Long-Range Interactions between Soft Colloidal Particles in Slit−Pore

1296

J. Phys. Chem. B 2007, 111, 1296-1303

Long-Range Interactions between Soft Colloidal Particles in Slit-Pore Geometries Sabine H. L. Klapp,*,†,§ D. Qu,‡ and Regine v. Klitzing‡ Stranski-Laboratorium fu¨r Physikalische und Theoretische Chemie, Sekretariat C7, Technische UniVersita¨t Berlin, Strasse des 17. Juni 115, D-10623 Berlin, Germany, Institut fu¨r Theoretische Physik, Sekretariat PN 7-1, Fakulta¨t II fu¨r Mathematik und Naturwissenschaften, Technische UniVersita¨t Berlin, Hardenbergstrasse 36, D-10623 Berlin, Germany, and Stranski-Laboratorium fu¨r Physikalische und Theoretische Chemie, Sekretariat TC9, Technische UniVersita¨t Berlin, Strasse des 17. Juni 124, D-10623 Berlin, Germany ReceiVed: September 13, 2006; In Final Form: December 1, 2006

Combining theoretical and experimental techniques, we investigate the structure formation of charged colloidal suspensions of silica particles in bulk and in spatial confinement (slit-pore geometry). Our focus is to identify characteristic length scales determining typical quantities, such as the position of the main peak of the bulk structure factor and the period of the oscillatory force profile in the slitpore. We obtain these quantities from integral equations/SANS experiments (bulk) and Monte Carlo simulations/colloidal probe-AFM measurements (confinement), in which the theoretical calculations are based on the Derjaguin-Landau-Verwey-Overbeck (DLVO) potential. Both in bulk and in the slitpore, we find excellent qualitative and quantitative agreement between theory and experiment as long as the ionic strength chosen in the DLVO potential is sufficiently low (implying a relatively long-ranged interaction). In particular, the bulk properties of these systems obey the widely accepted density scaling of ξ ∝ φ-1/3. On the other hand, systems with larger ionic strengths and, consequently, more short-ranged interactions do not obey such power law behavior and rather resemble an uncharged hard-sphere fluid, in which the relevant length scale is the particle diameter.

Introduction Colloidal dispersions are omnipresent in daily life. Therefore, the understanding and control of the interaction between colloidal particles is of great interest not only in scientific contexts but also for technical applications. Due to ongoing miniaturization, in addition to their volume properties, the structure and dynamics of colloidal dispersions in confined geometries become more and more important.1 In the present paper, we are particularly interested in the characteristic length scales determining structure formation of charged colloids in slit-pore geometries and in bulk. We investigate these questions using various theoretical and experimental tools. Our investigations are based on silica suspensions that present a suitable model system with tunable interactions. Confining colloidal dispersions in slit-pore geometries leads to the so-called structural forces caused by ordering of molecules, particles, or aggregates. These forces were shown experimentally for small molecules such as octamethylcyclotetrasiloxane2 and water,3 which were entrapped between two mica plates in a surface force apparatus. The approach of the plates toward each other leads to damped oscillatory forces that are explained by a layering of the molecules parallel to the mica surfaces.4 With increasing external force, the molecules are squeezed out of the slitpore layer by layer. The oscillation period is connected to the distance of the molecular layers. Oscillatory forces also occur in thin films of molten salt,5 liquid crystals,6 or colloidal particles.7-9 * To whom correspondence should be addressed. E-mail: sabine.klapp@ fluids.tu-berlin.de. † Fakulta ¨ t II fu¨r Mathematik und Naturwissenschaften. ‡ Stranski-Laboratorium fu ¨ r Physikalische und Theoretische Chemie, Sekretariat TC9. § Stranski-Laboratorium fu ¨ r Physikalische und Theoretische Chemie, Sekretariat C7.

A change of the interaction parameters is reflected in the characteristics of the force oscillations. For instance, atomic force microscope (AFM) measurements on a solution of silica, sulfonated polystyrene particles, or both entrapped between a silica microsphere and a flat silica plate show that with increasing diameter, increasing surface potential, and increasing concentration of the particles, the force oscillation becomes more pronounced (i.e., the minima become deeper and the maxima higher).10 Theoretically, structural forces have been studied using various statistical-mechanical methods, such as density functional theory (see, e.g., refs 11 and 12) and computer simulations.13 The studies have shown that theoretical tools can describe oscillatory forces in a variety of model systems, such as hard spheres,12,14 polar fluids,15 liquid crystals,16 and polyelectrolytes.17 The present paper deals with structure formation of charged particles in aqueous bulk solution and in thin liquid films. In bulk solution, the decisive quantity for structure formation is the structure peak, which is a measure for the mean particle distance. We determine this quantity experimentally by smallangle neutron-scattering (SANS) and theoretically by hypernetted chain (HNC) integral equations, in which the calculations are based on the well-known Derjaguin-Landau-VerweyOverbeck (DLVO) potential.18 In slit-pore geometry the relevant quantity is the period of the force oscillation, which we measure by AFM. The corresponding theoretical calculations are carried out using Monte Carlo (MC) simulations. The period of the force oscillation gives a measure for the layer thickness, that is, the particle distance just before the particles are pressed out. One central question of our study is the effect of the range of the interaction, which we can control by the silica density

10.1021/jp065982u CCC: $37.00 © 2007 American Chemical Society Published on Web 01/24/2007

Soft Colloidal Particles in Slit-Pore Geometries

J. Phys. Chem. B, Vol. 111, No. 6, 2007 1297

and the ionic strength, on the bulk structure peak, focusing on the appearance of power law behavior. Another point we wish to explore is the density dependence of the period of force oscillations in the slit pore, which has been discussed in the literature,19,20 and has generated some controversy. Connected to these questions is the mathematical description of the colloidal interactions, which we achieve by comparing theory and experiment. Model System and Parameters In the present work, we adopt an effective one-component description of the real colloidal solution. That is, we focus on the structure and ordering of the macroions (silica particles), whereas the “solvent particles” (counterions plus additional salt ions) are treated only implicitly. For a system with low macroion density, F (and moderate macroion charges Z), the corresponding effective interaction between two macroions with separation r can be derived on a rigorous statistical-mechanical basis,21 leading to the well-known DLVO potential,18

uDLVO(r) ) (Z˜ e0)2

exp(-κ(r - 2R)) 4π0r

(1)

where e0 is the elementary charge, 0 is the permeability of vacuum,  is the dielectric constant of the solvent, and R is the particle radius. In addition, Z˜ is the effective valency which (assuming that the colloidal concentration is low compared to that of the added salt) is given by

Z˜ )

Z 1 + κR

(2)

Finally, κ is the inverse Debye screening length, defined as

κ)

x

K

1

0kBT

(Fc(zce0)2 +

Fk(zke0)2) ∑ k)1

(3)

where kB is Boltzmann’s constant, T is the temperature, and Fc and zc are the number density and valency of the counterions, respectively. The remaining sum refers to the additional salt ions. Assuming univalent counterions (|zc| ) 1), the condition of charge neutrality between counter- and macroions requires Fc ) |Z|F. Equation 3 can then be rewritten as

κ)

x

e02 (ZF + 2INA) 0kBT

(4) 1/ ∑K F z2 2 k)1 k k

where we have introduced the ionic strength I ) of the additional salt, and NA is Avogadro’s constant. We note that due to the appearance of the Debye length, the DLVO potential depends on both the density of the macroions and on the temperature. The total fluid-fluid interaction potential used in the present calculations is given by

uff(r) ) uSS(r) + uDLVO(r)

(5)

where the first term represents an additional soft-sphere repulsion. The latter is defined as

uSS(r) ) 4SS(σ/r)12

(6)

with the particle diameter σ ) 2R. The soft-sphere interactions have been introduced purely for technical reasons. In fact, in

the actual calculations, the parameter ff has been set such that the SS interactions are essentially neglible against the DLVO repulsion (see below). To model the slit-pore confinement, we consider the silica solution to be squeezed between two plane, parallel, smooth, surfaces separated by a distance h along the z axis of the coordinate system and of infinite extent in the x-y plane. In the present work, we confine ourselves to the investigation of uncharged surfaces only. Specifically, we employ the fluidwall potential21

ufw(z) )

4π σ 9 σ 9 + fw 45 h/2 + z h/2 - z

[(

) (

)]

(7)

From the preceding equations, it follows that the physical properties of the colloidal model system depend on temperature (T), density (F), valency (Z), and diameter (σ) of the macroions, the ionic strength (I) of the additional salt, and the numerical parameters ff and fw characterizing the fluid-fluid and fluidwall repulsion, respectively. To mimic the experimental conditions (see Experimental Methods), the theoretical results presented in this work have been obtained with the parameters T ) 298 K and σ ) 26 nm. The valency has been set to Z ) 35, as estimated from the Grahame equation for the surface charge density, σ22

σ ) x80kBT sinh

( ) e 0ζ 2kBT

(8)

where the ζ potential is given by (about) -80 mV at an ionic strength of (about) I ) 10-4mol/L (see Particles and Suspensions). However, since the precise value of I is not known from the experiments, and to clarify the role of I for the system’s properties, we have performed calculations also at several other values of the ionic strengths in the range I ) 10-1-10-5 mol/ L. From a more physical point of view, variation of I allows us to explore the influence of the range of the repulsive interactions, which depend on the (inverse) Debye length and, therefore, also on I (see eq 4). The density of macroions has been characterized through the volume fraction φ ) Fσ3, and we have considered systems in the range φ ) 0.2-30 %. Finally, the repulsion parameters have been fixed such that ff/kBT ) fw/kBT ) 1.0. Integral Equations for the Bulk System In bulk experiments, the characteristic length describing the local structure in the fluid was deduced from the location of the first (“structural”) peak in the scattering intensity I(q) ∝ P(q)S(q), where P(q) and S(q) are the form factor and the structure factor, respectively, and q ) |q| is the length of the scattering vector. The relevant structural information is contained in S(q), which is related to the pair correlation g(r) via21

S(q) ) 1 + F

∫ dr exp[iq ‚ r](g(r) - 1)

(9)

In the present work, we calculate S(q) or, equivalently, the pair correlation function g(r) ≡ h(r) + 1 (with h(r) being the total correlation function) by integral equations theory. This involves simultaneous (numerical) solution of the exact Ornstein-Zernike equation23

h(r12) ) c(r12) + F

∫ dr3 h(r13)c(r32)

(10)

1298 J. Phys. Chem. B, Vol. 111, No. 6, 2007

Klapp et al.

TABLE 1: HNC Results for the Dimensionless Compressibility, Internal Energy, (Virial) Pressure, and Excess Chemical Potential of the Colloidal Solution at Volume Fraction O ) 5.0 % (Gσ3 ) 0.095) and Various Ionic Strengths Ia I [mol/L]

κσ

βχ/F

βU/N

βP/F

βµex

10-5 10-4 10-3

1.11 1.37 2.91

0.033 0.052 0.224

10.6 (10.6) 5.98 (5.95) 0.76 (0.75)

1.46 (1.46) 0.93 (0.93) 0.25 (0.25)

26.04 (25.16) 15.76 (15.22) 2.90 (2.84)

ensemble, P⊥ can be expressed as a statistical average involving the instantaneous normal component of the (virial) pressure tensor,13 that is

P ⊥ ) kB T

〈N〉µ,T Ah

-

〈∑ ∑ N

1 Ah

N

i)1 j*i

where c(r) is the direct correlation function, combined with an approximate closure relation relating the correlation functions to the pair potential. We choose the hypernetted chain approximation,21 defined as

g(r) ) exp[-βu(r) + h(r) - c(r)]

(11)

where β ) 1/kBT. Using the HNC correlations to calculate various thermodynamic quantities,21 such as compressibility, internal energy, chemical potential, and pressure, we have found a very good internal consistency. For example, the pressure data resulting from virial and free energy route agreed with each other within 0.5%, indicating high accuracy of the HNC results under the conditions considered in the present work. An alternative choice would be the Rogers-Young (RY) closure,24 which interpolates between the HNC and the Percus-Yevick approximation and has been widely used in the theory of macroion solutions (see, e.g., ref 25). Within our calculations, however, HNC and RY gave identical results. Moreover, the HNC thermodynamical data are in very good agreement with those from (canonical) MC simulations (see Monte Carlo Simulations). Some exemplary data are given in Table 1, indicating that an increase in the ionic strength, I (i.e., a decrease of the range of the repulsive interactions), at fixed volume fraction yields an increase in the (dimensionless) compressibility. At the same time, the total internal energy, the pressure, and the (excess) chemical potential decrease, reflecting the reduction of overall repulsion. Monte Carlo Simulations To investigate the spatially confined silica solutions, we performed MC simulations in the grand canonical (GC) ensemble, that is, at fixed temperature, T; fixed volume, V ) Ah of the simulation box (with A being the box area parallel to the surfaces); and fixed chemical potential, µ. The GCMC method assumes that the confined fluid is in contact with a bulk reservoir at the same chemical potential,26 whereas the particle number N (in the pore) can fluctuate. This situation corresponds precisely to the situation under which the present experiments are performed. To identify the chemical potentials of interest, we have first conducted MC simulations of the bulk systems in the canonical (NVT) ensemble, from which the chemical potential has been extracted using Widom’s test particle method.26 The resulting values have then been used to perform GCMC simulations at various surface separations h in the range 1.6σ e h e 4.5σ (≈ 40-120 nm). In all of these calculations, the inverse Debye length has been fixed at the value corresponding to the bulk system. For reasons explained in the Colloidal Films section, a central quantity in the present study is the normal pressure, P⊥, exerted by the fluid on the two confining surfaces. In the grand canonical

rij

∂r

〈∑

r)rij

〉

-

µ,T

|

∂ufw(z) zi Ah i)1 ∂z 1

a

The corresponding values for the dimensionless inverse Debye lengths κσ are given in the second column. MC data are shown in parentheses.

|

z2ij ∂uff(r)

N

z)zi

〉

(12)

µ,T

where 〈...〉µ,T denotes an average in the grand canonical ensemble. A second quantity of interest is the local density profile. Due to the planar, homogeneous character of the confining walls, the fluid is translationally invariant in the x and y directions such that the local density depends only on z. A statistical expression is then given by

F(z) )

〈N(z)〉µ,T A∆z

(13)

where N(z) is the average number of particles inside slices of thickness ∆z ) 0.05h. Experimental Methods Particles and Suspensions. Silica beads (Ludox TMA-34, deionized) were purchased from Aldrich (Taufkirchen, Germany). To remove the salt, the samples were dialyzed against Milli-Q water (Millipore, Billerica, MA) for 10 days. The tubes with a MWCO of 1000 were from Roth (Karlsruhe, Germany). The particle size was determined by scanning electron microscopy (SEM S-4000, Hitachi, Tokyo, Japan) at a primary electron energy of 20 keV and atomic force microscopy with tapping mode with a multimode nanoscope (Digital Instruments, Santa Barbara, CA). Both methods gave a particle diameter of ∼26 nm. The ζ potential was determined by electrokinetic measurements (Zeta Sizer, Malven). At an ionic strength of ∼10-4 mol/L, the value of the potential is about -80 mV. For most of the experiments, Milli-Q water was used as solvent. Only for small-angle, neutron-scattering experiments was D2O (99.9% D, euriso-top, Saarbru¨cken) the solvent. The weight percentage and the density of the solutions were determined by weighing the sample before and after drying (24 h at 400°). The density of the silica particles is 2 g/cm3 (determined by thermogravimetric measurements), corresponding to a conversion factor of 0.5 from weight percentage to volume percentage. SANS Measurements. The neutron experiments were carried out at the SANS-I-Spektrometers at the Paul-Scherrer Institut (Switzerland). To get enough contrast, D2O was used as the solvent for the Ludox particles. Colloidal-Probe AFM. The colloidal probe (CP) technique was first developed by Ducker et al.27 In our experiments, a silica particle is glued with epoxy to a tipless cantilever (Ultrasharp Contact Silicon Cantilevers, CSC12) produced by µMasch. The silica particles are produced by Bangs Laboratories, Inc., and all have a radius, R, of ∼3.35 µm. The tip is cleaned with plasma cleaning for 10 min right before each measurement cycle to remove all the organic components on its surface. The substrate used is a silicon wafer with a native SiO2 top layer, cleaned using the RCA method,28 and stored in Millipore water before usage. Just before each experiment, the substrate is taken out of the water and dried in a nitrogen stream.

Soft Colloidal Particles in Slit-Pore Geometries

Figure 1. Structure factor at ionic strength I ) 10-5 mol/L and various volume fractions, φ.

Then a drop of the Ludox solution is put onto the substrate, and the probing head is immersed in the solution. Force-vsdistance F(h) curves were measured with a commercial atomic force microscope MFP (molecular force probe) produced by Asylum Research, Inc. and distributed by Atomic Force (Mannheim, Germany). Because both the silica particles in the solutions and the interfaces are negatively charged, no adsorption occurs on the surfaces. For each solution, altogether 10-20 force-distance curves were measured at different lateral positions on the same substrate as well as on different substrates to ensure reproducibility and to get good statistics. Oscillatory force curves occur, and the period can be determined by the distance between two adjacent minima. The final result of the force period is the average period of 10-20 curves, and the error bars are from the standard deviation. Results and Discussion Bulk Structure. To understand the local structure in the bulk silica solution, we have performed several HNC calculations of the structure factor, S(q), as well as SANS measurements of the scattering intensity, I(q). These two quantities differ by the form factor, P(q), such that their full q-dependence is not directly comparable. However, since P(q) is almost constant in the q range of interest, we can safely assume that I(q) and S(q) exhibit their main (“structural”) peak at the same wavenumber, qmax. The corresponding length, ξ ≡ 2π/qmax, roughly coincides with the position of the main peak of the pair correlation, g(r), of the silica particles as a consequence of the fact that S(q) and g(r) are directly related by a Fourier transform (see eq 9).21 Due to this close connection, the length ξ ) 2π/qmax, indeed, characterizes the local structure of the system and can be used as a measure of the average spacing between two silica particles. A further interesting feature of the structure factor is the height of the main peak, S(qmax), characterizing the degree of the local (shell) structure around a given particle, that is, roughly speaking, the number of oscillations in g(r). Finally, the long wavelength limit S(q f 0) is proportional to the system’s compressibility.21 Theoretical results for S(q) at ionic strength I ) 10-5 mol/L and several volume fractions are plotted in Figure 1. At the small values of φ characterizing the experimental range explored in the present work, the structure factor has only one (main) peak, whose position shifts to higher q values with increasing φ. This is expected from the preceding remarks and reflects that the average spacing between two silica particles becomes smaller. Upon further increase in φ, one also observes the development of additional oscillations in S(q) and a strong increase in the height of the main peak. We attribute this

J. Phys. Chem. B, Vol. 111, No. 6, 2007 1299

Figure 2. Structure factor at φ ) 5.0% and various ionic strengths, I.

behavior, on one hand, to the increasing importance of packing effects when the system becomes denser and denser (an effect that would also be present without any charges). On the other hand, an increase in the silica volume fraction also implies an increase in charges in the system (in other words, an increase of κ; see eq 4), which induces stronger repulsion (on the average) and a more pronounced local ordering of the particles. We now consider the high-density limit of the structure factor in Figure 1. According to the so-called Hansen-Verlet criterion,29 a system freezes at S(qmax) ≈ 2.85. The present calculations (at I ) 10-5 mol/L) indicate that this is the case at φfreezing ≈ 44 %. Interestingly, this value is essentially independent of the ionic strength of the additional salt. This seems to be in contrast to experimental observations30 on the solidification of charged latex spheres in which a strong dependence of the ionic strength has been observed. We note, however, that the total charge (and size) of the latex spheres was much higher than that of the silica spheres considered in the present work, making the Coulomb interactions (of the latex spheres) more dominant.30 Therefore, a direct comparison of the systems is quite difficult. Within the present experiments, the validity of the theoretical predictions could not be checked, since the silica particles could not be stabilized in this high concentration regime. Nevertheless, at the relatively low concentrations considered, we did not find any evidence for crystallization. So far, results were presented for a very low ionic strength (I ) 10-5 mol/L) of the additional salt. In the following, we consider in more detail screening effects on S(q) induced by variation of the ionic strength. Indeed, the ionic strength does have a strong influence on S(q) at small volume fractions. This is demonstrated in Figure 2 where we have plotted S(q) for various values of I at an exemplary value of φ typical for the dilute regime. Starting from low salt concentrations (I ) 10-5 mol/L), it is seen that an increase in I has several effects. First, the height of the main peak decreases, reflecting that the local structure around a given particles becomes less pronounced. Second, the magnitude of S(q) in the limit q f 0, that is, the compressibility, increases, indicating a weakening of the repulsive interactions between the silica particles. This is consistent with the thermodynamic state data presented in Table 1, reflecting that, apart from the compressibility, pressure and average internal energy also decrease with increasing ionic strength. A further conclusion from the theoretical data in Figure 2 is that an increase in I shifts the position of qmax toward larger values, that is, smaller lengths, ξ, in real space. Compared to the effect of particle concentration, however, the influence of the ionic strength on qmax is much lower. A change of ∼1 order of magnitude in φ yields a shift in qmax of 0.1 nm-1, whereas even a change of 4 orders in I is not sufficient to get the same change in qmax. This weak sensitivity of the structure peak with respect to the ionic strength has also been observed in scattering

1300 J. Phys. Chem. B, Vol. 111, No. 6, 2007

Klapp et al. geometric arguments33 for isotropic fluids in three spatial dimensions, one would expect that ξ scales with the packing fraction as ξ ∝ φ-1/3. As seen from the double-logarithmic representation in Figure 3b, power law behavior of the form

ξ ) aφ-b

Figure 3. (a) Characteristic length as function of the volume fraction for various ionic strengths, I. Included are the experimental (SANS) data. (b) Double-logarithmic representation of the data corresponding to low I, together with the SANS data.

spectra of polyelectrolyte solutions.31 In all these systems, the addition of salt has two effects. One is the screening, that is, the increase in κ (see eq 4), which leads to a stronger damping of the oscillations. The other effect is ion condensation at the macroion, which reduces the effective charge (see eq 2). Having seen the effect of the additional salt, it is worthwhile to briefly come back to the density dependence of S(q) shown in Figure 1. Due to charge neutrality, an increase in macroion concentration implies a simultaneous increase in the counterion concentration and, consequently, in the screening effect (that is, an increase in κ, see eq 4). Therefore it is interesting to ask for the role of the counterion screening for the structure factor. For the studied system, a particle concentration of 0.5 vol % corresponds to 10-6 mol/L and 30 vol %, to 6 × 10-5 mol/L. Under the assumption that each particle carries 35 charges, the contribution by the counterions of the particles to ionic strength increases from 3.5 × 10-5 mol/L to 2.1 × 10-3 mol/L in the considered particle concentration regime. Hence, the screening length would be dominated by the counterions, and its value varies between 53 nm (for 0.5 vol %) and 6.9 nm (for 30 vol %). But obviously, the screening effect of the counterions is much weaker than the influence of the increase in charges that leads to a pronounced structure peak in Figure 1 for high particle concentrations. An analog experimental result was found for other charged macroions, such as polyelectrolytes, in which an increase in concentration or degree of charge leads always to a higher amplitude of the structure peak.32 The theoretical results for the length ξ ) 2π/qmax as function of both the volume fraction and the ionic strength are summarized in the two parts of Figure 3, where we have included the experimental (SANS) data. From Figure 3a, one clearly observes that the two theoretical curves obtained at low salt concentrations (I ) 10-5-10-4mol/L) are closest to the experimental data, both in magnitude of the characteristic length, ξ, and in its dependence on the volume fraction, φ. From simple

(14)

is, indeed, shown by the data for I ) 10-5 mol/L and for the SANS data, whereas the data at I ) 10-4 mol/L are less conclusive. Using the above power law as a fit formula, one obtains for the theoretical curves (including that for I ) 10-4) the values a ≈ 80, b ≈ 0.30 (I ) 10-5) and a ≈ 70, b ≈ 0.21 (I ) 10-4), respectively. The corresponding SANS parameters lie between these values, that is, aSANS ≈ 80, bSANS ≈ 0.25. Strictly speaking, the geometric scaling (b ) 1/3) should be restricted to weakly interacting (ideal gaslike) systems, such as the ones considered in the above-mentioned SANS experiments. Interestingly, however, the present theoretical data at I ) 10-5 mol/L can be fitted to the above power law up to volume fractions φ ≈ 20 % (see Figure 3b). Only at even higher concentrations do the functions ξ(φ) approach a limiting value given roughly by the particle diameter, σ ≈ 26 nm. Turning now to the theoretical model systems with high salt concentrations, one sees from Figure 3a that the density dependence of ξ(φ) is much less pronounced. Indeed, a fit to eq 14 turned out to be impossible for these systems; instead, ξ is close to the particle diameter for all volume fractions considered in Figure 3a. This is reminiscent of pure hard-sphere (HS) systems, where ξ (as obtained from the Verlet-Weiss fit34 of the exact hard-sphere structure factor) nearly coincides with the particle diameter throughout the fluid density range. It seems worth noting that in a very recent study of silica solutions, Waltz and co-workers19 presented and analyzed another measure (different from ξ) for a mean particle distance that is not based on the structure factor, but on first-order perturbation theory. According to this measure, even HS systems obey power law behavior up to high concentrations, indicating that the precise definition of the mean distance is crucial for its density dependence. Colloidal Films. Experimentally, the primary quantity from which we extract relevant length scales in the confined silica solutions is the force-distance profile, that is, the force, F, between a solid surface and that of a spherical nanoparticle (radius R) as function of the thickness, h, of the liquid film confined between them. When the diameter of the nanoparticle is sufficiently large as compared to the distance between the interfaces, which is the case in the present experiments (see the Colloidal-Probe AFM section), the two surfaces can be assumed to be locally planar (Derjaguin approximation22,35). As a consequence, the normalized force, F/R, becomes a function of the film thickness alone, that is, F/R ) F(h)/R. In typical colloidal suspensions, this function displays oscillatory character, and the period of these oscillations, dslit, characterizes the structure formation in the film. Within the GCMC simulation technique, we do not have direct access to the force profile, but rather, to a closely related quantity, the pressure, P⊥, exerted by the fluid onto two planeparallel confining surfaces separated by a distance h (see eq 12). Connection between P⊥ and F/R involves again the Derjaguin approximation,22,35 which allows one to derive the relation16,12

F(h) 2πR

∞

)

∫ dh′ (P⊥(h′) - Pbulk) h

(15)

Soft Colloidal Particles in Slit-Pore Geometries

J. Phys. Chem. B, Vol. 111, No. 6, 2007 1301

TABLE 2: Bulk Volume Fractions and Corresponding Dimensionless Chemical Potentials at Two Values of the Ionic Strength φbulk [%] βµ (I ) ) βµ (I ) 10-5)

5

8

10

15

20

30

15.2

17.0

12.0 18.7

17.1 22.0

22.7 26.8

34.9 40.1

10-4

where Pbulk is the pressure of a bulk fluid characterized by the same temperature and chemical potential. From eq 15, it follows that

( )

d F(h) (2π)-1 ) -(P⊥(h) - Pbulk) ≡ -f(h) dh R

(16)

reflecting that the normalized normal pressure, f(h), is essentially the deriVatiVe of the force profile. As a consequence, the maxima and minima of the force profile correspond to zeros in the function f(h), and the period of oscillations in the force profile, dslit, can be estimated numerically as the distance between two next-nearest zeros of the normalized normal pressure. We prefer this method to estimate dslit over a numerical integration of f(h), since we are primarily interested in the period of force oscillations (rather than in other characteristics of these functions). We also note that the functions f(h) obtained for the present systems could not be easily fitted to the standard shapes known for simple (e.g., Lennard-Jones-like) fluids.16 In the following, we focus on results obtained at ionic strengths I ) 10-5 mol/L and I ) 10-4 mol/L, since the corresponding model systems seem to be closest to the real colloidal suspension (see Figure 3). The corresponding chemical potentials used in the GCMC simulations are given in Table 2. Numerical results for the function f(h) are shown in Figure 4a, where we focus on the influence of the volume fraction φbulk of the bulk fluid with which the confined fluid is in thermodynamic equilibrium (I ) 10-4 mol/L). For all values of φbulk shown in Figure 4a, the functions f(h) have damped oscillatory character similar to what is known for simple fluids16 (for volume fractions φbulk < 10%, the oscillations vanish at this particular ionic strength). Furthermore, a comparison of the data indicates two main effects of an increase in φbulk on the function f(h): First, the amplitude of the oscillations increases, reflecting that the stratification of the film (i.e., the formation of fluid layers parallel to the surfaces) becomes more and more pronounced. This can also be seen more directly from Figure 4b, where we have plotted the local volume fraction φ(z) ) (π/6)σ3F(z) (see eq 13) of silica particles at a fixed thickness and various values of φbulk. Clearly, the density profiles become sharper when the corresponding bulk volume fraction increases. Turning back to Figure 4a, the second feature displayed by the pressure profiles is that an increase in φbulk yields a decrease in the period of the corresponding force profile, dslit. A similar observation has been made in an earlier MC study by Jo¨nsson et al.17 Furthermore, both features, the increase in the amplitude and the decrease of the period of the oscillations of the normal pressure, resemble the behavior observed in experimental measurements of the force-distance curves of confined silica solutions. This may be verified from Figure 5, where we have plotted various results from CP-AFM measurements at various volume fractions. While at φ ) 1.25%, an oscillation can be hardly detected, and the oscillatory forces are well-pronounced at a volume fraction of 1 order of magnitude higher. The comparison between the volume fractions of 4.1% and 12.3% shows that in addition to an increase in amplitude, the period

Figure 4. (a) Simulation results for the normalized normal pressure as a function of film thickness at ionic strength I ) 10-4 mol/L and various (bulk) volume fractions, φbulk. (b) Corresponding density profiles at film thickness h ) 2.8σ (73 nm).

Figure 5. Force curve of the silica particle suspension confined between two silica surfaces as measured by colloidal-probe AFM. The concentration of the respective volume phase is given in volume percentage.

of the force oscillation decreases from 50 nm (4.1%) to 40 nm (12.3%). At much higher silica concentrations (30 vol %), the oscillation period is on the order of the particle diameter36 or the effective diameter (geometric diameter plus twice the Debye length).20 This could be a hint for a starting crystallization at a lower particle concentration than predicted for the volume phase (see discussion in the Bulk Structure section). Beyond the influence of (bulk) volume fraction, it is also interesting to consider the role of the ionic strength (of added salt), I, on the oscillatory force and pressure profiles. To explore this point from the theoretical side, we have performed additional GCMC simulations at I ) 10-5 mol/L. A comparison of two of the resulting functions, f(h), with the corresponding ones at the larger ionic strength I ) 10-4 mol/L is shown in Figure 6a. The data indicate that addition of salt somewhat reduces the amplitude of the oscillations, which is consistent with the softening of the corresponding density profiles plotted in Figure 6b. On the other hand, the resulting period of the force oscillations, dslit, is more or less unaffected by a variation of I, at least at this particular volume fraction. Indeed, at lower volume fractions characterizing the experimental systems (φ

1302 J. Phys. Chem. B, Vol. 111, No. 6, 2007

Figure 6. (a) Normalized normal pressure as a function of film thickness at two volume fractions and two typical values of I. (b) Density profiles at film thickness h ) 2.8σ (73 nm).

Figure 7. Period of force oscillations for a silica particle suspension confined between two silica surfaces as obtained by MC simulations and by CP-AFM.

e12%, see Figure 5), the MC results become quite sensitive to the ionic strength in the sense that the oscillations actually vanish when there is too much salt in the system. Finally, we compare in Figure 7 theoretical (I ) 10-5/10-4 mol/L) and experimental (CP-AFM) data for the period of the force oscillations, dslit. From the theoretical side, only the calculations with I ) 10-5 mol/L gave reliable results at volume fractions below 10%, and even at this very low ionic strength, it was impossible to determine dslit in the range φ e 5 %. Nevertheless, in the concentration range accessible by both theory and experiment, the data in Figure 7 indicate very good agreement. This concerns not only the concentration dependence of the force period but also the absolute values of these quantities. Conclusion This paper deals with the structure formation of suspensions of silica particles in the volume phase and under confinement in a slit-pore geometry. Focusing on length-sensitive quantities,

Klapp et al. such as the position of the main peak of the structure factor (volume) and the period of the force oscillations (slitpore), we find excellent agreement between the experimental results from SANS (volume) and CP-AFM measurements (slit-pore), on one hand, and the theoretical results from HNC integral equations (bulk) and MC simulations, on the other hand. From a theoretical point of view, the present results indicate that the DLVO potential used to describe the silica interaction yields a very good description of the systems’s properties not only under bulk conditions (which was known before; see, e.g., refs 25, 37, 38) but also in the presence of spatial confinement (which is not at all obvious38). On the other hand, from an experimental point of view, the present agreement shows the importance of theoretical calculations to determine the ingredients of the interaction potential (such as the ionic strength) of the real system. Concerning bulk properties, a particularly interesting finding is that the commonly accepted power law ξ ∝ φ-1/3 for the average particle distance ξ ) 2π/qmax in three spatial dimensions (and isotropic particle distributions) describes the present theoretical (HNC) results only for low ionic strength, that is, for relatively long-ranged potentials characterized by small values of κ. These particular ionic strengths coincide with those estimated for the real system, where ξ, according to SANS measurements, does indeed respect the above power law. Increasing the ionic strength in the model system yields a decrease in the exponent of φ, and at high ionic strengths, ξ is close to the particle diameter throughout the fluid density range, similar to what is predicted for a (completely screened) hardsphere fluid (κ f ∞). A second effect of an increasing ionic strength is the reduction of oscillations (and peak heights) of the structure factor due to screening effects and a related partial disordering. On the other hand, an increase in the particle concentration (at fixed ionic strength) yields more structure in S(q) (reflecting a more pronounced local ordering in real space) and a reduction of ξ. The force oscillation period dslit characterizing the structure formation in slitpore geometry displays on a qualitative level the same dependence on the particle concentration and the ionic strength as S(q). We note that theoretical (MC) results in the experimentally investigated concentration range could only be obtained at low ionic strength; increasing the latter yields a reduction and eventually a vanishing of the oscillations. In the concentration range accessible for both methods, theory (MC) and experiment (CP-AFM) yield again results in excellent agreement. We also note that the resulting lengths characterizing the system’s properties in bulk and confined geometry have similar magnitudes (for a given concentration). On the basis of the results of the present work, it seems worthwhile to investigate in more detail the relation and interchange between bulk properties, on one hand, and properties in spatial confinement, on the other hand. Furthermore, one may expect that the properties of the confining surfaces have a significant effect on the structure formation. Simulations and experiments in this direction are underway. Acknowledgment. We thank Madeleine Kittner for fruitful discussions regarding the GCMC simulations of the colloidal films. Financial support from the Deutsche Forschungsgemeinschaft (DFG) via Sonderforschungsbereich 448 “Mesoskopisch strukturierte Verbundsysteme” (projects B6 and B10) is gratefully acknowledged. S.H.L.K. thanks for funding via the EmmyNoether-Programm (DFG). D.Q. thanks the Max Planck Society

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