Inverse density-functional theory as an interpretive tool for measuring
colloid-surface interactions in dense systems
Mingqing Lu, Michael A. Bevan, and David M. Ford
a͒
Department of Chemical Engineering, Texas A&M University, College Station, Texas 77843-3122
͑Received 28 March 2005; accepted 18 April 2005; published online 15 June 2005͒
Recent advances in optical microscopy, such as total internal reflection and confocal scanning laser
techniques, now permit the direct three-dimensional tracking of large numbers of colloidal particles
both near and far from interfaces. A novel application of this technology, currently being developed
by one of the authors under the name of diffusing colloidal probe microscopy ͑DCPM͒,istouse
colloidal particles as probes of the energetic characteristics of a surface. A major theoretical
challenge in implementing DCPM is to obtain the potential energy of a single particle in the external
field created by the surface, from the measured particle trajectories in a dense colloidal system. In
this paper we develop an approach based on an inversion of density-functional theory ͑DFT͒, where
we calculate the single-particle-surface potential from the experimentally measured equilibrium
density profile in a nondilute colloidal fluid. The underlying DFT formulation is based on the recent
work of Zhou and Ruckenstein ͓Zhou and Ruckenstein, J. Chem. Phys. 112, 8079 ͑2000͔͒. For
model hard-sphere and Lennard-Jones systems, using Monte Carlo simulation to provide the
“experimental” density profiles, we found that the inversion procedure reproduces the true
particle-surface-potential energy to an accuracy within typical DCPM experimental limitations
͑ϳ0.1kT͒ at low to moderate colloidal densities. The choice of DFT closures also significantly
affects the accuracy. © 2005 American Institute of Physics. ͓DOI: 10.1063/1.1929734͔
I. INTRODUCTION
A. Diffusing colloidal probe microscopy
Diffusing colloidal probe microscopy ͑DCPM͒ is an
emerging surface analysis technique currently being devel-
oped by one of the authors.
1
In DCPM, an ensemble of freely
diffusing colloidal particles are employed as ultrasensitive
probes of a nearby surface. Total internal reflection
2
- and
video
3
-based optical microscopy techniques, which combine
the scattering of an evanescent wave with standard image
capture and analysis algorithms, are used to monitor the
three-dimensional Brownian excursions of the colloidal par-
ticles as they sample spatial positions over time. Confocal
scanning laser microscopy
4
methods are also being devel-
oped to measure three-dimensional colloidal trajectories near
surfaces. The particle trajectories may then be analyzed as
time-averaged distribution functions using statistical me-
chanical interpretations to yield the relative potential energy
of a single colloidal particle as a function of xyz position
near the surface ͑the main goal of this paper͒. If the surface
is known to be a chemically and physically uniform plane,
only the surface normal direction z is important and the
analysis is consequently simplified.
Because DCPM exploits colloidal Brownian motion as a
natural gauge of potential-energy landscapes, it is inherently
capable of measuring energies and forces 10
3
times weaker
than the range accessible using “top down” methods employ-
ing external mechanical manipulation ͑i.e., scanning probes,
optical tweezers͒.
2
Preliminary work
1
has successfully
implemented DCPM to perform ensemble measurements in
model synthetic systems, and the technique is currently being
extended to measure specific equilibrium interactions be-
tween protein pairs covalently attached to metal nanopar-
ticles and planar substrate surfaces.
B. Theoretical requirements
The main goal of this paper is to develop a theoretical
approach that yields an accurate single-particle-surface po-
tential based on information extracted from particle distribu-
tion functions measured using DCPM. Density-functional
theory ͑DFT͒ is a logical choice for this purpose. One of the
key expectations underlying DFT is a unique correspondence
between a density profile and the underlying particle and
surface pair potentials.
5
Thus, we expect that if we extract an
equilibrium density profile
͑r͒ from the DCPM measured
particle trajectories, we can invert it to obtain the colloid-
surface potential
ext
͑r͒ using the fundamentals of DFT.
Subject to spatial and temporal sampling limitations, the
DCPM technique indeed produces histograms
1
that may be
normalized to produce density profiles. This paper focuses
on the development of an inverse DFT approach that is ͑1͒
accurate to within the inherent experimental limitations of
DCPM, typically on the order of 0.1kT, ͑2͒ fast enough to
report a surface potential within minutes on a computer
workstation, and ͑3͒ systematically adaptable to a range of
different particle and surface types.
a͒
Author to whom correspondence should be addressed. Fax: ͑979͒ 845-
6446; electronic mail: d-ford@chemail.tamu.edu
THE JOURNAL OF CHEMICAL PHYSICS 122, 224710 ͑2005͒
0021-9606/2005/122͑22͒/224710/6/$22.50 © 2005 American Institute of Physics122, 224710-1