Depletion Attraction Caused by Unadsorbed Polyelectrolytes

layers of two opposing surfaces overlap, dissolved polymer molecules are excluded from that region. ...... Aysen Tulpar, Robert D. Tilton, and Joh...
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Langmuir 1998, 14, 5106-5112

Depletion Attraction Caused by Unadsorbed Polyelectrolytes Edward S. Pagac,† Robert D. Tilton, and Dennis C. Prieve* Colloids, Polymers and Surfaces Program and Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213 Received January 13, 1998. In Final Form: June 9, 1998 Total internal reflection microscopy was used to measure the total interaction between a 6 µm glass sphere and a glass plate, separated by an aqueous solution containing 0.1-1.0 mM of KBr, when both surfaces are saturated with physisorbed polylysine. When the excess polylysine is completely removed from the solution, the sphere fluctuates around the secondary potential-energy minimum formed between double-layer repulsion and gravitational attraction. Subtracting gravity leaves a contribution from doublelayer repulsion which decays exponentially with distance; the decay length is virtually identical to the Debye length calculated for each ionic strength. However, the presence of as little as 10 ppm of unadsorbed 26 kDa polylysine (rod length of 45 nm) causes a measurable attraction, although the most probable separation distance without polymer (150 nm) is much larger than the size of the macromolecule. Increases in the attraction with unadsorbed polymer concentration and decreases in the attraction with increasing KBr concentration correlate with the calculated osmotic pressure for two different molecular weights of polylysine, indicating that the attraction arises from the depletion of the polyelectrolyte from the gap between the sphere and the plate.

Introduction The use of polyelectrolytes to alter interactions in colloidal systems is common in a wide variety of applications. These include wastewater treatment, paint formulation, and ore flotation. Adsorption of polyelectrolytes to colloidal particles or macroscopic surfaces changes the net surface charge and can result in charge neutralization or reversal. In addition, if the adsorbed polymer molecules extend far into solution, electrosteric stabilization or bridging flocculation may occur depending on the surface coverage. While the effects of adsorbed polymer on colloidal interactions have been well studied both theoretically and experimentally, the effects of unadsorbed polymer on colloidal interactions are not as well understood. If polymer molecules remain in solution, a depletion layer can form near an interface. This occurs when the net segmental adsorption energy is inadequate to compensate for the entropy losses that are incurred upon adsorption to the interface. A depletion layer can still form if an adsorbed polymer layer is present, because no more polymer can adsorb once the surface is saturated. This, in effect, changes the energy of additional adsorption and results in the formation of a depletion layer. When the depletion layers of two opposing surfaces overlap, dissolved polymer molecules are excluded from that region. This causes a difference in osmotic pressure between the bulk solution and the gap region, resulting in a net attractive force between the two surfaces which is known as the depletion force.1,2 The charge of the polymer can have a dramatic impact. For the same molar concentration, polyelectrolytes can have much larger osmotic pressures than neutral polymers because electroneutrality requires the counterions to * To whom correspondence should be addressed. E-mail: [email protected]. † Current address: PPG Industries, Inc., Coatings and Resins Research Center, 4325 Rosanna Drive, Allison Park, PA 15101. (1) Asakura, S.; Oosawa, F. J. Chem. Phys. 1954, 22, 1255. (2) Asakura, S.; Oosawa, F. J. Polym. Sci. 1958, 33, 183.

remain with the polymer; each counterion can then contribute as much to the osmotic pressure as each polymer molecule. When the salt concentration cs is much larger than the polyelectrolyte concentration cp, the osmotic pressure can be calculated from3

(

Π ) kT cp +

)

z2cp2 4cs

(1)

where z is the charge number of the polyelectrolyte and k is Boltzmann’s constant. In particular, this should be valid when cs >> |z|cp. Previous experimental studies of the depletion interaction have largely focused on phase separation of colloidal dispersions.4-7 Due to the small magnitude of depletion forces, very few direct experimental measurements have been reported. Richetti and Kekicheff8 were able to measure depletion forces between mica caused by cetyltrimethylammonium bromide (CTAB) surfactant micelles using a surface forces apparatus. These studies were done with relatively high concentrations (∼1-7 vol %) of micelles in order to increase the magnitude of the force; the interactions could not be detected at separation distances larger than 40 nm. Sober and Walz9 exploited the sensitivity of total internal reflection microscopy (TIRM)10-13 to measure depletion forces between a glass slide and a polystyrene sphere again caused by CTAB micelles, but at much lower concentrations (∼0.02 vol %). They were able to measure interaction energies on the (3) Hiemenz, P. Principles of Colloid and Surface Chemistry; Marcel Dekker: New York, 1977; Section 4.9. (4) Cosgrove, T.; Obey, T. M.; Ryan, K. Colloids Surf. 1992, 65, 1. (5) de Hek, H.; Vrij, A. J. Colloid Interface Sci. 1979, 70, 592. (6) de Hek, H.; Vrij, A. J. Colloid Interface Sci. 1981, 84, 409. (7) Gast, A. P.; Hall, C. K.; Russell, W. B. J. Colloid Interface Sci. 1983, 96, 251. (8) Richetti, P.; Kekicheff, P. Phys. Rev. Lett. 1992, 68, 1951. (9) Sober, D. L.; Walz, J. Y. Langmuir 1995, 11, 2352. (10) Bike, S. G.; Prieve, D. C. Int. J. Multiphase Flow 1990, 16, 727. (11) Walz, J. Y.; Prieve, D. C. Appl. Opt. 1993, 32, 1629. (12) Liebert, R. B.; Prieve, D. C. Biophys. J. 1995, 69, 66. (13) Pagac, E. S.; Tilton, R. D.; Prieve, D. C. Chem. Eng. Commun. 1996, 148-150, 105.

S0743-7463(98)00058-4 CCC: $15.00 © 1998 American Chemical Society Published on Web 08/07/1998

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Table 1. Summary of Polymer Characteristics polymer

Mw

N

Mw/Mn

za

Lb (nm)

PL179 PL25 PL1 PEO

179000 25700 1000 782586

861 124 4 17786

1.10 1.20

172 25 1

312 45 1.4

1.11

a

The osmotic charge number for polylysine is a known constant (0.2 × degree of polymerization, N) independent of electrolyte concentration.34 b End-to-end rod length calculated using 0.362 nm/monomer.21

order of kT and up to separation distances of 100 nm. Recently, Milling14 used an atomic force microscope to examine the interactions between a silica particle and a silica slide in the presence of the nonadsorbing polyelectrolyte sodium poly(styrenesulfonate). Depletion and structural forces due to the presence of dissolved polymer were measured up to separations of 150 nm. Polyelectrolyte concentrations exceeding 100 ppm were required to accurately measure the forces. In the current study, we report the direct measurement with TIRM of depletion forces between a microscopic glass sphere and a glass slide in the presence of polylysine, a polyelectrolyte which adsorbs irreversibly to glass. Experimental Section Materials. We purchased three poly-L-lysine hydrobromide polymers from Sigma (Mw ) 1000, 25700, 179000). The poly(ethylene oxide) (PEO) homopolymer (Mw ) 782586) was purchased from Pressure Chemical. Polymer characteristics are listed in Table 1. ACS grade potassium bromide was obtained from Fisher. All water was pretreated by reverse osmosis and further purified by ion exchange using a RGW-5 reagent grade water purification system (Photronix Corp., Medway, MA). Technique. The potential energy profile measurements were conducted using TIRM.10-13 Light scattering by a single spherical particle, illuminated by an evanescent wave, is monitored over a long period of time. The intensity I of light scattered at any instant in time is related to the instantaneous elevation h of the sphere above the plate according to11

I(h) ) I0e-βh where I0 is the intensity of a stuck particle and β-1 is the penetration depth of the evanescent wave. For a penetration depth of 100 nm, a 1% change in intensity corresponds to a 1 nm change in elevation. Over a long period of type, the sphere will sample different elevations according to a Boltzmann distribution

[

p(h) ) A exp -

]

φ(h) kT

where φ(h) is the potential energy of the sphere when it is located at h, kT is the thermal energy and A is a normalization constant. The probability density p(h) for elevation can be constructed from a histogram of the measured elevations. The reader is directed to refs 10-13 for a complete description of the TIRM protocol and analysis. Procedure. The measurements were conducted using borosilicate glass spheres (5.1 ( 0.5 µm diameter, Duke Scientific, Palo Alto, CA) and glass microscope slides (gold seal) obtained from Fisher. The microscope slides were cleaned by immersion in a saturated solution of potassium dichromate in 36 N sulfuric acid for 20 min at room temperature and then by thoroughly rinsing in water, soaking in 6 M hydrochloric acid for 20 min, rinsing again in water, soaking for 20 min in a 0.1 M sodium hydroxide solution, rinsing in water, and finally drying in a nitrogen jet. All experiments conducted in this study were carried out at 25 °C, and the measured pH was between 5 and 6. This resulted in the glass surfaces having a net negative charge (isoelectric pH (14) Milling, A. J. J. Phys. Chem. 1996, 100, 8986.

Figure 1. Potential energy profiles for a 5.5 µm glass sphere interacting with a glass slide in various salt concentrations in the absence of polymer. Solid curves are calculated from (7) using the values of the parameters in Table 2. Changing the ionic strength does not change the slope of the right-hand side of the profiles (slope is given by G). of silica is approximately 2-3) and the polylysine samples having a net positive charge (pK ∼ 10). Prior to measuring potential energy profiles in the presence of dissolved polylysine, the glass spheres and the glass slide were subjected to 24 h of contact with a 200 ppm solution of the polymer of the appropriate molecular weight. The potential energy profiles were first obtained in a 200 ppm solution of polylysine and the desired concentration of KBr. Previous studies15-17 have shown that polylysine irreversibly adsorbs within several minutes to negatively charged surfaces (including silica) at these bulk concentrations and yields surface concentrations of 0.3-0.5 mg/m2 with thin layers (1-3 nm). The solution was then replaced with a polymer-free solution of the same background KBr concentration using a syringe pump (Sage Instruments, Model 355, Cambridge, MA), holding the particle in place using radiation pressure as described by Walz and Prieve.18 Once a measurement was conducted in the polymerfree KBr solution, a new solution of a given polymer concentration (same background KBr) was then added and additional measurements were taken. This procedure was repeated until all the desired polymer concentrations were studied. This protocol enables measurement of potential energy profiles for the same particle levitated above the same location on the glass slide under various conditions. Thus a direct comparison between the measured profiles is possible.

Results and Discussion Effect of Ionic Strength without Polymer. Shown in Figure 1 are examples of typical potential energy profiles for a 5.5 µm diameter glass sphere interacting with a glass slide in the presence of various concentrations of KBr. There is no polymer present in solution or at the surfaces. The main contributions to the potential are electrostatic double layer repulsion and gravity. When the separation distance is several Debye lengths, we expect van der Waals attraction to be severely retarded and screened, so as to be negligible; we also expect double-layer repulsion to be well-modeled using linear superposition and Derjaguin’s approximations. Then, for a 1:1 electrolyte, the total (15) Pagac, E. S.; Prieve, D. C.; Tilton, R. D. Langmuir 1998, 14, 2333. (16) Furst, E. M.; Pagac, E. S.; Tilton, R. D. Ind. Eng. Chem. Res. 1996, 35, 1566. (17) Afshar-Rad, T.; Bailey, A. I.; Luckham, P. F.; MacNaughton, W.; Chapman, D. Colloids Surf. 1988, 31, 125. (18) Walz, J. Y.; Prieve, D. C. Langmuir 1992, 8, 3073.

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Pagac et al.

Table 2. Parameters used to Fit Eq 7 to Data in Figure 1 κ-1 (nm) CKBr (mM)

G (pN)

regressed

from (4)

0.2 0.6 0.8

0.23 ( 0.02 0.22 ( 0.02 0.25 ( 0.02

21.5 ( 0.03 11.5 ( 0.04 10.5 ( 0.03

21.0 12.2 10.8

potential energy profile is expected to obey

φ(h) ) B exp(-κh) + Gh

(2)

where

B ) 16a

( ) kT e

2

tanh

( ) ( ) eψ1 eψ2 tanh 4kT 4kT

(3)

 is the dielectric permittivity of water, a is the radius of the sphere, e is the elemental charge, and ψ1 and ψ2 are the Stern potentials of the sphere and the plate.

κ)

x

8πCe2 kT

(4)

is the Debye parameter, C is the total ionic strength, and

G)

4 3 πa (Fs - Ff)g 3

(5)

is the net weight of the sphere. Equation 2 has a single minimum at

κhm ) ln

κB G

(6)

The charge parameter B is difficult to determine independently. Fortunately, we can eliminate B between (2) and (6) to obtain the relative potential energy in terms of the relative separation distance h - hm:

φ(h) - φ(hm) G ) {exp[-κ(h - hm)] - 1} + kT κkT G (h - hm) (7) kT In other words, the shape of the PE profile is not affected by B. Increasing the charge on either the sphere or the plate will shift the minimum to larger separation distances hm according to (6), but it does not affect the shape. The solid curves shown in Figure 1 are best fits to (7) using the values of the parameters in Table 2. Each side of the profile reflects one of the main forces. The lefthand side is dominated by electrostatic repulsion (first term in (7)), whereas the slope at large separation distances (right-hand side) is a direct measure of the net weight G of the sphere. This has been used to accurately measure the weight of particles between 5 and 30 µm in diameter.19 Increasing the electrolyte concentration results in a shift toward smaller separation distances and an increasingly negative slope of the left hand side of the profile (decreasing Debye length κ-1). Note that changing the electrolyte concentration does not change the slope of the right hand side as indicated by the equality of values for the net weight of the particle, G, in Table 2. The solid curves fit the experimental data quite well; in particular, the apparent decay lengths obtained by regression agree well with the Debye lengths calculated from (4). G ) 0.23 pN corresponds to a 5.5 µm diameter sphere having an (19) Prieve, D. C.; Frej, N. A. Langmuir 1990, 6, 396.

Figure 2. Effect of various concentrations of PL179 on the potential energy profiles for a 6.0 µm glass sphere interacting with a glass slide in solutions containing 0.25 mM KBr. Increasing the PL179 concentration results in a narrowing of the profile and a shift to smaller separation distances which are caused by depletion attraction. The solid curve is from (7), which has been fit to the 0 ppm data using κ-1 ) 19.2 ( 0.2 nm and G ) 0.12 ( 0.01 pN.

apparent density of 1.27 g/cm3 , which is somewhat less than expected for glass. This smaller apparent weight arises because part of the actual weight is offset by upward radiation pressure, which is employed to retard the lateral motion of the particle during the experiment. Effect of Polymer Concentration. After allowing 24 h for the surfaces to come into equilibrium with a solution containing 200 ppm of polylysine, potential energy profiles were first measured between a 6.0 µm diameter glass sphere and a glass slide in the presence of 200 ppm of unadsorbed polylysine and 0.25 mM KBr. Then the solution is replaced with one containing 0.25 mM KBr, but no polylysine, while the particle is held in place using optical tweezers. The potential energy profile is again measured. This procedure is repeated several times using different concentrations of polylysine. Since polylysine adsorption is irreversible,15 the amount of polylysine adsorbed should be the same for all profiles. Figure 2 shows the resulting profiles for different polylysine concentrations. Since separation distances are obtained by comparing the intensity of a levitated particle with the intensity of the same particle when stuck, the separation distance is most likely that between the outer edges of the physisorbed polymer layers. In the absence of added polylysine, the potential energy profile shown in Figure 2 is in good agreement with (7) represented by the solid curve. Upon increasing the concentration of unadsorbed polymer, the profiles shift to smaller separation distances and the slope of the right hand side increases with increasing bulk polymer concentration. On the basis of Figure 1, this change in slope is not expected to arise solely from any increase in the ionic strength associated with the addition of the polyelectrolyte. Moreover, at these dilute concentrations, the density of the polylysine solution is not significantly different from that of the electrolyte solution, so the change in slope is not associated with any changes in the net weight of the particle. Finally, since the adsorbed polylysine layers on each surface are very thin (∼1-3 nm) compared to the separation distance typically encountered (>40 nm), the presence of steric forces, direct bridging forces, or midplane crossing bridging forces20 is

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Langmuir, Vol. 14, No. 18, 1998 5109

highly unlikely. Being unable to explain the observations using only double-layer repulsion and gravity, we suspect that another force (depletion attraction) is acting. Dynamic light scattering studies21,22 of similar polylysine samples suggest the polymer is in an extended rodlike conformation at all electrolyte concentrations encountered in this study (0.1-1.0 mM KBr). The characteristics of the polylysine samples are shown in Table 1. The end-to-end length of PL179 in a rodlike conformation is larger than all the separation distances experienced by the particle, which also suggests that it might be excluded from the region between the surfaces, thereby causing depletion attraction. This attraction has a profound effect on the particle-wall interactions even at bulk polymer concentrations as low as 5 ppm. Milling14 measured depletion forces between silica surfaces in the presence of sodium poly(styrenesulfonate) down to bulk concentrations of approximately 100 ppm using an atomic force microscope. Milling14 also found that the attractive minimum becomes deeper with increasing polymer concentration. The deepening of the potential energy well is not apparent in Figure 2 because the bottom of the well is chosen as the reference state for potential energy profiles determined with TIRM. To see what effect increasing the strength of depletion attraction might have on the TIRM profiles, we calculated some profiles of the total energy. The contribution from depletion attraction between a sphere and a plate was computed according to the theory of Scheutjens and Fleer23,24

Figure 3. Sketch illustrating effects of the depletion interaction on potential energy profiles predicted by adding (8) and (2) for several assumed values of the osmotic pressure Π. The depletion interaction results in a deepening of the attractive well. Increasing the magnitude of the depletion interaction results in the potential energy profiles becoming more narrow and shifting to smaller separation distances.

φdep )

{

1 -πΠ (a + ∆)(h - 2∆)2 + (h - 2∆)3 for 0 < h < 2∆ 3 0 for h > 2∆

[

]

(8) who developed this equation for neutral polymers, where Π is the osmotic pressure of the bulk solution, ∆ is the depletion layer thickness, and a is the radius of the sphere. This depletion energy is then added to the contributions from double-layer repulsion and gravity given by (2). The upper curve in Figure 3 corresponds to the potential energy profile in the absence of any depletion interaction (Π ) 0). When Π is increased, the profile narrows and the attractive minimum deepens and shifts toward smaller separation distances. Shown in the inset are the simulated potential energy profiles replotted with the minimum of each curve shifted to zero potential energy, as in Figure 2. Since TIRM only examines 0-5kT of the interaction, the effect of the depletion attraction on a potential energy profile is to narrow the profile and shift the minimum to smaller separation distances. More experimental results are illustrated in Figure 4 using a smaller polylysine (PL25). For an uncharged random-coil polymer, Fleer et al.25 identified the depletion thickness ∆ with the radius of gyration of the molecule (Rg ) [〈R2〉/6]1/2); thus the separation distance between (20) Dahlgren, M. A. G.; Waltermo, A° ; Blomberg, E.; Claesson, P. M.; Sjo¨stro¨m, L.; Torbjo¨rn, A° .; Jo¨nsson, B. J. Phys. Chem. 1993, 97, 11769. (21) Nemoto, N.; Matsuda, H.; Tsunashima, Y.; Kurata, M. Macromolecules 1984, 17, 1731. (22) Wilcoxon, J. P.; Schurr, J. M. J. Chem. Phys. 1983, 78, 3354. (23) Scheutjens, J. M. H. M.; Fleer, G. J. J. Phys. Chem. 1979, 83, 1619. (24) Scheutjens, J. M. H. M.; Fleer, G. J. J. Phys. Chem. 1980, 84, 178. (25) Fleer, G. J.; Scheutjens, J. M. H. M.; Vincent, B. Polymer Adsorption and Dispersion Stability; Goddard, E. D., Vincent, B., Eds.; ACS Symposium Series 240; American Chemical Society: Washington, DC, 1984; p 245.

Figure 4. Effect of various concentrations of PL25 on the potential energy profiles for a 5.9 µm glass sphere interacting with a glass slide in solutions containing 0.25 mM KBr. Increasing the PL25 concentration results in an increase in the magnitude of the depletion interaction. This results in a narrowing of the profile and a shift to smaller separation distances. The solid curve is from (7), which has been fit to the 0 ppm data using κ-1 ) 17.3 ( 0.3 nm and G ) 0.10 ( 0.01 pN.

two non-adsorbing plates below which polymer is largely depleted from that region, or 2∆, is on the order of the rms end-to-end distance 〈R2〉1/2. In bulk solution, rod-shaped molecules like polylysine undergo Brownian rotation. When all orientations are sampled over some interval of time, the molecular rod sweeps out a sphere whose diameter equals its end-to-end length. When confined between two plates whose separation h is less than the end-to-end length L, the rod-shaped molecule loses its ability to sample all orientations and thus suffers losses in entropy. So for rod-shaped molecules, 2∆ is expected to correspond roughly to L. For the PL25 used in Figure 4, L is estimated to be 45 nm (see Table 1), which is less than one-third of the most probable separation distance (150 nm) sampled by the sphere in a polymer-free solution. Yet very strong depletion attraction is evident as the polymer concentration is increased. This increased range of the depletion

5110 Langmuir, Vol. 14, No. 18, 1998

force may be due to the electrostatic repulsion between the similarly charged polymer and the surfaces. Electrophoresis experiments showed that polylysine adsorption to glass produces a net positive interfacial charge. Hoagland26 examined the interactions between rodlike polyelectrolytes and similarly charged surfaces. He found that for κL < 10, the depletion layer thickness can increase to two or three times the rod length. This is similar to the conditions in Figure 4 (κL ) 2.1). In addition, Asakura and Oosawa2 and Walz and Sharma27 have shown that both the range and magnitude of the depletion interaction can increase in the presence of charged macromolecules. Other results2,28 suggest that rodlike molecules will give rise to larger depletion forces than random coils. These effects arise from the increased osmotic pressure of charged species and the limited orientational freedom of ellipsoidal molecules near an interface, respectively. The profile in Figure 4 for 10 ppm bulk polymer concentration displays a significant amount of curvature above 200 nm. This effect was observed repeatedly in the 5-10 ppm concentration range and is also evident in the theoretical predictions of Figure 3. As the bulk concentration is increased past 10 ppm, the particle “falls” into a relatively deep depletion minimum and rarely samples higher energy positions. Further increases in the bulk polymer concentration cause the minimum to deepen, and as a result, the profiles become more narrow. Theory25,29 and experiments14,30 suggest that increasing the bulk polymer to high concentrations can result in depletion restabilization. While the focus of this work was not on this phenomenon, it is worth noting that experiments conducted at much higher concentrations (1000-5000 ppm) revealed that the particles were still levitated above the glass slide. This is surprising given the fact that the ionic strength of these high polymer concentration solutions should be large enough to screen electrostatic repulsion forces and thereby cause the particle to become adhered to the slide. This aspect will be more closely examined in future work. If the size of the macromolecule is sufficiently smaller than the gap, the depletion interaction disappears. This is illustrated in Figure 5. These results are for a 6.3 µm glass sphere interacting with a glass slide in the presence of varying amounts of the oligomer PL1 in a background salt concentration of 0.25 mM KBr. Rather than molar concentrations, the mass concentrations of PL1 were chosen to equal those used for PL179 and PL25 in order to hold the concentration of ionizable groups constant. The profiles in Figure 5 resemble those in Figure 1 which involve only electrostatic repulsion and gravity with no contribution from a depletion attraction. Increasing the PL1 concentration from 30 to 200 ppm decreases the apparent Debye length from 19.3 to 11.1 nm. This corresponds to apparent ionic strengths of 0.25 and 0.75 mM respectively. If we assume that each oligomer contains two ionizable groups, the calculated ionic strengths for 30 and 200 ppm PL1 in the presence of 0.25 mM KBr are 0.32 and 0.75 mM, which are reasonably close to those deduced from the experimental data. This suggests that the PL1 is simply acting as a multivalent ion. The slope of the right hand side (G) is within experimental error of being the same for the two PL1 concentrations. (26) Hoagland, D. A. Macromolecules 1990, 23, 2781. (27) Walz, J. Y.; Sharma, A. J. Colloid Interface Sci. 1994, 168, 485. (28) Mao, Y.; Cates, M. E.; Lekkerkerker, H. N. W. Physica A 1995, 222, 10. (29) Mattice, W. L.; Napper, D. H. Macromolecules 1981, 14, 1006. (30) Rawson, R.; Ryan, K.; Vincent, B. Colloids Surf. 1988/89, 4, 89.

Pagac et al.

Figure 5. Effect of various concentrations of PL1 on the potential energy profiles for a 6.3 µm glass sphere interacting with a glass slide in solutions containing 0.25 mM KBr. Increasing the PL1 concentration results in a decrease in the apparent Debye length and no change to the gravitational contribution.

To further investigate the effect of charge and shape on the depletion force, we measured the interactions between a glass sphere and a glass slide in the presence of a neutral random-coil polymer, poly(ethylene oxide) (PEO) (Mw ) 782 586) whose 2Rg ) 117 nm31 is between the rod lengths of PL25 and PL179. At 2000 ppm of this PEO, we observed no detectable depletion effect. These concentrations of PEO correspond to calculated osmotic pressures comparable to those calculated for the polylysine samples via eq 1. This suggests that the depletion layer thickness for a neutral polymer is smaller than for a polyelectrolyte. Hoagland26 predicted that charged rodlike polymers have a larger depletion layer thickness compared to neutral rodlike polymers of the same length under conditions of κL < 10. Effect of Ionic Strength with Polymer. The concentration of small ions can have a profound effect on the depletion attraction caused by polyelectrolytes. Figure 6 shows the potential energy profiles obtained for a 5.6 µm PLL-coated glass sphere interacting with a PLL-coated glass slide in a solution of 30 ppm of PL25 and in the presence of varying amounts of KBr. The profile corresponding to the lowest concentration of KBr (0.1 mM) is a very narrow parabola, suggesting very strong depletion attraction. As the concentration of KBr increases, the potential energy profile broadens, suggesting a weakening of the depletion attraction. Furthermore, the profile corresponding to the highest concentration of KBr (1 mM) resembles those with no free polymer present. In fact the apparent net weight G in 1.0 mM KBr with polymer (G ) 0.13 ( 0.02 pN) and the value measured in the absence of polymer (G ) 0.10 ( 0.02 pN) are almost the same. This weakening of depletion attraction with added salt is probably caused by a reduction in osmotic pressure as predicted by eq 1 for an increase in salt concentration cs. Although depletion attraction is weakening in response to the addition of salt, the most probable separation distance decreases because the double-layer repulsion is also weakened by the addition of salt. The same effect was observed by Milling14 using an atomic force microscope and the polymer sodium poly(styrenesulfonate). (31) Devenand, S.; Sesler, J. C. Macromolecules 1991, 24, 5943.

Depletion Attraction

Figure 6. Effect of various concentrations of KBr on the potential energy profiles for a 5.6 µm glass sphere interacting with a glass slide in solutions containing 30 ppm of PL25. Increasing the salt concentration decreases the magnitude of the depletion interaction and broadens the profile.

Figure 7. Correlation between the estimated depletion interaction (∆G) and the molar concentration of polymer. While individual sets of data might fall around a line having a slope of unity, the intercepts of these lines vary widely, indicating no overall correlation with the molar concentration of polymer.

Correlation with Osmotic Pressure. Asakura and Oosawa1,2 ascribed the depletion force to the difference in osmotic pressure between a solution of macromolecules and a solution without macromolecules. The size and nature (i.e. charge, conformation, and rigidity) of the macromolecule determines the separation distance 2∆ where the depletion force begins to act. However, once this separation distance is achieved the depletion force should simply equal the difference in osmotic pressure multiplied by the area of overlap of the depletion layers. To correlate our results, we assume that this difference in osmotic pressure is between a solution having the polymer concentration of the bulk and a solution having no polymer; this difference is approximated by (1). We use the increase in slope ∆G at large separation distances as an estimate of the depletion attraction. Shown in Figure 7 is a plot of the measured ∆G vs the molar concentration of the polyelectrolyte. For an ideal solution of a neutral polymer, the osmotic pressure would be directly proportional to the molar concentration of polymer, according to van’t Hoff’s law (first term of eq 1). While any one of the sets of data might be reasonably fit by a straight line with a slope of one, the intercepts of the straight lines for each set vary by 108. Clearly the molar

Langmuir, Vol. 14, No. 18, 1998 5111

Figure 8. Correlation between the estimated depletion interaction (∆G) and the osmotic pressure calculated from (1). While the correlation is far from perfect, it is much better than that of Figure 7.

concentration of polymer is not the only variable which affects the depletion attraction. Figure 8 shows the correlation between the measured ∆G and the osmotic pressure calculated from eq 1, using all the same data as in Figure 7, plus two additional sets. While there is still considerable scatter around a single straight line having a slope of unity, the breadth of scatter is orders of magnitude smaller than in Figure 7 (102 rather than 108), thus confirming the importance of polyelectrolyte charge and ionic strength on the osmotic pressure and depletion attraction. Of course, osmotic pressure is not the only factor affecting the depletion attraction. Another parameter appearing explicitly in (8) is the depletion layer thickness ∆. Electrostatic interactions between polyelectrolyte molecules and the surfaces can be expected to cause ∆ to depend on ionic strength, which in turn is affected by the concentration of both polyelectrolyte and salt. Moreover, the need for local electroneutrality of the polymer solutions can cause the small ions to distribute nonuniformly when the polymer is distributed nonuniformly between the gap region and bulk solution outside. Thus the ionic strength in the gap is not necessarily the same as the bulk solution. In short, the correlation in Figure 8 between osmotic pressure and the depletion force should show considerable scatter since several factors remain to be taken into account. These calculations do not account for any oscillatory forces that have been found to be present in other depletion force studies.8,9,14,32,33 These oscillatory forces have been attributed to changes in free energy upon packing of macromolecules in a confined space between interacting surfaces. While no substantial evidence of oscillatory forces were encountered in this study, a longer time was required for particles in polymer solutions to reach their equilibrium profiles than particles in the absence of polymer. This may be due to the slow squeezing out of the polymer molecules from the gap region as the particle approaches the surface. Oscillatory forces would likely be involved in this process. (32) Parker, J. L.; Richetti, P.; Kekicheff, P. Phys. Rev. Lett. 1992, 68, 1955. (33) Nikolov, A. D.; Wasan, D. T. J. Colloid Interface Sci. 1989, 133, 1. (34) Daniel, E.; Alexandrowicz, A. Biopolymers 1963, 1, 473.

5112 Langmuir, Vol. 14, No. 18, 1998

Conclusions The interactions between a colloidal glass sphere and a glass slide in the presence of polylysine have been measured using total internal reflection microscopy. The results show a strong depletion attraction at polyelectrolyte concentrations as low as 5 ppm. The depletion attraction increases with increasing polyelectrolyte concentration and decreasing background electrolyte. The magnitude of the interaction was shown to correlate with the osmotic pressure of the polyelectrolyte solution regardless of whether the change to the osmotic pressure was induced by altering the polymer or salt concentration. The range of the depletion interaction, and therefore the depletion layer thickness, can extend to several times the

Pagac et al.

rod length of the polylysine molecule. No significant depletion attraction was detected using PEO having a random-coil diameter comparable to the rod length of the PL at concentrations expected to yield similar osmotic pressures to those of the PL solutions. These results suggest that long-range electrostatic repulsion between the polyelectrolyte in solution and the similarly charged adsorbed layer on either surface contributes significantly to the range of the depletion attraction. Acknowledgment. This work was supported by the National Science Foundation under grants BES-9501145 and CTS-9623849. LA980058F