Immersion Depth of Positively versus Negatively Charged

Sep 12, 2012 - of charged nanoparticles that are trapped at an air−water interface. ... immersion depth with respect to complete charge reversal. Th...
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Immersion Depth of Positively versus Negatively Charged Nanoparticles at the Air−Water Interface: A Poisson−Boltzmann Model Ahis Shrestha,† Klemen Bohinc,‡ and Sylvio May*,† †

Department of Physics, North Dakota State University, P.O. Box 6050, Fargo, North Dakota 58108, United States Faculty of Health Sciences, University of Ljubljana, Zdravstvena 5, SI-1000 Ljubljana, Slovenia

‡

ABSTRACT: Electrostatic interactions affect the immersion depth of charged nanoparticles that are trapped at an air−water interface. Recent experiments indicate that upon adding salt negatively charged nanoparticles penetrate deeper into the aqueous phase, whereas positively charged nanoparticles exhibit opposite behavior. It has been proposed that this unexpected lack of invariance with respect to the nanoparticle’s charge reversal is caused by a negative surface potential of the air−water interface. To support this hypothesis, we have performed detailed calculations based on nonlinear Poisson− Boltzmann theory of individual spherical particles that are either negatively or positively charged and reside at the interface between air and water. The nanoparticles possess dissociable surface groups that become charged when exposed to the aqueous environment. We calculate the optimal immersion depth from a numerical minimization of the total free energy, which we express as the sum of a surface tension term and an electrostatic contribution. In all calculations we fix the surface potential at the air−water surface at −50 mV. In qualitative agreement with recent experiments, our model predicts opposite behaviors of negatively versus positively charged nanoparticles: adding salt increases/decreases the water immersion depth of negatively/positively charged nanoparticles.

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INTRODUCTION Interactions between colloids trapped at fluid interfaces have been the subject of intensive studies in recent years. Besides the practical relevance of colloid-decorated interfaces as emulsion stabilizers,1 colloidosomes,2 and bijels,3,4 these systems also pose interesting physical questions related to the ability of the interface to mediate many-body interactions between the trapped colloids.5 Interfacial deformation can arise simply due to the gravitational weight of colloids, typically with sizes above the capillary length (≈2 mm in water) but also for particles with sizes down to about 10 μm.6 A situation of considerable complexity arises when colloids carry electrical charges. Chargemediated forces then depend on various factors such as the presence of mobile salt ions, the dielectric mismatch across the interface, and the ability of the colloids to regulate their charge density.7−10 Even in the dilute limit (where the concentration of colloids at the fluid interface is small) and for sufficiently small colloids (i.e., up to the micrometer range, where gravitational effects can be neglected), there are interesting experimental observations that challenge our theoretical understanding of nanoparticle−interface interactions. In the following we briefly discuss such an experiment. Gehring and Fischer11 (see also previous work by Dhar et 12 al. ) have used particle tracking to observe the diffusion of individual, negatively or positively charged nanoparticles trapped at the air−water interface. The two types of nanoparticles were either carboxylate-modified or aminomodified, with sizes of about 100 and 200 nm, respectively. © 2012 American Chemical Society

The nanoparticle’s diffusion constant was determined as a function of added monovalent salt in the aqueous subphase, ranging from 0.1 μM to 10 mM of added sodium chloride. As discussed theoretically by Fischer et al.,13 the diffusion constant of the nanoparticles is related to their immersion depth at the air−water interface. On the basis of their diffusion data, Gehring and Fischer arrive at the conclusion that the addition of salt has a qualitatively different effect on the two oppositely charged nanoparticles: while the negatively charged particles become immersed deeper into the water, the positively charged particles are displaced away from the aqueous phase. This result appears surprising in light of the expected invariance of the immersion depth with respect to complete charge reversal. That is, replacing a negatively by a positively charged nanoparticle also changes the sign of the diffuse counterion layer and thus should not lead to a different salt dependence of the immersion depth. To reconcile their experimental results,11 Gehring and Fischer point out that the air−water interface carries an excess of OH− ions that leads to a more negative potential in the air as compared to the bulk water. The corresponding surface potential is of the order of about −50 mV,14 comparable in magnitude to the zeta-potential of the nanoparticles. It should be pointed out that the magnitude, and even the sign, of the potential drop across the air−water interface are not wellReceived: August 5, 2012 Revised: September 11, 2012 Published: September 12, 2012 14301

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known and remain a subject of research,15,16 but any nonvanishing surface potential breaks the charge reversal symmetry of replacing negatively by positively charged nanoparticles. It is reasonable to speculate that a reduction of the electrostatic screening (i.e., a reduction of the amount of salt in the system) has opposite effects on the immersion depth of positively versus negatively charged nanoparticles if a nonvanishing surface potential is present at the air−water interface. Consequently, Gehring and Fischer propose to incorporate the potential drop across the air−water interface into modeling the electrodipping of the nanoparticles.11 This is the subject of the present work. We study a model for the immersion depth of a single spherical nanoparticle at the air−water interface. The particle is assumed to carry either acidic or basic surface groups that become charged via deprotonation or protonation in presence of an aqueous medium but remain neutral when exposed to air. We assume the particle has an intrinsic nonelectrostatic propensity to associate with the interface due to its surface tensions with water and air. Additional electrostatic interactions will then modulate the immersion depth. We model the electrostatic interactions based on the classical nonlinear continuum Poisson−Boltzmann model, assuming (as Gehring and Fischer have suggested11) that the air−water interface carries a constant negative potential of −50 mV. To simplify the numerical calculations, we make a number of additional assumptions: (i) the air−water surface remains perfectly flat, (ii) the electrostatic field outside the aqueous region is negligibly small due to the large mismatch of the dielectric constants between air and water and due to the absence of charges on the air-immersed region of the nanoparticle, and (iii) hysteresis of the contact angle on the particle surface is negligible. With this, the equilibrium immersion depth is calculated from the minimum of the totalelectrostatic plus nonelectrostaticfree energy. On the basis of detailed numerical calculations, we demonstrate the salt dependence of the nanoparticle’s immersion depth to be qualitatively consistent with the experimental results from Gehring and Fischer:11 negatively charged nanoparticles penetrate deeper into to aqueous phase as salt is added, whereas positively charged nanoparticles exhibit opposite behavior.

Figure 1. Schematic illustration of an uncharged nanoparticle of radius R trapped at the air−water interface. The nanoparticle partially penetrates from the water into the air. This is characterized by the dimensionless variable s with −1 ≤ s ≤ 1. For s = 1 the particle is fully immersed in water, for s = 0 the particle’s equator coincides with the air−water interface, and for s = −1 the particle is fully immersed in air. Note that at equilibrium s = sopt = cos θc is related to the classical contact angle θc as indicated. The air−water, nanoparticle−air, and nanoparticle−water surface tensions are denoted by γaw, γna, and γnw, respectively.

− Fref T of transferring the nanoparticle from its fully waterimmersed state to the air−water interface can be expressed as γ − γnw γ ΔFT = na (1 − s) − aw (1 − s)(1 + s) 2 (1) 2 4 4πR In the absence of additional contributions to the free energy, ΔFT(s) would be minimized for the optimal (dimensionless) penetration depth s = sopt with γ − γnw s opt = na γaw (2) and the resulting minimal transfer free energy is ΔFT(sopt) = −πR2γaw(sopt − 1)2. Note that eq 2 is known as Young’s equation where sopt is related to the classical contact angle17 θc through sopt = cos θc. Partitioning into the interface obviously requires the air−water surface tension to be sufficiently large, γaw > |γna − γnw|. If the air−water surface tension is too small, the nanoparticle would remain fully immersed in one phase, in water for γnw < γna, and in air for γna < γnw. In the present study we find it convenient to simply chose γnw = γna, implying equatorial partitioning sopt = 0 in the complete absence of electrostatic interactions, irrespective of the air−water interfacial tension γaw. We now consider the spherical nanoparticle to carry electrical charges. In what follows, we account for the formation of the diffuse double layer of counter- and co-ions explicitly within the nonlinear Poisson−Boltzmann framework. Because of the system’s axial symmetry, it is convenient to use cylindrical coordinates (r, ϕ, z) with radial distance r, azimuthal angle ϕ, and longitudinal axis of symmetry z directed toward the aqueous phase such that the plane z = 0 coincides with the air−water interface (see Figure 2). As pointed out above, we assume the nanoparticle carries either acidic or basic surface groups that become charged (via deprotonation or protonation) when the particle is solvated. That is, the particle has a uniform surface charge density σ when residing in the bulk aqueous phase (s ≫ 1). It then carries its maximal number of charges

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THEORY Consider a sufficiently small (i.e., up to the micrometer range) spherical nanoparticle of radius R that is trapped at the air− water interface (see Figure 1). Assume a fraction (1 + s)R of the entire diametrical extension 2R of the nanoparticle is immersed into the aqueous region. Here, s is defined as dimensionless immersion depth with −1 < s < 1 for partial immersion (complete immersion into air for s ≤ −1 and complete immersion into water for s ≥ 1). For s = 0 the equatorial region of the nanoparticle coincides with the air−water interface. We initially assume the nanoparticle is uncharged and denote the surface tensions between air−water, nanoparticle−air, and nanoparticle−water by γaw, γna, and γnw, respectively. For a nanoparticle immersed completely into water (i.e., s ≥ 1) the 2 free energy is Fref T = γnw4πR + γawAaw, where Aaw is the total lateral extension of the air−water interface. Here and in the following we use the index “T” for “tension” to denote nonelectrostatic free energy contributions. For a partially immersed particle (i.e., −1 < s < 1) the interface remains flat and the free energy is FT = 2πR2[γnw(1 + s) + γna(1 − s)] + γaw[Aaw − πR2(1 − s)(1 + s)]. Hence, the free energy ΔFT = FT

N = 4πR2 14302

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specific salt effects) remain an area of active research.23 In the present work we will imposeas suggested by Gehring and Fischer11a fixed negative and uniform surface potential Φ0 = −50 mV, irrespective of the distance to the nanoparticle. We emphasize that using a uniform potential represents an approximation, corresponding to the thermodynamic limit that the surface charges giving rise to Φ0 can migrate freely (i.e., without invoking a free energy penalty) along the air−water interface. Ignoring the salt dependence of Φ0 constitutes an additional approximation that we justify by focusing on small amounts of added salt. Modeling the surface energetics more accurately is beyond the scope of the present work; it would lead to an appropriate fixed electrochemical potential. This would affect the results, but not likely in a qualitative manner. Within the Poisson−Boltzmann framework we express the local concentrations of monovalent salt cations and anions by n+ and n−, respectively. In the bulk n+ = n− = n0. Moreover, we introduce the dimensionless electrostatic potential Ψ = eΦ/ kBT, where Φ is the electrostatic potential and kBT denotes the thermal energy unit. Because, first, the air-exposed region of the nanoparticle is uncharged and, second, water has a much larger dielectric constant than both air and the interior of the nanoparticle, the electric field penetrating into air and nanoparticle is smallwe neglect it in the present work. We then need to focus only on the electric field inside the aqueous region V, i.e., associated with the diffuse layer of mobile salt ions in the vicinity of the nanoparticle and air−water interface. The mean-field electrostatic free energy F̃ el of a single nanoparticle at the air−water interface can be written according to the Poisson−Boltzmann model as17,24−26

Figure 2. Schematic illustration of a positively charged nanoparticle at the air−water interface. The water-exposed region of the nanoparticle carries a uniform charge density σ, corresponding to N(1 + s)/2 actual charges. Here, N = 4πR2|σ|/e is the maximal number of charges the nanoparticle can carry when fully solvated, and s is the dimensionless immersion depth (see Figure 1). The air−water interface has a constant negative surface potential Φ0. Monovalent salt anions and cations are present, each with bulk concentration n0. The system exhibits cylindrical symmetry about the z-axis. We have solved the Poisson−Boltzmann equation (see eq 10) within the aqueous solution (the shaded region), subject to fixing both σ and Φ0.

where e denotes the elementary charge. Upon partially penetrating into the air, the water-immersed region of the nanoparticle retains its constant surface charge density σ, whereas the air-exposed surface groups protonate/deprotonate into their uncharged state, implying a vanishing surface charge density of the nanoparticle’s air-exposed region. (Without becoming uncharged, the air-exposed nanoparticle region would induce long-ranged electrostatic interactions through the air.18 In addition, the presence of charges at the air-exposed region of the nanoparticle affects the immersion depth and leads to deformations of the air−water interface.19) It is important to point out that the free energy corresponding to the charging/discharging (via protonation or deprotonation) is independent of the salt content and proportional to the airexposed surface area of the nanoparticle; hence, it can be incorporated into γna. Figure 2 shows a schematic illustration of a single spherical nanoparticle, carrying a positive surface charge density σ at the water-immersed region. As indicated in Figure 2, the aqueous solvent region contains a symmetric 1:1 salt solution, i.e., monovalent anions and cations, each with bulk concentration n0. A crucial point of the present work is related to the electrostatic properties of the air−water interface. Experimental measurements and theoretical estimates of the surface potential Φ0 (i.e., the potential in air minus the potential in water) vary considerably, roughly between −3 and +1 V.15,16 For example, using classical molecular dynamics with polarizable force fields, Wick et al.20 obtained Φ0 = +0.5 V and a weak dependence on added salt (KCl). Detailed quantum mechanical molecular dynamics recently yielded Φ0 = +28 mV based on effective partial charge models of liquid water16 and Φ0 = −3.1 V based on explicit ab initio electronic charge density.21 The negative sign of Φ0 is in agreement with various experimental determinations as discussed by Kathmann et al.,21 including the Φ0 = −65 mV for the zeta-potential of an air bubble in deionized water as determined from electrophoretic mobility measurements.22 Determination of the accurate surface potential and its dependence on the presence of salt (including

Fel̃ = kBT

⎡

2

) ∫V dv⎢⎢⎣ (∇Ψ 8πl B

⎛ n ⎞ ⎛ n ⎞ + n+⎜ln + − 1⎟ + n−⎜ln − − 1⎟ ⎝ n0 ⎠ ⎝ n0 ⎠

⎤ + 2n0 ⎥ ⎥⎦

(4)

Here, the integration runs over the volume V of the aqueous space, with the three terms in the integrand corresponding to the energy stored in the electric field and the demixing entropy of the salt cations and anions. Note that lB is the familiar Bjerrum length (lB = 0.7 nm in water). This length is defined such that at a separation lB two elementary charges experience an interaction energy equal to the thermal energy kBT. We point out that F̃ el vanishes for a hypothetically uncharged system, where Ψ = 0 and n+ = n− = n0 everywhere. Hence, eq 4 represents the so-called charging free energy.17 We need to point out that F̃el represents the correct thermodynamic potential if all macroions have a fixed surface charge density. In our present system, this is the case for the nanoparticle, which carries a fixed σ within the water-immersed region. However, for the air−water surface the local surface charge density, σaw, is not fixed. Instead, we fix the dimensionless surface potential Ψ0 = Ψ(z=0) = eΦ0/kBT. (Note that at room temperature Ψ ≈ Φ/(25 mV), and hence, the fixed surface potential Φ0 = −50 mV corresponds to Ψ0 = −2.) It is therefore that we need to apply a Legendre transformation from F̃ el(σ,σaw) to Fel(σ,Ψ0), where Fel expresses the correct thermodynamic potential for fixed Ψ0 at the air− water surface. This Legendre transformation is given by25,27 Fel = Fel̃ − kBT

∫A

aw

14303

da Ψ

σaw e

(5)

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where in our particular case Ψ = Ψ0 = −2 is a constant everywhere at the air−water surface and where the integration runs over the air−water surface Aaw. Clearly, the thermodynamic potential Fel according to eqs 5 and 4 (and not F̃el) must be used in the present work to derive the Poisson−Boltzmann equation and compute the relevant free energy. The dimensionless potential Ψ fulfills the Poisson equation ΔΨ = −4πlBρ/e, where Δ denotes the Laplacian and where the local charge density ρ = e(n+ − n−) accounts for the presence of the salt ions. In addition, at the water-immersed surface region of the nanoparticle the fixed surface charge density σ determines the derivative of the dimensionless potential ⎛ ∂Ψ ⎞ σ ⎜ ⎟ = −4πlB ⎝ ∂n ⎠nw e

We briefly discuss the electrostatic reference free energy Fref el pl = Fsph el + Fel . It consists of two independent contributions: the pl charging free energies Fsph el and Fel of a single sphere in bulk water and an unperturbed planar surface, respectively. For the former an analytical solution is not known, but Fref el can be calculated numerically, based on solving the Poisson− Boltzmann equation Ψ″(r) + (2/r)Ψ′(r) = (1/lD2) sinh[Ψ(r)] in spherical coordinates, where r is the radial distance, subject to the boundary conditions, Ψ′(r)|R = −4πlBσ/e and Ψ′(r)|r→∞ = 0. Here, a prime denotes the derivative with respect to the argument. The corresponding free energy, based on the familiar 2 σ charging process,17 can be expressed as Fsph el = 4πR ∫ 0Φ(σ̅) dσ̅. For the latter (i.e., the charging free energy of an unperturbed planar surface, Fplel ) an analytical expression follows easily from the charging process

(6)

Felpl 1 =− kBTAaw e

along the normal direction of the nanoparticle−water interface, pointing into the electrolyte. Similarly, at the air−water interface Aaw, the relation (∂Ψ/∂z)z=0 = −4πlBσaw/e between the derivative of the dimensionless potential (∂Ψ/∂z)z=0 and the local surface charge density σaw holds. However, unlike σ, the surface charge density σaw of the air−water interface is not fixed. Instead, as discussed above, we fix the dimensionless surface potential Ψ0. The local ion concentrations n+ and n− appear in the free energy Fel as unconstrained variables. Minimization of Fel(n+,n−) subject to Poisson’s equation yields the familiar Boltzmann distributions n± = n0 exp(∓Ψ) and hence gives rise to the standard nonlinear Poisson−Boltzmann equation lD2ΔΨ = sinh Ψ where the Debye screening length lD is a measure of the salt concentration n0 through 1/lD2 = 8πlBn0. The boundary conditions for solving the Poisson−Boltzmann equation are determined by, first, fixing σ according to eq 6 and, second, keeping the surface potential Ψ(r , z = 0) = Ψ0 = −2

∫A

+

da Ψσ − nw

1 8πlBlD2

(9)

RESULTS AND DISCUSSION We have solved the standard nonlinear Poisson−Boltzmann equation ∂ 2Ψ 1 ∂Ψ ∂ 2Ψ 1 + + = 2 sinh Ψ 2 2 r ∂r ∂r lD ∂z

(10)

for the cylindrically symmetric potential Ψ(r,z) within the aqueous phase next to a single spherical nanoparticle at the air− water interface. This is illustrated in Figure 2, where the shaded region marks the aqueous phase. Recall our choice of the cylindrical coordinate system with its long axis normal to the air−water interface passing through the center of the nanoparticle, thus rendering eq 10 independent of the azimuthal angle ϕ. Numerical solutions of eq 10 are determined subject to the boundary conditions in eqs 6 and 7, accounting for the fixed surface charge density σ at the water-immersed region of the nanoparticle as well as for the fixed potential Ψ0 = −2 at the air−water surface. Our numerical method involves transforming the nonlinear Poisson−Boltzmann equation into a sequence of linearized differential equations that were solved using a Newton−Raphson iteration scheme.28 From the solutions we have computed the electrostatic contribution to the transfer free energy ΔFel as described above, following eq 8, and from that the total transfer free energy ΔF. We base all our calculations on a nanoparticle radius R = 10 nm, which we subsequently use as our unit length. Numerical results for ΔFel were computed for various values of the dimensionless numbers N = 4πR2|σ|/e, lD/R, and s, all for fixed Ψ0 = −2 and lB/R = 0.07. We point out that our choice lB/R =

da Ψσaw]

aw

∫V

4 sinh2(Ψ0/4) 2πlBlD

■

(7)

∫A

σ(Ψ̅0) dΨ̅0 = −

Here σ(Ψ0) = sinh(Ψ0/2)e/(2πlBlD) is the surface charge density as a function of the surface potential of the unperturbed air−water interface (i.e., in the absence of the nanoparticle). Equation 9 can be viewed as electrostatic contribution to the 2 total surface tension γtot aw = γaw − 4 sinh (Ψ0/4)/(2πlBlD) at the air−water surface. Here, the negative sign indicates that the fixed potential Ψ0 tends to lower the total surface tension. With increased salt content (i.e., smaller Debye length lD) the surface tension would eventually vanish, namely for lD* = 4 sinh2(Ψ0/ 4)/(2πlBγaw). To prevent this scenario from happening (which is a consequence of keeping Ψ0 strictly constant, even for small lD), we demand lD ≫ lD*. In this low salt regime, the surface tension at the air−water interface γtot aw ≈ γaw is independent of the salt concentration.

at the air−water interface constant. Also, in the bulk aqueous phase, the potential vanishes; Ψ(z→+∞) = 0. Using the Boltzmann distributions as well as Gauss’ law (for details of the derivation see the Appendix in ref 28), we can reexpress the free energy in eq 4 equivalently as Fel 1 = [ kBT 2e

∫0

Ψ0

dv [Ψ sinh Ψ − 2 cosh Ψ + 2] (8)

We point out that the minus sign in front of the second integral of eq 8 results from the Legendre transformation (see eq 5). After numerically solving the Poisson−Boltzmann equation (yielding Ψ), we can use eq 8 to calculate the electrostatic free energy. This applies to any immersion depth, i.e., to any value of s with −1 ≤ s ≤ 1. We can even consider the limit s → ∞, where the nanoparticle is fully immersed in the bulk of the aqueous phase. In this case, Fref el = Fel(s → ∞) defines the reference state with respect to which we measure the electrostatic contribution to the transfer free energy ΔFel = Fel − Fref el . (Recall that the transfer free energy refers to moving the nanoparticle from its fully water-immersed state to the air− water interface.) The total transfer free energy of the system is the sum of the nonelectrostatic and electrostatic contributions, ΔF = ΔFT + ΔFel. It adopts a minimum ΔF(sopt) with respect to the dimensionless immersion depth s = sopt. 14304

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0.07 recovers the Bjerrum length lB = 0.7 nm in water. Recall that N is the number of charges of a fully charged nanoparticle (i.e., when the nanoparticle is fully immersed in water; s ≥ 1). In principle, our calculations can be applied to a nanoparticle of any radius R, yet with a corresponding rescaling of the Bjerrum length lB. For, say R = 20 nm, our calculations would correspond to a Bjerrum length of lB = 1.4 nm and thus a solvent’s dielectric constant of 40. We have not carried out systematic calculations for scaled Bjerrum lengths other than lB/R = 0.07. Hence, in the present work we do not consider variations of the particle size R. Figure 3 shows our numerical results for ΔFel, derived for lD/ R = 2 (left diagram) and lD/R = 0.5 (right diagram) and three

differences between the two curves of each pair result from the nonvanishing surface potential Ψ0 = −2. Positively charged nanoparticles interact favorably with a surface of fixed negative potential, implying a more favorable electrostatic contribution ΔFel to the transfer free energy as compared to negatively charged nanoparticles. The behaviors of the functions ΔFel(s) are qualitatively different for positively and negatively charged nanoparticles. For our discussion below, the more relevant of the two diagrams in Figure 3 is the left one; i.e., lD/R = 2. Here, the electrostatic contribution ΔFel to the transfer free energy always becomes more favorable (i.e., ΔFel decreases) as a positively charged particle becomes immersed deeper (growing s) into the aqueous phase. This is due to the favorable interaction of the positive charges of the nanoparticle with the air−water surface. The behavior of a negatively charged nanoparticle is opposite: here a less deep immersion into the aqueous phase is preferred from an electrostatic point of view. We point out that ΔFel(s = −1), where the particle becomes immersed fully into air, corresponds to the negative value of the charging free energy Fsph el of a single sphere in bulk water. This charging free energy exhibits charge reversal symmetry; it is thus the same for positively and negatively charged nanoparticles. Hence, any two corresponding curves for oppositely charged nanoparticles in Figure 3 must approach each other in the limit s → −1. Recall the nonelectrostatic contribution ΔFT = −πγawR2(1 − 2 s ) to the total transfer free energy which, on its own, favors equatorial immersion of the nanoparticle into the air−water interface. We also recall that this equatorial preference is a consequence of our assumption γna = γnw. In the presence of electrostatic interactions, the optimal dimensionless immersion depth s = sopt corresponds to the minimum of ΔFel + ΔFT. As is evident from the left diagram of Figure 3, the electrostatic contribution to the transfer free energy shifts the minimum in opposite directions for oppositely charged nanoparticles. Positively charged particles benefit from the interaction with the air−water interface due to the negative surface potential and thus tend to be immersed deeper into the aqueous phase. In contrast, negatively charged particles are pushed toward the air. This behavior, which is qualitatively the same irrespective of the exact value of γawR2, is modulated by the salt content as expressed by the Debye length lD. We analyze this dependence in the following. In Figure 4, we show the optimal dimensionless immersion depths sopt for the different maximal numbers of surface charges N used in Figure 3 (marked by the same symbols) as a function of the scaled Debye length lD/R. The two diagrams in Figure 4 correspond to two different air−water surface tensions, γawR2 = 12 kBT and γawR2 = 48 kBT, as indicated. We have used two different values for the air−water surface tension to illustrate the influence of that parameter. In fact, our two choices span the most interesting region where salt-induced changes of the immersion depth of the nanoparticles are significant. Yet, we point out that with R = 10 nm both choices of γaw are much smaller than the nominal surface tension γaw = 12 kBT/nm2 of the bare air−water interface.29 It is also worth pointing out that the experiments by Gehring and Fischer11 were carried out in the presence of an uncontrollable amount of surfactants at the air−water interface. Yet, the surface concentration of the surfactants was still too small in the experimental work to significantly affect the air−water interfacial tension. We first discuss the curves in Figure 4 marked by the symbol ●, which refer to a neutral nanoparticle (N = 0). For

Figure 3. Electrostatic contribution to the transfer free energy ΔFel as a function of the dimensionless immersion depth s. The left and right diagrams are derived for lD/R = 2 and lD/R = 0.5; different curves correspond to different N = 0 (●), N = 25 (○), and N = 50 (◇). For the latter two cases, the lower and upper curves correspond to positively (“pos”) an negatively (“neg”) charged nanoparticles, respectively.

different values of N, all plotted as a function of the dimensionless immersion depth s. We begin our discussion of the results in Figure 3 with the curves marked by the symbol ●. These curves refer to uncharged nanoparticles; i.e., N = 0. Despite their lack of charges, these particles exhibit an electrostatic interaction with the air−water interface. This interaction is unfavorable (ΔFel > 0) and is mainly caused by the loss of surface area Aaw upon partitioning into the interfacial region. Recall that our condition of a nonvanishing surface potential Ψ0 reduces the surface tension γtot aw; a smaller interfacial area Aaw is then unfavorable from an electrostatic point of view. We also observe that ΔFel(s) is not symmetric with respect to s → −s, despite the change in Aaw being the same for s and −s. This is clearly visible for lD/R = 0.5; see the right diagram of Figure 3. This asymmetry can easily be explained: an uncharged nanoparticle that penetrates more into the aqueous phase displaces the electric field due to its low dielectric constant. Hence, the airimmersed state s < 0 is more favorable than the corresponding water-immersed state s > 0, implying ΔFel(−|s|) < ΔFel(|s|) as seen most clearly in the right diagram of Figure 3. Next we discuss the case of nonvanishing nanoparticle surface charge in Figure 3. A pair of two curves that join each other at s = −1 is displayed for each nonvanishing N (marked by the symbol ○ for N = 25 and by ◇ for N = 50). The lower and upper curves of each pair correspond to positively and negatively charged nanoparticles, respectively. Clearly, the 14305

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N = 50 that we have investigated in Figures 3 and 4 qualitatively reproduce the results by Gehring and Fischer.11 However, upon further increasing N the behavior changes qualitatively. Figure 5 shows sopt = sopt(N) at fixed lD/R = 2 and

Figure 4. Optimal dimensionless immersion depth sopt as a function of the scaled Debye length lD/R. The left and right diagrams are derived for γawR2 = 12 kBT and γawR2 = 48 kBT; different curves correspond to different N = 0 (●), N = 25 (○), and N = 50 (◇). For the latter two cases, the lower and upper curves correspond to negatively (“neg”) and positively (“pos”) charged nanoparticles, respectively. The vertical dashed line in each diagram marks l*D/R.

Figure 5. Optimal dimensionless immersion depth sopt as a function of N. Left and right diagrams correspond to γawR2 = 12 kBT and γawR2 = 48 kBT, as indicated. In both diagrams, the lower and upper curves correspond to negatively (“neg”) and positively (“pos”) charged nanoparticles. All results are computed for lD/R = 2.

sufficiently large Debye lengthfor both diagrams roughly lD/ R ≳ 2the optimal penetration depth is equatorial; s = 0. However, below lD/R ≈ 2 the neutral nanoparticle starts preferring to be immersed more in air than in water. Generally, the presence of a low dielectric body penetrating into the aqueous phase prevents the formation of an electric field in that region. The corresponding energy penalty becomes significant when the Debye length approaches the size of the nanoparticle. Hence, the nanoparticle becomes displaced toward the air. In addition, when the Debye length decreases, the total air−water surface tension γtot aw becomes smaller due to its electrostatic contribution and eventually vanishes for lD*. To avoid this unrealistic scenario, we focus on lD ≫ lD*. (Recall that this requirement is a consequence of fixing Ψ0, even for small lD.) We have marked the position of l*D/R in both diagrams of Figure 4 by a vertical dashed line. Next, we discuss charged nanoparticles, for which the salt dependence of their optimal immersion depth sopt is displayed in Figure 4; see the symbols ○ for N = 25 and ◇ for N = 50. For each nonvanishing N the lower and upper curves correspond to negatively and positively charged nanoparticles. Note the logarithmic scale of the lD/R-axis. For lD ≫ l*D both diagrams make the same qualitative prediction about the salt dependence of the immersion depth: adding salt displaces negatively charged nanoparticles deeper into the water and positively charged nanoparticles toward the air. This opposite behavior agrees qualitatively with the interpretation of the experimental results by Gehring and Fischer11 as discussed in the Introduction. As expected, the changes of the immersion depth with salt are more pronounced for γawR2 = 12 kBT (left diagram) than for γawR2 = 48 kBT (right diagram). Further increasing the air−water surface tension would render the salt dependence of the immersion depth even smaller. Given γawR2 = 1200 kBT (this corresponds to the nominal surface tension of water γaw = 12 kBT/nm2 and a nanoparticle radius R = 10 nm), we would obtain very small changes of the immersion depth indeed. For example, with lD/R = 10 and N = 50, the immersion depth of positively and negatively charged nanoparticles would be sopt = 0.003 and sopt = −0.009, respectively. Another interesting parameter is the maximal number of surface charges N on the nanoparticle. The values N = 25 and

two different air−water surface tensions, γawR2 = 12 kBT (left diagram) and γawR2 = 48 kBT (right diagram). Clearly, only for sufficiently small N are the nanoparticles displaced in different directions. For large N, the particles possess a large electrostatic charging free energy, when solvated in the aqueous phase. It is then favorable for the system to expel the particles into the air (sopt < 0), irrespective of the air−water surface potential Ψ0. We finally discuss what prevents us from attempting a quantitative (rather than merely qualitative) comparison with the experimental results of Gehring and Fischer.11 First, choosing R = 10 nm gives us access to broader and physically more interesting variations of the ratio lD/R. That is, our present study considers Debye lengths ranging from being much smaller (lD/R = 0.1) to being much larger (lD/R = 20) as compared to the particle size R = 10 nm, whereas in the experimental system the Debye length remains always smaller than the particle size (100−200 nm). Second, the surface charge density of the nanoparticles is not known. In fact, we only find opposite changes in the immersion depth sopt for oppositely charged particles if σR2 ≲ 4, corresponding to N ≲ 50 (see Figure 5). Third, the crucial quantity of the present model is the surface potential Φ0 at the air−water surface. As discussed in the Theory section, neither its magnitude nor its uniformity (i.e., Φ0 = constant) is well-established. The actual magnitude of the potential difference may be considerably larger than 50 mV.21 Indeed, with a significantly larger magnitude of Φ0, the relevant surface charge densities σR2 would increase. The correspondingly larger electrostatic free energies ΔFel would imply that larger values of the scaled air− water tension γawR2 would give rise to notable changes in sopt. Hence, the fact that the present work (which is based on Φ0 = −50 mV) yields interesting results only for small surface charge density and small air−water surface tension may be an indication that the actual surface potential Φ0 is more negative than −50 mV. In summary, we presented detailed mean-field electrostatic calculations addressing the immersion depth of nanoparticles that are trapped at the air−water interface. Our model predicts that as salt is added to the system, negatively charged particles 14306

dx.doi.org/10.1021/la303177f | Langmuir 2012, 28, 14301−14307

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penetrate deeper into the water whereas positively charged particles become displaced more toward the air. This finding is in qualitative agreement with experimental observations by Gehring and Fischer.11 The opposite behavior of oppositely charged nanoparticles is produced by a nonvanishing air−water surface potential. It is further promoted by a sufficiently small density of charges on the nanoparticle and by a sufficiently small air−water surface tension. We reiterate that our model relies on significant assumptions, including the value and constancy of the surface potential, the absence of charges at the air-exposed region of the nanoparticles, the flatness of the air−water surface, and the familiar approximations that come with using the classical Poisson− Boltzmann model. For example, a perfectly flat air−water surface may not represent the optimal state in the vicinity of the nanoparticle. Even from an electrostatic point of view alone, a surface deformation may be favorable if it increases the translational entropy of the counterions or reduces the electrostatic field energy. We neither find it straightforward to predict the type of deformation nor to find it through numerical calculations. However, any decrease in the electrostatic free energy due to a nonflat air−water surface must lead to a deeper immersion depth. In this sense, our present numerical results underestimate the true penetration depth into the aqueous phase. Consideration of a residual charge on the air-immersed part of the nanoparticle leads to a similar conclusion. Despite all these simplification, we find it likely that the present model captures important qualitative aspect of the physics that governs the partitioning of a charged nanoparticle at a dielectric interface.

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AUTHOR INFORMATION

Corresponding Author

*E-mail [email protected]; Ph (701) 231-7048; Fax (701) 231-7088. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors acknowledge support by the Slovenian Research Agency through Grant BI-US/11-12-046. S.M. thanks Thomas Fischer for valuable discussions.

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REFERENCES

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dx.doi.org/10.1021/la303177f | Langmuir 2012, 28, 14301−14307