-markable as it allows for a precise determination of thehard sphere packing fraction. Moreover, it provides a directquantitative theoretical understanding of the structure in ourexperimental system, which is crucial in establishing the fun-damental relation between structure and dynamics in simpleliquids and glass forming systems.

We thank Jürgen Horbach, Martin Oettel, and MichaelJuniper for useful discussions. A.L.T., D.G.A.L.A., andR.P.A.D. acknowledge the Engineering and Physical SciencesResearch Council (UK) (EPSRC) for financial support.

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