Kinetics of Silver Nanoparticle Deposition at PAH Monolayers

Article pubs.acs.org/Langmuir

Kinetics of Silver Nanoparticle Deposition at PAH Monolayers: Reference QCM Results Katarzyna Kubiak, Zbigniew Adamczyk,* and Magdalena Oćwieja Jerzy Haber Institute of Catalysis and Surface Chemistry, Polish Academy of Sciences, Niezapominajek 8, 30-239 Cracow, Poland S Supporting Information *

ABSTRACT: The deposition kinetics of silver nanoparticles on Au/SiO2 /PAH substrate was studied under in situ conditions using the QCM method and the ex situ SEM imaging. Because of low dissipation, the Sauerbrey equation was used for calculating the mass per unit area (coverage). Measurements were done for various bulk suspension concentrations, flow rates, and ionic strengths. It was shown that particle deposition for the low coverage regime is governed by the bulk mass transfer step that results in a linear increase of the coverage with the time. A comparison of QCM and SEM results showed that the hydration of the silver monolayers was negligible. This allowed one to derive a universal kinetic equation that describes the mass transfer rates in the cell as a function of the bulk concentration, flow rate, and diffusion coefficient. Measurements were also performed for longer times and for various ionic strengths where the deposition kinetics and the maximum coverage of particles were determined. The experimental data confirmed a significant increase in the maximum coverage with ionic strength. This was interpreted as due to the decreasing range of the electrostatic interactions among deposited particles. These results were adequately interpreted in terms of the extended random sequential adsorption (eRSA) model. Additionally, it was shown that the QCM data matched the ex situ SEM results, indicating that the monolayer hydration was also negligible for higher coverage range. These results derived for the model silver nanoparticle system can be exploited as reference data for the interpretation of protein adsorption kinetics where the dry mass is needed in order to assess the extent of hydration. air16,17 that allows one to study particle deposition and desorption (release) kinetics from various substrates15 is inaccurate for higher coverages. Therefore, it cannot be applied for fast transient processes of monolayer formation. Other surface oriented techniques such as UV−vis spectroscopy,18 scanning electron microscopy (SEM),19,20 and X-ray photoelectron spectroscopy (XPS),15 represent ex situ methods difficult to apply for kinetic studies. More precise, in situ information is provided by the evanescent wave spectroscopy21 and electrokinetic techniques, most often the streaming potential measurements.15 However, the electrokinetic methods are best suited for studying longlasting particle deposition or desorption (release) processes for a not too high surface coverage range. One can expect that the real-time deposition kinetics of silver nanoparticles can be conveniently studied by the quartz crystal microbalance (QCM) technique that is widely used for determining polyelectrolyte22−25 and protein adsorption at various substrates.26−28 This method seems particularly suitable for silver nanoparticles because of their crystallinity, high density, and shape stability that considerably reduces the

1. INTRODUCTION Silver nanoparticle monolayers deposited on solid substrates exhibit a wide spectrum of practical applications in medicine,1,2 catalysis,3,4 chemical analysis5,6 as well as in electronics7 and textile industry.8 The monolayers are deposited on various materials, in particular fibers or polymers,8,9 and applied in manifold consumer products such as clothes, laboratory and surgical gowns, and dressing bandages.1,10 They can also be exploited as analytical sensors in surface-enhanced Raman spectroscopy (SERS).11 In response to the extensive range of practical applications, many experimental works have been published in the literature focused on silver monolayer and film formation.12−15 However, despite an essential significance, the fundamental aspects of silver particle deposition mechanisms on solid substrates, especially the kinetics of these processes, have sparsely been studied.15 There are two main reasons for the deficit of such systematic works: (i) difficulties with obtaining pure and stable silver particle suspensions of high solid weight concentration and (ii) the limited number of experimental techniques for efficient measurements of transient nanoparticle deposition under in situ conditions. Whereas the first issue is recently solved by applying the controlled chemical reduction route for the synthesizes of charge stabilized particles,15 the second limitation has not been successfully overcome yet. For example, the atomic force microscopy (AFM) imaging in liquid or in © 2015 American Chemical Society

Received: December 23, 2014 Revised: February 16, 2015 Published: February 18, 2015 2988

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was added dropwise into the reduction mixture (400 mL) containing 3.33 mM trisodium citrate and 1.41 mM sodium borohydride. The reaction mixture was stirred for an hour, and after this period of time, the obtained yellow silver suspension was purified from the excess of ions using the membrane filtration method.15 The mass concentration of the stock silver suspensions after the purification was determined using the densitometer DMA 5000 from Anton Paar.21 The size distribution of particles was investigated by transmission electron microscope (TEM) using FEI Tecnai G2 microscope at 200kV equipped with high-angle annular dark field scanning transmission HAADF/STEM and EDAX energy dispersive X-ray (EDX) detectors (see the supporting information). The particle monolayer coverage was determined using a scanning electron microscope (SEM, JEOL JSM-7500F) working in a transmission mode (see the Supporting Information, Figure 1). The hydrodynamic diameter of particles and the stability of their suspensions were determined via the dynamic light scattering (DLS) measurements of the diffusion coefficient using the Nano ZS Zetasizer from Malvern. The electrophoretic mobility of particles was measured using the Nano ZS Zetasizer apparatus. Knowing the electrophoretic mobility, the zeta potential of particles was calculated using Henry’s equation according to the procedure described in previous work.15 Quartz sensors covered by the silicon dioxide (SiO2) layer used in all experiments had a fundamental frequency of 5 MHz. They were commercial product of Q-Sense, Gothenburg, Sweden. Before every measurement, the sensors were cleaned in a mixture of 95% sulfuric acid (H2SO4) and hydrogen peroxide (30%) in volume ratio 1:1 for 10 min. Afterward, they were rinsed by deionized water at 80 °C for 30 min and dried out in a stream of a nitrogen gas. The homogeneity and roughness of the sensors were examined by semi-contact mode atomic force microscopy (AFM) imaging carried out under ambient conditions. It was confirmed that the sensors were homogeneous and smooth exhibiting the root-mean-square roughness of about 1 nm (see the Supporting Information, Figure S2). A quartz crystal microbalance with dissipation monitoring was used for a real-time investigation of silver particle deposition kinetics. A freshly cleaned and dried sensor was installed in a QCM cell. Every measurement was started by obtaining a stable baseline for the pure electrolyte (NaCl) of controlled ionic strength and pH that was kept at 8 throughout the experiments. After the stabilization of the baseline, the cationic polyelectrolyte (PAH) solution of the concentration of 5 mg L−1 was flushed. When a stable frequency and dissipation signal of adsorbed PAH monolayer was obtained, the pure electrolyte solution was again flushed in order to remove loosely bound molecules. In the next step the silver particle suspension of controlled concentration was pumped. When the frequency shift became stable, the pure electrolyte solution was flushed once again in order to study the particle desorption. The concentration of silver particle suspensions in the measurements was changed in the range of 10−50 mg L−1 and the flow rate 6.67 × 10−4−2.66 × 10−3 cm3 s−1. Silver nanoparticle deposition measurements were performed according to the two different regimes: (i) the pure convection and (ii) the convection/diffusion. Under the latter regime, the flow was stopped after a controlled time, and afterward the silver particles deposition was continued under pure diffusion. In this way the maximum coverage of the particle monolayer was attained in a more controlled way because the shear-induced surface aggregation of particles was minimized. The determination of silver nanoparticle mass adsorbed was carried out using the Sauerbrey’s model, where the change in the frequency of the oscillating sensor is proportional to the change in the adsorbed mass, according to the equation24,25,32

hydration, water trapping, and energy dissipation effects that usually disturb the interpretation of QCM measurements. However, despite its potential significance, few works have been published that are focused on determining silver particle deposition using the QCM technique. Bandyopadhyay et al.29 measured the kinetics of silver particle deposition (10 nm in diameter) on aluminum surfaces modified by molecules of 4carboxythiophenol. It was observed that the monolayer mass increased linearly with the deposition time and after exceeding 5 min attained a constant value of 730 ng cm−2. This corresponds to the maximum coverage of the monolayer equal to 10.4%. However, no information about pH, ionic strength, and nanoparticle concentration in the suspension was given in this work. The formation of silver nanoparticle monolayers on silanized surfaces of glass, carbon, and gold was studied by Bright and coworkers30 using UV−vis spectroscopy and QCM. Citratestabilized nanoparticles of size 22 and 55 nm and other particles prepared by ethylenediaminetetraacetic acid (EDTA) reduction of the size 14 and 15 nm were used in this study. A significant difference in the deposition rate between citrate and EDTAstabilized nanoparticles was observed. The maximum mass of citrate-stabilized nanoparticles (55 nm in diameter) deposited on the substrate was equal to 10.5 μg cm−2, which corresponds to the coverage of 15%. However, in the case of EDTAstabilized nanoparticles (15 nm in diameter) the maximum mass of the monolayer was 17.2 μg cm−2, which corresponds to the coverage of 163% indicating formation of multilayer coverage (most probably silver mirror). Deposition of citrate-stabilized silver particles of size 24.9 nm on glass and silicon modified with poly(4-vinylpiridine) (P4PV) was studied using AFM and QCM by Kim et al.31 The Sauerbrey equation was used for calculating the mass of deposited particles and their surface concentration. In this way the kinetics of the monolayer formation was obtained. The results obtained by QCM were in agreement with the experimental data derived from AFM. Because of a the lack of systematic studies, the goal of this work is to determine mechanisms of silver nanoparticle deposition under flow controlled transport and quantitatively evaluate the kinetics of this process using QCM calibrated by the scanning electron microscopy (SEM). The experimental data are theoretically interpreted in terms of the extended random sequential adsorption model (eRSA) which has not been attempted before in the literature. This allows one to determine the universal dry mass transfer coefficient for the QCM cell. In this way one can correctly interpret protein adsorption kinetic data where the dry mass is needed in order to assess the extent of hydration.

2. EXPERIMENTAL SECTION Silver nitrate, trisodium citrate, sodium borohydride, sodium chloride, sodium hydroxide, and hydrochloric acid were commercial products of Sigma-Aldrich. Poly(allylamine hydrochloride) (PAH) having a molecular weight of 70 kDa was purchased from Polysciences. All the chemical reagents and solvents were commercial products and were applied without further purification. Ultrapure water, used throughout this investigation, was obtained using the Milli-Q Elix&Simplicity 185 purification system from Millipore SA Molsheim, France. The synthesis of silver nanoparticles was carried out according to the chemical reduction method using sodium borohydride as a reducing agent and trisodium citrate as a stabilizing agent. For this purpose, 200 mL of aqueous solution containing 1 mM silver nitrite

Δm = − C

Δf n

(1)

where Δm is mass change, Δf is the frequency change, n is the overtone number, and C is the mass sensitivity constant depending on 2989

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Langmuir Table 1. Physicochemical Characteristics of Silver Nanoparticles Used in this Work property [unit], symbol

value

specific density [g cm−3], ρp diffusion coefficient [cm2 s−1], D hydrodynamic diameter [nm], dH particle size [nm], dp geometrical cross-section area [nm2], Sg plasmon absorption maximum [nm], λp

10.49 3.27 × 10−7 ± 0.6 × 10−7 15 ± 3 15 ± 4 176 393

remarks

adsorbed at ionic strength of 10−2 M was positive and equal to 45 mV. Moreover, the changes in the zeta potential of the PAH monolayer within the time period of 3 h were rather minor amounting to only a few millivolts. The stability of PAH monolayers deposited at the Au/SiO2 substrate was also studied in this work using the QCM measurements. A typical experimental run is presented in Figure 1 as the dependence of the relative frequency shift on

the physical property of the sensor that, for a 5 MHz AT-cut quartz crystal, is equal to 0.177 mg m−2 Hz.24,25,32 The kinetics of silver particle deposition on the QCM crystal were also monitored ex situ using scanning electron microscopy (SEM). The surface concentration of particles was calculated from the SEM micrographs using an image-analysis software MultiScan Base.

3. RESULTS AND DISCUSSION 3.1. Bulk Particle and Substrate Characteristics. The weight concentration of silver particles in the stock suspension after the cleaning procedure was 123 mg L−1. This was determined by measuring the density of the native suspension and the effluent acquired from the membrane filtration as described in ref 33. The size distribution of particles was characterized by TEM which allowed one to produce a histogram of particle size distribution (see the Supporting Information Figure S1). The average particle size derived from the histogram was 15 ± 4 nm (see Table 1). Additionally, the hydrodynamic diameter of silver particles was determined by DLS via the diffusion coefficient measurements. The hydrodynamic diameter of the particles was equal to 15 ± 3 nm at pH 8 and ionic strength range 10−4−10−2 M. It was also confirmed by separate measurements that the hydrodynamic diameter of particles and the absorption maximum did not change for prolonged time periods (1000 h) for ionic strength range 10−4−0.01 M. This suggests that the suspension was stable over the time of the QCM deposition experiments lasting typically 5−40 min. On the other hand, the measurements of the electrophoretic mobility of the silver particles allows one to determine the zeta potential (ζs) using Henry’s equation and the number of the electrokinetic (uncompensated) charges from the Stokes− Lorenz relationship.15,35 The results collected in Table 2

Figure 1. Primary QCM measurements of PAH adsorption/ desorption at the Au/SiO2 sensor, expressed as the frequency change [Hz] for the third overtone vs the time (pH 8, bulk polyelectrolyte concentration 5 mg L−1).

the solution flushing time. As can be seen, the PAH monolayer formation time lasted 4 min. Afterward, the pure electrolyte was flushed through the cell, and the desorption run was recorded. Various electrolyte concentrations ranging between 10−4 and 10−2 M were used. As can be noticed, little desorption of PAH was observed during this period of time, especially for the lower ionic strength. These results indicate that PAH monolayers are a convenient platform for the deposition of silver particles. 3.2. Kinetics of Particle Deposition. Using the abovedescribed procedure, kinetic measurements were performed in order to determine the mechanism of particle deposition, in particular to determine the mass transfer coefficients and the hydration degree of the monolayers. A primary experimental run is shown in Figure 2 where the dependence of the relative frequency shift on the time is presented. After adsorption and rinse of the PAH monolayer, the silver particle suspension of the bulk concentration of 50 mg L−1 was pumped through the cell at the rate of 0.08 mL min−1 (1.33 × 10−3 mL s−1). A considerable decrease in the relative frequency shift was observed that confirmed an efficient deposition of silver particles. After the time 60 min the pure 10−3 M electrolyte was flushed through the cell over the time of 30 min. As can be

Table 2. Electrophoretic Mobility, Number of Elementary Charge, and Zeta Potential of Silver Nanoparticles for Various Ionic Strengths (pH 8, T = 298 K) ionic strength [M] 0.0001 0.001 0.01

κdp

μe [μm cm (V s)−1]

ζp [mV] Henry’s model

Nc

σ [e nm−2]

0.25 0.78 2.46

−3.69 ± 0.1 −3.38 ± 0.1 −3.04 ± 0.1

−70.2 −63.1 −54.1

−29 −27 −24

−0.0412 −0.0384 −0.0341

literature data34 measured by DLS for T = 298 K, pH 8.0, I = 10−4−0.01 M NaCl determined in this work from size distributions obtained from TEM micrographs calculated from geometry measured for pH 4.0−8.0, I = 10−4−0.01 M NaCl, and silver suspension concentration cb = 20−25 mg L−1

indicate that the particles exhibit negative electrophoretic mobility (zeta potential) for the above ionic strength range. Accordingly, for I = 10−4 M the zeta potential is equal to −70 mV, and for I = 0.01 M it increases to −54 mV. The electrokinetic properties of PAH molecules and the stability of their monolayers on mica were extensively studied using the streaming potential method in ref 35. It was shown that the zeta potential of a PAH monolayer of the coverage 0.4 2990

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Figure 2. Primary QCM measurement of silver particle deposition/desorption at the Au/SiO2 sensor, expressed as the frequency change [Hz] for the third overtone and the relative dissipation (right-hand axis) vs the time (pH 8, bulk suspension concentration 50 mg L−1, ionic strength of deposition 10−3 M, flow rate 1.33 × 10−3 mL s−1).

seen in Figure 2, practically no change in the QCM signal was observed which confirms a negligible desorption of particles. It is also interesting to mention that the dissipation function assumed values below 10−6 (see Figure 2, right-hand axis) for the entire deposition/desorption run that validates the use of Sauerbrey’s equation for calculating the adsorbed mass per unit area of the crystal, denoted by Δm. It is usually expressed in ng cm−2.26,27 However, in this work, the particle coverage is expressed in mg m−2 and denoted by Γ. It should be noted that Γ = 0.01Δm. The validity of Sauerbrey’s model is further confirmed by the fact that the frequency shift induced by particle deposition was independent of the overtone number (3−9, see the Supporting Information Figure 3). Therefore, for calculating the adsorbed mass, the third overtone was used that increases the sensitivity of measurements. The kinetics of silver particle deposition on PAH modified Au/SiO2 crystal determined in this way at pH 8, I = 10−3 M, and the flow rate of 1.33 × 10−3 cm3 s−1 is shown in Figure 3 for various bulk suspension concentration of 50, 30, and 10 mg L−1. A few interesting observations can be made based on these kinetic runs. First, one can estimate that the duration of the nonstationary (transient) deposition regime that is characterized by the variable slope of the ΔΓ vs the deposition time dependence is only 15 s. Afterward, for longer times, a perfect linearity of the coverage vs the time dependence is observed for all bulk suspension concentration. This behavior supports the hypothesis that silver particle deposition under the steady state was solely convection controlled. Moreover, the linearity of the kinetic runs (determined with a relative precision of ±1%) also shows that the hydration of the silver particle monolayer was minor and did not decrease with the particle coverage as previously observed for proteins.27,28,36 In order to quantitatively confirm this result, the coverage of silver particle monolayers on the crystal was independently determined by ex situ SEM imaging. It should be mentioned that because of a considerable electric conductivity of silver monolayers, no supporting layer evaporation was needed that improves the precision of these measurements. From the SEM micrographs of particle monolayers (see Figure 3b) the average number of

Figure 3. (a) Kinetics of silver particle deposition on PAH modified Au/SiO2 sensor (pH 8, I = 10−3 M, flow rate 1.33 × 10−3 cm3 s−1) determined by QCM for bulk suspension concentration of (1) 50 mg L−1, (2) 30 mg L−1, and (3) 10 mg L−1; the hollow symbols present the coverage obtained by direct counting of deposited particles: (1) 4%, (2) 8.5%, and (3) 12%. (b) SEM images of silver nanoparticle monolayer for different surface coverages.

particles per unit area (surface concentration) denoted by Ns was determined by an image analyzing software. Knowing Ns, the coverage Γ can be calculated from the linear dependence Γ = (πd p3ρp /6)Ns

(2)

where dp is the particle size and ρp is the silver particle density. As can be seen in Figure 3, the results derived from SEM (depicted as hollow points) agree with the QCM data. 2991

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this dependence in the linear region ⟨sl⟩ = 1.04 × 10−3 ± 5 × 10−5 L m−2 s−1. Knowing the slope, the mass transfer rate constant ⟨kc⟩ that is defined as Γ/(Δtcb) can be calculated from the simple expression

Another interesting feature of the kinetic runs shown in Figure 3 is that the slope of the linear dependencies increases proportionally to the bulk suspension concentration cb which also validates the bulk transport controlled regime of particle deposition. This conclusion is confirmed by other experiments shown in Figure 4, where the influence of ionic strength on the

⟨kc⟩ = 0.1⟨sl⟩

(3) −1

where the ⟨kc⟩ unit is cm s . Hence, for the above flow rate of 1.33 × 10−3 cm3 s−1, ⟨kc⟩ = 1.04 × 10−4 ± 5 × 10−6 L m−2 s−1. It should be mentioned that this parameter has a major significance for a quantitative analysis of the kinetics of mass transport in the QCM cell for other solutes, e.g., polyelectrolytes or proteins. This is so because for convection driven transport the average mass transfer rate constant ⟨kc⟩ can be expressed as37,38 ⟨kc⟩ = Cf Q1/3D2/3

(4)

silver particle deposition rate was studied at pH 8, flow rate 1.33 × 10−3 cm3 s−1, and bulk suspension concentration 30 mg L−1. It can be seen that for all ionic strengths (varied between 10−4 and 10−2 M) the Γ vs the time dependencies become linear after a short transition time and are characterized by an identical slope. The validity of the bulk transfer controlled deposition regime becomes more evident by plotting the results shown in Figures 3 and 4 as the dependence of Γ/cb on the deposition time t. As can be noticed (see Figure 5), the previously obtained data are transformed in this way to a universal relationship. It was calculated, by a the least-squares fitting that the average slope of

where Cf is a constant depending on the cell’s geometry but independent of the solute physicochemical parameters (such as the size and shape), Q is the volumetric flow rate, and D is the diffusion coefficient of the solute. The Cf constant is known in an exact analytical form for various flows of a practical significance, e.g., for the impinging jet flow, for the wall-jet axisymmetric flow, parallel-plate channel flow, etc.38 The validity of eq 4 for QCM cell flows was also confirmed by Zhang et al.,39 who studied adsorption and desorption of polymeric nanoparticles (size range 90−305 nm) loaded by a drug. These authors have also calculated theoretically the mass transfer coefficients (adsorption rate constants) using a bipolar coordinate system by neglecting the perpendicular diffusion component. However, their cell’s geometry was different than in the present case; therefore, a series of kinetic experiments were carried out where the influence of the flow rate was studied. The results collected in Figure 6 indicate that the mass transfer rate linearly increases with Q1/3 (for the volumetric flow rate flow varying between 6.67 × 10−4 and 2.66 × 10−3 cm3 s−1). This finding confirms that eq 4 can be used for predicting the mass transfer kinetics in the QCM cell. Therefore, using the known value of the diffusion coefficient of silver particles given in Table 1, one can calculate from eq 4 that the Cf constant is equal to 19.9 cm−4/3.

Figure 5. Universal plot showing the dependence of ΔΓ/cb on the silver particle deposition time t obtained for various bulk suspension concentrations and ionic strengths at pH 8 and flow rate of 1.33 × 10−3 cm3 s−1. The red line shows the average of all experimental runs.

Figure 6. Kinetics of silver particle deposition on PAH modified Au/ SiO2 sensor (pH 8, bulk suspension concentration 30 mg L−1, I = 10−3 M) determined by QCM for various flow rates: (1) 2.66 × 10−3 cm3 s−1, (2) 1.33 × 10−3 cm3 s−1, and (3) 6.67 × 10−4 cm3 s−1.

Figure 4. Kinetics of silver particle deposition on PAH modified Au/ SiO2 sensor (pH 8, flow rate 1.33 × 10−3 cm3 s−1, and bulk suspension concentration 30 mg L−1) determined by QCM for various ionic strengths: (1) 10−2 M, (2) 10−3 M, and (3) 10−4 M.

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Langmuir This result is significant because knowing Cf one can calculate the dry mass of any solute, for example protein, from the linear dependence Γd = 19.9Q1/3D2/3cbt

this behavior indicates that the deposition process was irreversible. One should mention that that the characteristic time of the monolayer formation decreased linearly with the bulk concentration of particles and was equal to 7, 11, and 25 min for cb = 50, 30, and 10 mg L−1. This is in accordance with the following formula derived from eq 5 by postulating that for t = tm, Γ = Γmx

(5)

Using this reference result, one can predict the degree of hydration defined as the ratio of the trapped water volume vw to the dry solute volume vd from the relationship Hv(t ) =

⎤ vw Γ(t ) ρ⎡ = ⎢0.0502 1/3 2/3 − 1⎥ ⎥⎦ ρw ⎢⎣ vd Q D cbt

tm = 0.0502 (6)

ρw Hv(t ) Γd = Γ ρ + ρw Hv(t )

Q

D

cb

(8)

As can be noted, the monolayer formation time decreases proportionally to the bulk suspension concentration. In order to gain more insight into the mechanism of particle deposition, the results shown in Figure 7 were theoretically interpreted in terms of the extended random sequential adsorption (eRSA) model. This is an universal approach that allows one to theoretically predict the blocking function and the maximum coverage of particles interacting via the screened Coulomb potential. It is convenient to express the results obtained from the RSA model in terms of the absolute (thermodynamic) coverage of particles defined as

where ρ is the solute density and ρw is the water density. Often the hydration degree is alternatively defined as24−28 H=1−

Γmx 1/3 2/3

(7)

It is interesting to mention that an advantage of eq 6 is that in order to calculate the dynamic hydration degree, one does not need any additional experimental data derived for example from ellipsometry or reflectometry.24−28 Although eq 6 is strictly valid for the linear adsorption regime, one can also asses the hydration degree for the nonlinear regime using appropriate extrapolation as proposed in refs 27 and 28. Therefore, the next series of experiments was devoted to measurements of particle deposition kinetics for longer times where it becomes nonlinear because of surface blocking effects. Such measurements were previously done for the silver particle deposition on PAH modified mica by using the AFM, SEM, and streaming potential methods.15,35,40 Analogous kinetic runs obtained in this work using the QCM cell are presented in Figure 7 (pH 8, I = 10−3 M, and Q = 1.33 × 10−3 cm3 s−1).

θ = SgNs = 1.5 × 10−7

1 Γ ρs dp

(9)

πd2p/4

where Sg = is the geometrical cross-section area of particles. Thus, the blocking function in the case of nearly spherical particles is given by15 2

3

B(θ ) = (1 + a1θ ̅ + a 2θ ̅ + a3θ ̅ )(1 − θ ̅ )3

(10)

where θ̅ = θ/θmx is the normalized coverage of particles, θmx = 1.5 × 10−7(1/ρpdp)Γmx is the maximum coverage of particles depending on the ionic strength,37,38,41−43 and a1−a3 are the dimensionless coefficients equal to 0.812, 0.426, and 0.0717, respectively. The maximum coverage can be calculated by numerical modeling using a Monte Carlo type algorithm.38,43 These results can be approximated with a satisfactory accuracy by the analytical formula: θmx = θ∞

1 (1 + 2h*/dp)2

(11)

where θ∞ is the maximum coverage for hard (noninteracting) particles equal to 0.547 for spheres44,45 and h* is the effective interaction range characterizing the repulsive double-layer interactions among particles, which can be calculated from the formula37,38

Figure 7. Kinetics of silver particle deposition on PAH modified Au/ SiO2 sensor (pH 8, I = 10−3 M, and flow rate 1.33 × 10−3 cm3 s−1) determined by QCM for bulk suspension concentration of (1) 50 mg L−1, (2) 30 mg L−1, and (3) 10 mg L−1. The dashed lines show the theoretical results calculated from the eRSA model and the dasheddotted line the results calculated from the usual RSA model by neglecting the coupling between bulk and surface transport.

2h*/d p =

⎧ ⎡ ϕ ⎤⎫ ⎪ 1 ⎪ ϕ0 ⎢1 + 1 ln 0 ⎥⎬ ⎨ln − ln ⎪ κdp ⎩ 2ϕch κd p 2ϕch ⎥⎦⎪ ⎢⎣ ⎭

(12)

where κ is the reciprocal double-layer thickness, ϕ0 = 16πεdpϕp(kT/e)2 tanh2(ζpe/4kT) is the characteristic interaction energy of particles, and ϕch is the scaling interaction energy. Using the parameters pertinent to the silver particles (see Table 2), one can calculate from eqs 11 and 12 that for the ionic strength of 10−3 M θmx = 0.12 which corresponds to Γmx = 12.9 mg m−2. This agrees with the experimental result shown in Figure 7.

An interesting feature observed in Figure 7 is that for longer deposition time a stationary value of the silver particle coverage Γmx is attained that is independent of the bulk suspension concentration and the flow rate. For the ionic strength of 10−3 M, Γmx = 12.9 mg m−2. As discussed in previous works,37,38,40 2993

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Langmuir If the blocking function is known, the particle deposition kinetics can be calculated by numerically integrating the governing mass transport equation discussed in ref 15

∫θ

θ

0

(ka − kc)B(θ′) + kc dθ′ = kct SgkanbB(θ′) − kdθ′

(13)

where ka and kd are the adsorption and desorption kinetic constants and nb = (6 × 10−6)/(πdp3ρp)cp is the bulk number concentration of the silver particle suspension. It should be mentioned that eq 13 represents a general solution of the convective mass transfer kinetics valid for arbitrary solute where the coupling between the bulk transport and surface transport steps is considered in an exact way. If the coupling is neglected, by assuming that ka = kc, eq 13 reduces to the simpler dependence:

∫θ

θ

0

1 dθ ′ = t SgkanbB(θ′) − kdθ′

Figure 8. Kinetics of silver particle deposition on PAH modified Au/ SiO2 sensor (pH 8, bulk suspension concentration 50 mg L−1, and flow rate 2.66 × 10−3 cm3 s−1) determined by QCM for various ionic strengths: (1) 10−2 M, (2) 10−3 M, and (3) 10−4 M. The lines show the results obtained for the convection/diffusion deposition regime. The full points show the SEM experimental results obtained for the mica/PAH system, and the hollow points show the SEM results for the Au/SiO2 system.

(14)

Assuming negligible desorption, i.e., kd = 0, the kinetics of particle deposition was theoretically calculated from eq 13 using a four-point numerical integration algorithm. As can be seen in Figure 7, these theoretical results calculated without using adjustable parameters adequately reflect the experimental data, including the maximum coverage that matched (within the error bounds) the experimental value of θ = 0.12 (Γ = 12.9 mg L−1). A slight deviation of experimental data from the theoretical model observed for intermediate adsorption time can be interpreted as due to the nonuniformity of mass transfer rate over the entire crystal area. The agreement of experimental and theoretical maximum coverage also confirms that the hydration of the silver particle monolayer for this higher coverage range is negligible. On the other hand, the theoretical results derived from eq 14 by neglecting coupling (shown as dashed-dotted lines in Figure 7) significantly underestimate the experimental data. This indicates that the eRSA model is more appropriate for predicting particle deposition kinetics. It is so because in this model a significant correlation between particle deposition attempts it considered in an exact way whereas in the classical RSA model no correlation is assumed. The validity of the eRSA model was also confirmed in the series of kinetic measurements performed for various ionic strengths (pH 8, bulk suspension concentration 50 mg L−1, and flow rate 2.66 × 10−3 cm3 s−1). The primary goal of this measurement was to quantitatively evaluate the role of the electrostatic interactions in particle deposition for the higher coverage range. The results collected in Figure 8 clearly indicate that the maximum coverage attained for the longer time of deposition θmx abruptly increases with ionic strength from 0.08 for I = 10−4 M to 0.31 for I = 10−2 M. It should be mentioned that in order to increase the precision of the maximum coverage determination a combined flow/diffusion deposition procedure was applied. Accordingly, the flow in the QCM cell was switched off after some time, and the particle deposition was continued under pure diffusion. This eliminates the possibility of the shear-induced surface aggregation of particles that could lead to multilayer coverage formation. As can be seen in Figure 8, the maximum coverage obtained in our in situ QCM studies agree with the ex situ SEM results obtained for both the Au/SiO2/PAH substrate and the mica/ PAH substrate. It is interesting to mention that the

experimental maximum coverages quantitatively agree with the theoretical results calculated from eqs 11 and 12. The strong increase in the maximum coverage with ionic strength, predicted by the eRSA model and experimentally confirmed, suggests that the blocking effects in the silver particle deposition processes are governed by the lateral electrostatic interactions among particles.

4. CONCLUSIONS It was shown that deposition kinetics of silver nanoparticles can be studied in situ using the QCM method. Because of low dissipation, the Sauerbrey equation could be used for calculating the mass per unit area (coverage). The results obtained for various bulk suspension concentrations, flow rates, and ionic strength proved that particle deposition is governed by the bulk mass transfer step which results in a linear increase of the coverage with the time. It was also confirmed that the hydration of the silver monolayers is negligible because of their high density and crystallinity. This allowed one to derive a universal formula, expressed by eq 5, that can be used for calculating the dry mass of any solute (e.g., proteins) if its diffusion coefficient is known. In this way the hydration (trapped water) parameter can be calculated in an ab initio manner without using additional reflectometric or ellipsometric measurements. This conclusion was further confirmed by kinetic measurements performed for longer times and for various ionic strengths that were adequately interpreted in terms of the extended RSA model. A significant increase in the maximum coverage with ionic strength was observed in these experiments that was interpreted as due to the decreasing range of the electrostatic interactions among deposited particles. Additionally, the QCM data matched the ex situ SEM results indicating that the monolayer hydration was also negligible for higher coverage range. The obtained results derived for the model silver nanoparticle system can be exploited as reference data for the interpretation of protein adsorption kinetics where the dry mass 2994

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is needed in order to assess the extent of the hydration and determine the true monolayer thickness.

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ASSOCIATED CONTENT

S Supporting Information *

Physicochemical characteristics of silver particles, QCM sensors, and QCM measurements. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*(Z.A.) Phone +48126395134; fax +48124251923; e-mail [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was financially supported by the EU Human Capital Operation Program, Polish Project No. POKL 04.0101-00-434/ 08-00, and supported by the NCN Grant UMO-2012/07/B/ ST4/00559 and partially financed from CMST COST Action CM1101. The authors are grateful to Dr Małgorzata Zimowska for the assistance in the SEM measurements. Also, stimulating discussions with Prof. Fouzia Boulmedais and Prof. Pierre Schaaf on the QCM measurements and interpretation of the results are acknowledged.

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REFERENCES

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