Effects of Particle Deformability on Interaction between Surfaces in

Two types of polymer particles, which were kindly donated by Mitsubishi .... The short dashed line is for Fc = FcR; the long dash line is for Fc = FcH...
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Langmuir 2003, 19, 110-117

Effects of Particle Deformability on Interaction between Surfaces in Solutions Ivan U. Vakarelski,† Akihiro Toritani,‡ Masaki Nakayama,‡ and Ko Higashitani*,† Department of Chemical Engineering, Kyoto University, Yoshida, Sakyo-ku, Kyoto 606-8501, Japan; and Resins & Plastics Development Center, Mitsubishi Rayon Co., Ltd., Otake, Hiroshima 739-0693, Japan Received September 19, 2002. In Final Form: November 6, 2002 Interactions between a deformable polymer particle and a solid surface of a mica plate in solutions were investigated using atomic force microscopy (AFM), where the particles have the same surface property but different bulk elasticity. Measurements were done in a solution of cetyltrimethylammonium bromide (CTAB), above its critical micelle concentration (cmc). Under these conditions positively charged layers were formed on both mica and particle surfaces, which induce a high repulsive potential between the approaching surfaces. However, above a critical pressure, the surfaces come in adhesive contact by destroying the adsorbed layers. It was found that the total repulsive force between the surfaces and the force required to adhere the particle to the surface increased significantly if the glass transition temperature of the particle was below the ambient temperature. This indicates that the elastic deformation of the particle as the surfaces are brought together greatly increases the total repulsion within the system. A quantitative interpretation of the data was carried out, using a simple model, which was derived from the theory for elastic body deformation developed by Hughes and White.1 Results of this study provide significant information on the stability of suspensions of low elastic modulus particles.

Introduction The kinetic effects of approaching surfaces in solutions on their interaction are of fundamental importance to understand the stability of colloidal systems. It was shown theoretically and confirmed experimentally that the stability of emulsions does depend on the deformability of droplets.2-5 However, the particle deformability is usually not considered as a factor to control the stability of dispersions, mainly because particles have a relevantly high elastic modulus and are not deformed by electrostatic repulsive forces in their coalescence. The growing industrial application of low elastic modulus particles evokes a need to investigate the significance of the surface deformation for systems in which the deformability of particles is between those of liquid droplets and solid particles. To study this role of particle deformability, interactions between a particle and a plate are compared, using polymer particles with glass transition temperatures well below and above the ambient temperature. The former particle, commonly known as an elastomer particle, is of much lower elastic modulus and higher deformability than the solid particles usually employed. Over the past decade AFM has been established as a powerful method for direct measurement of the interaction force between micron size particles and flat surfaces or other particles. These measurements are usually done between solid particles and surfaces. As well, there have * To whom correspondence should be addressed. † Kyoto University. ‡ Mitsubishi Rayon Co., Ltd.. (1) Hughes, B. D.; White, L. R. Q. J. Mech. Appl. Math. 1979, 32, 445. (2) Denkov. N. D.; Kralchevski, P. A.; Ivanov, I. B.; Vassilieff, C. S. J. Colloid Interface Sci. 1991, 143, 157. (3) Hofman, J. A. M. H.; Stein, H. N. J. Colloid Interface Sci. 1991, 147, 508. (4) Danov, K. D.; Denkov, N. D.; Petsev, D. N.; Ivanov, I. B.; Borwankar, R. Langmuir 1993, 9, 1731. (5) Hartley, P. H.; Griezer, F.; Mulvaney, P.; Stevens, G. W. Langmuir 1999, 15, 7282.

been advances in the measurement of interactions when one of the surfaces is a fluid, gas bubble6,7 or oil droplet5,8 and in other cases deformable substrates such as viscoelastic polymers9-11 or polymer particles.12,13 Recently, we have investigated the adhesive behavior of an elastomer particle on a mica surface in water.14 Through the attachment of an elastomer particle to an AFM cantilever of similar stiffness, both the particle surface interaction and the particle deformation can be measured simultaneously. Figure 1 schematically illustrates the fundamental differences in interaction forces caused by the particle deformability. The lower curve indicates the standard DLVO interaction between two nondeformable particles. In the case of the deformable particle, the surface is flattened by the repulsive force, so that the interaction area between particles increases at the same separation, which increases the total repulsion. A quantitative treatment for this phenomenon will be given in the theoretical section, with the consideration based on the repulsive interaction theory for elastic spheres developed by Hughes and White.1 Two types of polymer particles are used; one is a hard particle and the other is a soft particle coated with the (6) Butt, H. J. J. Colloid Interface Sci. 1994, 166, 109. (7) Ducker, W. A.; Xu, Z.; Israelachvili, J. N. Langmuir 1994, 10, 3279. (8) Mulvaney, P.; Perera, J. M.; Biggs, S.; Grieser, F.; Stevens, W. G. J. Colloid Interface Sci. 1996, 183, 614. (9) Aime, J. P.; Elkaakour, Z.; Odin, C.; Bouhacina, T.; Michel, D.; Curely, J.; Dautant, A. J Appl. Phys. 1994, 76 (2), 754. (10) VanLandingham, M. R.; McKnight, S. H.; Palmese, G. R.; Elings, J. R.; Huang, X.; Bogetti, T. A.; Eduljee, R. F.; Gillespie, J. W., Jr. J. Adhesion 1997, 64, 31. (11) Gilles, G. G.; Prestidge, C. A.; Attard, P. Langmuir 2001, 17, 7955. (12) Biggs, S.; Sprinks, G. J. Adhesion Sci. Technol. 1998, 12, 461. (13) Portigliatti, M.; Koutsos, V.; Hervet, H.; Le´ger, L. Langmuir 2000, 16, 6374. (14) Vakarelski, I. U.; Toritani, A.; Nakayama, M.; Higashitani, K. Langmuir 2001, 17, 4739.

10.1021/la026581i CCC: $25.00 © 2003 American Chemical Society Published on Web 12/11/2002

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Figure 2. Schematic drawing of particles and a mica surface with CTAB adsorbed layers. Figure 1. Schematic diagram of the interaction force, F, as a function of the surface separation, h, for both deformable and nondeformable particles of the same radius and surface properties.

same polymer as the hard one, which ensures the same surface property to both particles, despite their difference in bulk elasticity. Measurements are carried out between a particle and a flat mica surface in a cetyltrimethylammonium bromide (CTAB) solution above the cmc. In this case, the CTAB will form an adsorbed layer on both surfaces of the mica and particle, and creates a highly repulsive electrostatic force between them, by which effects due to the difference in deformability are strongly pronounced. Ionic surfactants are often used to increase the dispersion stability. CTAB was chosen because its adsorption kinetics are already well studied, including with the force measurement techniques.15-17 For reference experiments, measurements are also carried out between hard materials of a silica particle and a mica plate at the same CTAB concentration. Experimental Section

(Olympus) were used. The exact value of the spring constants was calculated by measurement of the loaded and unloaded cantilever resonance frequency.18 The particles were attached to the cantilever using high-temperature melting epoxy resin (Shell). The cantilever tip was broken in advance to allow for the attachment of the particle on the flat cantilever surface. Scanning electron microscope images of the cantilever tip with the attached particles as shown in our previous work14 were used to ensure that the epoxy did not contaminate the contacting surface and for the precise determination of the particle diameter. A description of the AFM force curve analysis is given elsewhere;19 here, we give a brief overview to establish the particle deformation measurement. The mica substrate is affixed to a metal plate, which moves vertically at a constant velocity through the control of the voltage of the piezo system. The cantilever deflection is detected by the voltage change of the split photodiode, which represents the position of the laser beam reflected from the rear of the cantilever. The deflection is then converted into the interaction force F between surfaces, using the spring constant of the cantilever. Raw data of the piezo displacement ∆ (V) to the cantilever deflection u (nm) for cantilevers with identical spring constant (2 N/m) with an attached silica particle and with an attached elastomer particle are shown in Figures 3 and 4. After the probe contacts the mica surface, there appears a linear region of ∆ versus u. The reciprocal slope bp ) ∂∆/∂u when the probe is in contact with the surface is lower for the elastomer particle case. This difference allows us to evaluate the effective elastic constant of the particle, by the following equation.

Two types of polymer particles, which were kindly donated by Mitsubishi Rayon Co., were used in the experiments. The first one is a composite polymer of 98 wt % poly(2-ethylhexyl acrylate) (2EHA) and 2 wt % poly(acryl methacrylate) (PAMA) whose surface is coated with a thin layer of poly(methyl methacrylate) (PMMA) of a few nanometers. The glass transition temperature for 2EHA is Tg ) -55 °C, which is well below the ambient temperature. This particle is referred as an elastomer particle in the text. The second type of particle is solely composed of PMMA of Tg ) 100 °C, which is referred to as a PMMA particle. These particles are spherical but usually polydispersed with diameters from 1.5 µm up to 5 µm. Two types of particles, the mica surface and adsorbed layers of CTAB, are drawn schematically in Figure 2. Monodispersed spherical particles of nonporous Strober silica of 8.0 µm in diameter were kindly supplied by Shokubai Kasei Co. PMMA and the silica particles are referred to in the text as solid particles. The CTAB was supplied from Tokyo Kasei Co. and used without further purification. CTAB solutions were prepared using Millipore purified water. Muscovite mica was freshly cleaved before use. The room temperature during experiments was kept approximately at 22 °C. The force measurements were done with a Digital Instruments Nanoscope III atomic force microscope (Santa Barbara, CA). Rectangular cantilevers with a spring constant of 2 N/m

u0 is the output voltage of the nondeflected cantilever, and ∆0 is a constant, which can be chosen to shift the force curve to some reference position. When the solid particles approach the surface, d is equivalent to the particle surface separation h. On the other

(15) Pashley, R. M.; Israelachvili, J. N. Colloids Surf. 1981, 2, 169. (16) Kekicheff, P.; Christenson, H. K.; Ninham, B. W. Colloids Surf. 1989, 40, 31. (17) Drummond, C. J.; Sender, T. J. Colloids Surf., A 1994, 87, 217.

(18) Cleveland, J. P.; Manne, S.; Bockec, D.; Hansma, P. K. Rev. Sci. Instum. 1993, 64, 1. (19) Ducker, W. A.; Senden, T. J.; Pasheley, R. M. Langmuir 1992, 8, 1831.

sp ) scbc/(bp - bc)

(1)

where sc and sp are the spring constants of the cantilever and the particle, and bc and bp are the reciprocal slopes when the bare cantilever or cantilever with attached particle is in contact with the surface. For the bare cantilever, we found bc ) 62 ( 2 nm/V, which was approximately the same ((2 nm/V) as that for the cantilever with an attached silica particle, since sp . sc. For the elastomer particle shown in Figure 4b, the maximum slope was bp ) 121 nm/V or sp ) 2 N/m. Knowing bc, the displacement of the cantilever end relative to the substrate surface, d, can be evaluated. The relationship of u versus ∆ was converted to that of F versus d using the following relations.

F ) sc(u - u0)bc

(2)

d ) (∆ - ∆0) + (u - u0)bc

(3)

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Figure 5. Schematic drawings of the particle on the cantilever before the interaction start (or F ) 0) and at the maximum load position (or F ) Fm). u0 is the AFM diode output voltage corresponding to F ) 0. d0 ) d0 (∆0) is the reference displacement of the cantilever end to the piezo flat, chosen at the maximum load position. schematically in Figure 5. A detailed explanation and usage of these equations will be given in the following sections.

Theory

Figure 3. ∆ vs u data for an 8 µm silica particle approaching a mica surface in a 1.5 mM CTAB solution: (a) maximum force less than the critical; (b) maximum force higher than the critical.

The classical Hertz model, for two elastic spheres in contact,20 assumes that spheres are pressed into contact within a “hard wall” interaction, that is, an infinitely steep repulsion between surfaces in contact. A more complete theory of the interaction between elastic spheres was introduced by Hughes and White1 in 1979, allowing spheres to interact via the surface forces of finite range. In this section, simplified expressions based on this theory are developed to interpret the experimental measurements obtained. Consider two elastic spheres of radii R1 and R2, Young’s moduli E1 and E2, and Poisson’s ratios ν1 and ν1. Then the effective radius R is determined by

1 1 1 ) + R R1 R2

(4)

and the effective elastic coefficient K is given by1

[

]

2 1 - ν22 1 3 1 - ν1 ) + K 4 E1 E2

(5)

In the case of a deformable sphere and a rigid flat, the following must hold: R2 f ∞, E2 f ∞, R ) Rsphere, and K ) Ksphere. Suppose p(h) is the disjoining pressure between two plates composed of the same material as spheres at the separation h. As the plates approach each other, the disjoining pressure will increase exponentially up to some critical value pc at the separation hc.

p(h) ) pc exp(-k(h - hc)) for h g hc

Figure 4. ∆ vs u data for a 1.8 µm elastomer particle approaching a mica surface in a 1.5 mM CTAB solution in the case of (a) maximum applied force less than the critical and (b) maximum applied force higher than the critical. hand, when the elastomer particle contacts the surface, the change in d equals the particle deformation δ. The relationship among, d, h, and δ, with respect to the reference states, is shown

(6)

where k-1 is the interaction decay length. The above has been established to be a good approximation for rapidly decaying interactions. For the case of spheres, the value of h indicates the nearest surface separation, as illustrated in Figure 6a. If h , R1 and R2, and k-1 , R1 and R2, the Derjaguin approximation holds.21 This indicates that the pressure (20) Hertz, H. In Miscellanious papers; McMillan&Co.: London, 1986. (21) Derjaguin, B. V. Kolloidzeitschrift 1934, 69, 155.

Effects of Particle Deformability

Langmuir, Vol. 19, No. 1, 2003 113 Table 1. Fc/FcH, Fc/FcR, and Fc/(FcH+ FcR) Ratio for Different λ Values, Obtained as Explained in the Text λ

Fc/FcH

Fc/FcR

Fc/(FcH + FcR)

0.1 1.0 2.0 3.0 4.0 5.0

2.56 × 102 4.22 1.95 1.39 1.22 1.14

1.05 1.73 3.20 5.14 8.00 11.68

1.05 1.22 1.21 1.09 1.06 1.04

and

δ)

a2 R

(11)

The Hertz solution is strictly valid for small deformation or when a , R. The distribution of the pressure within the contact circle is given by Figure 6. Schematic drawings of (a) a deformable particle approaching a flat rigid surface (the dashed line indicates the position of the surface when there is no interaction); (b) a rigid particle to a rigid surface; and (c) an elastic particle in Hertz contact with a rigid flat.

on the nearest surface of the sphere is equal to that between two flat plates at the same separation, which is given by eq 6. Another idealization introduced is that, once p > pc, the pressure sharply drops or the surfaces jump into adhesive contact. Substituting these conditions into Fc ) F(pc), the total force Fc necessary to adhere two surfaces will be estimated. The problem is mathematically complex, but the values of Fc were numerically solved against the following dimensionless parameter by Hughes and White.1 1/2

λ)

4 (2kR) pc 3 K

(7)

According to Hughes and White, the exact solution results in the Hertz theory when λ is large. For greater values of λ, the infinitely steep repulsion assumed by the Hertz theory is better satisfied and the exact solution approaches the Hertz theory prediction. On the other hand, the sphere becomes nondeformable when this parameter is small, that is, E f ∞. For these limiting cases, it is easy to obtain the analytical expression for Fc, as demonstrated below. Nondeformable Sphere of Small λ. In the case of a nondeformable or rigid sphere shown in Figure 6b, the Derjaguin approximation holds in the form

F(h) ) 2πR

∫h∞p(h) dh

(8)

Substituting eq 6 into this equation, the critical force for rigid bodies, FcR, at h ) hc is approximately given by the following equation.

FcR )

2πRpc k

(9)

Hertz Contact of Large λ. The Hertz theory gives the following relationships between the total load applied on the sphere F, the radius of contact area a, and the penetration δ, as shown in Figure 6c.

p(r) )

r 3Ka 12πR a

2 1/2

( ( ))

(12)

where r is the radial coordinate in the contact area, as shown in Figure 6c. If the condition p(0) ) pc is imposed, using the above relationships, the expression for the critical force evaluated by the Hertz theory, FcH, is given as follows. H

Fc

2 3 2π 3R pc ) 3 K2

( )

(13)

General Case. As discussed previously, the exact value of Fc can be calculated for a certain range of λ. In Table 1 the values of the ratios Fc/FcH are given by Hughes and White (Table 1 in ref 1). The expression of FcH/FcR is easily derived from eqs 9 and 13:

FcH R

Fc

)

π2 2 λ = 0.41λ2 24

(14)

The combination of this equation and the values of Fc/ FcH enables us to evaluate the ratios Fc/FcR and Fc/(FcR+ FcH) for various values of λ, as shown in Table 1. As expected, Fc f FcH with the increase of λ, and Fc f FcR with the decrease of λ. Therefore, Fc > FcH and Fc > FcR for values of λ in between. It is useful to know that Fc > (FcH + FcR) and Fc = FcH + FcR with a maximum deviation of less than 22% for any value of λ. To illustrate the results of this analysis for the case of our experiments, in Figure 7 is shown the increase of Fc with the particle softness increase:

Fc ) Fc(K) for R, pc, and k ) constant

(15)

As far as FcR is independent of K, Fc(K) will be proportional to Fc/FcR, which is already estimated in Table 1 as a function of λ. In this case, λ depends only on K; that is, λ ∼ 1/K. At small values of λ < 0.1, Fc does not depend on the particle softness and the value is given by eq 9. At λ > 10, eq 13 for FcH provides a good approximation for how Fc increases with K. For intermediate values of λ, either the value given by the exact solution or that given by the approximation Fc = FcH + FcR should be employed. Results and Discussions

Ka3 F) R

(10)

CTAB is a cationic surfactant which forms a bilayer on negatively charged hydrophilic surfaces at concentrations

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Figure 7. Fc/FcR as a function of λ at fixed R, k, and pc. As λ ∼ 1/K, the λ increase corresponds to the deformability increase. The short dashed line is for Fc ) FcR; the long dash line is for Fc ) FcH ∼ λ2; the solid line is for Fc ) FcR + FcH; and the / symbols are the theoretical values taken from ref 1.

above the cmc. It is well established that at the concentration 1.5 mM CTAB stable positively charged bilayers formed on both the surfaces of mica and silica with similar charge magnitudes.15-17 These surfaces inhibit contact not only because of the long-range electrostatic repulsion between the positively charged bilayers but also because of the strong short-range hydration repulsion between the CTAB headgroups. More recent studies using the AFM “soft imaging” technique revealed that the layer morphology may exist as wormlike or spherical micelle structures.22,23 CTAB is known to adsorb readily on polystyrene latex surfaces.24,25 But the adsorption process in this case is more complicated, since competing adsorption mechanisms are presentsthe electrostatic adsorption of surfactant headgroups to the charged surface and the hydrophobic adsorption of the surfactant tail to the polymer surface of particles. Nevertheless, the possible surface structures of the saturated adsorbed layer above the cmc are well characterized, such as, surface micelles, semiglobular micelles, or a single monolayer with the surfactant headgroups orientated predominantly toward the solution.24,25 Electorphoretic measurements indicate that the surface is strongly positively charged.25 It is presumed that similar adsorption behavior will occur on the PMMA surface, giving rise to the strong repulsion when the particle approaches to the mica surface. Due to the lack of data for the surface charge of the PMMA surface covered with CTAB, experiments were carried out in the following way: the interaction between a silica particle and a mica surface was measured in a 1.5 mM CTAB solution as a reference system from which the bilayer surface potential is evaluated. Then force measurements between the PMMA particle and mica were performed at the same CTAB concentration, from which the potential of the PMMA surface adsorbed by CTAB was estimated. Once the interaction pattern between the PMMA particle and the mica surface is known, experiments were performed using the elastomer particles covered with a thin layer of PMMA, to clarify the effect of the particle deformability on the interaction. The measuring procedure was as follows: initially the AFM liquid cell with a probe mounted inside was filled with pure water, and then a 1.5 mM CTAB solution was (22) Manne, S.; Cleveland, J. P.; Gaub, E. H.; Gaub, H. E.; Stucky, G. G.; Hansma, P. K. Langmuir 1994, 10, 4409. (23) Lamont, R. E.; Ducker, A. W. J. Am. Chem. Soc. 1998, 120, 7602. (24) Zhao, J.; Brown, W. Langmuir 1996, 12, 1141. (25) Xu, R.; Smart, G. Langmuir 1996, 12, 4125.

Figure 8. Interaction between a mica plate and (a) an 8 µm silica particle or (b) a 5 µm PMMA solid particle in the presence of 1.5 mM CTAB. The solid lines represent the DLVO force curves. For the silica/mica system the nonretarded Hamker constant is A12324 ) 0.5 × 10-20 J, and for the PMMA/mica systems it is A12324 ) 0.4 × 10-20 J.

injected immediately. Measurements were started 1 h after the solution injection. This will give the maximum strength to the CTAB bilayers, according to the previous results,17 as well as ensure that measurements were performed at the same stage of consolidation of adsorb layers. Silica Particle to Mica Surface in 1.5 mM CTAB. Figure 3 displays the raw data of the piezodisplacement with cantilever deflection for two different cases. In the first case (Figure 3a), the maximum applied load Fm is just below the force necessary to break the surfactant layers, Fc. Thus, the probe rebounds from the surface and there is no hysteresis between the approach and the retraction part. In the second case (Figure 3b), Fm is higher than Fc and so the surfactant layer is broken. This correlation is converted into the relation of the force F versus the separation h shown in Figure 8a, where the breakage of the adsorbed layer is more clearly observed as a sharp jump of the surfaces. Here it is assumed that the effect caused by the compression of CTAB remaining between the surfaces was insignificant. Hence, the constant compliance for hard surfaces is assumed to convert the data in Figure 3b into those of Figure 8a, following the standard procedure.19 Zero separation was set at the separation corresponding to the constant compliance region after the jump-in. The jump-in distance is about 3.6 nm, which is approximately equal to the thickness of two CTAB monolayers, that is, twice the thickness of a 1.7 nm layer.15 After the jump, the surfaces come into strong adhesive contact and a significant pulloff force, Foff, must be applied to separate them on retraction.

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Table 2. Summarized Particle and Interaction Characteristics Used for λ Estimation in Each Systema particle silica PMMA1 PMMA2 PMMA3 elast 1 elast 2 elast 3 a

diameter (µm) 8.0 2.5 4.0 5.0 1.8 3.1 4.0

Eb (MPa) 70 × 3 × 103 3 × 103 3 × 103 1.8 1.6 1.2

103

Fc/R (mN/m) 35-40 30-34 28-32 30-33 170-200 250-300 300-400

pc(kc-1 (nm)) (MPa) 6.0 (1) 2.5 (2) 1.3 (3.8) 1.8 (2.7) 0.61 (-) 0.54 (-) 0.44 (-)

((Fc/R)elast./(Fc/R)MMA)c

λ 10-3

2.7 × 2.2 × 10-2 1.1 × 10-2 1.9 × 10-2 5-8 7-10 9-12

6 9 11

All data are collected at V ) 100 nm/s. b Silica and PMMA from ref 31. c From the average experimental values.

Figure 8a indicates that the interaction is always repulsive before the jump-in. The theoretical predictions based on the DLVO treatment for identical surfaces26 are conducted through the fitting of the data in the region of long-range electrostatic interaction. The results for the constant charge and constant potential of surfaces are shown as solid curves in the figure. An approximate nonretarded Hamaker constant of 0.5 × 10-20 J was calculated for the mica/CTAB/water/CTAB/silica five-layer system through the method described by Israelachvili27 using the optical properties of a hexadecane layer to model those of the CTAB layers.15 The fitted curves for the longrange interaction gave a Debye length, k-1, of 9.4 nm, which is close to the theoretical value of 9.8 nm for this system,15 and the calculated surface potential, ψ0, of 98 mV, is in the range of literature values for surfaces covered with CTAB at the same concentration.17,28 It is clear that the data deviate from the DLVO theory at very short surface separation (3.5 < h < 5.0 nm), which is attributed to an additional short-ranged repulsion. This additional force is know as the hydration or structural force, and several theories have been proposed for its explanation.29,30 However, the force curve in this range can be empirically fitted with an exponential law with the decay length in this region kc-1 ) 1.1 nm. Two other experiments were repeated using different probes, and very similar results were obtained; that is, kc-1 is always about 1 nm, the Fc/R is in the range 35-40 mN/m, and the pull-off force, Foff/R, is about 30-45 mN/m. Substituting experimental values of kc-1 and Fc into eq 9, the critical pressures, pc, at which the CTAB layers are broken can be estimated to be 6.0 MPa, as shown in Table 2. This value slightly overestimates the actual pressure because, far from the surface, the decay length is 9.4 nm, and so the value of 6.0 MPa will be a good estimation of the upper limit of the pressure. The value of λ for this system is evaluated by substituting literature values of elastic modulus for silica31 and the mica32 and the values of kc and pc into eq 7, as shown in Table 2, which must be the upper limit of λ. It is indicate that, when λ ∼ 10-3 (that is, when the value of λ is much lower than 1), particles and walls are regarded as rigid. PMMA Solid Particle to Mica Surface in 1.5 mM CTAB. For the system of mica/CTAB/water/CTAB/PMMA, similar force curves were observed as shown in Figure 8b. The best fit with the DLVO theory for the data in the long-range interaction indicates that the adsorbed layer of CTAB on the PMMA surface generates a potential close (26) Chan, D. Y. C.; Pashley, R. M.; White, L. R. J. Colloid Interface Sci. 1980, 77, 283. (27) Israelachvili, J. N. Proc. R. Soc. London, A 1972, 39, 331. (28) Biggs, S.; Mulvaney, P. J. Chem. Phys. 1994, 100, 8501. (29) Israelachvili, J. N.; Wennerstrom, H. Nature 1996, 379, 219. (30) Paunov, V. N.; Binks, B. P. Langmuir 1999, 15, 2015. (31) Mark, J. E. Physical Properties of Polymer Handbook; AIP Press: Woodbury, NY, 1996; pp 332-333, 372. (32) Simmons, G.; Wang, H. Single Crystals Elastic Constants and Calculated Aggregate Properties, 2nd ed.; MIT Press: Cambridge, MA, 1971; p 321.

to that for the preceding mica/silica system. This does not guarantee the same structure of the adsorbed layer, but at least the charge density is close to that of the bilayer. Again, the interaction is repulsive until the jump-in. The jump-in distance is found to be between 2.5 and 3.5 nm for PMMA particles of three different sizes. Sometimes, the jump-in distance changed after numerous pressings. This might be attributable to the plastic deformation of the asperity of the particle surface.12 To avoid this effect, only the data given by the first piezocycle are accepted here. The major difference from the case of the silica particle is in the decay of F with h at 3.2 < h < 5.0 nm; the force decay is not so sharp. The values of kc-1 were found to be in the range 2.0-3.8 nm, as shown in Table 2, which might be due to the difference in the structure of the adsorbed layer or to the roughness of the particle surface. Despite the three different particles used, values of Fc/R were nearly the same within 28-34 mN/m. Values of pc and λ, which were evaluated by the same procedure as before, were of the same order (of 10-2) as those for silica, as given in Table 2. It is also found that the pull-off force is approximately equal to that for the silica/mica system, that is, Foff/R ) 40-45 mN/m. All these data indicate that the nondeformable approximation is acceptable for PMMA particles. Elastomer Particle to Mica Surface in 1.5 mM CTAB. The interaction between an elastomer particle and mica in a 1.5 mM CTAB solution was examined, and raw data for the cases Fm < Fc and Fm > Fc are shown in parts a and b of Figure 4, respectively. When Fm < Fc, the particle rebounds from the surface, as shown in Figure 4a. When Fm > Fc, surfaces come into adhesive contact, as estimated by the existence of the load-unload hysteresis and the pull-off force shown in Figure 4b. The transition to adhesive contact is not marked by a sharp jump-in of the probe, as it was for the previous systems. Nevertheless, it was possible to estimate the critical force by gradual increase of the maximum load until the adhesion appeared. The elastomer particle is not simply elastic but viscoelastic. However, during nonadhesive particle compression, the viscoelastic losses are negligible and the particle can be considered as linearly elastic.14,33,34 As for the pulloff in the adhesive case, a number of interesting effects, which were generated from the viscoelastic nature of particles, were observed14 but not discussed here because our attention is paid only to the interaction between repulsive surfaces. All the data given in Table 2 were obtained at the scan rate 100 nm/s. Since there was no significant change in the value of Fc/R even though the scan rate varied from 10 nm/s up to 1000 nm/s, we assumed that the hydrodynamic force effect was negligible for this range of approaching rates. As explained in the Experimental Section, the cantilever was calibrated prior to the attachment of the particle. (33) Maugis, D.; Barquins, M. J. Phys. D 1978, 11, 1989. (34) Greenwood, J. A.; Johnson, K. L. Philos. Magn. A 1981, 43, 697.

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Figure 10. Diamonds: elastomer particle F(d) data from Figure 9. Circles: solid PMMA particle F(h) data from Figure 7b shifted horizontally to coincide with the elastomer particle data. Squares: elastomer particle data shifted back horizontally by the corresponding force Hertzian penetration as shown in the inset. Figure 9. Data from Figure 4a transformed to F/R vs d data. The inset shows the encircled data. Good fit with the solid particle decay length indicates that at that separation a particle can be considered as nondeformable. At higher forces the data are fitted with the Hertz curve shown by the solid line.

Then, using the transformation given by eqs 2 and 3, raw data were converted into the scaled force F/R to the relevant displacement d. Figure 9 shows a F(d) force curve converted from the data in Figure 4a, where the particle approaches the surface in the region Fm < Fc. The situations of particles corresponding to the regions of the force curve are drawn below the graph, respectively. If δm ) δ (Fm), as it is shown in Figure 5, and we choose ∆0 ) δm in eq 3, then

d ) (δm - δ) + h

(16)

As shown in Figure 9, the value of F/R at d > 135 nm decays exponentially and k-1 ) 10 nm, which is nearly the same as that for nondeformable particles shown in Figure 8. This indicates that the deformation is negligible, that is, δ ≈ 0 and ∂d ) ∂h, in the range of sufficient separation. On the other hand, h ≈ 0 and ∂d ) -∂δ, when the particle is in contact with the surface. Now the data of the force curve at d < 70 nm are fitted well by the Hertz theory expressed by the solid curve, δ ∼ F2/3, in Figure 9, from which the elasticity modulus E of the particle is determined. In the intermediate region of 70 < d < 135 nm, ∂d ) ∂h - ∂δ. The data of Fc/R for elastomer particles of three different sizes are presented in Table 2, where the following differences are clarified compared with the case of PMMA hard particles: (1) values of Fc/R are much larger than those for PMMA particles, and (2) the value of Fc/R increases clearly with the particle size, which was not the case of PMMA particles. In principle, the Hertz theory is not applicable for the relatively large deformation of particles given above, because the theory is valid only at a , R.14,35,36 Neverthe(35) Deruelle, M.; Hervet, H.; Jandeau, G.; Leger, L. J. Adhesion Sci. Technol. 1998, 12, 225. (36) Shull, R. K.; Dongchan, A.; Chen, W. C.; Flanigan, C. M.; Crosby, A. J. Macromol. Chem. Phys. 1998, 199, 489.

less, the apparent values of E are determined using eqs 10 and 11, because the data of force curves are fitted well with the theory, as illustrated in Figure 9. It is clear that the values of E obtained decrease with the particle size, although they are in the range expected for elastomer materials.31,37 This indicates that the apparent stiffness of particles increases slightly with decreasing particle size, as observed before.14,35 Substituting values of E and Fc into eq 13, the critical pressure, pc, is roughly estimated as given in Table 2. The value decreases gradually with particle size, but all the values are apparently smaller than those for PMMA particles. If the CTAB layer of the same characteristics is formed on both the surfaces of elastomer and PMMA particles, the value of pc should be independent of the kind of particles. The reason for the difference of pc for elastomer and PMMA particles is not clear at present. This may be due to the approximate character of our considerations, but it may mean that the critical pressure is lower for adsorbed layers pressed in a larger film. More sophisticated experiments are needed to clarify the reason. Values of λ for elastomer particles are evaluated as shown in Table 2, using the values of E and pc and assuming that the average value kc for the PMMA is applicable. Despite these approximations, the magnitude is greater than unity and increases with increasing value of Fc/R. It should be noted that the increase of Fc/R with the particle radius coincides with eq 13. The data at d > 110 nm in Figure 9 are replotted in the logarithmic scale in Figure 10. Then the data in Figure 8b are drawn on the same figure by shifting horizontally such that the forces at d > 135 nm coincide between both series of data, because the electrostatic force between surfaces must be the same at the separation where the effects of particle deformation are negligible. On the other hand, the data at d < 80 nm in Figure 9 are well expressed by the Hertz theory as explained above, which indicates that two surfaces may be in Hertzian contact, and h ) 0, at least around d ∼ 0. Then, the penetrated distance, δH, after the surface contact may be estimated by the following (37) Sperling, L. H. Introduction to Physical Polymer Science; WileyInterscience: New York, 1992; Chapter 8.

Effects of Particle Deformability

Langmuir, Vol. 19, No. 1, 2003 117

elastomer particle interaction begins to deviate form rigid particle approximations. From the data in Figure 10 it is shown that the deviations are significant once F/R > 3 mN/m. Then, formally taking Fc/R ) 3 mN/m and k-1 ) 10 nm in eqs 6 and 8, we found λ ) 0.26. As it is shown in Figure 5 according to the proposed theoretical model, this is a very reasonable value for the limit to which the particle can be considered nondeformable.

Figure 11. Same data as in Figure 10 with zero separation offset to the adsorbed layers contact position. The solid line is schematically showing the elastomer particle F(h) dependence.

equation given by the Hertz theory.

δH )

R1/3F2/3 K2/3

(17)

When the contribution of this Hertzian penetration is subtracted, the force curve of F/R versus d is expressed as that in Figure 10. It is clear that the vertical force curve around d ) δm ∼ 123 nm indicates that two surfaces contact, that is, that h ) 0. It is important to point out that this position coincides approximately with the position where the surfaces of a PMMA particle and a mica plate with CTAB layers contact each other before the jump-in, that is, h ) 0. This consistency implies that the treatment of the data of the elastomer particle given above is adequate. Far from the surface, δH > δ and the data from which the Hertzian penetration is subtracted should be considered only as the upper limit data. Figure 11 depicts these force curves plotted against h. It is clear that the force curve for the elastomer particle against h, F(h), must satisfy the following: F(h) coincides with the force curve for the PMMA particle at d > 135 nm and with the curve from which the contribution of the Hertzian penetration is subtracted at h ) 0. In other words, F(h) approaches the force curve for nondeformable particles, at large separation distances, and as h decreases, F(h) approaches the force curve for Hertzian particles, as shown by the solid curve in the figure. It is important to know the value of λ at which the deformation of particles becomes significant or where the

Conclusions In this work repulsive interaction between particles of different deformability in solution was studied. The following was clarified. (1) It was confirmed that, at the concentration 1.5 mM CTAB, highly charged layers were formed on mica and silica surfaces. It was found that an adsorbed layer of similar charge was formed on the PMMA polymer particle surfaces. The combination of the long-range double layer interaction with the short-range hydration repulsion resulted in an increasingly strong repulsive force between the surfaces until the adsorbed layers came into contact. (2) In all of the systems studied, the particle could be pressed to the surface with some force higher than a critical force, Fc, necessary to destroy the adsorbed layers and to bring the surfaces into adhesive contact. If the maximum applied force on the particle was lower than this critical force, the particle rebounded from the surface. (3) The main experimental finding was that, in the case of the micron size elastomer particles, the scaled critical force, Fc/R, is up to more than 10 times higher than that in the case of the solid particles. This effect is important when considering the dispersion stability of low elastic modulus particles. (4) A quantitative treatment of the deformation effect on the total repulsive interaction between approaching elastic spheres was done using the Hughes and White theory.1 The deformation effects were expressed as a function of the parameter λ, which is a dimensionless combination of the particle radius and elastic modulus and the repulsive force magnitude and decay length. (5) This is the first experimental attempt to confirm the theory for repulsive interaction1 using micron size elastic particles. The theoretical predictions seem to account for the general behavior of the systems, as the scaled critical force increases, when the estimated values of λ are higher than 1. A further extension of the theory to include the case of large deformation is needed for more precise prediction of the micron size deformable particle interaction. LA026581I