7.2. Osmotic pressure[[Student version, January 17, 2003]] 223
glass cover slip100xfocal planea
Figure 7.4: (Schematic; experimental data.) (a)Experimental setup of an experiment to measure depletion
interactions. A microscope looks at the central plane of a rigid vesicle containing a polystyrene sphere (the “sheep”)
of radius 0.237μm.(b)Histogram of the measured location of the large sphere’s center over 2000 observations. The
solvent in this case contained no smaller objects. Instead of displaying frequencies by the height of bars, the figure
instead shows how often the sphere was found in each location by the color of the spot at that position; lighter colors
denote places where the sphere was more often found. The dashed line represents the actual edge of the vesicle; the
sphere can come no closer than its radius. (c)As(b), but this time the vesicle contained a suspension of smaller,
0.04μmspheres (“sheepdogs”) with volume fraction about 30%. Though the “sheepdogs” are not optically visible,
they cause the “sheep” to spend most of its time clinging to the wall of the chamber. [Digital image kindly supplied
byA. Dinsmore; see Dinsmore et al., 1998.]
Now let the two surfaces come together. If their shapes match, then as they come together their
depletion zones merge and finally disappear (Figure 7.3b). The corresponding reduction in free
energy gives an entropic force driving the surfaces into contact. The effect does not begin until the
twosurfaces approach each other to within the diameter 2aof the small particles: The depletion
interaction is of short range. Even if the two surfaces’ shapes do not match precisely, there will
still be a depletion interaction, as long as their shapes are similar on the length scale of the small
particles. For example, when two big spheres meet (or when a big sphere meets a flat wall), their
depletion zones will shrink as long as their radii are much bigger thana,because they look flat to
the small spheres.
Wecan also interpret the depletion interaction in the language of pressure. Figure 7.3b shows
asmall particle attempting to enter the gap, but instead bouncing away. It’s as though there were
asemipermeable membrane at the entrance to the gap, admitting water but not particles. The
osmotic pressure across this virtual membrane sucks water out of the gap, forcing the two large
particles into contact. The pressure is the change of free energy per change of volume (Equation 7.2).
As we bring the surfaces into contact, the volume of the depletion zone between them shrinks from
2 aAto zero. Multiplying this change by the pressure dropckBTin the zone gives
(∆F)/A=ckBT× 2 a. (7.10)The rearrangement of a thin layer around a huge particle may seem unimportant, but the total
effect of the depletion interaction can be considerable (see Problem 7.5).
A. Dinsmore and coauthors gave a clear experimental demonstration of the depletion interaction
(Figure 7.4). They prepared a vesicle containing one large particle, about a quarter of a micrometer
in radius, and a solution. In one trial the solution contained a suspension of smaller particles, of
radius 0.04μm;inanother trial these particles were absent, with everything else the same. After
carefully arranging to eliminate all other attractive forces between spheres (for example, electrostatic
repulsion), they found a dramatic effect: The mere presence of the small “sheepdog” particles forced