By tracking the Brownian trajectory of a sphere of 1 p,m radius in a suspension of 35 nm radius beads, we probe the depth of the entropic potential experienced by a large sphere close to the cell wall. This potential is caused by the pressure exerted by the Quid of small beads when they cannot fit in the gap between the large bead and the wall. We find that the potential depth is of order kT. Its dependence on the volume fraction of small spheres is significantly overestimated by a theory which treats the small beads as an ideal gas.

1. VOLUME 73, NUMBER 21 PH YS ICAL REVIEW LETTERS 21 NovEMBER 1994
Direct Observation of the Entropic Potential in a Binary Suspension
P. D. Kaplan, ' Luc P. Faucheux, and Albert J. Libchaber'
'NEC Institute, 4 Independence TVay, Princeton, New Jersey 08540
Physics Department, Princeton University, Princeton, New Jersey 08542
(Received 10 May 1994)
By tracking the Brownian trajectory of a sphere of 1 p, m radius in a suspension of 35 nm radius
beads, we probe the depth of the entropic potential experienced by a large sphere close to the cell
wall. This potential is caused by the pressure exerted by the Quid of small beads when they cannot fit
in the gap between the large bead and the wall. We find that the potential depth is of order kT. Its
dependence on the volume fraction of small spheres is significantly overestimated by a theory which
treats the small beads as an ideal gas.
PACS numbers: 05.40.+j, 65.50.+m, 82.70.Dd
An object bombarded on all sides by small particles
experiences random instantaneous forces and zero average
force unless the population of particles on one side has
been reduced. This idea was first used in an eighteenth-
century attempt to explain gravitation [1]and in the 1950s
to explain the attraction between large particles in a fluid
of smaller particles when the gap between large particles
is too small for the small ones to fit [2,3]. This effect
contributes to the thermodynamics of any mixture of
macromolecules with significantly different sizes. In the
colloid and polymer literature this is referred to as the
depletion force [4—
6]. In the biological literature it is
known as macromolecular crowding, and it significantly
affects reaction rates between large macromolecules in the
crowded cellular environment [7]. The recent literature on
the possible existence of phase separation in binary hard-
sphere mixtures depends on this entropic effect [8—
13].
In this Letter, we present the first direct observation of the
entropic potential between a large particle and a wall.
The free energy of a hard-sphere system is entirely
entropic, depending on the volume accessible to the
particles. A potential arises from the change in the
volume available to the small spheres when the large
sphere of radius RL is pushed against the wall. This is
because the center of the small bead is excluded from
both a shell around the large bead and a layer along the
wall [8] of thickness Rs, where Rs is the small bead
radius (Fig. 1). When the large bead is in contact with the
wall some of the excluded volumes overlap. The volume
available to the small bead increases, and the total free
energy of the system decreases.
The sample was a suspension of 35 nm radius polysty-
rene beads to which we added a very small number of 1
or 1.25 p,m radius silica beads. The repulsive Coulomb
interactions which prevent the beads from sticking to the
coverslip and to each other are exponentially screened,
decaying over a few Debye lengths. We add salt (0.01M
NaC1) to reduce the Debye length to 3 nm, making the
interparticle potential essentially hard sphere [8,14]. All
glassware was cleaned with surfactant [15] and rinsed in
deionized water. Each sample cell was constructed from a
microscope slide, a 50 p,m mylar spacer, a nuinber 1 cov-
erslip, and fast epoxy. The surface of the coverslip was
imaged by an inverted microscope with an oil immersed
100X objective (see Fig. 2). In this experiinent, unlike
most experiments in optical microscopy, the coverslip is
a central part of the system; we are studying the entropic
potential between the coverslip and the large bead. This
potential depends only on the height of the bead. Note
that the bead can diffuse parallel to the coverslip with-
out altering the overlap of excluded volumes. Using the
techniques described elsewhere [16],the computer moni-
tors the horizontal but not the vertical position of the large
bead by processing images at 15 to 30 Hz. For each sam-
ple, between 5 and 20 time series consisting of 10000
points were obtained.
The time series contains information about vertical dis-
placements because the horizontal diffusion coefficient is
a function of the vertical position of the bead [16]: When
the bead is close to the wall, its diffusion coefficient and
average horizontal displacements are smaller than when
FIG. 1. The entropy of the small beads in a binary mixture
depends on their accessible volume. The center of each small
bead is excluded from a layer of thickness R& around the large
bead and along the wall. When the large bead is moved against
the wall, the excluded volumes overlap, the volume available to
the small beads increases, and the free energy of the system
decreases. The overlap in the excluded volumes responsible
for this entropic potential is shaded.
0031-9007/94/73(21)/2793(4)$06. 00 1994 The American Physical Society 2793

2. VOLUME 73, NUMBER 21 PHYSICAL REVIEW LETTERS 21 NovEMBER 1994
Z J
bjectiv
Camera Coxnputer
FIG. 2. The experimental setup: a large sphere (radius 1 and
1.25 p, m) is suspended in an aqueous suspension of 35 nm
radius particles. The charge coupled device camera records
the horizontal bead displacement, which is then digitized by
the computer.
it is further from the wall. The diffusion coefficient
decreases in the vicinity of the wall because there is less
space for the fiow of water around the moving bead.
The time series of the horizontal bead displacement
along one horizontal axis (Fig. 3) suggests that the motion
of the bead can be described by a two-state model: The
bead is either "trapped" close to the wall or "free" in
solution with a larger horizontal diffusion coefficient than
in the trapped state. We see this in Fig. 3. There are
periods during which the displacement of the bead is
significantly smaller than the average displacement. This
is the signature of a bead trapped against the coverslip
by the entropic potential. Three of these periods are
indicated in Fig. 3 by a dark line above the time series.
Identification of the state of the particle at any point in
the time series is uncertain; it is only on average that
the horizontal displacement is smaller when the bead
is trapped.
By analyzing statistics of the time series as a whole, we
clearly identify the effects of trapping. Once the bead is
trapped, the mean escape time from the potential well of
depth BF is
c5FjkT
~ —~Fe
where k is the Boltzmann constant, T is the temperature,
and rF is the inverse frequency of escape attempts [17].
Measurement of the mean escape time, known as the
Kramers' time, directly gives access to the depth 6F.
Similarly, once the bead escapes from the well, it will
become trapped again after a characteristic time that
we assume is longer than v. As we cannot identify
individual trapping events from the time series, we
resort to a statistical analysis, examining the probability
P(hx, ht) to measure a horizontal displacement Ax of
the bead after a time At. For a particle with a single
diffusion coefficient, P(b, x, At) is a simple Gaussian, a
straight line when plotted logarithmically against Ax .
Over intervals shorter than the mean escape time r, the
bead in this experiment is either trapped or free. The
probability P(hx, b, t) should then be the sum of the two
Gaussians corresponding to the trapped and free diffusion
coefficients [see Fig. 4, inset (a)]. On the other hand,
over intervals At much larger than v, the fraction of time
during which the bead is trapped (r/Ar) tends to zero,
and the probability distribution should approach a single
Gaussian characterized by an average diffusion coefficient
[see Fig 4, in. set (b)]. These qualitative features of the
probability distributions P(hx, ht) shown in the insets of
Fig. 4 are consistent with this two-state model.
To test this model, we fit the distribution P(b,x, b, r)
by one and by two Gaussians. The accumulated square
errors of the fits are, respectively, gz(I) and g (2). We
compare these fits in Fig. 4 by plotting the value of the
relative error y = g2(1)jg2(2) —1 as a function of b, t
Large values of y at small offset times At indicate that
P(hx, b, t) is much better described by two Gaussians than
21.0-
20.5—
20.0—
195—
18.0—
I
0
I
50
I
I
I
I
II
I
II
I
I'I
II
I
'aII
I'
100 1 50
t (sec)
I
200 250
I
300
FIG. 3. Time series of the horizontal displacement in microns
along one direction of 1 p,rn radius silica bead in a solution
of 35 nm radius polystyrene spheres (volume fraction is 0.28)
as a function of time in seconds. Three periods of smaller
than average displacement are indicated by a thick bar above
the time series. The bead is probably trapped by the entropic
potential during these periods.
20 ~
0 3
I
dt {sec)
FIG. 4. We fit the probability distribution P(hx, br) (insets)
with one and with two Gaussians. The relative error in
these fits, y = g'(I)/g'(2) —1, is plotted as a function of the
offset time At in seconds for a 1 p,m radius silica sphere in
a solution of small spheres (volume fraction is 0.28). The
solid line corresponds to the fit of y by a Lorentzian. Insets
show the probability P(hx, b, t) as a function of the horizontal
displacement Ax2 in microns squared for (a) d, t = 2.15 and
(b) At = 24.6 s.

3. VOLUME 73, NUMBER 21 PHYSICAL REVIEW LETTERS 21 NOVEMBER 1994
by a single one. We estimate the transition time ~ from a
two Gaussian distribution to a single Gaussian distribution
by fitting the relative error y with a Lorentzian A/[I +
(/J. t/r) ] (solid line in Fig. 4). We choose a Lorentzian
because it is a simple function with a single characteristic
time; there are other equally good functions we could
have chosen. To check that the time 7- obtained in this
fit is a measure of the Kramers' time, we used the same
analysis on time series generated by computer simulations
of a two-state random walker in which the Kramers'
time is known. The time recovered by this analysis was
within 10% of the actual mean escape time as long as the
Kramers' time is much smaller than the capture time.
The potential depth is equal to kT In(r/rF) from
Eq. (1). An estimate of rF is the time rD = Rs/2D
it takes for the large bead to diffuse in the vertical
direction to the edge of the well (a small bead diameter)
where D is the diffusion coefficient in the vertical
direction. The diffusion coefficient D is not equal to
the Stokes-Einstein value Dp = kT/6rrripRI, , where rip
is the bulk viscosity of water. When the large bead is
a distance Rs from the wall, the diffusion coefficient
is reduced by a factor of Rs/RL due to hydrodynamic
interactions with the wall [18]. Measurements of the
bulk viscosity of hard-sphere suspensions indicate that
the viscosity increases by a factor of (I —Ps/0. 63)
where Ps is the volume fraction filled by small beads
[18]. Combining these factors gives the diffusive time
as rn = Rs/2Dp(Rs/RL)(1 —@s/0.63)2 which is of the
order of 0.01 s. We plot in Fig. 5 the values of ln(r/rD)
as a function of Ps for the different samples studied.
The smallest volume fraction for which we could detect
trapping is Ps = 0.14. At lower volume fractions the
Kramers' time becomes comparable to our sampling rate.
To better understand the measurements, we calculated
the potential assuming that the Quid of small beads
is an ideal gas (straight line in Fig. 5). Within this
approximation, BF is proportional to the overlapping
excluded volumes shown in Fig. 1, and is equal to BF =
3kTPs(RL/Rs) [8]. The physical limitation of the ideal
gas approximation is that it allows small beads to overlap
each other. The dependence of the observed potential on
@s deviates from the above ideal gas law. Note that
in this ideal gas model we use effective hard sphere
radii which are the actual radii of the beads plus a few
Debye lengths (3 nm) to account for screened Coulomb
repulsion.
Our calculation of ro is only an estimate of rF.
Any correction to rD will result only in a vertical
shift of the data and will not change the observed
slope of the data in Fig. 5. In order to check that
rD is a reasonable estimate of rF, we calculated the
potential within the Percus-Yevick (virial) approximation
[19] and found that ro/rF —2.5. We used the Percus-
Yevick approximation because it has an analytic solution
unlike the more refined theories used in Ref. [10]. The
measurements reported here highlight the need for an
explicit presentation of the entropic potential calculated
within these more sophisticated closure relations.
In conclusion, we have directly observed the entropic
potential between a large sphere and a wall in a suspen-
sion of small beads by tracking the Brownian trajectory
of the large sphere with video microscopy. The potential
is of the order of a few kT and deviates strongly from a
theory which treats the small spheres as an ideal gas.
We thank Arjun Yodh, David Pine, and Raymond
Goldstein for enlightening discussions.
4
3
~+ae
II
4
1-
0.1 0.2 0.3
FIG. 5. The potential depth BF in units of kT as a function of
the volume fraction of the small spheres Ps. The solid straight
line corresponds to the ideal gas result for a large sphere radius
of I p,m. The squares and triangles are the experimental values
of In(r/ro) for spheres of radius 1 and 1.25 y,m, respectively.
Note that the time rD is an estimate within a factor of order
one of the large bead's inverse frequency of escape attempts,
rF. Corrections to ~& could shift the data vertically by as much
as a few kT.
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