VOLUME 78, NUMBER 20

PHYSICAL REVIEW LETTERS

19 MAY 1997

Direct Measurement of Infinitesimal Depletion Force in a Colloid-Polymer Mixture
by Laser Radiation Pressure
Y. N. Ohshima,1 H. Sakagami,1 K. Okumoto,1 A. Tokoyoda,1 T. Igarashi,1 K. B. Shintaku,1 S. Toride,1 H. Sekino,1K. Kabuto,2 and I. Nishio11Department of Physics, Aoyama-Gakuin University, Setagaya, Tokyo, 157 Japan
2Photon Force Research Laboratory, Fuji, Shizuoka, 417 Japan(Received 1 May 1995; revised manuscript received 3 February 1997)

It is widely believed that depletion force causes formation of colloidal crystal or flocculation in thestable colloidal suspension mixed with nonadsorbing polymers. This force works between a colloidal
particle and plane, which can be regarded as a sphere with infinite radius. We report a novel techniquethat enables us to measure the depletion force working between a colloidal particle and a glass
surface using laser radiation pressure. We found that the observed force almost coincides with thepredicted one. [S0031-9007(97)03137-2]
PACS numbers: 82.70.Dd, 36.20.– r, 42.62.HkWhen electrolyte composition surrounding latex particle is altered properly, e.g., reduction of the ionicstrength by dialysis, latex particles form ordered phase[1]. In this case, it is widely believed that the crystal
formation is due to the interparticle repulsive force andcounted as one of the examples of Alder phase transition[1,2]. By adding adequate amounts of nonadsorbing
polymers [3] (or smaller latex particles [4]), the system
also undergoes phase separation or flocculation and forms
dense ordered phase as well as dilute disordered phase.
This phenomenon is not based on an interaction between latex particle and polymer such as bridging wherepolymers role like glue, but is caused by a new interparticle attractive force called “depletion” force. Thishypothesis [5] has been investigated theoretically [6] orexperimentally [3–7] by many groups from some angle
and technique. A brief outline of this effect—depletioneffect—is mentioned in the following sentences.
In dilute solution, polymers behave like hard sphere andrefuse to interpenetrate each other. In the mixture of suchpolymers and latex particles, there exist the regions where
the center of mass of the polymer cannot get into and isreferred to depletion region (see Fig. 1). These regions
exist surrounding each latex particle as well as along
the hard wall. The thickness of the layer of depletion
region is equal to the effective radius of polymer that may
coincide with the hard-sphere radius of polymer describedabove. When a latex particle approaches another latex
particle or hard wall, the depletion regions overlap eachother. Then, the total volume of the depletion region in
the system decreases as much as the overlapped volume.
As a result, the entropy of polymer solution increases andfree energy decreases. This discussion suggests that theentropic attractive force— depletion force—is generated
between particles as well as between particle and planesuch as the internal wall of the cell.
The increase of free energy DF is written as
2NkB T lnfsV 1 DV dyV g, where N is the total number
0031-9007y97y78(20)y3963(4)$10.00

of polymers, kB the Boltzmann constant, T the absolutetemperature, V the original free volume for the centerof mass of the polymers, and DV is the volume of theoverlapped depletion region. Here we have assumed
that dilute polymer solution is an ideal solution [8]. AsDV is a function of the gap d between two particles (orthe distance between particle and wall), depletion force
f is given as f ­ 2≠DFy≠d ­ sNkB TyV d s≠DV y≠dd.
Here we have assumed that DV yV ø 1. Then, f can bewritten as
NkB T(1)Sosm ­ PSosm .
VIn fact, NkB T yV is equal to the osmotic pressure P ofpolymer solution. We define ≠DV y≠d as the area Sosm
on which osmotic pressure presses, and this equationsuggests that depletion force is analogous to osmoticpressure. Sosm is the area of the cross circle of two
surfaces of depletion regions represented in Fig. 1.
The measurement of such microscopic interparticle
force has been almost impossible due to the difficulty off ­FIG. 1. Illustration of depletion effect. See the third andfourth paragraphs of text.

© 1997 The American Physical Society

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PHYSICAL REVIEW LETTERS

control of the particle position as well as the extremely
weak strength of the force but few examples [9]. We
have taken notice of radiation pressure—the pressing
force given by light to object—and have succeeded inapplying it to the direct measurement of depletion force.
It is of importance to point out that milliwatts of focusedlaser light can exert sufficient radiation pressure force on
micrometer sized particles [10].
Because of a few technical problems, we investigateddepletion force not between two latex particles but between latex particle and wall of cell in this study. As
described above, depletion force is generated where depletion region exists, e.g., between sphere and plane, andprovides us a good estimation of depletion force between
latex particles (see Fig. 1).
Figure 2 shows schematic representation of experimental setup and technique (see the figure caption for detailsof method of measurement). We investigated the systemthat consists of poly (ethylene oxide) (PEO) and few polystyrene latex particles in water. We set the number concentration of latex particles about 1 particleys50 mmd3 .
Polystyrene latex (1.0 mm in diameter) was obtained fromSekisui Chemical Co., Ltd. PEO (Mw ­ 2 3 106 and1 3 106 ) with the inhibitor BHT were obtained from

FIG. 2. (a) Schematic diagram of basic apparatus. (b) A latex
particle irradiated by a focused laser in the sample cell; thesample cell was placed on a XY stage. Internal cell bottom
surface was observed through objective lens by a CCD camera.
By moving the optical cell using the XY stage, the latex particle
trapped on the wall of the cell was detected using the focusedand attenuated Ar ion laser beam as an illumination. At thismoment, we were able to detect two-dimensional Brownian
motion of the particle as the fluctuation of the scattered lightfrom the particle. This Brownian motion was easily overcomeby the slight increase of the laser intensity that trapped theparticle at the center of the beam axis; since the force thatradiation pressure gives on the particle has the components
toward as well as along the center of the beam axis [10]. Then,
increasing the laser power gradually (under computer control),when the power amounted to the certain value, the trapped
particle on the internal wall of the cell by the depletion force
was torn off and blown away in solution by radiation pressure.
The value of laser intensity at that moment was regarded asa measure of the strength of the force working between theparticle and wall, that is, depletion force.

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19 MAY 1997

Aldrich Chemical Company Inc. We chose the PEO withinhibitor because PEO has a tendency to react with O2
from the air to form colored material [11]. We have usedthe value of radius of gyration, Rg , of PEO using Mw
dependence determined by J. C. Selser et al. [12] (Rg ­0.101 mm for Mw ­ 2 3 106 and Rg ­ 0.0677 mm forMw ­ 1 3 106 ). We have performed the measurement
for each Mw of PEO and found essentially the same results if scaled to Mw properly. Here, only the result forMw ­ 2 3 106 is shown in this report due to the limitation of space.
After a given amount of PEO was added to this dilute
latex solution, the sample solution was transferred into
the optical cell on the experimental setup and was kept for
one hour before the measurement to reduce the convectionin the cell. Optical cell was constructed of normal microslide glass (MATSUNAMI GLASS Ind., Ltd. S-1112)
and was disposed after each experiment. In the water,
this glass surface carries faint same negative charge aslatex particle [13]. All measurements were performed atthe room temperature.
It is important to study the trapping energy in comparison with the thermal energy kB T . In case offlat wall and spherical particle, we have DFmax ­321 s4y3drpol
g, where rlat is the2skB TNyV dpf4rlat rpolradius of the latex particle and rpol is the effective radiusof the polymer. We have used the radius of gyration Rg
as rpol in this report. From these simple estimations,
we can calculate the concentration at which DFmax goesbeyond kB T and it is found to be 0.0241 mM. It wasindeed very hard to detect trapped particles below and justabove this concentration. In addition, no trapped particle
was found at all for concentration of PEO ­ 0.
Figure 3 represents the raw data of the concentration
dependence of the minimum laser intensity required toblow the particle off the wall of the cell (in short, blowintensity). Here, each circle represents one latex particle
studied. Since we can draw only a rough tendency as it is,
distributions of blow intensity on each concentration wereplotted, e.g., in the Fig. 3 inset. We find that most ofsuch a histogram have a clear maximum or specific cutoff
value. Rather large variance (around the mode value) may
be partially due to the fact that DFmax on a given rangeof concentration is ,1 3 times of kB T as well as due tothe condition of the glass surface being not so flat for themicrometer sized particle (maximum unevenness of thesurface is 0.9 mm) [13].
We found that laser output of 1 mW gave radiationpressure force of 0.0395 6 0.0023 pN for a latex particle
on a glass surface (see [14] for details). Mode or cutoff
values of the intensity are plotted in Fig. 4. As clearlyseen here, these bars are fairly proportional to the polymer
concentration up to the certain concentration cy . Making
an analogy between depletion force and osmotic pressure,
this part of the result is easily understood. See Eq. (1).
Sosm is defined above as the area of cross circle of two

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PHYSICAL REVIEW LETTERS

FIG. 3. Dependencies of the minimum laser intensity required
to blow the particle away (in short, blow intensity) on
PEO concentration. Each circle represents the result of one
measurement. Over 97% of the events are plotted in this figure.
The number of the data is about 100 on each concentration.
Inset: typical histogram of the blow intensity at 0.0667 mM
with clear maximum.

surfaces of depletion regions and corresponds to the area
under osmotic pressure. In the case of a spherical particle
and flat wall, we found that Sosm can be written as
Sosm ­ 4prlat rpol(2)

and is constant as far as rlat and rpol are constant.
Theoretical strength of the depletion force predicted by
Eqs. (1) and (2) are presented by dotted lines as 1.55spNd
per smMd in Fig. 4. Solid line is obtained by the leastsquares method for the center of vertical bars up to cy .

FIG. 4. Concentration dependencies of the depletion force.
The length of each vertical bar corresponds to the width of
the bar of mode or cutoff value in each histogram (see Fig. 3inset). The scale of the horizontal subaxis is defined as the unitthat corresponds to the concentration where stacking energy ofthe depletion force is equal to kB T . C y shows the maximum
concentration up to where the depletion force is proportional topolymer concentration.

19 MAY 1997

Small nonzero (negative) intersection of the solid line
and ordinate might suggest the existence of Coulombic
repulsive force between glass plate and latex particles.
However, we also studied the system with 1 and 2 mMof salt (NaCl) and obtained similar results within the experimental errors. Thus, this nonzero intersection might
be due to a systematic error associated with the measurement of very weak forces. As the observed force depends
strongly on the polymer concentration but not on the ionicstrength up to 2 mM, we can conclude that we have observed the depletion force. It might be of interest to pointout here that, above 5 mM of NaCl, we found strongly
stuck particles to the glass surface that indicated the vander Waals interaction.
The ratio of slope of observed force (solid line) topredicted force (dotted line) p is 1.88. (We found thatp is 1.77 for Mw ­ 1 3 106 .) This discrepancy might
be due to the following possibilities.
(I) The error of the estimation of depletion region (isequivalent to that of effective radius, rlat or rpol ). (i) The
repulsive interaction between glass wall and PEO (orbetween latex particle and PEO) that might increase thethickness of the layers of depletion region along the glasswall and, as a result, increase the depletion force. (ii) In
the estimation of depletion force we have used Rg asrpol . It is important to point out that Rg might not always
coincide to rpol [7], since rpol should be the radius whenpolymer behaves like hard sphere. (iii) Physisorption of
the polymers to the latex particle that might increase rlat .
(II) If the bottom part of the latex sphere that ispressed to the glass wall by the depletion force becomesflat and the vertical height becomes 5% less than theoriginal diameter, Sosm increases as much as 20%. Suchdeformation of shape of latex particle also increases
apparent depletion force.
(III) Another attractive force depending on the PEO
concentration exists. Despite this discrepancy, we would
like to stress the fact that the observed strength ofdepletion force almost coincides with the predicted one.
The remaining problem is the sudden decrease of thedepletion force above c y . To understand these results,
it is very important to point out that polymer solution
has a characteristic length scale called correlation length
j. j is equal to hydrodynamic radius and proportional
to rpol and Rg for the dilute solution where polymer
is independent of each other. However, above certain
concentration, the polymer starts to overlap each other.
Such concentration is referred to as cp that gives theboundary between dilute solution and semidilute solution[8]. For semidilute solution, because of the physicalinterpolymer entanglement, each polymer is divided into
smaller units called blobs. The size of these blobs
determines the characteristic length scale of the solution.
In other words, polymer behaves as the sphere with aradius of j. Thus, rpol in Eq. (2) must be replaced withcorrelation length j.
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FIG. 5. Dependencies of the correlation length j of PEO
on concentration. These plots are obtained by quasielastic
light scattering spectroscopy. Measurements were done at
1yjk$s j ­ 0.1291 mm. The concentration where j begins todecrease gives us cp of polymer solution where polymers start
to overlap.

Figure 5 represents the observed j of the polymer
solution. As clearly seen here, sudden decrease of joccurs at about 0.1 mM. This concentration corresponds
to cp for this system. Above cp , P might increase but
the Sosm decrease resulted in the decrease of the depletion
force. It is of great importance to point out that cy wheredepletion force begins to decrease suddenly is in good
agreement with c p where polymers start to overlap.
These results clearly show that the force which traps
latex particles onto the glass surface is mainly the depletion force. Moreover, this conclusion provides us withstrong evidence that depletion force causes the formationof colloidal crystal or flocculation in the stable colloidal
suspension mixed with water-soluble polymer. Here wewant to emphasize that the force working to micrometer
sized particle is measured directly and the observed value
almost coincides with the predicted one.
This technique that makes the measurement of various
infinitesimal forces on a particle possible can be applied
to the studies of interaction between charged particle andsurfaces. For example, by using antibody coated latexparticles and antigen coated surfaces, this technique canbe available to the study of antigen antibody interactions.
We would like to express our gratitude to Professor Sei
Hachisu for the introduction of this subject to us. We
thank Professor Toyoichi Tanaka and Professor George
B. Benedek for helpful discussions, and appreciate Mr.
Tatsuo Fukui, Mr. Satoru, P. Shibata, and Mr. Keita
Sakurai for their help. I. N. would like to thank Dr.
Gennosuke Yumi for his technical advice. This research
was partially supported by a research grant from AGU.

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19 MAY 1997[1] S. Hachisu, Y. Kobayashi, and A. Kose, J. Colloid
Interface Sci. 42, 342 (1973).
[2] B. J. Alder and T. E. Wainwright, Phys. Rev. 127, 359(1962).
[3] P. R. Sperry, H. B. Hopfenberg, and N. L. Thomas,J. Colloid Interface Sci. 82, 62 (1981).
[4] A. D. Dinsmore, A. G. Yodh, and D. J. Pine, Nature(London) 383, 239 (1996).
[5] S. Asakura and F. Oosawa, J. Chem. Phys. 22, 1255
(1954).
[6] H. N. W. Lekkerkerker, W. C. K. Poon, P. N. Pusey,A. Stroobants, and P. B. Warren, Europhys. Lett. 20, 559(1992); W. C. K. Poon and P. B. Warren, Europhys. Lett.
28, 513 (1994).
[7] X. Ye, T. Narayanan, and P. Tong, Phys. Rev. Lett. 76,4640 (1996).
[8] P. G. de Gennes, Scaling Concepts in Polymer
Physics (Cornell University Press, New York, 1979),
Chap. 3.
[9] F. L. Calderon, T. Stora, O. M. Monval, P. Poulin, andJ. Bibette, Phys. Rev. Lett. 72, 2959 (1994).
[10] A. Ashkin, Phys. Rev. Lett. 24, 156 (1970).
[11] Aldrich Chemical Company Inc. (private communication).
[12] K. Devanand and J. C. Selser, Macromolecules 24, 5943
(1991).
[13] MATSUNAMI GLASS Ind.,Ltd. (private communication).
[14] The common method of obtaining the conversion factor
a of the laser intensity to force was decided by themeasurement of the terminal velocity of the same latex
particle illuminated by the same laser beam in waterusing the Stokes’ law. However, a is not applicableto the particle next to the glass wall. Because therefractive indices of glass of the cell (1.523) and latex
particle (,1.6) are very close compared to that of water(1.33); radiation pressure force for latex particle on aglass wall might be smaller than the force observedin water. To correct this effect, we performed thefollowing experiment. In experimental setup represented
in Fig. 2(a), we set another laser beam irradiated fromabove as well as beneath the particle on the glass
plane in aqueous solution without polymers. The laser
beam from above had the role to trap the particle
instead of the depletion force and had fixed output.
Slightly increasing the output of laser from beneath,we measured the point that the force from beneathexceeded the force from above. From the ratio b ofthese two radiation pressure forces, we were able to getreal conversion factor aysb 1 1d. This is calculatedon the assumption that the reflection from the particle
surface next to the glass surface can be neglected (largest
conversion factor is ayb). As a result, we found thatlaser output of 1 mW gave radiation pressure forceof 0.0395 6 0.0023 pN for a latex particle on glass
surface.