Anisotropic Particle Deposition Kinetics from Quartz Crystal Microbalance Measurements: Beyond the Sphere Paradigm

Deposition kinetics of polymer particles characterized by a prolate spheroid shape on gold sensors modified by the adsorption of poly(allylamine) was investigated using a quartz crystal microbalance and atomic force microscopy. Reference measurements were also performed for polymer particles of a spherical shape and the same diameter as the spheroid shorter axis. Primarily, the frequency and dissipation shifts for various overtones were measured as a function of time. These kinetic data were transformed into the dependence of the complex impedance, scaled up by the inertia impedance, upon the particle size to the hydrodynamic boundary layer ratio. The results obtained for low particle coverage were interpolated, which enabled the derivation of Sauerbrey-like equations, yielding the real particle coverage using the experimental frequency or dissipation (bandwidth) shifts. Experiments carried out for a long deposition time confirmed that, for spheroids, the imaginary and real impedance components were equal to each other for all overtones and for a large range of particle coverage. This result was explained in terms of a hydrodynamic, lubrication-like contact of particles with the sensor, enabling their sliding motion. In contrast, the experimental data obtained for spheres, where the impedance ratio was a complicated function of overtones and particle coverage, showed that the contact was rather stiff, preventing their motion over the sensor. It was concluded that results obtained in this work can be exploited as useful reference systems for a quantitative interpretation of bioparticle, especially bacteria, deposition kinetics on macroion-modified surfaces.


■ INTRODUCTION
Experimental studies of particle deposition kinetics furnish essential information about their interactions with interfaces, especially the adhesion strength, which is a crucial issue for colloid science, biophysics, medicine, soil chemistry, etc.This knowledge can be used to control and optimize various practical processes, such as filtration, flotation, protective coating formation, paper making, catalysis, and bioengineering.In addition, results acquired for colloid systems under welldefined transport conditions can serve as useful reference results for the interpretation of macromolecule and bioparticle adsorption phenomena, especially virus and bacteria attachment to abiotic surfaces.
It should be underlined that the shape of colloid silica particles, 1−4 carbon nanotubes, 5,6 polymer microparticles, 7−9 or macroion molecules 10−13 resembles cylinders or prolate spheroids.Similarly, the anisotropic molecular shape is common among biocolloids, such as DNA fragments, 14−16 proteins, 17,18 viruses, 19−22 and bacteria, 23−26 comprising the common bacterial strains, such as Escherichia coli, Legionella pneumophila, Sphingopyxis alaskensis, or Hylemonella gracilis. 27,28cause of its significance, nano-and microparticle adsorption kinetics was extensively studied by numerous experimental techniques, such as atomic force microscopy (AFM), 29−31 scanning electron microscopy (SEM), 32−35 ellipsometry, 36−38 reflectometry, 39−41 surface plasmon resonance (SPR), 42,43 and electrokinetic methods. 44,45n comparison to these techniques, the widely used quartz crystal microbalance (QCM) method exhibits pronounced advantages, enabling sensitive, in situ measurements of deposition/desorption kinetics under flow conditions for particle size ranging from nanometers 46−53 to a few micrometers. 54,55However, these investigations were mainly focused on spherical particles, with only a few studies performed for anisotropic particles.In refs 56−59, the deposition of liposomes was investigated and theoretically interpreted in terms of lattice Boltzmann numerical modeling.It was predicted that the absolute value of the negative frequency shift decreased with the coverage and axis ratio of the liposomes modeled as oblate spheroids with the size of 78 nm.
Scarce experiments were reported for colloid particles of an elongated shape.In ref 60, the deposition of rod-shaped polystyrene particles with the aspect ratio of 2:1 and 4:1 at silica sensors modified by adsorption of poly-L-lysine (PLL) was studied by a QCM.The frequency shift acquired for the third overtone was used to determine the initial deposition rate of the particles as a function of the ionic strength (varied between 0.05 and 50 mM).It was confirmed that the deposition rate was considerably lower for the spheroidal particles compared to the spherical particles, independent of the ionic strength.
Similar measurements were performed in ref 61, where the deposition kinetics of native and stretched 200 nm in diameter polystyrene particles at alginate and harpeth humic-acid-coated silica sensors was investigated.A significant decrease in the absolute value of the frequency shifts was observed for the more rough sensor coated by harpeth humic acid, especially for the stretched particles characterized by the aspect ratio of 1.6.However, these interesting experimental results were not interpreted by any theoretical model.
In a recent work, 62 the adsorption kinetics of positively charged polymer particles of a spheroidal shape at bare gold and silica sensors was investigated using the QCM method.It was confirmed that, for such particles, the hydrodynamic slip effect played a significant role, decreasing manifold the QCM signals compared to firmly attached particles with respect to both the frequency and dissipation shifts.
−28 In refs 69 and 70, adsorption kinetics of fibrinogen, whose molecule has an elongated shape with the length to width aspect ratio exceeding 7, was studied by QCM.The experimental results were interpreted in terms of a coarsegrained Monte-Carlo-type theoretical modeling, which allowed the determination of the ratio of the real coverage to that calculated from the Sauerbrey equation.
Interesting studies based on the QCM were also reported for bacteria strains, inter alia E. coli, Salmonella, 25−27 and soil bacteria. 71,72Whereas in most of the works a negative frequency shifts were measured, in ref 26, dealing with Lactococcus lactis, a positive, albeit rather small, equal to 20 Hz frequency shift (third overtone, polystyrene-coated sensor) was reported.However, no attempt was undertaken to quantify the obtained results and to elucidate physical mechanisms of bacteria attachment.
One can argue that the scientific value of these tedious experiments could be enhanced if appropriate reference results obtained for well-characterized particles and sensor surfaces were available.Therefore, considering the lack of such experimental data, the main goal of this work was to determine the mechanism of prolate spheroidal particle deposition at solid/electrolyte interfaces with the main focus on assessing the feasibility of the QCM measurements to furnish quantitative kinetic results.Experiments were performed for polymer particles bearing a negative charge analogous to most bioparticles, especially bacteria. 27,28To facilitate a reliable interpretation of the results, the gold sensors were thoroughly characterized in terms of topographical properties, comprising the average surface height, the root mean square (rms), and the skewness.Analogous deposition kinetic measurements were carried out for spherical particles of the size matching the shorter spheroid axis.The real, often referred to as the dry, particle coverage was determined by atomic force microscopy (AFM) and in a continuous manner, applying a hybrid random sequential adsorption (RSA) modeling.
This enabled us to quantitatively determine the complex impedance of the sensor as a function of the ratio of the particle size to the hydrodynamic boundary layer thickness for a broad range of coverage.In the case of spheroidal particles, the real and complex impedance components were equal to each other and linearly decreased with the square root of the overtone number.This was interpreted as the effect of a hydrodynamic, lubrication-like contact with the sensor, enabling a sliding motion of the particles.In contrast, the spherical particle contact was rather stiff, preventing their motion over the sensor.
−76 ■ EXPERIMENTAL SECTION Materials.All chemical reagents comprising sodium chloride, sodium hydroxide, and hydrochloric acid were commercial products of Sigma-Aldrich and were used without additional purification.Ultrapure water was obtained using the Milli-Q Elix and Simplicity 185 purification system from Millipore.
The particle chemical composition was characterized by X-ray photoelectron spectroscopy (XPS), performed using the PHl 5000 VersaProbe, scanning electron spectroscopy for chemical analysis (ESCA) microprobe (ULVAC-PHI, Japan/U.S.A.) instrument at a base pressure below 5 × 10 −9 mbar.The particle morphology and size distribution were characterized by scanning electron microscopy (SEM) using a JEOL 5500LV apparatus (Akishima, Japan).
Except for the spheroidal P(S/PGL), negatively charged polystyrene particles supplied by Invitrogen were used as a control solute in the deposition kinetics measurements carried out by a QCM.
Freshly cleaved mica sheets were used in the AFM measurements of spheroid adsorption kinetics under diffusion conditions.
The poly(allylamine) (PAH) aqueous solutions prepared from the crystalline powder supplied by the PSS Polymer Standards Service were used for the modification of the mica and QCM sensors.
Gold substrate plates used in the streaming potential measurements were prepared using the Si/SiO 2 wafers as the support coated with the gold layer (approximately 100 nm thick) by thermal evaporation (see the Supporting Information).
Gold sensors with a 5 MHz fundamental frequency used in the experiments were supplied by Quartz Pro, Jarfalla, Sweden.Before measurements, the sensors were cleaned in a mixture of 96% sulfuric acid (H 2 SO 4 ) and hydrogen peroxide (30%) in a volume ratio of 1:1 for 10 min.Afterward, the sensors was rinsed by ultrapure water and boiled at 80 °C for 30 min, rinsed again with ultrapure water, and dried out in a stream of nitrogen gas.
The topography of the sensors was determined by AFM imaging carried out under ambient conditions in semi-contact mode.All of the relevant parameters were determined (see the Supporting Information).The rms of the gold sensors was equal to 1.4 ± 0.1 nm, and the roughness correlation length and wavelength were 50 ± 5 and 70 ± 5 nm, respectively.
Methods.The bulk concentration of particles in the stock suspension was determined by the dry mass method.Before each deposition experiment, the stock suspension was diluted to the desired concentration, typically equal to 40−200 mg L −1 by pure NaCl solutions with the pH adjusted to 5.6.
The diffusion coefficient of the particles was determined by dynamic light scattering (DLS) using the Zetasizer Nano ZS instrument from Malvern.The hydrodynamic diameter was calculated using the Stokes−Einstein relationship.The electrophoretic mobility of particles was measured by the laser Doppler velocimetry (LDV) technique using the same apparatus.The zeta potential was calculated using the Ohshima formula 78 (see the Supporting Information).
The QCM measurements were carried out according to the standard procedure described in ref 52 using the Q-Sence QCM Instrument (Biolin Scientific, Stockholm, Sweden).First, a stable baseline of a pure electrolyte with a fixed concentration (either 1 or 10 mM NaCl and pH 5.8) was attained in the QCM-D cell for a defined flow rate (typically 1 × 10 −3 cm 3 s −1 ).Next, a PAH solution of the bulk concentration equal to 5 mg L −1 was flown through the cell until a stable signal were obtained.Afterward, the particle suspension of a fixed concentration was flushed at the same flow rate.After a prescribed time, the desorption run was initiated, where a pure electrolyte solution of the same pH and ionic strength was flushed through the cell.
The deposition kinetics of particles was determined using the AFM method as previously described in ref 52.Accordingly, after completion of a QCM adsorption run, the sensor was removed from the suspension and imaged under ambient conditions by AFM using the NT-MDT Solver BIO device with the SMENA SFC050L scanning head.The number of particles per unit area (typically 1 Table 1.Physicochemical Characteristics of the Spheroidal Particles (SA200) and Polystyrene Spherical Particles (L200) Investigated in This Work Langmuir μm 2 ) denoted hereafter by N, was determined by a direct counting of over a few equal-sized areas randomly chosen over the sensor with the total number of particles exceeding 2000.
The direct AFM enumeration method was also used to quantify the adsorption kinetics of the spheroidal particles on mica.The aim of these experiments was to determine, independent of the dry weight method, the real particle concentration in the suspension after the dilution step and to determine the maximum particle coverage used as a scaling variable in the RSA modeling.
The zeta potential of bare and PAH-covered plates was determined via streaming potential measurements performed according to the procedure described in ref 8 applying the Smoluchowski formula, where the correction for the surface conductivity was considered (see the Supporting Information).
All experiments have been performed at the temperature of 298 K.The deposition kinetics of particles was theoretically interpreted in terms of a hybrid approach, where the bulk particle transport was described by the convective−diffusion equation with the nonlinear boundary condition derived from the random sequential model (a detailed description of this approach is given in the Supporting Information).In an analogous way, the L200 particles were characterized by AFM, DLS, and LDV measurements.Their diffusion coefficient was equal to 2.4 × 10 −12 cm 2 s −1 (for the pH range of 5−6 and the above NaCl concentration range).This corresponds to the hydrodynamic diameter of 200 nm, which coincides within error bounds with the shorter axis of the spheroidal SA200 particles.The zeta potential of the particles was equal to −64 and −59 for the NaCl concentration range of 1−10 mM and pH 5−6.
The zeta potential of gold substrates was equal to −58 and −47 mV at 1 and 10 mM NaCl concentration, respectively, at pH equal to 5−6, whereas the zeta potential of PAH-covered gold and silica substrates was in both cases equal to 60 and 40 mV for this NaCl concentration range and pH 5−6.
Small Particle Coverage Limit.A series of QCM kinetic runs yielding the frequency and dissipation shifts were performed for the SA200 and L200 particles under the same physicochemical conditions, i.e., 1 mM NaCl, pH 5.6, volumetric flow rate of 10 −3 cm 3 s −1 , and various bulk suspension concentrations.After completion of the run, the real particle coverage was determined by AFM, as described above.This parameter was used as a scaling variable for the hybrid RSA modeling, which furnished the particle mass coverage versus the adsorption time dependencies in a continuous manner (see the Supporting Information).Such a procedure, previously applied in ref 52, enabled a reliable determination of the impedance and other derivative functions used in the literature for the interpretation of the QCM measurements. 65,68,73,74rimary results derived from the QCM kinetic measurements are shown in Figure 1 for the short time period up to 10 min.In panels a and b, the normalized frequency shifts −Δf/n o and the bandwidth shifts f D D (where f F is the fundamental frequency of the sensor of 5 × 10 6 Hz and ΔD is the dissipation shift) are plotted as a function of time for the spheroidal particles and for various overtones n o (1−11).Analogously, in panels c and d, the dependencies of the frequency and bandwidth shifts for the spherical particles are shown.Although quantitatively different, in all cases, the frequency and bandwidth shifts can be approximated by linear functions of the deposition time.This facilitates a precise calculation of the complex impedance ΔZ* in the limit of low particle coverage considering that it is defined as the ratio of the tangential stress (i.e., the force per the surface area of the sensor) to the surface velocity 55,75 where ΔF i * is the excess complex force transferred to the sensor because of the particles present in its vicinity (but not necessarily adsorbed), ΔS is the surface area of the sensor, and V i * is the sensor complex velocity.
−76 However, the impedance can be experimentally determined using the above QCM kinetic data, taking advantage of the following definitions 53,59,74 where Z Q is the acoustic impedance of quartz equal to 8.8 × 10 6 kg m −2 s −1 .For a purely inertia load, the impedance is given by where ω = 2πf F n o is the angular velocity of the sensor oscillations, n o is the overtone number, m pl is the single-particle mass, N p is their number at the surface, and is the net mass of the particle layer per unit area referred to as the mass surface coverage or just coverage.
It should be mentioned that eq 4 is valid for an arbitrary particle shape, size, and coverage.Using this inertia load impedance, i.e., ωΓ, as a scaling variable, 55,73 one obtains the following expressions connecting the normalized impedance with the frequency and bandwidth shifts: where is the QCM coverage referred to often as the "wet" mass is the Sauerbrey constant equal to 0.177 mg m −2 Hz −1 for f F = 5 × 10 6 Hz, and is the acoustic ratio. 73,74ssuming that Γ Q and Γ are a linear function of the time, which was the case for the runs shown in Figure 1, eq 6 can be converted to the following form: where s lQ and s l are the corresponding slopes of Γ Q and Γ on the deposition time dependencies, respectively.In Figure 2, the dependencies of the imaginary and real impedance components on the b/δ parameter are shown for spheroidal (panel a) and spherical particles (panel b), where is the hydrodynamic boundary layer thickness and v is the dynamic viscosity of the fluid.Considering that, for the anionic spheroids, b = 103 nm, the b/δ parameter in experiments whose results are shown in Figure 2a varied between 0.433 and 1.44.One should also mention that, because of the roughness of the gold sensor (a thorough analysis of the sensor topography is presented in the Supporting Information), the energy minimum distance h m of the spheroids was equal to 5 nm.This gives 0.05 for the h m /b parameter pertinent to these experiments.
The average values of Z̅ im calculated from the experimental results shown in Figure 2a were equal to 2.1 ± 0.1 and 0.67 ± 0.05 for b/δ of 0.433 and 1.44, respectively.The average values of Z̅ re were equal to −2.1 ± 0.1 and −0.70 ± 0.05 for b/δ of 0.433 and 1.44, respectively, which practically coincide (in absolute terms) with Z̅ im .This regularity was also observed for the intermediate values of the b/δ parameter, and for the cationic (positively charged) spheroid deposition on the bare gold sensor previously reported in ref 62, see Figure 2a.This indicates that, for spheroidal particles, the complex impedance is proportional to 1 − i. Considering this, the experimental results were fitted by the following function: with the dimensionless constant C z equal to 0.96.One should underline, however, given the soft hydrodynamic contact of the particles with the sensor, that this constant is expected to depend upon the h m /b parameter.
As seen in Figure 2a, eq 12 well reflects the experimental data for the entire range of the b/δ parameter.
Analogous results obtained for the spherical particles, where the range of b/δ was practically the same as for spheroids, are shown in Figure 2b.One can notice that the Z̅ im impedance component was considerably larger that for spheroids and was equal to 5.0 ± 0.1 2.1 ± 0.05, for b/δ of 0.433 and 1.44, respectively.Therefore, in comparison to spheroids, it did not decrease below unity, which was marked as the dotted line in Figure 2b.On the other hand, the real impedance component was practically independent of b/δ within experimental error bounds and assumed an average value of −2.0 ± 0.2.Such a behavior was predicted in ref 74 for a stiff contact of the particles with the sensor, although theoretical data pertinent to combined hydrodynamic and inertia forces are not available.However, some theoretical predictions for a stiff contact can be derived from the model of a freely oscillating sphere formulated by Tarnapolsky and Freger, 55 yielding the following expression for Z̅ im valid for b/δ larger than unity: The theoretical predictions derived from eq 13, shown in Figure 2b as the dashed line, overestimate the experimental data, although the general trend is well-reflected for the entire range of the b/δ parameter.A better fit can be achieved using the following formula: where C zs = 1.8.It is also interesting to mention that the acoustic ratio −Z̅ re / Z̅ im calculated using eq 9 considering that Z̅ re = −2 agrees within experimental error bounds with the acoustic ratio defined in ref 74 (considering the scaling factor of 1 / 2 ).
Moreover, using the above fitting functions, one can formulate the following expression enabling direct calculations of the real spheroidal particle coverage using the experimental frequency and bandwidth shifts: where and δ 1 is the boundary layer thickness for the first overtone.In our case, C z was equal to 0.96 for h m /b = 0.05.It should be noticed that, according to eq 15, the real particle coverage is proportional to −Δf/n o 1/2 .Analogously, using the experimental bandwidth, one obtains the formula For spherical particles the stiff contact regime, one has where With the advantage of the fact that eqs 16 and 18 are valid under the linear kinetic regime, they can be directly used to determine the mass transfer constant, a parameter of basic significance for every kinetic study. 44,79,80Considering the definition of the mass transfer constant, one can derive the following expression for the spheroids and spheres, respectively: where c b is the bulk mass concentration of the particles, expressed in mg L −1 .Additionally, knowing the Z̅ im impedance component, one can calculate the correction function H often used in the literature 52,65,66,69,70,74 for the interpretation of experimental results derived from QCM measurements and defined as Thus, knowing H, one can calculate the real particle coverage as The dependencies of H on the b/δ parameter for spheroids and spheres are shown in panels a and b of Figure 3, respectively.In the former case, the function was equal to 0.48 at b/δ = 0.433 and then monotonically decreased, attaining negative values for b/δ larger than 0.8.This trend is wellreflected by the following fitting function derived using eq 12 for spheroids: In the case of spheres, the function was equal to 0.80 at b/δ = 0.433 and then slowly decreased, attaining 0.54 at b/δ = 1.44.Using eq 14, the following formula was derived to interpolate the experimental data: As mentioned, all of these formulas are strictly valid for the low coverage regime, where the frequency and bandwidth shifts remain a linear function of the deposition time.In the next section, these results are extended to the arbitrary particle coverage, exploiting the long-time kinetic data derived from the QCM measurements.
Analysis of the Long-Time Kinetic Data.The long-time kinetic runs performed under identical conditions as before (see legend to These primary experimental data were converted to the QCM coverage calculated as −C s (Δf/n o ), yielding the kinetic runs shown in Figure 5 for spheroids and spheres.In both cases, the QCM coverage, which significantly exceeded the real coverage for the first overtone, systematically decreased for larger overtones.Interestingly, in the case of spheroids, the kinetic curve derived from QCM coincided for the fifth overtone with that pertinent to the real coverage derived from the RSA calculations.It should be mentioned that the RSA kinetics was calibrated using the AFM coverage determined after finishing each measurement, comprising the desorption step, using a direct counting procedure.The maximum AFM coverage was equal to 100 and 78 mg m −2 for spheroids and spheres, respectively.
Using the experimental results shown in Figure 5, the imaginary and real impedance components can be calculated from eq 6 as a function of the deposition time (with the real particle coverage as a scaling parameter).To facilitate the analysis of obtained results, the kinetic dependencies obtained in this way are converted to the dependencies of Z̅ im /Z̅ im 0 and Z̅ re /Z̅ re 0 on the real particle coverage (where Z̅ im 0 and Z̅ re 0 are the above determined impedances in the limit of low particle coverage).The results shown in Figure 6 for spheroids confirm that the normalized imaginary and real impedances were close to unity for all overtones and for the entire range of particle coverage.This is a significant result showing that the previously derived formula describing the impedance components, i.e., eq 12, remains valid for an arbitrary coverage of spheroids.As a consequence, eqs 15 and 16 can be used to calculate the deposition kinetics, expressed in terms of the real coverage, using the experimental frequency or bandwidth shifts acquired for various overtones.
The above results obtained for spheroids are consistent with a non-localized deposition mechanism, where the particles may undergo sliding motions in the direction parallel to the sensor surface without physically contacting the sensor surface. 8,52A thorough analysis of the interactions of a spheroid with the rough substrate (Supporting Information) showed that this is physically feasible because of the presence of a deep energy minimum appearing at the distance of 5 nm.This prevented the diffusion of the captured particles in the perpendicular direction over the distance larger than a fraction of a nanometer.On the other hand, the particle motion in the parallel direction was feasible because the spheroidal particle length significantly exceeded the average distance between the roughness peaks, estimated to be about 200 nm.In effect, the particles were deposited at a few peaks (see Figure S6a of the Supporting Information) and could freely execute a sliding motion along the surface.Thus, their interactions with the surface were purely hydrodynamic of lubrication-like type, controlled by the smallest distance between the particle and the sensor surfaces, which appeared close to the center of the particle.Because these hydrodynamic interactions were confined to a small space underneath the particles, the dependence of the impedance upon the coverage is expected to be rather negligible.
It should be mentioned that an analogous behavior was recently reported in ref 81.It was shown that spherical particles (of the size between 25 and 100 nm) linked to the sensor by a molecular receptor showed a perfectly linear increase in the normalized frequency shift with the dimensionless coverage θ

=
. Experimental conditions are the same as in Figure 4.
Langmuir up to 0.5.This is tantamount to the fact that the normalized sensor impedance was independent of the particle coverage, which was observed in Figure 6 for spheroids.On the other hand, for spherical particles, whose size was comparable to the average distance between roughness peaks, their perpendicular and parallel motions were prohibited because this would require a significant increase in the energy (see the Supporting Information).Therefore, their contact with the surface was rather stiff, enabling the transfer of the inertia force appearing because of the sensor surface periodic acceleration.
Because of such stiff contact of spherical particles, the dependence of the normalized imaginary impedance component upon the real particle coverage, shown in Figure 7, was more complicated in comparison to this dependence found for spheroids.For the first to fifth overtones, it monotonically decreased with the real particle coverage, but at higher overtones, these dependencies were non-monotonic, exhibiting a maximum at the coverage of ca.42 mg m −2 (θ = 0.30).Unfortunately, in this case, no universal formula can be derived, furnishing the real particle coverage for arbitrary frequency shifts.The range of applicability of eq 15 for spheroids and eq 18 for spheres can be estimated analyzing the results shown in panels a and b of Figure 8 as the dependencies of the real particle coverage on −Δf/n o 1/2 .As expected for spheroids, eq 15 yields adequate results for the wide range of −Δf/n o 1/2 and for all overtones.On the other hand, for spheres, eq 18 only yields an adequate fit for the fifth overtone and −Δf/n o 1/2 smaller than 1500 Hz.For other overtones, the range of applicability of eq 18 is rather limited.One should also consider that the choice of an adequate overtone, yielding the best results, cannot be done a priori without performing a thorough analysis of experimental data.Therefore, the results obtained for spheres suggests that, at the present time, the deconvolution of the QCM signal aimed at the calculation of the real coverage is only feasible in the limit of low-frequency shifts.
■ CONCLUSION Deposition kinetics of anionic spheroidal particles at polycation-modified gold sensors was determined applying a QCM supplemented by AFM and analyzed in terms of theoretical modeling based on the hybrid RSA approach.Analogous measurements were carried out for spherical particles of the same diameter as the spheroid shorter axis.
A thorough analysis of the kinetic data enabled determination of the complex impedance for various overtones, which corresponded to the b/δ parameter between 0.433 and 1.44 and for a broad range of the real particle coverage.
It was shown that the frequency shifts for all overtones were linear functions of the particle coverage.These results were explained in terms of a hydrodynamic, lubrication-like contact of particles with the sensor, enabling their sliding motion.The feasibility of such a non-localized deposition mechanism of spheroids was confirmed by the analysis of their interactions with the sensor surface whose topography was thoroughly characterized by AFM.Considering the experimental results, a Sauerbrey-like equation, eq 15, was derived enabling a facile determination of the real spheroidal particle coverage using the frequency or bandwidth (dissipation) shifts derived from experiments.This equation can also be used to determine the mass transfer constant of the particles in the QCM cell.
On the other hand, for spherical particles, the imaginary impedance significantly exceeded unity for all overtones and arbitrary coverage range and the frequency shifts were nonlinear functions of the particle coverage, which was interpreted as the effect of a stiff contact.On the basis of these results, eq 18 was derived enabling calculation of the real particle coverage and as a consequence of the mass transfer rate constant.However, it was shown that this equation only yields precise data for low-frequency shifts.
One can expect that the results obtained in this work can be used as useful reference systems for a quantitative interpretation of bioparticle, especially bacteria, deposition phenomena, significantly extending the applicability range of the QCM technique.

■
RESULTS AND DISCUSSIONBulk Particle and Substrate Characteristics.The relevant physicochemical characteristics of the spheroidal SA200 and spherical L200 particles used in the QCM kinetic measurements are collected in Table1.The former particles were characterized by the dimension 2a × 2b × 2b of 1010 × 205 × 205 nm (derived from AFM) and 1020 × 215 × 215 (derived from SEM, where a ca. 10 nm thick layer was sputtered), with a low dispersity of ca.5%.Hence, their axis ratio As = a/b and the cross-section area in the side-on orientation S g were equal to 4.93 and 0.163 μm 2 .The diffusion coefficient of the particles directly measured by DLS was equal to 1.1 × 10 −12 m 2 s −1 (for the NaCl concentration of 1−10 mM).This yields the hydrodynamic diameter of 430 nm calculated from the Stokes−Einstein formula.The zeta potential of the particles derived from the LDV measurements varied between −49 and −36 mV for the NaCl concentration range of 1−10 mM and pH 5−6.

Figure 2 .
Figure 2. Imaginary and real parts of the sensor impedance in the limit of low particle coverage determined from the experimental kinetic data.(a) Spheroidal particles: the points show experimental results derived from measurements carried out for anionic spheroids under various bulk concentrations and deposition times, and the square points show the previous results obtained in ref 62 for cationic spheroids on a bare gold sensor.Experimental conditions: PAH-modified gold sensor, 1 mM NaCl, pH 5.6, and volumetric flow rate of 10 −3 cm 3 s −1 .The solid line shows the theoretical values of the impedances calculated from eq 12.(b) Spherical particles (L200): the points show experimental results derived from measurements carried out under various bulk concentrations and deposition times.Experimental conditions: PAHmodified gold sensor, 1 mM NaCl, pH 5.6, and volumetric flow rate of 10 −3 cm 3 s −1 .The dashed/dotted line 1 shows the theoretical values calculated from eq 13 using the Tarnapolsky and Freger 55 model pertinent to large b/δ; the solid line 2 shows the results calculated from the fitting function eq 14; the dotted line 3 shows the results pertinent to the purely inertia load; and the dashed line 4 shows the zero impedance.

Figure 3 .
Figure 3. Dependence of the H function in the limit of low particle coverage upon the b/δ parameter.(a) Spheroidal particles: the solid line denotes the theoretical data calculated using eq 24.(b) Spherical particles: the solid line denotes the theoretical data calculated using eq 25.Experimental conditions are the same as in panels a and b of Figure 2.

Figure 1 )
are shown in Figure 4.In panels a and b, the normalized frequency shifts −Δf/n o and bandwidth shifts D are shown for spheroidal particles and various overtones.Analogously, in panels c and d, the dependencies of the frequency and bandwidth shifts for spherical particles are shown.It is worth mentioning that no change in the signals was observed after the initiation of the desorption run, where a pure electrolyte of the same ionic strength was flushed through the QCM cell.

Figure 5 .
Figure 5. Kinetics of particle deposition expressed as the dependence of the QCM coverage calculated using the Sauerbrey constant for the overtones 1−11.The points represent the real (dry) particle coverage derived from AFM, and the dashed lines denote the theoretical results derived from the hybrid RSA model.(a) Spheroidal particles.(b) Spherical particles.Experimental conditions are the same as in Figure 4.The arrows show the initiation of the desorption run.

Figure 6 .
Figure 6.Dependence of the (a) imaginary and (b) real impedance components (normalized using the initial impedances) on the real particle coverage for spheroidal particles.The upper horizontal axis shows the dimensionless particle coverage b 3 4 p

Figure 7 .
Figure 7. Dependence of the imaginary impedance components upon the real particle coverage for spherical particles.The upper horizontal axis shows the dimensionless particle coverage b 3 4 p=.Exper-

Figure 8 .
Figure 8. Real spheroidal particle coverage versus −Δf/n o 1/2 for different overtones.(a) Spheroidal particles: the dashed line shows the results calculated from eqs 15 and 16.(b) Spherical particles: the dashed line shows the results calculated from eqs 18 and 19.Experimental conditions are the same as in Figure 4.