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Probing Colloid−Substratum Contact Stiffness by Acoustic Sensing in a Liquid Phase Adam L. J. Olsson,† Henny C. van der Mei,† Diethelm Johannsmann,*,‡ Henk J. Busscher,† and Prashant K. Sharma*,† †

Department of Biomedical Engineering, University Medical Center Groningen and University of Groningen, P.O. Box 196, 9700 AD Groningen, The Netherlands ‡ Institute of Physical Chemistry, Clausthal University of Technology, D-38678 Clausthal-Zellerfeld, Germany ABSTRACT: In a quartz crystal microbalance, particles adhering to a sensor crystal are perturbed around their equilibrium positions via thickness-shear vibrations at the crystal’s fundamental frequency and overtones. The amount of adsorbed molecular mass is measured as a shift in resonance frequency. In inertial loading, frequency shifts are negative and proportional to the adsorbed mass, in contrast with “elastic loading”, where particles adhere via small contact points. Elastic loading in air yields positive frequency shifts according to a coupled resonance model. We explore here the novel application of a coupled resonance model for colloidal particle adhesion in a liquid phase theoretically and demonstrate its applicability experimentally. Particles with different radii and in the absence and presence of ligand−receptor binding showed evidence of coupled resonance. By plotting the frequency shifts versus the quartz crystal microbalance with dissipation overtone number, frequencies of zero-crossing could be inferred, indicative of adhesive bond stiffness. As a novelty of the model, it points to a circular relation between bandwidth versus frequency shift, with radii indicative of bond stiffness. The model indicates that bond stiffness for bare silica particles adhering on a crystal surface is determined by attractive Lifshitz−van der Waals and ionicstrength-dependent, repulsive electrostatic forces. In the presence of ligand−receptor interactions, softer interfaces develop that yield stiffer bonds due to increased contact areas. In analogy with molecular vibrations, the radii of adhering particles strongly affect the resonance frequencies, while bond stiffness depends on environmental parameters to a larger degree than for molecular adsorption.

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local pressure at the most heavily loaded asperities is usually beyond the yield strength, resulting in plastic deformation,12,2 which is usually not true for colloidal particles. Particle deformation is a minor issue for silica particles, but of course is present for softer objects such as biological cells. Given the importance of colloid adhesion and the challenges associated with it, dedicated instrumentation is needed and has been developed. The atomic force microscope (more specifically, a modification called a colloidal force probe13,14) is widely employed to directly measure the forces of adhesion. The tangential stiffness of a contact can be determined with lateral force modulation.15 Note that the colloidal force probe in its routine way of operation applies a vertical force. In technical and in biological environments, particle removal usually is achieved by a tangential force as for instance exerted by liquid flow, which induces rolling or sliding. Rolling friction has been investigated with AFM,16 but these studies are tedious and difficult to apply on a routine basis. Techniques to study debonding under tangential loading include centrifugation,17 rotating disks,18 and flow displacement devices.19 Flow

ontrol of colloidal particle adhesion, including adhesion of biocolloids such as bacteria, yeasts, mammalian cells, and diatoms to flat substrata, is important in different areas of science and engineering. Immobilized colloids are used for high-density data storage,1 in sensing,2,3 and in thermoelectric devices,4 while bacterial adhesion to surfaces can be detrimental in many environmental, industrial, and biomedical applications.5,6 The energy of adhesion affects both the efficiency of an intended colloidal particle immobilization and the force needed for detachment, which is of particular importance in the context of cleaning.7,8 Contact-mechanical behavior of colloidal particle adhesion characteristically differs from contact mechanics on the molecular level9 and from contact mechanics on the macroscopic level.10 On one hand, colloidal particles rarely form “single asperity contacts” as are achieved in suitably designed atomic force microscopy (AFM) experiments using nanoscopic tips for contact.11 An important input parameter in all models is roughness. On the other hand, friction and adhesion of colloids is not a classical multiscale problem, as in macroscopic environments.12 The contact diameter of colloidal particles is typically between a few nanometers and about 100 nm. Corresponding to this limited range of spatial scales, there is a limited range of local pressures. In macroscopic contacts, the © 2012 American Chemical Society

Received: February 6, 2012 Accepted: April 11, 2012 Published: April 11, 2012 4504

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air because in liquids the torsional resonances are overdamped. In related work, Jersch and co-workers have integrated quartz crystal tune-fork resonators into the atomic force microscope.31 More generally speaking, the measurement of forces and stiffnesses based on the detuning of a resonance is quite common in dynamic AFM measurements.32 However, this technique is not usually applied to colloid−substratum bonds. The aim of this study is to extend the use of a coupled resonance model to the analysis of the QCM’s response to colloidal particle adhesion in a liquid phase. We interpret the experimental results using two simple parameters: the frequency of zero crossing and the radius of the circle seen in polar diagrams of bandwidth versus frequency shift. Neither of the two parameters have been used previously to interpret QCM data. From these parameters, we infer changes in the bond stiffness at different ionic strengths and for two particle radii. The influence of ligand−receptor binding is evaluated by adsorbing biotin and streptavidin to the sensor surface and to the particle surface, respectively.

displacement devices yield detailed information about the kinetics of (bio)colloidal particle adhesion, but provide only an estimate of the adhesion force.20 In our study we have applied a quartz crystal microbalance (QCM) to investigate colloidal particle adhesion. The QCM is widely known to the scientific community as a film thickness monitor. The QCM consists of an acoustic sensor surface and an AT-cut quartz crystal, which undergoes a thickness-shear vibration at its fundamental frequency (often 5 MHz) and at its overtones. The amount of adsorbed mass is inferred from the shift in the resonance frequency, Δf. In the simplest case, first described by Sauerbrey, the frequency shift is negative and proportional to the mass per unit area,21 and we term this “inertial loading”. Inertial loading is the opposite of “elastic loading”, which occurs when particles with a diameter above 1 μm adhere to the sensor surface via small points of contact. Elastic loading sometimes yields positive frequency shifts (Δf > 0) as reported for adhesion of colloidal particles,22,23 bacteria,24 diatoms,25 and protozoan oocysts.26 Elastic loading gives insight into the properties of the link between the adsorbed object and the substratum. Dybwad showed in 1985 that positive frequency shifts of adhering particles can be explained by coupled resonances, in which the adhering particles form resonators themselves.27 A particle on a plate is represented as a point mass m linked to the substratum by a spring with a stiffness k with a “particle resonance frequency” ωP. If the particle resonance frequency is much below the frequency of the quartz crystal, the particles are essentially clamped in space by inertia. The bonds exert a restoring force onto the vibrating crystal, thereby increasing its effective stiffness and, as a consequence, its resonance frequency. If the resonance frequencies of the adhering particles and the quartz crystal are similar, one observes a transition from negative frequency shifts at lower overtones (Δf being dominated by inertia) toward positive frequency shifts at higher overtones (Δf being dominated by the link). We term this crossover frequency the “frequency of zero crossing”, f ZC. At f = f ZC, the particles absorb a large amount of the oscillation energy, leading to a maximum in the dissipation signal.28,29 It is unlikely, however, that in real cases all particles adhere with the same bond stiffness.13 Importantly, the particles contacting the resonator are only slightly perturbed around their equilibrium positions. The contacts are not ruptured, which makes the technique nondestructive. The stiffness as derived from QCM experiments must not naively be identified with the rupture forces as determined with the atomic force microscope or flow displacement devices. The contact stiffness is the ratio of force and displacement in the limit of small displacement, where the contact behaves as a Hookean spring, whereas debonding occurs at much larger forces. Rupture usually involves a cascade of events (most of them irreversible), whereas small-scale deformation can usually be described in the frame of equilibrium thermodynamics. In the literature there are a few studies in which the atomic force microscope was used as a resonator exerting tangential stress onto the sample. Reinstadtler et al. employed the atomic force microscope cantilever as a torsional resonator,30 where the tip exerts a tangential force onto the substratum. The interaction with the substratum changes the frequency and the bandwidth of this resonance in complete analogy with QCMbased measurements. This technique has so far only been applied to molecular-scale contact problems and works best in

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MODELING In the coupled resonance model (see also refs 28 and 33), an adhering particle is modeled by an equivalent mechanical circuit

Figure 1. (a) Mechanical equivalent of a coupled resonator containing a point mass (m), an elastic spring (k), storing oscillation energy, and a dashpot (ξ) causing dissipation or loss of the oscillation energy. (b) Shifts of resonance frequency, Δf (full line), and bandwidth, ΔΓ (dashed line), according to eq 5. At ω = ωP = 2πf ZC the frequency shift turns from negative to positive. We denote this frequency the “frequency of zero crossing”, f ZC. γ is the width of the line. The circles and triangles were inserted to remind the reader that only certain discrete values of ω (corresponding to the harmonics of the crystal) can be interrogated with a QCM. (c) Change in bandwidth (ΔΓ) versus frequency shift (Δf) (“polar diagram”). One finds a circle with radius RPD. In this example, all particles are assumed to have the same contact stiffness. We assume a homogeneous absorption line with γ = 0.1ωP.

containing a point mass, a spring, and a dashpot (Figure 1a), yielding a “particle resonance frequency” ωP given as ωP* = 2πf P* =

k* m

(1)

in which m is the equivalent mass and k* is the stiffness of the link. The subscript “P” refers to the particle, and asterisks denote complex variables. The spring constant, k*, is complex because the adhering particle dissipates energy when set in movement by the resonator. The link is represented by a spring and a dashpot acting in parallel. k* can be written as k* = k + iωξ

(2)

where ξ is the drag coefficient of the dashpot, which quantifies the dissipation of energy. In liquids the dissipation is mostly due to viscous drag from the bulk. The dashpot may encompass 4505

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Equation 4 simplifies in its limiting cases of low (ω ≪ ωP) and high (ω ≫ ωP) frequencies. When the bond between the adhering particle and the sensor surface is stiff (k ≫ mω2), the particle essentially moves with the resonator, thereby increasing its mass. This limiting case describes inertial loading in the Sauerbrey sense.20 In the opposite limit of weak elastic bonding (k ≪ mω2), the frequency shift is positive and equal to ( fosc f FNPk)/(ωπZq), describing elastic loading. In the intermediate situation the stiffness is comparable to mω.2 Understanding this situation becomes easier when eq 4 is rewritten as

viscoelasticity at the link, but it is difficult to disentangle dissipation at the link from dissipation in the bulk. For simplicity, we assume both k and ξ to be independent of frequency within the limited range of frequencies covered by the QCM (5−65 MHz). Note that the frictional force exerted by the contact does depend on the frequency because of the prefactor ω in eq 2. Over larger frequency ranges, k usually does depend on the frequency. The frequency dependence (also termed “viscoelastic dispersion”) is particularly important when polymers are involved. Note that the coupled resonance model as such does not make a statement on howexactly an adhering particle moves. In particular, it leaves open to what extent bending and tangential displacement contribute to the mode of vibration. From eq 1 it follows that the resonance frequency of the coupled resonator, ωP, is a complex variable, and we therefore briefly expand on the use of complex resonance frequencies. Importantly, we also describe the behavior of the QCM itself in terms of complex resonance frequencies and shifts, f * = f + iΓ and Δf * = Δf + iΔΓ. The imaginary part of the resonance frequency, Γ, is a measure of energy dissipated per cycle. If the resonance is sufficiently sharp as in the QCM, Γ is equal to the half-bandwidth at half-height (“bandwidth”) of the resonance curve. Γ is related to the more routinely used dissipation parameter, D, by Γ = Df /2

fF Δf * Δf + iΔΓ mω(ωP 2 + iωγ ) = = fosc 2 NP NP NPπZq ω − ωP 2 − iωγ

where the parameter γ with the dimension of frequency is defined as γ = ξ/m

(6)

Equation 5 describes a Lorentzian, which is conceptually important. Figure 1 graphically displays Δf and ΔΓ versus ω. The dependence of Δf and ΔΓ on frequency amounts to an acoustic absorption line. For γ ≪ ωP, γ is the full bandwidth at half-maximum of the absorption line. As in optical spectroscopy, the experimentally determined line width may be different from γ in the case where the adhering particles vary in their surface properties, as will be shown for the present collection of particles used. The absorption line then is smeared out, and what is determined experimentally is the “heterogeneous linewidth”, γhet. To quantitatively reproduce the data, we use

(3)

In calculating the shifts in frequency and bandwidth induced by the presence of adhering particles, we follow the small load approximation, which states that the shift of the complex resonance frequency of the QCM is proportional to the ratio of the area-averaged tangential stress and the tangential velocity at the crystal surface.34 Note that both stress and velocity are to be understood as complex amplitudes of the corresponding oscillatory quantities. The stress/velocity ratio therefore is a complex number as well (see also ref 33). Calculating the tangential force exerted by the coupled resonator onto the crystal sensor surface and inserting this force into the small load approximation gives28 f Δf * Δf + iΔΓ mω(k + iωξ) = = F fosc πZq NP NP mω 2 − (k + iωξ)

(5)

f ω Δf * = F fosc m NP NPπZq

∫0

ωP 2 + iωγ

∞ 2

ω − ωP 2 − iωγ

g (ωP)dωP (7)

where g(ωP) is the distribution function of values for ωp. It turns out that quantitative fitting of the data is possible, but that simple distribution functions such as Gaussian or Poissonian distributions do not always suffice. Given that the shape of the distribution function is a priori unknown, the parameter space for fitting becomes impracticably large. To compare results of different experiments without quantitative fitting, we have identified two qualitative parameters which can be readily extracted from an experiment without modeling. These are the frequency of zero crossing, f ZC, and the radius of the circle in a polar diagram of bandwidth versus frequency shift, RPD. The frequency of zero crossing is the frequency at which Δf changes sign from being negative at f < f ZC to being positive at higher harmonics. Experimentally, f ZC will not usually coincide with one of the harmonics and needs to be determined by interpolation. It might also be determined by extrapolation in the case where f ZC falls slightly outside the window of observable frequencies in the QCM, usually ranging from 5 to 65 MHz. For a homogeneous absorption line with γ ≪ ωP, the meaning of f ZC is readily understood from Figure 1b: f ZC is close to ωP/(2π) = 1/(2π)(k/m)1/2. This relation can be extended to the heterogeneously broadened case by

(4)

in which f F is the fundamental frequency (5 MHz), Zq is the acoustic impedance of AT-cut quartz (8.8 × 106 kg m−2 s−1), NP is the number of adhering particles per unit area, and fosc is the oscillator strength, a new numerical factor with 0 < fosc < 1. It accounts for the fact that some of the interfacial stress may act in the normal direction, while the QCM only responds to tangential stress. The name “oscillator strength” for the parameter fosc was in fact taken from optical spectroscopy where resonances couple to the exciting field (electromagnetic dipole radiation) to a variable degree. If there is no coupling at all, the respective transition is called “forbidden”. In the case of the QCM, the exciting field is the tangential force exerted by the substratum. Only those modes of particle motion are allowed which themselves exert a tangential force onto the crystal. Even for those allowed modes, the coupling strength is variable. It is unity for a particle rigidly attached to the substratum, but one has fosc < 1 for more complicated displacement patterns. Equation 4 applies to all frequencies, ω, but with any given sensor crystal, one can only interrogate frequencies equal to one of the overtones of the crystal.

fZC =

1 1 ωP,eff = 2π 2π

keff m

(8)

in which the subscript “eff’” stands for “effective”. The relation between keff and the average spring constant k depends on the distribution g(ωP). Since g(ωP) is not known, quantitative 4506

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Figure 2. Δf/NP and ΔΓ/NP as a function of the overtone number n (a, c, e) and their relation as expressed in a polar diagram (b, d, f) for 0.5 μm radius silica particles adhering to differently coated sensor crystals and at various ionic strengths. The corresponding frequencies are obtained by multiplying the fundamental frequency (5 MHz) by the overtone number n. Data are averaged values over two separate measurements. (a, b) Silica particles on a silica surface. (c, d) Silica particles on a biotinylated surface. (e, f) Streptavidin-coated silica particles on a biotinylated surface.

analysis is difficult. If the effective mass, m, remains constant between experiments, an increase in f ZC translates to an increase in bond stiffness, and vice versa. Interestingly, there is a second qualitative parameter available for analysis besides f ZC, which is the radius of the circles found in polar diagrams of ΔΓ = Im(Δf *) versus Δf = Re(Δf *) (Figure 1c). Displaying QCM results in this way has been proposed before,34 but the occurrence of a circle and the analysis of the circle’s radius in terms of the contact stiffness are novel. Even in cases where the apex of the circle (where ΔΓ is at its maximum and Δf changes sign) is not covered by the window of observable frequencies in the QCM, its radius, RPD, may sometimes still be determined. The meaning of RPD is clear for homogeneous acoustic absorption lines with γ ≪ ωP. From eq 5

2RPD ≈

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NPπZq

fosc

k γ

keff γhet

(10)

MATERIALS AND METHODS

Particle Preparation. Bare and streptavidin-coated silica particles with radii of 0.5 and 2.5 μm (Bangs Laboratories, Inc., Fisher, IN) were washed twice by centrifugation in 10 mL of ultrapure water and diluted to a final concentration of 2 × 108 and 2 × 106 particles/mL, respectively. Preparation of QCM Sensor Surfaces. Silica-coated sensor crystals (Q-sense, Gothenburg, Sweden) were cleaned by immersion into, and sonication in, (w/v) sodium dodecyl sulfate (SDS) for 15 min, sonication in ultrapure water at 30 W for 2 min, and a final, 15 min UV/ozone treatment. This yielded a hydrophilic silica surface with a 0° water contact angle. Gold-plated sensor crystals (Q-sense) were biotinylated by overnight immersion in a 0.1 mM solution of biotinylated polyethylene glycol (PEG) alkanethiol (Nanoscience Instruments, Phoenix, AZ) dissolved in ethanol (100%). The PEG

ΔΓ(ω = ωP) NP fF mωP(ωP 2 + iωPγ ) ≈ fosc NPπZq iωPγ fF

NPπZq

fosc

We have gone through numerical simulations using Gaussian distributions for g(ωP) in eq 7 and found eq 10 to be confirmed. The main use of eq 10 is in the comparison of different data sets. If one assumes γhet and fosc to remain constant between experiments, an increase in RPD implies an increase in bond stiffness and vice versa.

2RPD,homo ≈

≈

fF

(9)

Equation 9 can be extended to the heterogeneous case by replacing γ with γhet and replacing k by keff: 4507

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Figure 3. Same as Figure 2, but for particles with a radius of 2.5 μm.

absence of particles, for each ionic strength. These frequency and bandwidth values were then deducted from the frequency and bandwidth values obtained in the presence of particles. The shifts in frequency (Δf) and bandwidth (ΔΓ) are all normalized to the number of particles per unit area (Δf /NP and ΔΓ/NP) to allow for a comparison between experiments with variable NP.

chain between the alkanethiol and the biotin moiety was six repeat units long. The chain acts as a flexible spacer. Prior to deposition, the crystals were cleaned by immersion in a 3:1:1 mixture of ultrapure water, NH3, and H2O2 (Merck, Darmstadt, Germany) at 75 °C for 10 min, followed by 10 min of UV/ ozone treatment and a final 70% ethanol rinse. Water contact angles on the cleaned gold surfaces were 86° ± 2°. QCM Measurements. Adhesion of silica particles to the QCM crystal surfaces was followed in a window-equipped QCM device (E1, Q-sense) using a metallurgical microscope. Silica particles suspended in ultrapure water were introduced into the QCM chamber and allowed to settle on the crystal surface, but due to electrostatic repulsion particles could not adhere strongly enough to become immobilized. Their Brownian motion remained clearly visible under the microscope. Strong adhesion of the particles was enforced by changing the ionic strength of the buffer to 150 mM KCl. The frequency and bandwidth were monitored until they became stable, which usually occurred within 3 min. The number of particles per unit area, NP (particles/cm2), was determined from images captured from the center of the QCM crystal with a charge-coupled device (CCD) camera. The number of particles was kept low enough to prevent clustering; most adhering particles were well separated from each other. In the analysis we therefore assume that interparticle interactions can be neglected. After the particle behavior was monitored at 150 mM KCl for 3 min, the ionic strength was decreased stepwise to 5 mM KCl while QCM measurements were taken as described above between each step. Prior to each measurement, frequency and bandwidth shifts were first measured in the

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RESULTS Figures 2 and 3 summarize the QCM with dissipation (QCMD) responses to the adhesion of particles of different radii and with and without a streptavidin coating to a sensor crystal with and without a biotinylated coating and at different ionic strengths. On the left-hand side, they are expressed as a function of the overtone number, n, while the right-hand side shows the corresponding polar diagrams. For silica particles with a radius of 0.5 μm adhering to a silica surface, Δf is negative at low n, but changes sign with increasing overtone number. Both the magnitude of Δf and the overtone order at which Δf changes sign gradually increase as the ionic strength increases (Figure 2a). The polar diagrams (Figure 2b) display the characteristic circles. The circle radii gradually increase with increasing ionic strength. Importantly, the dependence of ΔΓ on overtone order does not reveal sharp lines as shown in Figure 1b, but the lines are broad, and in many cases we only see an increase of ΔΓ with n and a plateau. Parts c and d of Figure 2 represent the QCM-D responses for 0.5 μm silica particles on biotinylated surfaces showing an overall appearance similar to that observed for adhesion to the silica surface. However, adhesion to biotinylated sensor crystals 4508

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causes the zero-crossing frequencies to move to higher overtone numbers as compared with particle adhesion to bare silica (compare parts a and c of Figure 2). Since the window of observable frequencies only extends to 65 MHz, no exact zerocrossing frequency can be extracted, resulting in polar diagrams with only partial circles. Importantly, their radii are much larger than for adhesion to silica (compare parts b and d of Figure 2). The polar diagrams suggest that there is an f ZC just above the 13th overtone at the highest ionic strength. f ZC shifts to higher frequencies as the ionic strength decreases, and concurrently the diameters of the circles increase. For streptavidin-coated particles adhering to biotinylated crystals (Figure 2e,f), the zero-crossing frequencies are located toward higher overtones, with f ZC potentially far above the 13th overtone. The diameters of the circles in the polar diagrams are even larger than on a bare or biotinylated sensor with bare particles. The streptavidin−biotin interaction yields only a quarter of a circle in the polar diagram, and its diameter remains constant regardless of the ionic strength. In the case of particles with a 2.5 μm radius (Figure 3), their larger mass moves the zero-crossing frequency toward lower frequencies in line with eq 8. In most cases they are below 5 MHz and therefore outside the window of observable frequencies. For silica particles adhering to the silica surface (Figure 3a,b), Δf is positive at all overtones and slightly increases with the overtone number. In contrast to the smaller particles, ΔΓ attains its highest value at the fundamental frequency and decreases with the overtone number, indicating that f ZC is below the fundamental frequency. The polar diagram (Figure 3b) extends over too small of a range to be meaningful. Biotinylation of the sensor crystal (Figure 3c,d) moves the coupled resonance curve toward higher overtones. The data suggests a zero-crossing frequency slightly below the lowest harmonic at 5 MHz. The polar diagrams display almost half of their characteristic circles. Note that, by comparison with Figure 2d (0.5 μm particles), the larger particles yield a larger RPD. Also, RPD decreases with increasing ionic strength, while it increased with ionic strength for the smaller particles (Figure 2d). For streptavidin-coated particles adhering to biotinylated surfaces (Figure 3e,f), f ZC is located between the fundamental and the third overtone. The polar diagrams almost yield full circles (Figure 3f), with diameters decreasing with increasing ionic strength, similar to what is observed for silica particles on a biotinylated surface (Figure 3d).

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Figure 4. Quantitative modeling of the QCM-D response to the adhesion of silica particles with a radius of 0.5 μm to a bare silica surface at 150 mM. (a) Experimental data and fit according to eq 11 indicated by the solid lines with fixed parameters γ = 2π 3 MHz and ωP,c = 2π 1 MHz, yielding σ = 2π 37 MHz and foscm = 0.19 pg. (b) Distribution of particle resonance frequencies g(ωP) resulting from the fit of the data in Figure 2a.

f ω Δf * 1 = F fosc m NP NPπZq 2π σ

∫0

∞

⎛ (ω − ω )2 ⎞ ̅P P,c ⎟⎟dω̅ P exp⎜⎜ − 2 − iωγ ⎝ 2σ ⎠

(ω̅ P 2 + iωγ ) ω 2 − ω̅ P 2

(11)

where ωP,c is the center of the Gaussian and σ is its standard deviation. We fixed the parameters γ and ωP,c at γ = 2π 3 MHz and ωP,c = 2π 1 MHz. Using these fixed parameters, σ and foscm were determined by χ2 minimization, which resulted in σ = 2π 37 MHz and foscm = 0.19 pg. Note that the parameter m is not an output of the modeling, and only the product foscm is accessible. Since m is not known, one cannot convert values of ωP to values of the contact stiffness by means of eq 1. The appearance of the line in Figure 4a was rather insensitive to the values of ωP,c and γ as long as these were much smaller than the width of the Gaussian distribution, σ. The errors bars in γ and ωP,c certainly are larger than the values themselves, but no good match was achieved unless both were smaller than σ. We modeled numerous other data sets and found the latter statement to hold in general. The experimental data all represent absorption lines which are heterogeneously broadened. In Figure 5 we further corroborate heterogeneous line broadening by an argument which does not involve modeling. From Figures 2 and 3, we can conclude that the line width of the coupled resonance is larger than f ZC. If this line broadening goes back to dissipation only (γ > ωP), the oscillation would be “overdamped”. Importantly, large damping tilts the absorption line. For γ > ωP, the asymmetry becomes so strong that Δf does not even become positive at ω ≫ ωP, but instead Δf remains negative for all frequencies (Figure 5a) and only a hemicircle is found in the polar diagram (Figure 5c), which contradicts the experimental findings. In our experimental data, we do see a crossover from negative to positive Δf (Figures 2 and 3), and such a behavior can only be reproduced with a superposition of lines with low γ but variable ωP. Most likely, the large spread in values for ωP goes back to a spread in values of the contact stiffness, k, which is illustrated in Figure 5g in terms of a mechanical equivalent circuit. In purely mathematical terms, the

DISCUSSION

We have utilized a QCM-D to investigate the adhesive bond stiffness between the sensor surface and adhering particles with different radii and in the absence and presence of ligand− receptor binding in a liquid phase at different ionic strengths. All particle−substratum combinations showed evidence of a coupled resonance in a liquid phase. Before addressing the various trends in the dependence of f ZC and RPD on the experimental parameters, we elaborate on the quantitative application of eq 7 to the data. In Figure 4 we show for one selected example (bare silica particles on a bare silica surface at 150 mM, black dots in Figure 2a) that quantitative modeling is possible. We started from eq 7 and inserted a Gaussian for the distribution function g(ωP): 4509

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Figure 6. Frequencies of zero crossing (a, c) and the corresponding radii of the circles in polar diagrams (b, d) as a function of the ionic strength. Note that Figures 2 and 3 do not in all cases allow reliable values to be read for f ZC and RPD. Key: (a, b) 0.5 μm particles, (c, d) 2.5 μm particles, (squares) silica particles/silica surface, (circles) silica particles/biotinylated surface, (triangles) streptavidin-coated particles/ biotinylated surface. Figure 5. Comparison of homogeneous and heterogeneous line broadening. (a−c) Homogeneous case as given by eq 5 with γ = 0.1ωp (black) and γ = 1.1ωp (red). (d−f) Heterogeneous case as described by eq 11 with 2σ = 0.1ωp (black) and 2σ = 1.1ωp (red). (g) Mechanical equivalent circuit of particle adhesion with different adhesive bond stiffnesses, giving rise to heterogeneous line broadening.

Consistently, the radii of circles in the polar diagrams increase with the ionic strength (Figure 6b), which implies an increasing stiffness as long as the heterogeneous line width remains constant. This result can be explained on the basis of the Derjaguin−Landau−Verwey−Overbeek (DLVO) theory. The silica particles are held on the silica surface by a balance between attractive Lifshitz−van der Waals forces and electrostatic repulsion, while salt may screen electrostatic repulsion. Consequently, the attractive Lifshitz−van der Waals forces pull the particles tighter toward the surface, thereby increasing the contact area and the bond stiffness. For the 2.5 μm silica particles on the silica surface, f ZC has moved to lower values (Figure 3a), which is in line with the higher mass of the 2.5 μm particles as compared with the 0.5 μm silica particles. Unfortunately, the values of f ZC are so far below 5 MHz that extrapolation is impossible. The polar diagram in Figure 3b can therefore not be interpreted. The bond stiffness increases when the crystal surface becomes softer by biotinylation (compare parts a and c of Figure 2 as well as parts a and c of Figure 3). It feels counterintuitive at first that a “soft” biotinylated layer forms a stiffer contact than observed for particle adhesion to a “hard” bare silica surface. The explanation lies in the fact that deformation of soft surfaces creates a larger contact area upon adhesion of the silica particles as compared to that of hard surfaces. A larger contact eventually creates a stiffer bond, even if the contacting material is less stiff than silica. Another interesting aspect is the comparison of the circles in Figures 2d and 3d. This is the same type of bond (silica particles on a biotinylated surface), but with two different particle radii. Note that mass does not enter eq 10, and therefore, the increase in RPD when the 0.5 μm particles are replaced with the 2.5 μm particles reflects the bond stiffness alone (assuming constant values of fosc and γhet; see eq 10). This increase in contact stiffness is in line with an increase in contact area. The bond stiffness further increases upon coating of the particles with streptavidin to create a complete ligand−receptor bond with the biotin group (Figures 2c,e and 3c,e). For the 0.5 μm particles, f ZC is above 65 MHz, and no value can be determined by extrapolation. RPD is large and approximately constant, although the curvatures of the quarter circles can only

spread can equally well originate from variability in m, but since the size is the same for all particles, within 15% according to the manufacturer, the effective mass is narrowly distributed. In general, a certain spread in the adhesion strength between colloidal particles and a surface has been reported before (see, for instance, ref 13 for a review). However, AFM-based instrumentation typically yields rupture forces under vertical pulling, rather than a tangential contact stiffness. We find it interesting that the distribution in Figure 4 is even wider than what is typically reported for detachment forces. A “wide” distribution in the latter context would denote a standard deviation about half as large as the mean (see, for example, Figure 26 in ref 13). In Figure 4, the width of the distribution g(ωP) applied is about as large as the mean. The reason underlying the large variability in stiffness is nanoscale roughness. Given that contact stiffness depends on the contact size and that, furthermore, roughness inevitably entails a large number of different geometrical configurations, heterogeneous broadening is not a surprise. Because of roughness, the familiar models of contact mechanics (such as the Derjaguin-MullerToporov (DMT) model35,36) cannot be applied. In terms of theoretical modeling, inclusion of roughness into contact mechanics is just emerging.37 We now turn to the more qualitative trends apparent in Figures 2 and 3. Following eqs 8 and 10, the frequencies of zero crossing and the radii of the circles in the polar diagrams are indicators of bond stiffness, and it is shown that an influence of ionic strength on bond stiffness is always there. It is summarized in Figure 6 in terms of the zero-crossing frequencies ( f ZC, left) and the radii of the circles in the polar diagrams (RPD, right). For 0.5 μm silica particles adhering to a bare silica surface, the bond stiffness increases with increasing ionic strength, as evidenced by the shift in f ZC toward higher values (Figure 6a). 4510

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be read with poor accuracy. For the 2.5 μm particles, their larger mass has pulled the zero-crossing frequency back into the accessible range. Interestingly, the bond stiffness decreases with increasing ionic strength in the case of ligand−receptor binding for the 2.5 μm silica, as revealed by the trends in both f ZC and RPD (triangles in Figure 6c,d). This observation cannot be explained with the DLVO theory, because at elevated ionic strength, the particles are pulled toward the surface, which should result in increased stiffness. A possible explanation arises from the fact that the solubility of polyethylene in water decreases with increasing ionic strength.38 At high ionic strength, the flexible spacer linking the biotin groups with the alkanethiol will collapse. The biotinylated surface becomes stiffer, but the contact nevertheless softens because the harder material leads to a smaller contact size. The same argument was invoked above to explain the bond stiffening caused by addition of a soft biotin layer to the resonator surface.

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SUMMARY AND CONCLUSIONS Depending on the particle size, coatings of particles and crystal surfaces by ligand−receptor molecules, ionic strength, and overtone number, frequency shifts, Δf, induced upon particle adhesion from suspension to sensor surfaces in the QCM were positive or negative. The results can be explained using a coupled resonance model, in which adhering particles are considered as resonators on their own. In the proposed model, two novel parameters were identified reflecting the stiffness of the bond between adhering particles and the sensor surface. These are the frequency of zero crossing, f ZC, in a graph of the frequency shift versus the overtone number and the radius of the circle in a polar diagram of the line width versus frequency shift, RPD. The bond stiffness between silica particles adhering to a silica crystal increased with the ionic strength. Salt screens the electrostatic repulsion, which causes the Lifshitz−van der Waals attraction to pull the particles more strongly to the surface. Interestingly, this trend was reversed when adhesion occurred across biotin−streptavidin bonds. We attribute this to the soft spacer layer between the crystal and the biotinylated layer collapsing and hardening with increasing ionic strength. A harder surface implies a smaller contact area, which weakens the bond. In analogy with molecular vibrations, the radii of adhering particles strongly affect the resonance frequencies, while bond stiffness depends on environmental parameters to a larger degree than for molecular adsorption. Quantitative modeling of the coupled resonances using the model forwarded is possible as well, but our work indicates for the first time ever that a certain distribution of bond stiffnesses must be taken into account to adequately explain the QCM-D response to particle adhesion on the basis of a coupled resonance model.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] (P.K.S.); johannsmann@pc. tu-clausthal.de (D.J.). Notes

The authors declare no competing financial interest.

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REFERENCES

(1) Hayward, R. C.; Saville, D. A.; Aksay, I. A. Nature 2000, 404 (6773), 56−59. 4511

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(38) Willauer, H. D.; Huddleston, J. G.; Rogers, R. D. Ind. Eng. Chem. Res. 2002, 41 (11), 2591−2601.

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