Two-dimensional hard disks are a fundamentally important many-body model system in classical statistical 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 comparison between the full set of radial distribution functions and the contact values of a two-dimensional binary colloidal hard sphere model system and those calculated using fundamental measure theory. We find excellent quantitative agreement between our experimental data and theoretical predictions for both single component and binary hard disk systems. Our results provide a unique and fully quantitative mapping between experiments and theory, which is crucial in establishing the fundamental link between structure and dynamics in simple liquids and glass forming systems.

The radial distribution function, g(r), is central to the statistical mechanics of the liquid state and is proportional to the probability of finding a particle in an infinitesimal shell at a distance r from another particle. The function quantifies the average structure of the system and, where pair interactions may be assumed, provides direct links to many important quantities, such as the virial equation of state and the excess internal energy.1 In addition, dynamic quantities like the diffusion coefficient and the intermediate scattering function are often directly linked to structural measures such as the contact value of g(r) or the structure factor.2–4 However, to exploit these fundamental relations, it is imperative to have a full quantitative understanding of the structure in the system.

In most simple fluids, the structure is largely determined by the repulsive part of the potential5–8 and the simplest model fluid with a purely repulsive potential is a fluid consisting of thermal hard spheres. The hard sphere system is of paramount importance because its free energy is governed by entropy, which allows for a detailed study of the behavior of entropic statistical systems.9 In addition, hard spheres often serve as the starting point for the typical theoretical treatment of a simple fluid, where interactions are separated into a hard core reference system plus a perturbation.7,10

Despite the fact that accounting for hard core repulsion is difficult in theoretical approaches, the radial distribution functions of hard sphere and hard disk systems have attracted a vast amount of attention from theory11–15 and in computer simulations.16–20 Experimentally, colloids are often used as a hard sphere model system21,22 and experimental determination of the radial distribution function of colloidal systems was achieved first in scattering experiments,23 and later directly by confocal microscopy.24–26 Measurements of the 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 to theory is not reported, not least due to the lack of an accurate theory. Recently, an accurate fundamental measure theory (FMT) for 2D hard disk mixtures has been developed and employed for single component systems. Nevertheless, a comprehensive quantitative mapping between experimental data and theory for both single and binary hard disk systems has been lacking, despite their importance as a reference system for simple liquids and glass forming systems.3,29,33

In this communication, we present a direct comparison between the radial distribution functions and their contact values obtained from experiments on a two-dimensional binary colloidal hard sphere model system and those calculated using fundamental measure theory for binary hard disks. We consider systems of a variety of compositions and over a large range of total packing fractions,

$\phi _t=(\pi /4)\sum _i \sigma _i^2 \rho _i$
ϕt=(π/4)iσi2ρi⁠, with σi the diameter and ρi the number density of component i.

Our colloidal system consists of carboxylic acid functionalized melamine formaldehyde particles (Microparticles GmbH) with hard sphere diameters of σs = 2.79 μm and σl = 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 electron microscopy. The particles are allowed to sediment onto the base of a glass sample cell with a height of 200 μm, which is cleaned before use with a 2% solution of Hellmanex. The particle mass density of 1.57 g/ml results in a gravitational length of 0.07 μm and 0.02 μm for the small and large particles, respectively. As this is a fraction of the size of the particles, the out of plane fluctuations are negligible and the system is structurally two-dimensional.

The colloidal system is imaged using an Olympus CKX41 inverted bright-field microscope equipped with a PixeLink CMOS camera (1280 × 1080 pixels). Particle coordinates are subsequently obtained from the microscopy images with an error of 12 ± 10 nm using standard particle tracking software.39 At the highest measured total packing fraction ϕt for each of the different compositions, there are typically between 3000 and 4000 particles in each frame. Large and small particles are readily distinguished based upon the integrated brightness of the features found. The total packing fraction is varied over a range from approximately ϕt = 0.05 to 0.76 for six different systems with relative packing fractions, q = ϕlt, of approximately 0, 0.17, 0.37, 0.50, 0.71, and 1. Here, ϕl is the packing fraction of the large particles. Figure 1(a) shows a state diagram with all the experimental compositions studied and Fig. 1(b) shows a typical snapshot of the binary system at q = 0.71 and ϕt = 0.67.

FIG. 1.

(a) State diagram of the packing fractions of large (ϕl) versus small (ϕs) particles, showing the different compositions studied. The dashed lines indicate state points with a similar composition q = ϕlt. (b) A typical microscopy image of the binary colloidal system at q = 0.71 and ϕt = 0.67.

FIG. 1.

(a) State diagram of the packing fractions of large (ϕl) versus small (ϕs) particles, showing the different compositions studied. The dashed lines indicate state points with a similar composition q = ϕlt. (b) A typical microscopy image of the binary colloidal system at q = 0.71 and ϕt = 0.67.

Close modal

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

\begin{equation}x_i x_j g_{ij}(r)= \frac{1}{\rho }\left<\frac{1}{N}\sum ^{N_i}_{\mu =1}\sum ^{N_j}_{\nu \ne \mu } \delta ({\bf r}+{\bf r}_\mu -{\bf r}_\nu )\right>,\end{equation}
xixjgij(r)=1ρ1Nμ=1NiνμNjδ(r+rμrν),
(1)

with N the total number of particles, ρ = N/A the total number density of the system (A is the surface area), xi, j = Ni, j/N the fraction 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 component system particles μ and ν are by necessity of the same species, but in a mixture we can define radial distribution functions for i and j representing the same or different particle species. For the binary system considered here, this results in the following three partial radial distribution functions, gss(r), gll(r), and gls(r) = gsl(r), which depend upon correlations between like or unlike particles. The experimental radial distribution functions are averaged over at least 200 frames.

In order to compare the experimentally measured radial distribution functions with theoretical predictions we calculate gij(r) within the framework of classical density functional theory40 (DFT) using the so-called test particle route. In this method one fluid particle of component i = s, l is fixed at the origin of the frame of reference and thereby turned into an external field for the rest of the system. The inhomogeneous density distributions

$\rho _j^{(i)}(r)$
ρj(i)(r) of species j = s, l, of the fluid mixture, subjected to an external field of a fixed particle of species i, are calculated numerically by minimizing the density functional. From the density distributions one can obtain the radial distribution functions directly:

\begin{equation}g_{ij}(r) = \frac{\rho ^{(i)}_j(r)}{\rho _j}.\end{equation}
gij(r)=ρj(i)(r)ρj.
(2)

Since we are interested in the radial distribution functions of fluid mixtures, we require a DFT for hard-disk mixtures. Fundamental measure theory41,42 is a reliable and versatile DFT approach for hard-body mixtures. Despite this, an accurate FMT functional for hard-disk mixtures has only recently been constructed and, up to now, only been applied to one-component systems.43 

To compare the experimental data to our FMT calculations the size ratio, α = σls, of the particles in the mixture must be considered. For the experimental system in 3D the size ratio is α = 4.04/2.79 ≈ 1.45, but at the base of the sample cell the centers of the small and large particles are not in the same plane. This makes the mixture slightly non-additive, i.e., σls = (1/2)(σll + σss)(1 + Δ) with Δ ≠ 0. By projecting the centers of a large and a small particle in contact onto the base plane one finds an effective size ratio of αeff ≈ 1.41, corresponding to a small negative value of Δ ≈ −0.017. In our FMT calculations for additive mixtures we assume a size ratio in between these two values and set αDFT = 1.43. We note that αDFT = 1.45 (1.41) slightly improves the agreement between experiments and theory for systems rich in large (small) particles. The deviations from the results for αDFT = 1.43 are, however, small, indicating that the small degree of non-additivity has little effect.

First, we present radial distribution functions for a one-component fluid, which corresponds to either only small particles (q = 0) or only large particles (q = 1). Figure 2 shows the g(r) for various total packing fractions ϕt as a function of the dimensionless radial distance r/σ. The experimental data (symbols) and theoretical predictions (lines) are in excellent agreement for all details, i.e., the contact value, the wavelength of oscillation, the decay (correlation) length, for all values of ϕt without any adjustable parameters. The high level of agreement between our experimental data and FMT calculations over the whole range of packing fractions confirms the experimental values of particle diameters, the packing fractions and shows that the colloidal system is an almost perfect model system for hard disks.

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).

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).

Close modal

More challenging, both for experiment and theory, is the determination of the partial radial distribution functions, gij(r), for a binary mixture. In Fig. 3 we show the three partial radial distribution functions gll(r), gls(r) = gsl(r), and gss(r) for various combinations of the packing fractions of the large (ϕl) and small (ϕs) particles as obtained from the experiments (symbols) and theory (lines). The agreement between experiments and theory is excellent for all compositions, even at rather high values of the total packing fraction ϕt. Both the experimental data and theoretical results show the same variation of the radial distribution functions arising from complicated packing effects for different parts of the ϕls plane (Fig. 1(a)). Note that for a binary mixture with two distinct diameters, a discontinuous change of the wavelength of oscillations in the asymptotic regime of the radial distribution functions gij(r), the so-called structural crossover, was predicted46,47 based on the theory of asymptotic decays of correlation functions, and found experimentally.48 For the size ratio used here, however, the diameters are too similar to expect structural crossover.

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 and dashed-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 have shifted gls(r) and gll(r) vertically.

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 and dashed-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 have shifted gls(r) and gll(r) vertically.

Close modal

Finally, we consider the contact values of the radial distribution function, gijij), for both the single component and binary systems. The experimental determination of the contact values, however, is rather subtle due to the fact that the measured g(r) is the convolution of the “true” g(r) with the finite bin size used in the measurement. As a result, the first peak is shifted to a slightly larger distance and the height of the first peak, g1, is smaller than the “true” contact value. The height of the first peak also depends on the bin size49 and the positioning 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 centering positions of the bins. As expected g1 decreases with the bin size, but is relatively independent of both the bin size and the centering position of the bins for bin sizes in the range of 0.02–0.05 μm. We therefore computed the distribution functions using a bin size of 0.0425 μm, which is chosen to be at the upper end of this range to optimise statistics.

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 (solid line).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 represent the predictions from Santos et al.45 for gijij) for compositions ranging from q = 0.17 to 0.71: gss (blue), gls (orange), and gll (green).

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 (solid line).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 represent the predictions from Santos et al.45 for gijij) for compositions ranging from q = 0.17 to 0.71: gss (blue), gls (orange), and gll (green).

Close modal

Next, the contact values gijij) are determined from our experimental radial distribution functions by fitting the decay of the first peak with an exponential function and extrapolating back to the hard disk diameter.49–53 The resulting contact values for the single component fluid and the binary systems are presented in Figs. 4(a) and 4(b), respectively. For the single component system our FMT gives an expression for the contact values equivalent to that from 2D scaled particle theory,44 and the agreement with the experimental data is excellent. In Fig. 4(b), we show the contact values for the three partial radial distribution functions of the binary system, gll, gls, and gss, for all studied compositions (q = 0.17, 0.37, 0.5, and 0.71). Our FMT does not yield a closed expression for the contact values and scaled particle theory54 does not agree well with contact values from simulations.55,56 We therefore compare our data to the prediction of Santos et al.,45 which agrees well with simulation data.56 As the contact values are weakly composition dependent, we represent the prediction of Santos for each gijij) as a coloured region, where the width accounts for the composition dependence. Again excellent agreement with our experimental data is observed, further corroborating the quantitative mapping between our colloidal system and two-dimensional hard disks.

In summary, we have presented a comprehensive comparison of the radial distribution functions of a two-dimensional binary colloidal hard sphere model system and fundamental measure theory calculations. We find excellent agreement between the experiments and fundamental measure theory for binary hard disks. The contact values of the radial distribution function also show excellent comparison to theoretical expressions, both for the single component and binary systems. Furthermore, our results confirm that the very small degree of non-additivity in the experimental system, which cannot be treated in the present theory, has little influence on the results. The unique and fully quantitative mapping between our experimental colloidal system and theory is remarkable as it allows for a precise determination of the hard sphere packing fraction. Moreover, it provides a direct quantitative theoretical understanding of the structure in our experimental system, which is crucial in establishing the fundamental relation between structure and dynamics in simple liquids and glass forming systems.

We thank Jürgen Horbach, Martin Oettel, and Michael Juniper for useful discussions. A.L.T., D.G.A.L.A., and R.P.A.D. acknowledge the Engineering and Physical Sciences Research Council (UK) (EPSRC) for financial support.

1.
J.-P.
Hansen
and
I. R.
Macdonald
,
Theory of Simple Liquids
, 4th ed. (
Academic Press
,
2013
).
2.
X.
Ma
,
W.
Chen
,
Z.
Wang
,
Y.
Peng
,
Y.
Han
, and
P.
Tong
,
Phys. Rev. Lett.
110
,
078302
(
2013
).
3.
T.
Kawasaki
and
H.
Tanaka
,
J. Phys.: Condens. Matter
23
,
194121
(
2011
).
4.
6.
H.
Longuet-Higgins
and
B.
Widom
,
Mol. Phys.
8
,
549
(
1964
).
7.
J. D.
Weeks
,
D.
Chandler
, and
H. C.
Andersen
,
J. Chem. Phys.
54
,
5237
(
1971
).
8.
S.
Amore
,
J.
Horbach
, and
I.
Egry
,
J. Chem. Phys.
134
,
044515
(
2011
).
9.
S.
Sastry
,
T. M.
Truskett
,
P. G.
Debenedetti
,
S.
Torquato
, and
F. H.
Stillinger
,
Mol. Phys.
95
,
289
(
1998
).
10.
J. D.
Weeks
,
D.
Chandler
, and
H. C.
Andersen
,
J. Chem. Phys.
55
,
5422
(
1971
).
11.
12.
M.
Wertheim
,
Phys. Rev. Lett.
10
,
321
(
1963
).
13.
F.
Lado
,
J. Chem. Phys.
49
,
3092
(
1968
).
14.
Y.
Uehara
,
T.
Ree
, and
F. H.
Ree
,
J. Chem. Phys.
70
,
1876
(
1979
).
15.
L.
Verlet
and
D.
Levesque
,
Mol. Phys.
46
,
969
(
1982
).
16.
D. G.
Chae
,
F. H.
Ree
, and
T.
Ree
,
J. Chem. Phys.
50
,
1581
(
1969
).
17.
W. W.
Wood
,
J. Chem. Phys.
52
,
729
(
1970
).
18.
W. W.
Wood
and
J. D.
Jacobson
,
J. Chem. Phys.
27
,
1207
(
1957
).
19.
B. J.
Alder
and
T. E.
Wainwright
,
J. Chem. Phys.
27
,
1208
(
1957
).
20.
J. G.
Kirkwood
,
E. K.
Maun
, and
B. J.
Alder
,
J. Chem. Phys.
18
,
1040
(
1950
).
21.
P.
Pusey
and
W. V.
Megen
,
Nature (London)
320
,
340
(
1986
).
22.
C.
Royall
,
W.
Poon
, and
E.
Weeks
,
Soft Matter
9
,
17
(
2013
).
23.
W. V.
Megen
and
I.
Snook
,
Adv. Colloid Interface Sci.
21
,
119
(
1984
).
24.
A. V.
Blaaderen
and
P.
Wiltzius
,
Science
270
,
1177
(
1995
).
25.
R. P. A.
Dullens
,
D. G. A. L.
Aarts
, and
W. K.
Kegel
,
Proc. Natl. Acad. Sci. U.S.A.
103
,
529
(
2006
).
26.
C. P.
Royall
,
A. A.
Louis
, and
H.
Tanaka
,
J. Chem. Phys.
127
,
044507
(
2007
).
27.
C. A.
Murray
and
D. V.
Winkle
,
Phys. Rev. Lett.
58
,
1200
(
1987
).
28.
M.
Brunner
,
C.
Bechinger
,
W.
Strepp
,
V.
Lobaskin
, and
H.
von Grunberg
,
Europhys. Lett.
58
,
926
(
2002
).
29.
M.
Brunner
and
C.
Bechinger
,
Europhys. Lett.
63
,
791
(
2003
).
30.
S.
Behrens
and
D.
Grier
,
Phys. Rev. E
64
,
050401
R
(
2001
).
31.
M.
Quesada-Perez
,
A.
Moncho-Jorda
,
F.
Martinez-Lopez
, and
R.
Hidalgo-Alvarez
,
J. Chem. Phys.
115
,
10897
(
2001
).
32.
G.
Kepler
and
S.
Fraden
,
Phys. Rev. Lett.
73
,
356
(
1994
).
33.
A.
Marcus
and
S. A.
Rice
,
Phys. Rev. Lett.
77
,
2577
(
1996
).
34.
B.
Cui
,
B.
Lin
, and
S. A.
Rice
,
J. Chem. Phys.
114
,
9142
(
2001
).
35.
B.
Cui
,
B.
Lin
, and
S. A.
Rice
,
J. Chem. Phys.
119
,
2386
(
2003
).
36.
M.
Carbajal-Tinoco
,
F.
Castro-Román
, and
J.
Arauz-Lara
,
Phys. Rev. E
53
,
3745
(
1996
).
37.
N.
Hoffmann
,
F.
Ebert
,
C.
Likos
,
H.
Löwen
, and
G.
Maret
,
Phys. Rev. Lett.
97
,
078301
(
2006
).
38.
O.
Marnette
,
E.
Perez
,
F.
Pincet
, and
G.
Bryant
,
Colloids Surf., A
346
,
208
(
2009
).
39.
J.
Crocker
and
D.
Grier
,
J. Colloid Interface Sci.
179
,
298
(
1996
).
41.
42.
R.
Roth
,
J. Phys.: Condens. Matter
22
,
063102
(
2010
).
43.
R.
Roth
,
K.
Mecke
, and
M.
Oettel
,
J. Chem. Phys.
136
,
081101
(
2012
).
44.
E.
Helfand
,
H. L.
Frisch
, and
J. L.
Lebowitz
,
J. Chem. Phys.
34
,
1037
(
1961
).
45.
A.
Santos
,
S. B.
Yuste
, and
M.
Lo´pez de Haro
,
J. Chem. Phys.
117
,
5785
(
2002
).
46.
C.
Grodon
,
M.
Dijkstra
,
R.
Evans
, and
R.
Roth
,
J. Chem. Phys.
121
,
7869
(
2004
).
47.
C.
Grodon
,
M.
Dijkstra
,
R.
Evans
, and
R.
Roth
,
Mol. Phys.
103
,
3009
(
2005
).
48.
J.
Baumgartl
,
R. P. A.
Dullens
,
M.
Dijkstra
,
R.
Roth
, and
C.
Bechinger
,
Phys. Rev. Lett.
98
,
198303
(
2007
).
49.
S.
Luding
and
O.
Strauß
,
Granular Gases
(
Springer
,
2001
).
50.
M.
Barošová
,
M.
Malijevský
,
S.
Labik
, and
W.
Smith
,
Mol. Phys.
87
,
423
(
1996
).
51.
M.
Gonzalez-Melchor
,
J.
Alejandre
, and
M.
Lopez de Haro
,
J. Chem. Phys.
114
,
4905
(
2001
).
52.
D.
Cao
,
K.-Y.
Chan
,
D.
Henderson
, and
W.
Wang
,
Mol. Phys.
98
,
619
(
2000
).
53.
J.
Tobochnik
and
P. M.
Chapin
,
J. Chem. Phys.
88
,
5824
(
1988
).
54.
J. L.
Lebowitz
,
E.
Helfand
, and
E.
Praestgaard
,
J. Chem. Phys.
43
,
774
(
1965
).
55.
C.
Barrio
and
J. R.
Solana
,
J. Chem. Phys.
115
,
7123
(
2001
).
56.
S.
Luding
and
A.
Santos
,
J. Chem. Phys.
121
,
8458
(
2004
).