Silver particle monolayers — Formation, stability, applications

    Silver particle monolayers - formation, stability, applications Magdalena O´cwieja, Zbigniew Adamczyk, Maria Morga, Katarzyna Kubiak PII: DOI: Reference:

S0001-8686(14)00215-2 doi: 10.1016/j.cis.2014.07.001 CIS 1455

To appear in:

Advances in Colloid and Interface Science

Received date: Revised date: Accepted date:

30 May 2014 30 June 2014 1 July 2014

Please cite this article as: O´cwieja Magdalena, Adamczyk Zbigniew, Morga Maria, Kubiak Katarzyna, Silver particle monolayers - formation, stability, applications, Advances in Colloid and Interface Science (2014), doi: 10.1016/j.cis.2014.07.001

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Silver particle monolayers - formation, stability, applications

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Magdalena Oćwieja, Zbigniew Adamczyk*, Maria Morga, Katarzyna Kubiak

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E-mail addresses: [email protected] (M. Oćwieja), [email protected], (Z. Adamczyk), [email protected] (M. Morga), [email protected] (K. Kubiak)

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Jerzy Haber Institute of Catalysis and Surface Chemistry, Polish Academy of Sciences, Niezapominajek 8, 30 - 239 Cracow, Poland.

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*Corresponding author Zbigniew Adamczyk Jerzy Haber Institute of Catalysis and Surface Chemistry Polish Academy of Sciences Niezapominajek 8 30-239 Krakow, Poland phone: +48126395134 fax: +48124251923 e-mail: [email protected]

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ACCEPTED MANUSCRIPT Contents Abstract 1. Introduction

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2. Nanoparticle synthesis methods

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3. Experimental techniques and methods

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4. Silver particle monolayers and films 4.1. Bulk particle characteristics 4.2. Kinetics of particle deposition

4.3. Stability of monolayers and particle release kinetics

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4.4. Particle bilayers 5. Applications

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6. Concluding remarks Acknowledgments

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References

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Graphical abstract

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ABSTRACT

Keywords:

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Formation of silver particle monolayers at solid substrates in self-assembly processes is thoroughly reviewed. Initially, various silver nanoparticle synthesis routs are discussed with the emphasis focused on the chemical reduction in aqueous media. Subsequently, the main experimental methods aimed at bulk suspension characterization are critically reviewed by pointing out their advantages and limitations. Also, various methods enabling the in situ studies of particle deposition and release kinetics, especially the streaming potential method are discussed. In the next section, experimental data are invoked illustrating the most important features of particle monolayer formation, in particular, the role of bulk suspension concentration, particle size, ionic strength, temperature and pH. Afterward, the stability of monolayers and particle release kinetics are extensively discussed. The results obtained by the ex situ AFM/SEM imaging of particles are compared with the in situ streaming potential measurements. An equivalency of both methods is demonstrated, especially in respect to the binding energy determination. It is shown that these experimental results can be adequately interpreted in terms of the hybrid theoretical approach that combines the bulk transport step with the surface blocking effects derived from the random sequential adsorption model. It is also concluded that the particle release kinetics is governed by the discrete electrostatic interactions among ion pairs on particle and substrate surfaces. The classical theories based on the mean-field (averaged) zeta potential concept proved inadequate. Using the ion pair concept the minor dependence of the binding energy of particles on particle size, ionic strength, pH and temperature is properly explained. The final sections of this review is devoted the application of silver nanoparticles and their monolayers in medicine, analytical chemistry and catalysis.

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Applications of silver nanoparticles, chemical synthesis of silver particles, deposition of silver particles, kinetic of silver particle deposition, modeling of silver particle deposition, release kinetics of silver particles, silver particle monolayers and bilayers, synthesis methods of silver particles

Highlights:  Formation of silver particle monolayers at solid substrates is self-assembly processes is thoroughly reviewed.  The essential role of electrostatic interactions described in terms of the ion pair concept is proven.  The validity of the hybrid phenomenological approach combined with the random sequential adsorption model is confirmed.

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ACCEPTED MANUSCRIPT 1. Introduction

Silver nanoparticles and their monolayers on solid substrates have a wide spectrum of

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practical applications in biology [1-3], medicine [4-7], chemical analysis [8-10], catalysis

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[11-14], electronic [15,16], cosmetic [17], pharmaceutical and textile industry [18-20], etc.

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Specifically, thanks to the antibacterial [21-25], fungicidal [26-28] and virucidal [29-30] properties, silver nanoparticles play an important role in medicine, where they are used in the form of liquids or aerosols for decontamination and disinfection [3,31]. They also

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constitute an important ingredient in creams and dental amalgamat [4,32,33]. They are also used to modify surfaces of various materials, in particular fibers or polymer [34-35], applied in manifold consumer products such as clothes, laboratory and surgical gowns, dressing

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bandages [36,37], etc. Excellent antimicrobial activity of silver particles make them applicable in the products of daily use such as cosmetics, toothpaste, soaps, detergents, paints, foil, food containers, textiles and water decontaminants [17, 38-39].

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Silver nanoparticle serve as analytical sensors in SERS spectroscopy [40-43], metal-

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enhanced fluorescence (MEF) [9,44], in immunosensing of biological probes and markers [45,46].

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The high surface area and surface energy make the silver nanoparticles a valuable material for catalysis with a particular importance in selective oxidation reactions [12,47,48] and hydrogenation of organic compounds [49,50]. The literature describes the catalytic

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activity of the silver nanoparticles in selective oxidation of ethylene to ethylene oxide [47], styrene oxidation [12] oxidation of methanol to formaldehyde [49], the formation of aminophenols in nitrophenols reduction [11,50], hydrogenation of acrolein [13], low temperature oxidation of ammonia to nitrogen [51], obtaining glycol by the oxidative dehydrogenation of glyoxal [52], synthesis of quinoline derivatives or selective reduction of nitrogen oxides to nitrogen [53]. Silver nanoparticles are also used in electronics as the conductive paths [15], a data storage devices, photonic and antireflective materials, optical fiber, transistors, electronic connectors or integral capacitors [54,55]. In response to the extensive range of practical applications, a plethora of experimental works have been published in the literature focused on silver particle synthesis [56-65], monolayer and film formation [66-72], practical applications as biocidal materials [3,4-7,21-

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ACCEPTED MANUSCRIPT 30] in chemical analysis [8-10, 45,46] and catalysis [11-14,47-53]. These issues have been discussed in the review works [23,73-75]. However, despite their significance, the more fundamental aspects of silver particle

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deposition mechanisms on solid substrates, especially the kinetics of these processes, have not

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been considered in these reviews. This is so, because only recently systematic studies on silver particle monolayer and bilayer formation, stability and particle release kinetics,

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performed by direct experimental methods have been published [76-81]. Therefore, in this work, attention is focused on a critical review of the kinetic aspects of silver nanoparticle deposition from stable suspensions leading to self-assembly monolayers of well controlled

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density and structure. Such monolayers are subsequently used for deriving information about particle release kinetics and mechanisms, quantitatively analyzed in term of theoretical

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approaches. This allows one to determine the binding energy of particles under various physicochemical parameters such as particle size, charge, ionic strength, pH and temperature that is significant for basic sciences and for practice.

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Accordingly, in the first part of this review, silver nanoparticle synthesis methods are

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briefly discussed with the emphasis focused on the chemical reduction methods in aqueous media. Subsequently, the main experimental methods aimed at bulk suspension

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characterization are critically reviewed by pointing out their advantages and limitations. Also, specialized methods devoted to particle monolayer and film studies including the in situ electrokinetic methods are discussed. In the next section, selected experimental data are invoked illustrating the most important features of particle monolayer formation, in particular,

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the role of bulk suspension concentration, particle size, ionic strength, pH and the surface modification of the substrate by cationic polyelectrolytes. Afterward, the stability of monolayers and particle release kinetics are extensively discussed. The final sections of this review are devoted to a particle bilayers and applications of silver particles in medicine, analytical chemistry and in catalysis. Rather to present a detailed collection of experimental data, the review is focused on proving the unity of particle deposition phenomena occurring in various scales, quantitatively interpreted in terms of phenomenological theoretical models by considering the decisive role of electrostatic interactions.

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ACCEPTED MANUSCRIPT 2.Nanoparticle synthesis methods Silver nanoparticle synthesis is usually based on the dispersion and condensation

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techniques that corresponds in recent nomenclature to the top-down methods, which consists

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in the miniaturization of larger structures and the bottom-up methods exploiting the aggregation and self-organization processes. The analysis of literature data indicates that the

nanoparticles [82]. However,

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bottom-up technique are predominantly used for the synthesis of silver and other noble metal the top-down techniques

exploiting physical dispersion

methods also exhibit some advantages. The most extensively used is the method of laser

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ablation in liquid phases developed by Cotton-Chumanov [56] and Fojtik [57]. Nanoparticles are produced from silver plates immersed in an aqueous solutions upon illumination by high-

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energy laser beam. The mechanisms of nanoparticle formation proposed by Mafune [83,84] postulates that the metal plate absorbs a large part of the laser impulse energy, that results in the formation of a hot plasma containing a high concentration of silver atoms and ions. The

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liquid serves as a cooling medium, therefore, its physicochemical properties affect the rate of

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nanoparticle formation, their shape, size and polydispersity. The laser ablation technique in liquid phases was developed in response to the demand for colloidal suspension free of

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contaminants and pollutions stemming from chemical compounds used during the synthesis. Hence, the preparation of chemically pure silver nanoparticles [85] is one of the most important advantage of this technique. Additionally, the ability to control the parameters of process such as wavelength of laser, energy and pulse duration allows one to obtain particles

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of well-defined sizes as small

polydispersity. Some studies show that the size of

nanoparticles increases linearly with the power of the laser beam whereas the efficiency of the process decreases for shorter impulse time [83]. Additionally, the ablation efficiency depends on the electron beam focusing although the published reports are sometimes contradictory [56,86,87] and require additional validation [88]. Most of the works devoted to the use of the laser ablation technique indicate that the stability of the silver nanoparticle suspensions depends on the presence of additional chemical compounds, which are dissolved in cooling medium. For example, the work of Chumanov [56] showed that the silver nanoparticles can be obtained in pure water however their polydispersity remains high. Therefore, the preparation of silver suspensions using laser ablation is often conducted in organic solvents. As demonstrated in various works [85,89-93], the appropriate choice of organic solvents allows one to modify the surface properties of obtained nanoparticles [88,93]. 7

ACCEPTED MANUSCRIPT Although the preparation of silver nanoparticles in the laser ablation process is improved, the influence of the physicochemical properties of solvents on size distribution and stability of nanoparticles is still unexplained [85,88,94]. Additionally, the lack of efficient

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stabilization of nanoparticles in pure water forces one to carry out the laser ablation processes

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in the presence of high-molecular weight compounds and surfactants [83,84,95,96] that diminishes the purity of obtained suspensions. In this way, the main advantage of this method

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is neutralized.

Silver nanoparticles are also obtained by electrochemical methods. In the classical Bregid‘s method [58] colloidal suspensions are prepared using an electric arc that is

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constructed from two silver electrodes immersed into a liquid solution [97,98]. In order to prepare nanoparticle suspensions, the electrodes are alternately connected and disconnected

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from a high voltage source that causes evaporation of the metal. Various forms of metal atoms and clusters obtained in the electric arc undergo nucleation and aggregation that results in formation of nanoparticles. This method of synthesis, analogously to laser ablation, can be

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conducted in aqueous as well as in organic solvents [97,98]. However, major limitations of

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this process is a low yield and inability to control morphology and size of nanoparticles [99]. Another electrochemical method was developed by Reetz [59]. where nanoparticles

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are produced in a special electrochemical cell (electrolyzer). The silver anode of this cell is gradually dissolved under the applied external electric field and the silver ions are reduced at the platinum cathode forming atoms and clusters which subsequently condense to nanoparticles. The advantage of this method is that it works at room temperature. Moreover,

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the changes in the current density enable one to control the size of nanoparticles [100]. An important issue is the selection of aprotic solvent that prevents the passivation of the electrodes. Additionally, the solvent used in the process should contain a small amount of auxiliary electrolyte that can improved the stability of colloidal system but it should not cause the deposition of nanoparticles on the cathode. As demonstrated in the work of RodriguezSancheza [101], if the type of electrolyte and the cathode material is improperly chosen, the electrochemical process can be completely inhibited. The problems with the electroreduction of silver ions produced at the anode as well as undesirable passivation of electrodes were eliminated in the improved method developed by Yin et al. [102]. The silver anode was replaced by the platinum electrode, the solution used in the electrolyzer contained silver nitrate and poly(vinylpirolidone) (PVP) and the system was sonicated or mixed during the process. It was shown that the organic stabilizer (PVP) played a major role in the process that could be carried out in aprotic solvent and the deposition of 8

ACCEPTED MANUSCRIPT silver at the electrode was significantly decreased. Recently, the methodology of silver nanoparticle preparation developed by Yin et al. [102] with various modification is widely used [103-106].

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Among the bottom-up approaches of major practical significance one should mention

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the sonochemical, fotochemical and chemical methods (see Table 1). In these processes, the formation of silver nanoparticles proceeds via condensation of reduced forms of silver

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stemming from organic or inorganic silver salts or complexes [107,108].

Table 1. Various methods of silver nanoparticle preparation. After Ref. 108. Energy source

electrochemical

electrons

sonochemical

ultrasonic wave

photochemical

ultrafiolet, visible light

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microwaves

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radiolysis

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Preparation method

reducing agent reducing agent

direct reduction by applying a negative potential Radicals (?) radicals, reductive elimination of ligands from the metal precursors hydrated electrons, radicals assisted by homogenous heating Chemical potential difference

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chemical

gamma radiation

Reduction mechanism

The first mentioning about using the ultrasound for the preparation of silver nanoparticles comes from the work of Nagat et al. [60]. A preliminary mechanism of this process was proposed by Salkar et al. [109]. It was shown that silver nanoparticles can be obtained by sonochemical reduction of an aqueous silver nitrate solution in an atmosphere of argon-hydrogen. Based on the electron paramagnetic resonance measurements (EPR) it was demonstrated that during the sonication in the aqueous solution, radicals stemming from a homolytic decomposition of water appear that could serve as the proper reducer of silver ions. It was also postulated that the hydrogen radicals that reduce silver ions, acquire a positive charge and transforme into hydronium ions. This assumption was supported by the observed decrease in pH of the reaction medium. However, by considering that the energy density of the ultrasound field is significantly too low for inducing the decomposition of water, the true reducing agents are probably the chemical contaminants dissolved in water.

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ACCEPTED MANUSCRIPT Accordingly, the role of the ultrasound field consists in increasing the temperature of the medium that accelerates the reduction of silver. Silver nanoparticles are also prepared using various types of electromagnetic radiation

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(see Table 2) that is referred to as photochemical methods. Silver ions are reduced by radicals

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generated from solvent molecules during irradiation [61]. Often ultraviolet and visible light is used during synthesis [61,110,111]. However, as shown in Refs. [61,110-113] the chemical

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structure of solvents and additional compounds used in the liquid medium play a significant role in these methods because they influenced the size distribution and the shape of nanoparticles.

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A special case of these methods is the radiolysis process where a high-energy gamma radiation is used. It was shown that as a results of water radiolysis, both radicals and hydrated

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electrons are produced, that are responsible for the reduction of silver ions [114-116].

ultraviolet

Wavelength

Photon energy

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visible light

Frequency

300 MHz-300 GHz

1 m -1 mm

1.24 μeV-1.24 meV

400 THz-789 THz

780 nm -380 nm

1.6 eV-3.4 eV

789 THz-30 PHz

380 nm- 10 nm

3.4 eV-124 eV

>60 EHz

< 5 pm

>250 keV

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Electromagnetic wave microwaves

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Table 2. Types of electromagnetic radiation used in the synthesis of silver nanoparticles.

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gamma radiation

The use of microwaves in silver nanoparticle synthesis is also quite common [62,117120]. Although microwave energy is too low to initiate chemical reactions, it causes a fast and efficient heating of the reaction mixture [121,122]. This is beneficial for the chemical reaction because the heat facilitates a homogenous nucleation and aggregation of nanoparticles significantly reducing the synthesis time [108]. In practice, the synthesis supported by microwave allows one to obtain monodisperse silver nanoparticles of controlled size and morphology [62,117-120]. It is worthwhile mentioning, that in the above methods of silver nanoparticle synthesis various chemical compounds are used [56-65,83-120]. These chemical reagents are introduced in the form of either silver precursors (soluble silver salts) and stabilizing agents that can play the role of reducing agents.

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ACCEPTED MANUSCRIPT In the first step of the chemical processes the silver ions, are reduced to neutral atoms according to the scheme: Ag+ + Red → Ag0 + Re+

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(1)

where: Red denotes the unreacted form of reducing agent and Re+ denotes the oxidized form

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of reducing agent.

The driving force of this reaction is the difference between the redox potentials (ΔE) of the two reactions: oxidation of the reducing agent and the reduction of silver ions [123].

dependence

neeE kT

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lnK e 

through the

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The values of ΔE is connected with the reaction equilibrium constant Ke

(2)

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where: ne is the number of electrons involved in the redox process, e is the elementary charge, k is the Boltzmann‘s constant, T is the absolute temperature.

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As one can deduce from Eq.(2), at standard conditions, the silver ion reduction proceeds if ΔE is positive, which means that the reduction potential of the reducing agent

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should be more negative than the standard reduction potential of the metal precursor. In practice, this differences should be larger than 0.3-0.4 V [107], compared to the standard reduction potential of silver ions equal to 0.8 V. However, in the case of complexed forms of

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silver ΔE decreases, that is related to the stability constant of silver compounds (Table 3). As shown in Ref. [123], ΔE is also influenced by pH. The increase in pH by using the ammonia solution causes a decrease in the redox potential, which is related to formation of complex between silver ions and ammonia ligands (Table 3). Moreover, the change in pH affects the changes in the oxidation reactions. One of the commonly used reducing agents - sodium borohydride under alkaline conditions exhibit the standard potential E0= -1.24 V but at acidic conditions this value increases to -0.48 V. Therefore, one can expect that sodium borohydride acts more efficiently at high (basic) pHs [107].

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ACCEPTED MANUSCRIPT Table 3. Changes in the redox potential of Ag ion (Eo) as a result of complex formation. From Ref. [123]. - log Kβ

E0 [V]

Ag ++ e- → Ag0

-

+0.80

[Ag(NH3)2] ++ e- → Ag0 -

0

7.2

[Ag(SO3)2] + e → Ag + 2SO3

2-

8.7

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3-

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Redox system

+0.38 +0.29

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[AgI4]2- +e-→ Ag0 + 4I-

15.0

-0.09

[Ag(CN)3] 2-+ e- → Ag0 +3CN-

22.2

-0.51

+0.01

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[Ag(S2O3)2]3-+ e- → Ag0 + 2S2O32-

The changes in the ΔE which can be realized by a proper selection of a silver salt,

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reducing agents and pH affect the properties of silver nanoparticles. As can be noticed by analyzing Eq. (2) when the value of ΔE increases, the reaction equilibrium is shifted toward the formation of large numbers of reduced silver forms that results in the appearance of

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nanoparticles of smaller sizes [123,124].

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The most frequently used precursors in the chemical reduction are silver nitrate [82,125-128], silver acetate [129,130], silver chlorate [110,131,132] and silver citrate [130,

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132,133]. The dominant use of silver nitrate is attributed to its low cost and chemical stability when compared to other types of silver salts or complexes [82]. As far as the reducers used in silver nanoparticle synthesis are considered, sodium

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borohydride is one of the most popular inorganic compound, which was introduced by Creighton in 1979 [125]. The mechanism of reduction using sodium borohydride was experimentally determined by van Hyning et al. [131,134,135]. Trisodium citrate is another commonly used compound [82,136] although it represents is weaker reducer than sodium borohydride. Therefore, the synthesis that is carried out in its presence should be performed at elevated temperatures. In the course of the reaction, citrate anions are converted to acetone dicarboxylic acid, that possess a structure prone to oxidation [107,137]. Among various types of organic reducers a special significance have compound whose chemical structure involves hydroxyl or carboxyl groups such as alcohols, aldehydes, carbohydrates or their derivatives such as tannins [138,139]. Methanol and ethanol [117,140,141] are the most frequently used alcohols. Formaldehyde is the most popular among aldehydes [142,143]. Interesting physichochemical properties of silver nanoparticles 12

ACCEPTED MANUSCRIPT are also obtained using Tollens process that exploits carbohydrates such as glucose, galactose,

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maltose and lactose as reducers [144,145].

Fig 1. TEM micrographs of spherical silver nanoparticles obtained using: a) laser ablation method ( prepared by 532 nm laser light with laser intensity of 120 mJ/pulse and irradiation time of 10 minutes, in the presence of dodecylsulfuric acid sodium salt (SDS), from Ref. [95], b) electrochemical synthesis (in the presence of poly(N-vinylpyrrolidone) (PVP), form Ref. [102], c) photochemical methods (γ-ray irradiation of acetic water solution of silver nitrate in the presence of chitosan, from Ref. [116]), d) chemical reduction method with use of trisodium citrate as reducing and stabilizing agent, from Ref. [64]. A variety of other reducers have been applied in the synthesis of silver nanoparticles [82,146-153] including such exotic substances as various plant extracts [154-156] pertinent to the green chemistry route. It should be mentioned that during the chemical preparation, the electrochemical potential of the redox reaction can be controlled via the type of used reagents, their stechiometric ratio, pH and the temperature. Hence, the synthesis of silver nanoparticles of 13

ACCEPTED MANUSCRIPT specific morphology and size is more efficient and reproducible in comparison with the topdown methods. According to commonly accepted mechanism [123], silver atoms generated during the

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reduction process aggregate to the form of clusters often referred to as embryos [123,124], see

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Fig. 2. These embryos of various sizes further grow and attain the stage of nuclei [123]. It is worthwhile mentioning that the size and the number of nuclei depend on many

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physicochemical parameters such as the precursor salt concentration, the temperature and the solvent viscosity. Starting from the nuclei phase one can distinguish three various mechanisms of their grow leading to nanoparticle formation (Fig. 2), [107,123] Growth by the diffusion of neutral atoms and their addition

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Growth by nuclei aggregation

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Autocatalytic growth of nuclei

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The growth mechanism influences the properties of the nanoparticles such as their shape, size, and crystallinity [107,123]. In a few works it was suggested that the diffusional

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growth causes formation of crystalline nanoparticles of regular shape and narrow size

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distributions [123,124]. On the other hand, the formation of nanoparticles by aggregation of nuclei promotes the formation of polycrystalline nanoparticles that exhibit irregular shapes

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and broad size distribution [107].

It is also established that the growth mechanisms can be controlled by using appropriate reducer composition [107,123]. For example, strong reducers such as sodium

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borohydride, cause a rapid reduction of silver ions resulting in a large number of nuclei. This in turn promotes formation of small nanoparticles. The reduction of silver ions is slow when a weak or mild reducer such as trisodium citrate or ascorbic acid are applied that results in creation of a small number of nuclei. Thus, their autocatalytical growth is more probable [107,157].

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Fig. 2. A schematic representation of the process of metallic nanoparticle formation. After Ref. [107].

Taking into account this regularities, a new two-step synthesis of silver nanoparticles was developed [158-160] referred to as the seed-mediated growth. It is based on the application of two types of reducers and two different precursors. In the first step of the synthesis small nanoparticles (seeds) are obtained by reduction of silver ions by the strong reducer. In the second step the obtained seed suspension is mixed with the fresh silver ionic solution and another reduction step is initialized by a weak reducer. It was demonstrated that by controlling the concentration of seeds and silver ions, nanoparticles of well-defined sizes and shapes can be prepared in simple and efficient way [160]. It is also interesting to mention that the solvents used in nanoparticle syntheses also play a significant role [82,107,161]. Their role can be so important that often the methods of the synthesis are classified according to the type of solvents used in the reaction, for example: 15

ACCEPTED MANUSCRIPT Synthesis in aqueous solutions

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Synthesis in organic solvents

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Preparation in reverse micelles

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Polyol synthesis

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Approximately 80% of silver nanoparticle syntheses are carried out using water as

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a dispersing medium (see Fig. 3) [82]. This is so because syntheses in organic solvents are more difficult to perform as a result of a lower solubility of silver salts and other chemical compounds, a lower dissociation degree of reagents and hampered transfer of electrons

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especially in media of lower polarity [124]. Therefore, the reactions are carried out in strongly polar solvents [162] that exhibit dual role as a dispersing medium and a reducer [163,164]. It is also demonstrated, that the problem of a limited solubility of the reducer in organic

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solvents can be solved introducing it into the reaction mixture as a gas, e.g., hydrogen [165].

Fig 3. The frequency distribution of solvents used in silver nanoparticle syntheses. From Ref. [82].

Silver nanoparticles of small sizes are also produced in two-phase systems of polar and non-polar liquids stabilized by amphiphilic compounds [166] and in reverse micelles [167-169]. Presently, two different methodologies of particle synthesis in reverse micelles are used [169]. The first is based on mixing of two emulsions of which one contains the dissolved 16

ACCEPTED MANUSCRIPT precursor of silver ions and the second solutions comprises the reducer

The second

methodology consists in supplying the reducer to the reaction mixture in the form of a gas. In these methods the final size of nanoparticles is practically the same as the size of micelles

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that in turn can be controlled by selection of organic solvents, surfactants and co-surfactants

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[169,170].

In contrast to the syntheses in reverse micelles, the polyol synthesis allows one to

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obtain nanoparticles of larger size [171]. This method, developed by Fiévet and coworkers is a simple and versatile route of preparing metal nanoparticle suspensions [172]. The reduction of inorganic silver salts is carried out at elevated temperatures (140o-160o C) in various polyol

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solutions, most often the polyethylene glycol [173-175] that plays also the role of a reducer. Additionally, polymers or surfactants are introduced to the reaction mixture in order to control

b)

c)

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a)

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the particle size and morphology [175,176].

Fig. 4 a) TEM micrograph and b), c) SEM images of silver nanoparticles obtained during a controlled polyol synthesis. From Refs. [172] and [175].

3. Experimental techniques and methods

The experimental methods discussed in this section can be divided into the bulk and surface oriented categories. The former techniques, such as X-ray diffraction (XRD), ultraviolet-visible spectroscopy (UV-Vis), infrared spectroscopy (IR), dynamic light scattering technique (DLS), microelectrophoresis, atomic absorption spectrometry (AAS) etc. are mainly used for the determination of basic physicochemical parameters of suspensions and particles such as the chemical composition [137,177], crystallinity [61,109, 116,117 178], concentration of suspension [78,79,179-181], electrokinetic charge (zeta potential) stability 17

ACCEPTED MANUSCRIPT and aggregation kinetics [79-81,179,182]. On the other hand, the electron microscopy can be used for determining the size, shape and morphology of particles under ex situ conditions after depositing the particles on a conductive support [57,62, 64,83,102,103,116,157,172 ].

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The most reliable information about the particle size and morphology can be derived

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from in situ atomic force microscopy (AFM) imaging in liquid environments [183,184]. An additional advantage of this technique consist in the fact that the monolayer density and

particle

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particle distribution can be directly determined. This allows one to perform reliable studies of deposition and desorption (release) kinetics from various substrates [66,68,77-

81,179]. Similar, in situ studies can also be performed by various gravimetric techniques,

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most often the quartz micro balance (QCM) as well as by ellipsometry and reflectometry [67,69,76,185]. However, in these cases, the information about monolayer density is averaged

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from macroscopic surface areas that often prohibits a quantitative analysis of the kinetics aspects of particle deposition. Additionally, the range of substrates that can be effectively studied using these techniques is rather limited, for example it is not possible to use mica.

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Other surface oriented techniques such as UV-Vis spectroscopy [67,68] infrared

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spectroscopy (IR) [72], X-ray photoelectron spectroscopy (XPS) [66], scanning electron microscopy (SEM) [187,188] also represent the ex situ methods, providing average

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information from macroscopic areas (with the exception of SEM). Precise in situ information is provided by the electrokinetic techniques, most often the streaming potential measurements that are particularly suitable for studying particle desorption (release) kinetics under diffusion or flowing conditions.

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In order to determine the structure and crystallinity degree of nanoparticles the X-ray diffraction is primarily used [61,109,116,117,178,189]. This is a powerful technique used to uniquely identify the crystalline phases of the materials and to measure their structural properties (strain state, grain size, epitaxy, phase composition, preferred orientation, and defect structure). XRD is also used to determine the thickness of thin films and multilayers, and atomic arrangements in amorphous materials (including polymers) and at interfaces [189]. The method is based on the use of diffraction patterns resulting from the interference strengthening of the X-rays reflected by lattice plane of an analyzed material. In X-ray diffraction, the Bragg-Brentano geometry is widely used for preferentially and randomly oriented polycrystalline films. The specimen is rotated at one half the angular velocity of the detector. Since the incident and diffracted X-rays make the same angle to the specimen surface, structural information is obtained only about (hkl) planes parallel to this surface. When the receiving slits, the specimen, and the focal point lie on a circle, the diffracted 18

ACCEPTED MANUSCRIPT X-rays are approximately focused on the receiving slits (parafocusing), which considerably improves the sensitivity. The advantages of XRD technique is its high selectivity, rapid measurement performance and relatively small quantity of material needed for analysis [189].

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Bulk properties of particle suspensions can also be characterized by using the UV-Vis

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spectroscopy that explores interactions between ultraviolet and visible electromagnetic radiation and a sample [186]. This is a useful technique providing both qualitative and

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quantitative information on the investigated materials. In the case of silver nanoparticles exhibiting plasmon resonance, this technique is commonly used to estimate the size, shape and polydispersity of colloidal particles. The absorption spectrum of spherical silver

NU

nanoparticles exhibits a characteristic maximum at the wavelength of λ = 400 nm, which can be hypso- or bato-chromatically shifted depending on the size of nanoparticles [186,190].

MA

The presence of additional peaks in the spectrum indicates the presence of non-spherical particles, and on the basis of the full width at half maximum of the spectrum the polydispersity of a nanoparticle suspension can be estimated. Changes in the intensity and

D

position of the absorption spectra are often used to determine the stability of nanoparticles

TE

under various physical factors such as the ionic strength of the electrolyte. The UV-Vis spectroscopy can also be applied to determine the concentration of the

CE P

particles in their suspensions. Beer‘s law states that absorption is proportional to the concentration of absorbing species (this is only true for dilute solutions) and Lambert‘s law states that the fraction of radiation absorbed is independent of the intensity of the radiation [186,190]. However, in practice, the UV-Vis measurements cannot be used as an absolute

AC

technique for determining the suspension concentration since they require a proper calibration using more specialized methods such as atomic adsorption spectrometry and densitometry [77,78]. The densitometry method is especially suitable for silver particle suspensions because of their high density ρp of 10.49 g cm-3 that allows one to perform precise measurements for the concentration range of 10 mg L-1 and above [78,79,179]. In this method, the oscillation period of a U-tube shaped cell containing the suspension is electronically measured. The period is monotonically parabolic function of the mass of the suspension. Knowing the mass and the volume of the cell one can calculate the density of suspension (ρs) [78,79,179]. An additional density measurement is performed for the supernatant (effluent) solution acquire, e.g., by a membrane filtration or centrifugation. Knowing both densities one can calculate the effective bulk concentration of the suspension from the following formula:

19

ACCEPTED MANUSCRIPT w

 p   s  t   s   p  t 

(3)

T

where ρt is density of the supernatant (effluent) solution.

IP

The in situ information about particle size, suspension stability and aggregation kinetics can be derived from the DLS measurements [77-79,179,183]. It is a non-invasive,

SC R

well-established technique for measuring the diffusion coefficient of molecules and particles. The principle of this method is based on the fact that particles in a dispersion undergo a random Brownian motion resulting from collisions with atoms or molecules [183,191]. Due

NU

to Brownian motion, the scattered light fluctuates at a rate that is dependent on diffusion coefficient of species under the investigation. The change in the scattered light intensity

MA

fluctuations is associated with both the amount of the particles and their diffusion coefficients. The analysis of these intensity fluctuations yields the autocorrelation function, which enables one to determine the diffusion coefficient D [191]. Knowing D one can calculate the

kT 3 D

(4)

CE P

dH 

TE

D

hydrodynamic diameter of the particle from the Stokes-Einstein relationship:

where, η is the dynamic viscosity of the solvent and dH is the hydrodynamic diameter. As can be deduced from Eq.(4) the hydrodynamic diameter can be interpreted as the

AC

diameter of an equivalent sphere having the same diffusion coefficient as the tested particle. Obviously, for spherical particles, the hydrodynamic diameter corresponds to its geometrical size. However, for particles of an anisotropic shape, especially elongated

ones, the

hydrodynamic diameter does not correspond to any of their geometrical dimensions. As a result, the shape of anisotropic particles cannot be deduced from the DLS measurements alone that is one of the main disadvantages of this technique. Moreover, the average particle size is calculated using a cumulative analysis that is well suited only for particles with a narrow and monomodal distributions, hence the DLS technique is not suitable for performing measurements for polydisperse (bimodal) systems [183,191]. Also the measurements are precise enough for a narrow range of the suspension bulk concentration (for silver nanoparticles typically 20 - 100 mg L-1) that is highly dependent on the system properties, mainly the particle size. Both too low and too high suspension concentration may lead to considerable errors in the measured diffusion coefficient. 20

ACCEPTED MANUSCRIPT On the other hand, the electric properties of particles, especially their electrokinetic charge that is a direct measure of their interaction among themselves and with substrate surfaces can be determined by the micro-electrophoretic technique [192,193]. In this method

T

the average translation velocity of particles U is measured in an uniform electric field E

IP

externally applied via two electrode system. In modern devices the laser Doppler velocitometry technique is used for determining U . Knowing the velocity one can calculate

U E

(5)

NU

e 

SC R

the electrophoretic mobility of particles μe from the constitutive dependence [192,193]:

Although the electrophoretic mobility is one of few parameters directly accessible experimentally, it cannot be used for quantitatively calculating the interactions of particles.

MA

The proper parameter for calculating the interactions is the zeta potential of particles  p defined as the electric potential in the slip plane. The zeta potential is calculated using the

   f ( d p ) e

CE P

p 

TE

D

Henry‘s equation:

(6)

where  is the electric permittivity of the solution, f(dp ) is the Henry‘s function of the

AC

dimensionless parameter, dp, 

1

 kT   2   2e I 

1/ 2

is the thickness of the electric double layer and

I is the ionic strength.

For spherical particles the f  d H  function can be analytically expressed for  d H  1 , in the following form [192] 2 3 4 4 6              e  t   2 f    [1  5    dt ] e 3 16 48 96  8 96   t

where  

(7)

 dH . 2

It should be noted that in the limit of   0 , the f  d H  function approaches 2/3. On the other hand, for  d H  1 the analytical approximation for f  d H  becomes 21

ACCEPTED MANUSCRIPT

3





25

 

2



220

 

(8)

3

T

f    1 

IP

Accordingly, for    , the f  d H  function approaches unity that corresponds to the

SC R

Smoluchowski‘s limit.

As far as surface characterization techniques are concerned, the AFM and various electron microscopy methods are the most widely used [66,68,70-72, 76-79,183,187,188,194]. Atomic force microscopy (AFM) is a technique that can be used for determining both

NU

the particle size, morphology [183] and the density of monolayers [184]. It can reflect the shape of a sample in three dimensions, down to the nanometer scale. Moreover, the advantage

MA

of this technique is the ability to examine non-conductive, soft, and live biological materials without sputtering a subsidiary layer. The sample is studied usually in air, but it can also be examined in liquid environments as well as under vacuum [184]. Atomic force microscopy

D

belongs to the family of scanning probe microscopy (SPM), which make use of various

TE

interactions between a tip and examined sample. The force, which occurs between the tip and the surface of a sample, make the cantilever twisting or bending. A four segment photodiode

CE P

detects the deflection of the cantilever through a laser beam focused on, and reflected from the rear of the cantilever. A computer processes the electrical differential signal of a photodiode obtained from each point of the surface and generates a feedback signal for the

AC

piezo-scanner to maintain a constant force on the tip. This information is transferred into a topographic image of the surface [184]. However, the disadvantage of the AFM method is that it often works under dry conditions and the sample imaging is limited to low coverage, depending on size of the particles. The electron microscopy, especially the transmission electron microscopy (TEM) can also be used for precisely determining the size and shape of the particles under ex situ (vacuum) conditions. In the TEM microscopy the electrons are passing through a thin specimen. The analysis provides information about both the material crystallography and its chemical composition [187,188]. It is also the best way to imagine the particles in order to determine their dimension and their size distribution. The design of conventional electron microscopes imposes a number of restrictions on the nature of the specimens that can be imaged. One of the most obvious drawbacks of the method is that the specimens have to be 22

ACCEPTED MANUSCRIPT stable under vacuum conditions. Additionally, the sample cannot be sensitive to electron beam (cannot be thermally- or photon-sensitive) [187,188,194]. Therefore, a more flexible in determining the particle monolayer density and

T

composition is the scanning electron microscopy (SEM). Contrary to TEM (transmission

IP

electron microscopy) that furnishes information about the internal structure of thin specimens, the SEM technique is primarily used to study the surface, or near surface structure of bulk

SC R

specimens. The electron source, usually the tungsten thermionic emission or field emission gun is used. In the SEM technique, the electrons are accelerated to an energy usually between 1 keV to 30 keV. Moreover, for bilayer formation investigation, the backscattered electrons

NU

are especially useful to determine the coverage of nanoparticles in both layers [187,188]. Backscatter electron signals provide image contrast as a function of elemental

MA

composition (COMPO mode), as well as, surface topography (TOPO mode). However, since the backscattered electrons are high-energy electrons, they can escape from much deeper than secondary electrons, thus the surface topography is not as accurately resolved as for

D

secondary electron imaging. The production efficiency for backscattered electrons is

TE

proportional to the sample material' s mean atomic number, which results in image contrast as a function of composition. Thus, higher atomic number material appear brighter than low

CE P

atomic number material in a backscattered electron image. The optimum resolution for backscattered electron imaging is about 5.5 nm [188]. However, a major disadvantage of the SEM technique consist in the fact that the monolayers made of poorly conductive particles have usually be coated by thin films of

AC

a conductive material, such as Cr, Au, Pt, Pd. This prevents charge build up on the specimen, and the associated image distortion [187,188]. Particle deposition processes are often studied using indirect experimental methods such as ellipsometry and reflectometry , total internal reflection fluorescence, UV-Vis spectroscopy, or quartz crystal micro-gravimetry (QCM) [66-72]. However, a disadvantage of these methods is that they are only sensitive for higher coverage, preferably close to the saturation coverage. Additionally, they cannot furnish any reliable information about the electrical state of the monolayers that is often important in view of their applications as efficient substrates for protein immobilization pertinent to biosensing applications. Such measurements can be performed using the electrokinetic techniques, most often the streaming potential method. It is the most convenient for in situ studies of nanoparticle mono- and multilayer formation at solid substrates and for particle desorption kinetics [80,81,179,195-197]. The main advantage of the streaming potential method is its high 23

ACCEPTED MANUSCRIPT sensitivity that allows one to perform measurements at low monolayer density [197,198]. Additionally, particle deposition and desorption experiments can be performed under wellcontrolled transport conditions using channels or capillaries made of the appropriate substrate

T

[195-198]. Because of the macroscopic fluid motion in the cell (usually driven by the

adjacent to solid/electrolyte interface appears.

This flux

IP

hydrostatic pressure difference) a convective flux of ions from the thin double-layer region is called the streaming current

SC R

[198,199]. This causes a charge separation that leads to the appearance of an electrostatic

potential difference Es (measured by using a pair of reversible electrodes) called the streaming potential proportional to the streaming current. The proportionality coefficient is

NU

the overall resistance of the cell Re that depends on the electric resistance of the medium, the geometry of the cell and the surface conductivity. Usually, in order to increase the precision

MA

of the measurements, the dependence of Es on the hydrostatic pressure difference Pc is determined. Knowing this quantity the zeta potential of the surface is calculated from the

TE

 Es ( )  Re Pc

(9)

CE P

c 

D

Smoluchowski‘s equation [195-199]:

It should be mentioned that for higher electrolyte concentration (of the order of 0.01 M and above) the surface conductivity plays a minor role and the overall electrical resistance is

AC

dominated by the bulk electrolyte resistance. This considerably increases the precision of the measurements.

The presence of the particles deposited on the surface of the cell perturbs the streaming potential in two ways [198-199] (i) by damping the local fluid flow in their vicinity and (ii) by supplying an additional contribution of ion flux from the double-layer region adjacent to their surfaces. These effects are accounted for by introducing the correction functions Fi , Fp calculated in Refs. [80,81,198] by numerically solving the hydrodynamic flow near surfaces covered by particles using the multipole expansion method. The constitutive dependence derived in this way has the form:

 ( )  Fi ( ) i  Fp ( ) p

(10)

24

ACCEPTED MANUSCRIPT where ζ(  ) is the zeta potential of the particle covered substrate calculated from the Smoluchowski‘s equation, Eq.(9), ζi is the zeta potential of bare surface, ζp is the zeta potential of particles in the bulk derived from micro-electrophoretic measurements,  = Np Sg

T

is the particle coverage, Sg is the characteristic cross-section area of the particle and Np is the

IP

surface concentration of particles.

The dimensionless functions Fi   , Fp   depend in the general case not only on the

SC R

particle coverage but also on the electrical double-layer thickness κ -1.

As shown in Ref. [198] the Fi   , Fp   functions can be approximated by the

NU

following analytical expressions:



1  1 e 2

2C p Fi



(11)

D

Fp ( ) 

MA

Fi ( )  eCi

TE

In the case of thin double layers, the Ci, Cp constants approach the limiting values of

CE P

Ci0  10.2 and C p0  6.51 , respectively [198].

From Eqs.(10-11) one can deduce that the limiting zeta potential for a high coverage

  (12)

p 2

AC

of particles is given by the simple relationship:

Additionally, by realizing that the Ci0 and C p0 constants assume values much higher than unity, one can predict that the zeta potential of interfaces covered by particles is sensitive to their coverage, especially if they bear opposite charge. As a result, the streaming potential technique allows for in situ measurements of particle or macromolecule coverage with a precision of 1% of a monolayer. Such a high precision is unprecedented by any other method. The streaming potential measurements allow one to determine the coverage of particles in situ just by measuring in a simple way the electric potential difference. In the general case the coverage can be precisely calculated by a numerical inversion of Eq.(10) if 25

ACCEPTED MANUSCRIPT the zeta potential of the bare substrate and the bulk zeta potential of particles are known. It was shown in Ref. [80,81] that a sufficient precision is attained by an iterative analytical

IP

1   p ln Ci  i   p

(13)

SC R

 

T

inversion of Eq.(10) that results in the approximate formula:

However, it should be mentioned that the precision of the particle coverage determination from Eq.(13) via the streaming potential measurements remains high for

NU

particle coverage below 0.3. Another disadvantage of this method is that the range of substrate surfaces that can be effectively studied is rather limited. Hence, the streaming

MA

potential method is particularly suited for determining particle desorption (release) kinetics for mica, silicon, quartz, glass and similar surfaces.

D

4. Particle monolayers and films

TE

4.1. Bulk particle characteristic

CE P

The citrate stabilized silver sols show a characteristic yellow colour that is attributed to the surface plasmon excitation of free electrons in the metallic particles [186]. This is quantitatively confirmed by the extinction spectrum in the UV-Vis region, shown in Fig. 5a.

AC

As can be observed, there appears a single, symmetric peak at the wavelength of 394 nm that suggests that the particles are nearly spherical and that the suspension is monodisperse with negligible content of aggregates [186]. Additionally, it was determined in Ref.[179] that the height of the peak monotonically increased with the concentration of the suspension independently measured by the densitometer, whereas its position remains fixed (Fig. 5b). By exploiting this finding, a robust method for determining the silver suspension density can be envisaged. This is confirmed by the experimental results plotted in Fig. 5a, where the dependence of the peak absorbance on the suspension concentration expressed in mg L-1 is plotted. As can be noticed, a linear dependence is obtained for a wide range of the suspension concentration (up to 25 mg L-1) that can be exploited for a more efficient determination of the unknown suspension concentration compared to the densitometry method.

26

ACCEPTED MANUSCRIPT The chemical composition (purity) and crystallinity of silver suspensions can be determined by XRD. An example of the relevant XRD pattern obtained in Ref.[200] for the same citrate stabilized silver suspension is shown in Fig. 6. The five strong Bragg reflections

T

appeared in 38.1o, 44.1 o, 64.6 o, 74.6 o, 81.6 o are pertaining to the (111), (200), (220), (311),

a)

b)

4

3

MA

2

A

A

3

NU

4 25 mgL-1 20 mgL-1 15 mgL-1 10 mgL-1 5 mgL-1 1 mgL-1

0 300

400

500

TE

D

1

600

 [nm]

SC R

consisted of pure silver and exhibit a polycrystalline structure.

IP

(222) planes of a face centered cubic (fcc) lattice of silver. This indicated that the particles

2

1

0 700

0

5

10

15

20

25

30

-1

cb [mg L ]

AC

CE P

Fig 5. a) The extinction spectra of the silver nanoparticle suspensions for various bulk concentrations. The peak of the maximum spectrum extinction occurs at λmx=394 nm. b) The dependence of the extinction at the maximum on the silver suspension bulk concentration (unpublished data).

5000

4000

CPS

3000

2000

1000

0 20

40

60

80

2 (degree)

Fig 6. The X-ray diffraction pattern of silver nanoparticles powder sample. The Bragg reflections appeared in 38.1o, 44.1 o, 64.6 o, 74.6 o, 81.6 o. The X-ray source was Cu Kα. From Ref. [200]. 27

ACCEPTED MANUSCRIPT

On the other hand, the shape, and size distribution of silver particles can be most directly determined by the TEM imaging. An typical micrograph showing citrate stabilized

T

silver particles deposited on a cupper grid is shown in Fig. 7a. As can be seen, the particles

IP

are nearly spherical and fairly monodisperse. In order to quantitatively determine their size distribution, a histogram is first obtained as described in Ref. [78,200]. Hence, using the TEM

SC R

micrographs, the deposited particle diameters are measured as the average value from two perpendicular directions and from the surface area of particles. In this way a plot of the frequency (probability) vs. the particle size is obtained that is presented in the form of

NU

a histogram in Fig. 7b. The average silver particle diameter of 15 nm with a standard

MA

deviation of 4 nm is obtained from the histogram shown in Fig. 7b [200].

D

a)

0.30

TE

0.25

0.20

Frequency

CE P AC

b)

0.15

0.10

0.05

0.00 0

10

20

30

40

d [nm]

Fig 7. a) The electron micrograph of silver particles, b) histogram of the size distribution of the particles. The average diameter of particles was equal to 15± 4 nm. From Ref [200]. Analogous size measurements for other types of silver particles are presented in Ref. [57,77,78,83,103,179,201]. Independently, the hydrodynamic size of silver particles is determined by the dynamic light scattering (DLS) where, originally diffusion coefficient is derived from the autocorrelation function of scattered light intensity fluctuations caused by the Browian motion

28

ACCEPTED MANUSCRIPT of particles. Knowing the diffusion coefficient, one can determine the hydrodynamic diameter of a particle from the Stokes-Einstein relationship (Eq. 4) From a physical point of view, the hydrodynamic diameter of a particle of an arbitrary

T

shape can be interpreted as the diameter of an equivalent sphere having the same

IP

hydrodynamic resistance coefficient (diffusion coefficient). Obviously, for spherical particles, the hydrodynamic diameter corresponds to its true physical size. The advantage of using the

SC R

hydrodynamic diameter over the diffusion coefficient is that it is independent of the temperature, viscosity and other parameters characterizing the solvent. Hence, it is a proper parameter for studying suspension aggregation processes. This is illustrated in Fig. 8 where

NU

the changes in the dependence of the normalized hydrodynamic diameter of the citrate stabilized silver suspensions on ionic strength (regulated by the addition of NaCl) is

MA

presented. The basic physicochemical characteristics of these suspensions are collected in Table 4 [79]. As can be seen in Fig. 8, the normalized hydrodynamic diameter remains constant for ionic strength up to 3x10-2 M indicating that the suspensions are stable.

D

A significant increase in the hydrodynamic diameters is only observed for ionic strength

TE

approaching 0.05 M, especially for the suspension of the largest particles size of 54 nm. Therefore, this can be treated the critical ionic strength indicating the beginning of

CE P

particle aggregation. In this way, using the routine DLS measurements, one to quite precisely

AC

determine the range of stability of silver suspensions.

55 50 45 40

dH/d0

35 30 25 20 15 10 5 0 10-5

10-4

10-3

10-2

10-1

I [M]

Fig. 8. The dependence of the normalized hydrodynamic diameter of silver nanoparticles dH /do on ionic strength, determined by DLS for silver suspension concentration 100 mg L-1, 29

ACCEPTED MANUSCRIPT

T

pH 5.8-6.0, T=298 K, the average particle size: (●) 15 nm, (□) 28 nm, (▲) 54 nm. The solid line represents non-linear fit of experimental data. After Ref. [79].

S1

S2

Value

Specific density [g cm-3]

10.49

NU

Property [unit]

17±5

29 ±5

58±6

Calculated from Eq.(4)

16±6

29 ±5

57±8

From size distribution obtained from AFM images

MA

9.08 x10-7

D AC

Particle size [nm]

Literature data [202]

1.75x10-7

TE

CE P

Hydrodynamic diameter [nm]

Remarks

Determined by DLS for T = 298 K, pH 5.8 I = 3x10-2 - 0.03 M NaCl

3.27x10-7

Diffusion coefficient [cm2 s-1]

S3

SC R

Sample

IP

Table 4. Physicochemical characteristics of silver nanoparticles. From Ref. [79].

Particle size [nm]

15±4

28 ±4

54±10

From size distributions obtained from TEM micrographs

Particle size [nm]

15.0

28.4

-

Calculated from adsorption kinetics [179,200]

Geometrical cross-section area Sg [nm2]

176

615

2290

Calculated from geometry

Total geometrical area 4Sg [nm2]

706

2460

9160

Calculated from geometry

Plasmon absorption maximum

394

400

425

Measured for pH 5.8 30

ACCEPTED MANUSCRIPT I = 10-2 M NaCl and silver sol concentration cb=20-25 mg L-1

[nm]

T

Except for the hydrodynamic diameter, the electrophoretic mobility, denoted by μe is

IP

another parameter of basic significance that characterizes the electrokinetic charge of particles

SC R

and consequently their interactions among themselves and with substrate surfaces. It is defined as the ratio of the translation velocity of particles to the strength of the electric field. It should be mentioned that the electrophoretic mobility is one of few parameters that can be directly measured by using micro-electrophoresis as above described. It is mainly controlled

NU

by the ionic strength, pH, temperature and to a lesser extent by the electrolyte composition. Therefore, this parameter is often determined in order to predict the stability range of silver

MA

particle suspensions and to interpret deposition kinetics measurements [79,179]. In Fig. 9 the dependence of electrophoretic mobility on pH obtained in these measurements is plotted for various silver particle sizes listed in Table 4. As can be observed, μe remains negative for all

D

ionic strengths and pHs up to 9. This indicates that the particles exhibit a negative

TE

electrokinetic charge. Moreover, is evident from the results shown In Fig. 9 that

the

electrophoretic mobility significantly increases with ionic strength for each size of

CE P

nanoparticles and decreases with pH. Thus, in the case of the 15 nm particles (suspension S1), for I = 10–4 M and – 2.7 μm cm (V s)-1 for

at pH 6.2, μe = – 3.52 μm cm (V s)-1

I =10-2 M. Analogously, at pH 9, μe = – 3.75 μm cm (V s)-1 for I = 10–4 and – 3.2 μm cm

AC

(V s)-1 for I =10-2 M. Similar trends are also observed for larger particle sizes of 28 and 54 nm (see Figs. 9 part b and part c).

a)

e [(mcm)(Vs)-1]

-2.0

-2.5

1 -3.0 2 3 -3.5

-4.0 2

3

4

5

6

7

8

9

10

pH

31

IP

T

ACCEPTED MANUSCRIPT

SC R

b)

-1.5

NU

-2.5 1 -3.0

2

-3.5

3

MA

em cm) (Vs)-1]

-2.0

-4.0

-4.5 3

4

5

6

7

9

10

TE

pH

c)

CE P

-1.0 -1.5 -2.0 -2.5 -3.0 -3.5

1

AC

e [mcm(Vs)-1]

8

D

2

-4.0 -4.5 -5.0 -5.5

2 3

-6.0 -6.5 2

3

4

5

6

7

8

9

10

pH

Fig 9. The dependence of the electrophoretic mobility of silver particles on pH for: a) particle size 15 nm (S1), b) particle size 28 nm (S2), and c) particle size 54 nm (S3). Curves 1 (■) , I = 10-2 M, curves 2 (●), I = 10-3 M, curves 3 (▲), I = 10-4 M. The solids lines denote a non-linear fits of experimental data (unpublished data). As can be noticed, for given ionic strength, the electrophoretic mobility significantly increases (becomes less negative) for particles of larger size.

32

ACCEPTED MANUSCRIPT Using

the

electrophoretic

mobility

one

can

calculate

the

electrokinetic

(uncompensated) charge of particles from the Lorentz – Stokes relationship [79,179]:

Nc  17.8 d H e

T

(14)

IP

where Nc is the number of elementary charges per particle (it should be remembered that

SC R

e = 1.602·10-19 C), η is expressed in g (cms )-1, dH is expressed in nm and μe in μm cm (V s)-1. It should be mentioned that the accuracy of Nc determination from Eq.(14) is the largest for low ionic strength [79,179].

Using the above electrophoretic mobility data one can calculate from Eq.(14) that for

NU

I of 10-4 M and pH 5.8, Nc equals to -31, -64 and -144 for the S1, S2 and S3 samples, respectively (see Table 5). As can be noticed, Nc monotonically increases with the particle

MA

size and ionic strength becoming -26, -47, and -114 for the S1, S2 and S3 samples, respectively and ionic strength of 10-2 M.

In Table 5 the two-dimensional electrokinetic charge densities calculated as

D

σe =Nc / 4Sg (where Sg is the geometrical cross-section area of a particle expressed in nm2)

TE

are also given. As can be noticed, contrary to the number of charges, σe significantly increases with the particle size. This means that the absolute charge density is much lower for larger

CE P

particles. Accordingly, for the S1 sample (particle size 15 nm ) σe = -0.0368 e nm-2 and for the S3 sample (particle size 54 nm ), σe = -0.0125 e nm-2 (for I of 10-2 M and pH 5.8 ).

Silver particle average size [nm]

AC

Table 5. The electrophoretic mobility, number of elementary charge and zeta potential of silver nanoparticles for various ionic strength (pH 5.8, T = 298 K). From Ref. [79]. Ionic strength [M]

κdp

μe μm cm (V s)-1

p [mV] Henry‘s model

p [mV] Smoluchowski‘s model

Nc

σ [e nm-2]

15

0.0001 0.001 0.01 0.03

0.25 0.78 2.46 4.26

-3.52 -3.23 -3.00 -2.73

-74.0 -66.5 -58.9 -51.1

-49.6 -45.5 -42.3 -38.5

-31 -29 -26 -24

-0.175 -0.163 -0.147 -0.136

28

0.0001 0.001 0.01 0.03

0.46 1.45 4.59 7.95

-3.86 -3.25 -2.84 -2.72

-80.5 -65.5 -52.7 -47.1

-54.4 -45.8 -40.1 -38.3

-64 -54 -47 -44

-0.104 -0.0877 -0.0762 -0.0714

54

0.0001 0.001

0.89 2.8

-4.54 -4.28

-93.1 -81.8

-63.9 -59.3

-144 -135

-0.0630 -0.0590 33

ACCEPTED MANUSCRIPT 0.01 0.03

8.86 15.34

-3.61 -2.79

-61.6 -44.7

-50.8 -39.3

-114 -86

-0.0498 -0.0376

Additionally, using the electrophoretic mobility data, one can calculate the zeta

T

potential of particles, a parameter commonly used to characterize nanoparticle suspensions

IP

and to predict their stability [79,179,182,200]. As above discussed, it is usually calculated

SC R

using Henry‘s equation (Eq. 6). The zeta potential of the S1-S3 samples calculated from the Henry‘s and Smoluchowski‘s model are collected Table 5.

The above results and many other data discussed in the literature [79] confirm that citrate stabilized silver particles are negatively charged for the wide pH range of 3-11 and

NU

arbitrary ionic strength. Because of the high electrostatic charge, the suspensions are stable for the ionic strength up to 3x10-2 M. Since the suspension stability increases with pH, therefore,

D

4.2. Kinetics of particle deposition

MA

the optimum conditions for their storage are pH > 5 and I < 10-2 M.

TE

Because many silver nanoparticle applications involves their monolayers deposited at solid substrates, this issue has extensively been studied in the literature [66-72,79,179,202-

CE P

219]. In order to systematize these experimental data, we first present in this section a general overview of the relevant works devoted to silver particle monolayer formation on various substrates. Subsequently, attention is focused on the works where the kinetic aspects of monolayer formation were a priority.

AC

Monolayers of silver nanoparticles were usually deposited on glass, quartz, silicon and ITO surfaces [66-68,203-213]. In a few recent studies [77-81,179,200,220] mica was used as a model substrate allowing one to precisely measure particle deposition and release kinetics. Most of the substrate surfaces were modified by adsorption of cationic polyelectrolytes or silans in order to promote an efficient deposition of negatively-charged silver nanoparticles, usually stabilized by citrate anions [137]. For sake of convenience, the substrates and other experimental conditions are listed in Table 6. As can be noticed, in most of these works, attention was focused on qualitatively determining the morphology of the monolayers using SEM and AFM imaging. Significantly less effort was devoted to quantitatively determine particle deposition kinetics and the stability of monolayers that could shed light on the mechanisms of monolayer formation.

34

IP

T

ACCEPTED MANUSCRIPT

AC

CE P

TE

D

MA

NU

SC R

Table 6. Selected works concerning silver nanoparticle deposition on solid surfaces.

35

ACCEPTED MANUSCRIPT -

unknown

parameter,

AFM

–

atomic

force

microscopy,

APS

-3-(aminopropyl)triethoxysilane,

Supporting layers

Synthesis/ type of reagents

Size of nanoparticles

Deposition time

MPS

-

-

days

SEM

-

[203]

TSC

40 nm

< 24 h

XPS,SEM, AFM

10-15%

[66]

MPS

-

100 nm

days

SEM

-

[204]

APTMS

TSC lub EDTA

15 - 55 nm

few hours

UV-vis, QCM

15%

[67]

PDDA

NaBH4, graphite

10 nm

4 min

CV

-

[205]

PVP

-

100 nm

all night

SEM

-

[206]

PAH

NaBH4, PAA

10 nm

-

[207]

glass

dendrimers

TSC

20 nm

-

[208]

glass

APS

-

4 nm

-

[70]

quartz

PDDA

-

[68]

glass

MPS, RD

-

[209]

glass

PDDA

-

[210]

glass Al/Al2O3 gold

-

NaBH4, MPSA TSC TSC,CTAB, MUA PVP, GL

-

[211]

CTP

NaBH4

PDDA PVP

quatz

T

IP

SC R

NU

coverage

[Ref]

14 - 30 nm

< 24 h

10 and 38 nm

<1h

QCM, SEM

10%

[69]

NaBH4, NaI, MAA

20-70 nm

< 24 h

AFM

-

[212]

TSC

24.9 nm

< 6h

QCM, AFM

24 %

[76]

NaBH4, TSC

7 nm

18 h

7%

[213]

NaBH4, TSC

-

3 gh

-

[214]

NaBH4, TSC

5 and 8 nm

24 h

AFM UV-vis, AFM AFM

-

[215]

-

TSC

-

all night

AFM, UV-vis, IR

-

[216]

PDDA

PMA

< 100 nm

2 min

SEM

-

[71]

ATPMS

TSC

18 nm

3h

-

[217]

CHPTAC

-

-

5 min

AFM, CV SEM, IR, ICP-OES, UV-vis

-

[72]

Ti/TiO2

APS

Ti/TiO2 Polyurethane foams nylon, silk silicon

-

AC

MPS

cotton

Method of analysis

UV-vis UV-vis, AFM UV-vis, AFM UV-vis, AFM, CV AFM UV-vis, AFM AFM

glass, silicon glass

3 min < 2h

MA

glass, quartz quartz, silicon ITO

-

8 nm

15 min

30 nm

< 10h

1 - 5 nm

24 h

D

glass, carbon, gold

dendrimers

TE

glass, quartz glass, ITO, silicon glass

CE P

Solid substrate

APTMS TSC 50-60 nm < 90 min [218] APTMS, TSC 50-60 nm 48 h UV-vis [219] glass MPTMS PEI NaBH4, PVA 25 nm 26.5 h AFM, SEM 21% [77] mica PEI 15% NaBH4, TSC 15 nm 26.5 h AFM, SEM [78] mica PAH 39% PAH NaBH4, TSC 28 nm 11 h AFM, SEM 33% [179] mica PAH NaBH4, TSC 54 nm 11 h AFM, SEM 36% [79] mica ATPMS – 3-aminopropyltrimetoxysilane, CHAPTAC - 3-chloro-2-hydroxypropyl trimethylammonium chloride, CTP – 4carboxythiphenol, CTAB – cetyltrimethylammonium bromide, CV –cyclic voltammetry, EDTA -ethylenediaminetetraacetic acid, GL – glucose, ICP-OES – inductively coupled plasma optical emission spectrometry, ITO - indium tin oxide, MMA mercaptoacetic acid, MPS – (3-mercaptopropyl)trimetoxysilane, MPTMS -3 -mercaptopropyltrimetoxysilane, MPSA - 3mercapto-1-propanesulfonic acid, MUA - 11-mercaptoundecanoic acid, PAA –polyacrylic acid, PAH – poly(allylamine hydrochloride), PDDA - poly(dialyldimethylammoniumchloride, PEI- polyetyleneimine, PMA –polymetacrylic acid,), PVA – polyvinylalcohol, PVP –polyvinylpyrrolidone, QCM – quartz crystal microbalance, RD – rhodamine, SEM – scanning electron microscopy, TSC –trisodium citrate, XPS – X-ray photoelectron spectroscopy. glass

36

ACCEPTED MANUSCRIPT In the work of Bar et al [66] deposition of silver and gold citrate stabilized particles of the size 25 to 30 nm on glass, silicon and ITO modified by dendrimers was studied under diffusion-controlled transport. The deposition time ranged from 5 minutes to 24 hours.

T

The coverage of particles was determined from AFM and SEM images by a direct counting

IP

procedure. The maximum surface concentration of silver particles obtained in this work was 22 (±2)·109·cm-2 ( 220 μm-2). This corresponds, for the 25 nm particles to the dimensionless

SC R

coverage of 10.8%. For 30 nm particles the maximum coverage was 15.5%. However, in this work the physicochemical parameters of the deposition process such as pH, ionic strength and suspensions concentration were not specified.

NU

Bandyopadhyay et al. [69] measured by QCM the kinetics of silver particle deposition on aluminum surfaces modified by molecules of 4-carboxythiofenol. The particles of the

MA

average size 10 nm, were obtained by reduction of silver sulfate(VI) by sodium borohydride. It was observed that the monolayer mass increased linearly with the deposition time and after exceeding 5 minutes attained a constant value 730 ng cm-2 (Fig. 10). This corresponds

D

to the maximum coverage of the monolayer equal to 10.4 %. It is worth mentioning, that

TE

besides the QCM measurements, the monolayers were also characterized using microscopic methods (SEM). However, no information about pH, ionic strength and nanoparticle

AC

CE P

concentration in the suspension was given in this work.

Fig. 10. Quartz crystal microbalance measurements of mass change with time for silver nanoparticles deposited in the 4-CTP SAM-covered (■) alluminium and (●) gold surfaces. The solid line has been fitted to Ag particle adsorption kinetics on the SAM-covered Al surfaces. From Ref. [69]. The formation of silver nanoparticle monolayers on silanized surfaces of glass, carbon and gold was also described in the work of Bright and coworkers [67]. Citrate-stabilized 37

ACCEPTED MANUSCRIPT nanoparticles with an average size 22 and 55 nm and particles prepared by ethylenediaminetetraacetic acid (EDTA) reduction with size 14 and 15 nm were used in this study. The kinetics of deposition under diffusion-controlled transport was monitored using

T

UV-vis spectroscopy for wavelengths of 408 and 410 nm. It was shown that the rate of

a

IP

nanoparticle deposition is proportional to the square root of deposition time. Unexpectedly, significant difference in the deposition rate between citrate and EDTA stabilized

SC R

nanoparticles was observed. This was attributed to the difference in the stabilizing agents used in the synthesis. Additionally, in this work, particle deposition was monitored using QCM measurements. The experimental results were presented as the dependence of changes in the

NU

frequency on time upon exposure of the dendrimers-dervitized crystal upon exposure to colloidal silver. Analogously as in the case of UV-vis measurements, the rate of monolayer

MA

formation was faster for EDTA-stabilized nanoparticles. The maximum mass of citratestabilized nanoparticles (55 nm in diameter) deposited on the substrate was equal to 10,5 μg·cm-2 that corresponds to the coverage of 15%. However, in the case of EDTA-

D

stabilized nanoparticles (15 nm in diameter) the maximum mass amass of the monolayer was

TE

17,2 μg·cm-2 that corresponds to the coverage of 163% indicating formation of multilayer coverage (most probably silver mirror).

CE P

Analogous studies of deposition kinetics of citrate-stabilized nanoparticles (50 and 60 nm in diameter) on glass modified with 3-aminopropyltrimethoxysilane (APTMS) or 3-mercaptopropyltrimethoxysilane (MPTMS) were carried out by Park et al. [219]. The UV-vis technique was used in order to monitor nanoparticle deposition. The maximum

AC

time of particle deposition was 48 hours but the experimental conditions, especially the suspension coverage, were not specified. Therefore, the coverage of particles cannot be quantitatively determined but the maximum absorbance of 2.782. Pallavicini et al. [213] studied deposition of stabilized)

on

glass

modified

with

7 nm silver nanoparticles (citrate

3-aminopropyltrimethoxysilane

(APTMS)

or

3-mercaptopropyltrimethoxysilane (MPTMS) at the elevated temperature of 303 K during 18 hours. The coverage of particles was determined using AFM imaging analogously as in Ref. [66]. The maximum surface concentration was 1.9·102 μm-2 that corresponds to the coverage of 7%. Deposition of citrate-stabilized nanoparticles with size of 24.9 nm on glass and silicon modified with poly(4-vinylpiridine) (P4PV) was studied using AFM and QCM by the Kim‘s group [76]. The Sauerbrey‘s equation was used for calculating the mass of deposited particles and their surface concentration. In this way the kinetics of the monolayer formation was 38

ACCEPTED MANUSCRIPT obtained. The results obtained by QCM were in agreement with the experimental data derived from AFM, see Fig 11 b.

b)

TE

D

MA

NU

SC R

IP

T

a)

CE P

Fig. 11. a) AFM images (1 μm x 1 μm scales) taken after soaking a P4VP film for certain period of time in Ag sol. b) Number of Ag nanoparticles (NPs) per 1 μm2 of P4VP: (Δ) counted from AFM images and (○) estimated from QCM data, drawn versus the adsorption time. From Rf. [76].

AC

Although the plateau value of the monolayer coverage was not obtained, the QCM measurements presented in Fig. 11 part b indicate that the highest maximum coverage was 500 μm-2 that corresponds to 24%. However, analogously as in previous works, the bulk concentration of particles was not determined and pH and ionic strength of the suspension was not controlled. Considering the deficit of reliable experimental data, in Refs. [77-81,179,200] extensive studies of silver particle deposition kinetics were performed with the aim of quantitatively evaluate mechanisms of these processes. In these works mica modified by controlled adsorption

of

cationic

polyelectrolytes

was

used

as

the

model

substrate.

The following polyelectrolytes were studied (a) poly(allylamine hydrochloride) hereafter referred to as PAH (molar mass of of 70 kDa), (b) poly(ethyleneimine) hereafter referred to as PEI (molar mass molecular weight 75 kDa), and (c) poly(diallyldimethylammonium chloride)

39

ACCEPTED MANUSCRIPT hereafter referred to as PDDA (molar mass 100-200 kDa). The structural formulae of these

SC R

IP

T

polyelectrolytes are schematically shown in Fig. 12.

NU

Fig. 12. A schematic representation of the structural formula of the cationic polyelectrolytes used for mica modification.

MA

Bulk physicochemical properties of the polyelectrolytes for various ionic strengths and a wide pH range were thoroughly characterized using DLS. The hydrodynamic diameter calculated using the Stokes-Einstein relationship, Eq.(4) indicates that the PAH and PDDA

D

molecules assume extended conformations for the ionic strength of 10-4 to 10-2 M and pH up

TE

to 9, whereas the PEI molecules assumes a compact conformation that reflects its branched structure suggested by the manufacturer. mobility of

these polyelectrolytes,

CE P

The electrophoretic

determined by micro-

electrophoresis for various ionic strength as a function of pH is shown in Fig. 13 [220].

AC

a)

100

1

5

2

80

3

4

4 40

2

20

1

[mV]

e [m cm (V s)-1]

60 3

IEP

0

0

-1

-20

-2

-40 3

4

5

6

7

8

9

10

pH

40

ACCEPTED MANUSCRIPT

b)

60

40

2

IEP.

1

 [mV]

4

20

-1

NU

0

0

-20 4

5

6

7

pH

8

9

10

11

MA

3

c)

100

4

D

5

TE

1

2 3

3

2

4

1

0

-1 4

5

6

AC

3

CE P

e [mm cm(V s)-1]

IP

3 3

7

8

80

60

40

 [mV]

e [m cm (V s)-1]

80

2

SC R

1 4

T

5

20

0

-20 9

10

pH

Fig. 13. The dependence of the micro-electrophoretic mobility µe [µm cm(Vs)−1] on pH determined experimentally for: a) PAH, b) PEI, c) PDDA solution: 1) (♦) I = 10-4 M, 2) (□) I = 10-3 M, 3) (●) I = 10-2 M, 4) (▲) I = 0.15 M. The solid lines are nonlinear interpolations of experimental data. From Ref. [220]. As can be seen, in all cases the mobility is highly positive for the low pH range, attaining at pH 5.5 and I = 10-2 M 4.04, 3.09 and 2.5 μm cm s-1 V-1 for PAH, PEI and PDDA, respectively. For higher pH the electrophoretic mobility of PAH and PEI monotonically decreases with pH and vanishes at pH 10 and 10.8 respectively, that can be identified as isoelectric points of the polyelectrolytes. However, no isoelectric point was observed in the case of the strong PDDA polyelectrolyte, whose molecules remain positively charged for pH

41

ACCEPTED MANUSCRIPT up to 10.5 and all ionic strengths (see Fig. 13). This is the result of the presence of the quaternary ammonium ionic groups [221]. Knowing the electrophoretic mobility and the hydrodynamic diameter, the electrokinetic

T

(uncompensated) charge of the polyelectrolyte molecules under various conditions was

IP

calculated from Eq.(14) in Ref.[220]. The results are shown in Fig. 14 as the dependence of

SC R

Nc on pH . 80 70

1) PAH

60

40

NU

Nc

50

2) PDDA

30

MA

3) PEI

20 10 0 3

4

5

6

7

8

9

10

D

pH

CE P

TE

Fig. 14. The number of elementary electrokinetic charges per molecule Nc vs. pH calculated from the Lorentz – Stokes relationship Eq. (14). 1) (■) I = 10-3 M, (●) I = 10-2 M PAH, 2) (◊) PDDA I = 10-2 M NaCl, 3) (▲) PEI I = 10-2 M NaCl. The dashed and solid lines are nonlinear interpolations of experimental data. As can be noticed, the electrokinetic charge is the highest for PAH varying between 67 at pH 4 and 38 at pH 9 (for ionic strength of 10-2 M). The PDDA and PEI molecules are

AC

characterized by considerably lower charge (see Fig. 14). It should be mentioned that the electrokinetic charge determines the interactions of polyelectrolyte molecules with mica that leads to formation of stable polyelectrolyte monolayers as below discussed. The adsorption kinetics of the polyelectrolytes and formation of their monolayers on mica was studied in Refs. [220,222-224] using the streaming potential method. Polyelectrolyte monolayers were adsorbed in situ under diffusion-controlled conditions from suspensions of appropriate concentrations (typically 1– 5 mg L-1). Using the measured streaming potential values, the effective zeta potential of polyelectrolyte covered mica was calculated from the Smoluchowski‘s formula (Eq. 9). In Fig. 15 the dependence of the zeta potential on ionic strength of bare mica and mica covered by the saturated polyelectrolyte monolayers is plotted. As can be seen, bare mica exhibits negative zeta potential for the entire range of ionic strength equal to -100 mV,

42

ACCEPTED MANUSCRIPT -62 mV and -50 mV for I = 10-4, 10-2 M, 0.15 M, respectively (at pH 5.5). The negative zeta potential of mica is converted into positive by a controlled adsorption of the polyelectrolyte molecules. The highest zeta potential is obtained for PAH, equal to 73, 48 and 32 mV for

T

I = 10-4, 10-2 0.15 M, respectively (at pH 5.5). Significantly lower zeta potentials were

IP

observed in the case of PEI and PDDA (see Fig. 15). Therefore, it was suggested in Ref. [220] that PAH monolayers on mica should be the most appropriate for promoting the deposition of

80 60

SC R

negatively charged nanoparticle.

1 2 3

NU

40

0 -20

MA

 [mV]

20

-40 -60 -80 -100

4

10-3

10-2

I [M]

10-1

TE

10-4

D

-120

CE P

Fig. 15. The dependence of the zeta potential of bare and polyelectrolyte-covered mica on ionic strength, determined by the streaming potential method: 1) (■) PAH-covered mica, 2) (▲) PEI-covered mica, 3) (◊) PDDA-covered mica, 4) (●) bare mica. The solids lines are nonlinear interpolations of experimental data. All the experiments were carried out at pH 5.5. From Ref. [220].

AC

In order to determine the stability of these monolayers, thorough desorption experiments were carried out in Ref.[78,220] using the streaming potential method. The polyelectrolyte monolayers of a desired coverage were first adsorbed in situ under diffusion-controlled conditions as above described. Afterward desorption runs were performed by flushing the cell with pure electrolyte under a moderate flow rate of 0.03 mL s-1. It was demonstrated that the monolayer zeta potential decreased with time but remained highly positive for the rinsing time up to 10 hours. Even less significant variations in the zeta potential of PAH monolayers were observed in the case of the diffusion-controlled desorption. Using the procedure of controlled polyelectrolyte adsorption on mica, systematic measurements were carried out in Refs.[78-80, 179,200] focused on evaluating the kinetics of silver nanoparticle deposition. In the first work of this series [78] the validity of the diffusioncontrolled deposition mechanism was quantitatively determined. The surface concentration of the deposited particles as a function of time was determined by means of AFM and SEM 43

ACCEPTED MANUSCRIPT imaging of monolayers under dry conditions (a typical monolayer is shown in the inset of Fig. 16a). These images were quantitatively analyzed by counting the number of deposited particles over equally-sized surface areas randomly chosen over the mica substrate. Usually,

T

up to ten various areas were considered with the overall number of counted particles

IP

numbering about 1000, which ensured the precision of these measurements better than 3%. In this way, the average number of particles per unit area (surface concentration), hereafter

SC R

denoted by Ns, was determined. Hence, the primary result of these experiments were the dependencies of Ns on the time t obtained for various bulk concentrations of the silver sol [78,200].

t

NU

The kinetic runs were expressed in terms of the of the square root of deposition time 1/2

rather than the primary time variable t. This is so, because for a not too high coverage,

MA

where the surface blocking effects remain negligible, the kinetics of particle deposition is described by the theoretical formula [225] (see Appendix, Eq. (A20)):

1/2

(15)

TE

D

D N S  2   t1/2 nb  

CE P

where D is the diffusion coefficient of the particle and nb is the bulk number concentration of particles connected with the weight concentration cp (expressed in mg L-1) via the linear dependence:

AC

6 106 nb  c  d m3  p p

(16)

44

ACCEPTED MANUSCRIPT a)

T

1

IP

1000 2

500 3 4 5 0 2

4

t

1/2

6

8

1/2

[min ]

MA

b)

20

-1

NS/cb [m *L*mg ]

10

NU

0

SC R

Ns [m-2]

1500

-2

D

15

TE

10

0 0

2

CE P

5

4

6

8

10

AC

t1/2 [min1/2]

Fig. 16. The kinetics of silver particle deposition on PAH modified mica for bulk suspension concentration: 1) ( ) 100 mg L-1, 2) (♦,◊), 50 mg L-1, 3) (■,□) 20 mg L-1, 4) (●) 10 mg L-1, 5) (▼) 5 mg L-1. a) The dependence of Ns [μm-2] on t1/2 [min1/2]. The solid line denotes the linear regression fits. b) The dependence of the reduced surface concentration Ns/cb [μm-2 L mg-1] on the square root of adsorption time t1/2 [min1/2]. Full symbols - AFM measurements, hallow symbols - SEM measurements. From Ref. [200]. Kinetic runs obtained for a broad range of bulk silver particles concentration range, 5-100 mg L-1 (deposition conditions I=10-2 M, pH 5.5 and T=293 K) are shown in Fig. 16 part a. For t1/2 < 10 min1/2 ( adsorption time t < 100 min.) the surface concentration of particles linearly increases with the square root of the time with the slope proportional to the bulk concentration. It is interesting to mention that the results obtained by AFM and SEM agree with each other for the entire range of adsorption time that supports the validity of both techniques to study deposition kinetics of silver particles. 45

ACCEPTED MANUSCRIPT In order to study this effect in more detail, the experimental results shown in Fig. 16a were expressed in the normalized form as the dependence of Ns /cb [μm-2 mg-1 L] on the square root of adsorption time t1/2 [min1/2]. As seen in Fig. 16b, a universal dependence is obtained sD   (

NS ) / t1/2 cb

IP

T

this way having the form of a straight line with the slope of 2.45 mg–1 L m-2 min.-1/2 = 2.6x1013 cm g-1 s-1/2 .

SC R

This confirms the proportionality of the particle adsorption rate to the bulk suspension concentration predicted by Eq.(15). Knowing the slope, one can experimentally determine the

   

2/7

 kT     3 

1/ 7

(17)

MA

 12 dm      2s D  p

NU

diameter of silver particles from the dependence derived in Ref. [78]

Considering this slope and noting that ρp = 10.49 g cm-3, η = 0.01 g (cm s )-1 at

TE

AFM and TEM (see Table 4).

D

T = 293 K, one obtains from Eq. (17), dm = 15 nm that agrees with the values determined by Additional kinetic runs were performed for a broader range of time with the aim of

CE P

determining the role of the bulk concentration of suspensions [200]. Typical results obtained for I = 10-2 M, pH 5.5 and the bulk silver concentrations of 20, 50 and 100 mg L-1 are shown in Fig. 17. For the sake of convenience, these results are expressed both in terms of the

AC

surface concentration Ns and the coverage calculated as θ = Sg Ns. As can be observed in Fig. 17, the particle coverage, after an initial linear increase (in respect to t1/2) attains practically the same maximum coverage θmx equal to 0.28, for all bulk silver suspension concentrations. This confirms a minor reversibility of the deposition process for this range of bulk suspension concentration (20 – 100 mg L-1).

46

ACCEPTED MANUSCRIPT

0.3 2

3

T

1

0.2 1000

IP



Ns [m-2]

1500

0.0 0 0

10

20

30

40

NU

t1/2 [min1/2]

SC R

0.1 500

MA

Fig. 17. The kinetics of silver particle adsorption at mica determined for various bulk suspension concentrations using AFM (full points) and SEM (hallow points). Particle deposition conditions: pH 5.5, T = 293 K. The points denote experimental results obtained for: 1) ( ) 100 mg L-1, 2) (♦,◊), 50 mg L-1, 3) (■,□) 20 mg L-1. The solid line denotes the theoretical results calculated from the RSA model. From Ref. [200].

D

It is interesting to observe that the time of monolayer formation, characterized by the

TE

maximum coverage θmx, decreases rapidly with the bulk suspension concentration. Thus, from the experimental data shown in Fig. 17 one can estimate that the characteristic time tm is

CE P

equal to 1000, 200 and 35 minutes for cb = 20, 50 and 100 mg L-1, respectively. These experimental data are in accordance with the theoretical prediction that can be derived from Eq.(15) by assuming that for t = tm , θ = θ mx. In this way one obtains the equation:

m

d3 cb2

AC

tm 

 2

4S g Dnb2

where cm 

 cm

(18)

2 2  2  mx  p

3

kT

If the particle diameter is expressed in nm and cb in mg L-1 the constant becomes:

cm  3.29 109

2  mx  p2

kT

(19)

2 For silver particles in aqueous media at the temperature of 293 K, cm  8.96 104mx [mg2 L -2 nm-3].

47

ACCEPTED MANUSCRIPT a) 2000

0.35

1800 0.30 1600 0.25

1200

0.20

1000

T

 0.15

800 600

IP

-2

NS [m ]

1400

0.10

400

0

SC R

0.05 200

0.00

0

5

10

15

20

t1/2 [min1/2]

NU

b)

25

0.35

500

0.30

0.25

MA

300

0.15

0.10

D

200

0.20



NS [m-2]

400

0 0

5

TE

100

10

0.05

0.00 15

20

25

t1/2 [min1/2]

CE P

c) 180 160 140

0.35 0.30 0.25

100

0.20



NS [m-2]

AC

120

0.40

80

0.15

60

0.10

40

0.05

20 0

0.00 0

5

10

15

20

25

t1/2 [min1/2]

Fig. 18. The kinetics of silver particle adsorption at mica modified by the PAH monolayer determined for pH 5.8-6.2, I=10-2 M, T= 293 K and silver particles with average size: a) (●,○) 15 nm, cb = 50 mg L-1, b) (■,□) 28 nm, cb= 150 mg L-1, c) (▲, ) 54 nm, cb = 400 mg L-1. The points denote experimental result obtained using AFM (full points) and SEM (hollow points). The solid line denotes the theoretical results calculated from the RSA model. The insets show an AFM images (scan size 2 μm x 2 μm) of silver monolayers a) Ns=506 μm-2, θ =0.09, b) Ns=276 μm-2, θ =0.17, c) Ns=27 μm-2, θ =0.17. The insets show the corresponding particle monolayers. From Ref. [79].

48

ACCEPTED MANUSCRIPT As can be noticed, the characteristic monolayer formation time is proportional to the cube of the particle size and inversely proportional to the square of the weight concentration of particles in the suspension. This means that obtaining high density monoalyers for larger

T

particles is only feasible for increasing the weight concentration cb. This effect is illustrated in

IP

Fig. 18 where the kinetic runs obtained for various silver particle sizes (15, 24 and 54 nm) are

SC R

plotted for I=10-2 M, pH 6 and T=293 K [79]. Analogously as before (see Fig. 17) a linear increase in the particle surface concentration (or coverage shown in the l.h.s. axis in Fig. 18) with t1/2 is observed. Afterward, after reaching the characteristic monolayer formation time of

NU

about 225 minutes the maximum coverage θmx is attained. This characteristic time, is practically fixed for all particle sizes because the weight concentration of the suspensions is increased for larger particles. Hence, the results shown in Fig. 18 are in accordance with

MA

theoretical prediction derived from Eq. (19).

It can be concluded that the results shown in Fig. 16-18 are useful because they confirm the validity of Eqs.(18-19) that allows one to predict the time needed to produce

D

dense silver particle monolayers under diffusion transport.

TE

It should also be mentioned that experimental results shown in Figs. 16-18 are successfully interpreted in terms of the random sequential adsorption (RSA) model discussed

CE P

in detail in Appendix A. This is an universal approach allowing one to theoretically predict the maximum coverage of particles interacting via the screened Coulomb potential for various particle shapes and sizes. In the case of nearly-spherical particles, the maximum coverage is

AC

only a function of the κd parameter and can be approximated for κd > 1 by the analytical formula:

 mx  

1

1  2h

*

/d



2

(20)

where θ∞ is the maximum coverage for hard (non-interacting) particles equal to 0.547 for spheres [225] and h* is the effective interaction range characterizing the repulsive doublelayer interactions among particles, which can be calculated from the formula

2h* / d p 

    1 1 {ln 0  ln 1  ln 0 } d 2ch 2ch   d

(21)

49

ACCEPTED MANUSCRIPT   pe   kT   is the characteristic interaction energy of particles where 0  16d p   tanh 2   e   4kT  2

and ch is the scaling interaction energy, close to the kT unit [225] .

T

Additionally, from the RSA model, the blocking function can be derived that is used

IP

as the boundary condition for the bulk transport equation (see the Appendix). Subsequently,

SC R

this transport equation can be solved by standard numerical procedures [225]. In this way theoretical prediction, depicted by solid lines in Figs. 17-18 are derived for various deposition conditions without using any adjustable parameters. As can be seen, these results adequately reflect the entire kinetic runs, especially the abrupt decrease in the deposition rates reaching

NU

the monolayer formation time. This effect, commonly observed for other nanoparticle [226229] and protein [230-232] systems appears because the bulk transport is much slower

MA

than the surface boundary layer transport except for the particle coverage approaching the maximum coverage. In the latter case, the surface blocking function rapidly vanishes (see Appendix), that makes the surface boundary layer transport much slower than the bulk

D

transport.

TE

Additionally, as can be seen in Fig. 18, the RSA model adequately reflects the increase in the maximum coverage of particles with the particles size from 0.29, to 0.36 for the particle

CE P

size of 15, and 54 nm, respectively. This effect is caused by the increase in the κd parameter that, according to Eq.(21), reduces the normalized interaction range h*/d. Obviously, the range of the repulsive electrostatic interactions can also be reduced by

AC

increasing the ionic strength of the suspension that should result in the increase in the maximum coverage. Because this issue is of a practical significance for preparing silver particle monolayers of a higher density, it has been systematically studied in Refs. [79,179]. Some of the results, obtained for the 15 nm particles (S1 sample) pH 5.5 and various ionic strength are shown in Fig. 19 [200]. As can be seen, the initial deposition rate of particles is little affected by the change in ionic strength that confirms the validity of the bulk transport controlled regime. On the other hand, the maximum coverage of particle monolayers θmx increases abruptly with ionic strength from 0.15 (I = 10-4 M) to 0.34 (I = 3x10-2 M). This is interpreted as due to the decreasing lateral electrostatic repulsion among particles in the monolayer caused by increased ion concentration (which decreases the thickness of the electric double-layer).

50

ACCEPTED MANUSCRIPT

2500 0.4 11

2000

0.3

T



-2

NS [m ]

22

1500

IP

0.2

33

1000

4

0

SC R

0.1

500

0.0

0

10

20

t

1/2

[min]

30

1/2

MA

NU

Fig. 19. The kinetics of silver particle deposition at mica modified by the PAH monolayer, determined for various ionic strengths using AFM (full points) and SEM (empty points) methods. Particle deposition conditions: pH 5.5, T = 293 K, the sol concentration 50 mg L-1. The points denote experimental results obtained for: 1) (●,○), I=3x10-2 M, 2) (♦, ) I=10-2 M , 3) (■,□), I=10-3 M, 4) (▲, ), I=10-4 M, and the solid lines denote the theoretical results calculated from the RSA model. From Ref. [200].

D

This is clearly seen in Fig. 20 where the dependence of the maximum coverage on ionic

TE

strength or the κd/2 parameter (upper axis) is presented. It is interesting to mention that the experimentally observed increase in the maximum coverage with ionic strength is adequately

0,25 0.6

0.4

mx

2,50

d/2

AC

0.5

CE P

reflected by the RSA model (solid line in Fig. 20).

0.3

0.2

0.1

0.0 10-4

10-3

10-2

10-1

I [M]

Fig. 20. The dependence of the maximum coverage of the silver particle monolayer on PAH covered mica on the ionic strength and the κd / 2 parameter (upper axis). The points are the experimental results obtained by SEM imaging for particle size 15 nm (S1 sample), pH 5.5, T = 293 K, the sol concentration 50 mg L-1. The solid line denote the theoretical results calculated from the RSA model (unpublished data).

51

ACCEPTED MANUSCRIPT Therefore, the results shown in Fig. 19-20 can be exploited as reference data for producing silver nanoparticle monolayers of a well defined coverage that can be regulated by the deposition time, bulk suspension concentration, and ionic strength. The optimum

T

conditions for obtaining the most dense monolayers having the maximum coverage of 0.35

IP

are achieved for ionic strength above 10-2 M and particle size above 20 nm. Additionally, these experimental results can be used for calibration of indirect

SC R

experimental techniques such as QCM (where the amount of trapped water can be assessed [67,69,76]) and the streaming potential [80,179]. This is vital because these methods are more economical and less time consuming compared to the tedious AFM and SEM imaging

NU

followed by enumeration step.

The effectiveness of the streaming-potential method working under in situ conditions is

MA

illustrated in Fig. 21, where the variation in the zeta potential of mica upon adsorption of the supporting PAH monolayer and silver particle monolayer are shown [80]. The coverage of silver particles is determined by the ex situ AFM and SEM imaging that allows one to

D

quantitatively interpret these results in terms of the 3D electrokinetic above model discussed

TE

(Eqs. 10,11). In this model, the adsorbed polyelectrolyte molecules and silver particles are treated as isolated entities exhibiting a 3D charge distribution [198, 233-235]. As can be seen in Fig. 21, the formation of silver particle monolayers results in an abrupt decrease in the

CE P

surface zeta potential. The slope of the dependence of  on considerably exceeding 10 for

 below 0.1. The inversion in sign of the zeta potential is observed for s = 0.08. For higher coverage of silver particles (s> 0.1), the zeta potential variations become rather minor and

AC

for s > 0.25 the zeta potential attains asymptotic values of -32 mV, -27 mV, and -23 mV for ionic strengths 10-3 M, 10-2 M, and 0.15 M NaCl, respectively. Thus, it was confirmed in these measurements that the limiting zeta potential for the high coverage range approaches 1/√2 = 0.71 of the bulk zeta potential of the nanoparticles in accordance with the theoretical predictions. Since the validity of the 3D electrokinetic model is confirmed in these experiments, the streaming potential measurements can be used for a robust in situ determination of silver particle deposition and their release kinetics. The particle coverage vs. the time dependencies are obtained by converting the measured streaming potentials to the particle coverage using Eq.(13). Experiments results obtained in this way are discussed in the next section.

52

ACCEPTED MANUSCRIPT a) [m-2]

N

PAH

0

600

1200

1800

Ns [m-2]

2400

3000 0

100

200

300

400

500

120 100 80

PAH monolayer

T

60

 [mV]

40

IP

20 0 -20

SC R

-40 silver monolayer

-60 mica -80 -100 0.1

0.2



0.3

0.4

0.0

0.1

0.2

s

PAH

0.3

b) N

PAH

0

600

[m-2]

Ns [m-2]

1200 1800 2400 3000 0

100

200

300

400

500

120

MA

100 80 PAH monolayer 60

20 0

D

 [mV]

40

-40

TE

-20

silver monolayer

mica

-60 -80 -100

0.1

0.2

0.3



PAH

c) 600

100 80 60

0.0

N [mm-2] PAH 1200 1800 2400 3000 0

AC

0

0.4

0.1

s

CE P

0.0

0.2

0.3

Ns [m-2]

100

200

300

400

500

PAH monolayer

40

 [mV]

NU

0.0

20 0 -20 mica

-40

silver monolayer

-60 -80 0.0

0.1

0.2



0.3

PAH

0.4

0.0

0.1

s

0.2

0.3

Fig. 21. The dependence of the zeta potential of mica on the coverage of PAH and silver nanoparticles. The points denote experimental results obtained from the streaming potential measurements. The deposition conditions I = 10-2 M NaCl, pH 6, and T = 298 K. Streaming potential measurements carried out at pH 5.5 and various ionic strengths: a) I = 10-3 M NaCl, b) I = 10-2 M NaCl , c) I = 0.15 M NaCl. The solid lines denote the exact theoretical results calculated from the 3D electrokinetic model, Eqs.(10,11). From Ref. [80]. 53

ACCEPTED MANUSCRIPT 4.3. Stability of monolayers and particle release kinetics The agreement between experimental data and theoretical results shown in Figs. 17-19 confirms that for such relatively high bulk suspension concentrations, silver particle

T

deposition on PAH modified mica was practically irreversible. Only for lowest bulk

IP

suspension concentration of 20 mg L-1 the maximum coverage of the monolayer slightly

SC R

decreased that may suggest a partial reversibility of deposition. In order to quantitatively study this effect in Refs.[79] through particle desorption experiments were performed

TE

D

MA

NU

according to the procedure schematically shown in Fig. 22.

CE P

Fig. 22. A schematic representation of silver particle desorption measurements.

In the first step, a silver particle monolayer of a defined initial coverage is deposited on PAH modified mica sheets as above described. After rinsing, the sheets are immersed in

AC

pure electrolyte solution of controlled ionic strength and pH. The silver particles are allowed to desorb under diffusion-controlled transport in a thermostated cell for a prescribed period of time. Their coverage is determined in a discontinuous way by ex situ SEM and AFM imaging followed by particle enumeration by an image analyzing software. In another version of the desorption (release) experiments, particle monolayers are formed in situ in the streaming potential cell that allows one to precisely control their coverage. Afterward, particle desorption is carried out under diffusion or flow conditions of controlled intensity. The change in particle coverage is followed in situ via the streaming potential measurements [80]. A typical kinetic desorption run obtained for a low initial coverage of silver particle monolayer equal to θo = 0.01 is presented in Fig. 23 ( particle size 15 nm, pH 5.5, T = 293 K, I = 10-2 M). As can be noticed, the fraction of silver particle released after 400 minutes is of about 20% and after 3600 minutes 50%. As can be seen, the experimental results shown in 54

ACCEPTED MANUSCRIPT Fig. 23 are adequately reflected by theoretical results obtained by numerically integrating the desorption transport equation derived in the Appendix (Eq. (A22))

'

1/2

2  Dt  =  Ka   

T



0

B( ' )d '

(22)

IP



SC R

where θ0 is the initial coverage of particles in the monolayer, B(θ) is the surface blocking function known in analytical form for spherical particle (see Appendix) and Ka is the equilibrium adsorption constant.

NU

The best-fit value of the equilibrium adsorption constant determined in this way was 0.47 cm.

MA

For a low coverage range, by considering that B(θ) remains constant and equal to B0, Eq.(22) can be integrated and converted to the useful logarithmic form:

  2  Dt  ln       K a B0     0 

(23)

TE

D

1/2

CE P

Eq. (23), which it is more convenient than Eq.(22), has a practical significance because it can be directly used to determine the equilibrium adsorption constant of particles. This can be done by plotting the experimental dependence of ln (θ/θo) on the square root of the desorption

AC

time t1/2. Consequently, knowing the slope of this dependence sl = Δln(θ / θo) /Δt1/2 on can calculate the adsorption constant from the dependence

1/2

2 D Ka     sl B0   

(24)

As can be seen in Fig. 23 part b, Eq.(23) well reflects experimental data for the entire range of time allowing to precisley determine the adsorption constants, which was 0.48 cm.

55

ACCEPTED MANUSCRIPT a)

T

0.010

IP

0.008

0.004

0.002

0.000 10

20

30

40

50

60

70

80

90

t1/2 [min1/2]

110

MA

b)

100

NU

0

SC R



0.006

0.2 0.0 -0.2

D

-0.6 -0.8

TE

ln(/0)

-0.4

-1.0

CE P

-1.2 -1.4 -1.6 0

10

20

30

40

50

60

70

80

90

100

110

t1/2 [min1/2]

AC

Fig. 23. The kinetics of silver particle desorption determined by AFM (full points), pH 5.5, I = 10-2 M T = 293 K and initial coverage of particles θ0 =0.01. a) The dependence of θ/ θo on t1/2 . The points denote experimental data obtained by AFM for pH 5.5, T = 293 K, I = 10-2 M, the initial coverage of particles θ0 = 0.01 and the solid line denotes the theoretical results calculated from the RSA model by the numerical integration of Eq.(A22) with Ka = 0.47 cm. b) The kinetics of expressed as the dependence of ln (θ/ (θo) on t1/2 . The points denote experimental data obtained by AFM for pH 5.5, T = 293 K, I = 10-2 M, the initial coverage of particles θo = 0.01. The solid line denotes the linear fit of experimental data having the slope of -0.0105 min-1/2 . From Ref. [79].

As shown in the Appendix, knowing the equilibrium adsorption constant one can determine the energy minimum depth using the equation derived in the Appendix   kT K a   m   m

1/2

 m / kT  e 

(25)

56

ACCEPTED MANUSCRIPT where δm is the characteristic thickness of the energy minimum between the particle and the interface.

IP

     

(26)

SC R

 Ka ln Ka 1   m m / kT   ln  ln  m 2    

T

Eq. (25) can be iteratively solved, which results in the approximate expression

Using the above value of the adsorption constant of 0.48 cm and assuming the typical

NU

value of the energy minimum δm = 0.5 nm (5x10-8 cm), it was calculated from Eq. (26) that

m = -16.8 kT [79,200].

MA

In Ref. [200] desorption runs were also performed for higher initial coverage of silver particles monolayers. For θo = 0.22 the best-fit value of the equilibrium adsorption constant was 1.2 cm that gives m =-17.8 kT, which is slightly lower than previously determined for the

D

low coverage range. This difference was interpreted in Ref. [200] as due to the polydispersity

TE

of silver particles assuming that for longer adsorption times, the larger particles will

CE P

preferably deposited because they have lower binding energy, replacing the smaller particles.

0.4

1

2

AC



0.3

0.2

0.1

0.0 0

20

40

60

80

100

120

140

cb [mg L-1]

Fig. 24. The isotherm of silver particle adsorption determined using the RSA model. Curve 1 is calculated using the experimental value of Ka = 1.2 cm and curve 2 for Ka = 0.48 cm. The points denote experimental results obtained for pH 5.5, I = 10-2 M, T=293 K. From Ref. [200]. It is also interesting to mention that knowing the equilibrium adsorption constant Ka, one can obtain the entire isotherm equation of silver particle deposition using the RSA 57

ACCEPTED MANUSCRIPT blocking function (see the Appendix). The isotherm obtained in this way is shown in Fig. 24. It is seen that the theoretical results are in agreement with experimental data obtained from particle deposition measurements where the maximum coverage was studied as a function of

T

the bulk suspension concentration. However, for cb < 10 mg L-1, these direct kinetic

IP

measurements become impractical because of the excessive experimental time of the completion of the monolayer (see Fig. 17). Therefore, for such a low bulk suspension

SC R

concentration range, the equilibrium coverage of silver nanoparticles can only be predicted from the isotherm equation (see the Appendix, Eq. (A5)).

In order to get more insight into silver particle release mechanisms, extensive

NU

measurements were performed in Ref.[79] where the role of particle size, ionic strength and

MA

the temperature was systematically studied.

1.2

1

D

1.0

TE



0.8

0.6

0.2

0.0 0

20

CE P

0.4

40

60

80

2

3

100

120

AC

t1/2 [min1/2]

Fig 25. The kinetics of particle desorption expressed as the dependence of θ/θo on t1/2. The initial coverage of particles θo = 0.05. The points denote experimental data obtained by AFM for I =10-2 M NaCl, pH 6, T=298 K and particles: 1) (▲) 54 nm, 2) (■) 28 nm, 3) (●) 15 nm. The solid line denotes the theoretical results calculated from the RSA model by the numerical integration of Eq.(22). From Ref. [79]. Typical release kinetics runs obtained for the initial coverage of monolayers equal to

o = 0.05 (pH 6, I =10-2 M) and various particle sizes are presented in Fig. 25 [79]. As can be seen, for a fixed ionic strength, the release rate significantly decreases with particle size. Thus, for the S1 sample (15 nm) the residue coverage of the monolayer after the time of 240 hours (this corresponds to t½ = 120 min ½ ) is 30% of the initial value, whereas for the S3 sample (54 nm) it remains at the level of 90%. This means that in the latter case particle desorption was practically negligible. As can be seen, the experimental data shown in Fig. 25 58

ACCEPTED MANUSCRIPT are quantitatively interpreted in terms of the theoretical solutions derived by a numerical integration of Eq.(22) using the following values of adsorption constants: 0.48 cm (S1 sample), 1.9 cm (S2 sample) and 4.0 cm (S3 sample). For sake of convenience these data

T

are collected in Table 7.

10-2

m [kT]

Ka [cm]

m [kT]

Ka [cm]

m [kT]

15 (S1)

3.0

-18.8

0.48

-16.9

0.39

-16.7

28 (S2)

5.0

-19.3

1.9

-18.3

1.1

-17.7

54 (S3)

24

-20.8

4.0

-19.1

1.0

-17.7

TE

D

MA

Ka [cm]

dp [nm]

0.1

NU

10-4

I [M]

SC R

IP

Table 7. Equilibrium adsorption constants Ka and energy minimum m for given ionic strength. From Ref. [79].

Using these adsorption constants the energy minimum depth was calculated in Ref.

CE P

[79] by a numerical inversion of Eq.(25) (see Table 7). The difference between the S1 sample (particle size 15 nm ) and the S3 sample (particle size 54 nm) for I=10-2 M, is only -2.2 kT, i.e., 13%. As discussed in Ref.[79] this is in conflict with the mean-field DLVO theory, where

AC

it is assumed that both the van der Waals and electric double-layer (edl) interactions for the particle/interface system are always proportional to the particle size. Hence, the ratio of the energy minima for the S1 and S3 samples should be 54/15= 3.6 whereas it is only 0.13. As shown in Ref. [79] the net interaction energy for the S1 sample stemming from the DLVO theory, where dp =15 nm, T= 298 K, I = 10-2 M, κ

-1

= 3.04 nm,  = 40 mV, p = -50 mV,

A123 = 7.5x10-20 J should be -75 kT at the distance h = 0.5 nm. This exceeds more than four times the experimentally found energy minimum depth of -16.9 kT. For the S3 sample (dp = 54 nm) and the same conditions one obtains – 270 kT for the net energy minimum, which is 14 times lower than experimentally determined (-19.1 kT). For the separation distance h = 10 nm one obtains in the case of S3 the net energy equal to -20.3 kT, which seems reasonable, but for the S1 sample one obtains -5.6 kT, which is more than three times higher than the experimental value.

59

ACCEPTED MANUSCRIPT Therefore, it was concluded that the equilibrium distance between silver particles and the interface is comparable or larger than 10 nm, where the van der Waals interactions become negligible. Additionally, the edl interactions are not properly described by the mean-

T

field model that predicts considerable changes of the interaction energy with particle size.

IP

In order to study this problem in more detail, additional series of experiments was performed

SC R

in Ref.[79] with the aim of determining the role of ionic strength in particle release processes. 0.2

0.0

1

2

-0.4

3

NU

ln(

-0.2

-0.6

-1.0

-1.2 0

20

40

60

80

100

120

D

t1/2 [min1/2]

MA

-0.8

CE P

TE

Fig. 26. The kinetics of silver particle desorption expressed as the dependence of θ/ θo on t1/2, the initial coverage of particles θo = 0.05. The points denote experimental data obtained by AFM for pH 6, T=298 K for various ionic strength ( ) I=10-4 M NaCl, (■) I=10-2 M NaCl, (□) I=10-1 M NaCl and particle size: 28 nm (Sample 2). The solid line denotes the theoretical results calculated from the RSA model by the numerical integration of Eq.(22). From Ref. [79].

AC

Typical kinetic runs obtained for a fixed initial coverage of monolayers equal to θ0= 0.05 (pH 6, I =10-2 M) and various particle sizes are presented in Fig. 26 for the S2 sample. As can be seen, the release rate systematically increases with ionic strength. Quantitatively, this effect can be analyzed by comparing the minimum energy depths, calculated by numerical inversion of Eq.(25). As shown in Table 7, m increases with ionic strength but the differences are rather minor, equal to -1.7 kT between ionic strength of 10-4 M and 0.1 M. This disagrees with predictions stemming from the DLVO theory that predicts much larger changes in the energy minimum depth with ionic strength, exceeding three orders of magnitude [79]. This suggest that the electrostatic component of the interaction energy does not vanish exponentially with the distance as predicted by DLVO but remains practically constant. This discrepancy was accounted for in Ref.[79] by introducing the discrete interaction model where the net energy is solely controlled by the Coulomb interactions among charges 60

ACCEPTED MANUSCRIPT located at the adsorbed PAH polymer chains and the silver particles (see Fig. 27). Consequently, in this configuration, the van der Waals interactions are negligible and the electrostatic interactions are governed by the finite number of ion pairs, strictly related to the

T

number of charges on the silver particle. Therefore, the interactions between the ion pairs are

IP

governed by the Coulomb law, i.e.,

SC R

e2  m   Ni 4 dim

(27)

NU

where Ni is the number of ion pairs in the interaction zone and dim is the minimum distance

CE P

TE

D

MA

between ion pairs.

AC

Fig. 27. A schematic view of silver nanoparticles immobilization on PAH covered mica. From Ref. [79]. As can be deduced, Eq.(27) reflects the main features of the particle release processes, i.e., the fact that the interaction energy does not explicitly depend on the particle size and ionic strength. Additionally, it is predicted from Eq. (27) that the energy minimum depth should increase with the temperature, because the dielectric constant significantly decreases with the temperature [236]. In order to check this hypothesis, a series of experiments was performed in Ref. [79] where the role of temperature in the particle release was studied. Results of these experiments obtained for the S2 sample (particle size 28 nm) and ionic strength of 10-2 M are shown in Fig. 28. As expected, the increase in the temperature significantly enhances the release kinetics of nanoparticles. Thus, at 293 K (20 oC) only 10% of particles is released after 3600 minutes (60 hours) whereas at 358 K (85oC), the fraction of particle released after this time amounts to 80%. This is reflected in decreasing adsorption constant calculated from Eq.(26), which is equal to 1.9 cm at the temperature of 20oC and 0.28 cm at 61

ACCEPTED MANUSCRIPT 85oC. However, the corresponding changes in the energy minimum depth are rather minor. Thus, for 20oC (To = 293 K), m =-18.3 kT0 and for T = 358 K, m = -19.6 kT0. Similar increase in the energy minimum depths with the temperature were observed for other

T

nanoparticle sizes [79]. This indicates that Eq.(27) is a valid approximation of the net

SC R

IP

interaction energy for the silver nanoparticle/PAH covered mica system.

1.2

1.0 1

0.6 2

MA

0.4

NU



0.8

0.2

3

0.0 20

40

60

80

100

TE

t1/2 [min1/2]

D

0

CE P

Fig 28. The kinetics of silver particle desorption expressed as the dependence of θ/ θo on t1/2, the initial coverage of particles θo = 0.05. The points denote experimental data obtained by AFM for pH 6-6.2, I=10-2M NaCl for various temperatures: 1) ( ) T=298 K, 2) (■) T=323 K, 3) (□) T=358 K and particle size 28 nm. The solid line denotes the theoretical results calculated from the RSA model by the numerical integration of Eq.(22). From Ref. [79].

AC

Although the SEM and AFM based methods of determining particle release kinetics are precise and reliable, they are tedious and the particle imaging is done under ex situ conditions. Therefore, in Ref. [80] extensive measurements were performed where the release kinetic derived from the streaming potential and AFM/SEM methods was compared for various ionic strengths and pHs. In these measurements, the silver particle monolayers of a defined initial coverage were produced at pH 5.5 and I =10-2 M, directly in the electrokinetic cell. Next, the cell was flushed with pure electrolyte of appropriate ionic strength (10-4 – 0.1 M NaCl) and pH (3.5-9) and the particles were allowed to desorb under diffusion-controlled transport for a desired period of time up to 200 hours. The coverage of the nanoparticles remaining on the surface as a function of time was determined in situ by direct streaming potential measurements. After these measurements, the coverage of particles was also determined from SEM micrographs by the direct enumeration procedure above discussed. Typical release kinetics runs obtained for the initial coverage of silver particles equal to 62

ACCEPTED MANUSCRIPT 0.20 are shown in Fig. 29 part a where the ionic strength effect is illustrated and in Fig. 29 part b where the role of pH is presented. As can be seen, in all cases the results obtained by streaming potential and SEM methods agree with each other within the experimental error

T

bounds.

IP

a)

SC R

1.0

1 2

0.8

/0

3 0.6

NU

0.4

0.2

0

10

20

30

40

50

60

70

80

90

100

t1/2 [min1/2]

110

TE

D

b)

MA

0.0

1.0

CE P

/0

0.8

0.6

0.4

0.0 0

10

AC

0.2

20

30

40

50

60

70

80

1 2 3

90

100

110

t1/2 [min1/2]

Fig. 29. The kinetics of silver particle release (S2 sample, particle size 28 nm) expressed as the dependence of o on t1/2for a) ionic strengths: 1) (■) 10-4 M, 2) (●) 10-2 M, 3) (▲) 0.1 M NaCl (pH 6) and for b) pH 1) (■) 9.0, 2) (●) 5.5 3) (▲) 3.5 (I = 10-2 M, T=298 K). The full points denote experimental data obtained by the streaming potential method (the initial coverage of particles o=0.20). The hollow points denote SEM, ex situ measurements. The solid lines denote the theoretical results calculated from the RSA model by the numerical integration of Eq.(22). From Ref. [80] This is reflected in the energy minimum calculated from Eq.(26) and listed in Table 8. As can be noticed, the difference in the energy minimum calculated from the ex situ (SEM) and in situ (streaming potential) methods are confined to one kT unit. As discussed in Ref.

63

ACCEPTED MANUSCRIPT [ 80] the minor changes in the energy minimum depths with pH agree with the discrete charge interaction model.

pH 5.5

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pH 3.5

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T

Table 8. Equilibrium adsorption constants Ka and energy minimum depths m for the S2 silver particle suspension (particle size 28 nm), I = 10-2 M, various pHs. (SP – means streaming potential, SEM –scanning electron microscopy). From Ref. [80]. pH 9.0

m [kT]

Ka [cm]

m [kT]

Ka [cm]

m [kT]

in situ SP measurements

1.3 ± 0.3

-18

4.1 ± 0.8

-19

4.2 ± 1

-19

ex situ SEM measurements

1.3 ± 0.3

-18

1.9 ± 0.2

-18

3.0 ± 0.6

-19

MA

NU

Ka [cm]

Summing up the results discussed in this section one can conclude that that the direct

D

enumeration of particles by AFM and SEM is an efficient tool for studying silver

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nanoparticles deposition and release kinetics. Using these methods the role of particle size, ionic strength, temperature and pH can be systematically studied. The obtained results are

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quantitatively interpreted in terms of the RSA approach. This allowed one to determine the equilibrium adsorption constants Ka for various physicochemical conditions and the energy minimum depth that is impractical using the classical adsorption methods.

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In contrast to predictions of the mean-field DLVO theory, the energy minimum is little dependent on particle size, ionic strength and pH. Thus, for the typical desorption conditions of I = 10-2 M, pH 5.8, m varied between -16.9 and -19.1 kT for the particle size of 15 and 54 nm, respectively. The decrease in m with the temperature is also observed, which contradicts the mean field theory. These experimental evidences indicate that the role of van der Waals interactions in the silver nanoparticle release processes from PAH covered mica is negligible. The kinetics of this process is governed by discrete electrostatic interactions among ion pairs and cannot be properly described by mean-field theories of the electrical double layer. It is also demonstrated that it is feasible to quantitatively monitor the formation of silver particle monolayers and particle releases kinetics in situ using the streaming potential measurements. This is considerably more convenient than using the tedious ex situ SEM/AFM measurements. 64

ACCEPTED MANUSCRIPT

4.4. Particle bilayers

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Despite major significance of such systems, few experimental results are reported in the

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literature dealing with bilayers or multilayers involving silver nanoparticles. One of the exemptions represents the work of Morga et al. [81] focused on the systematic studies of the

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hematite/silver bilayers deposited on the mica support. The coverage of nanoparticles in both layers was quantitatively determined by a direct enumeration using SEM micrographs and AFM images. Additionally, the formation of the bilayers was characterized by streaming

NU

potential measurements carried out under in situ conditions. The role of the mica substrate properties, the supporting layer (hematite) coverage and its zeta potential was determined.

MA

The advantage of the hematite/silver bilayer system is that the both nanoparticles are oppositely charged for a wide pH range up to 9 (isoelectric point of hematite), see Fig. 30. This facilitates an electrostatically driven self-assembly of silver nanoparticles without using

D

any anchoring layer as was done in the above discussed works [77-80,179,200]. Therefore,

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the results obtained for such ‗pure‘ nanoparticle bilayers can be exploited as reference data for interpretation of experiments performed using the layer by layer (LbL) technique that is

3.0 1

30

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e [m cm (V s)-1]

2.0

45

1.0

0.0

15 0

 [mV]

[237-240]

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widely used for preparation of polyelectrolyte, protein, and particle films on solid substrates

-15

-1.0

-30

-2.0 2

-45

-3.0

-60 -4.0 2

3

4

5

6

7

8

9

10

11

12

pH

Fig. 30. The dependence of electrophoretic mobility, e, and the zeta potential (r.h.s axis) of the hematite (particle size 24 nm) and silver (particle size 28 nm) on pH., I = 10−2 M NaCl and T = 298 K. The points denote experimental results obtained for 1 (■) hematite, 2 (●) silver nanoparticles. The solid lines represent nonlinear fits of experimental results. From Ref.[81]. 65

ACCEPTED MANUSCRIPT The first step of the bilayer formation measurements was evaluating the kinetics of hematite deposition on bare mica under diffusion conditions using the ex situ AFM and SEM imaging. The latter method is more accurate for higher coverage range. In Fig. 31, a typical

T

kinetic curve, obtained for the bulk hematite suspension (particle size 22 nm, concentration of

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20 mg L-1, pH 5.8, ionic strength 10-2 M), is shown as the dependence of the coverage θon the square root of deposition time t1/2. It can be noticed that for the deposition time below 900

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minutes (t1/2 < 30) a linear increase in the hematite coverage with t1/2 is observed analogously as above discussed for the silver PAH modified mica case (see Figs. 17-18). For the longer

NU

time, the maximum coverage of 0.36 is attained.

0.5

1200

1000

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0.4

600

0.3



Nh [m-2]

800

D

0.2

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400

200

0 10

20

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0

t

1/2

30

0.1

0.0 40

50

1/2

[min

]

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Fig. 31. The kinetics of hematite particle deposition (particle size 22 nm) on mica determined by SEM (squares) Particle deposition conditions: pH 5.8, I =10-2 M, bulk suspension concentration 20 mg L-1 (□).The full points (●), determined by AFM measurements, represent results obtained for monolayer formed in the channel. The solid line denotes the theoretical results calculated from the RSA model for diffusion-controlled transport. The inset shows the AFM image of a hematite particle monolayer of the coverage 0.1. From Ref. [81]. As above discussed, the saturation of the coverage observed in Fig. 31 is caused by the repulsive electrostatic interactions among particles and can be accounted for by the RSA model (solid line in Fig. 31). After establishing proper conditions for preparing hematite monolayers of desired and controlled coverage, extensive measurements of the silver nanoparticle deposition on such heterogeneous surfaces were performed in Ref.[81]. Since the AFM method is of a limited utility because of too small differences in particle sizes, in order to determine silver particle coverage the SEM imaging with BSE detector working in the COMPO mode was used. Using EDS detector the qualitative information concerning the elements in the bilayers can be acquired. 66

ACCEPTED MANUSCRIPT By using this technique, a series of calibrating experiments was performed, aimed at determining the dependence of the maximum coverage of silver nanoparticles θsmax on the coverage of the hematite supporting monolayerθh.

T

0.40

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0.35

0.25

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s

max

0.30

0.20 0.15 0.10

0.00 0.00

0.05

0.10

0.15

h

0.20

0.25

0.30

0.35

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0.05

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D

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Fig. 32. The dependence of the maximum coverage of the silver nanoparticles θsmx on the coverage of the hematite supporting monolayerθh. The points denote experimental results obtained by SEM BSE analysis. The solid line shows the non-linear fit of experimental data and the dashed line presents the theoretical results predicted for the deposition of one silver particle on one hematite particle, i.e., θsmx = (as/ah)2 θh (as and ah are the diameters of the nanoparticles). Deposition conditions: pH 5.8, I = 10-2 M, T = 298 K. From Ref. [81]. The results of these measurements are shown in Fig. 32. As can be observed, the coverage

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of silver nanoparticles increases abruptly with the hematite coverage and the initial slope is 1.5. This indicates that one hematite particle coordinated on average more than one silver particles in the limit of a low supporting monolayer coverage. It is interesting to mention that

AC

analogous effect was previously reported for bilayers formed by positive and negative latex particles deposited on mica whose size was 503 and 810 nm, respectively [241]. For θ0.2, the increase in the silver particle coverage becomes rather moderate and finally the maximum coverage of 0.35 is attained. This is practically equal to the coverage of the supporting hematite monolayer (see Fig. 31). The results shown in Fig. 32 are used for the interpretation of the streaming potential measurements that are less tedious and can be performed under in situ conditions. The main goal of these measurements was determining the influence of the upporting hematite layer coverage on the zeta potential of bilayers. The measurements were performed as follows: in the first stage, the supporting hematite monolayer of a desired coverage was formed in the channel. Afterward, silver monolayers were deposited under diffusion-controlled transport using suspensions of an appropriate concentration. The coverage of silver particles was adjusted by changing the deposition time. The coverage of the bilayer, formed under in situ 67

ACCEPTED MANUSCRIPT conditions, was also controlled by a direct enumeration (SEM, COMPO mode micrographs) of particles after streaming potential measurements. In this way, the dependencies of the zeta potential of bilayers, on the silver particle coverage were obtained for a fixed hematite

T

coverage [81]. Results of these experiments, performed for I = 10-2 M, pH 5.8, and the

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supporting hematite layer coverage of 0.35 are shown in Fig. 33. As can be seen, an abrupt decrease of the zeta potential from the initial positive value of 28 mV ( hematite monolayer)

hematite/silver bilayer. Ns [m-2]

60

0

250

500

750

0

1000

200

400

600

bulk hematite

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40

 [mV]

20

0

D

-20

-40

bulk silver

mica

0.1

0.2

h

0.3

0.4

0.0

0.1

0.2

s

0.3

0.4

0.5

CE P

0.0

TE

-60

800

NU

Nh [m-2]

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is observed upon silver particle deposition that leads to final value of -36 mV for the complete

AC

Fig. 33. Formation of the hematite/ silver monolayer on mica shown as the dependence of the zeta potential on the particle coverage (θhhematite, θs-silver). The solid lines show theoretical results calculated from Eqs.(10,11). Silver nanoparticle deposition was carried out for pH 6, I = 10-2 M, T = 298 K. From Ref.[81].

It can also be observed in Fig. 33 that variations in the zeta potential of the bilayer with the silver coverage are more pronounced than it case of hematite monolayer deposition on bare mica (lhs. part of Fig. 33). This leads to the inversion of the zeta potential at silver coverage as low as 0.07. As suggested in Ref. [81] this behavior can be exploited for a sensitive in situ monitoring of the hematite/silver bilayer formation that is more convenient that the SEM measurements. It should also be mentioned that the results shown in Fig. 33 (bilayer formation) are adequately interpreted in terms of the 3D electrokinetic model using Eqs. (10,11) with a higher value of the Cp constant of 10 [81]. This reflects a more intense outflow of charge from the silver particles attached to the hematite particles because of a higher shear rate prevailing at larger distances from the interface. This effect was previously confirmed in Ref. 68

ACCEPTED MANUSCRIPT [241,242]. However, the final value of the zeta potential of the hematite/silver bilayer is practically the same as for the PAH/silver system, i.e., 0.71 ζ = -36 mV (see Fig. 21b). This result has a significance for basic science showing that the zeta potential of bilayers in the

T

limit of high coverage attains 0.71 of the bulk zeta potential of the particle forming the

IP

external layer (silver in this case).

Additionally, in Ref.[81] the influence of the supporting layer coverage on the final zeta

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potential of the bilayer was studied. This is clearly seen in Fig. 34 where the zeta potentials of hematite monolayers (denoted as layer number 1) and silver particle monolayer (denoted as layer number 2) are shown. It should be mentioned that this form of presentation is often used

NU

by analyzing the LbL results of the polyelectrolyte [223,243] and particle [244] multilayer

MA

formation on solid substrates and colloid carrier particles [245,246].

40

20

1

D

2

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[mV]

0

3

-20

-40

-60

1

AC

0

CE P

4

2

layer number

Fig. 34. Formation of the silver nanoparticle monolayer on hematite covered mica shown as the dependence of the zeta potential on the layer number ( the deposition conditions: pH 6, I = 10-2 M, T = 298 K). 1) (●) θh = 0.35; 2) (○) θh = 0.2; 3) (■) θh = 0.07; 4) (□) θh = 0.03. From Ref. [81]. It was concluded in Ref. [81] that nanoparticle bilayer formation on mica can be efficiently monitored under in situ conditions using the streaming potential measurements, quantitatively interpreted in terms of the 3D electrokinetic model. These results obtained in this system can be used as reference data for the interpretation of polyelectrolyte film formation in the layer by layer (LbL) processes and protein adsorption pertinent to the antigen/antibody interactions. Beside significance for basic sciences, these measurements enabled one to develop a robust method for preparing nanoparticle bilayers of well controlled properties having potential applications in catalytic processes. 69

ACCEPTED MANUSCRIPT 5. Applications

Besides the above described significance for basic sciences as model colloid systems, find

a wide spectrum of practical applications in biology, medicine,

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silver particles

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pharmaceutical and textile industry, in chemical analysis, catalysis, electronics, etc. It is well established that silver nanoparticles exhibit strong biocidal effects against

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various micro-organisms, e.g., viruses, bacteria, fungi [21-30]. Therefore, silver nanoparticles are important material for producing medical devices and supplies such as wound dressings, scaffold, skin donation, recipient sites, sterilized materials in hospitals, medical catheters,

NU

contraceptive devices, surgical instruments, bone prostheses, artificial teeth, and bone coating [4-7,31-33,36,37]. Additionally, a wide use of silver particles is observed in consumer

MA

products such as cosmetics, lotions, creams, toothpastes, laundry detergents, soaps, surface cleaners, room sprays, toys, antimicrobial paints, home appliances etc. [17,38,39]. Among many interesting papers concerned with these biocidal applications we shall

D

mention the following ones.

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Panáĉek et al. [247] examined the antibacterial activity silver particles obtained in the chemical reduction method using various saccharides against seven strains of gram-

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positive and gram-negative bacteria. The results confirmed that silver particles synthesized using disaccharidies (maltose and lactose) exhibit a higher antibacterial activity than those synthesized using monosaccharidies such as glucose or galactose. Unfortunately, the investigated silver particles were of various size (varying between 25 and 100 nm) that

AC

prohibits to separate effects stemming from the particle size and surface chemistry. Martínez-Castañón et al. [22] demonstrated that antimicrobial activity of silver nanoparticles obtained by the reduction of silver nitrate involving gallic acid was correlated with the particle size. The study conducted for Escherichia coli and Staphylococcus aureus confirmed that the increase of particle size (from 7 nm to 89 nm) causes a decrease in their antibacterial activity. Similar effects were also observed by other researchers [248-250]. In these works it was argued that the nanoparticles of the size smaller than 10 nm exhibit the best activity because large number of them get attached to the bacterial cell membrane and also penetrate inside the bacteria [18]. The antibacterial properties of silver particles obtained by the reduction of silver nitrate with sodium borohydride were investigated by Ruparelia et al. [251] using four strains of Escherichia coli and three strains of Staphylococcus aureus. The minimum inhibitory concentration (MIC) for Escherichia coli MTCC 1687 was three time higher than 70

ACCEPTED MANUSCRIPT for Escherichia. coli MTCC 443. Hence, the strain specificity was confirmed

in this

investigation. Mohan et al. [252] using the inhibition zone test, showed that monodisperse silver

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nanoparticles (average size 10 nm) obtained by the green synthesis using dextrose and gelatin

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showed a pronounced antibacterial activity against Pseudomonas aeruginosa that was in some cases even better than antibiotics such as ciproflaxin and imipenem. The MIC value of

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the particles was equal to 6 mg L-1 in this case and for Escherichia coli the MIC value was of 10.6 mg L-1. It was also shown, contrary to previous literature reports that antimicrobial activity of the silver particles increased with their size.

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Kumar et al. [253] used the method of green biogenic synthesis to obtain silver nanoparticles (using Alternanthera dentata) that exhibited antimicrobial activity against

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Escherichia coli, Pseudomonas aeruginosa, Kliebsella pneumonia and Enterococcus faecalis. As experimentally confirmed, the particles of the size 10-100 nm were highly toxic for the bacteria strains at the concentration of 50 mg L-1. The antibacterial efficiency was more

D

pronounced against gram negative bacteria than gram positive bacteria.

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Antibacterial activity of silver and gold nanoparticles of the sized 12 nm and 14 nm, respectively was compared in the work of Bindhu and Umadevi [254]. Both noble metal

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nanoparticles were synthesized in green biogenic synthesis using Solanum lycopersicums extract as the reducer. Their activity was tested using Staphylococcus aureus and

the gram positive pathogen

the gram negative Pseudomonas aeruginosa. A significant

inhibition of the bacteria growth was observed, especially for silver nanoparticles who

AC

showed higher activity.

Antibacterial activity of silver nanoparticles of the size of 85 nm, obtained by green synthesis using aqueous extract of Delphinium denudatum was described in the work of Suresh et al. [255]. In two antibacterial tests: the agar well diffusion method and the microbroth dilution method, activity of the silver nanoparticles against Staphylococcus aureus ATCC 6538, Bacillus cereus NCIM 2106, Escherichia coli ATCC 8739, Pseudomonas aeruginosa ATCC 9027 was tested as a function of the nanoparticle bulk concentration. However, a rather high MIC values in the range of

250-500 mg L-1 were determined in

these experiments. Furthermore, the authors showed that the nanoparticles exhibited a considerable larvicidal activity against second instar larvae of dengue vector Aedes aegypti with a LC50 value of mg L-1. Bankura et al. [256] presented eco-friendly method of synthesis of silver-gold alloy nanoparticles (10 nm) using dextran polysaccharide as a reducing and stabilizing agent. 71

ACCEPTED MANUSCRIPT The authors showed, using the agar diffusion method that the Ag-Au alloy nanoparticles exhibit a high antibacterial activity against four bacterial strains: Bacillus subtilis [MTCC 736], Bacillus cereus [MTCC 306], Escherichia coli

[MTCC 68] and Pseudomonas

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aeruginosa [MTCC8158]. The nanoparticles showed at the concentration of 100 mg L-1

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a higher antimicrobial activity against Bacillus subtilis and Bacillus cereus than against the two other bacterial strains. The authors also confirmed that the antibacterial activity of silver

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nanoparticles is higher than Au nanoparticles reported earlier Kaittanis et al.[257], but not as efficient as that of dextran coated silver nanoparticles reported by Bankura et al. [256]. However, despite many literature reports on biocidal silver nanoparticle activity, the

NU

underlying physical mechanisms of these processes is not fully understood, in particular it is not known which factors play a decisive role, either the geometrical (such as particle size

MA

and morphology) or the surface chemical factors. Therefore, in Ref. [258] systematic

studies on the bactericidal effect of silver

nanoparticles having similar size (13-17 nm) and nearly spherical shape but varying in their properties were conducted. The

silver particle suspensions were obtained by

D

surfaces

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a chemical reduction of silver nitrate in the presence of various low molecular mass species both inorganic: sodium hypophosphite and sodium hexametaphosphate or organic phenolic

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compounds: gallic acid and tannin. The antimicrobial activity of these suspensions was quantitatively investigated by determining the MBC against

two different strains of

Escherichia coli ER 2566 and Escherichia coli K12. All suspensions exhibited a pronounced bactericidal effect against these strains. The most active against the K12 strain were the -1

AC

suspensions synthesized

in the presence of tannin and gallic acid, exhibiting MBC of

1–5 mg L . On the other hand, in the case of the tetracycline-resistant Escherichia coli strain, the highest activity (MBC of 10 mg L-1) was observed for the sol synthesized using sodium hypophosphate and sodium tripolyphosphate. Therefore, it was concluded the surface chemistry, that can be varied during the particle synthesis by selecting a proper combination of reducing and stabilizing agents, plays an important role, significantly enhancing the biocidal properties of silver particles. Moreover, these experimental proved that negatively charged silver nanoparticles can interact with

negatively charged bacterial cells that

contradicts the conclusions formulated in some previous works [259,260]. Summing up the extensive literature data one can formulate the conclusion that the most probable mechanism of biocidal activity of silver particles consists

of two steps

(i) a physical attachment of particles to the biomaterial and (ii) release of silver ions and other active chemical compounds present as stabilizers at the particles‘ surface. A direct penetration 72

ACCEPTED MANUSCRIPT of silver particles through the cell membrane inside the bacteria seems less probable. The overall activity of particles is, therefore, a product of the rates of these two processes. This means, that the particle size should be rather low (in the range of 10 nm) and the suspension

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should be stable because this increases the transport rate of particles to the biomaterial

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surfaces (that is much slower for larger particles and aggregates). Additionally, the particles should form a stable, long lasting

contact with the biomaterial that facilitates the

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accumulation of silver ions in the immediate vicinity of biosurfaces (bacteria walls). Besides these biocidal applications, silver nanoparticles are also often used in the surface-enhanced Raman spectroscopy (SERS) [10,40-43,56,87,212] and methods based on

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metal-enhanced fluorescence (MEF) [9,44]. The SERS technique consists in measuring the radiation scattered by analyte particles adsorbed on the metal surface when the signal of

MA

measured radiation is greatly enhanced in comparison to classical Raman measurement. Change in surface Raman extinction determined using the SERS technique provides information about the electronic structure of the adsorbate, and also significantly reduces the

D

threshold concentrations, which allows the measurement for dilute samples of analytes (of the

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order of 10-14 mol dm-3 [82,136,261]). SERS is a rapidly developing research method in which silver nanoparticles are one of the most commonly used materials [10,40-43,56,82,87,212]. In

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the literature one can find information about their use for the detection of trace amounts of pesticides, proteins, simple organic and inorganic compounds [82,136,261], the immunoassay labeling [45,46] and genetic diagnosis, including DNA sequencing [82,136,261]. Yang et al. [210] stated that the key to the wider application of Raman spectroscopy

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using roughened metallic surfaces is the development of highly enhancing substrates for analytical purposes. They developed a simple strategy for self-assembling silver nanochains on glass substrates for sensitive SERS substrates. The chain length of silver nanoparticles was controlled by adjusting the concentration of cetyltrimethylammonium bromide (CTAB) and 11-mercaptoundecanoic acid (MUA). CTAB with appropriate concentration served as the ―glue‖ that links the {100} facets of two neighboring Ag nanoparticles. MUA is found to be effective in ―freezing up‖ the aggregation of Ag short chains and preventing them from further aggregating into a long chainlike network structure. It was stated that surface plasmon bands can be tuned over an extended wavelength range by controlling the length of the nanochains. The Ag monolayer, mainly composed of four-particle nanochains, exhibited the maximum SERS enhancement factor of around 2.6 x108, indicating that a stronger SERS enhancement can be obtained in these interstitial sites of chainlike aggregated Ag nanoparticles. 73

ACCEPTED MANUSCRIPT In the other work of Yang et al. [262] the influence of various architectures of silver particles, from nanorods, triangular plates, hexagonal plates, and nanocubes on surface plasmon resonance was investigated. It was shown that these nanoparticles exhibit tunable

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surface plasmon resonance properties from the visible to near-infrared regions.

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The nanoparticles were also self-assembled on glass substrates and evaluated as potential surface-enhanced Raman scattering (SERS) substrates using trans-1,2-bis(4-pyridyl)ethylene

SC R

molecules. Triangular silver nanoparticles exhibited the largest enhanced SERS properties and were qualified as highly-sensitivity substrates for SERS-based measurements. Lee and Meisel [136] studied the interactions of an adsorbed dye with the energy

NU

states of the conduction band or surface plasmons of the gold and silver particles (20 nm). The adsorption of dyes was carried out in aqueous solutions. Highly efficient SERS scattering

MA

was observed from a carbocyanine dye for both suspensions. Once adsorbed onto the colloidal particle, the dye exhibited a strong surface enhancement of Raman scattering. In the case of carbocyanine dyes, the SERS could be easily observed at submicromolar dye concentrations

D

and thus rivals in its sensitivity conventional spectrophotometry.

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The SERS effect for dyes was also examined in the work of Nickel et al. [261] who obtained silver particles (40-70 nm) by a chemical reduction with hydrazine as a support for dye adsorption and SERS measurements. It was revealed that for silver precursor

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concentration of 1.5x10-4 M, the SERS spectra of dyes such as nile blue A could be recorded from a solution with concentrations as low as 10-10 M, whereas no SERS signal was observed for dye concentrations higher than 10-4 M. It was confirmed in this study that silver

AC

suspensions can be used for a qualitative detection of certain organic compounds. Silver nanoparticles and nanocomposites are also widely used in various catalytic processes [11-14,50,53]. Jiang et. al [263] investigated the catalytic effect of silver particles supported on silica spheres in the reduction of dyes (rose bengal, methylene blue, eosin). Tethering silver nanoparticles onto silica spheres effectively protects metal particles from aggregation. It was shown that surfactants and electrolytes have a profound influence on the catalytic activity of silver particles. A distinct decrease in the reduction rate of the dyes was observed upon the addition of surfactants (CTAB, SDS) due to the reduced accessibility of the particle surface. In the work of Gangula et. al [264], the green method for synthesizing gold (27 nm) and silver (64 nm) particles from the stem extract of Breynia Rhamnoides for catalytic purposes was described. The catalytic properties of obtained particles were compared with commercially available (chemically grown) nanoparticles. The dependence of the reaction 74

ACCEPTED MANUSCRIPT rates on various parameters such as the particle concentration, the nature of metal (gold, silver) in the catalytic conversion of 4-nitrophenol (4-NP) to 4-aminophenol (4-AP) was investigated. It was revealed that the conversion of 4-NP to 4-AP was the highest for gold

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nanoparticles (K = 9.19 x 10-3 s-1) compared to silver nanoparticles (K = 4.06 x 10-3 s-1). The

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slower reaction rate observed for silver was attributed to their relatively large size and the formation of surface oxide layer.

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The combined action of silver nanoparticles with NaBH4 in catalytic reduction of aromatic nitro compound was studied by Pradhan et al. [11]. It was confirmed that silver nanoparticles are the active component allowing the electron transfer from BH4− ion to the

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nitro compounds such as 2-nitrophenol (2-NP), 4-nitrophenol (4-NP) and 4-nitroaniline (4NA).

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Catalytic activity of silver particles was also described by Shiraishi and Toshima [47] who investigated the oxidation of ethylene. In this work colloidal silver particles of the size of 3.1 nm prepared by ethanol reduction of silver perchlorate in the presence of poly(N-vinyl-2-

D

pyrrolidone) (PVP) were used. Oxidation of ethylene catalyzed by PVP-protected silver

TE

colloids was performed in ethanol/water (1/1 v/v) at 90–95oC under 1 atm of ethylene/oxygen (2/1). It was shown that the catalysts in the form of colloidal silver had the higher catalytic

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activity than a commercial silver catalyst for oxidation of ethylene. It was also shown that the addition of alkali metal ions (cesium) markedly increased the catalytic activity of the colloidal silver catalyst.

Liang et. al. [265] and Mitsudome et. al.[266], studied alcohol oxidation using silver

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nanoparticle catalysts prepared by depositing silver nanoparticles on TiO2 nanotubes. The electrocatalytic properties of this system in the reaction of ethanol oxidation in alkaline media were investigated by cyclic voltammetry. It was shown that the catalyst exhibited a pronounce activity with the silver nanoparticles serving as the main dehydrogenation sites. On the other hand, Mitsudome et. al. [266] produced a highly efficient and reusable Ag/hydrotalcites catalyst for the heterogeneous dehydrogenation of alcohols under oxidantand additive free conditions. Silver catalysts have also become increasingly important in the selective oxidation of olefins for the synthesis of industrially interesting products such as epoxides and aldehydes [267]. Many works have been reported concerning the selective catalytic reduction of NOx to N2 by hydrocarbons (HC-SCR) using silver catalysts [53,268-273]. For example, Lindfors et al. [53] have demonstrated that the silver/alumina system is a very promising catalyst fin 75

ACCEPTED MANUSCRIPT this reaction. They investigated the influence of metal contents and role of the support, the concentration of the hydrocarbons on the reactions rate. It was revealed that the highest NOx reduction activity was observed for the silver /alumina catalysts having the silver loading of

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2 wt% [269-273]. For higher silver content, above 3 wt%, a sharp decrease in the catalyst‘s

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activity was observed. Miyadera [273] explained this by suggesting that that for higher silver loading the catalyst becomes more active in the reaction of total hydrocarbon oxidation that

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results in lower NOx conversion. Shimizu et al. [270] explained the role of silver loading by suggesting that below 2 wt%, mostly Ag+ ions are present at the alumina surface whereas at higher silver loading, Agn clusters are predominant. It was also further stated that the Ag+

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ions are responsible for the selective reduction of NO to N2, while the Agn clusters are responsible for the hydrocarbon combustion and N2O formation. In the review of Burch et al.

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[268] the properties of this silver/ alumina catalytic system have been thoroughly discussed. Pronounced catalytic activity of silver particles in the oxidation reaction of styrene was also observed [12,274,275]. Xu et. al. [274] studied the effect of the shape of silver

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nanoparticles on their catalytic activity. Truncated triangular silver nanoplates with well-

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defined planes, synthesized by a solvothermal approach were compared with that of cubic and near-spherical silver nanoparticles in the oxidation of styrene in colloidal solution. It was

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found that the crystal faces of silver nanoparticles play an essential role in determining the catalytic oxidation properties. The silver nanocubes having {100} crystal faces, the truncated triangular nanoplates and near-spherical nanoparticles predominantly exposed the most stable {111} crystal faces were studied. It was revealed that the rate of the reaction over the

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nanocubes was more than 14 times higher than that on nanoplates and four times higher than that on nearly spherical nanoparticles. Chimentao et. al. [12] investigated the influence of silver nanoparticle morphologies on the catalytic process of the selective oxidation of styrene in the gas phase. It was revealed that the both the activity and selectivity were strongly dependent on the morphology factor. The authors reported the synthesis of Ag nanoparticles using poly(vinylpyrrolidone) (PVP) as the template. Two morphologies of nanoparticles: silver nanowires and nanopolyhedra were obtained and investigated (see Fig. 35). Also, it was confirmed that the morphology and the chemical composition of the silver catalyst determined the activity and selectivity for the styrene oxidation reaction. It was found that the catalytic performance of the Ag nanowires for the selective oxidation of styrene can be improved by increasing the basic character of the catalyst, as well as the oxygen to styrene molar ratio. The synthesis of silver nanoparticles with a shape controller has potential applications for the selective oxidation of olefins. 76

Fig. 35.

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SEM images of silver catalysts: a) nanowires, b) nanopolyhedra, c) and

d)

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irregularly shaped particles studied in Ref. [12].

However, despite a considerable experimental effort, no general mechanism of silver

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particle action in these reactions has emerged. Also, none of the catalytic systems have been used in larger scale technical applications. Therefore, much more systematic work is needed in order to unequivocally prove

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what is the true active component in these catalytic reactions: the neutral silver atoms, oxidized forms of silver and the stabilizers used during the synthesis, silver ions or just the catalytic support.

Silver nanoparticles are also an important material for electronics to form conductive paths, data storage devices, anti-reflective materials. They are applied in fiber optics, single electron transistors, electronic connectors or integral capacitor banks [15,16,276,277]. Additionally, silver particles may be used in textile industry for coloration of fiber. For example, in the work of Tang et. al. [278], silver particles were assembled on cotton fibers to realize the coloration. The assembly of silver nanoparticles was achieved by linking of poly(diallyldimethylammonium chloride) (PDDA) at room temperature. The coloration was completed through electrostatic interaction between the PDDA treated cotton surface and the anisotropic silver nanoparticles in the reaction system. It was shown that the cotton colorized

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ACCEPTED MANUSCRIPT with silver nanoparticles showed reasonably good color resistance to washing, which will facilitate the practical application of this coloration process.

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5. Concluding remarks

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The chemical reduction method in aqueous media allows one to produce high purity

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charge stabilized silver suspensions of particle size that can be varied in broad limits by the synthesis conditions. Since the suspensions are stable for a broad range of ionic strength, pHs and bulk concentration, they can be efficiently used for the preparation of monolayers on solid substrates.

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Formation of these monolayers, their stability and particle release kinetics can be quantitatively studied by AFM and SEM imaging. In this way, the role of basic parameters

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such as the bulk suspension concentration, particle size, ionic strength, temperature and pH can be evaluated.

It is shown that the monolayer density and structure are governed by the

lateral

D

electrostatic interaction among deposited particles. Therefore, the maximum coverage that is

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experimentally accessible for higher ionic strength and particle size above 20 nm does not exceed 0.4. For smaller particle sizes and lower ionic strength, this value is considerably

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smaller.

The experimental kinetic results and the maximum coverage data obtained for various particle sizes and

ionic strength can be adequately interpreted in terms of the hybrid

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theoretical approach that combines the bulk transport step with the surface blocking effects derived from the random sequential adsorption model. This model also allows one to properly interpret particle release kinetics and to determine the equilibrium adsorption constant as a function of ionic strength for various particle size. Knowing the adsorption constant, the binding energy of particles with substrate surfaces

m can be calculated with a high precision. Thus, for the typical desorption

conditions of I = 10-2 M, pH 5.8, m varies between -17 and -19 kT for theparticle size of 15 and 54 nm, respectively. This minor dependence of the binding energy on particle size, ionic strength and temperature experimentally observed is properly explained using the ion pair concept for describing the electrostatic interactions of the silver particles with substrates. The classical theories based on the mean-field (averaged) zeta potential concept proved inadequate.

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ACCEPTED MANUSCRIPT It is also demonstrated that it is feasible to quantitatively monitor the formation of silver particle monolayers, bilayers and particle releases kinetics under in situ conditions using the streaming potential measurements that is considerably more robust than the

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AFM/SEM method.

suspensions and monolayers find

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Beside the significance for basic science as model colloid systems, the silver particle a wide spectrum of practical applications in biology,

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medicine, analytical chemistry and catalysis.

Acknowledgments

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This work was financially supported by the Research Grant: POIG 01.01.02-12-028/ 09-00.

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Appendix

Theoretical description of particle deposition/desorption based on the RSA model

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A quantitative description of particle deposition and release kinetics can be achieved

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in terms of the random sequential adsorption (RSA) approach. This model was successfully applied for describing irreversible adsorption (deposition) of colloid microparticles

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(polystyrene latexes) [229], nanoparticles [226-228] and proteins [230-232]. The method is extensively discussed in the book [225] and reviews [234,235]. According to the generalized RSA model, the constitutive equation describing the

ja =

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adsorption flux of particles to interfaces for an arbitrary adsorption mechanism is given by k 1 d  ka n( a ) B(   d  S g dt Sg

(A1)

where ja is the adsorption flux, S g is the characteristic cross-section area of the particle, t is the adsorption time, ka, kd are the adsorption and desorption kinetic constants, n(a) is the concentration of particles at the adsorption boundary layer of the thickness a, and B(θ) is the surface blocking function. For spherical particles the RSA blocking function is described by the expression

2

3

B( )  (1  a1  a2  a3 )(1   )3

(A2)

79

ACCEPTED MANUSCRIPT where    /  mx is the normalized coverage of particles, θmx is the maximum coverage of particles usually depending on the ionic strength [225,235], and a1 – a3 are the dimensionless coefficients equal to 0.812, 0.426 and 0.0717, respectively.

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The kinetic adsorption and desorption constant occurring in Eq.(A1) equations can be

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expressed in terms of physical parameters characterizing the transport conditions, such as the particle diffusion coefficient and the specific energy distribution governed by the depth of the

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primary minimum m , the energy barrier height b etc. [225,235]. Assuming a regular parabolic-like energy distribution near the minimum and the maximum (barrier), one can formulate the following expressions for the kinetic adsorption and desorption constants D  ka   b  eb / kT  a  kT 

(A3)

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1/2

1/2

D

k    kd  a  m  em / kT  m   kT 

energy minimum distance.

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where δm is the minimum distance between the particle and the interface, identified with the

D  1    a 1  ln a  2 m  

(A4)

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ka 

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For the barrier-less adsorption regime, b = 0, the adsorption constant is given by [225]

where  m is the characteristic distance from the interface where the interaction energy particle/interface becomes negligible. Eq.(A1) is nonlinear and it is coupled with the bulk transport equation. Therefore, it can be solved in the general case using only numerical methods. However, useful analytical solutions can be derived from Eq.(A1), in some limiting cases. For example, under equilibrium conditions, where the adsorption flux vanishes, Eq.(A1) reduces to the isotherm equation

e  S g K a nb B(e )

(A5)

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where  e is the equilibrium coverage of particles, Ka = ka / kd is the equilibrium adsorption constant and nb is the equilibrium concentration of particles in the bulk.

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Using Eq.(A5) with the RSA blocking function given by Eq.(A2), for fitting

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experimental data one can, in principle, determine the equilibrium adsorption constant. However, this becomes an impractical method in the case of strong adsorption (high affinity the bulk concentration range of particles becomes very low, and

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isotherms) because

consequently, the time of attaining equilibrium conditions excessively long. Another limiting analytical solution can be derived from Eq.(A1) in the case of a

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quasi-stationary transport, where the particle concentration n(a) remains in a local equilibrium with the surface coverage. This stems from the fact that the relaxation time of the

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bulk transport is usually much longer than the characteristic time of surface coverage variations [225]. In this case the particle flux to the adsorption layer is given

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K B( )  K d kc nb ( K  1) B( )  1

(A6)

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ja 

D

expression

by the

where K = ka / kc are the dimensionless coupling constants, Kd = kd/Sgkcnb is the dimensionless desorption constant, and kc is the bulk transfer rate constant, known in analytical form for

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many types of flows and interface configurations [225]. The dependence of the particle coverage on the time is obtained by integration of Eq.(A6) that which yields the expression 



0

( K  1) B( ' )  1 ' d  t / tch   K B( ' )  K d '

(A7)

where θo is the initial coverage of particles, tch = 1 /Sg kcnb is the characteristic time of particle monolayer formation under convection transport conditions and τ is the dimensionless time. Eq.(A7) can also be written in a more convenient, dimensional form 

(ka  kc ) B( ' )  kc '  Sg ka nb B( ' )  kd kc ' d  t 0

(A8)

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Eqs.(A7,A8) represent general solutions for particle adsorption/desorption kinetics under convection driven transport. Useful limiting forms of Eq.(A7) can be derived for large K, where the overall particle

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transfer rate is controlled by the bulk transport, i.e., if the inequality (K – 1) B (  ) >> 1, is

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fulfilled. In this case one obtains

 = t/ tch = τ = Sg kc nb t

(A9)

 

( K  1) B( ' )  1



0

'

d '  kd t

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

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On the other hand, for particle desorption runs, where nb → 0, Eq.(A8) becomes

(A10)

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However, a disadvantage of this method of determining the equilibrium adsorption

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constant is that one has to explicitly know the desorption constant kd that remains a fitting parameter.

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In the limit of low initial coverage by considering that the blocking function assumes a constant value close to unity denoted by B0 and ka << kc, Eq.(A10) can be analytically integrated giving the explicit expression: kc t K a B0

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  0e



(A11)

This equation indicates that the plot of ln (  /  0) on the desorption time t should give a straight line with the slope s1  

kc . Therefore, knowing kc, the equilibrium adsorption K a B0

constant can be determined from the simple expression Ka  

kc s1 B0

(A12)

A more complicated situation arises for the diffusion-controlled transport of particles. In this case the constitutive expression for the flux, Eq.(A1), cannot be integrated directly because the flux from the bulk to the interface and the concentration n(a) remain 82

ACCEPTED MANUSCRIPT non-stationary for at times. In this case, in order to explicitly evaluate particle adsorption kinetics, one has to solve the bulk mass transport equation, which assumes in the case of one-

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dimensional transport, the following form [225,235]:

(A13)

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n 2n D 2 t z

where n is the particle number concentration at the distance z from the interface, and D is the diffusion coefficient of the particle.

for z  

(A14)

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n  nb

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The bulk boundary condition for Eq.(A13) assumes the form:

On the other hand, the boundary condition at the edge of the adsorption layer should

k n 1 d   ka n( a ) B(   d at z   a z S g dt Sg

(A15)

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D

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D

reflect the continuity of the particle flux, which can be expressed using Eq.(A1) in the form

Because Eq.(A15) is non-linear in respect to the coverage, the entire boundary value

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problem described by Eqs.(A13-A15) also becomes non-linear and can only be solved numerically, using the efficient finite-difference method as discussed previously [225,235]. In order to perform such numerical calculations Eqs.(A13-A15) are usually converted to dimensionless forms.

However, useful limiting solution can be derived from Eqs.(A13-A15) for the quasistationary transport where the diffusion relaxation time in the adsorption layer is much smaller than the bulk transport relaxation time. In this case, the concentration n( a ) can effectively be treated as a constant and consequently Eq.(A13) can be integrated for a onedimensional transport to a planar surface. This yields the following expression for the flux of particles

1 d D ja =  [nb  n( a )]   S g dt  t 

1/2

(A16)

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where n(a) is connected with the coverage vie the constitutive dependence



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S g K a B( )

(A17)

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n( a ) 

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Thus, in the case of irreversible adsorption , where Ka→∞ , n(a) → 0, Eq.(A16) becomes

1 d  D     nb S g dt  t 

(A18)

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1/2

ja =

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By integrating this dependence, one obtains the explicit expression for the particle coverage as a function of time

D

1/2

nb

(A19)

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 Dt    

  2S g 

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As can be noticed, in this case, the coverage of particles is a linear function of the square root of the time, rather than the time, as is the case for convection controlled transport.

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The surface concentration of particles is given by the expression

1

 Dt  2 N  2  nb  

(A20)

On the other hand, in the case of desorption runs, where nb = 0, Eqs.( A16) becomes d  D   dt K a B( )   t 

1/2

(A21)

Integrating Eq.(A21), one obtains

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

0

B( ' )d '

'

1/2

2  Dt  =  Ka   

(A22)

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In general, due to the non-linear character of the blocking function, Eq.(A22) can only be

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evaluated numerically. However, for a low coverage range, assuming as before that B(θ)

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remains constant and equal to B0, Eq.(A22) can be integrated to the form

  2  Dt  ln       K a B0     0 

(A23)

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1/2

Eq.(A23) can be used for a robust determination the equilibrium adsorption constant

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of particles by plotting the experimental dependence of ln (θ/θo) on the square root of the desorption time t1/2. Knowing the slope of this dependence sl= Δln(θ/θo) /Δt1/2 , the adsorption

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constant can be calculated as

1/2

(A24)

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2 D Ka     sl B0   

Knowing the equilibrium adsorption constant Ka = ka/kd one can determine the energy

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minimum depth by observing that [225]

  kT K a   m   m

1/2

 m / kT  e 

(A25)

Eq. (A25) can be iteratively solved, which results in the approximate expression  Ka ln Ka 1   m m / kT   ln  ln  m 2    

     

(A26)

By considering Eq.(A4) one knowing Ka one can calculate the kinetic desorption constant the dependence

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 K a a 1 

D 1    ln a  2 m 

(A27)

Ka one can obtain the

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Moreover, knowing the equilibrium adsorption constant

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kd = ka / Ka

isotherm of particle adsorption by numerically solving Eq.(A5). This is more efficient than

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any direct measurements of the equilibrium coverage vs. the bulk suspension concentration.

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