Article Cite This: Langmuir XXXX, XXX, XXX−XXX
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Finite-Size and Solvent Dependent Line Tension Effects for Nanoparticles at the Air−Liquid Surface Hiroki Matsubara,† Jo Otsuka,† and Bruce M. Law*,‡ †
Department of Chemistry, Faculty of Sciences, Kyushu University, 744 Motooka, Nishi-ku, Fukuoka 819-039, Japan Department of Physics, Kansas State University, Manhattan, Kansas 66506-2601, United States
‡
S Supporting Information *
ABSTRACT: The line tension for a nanoparticle (NP) at the air−liquid surface can be determined by examining the variation in NP solution surface tension with bulk NP concentration. In this publication the variation in line tension with liquid solvent is examined for the homologous series of liquids from n-decane through to n-octadecane. Finite-size line tension effects are also studied by examining the variation in line tension with NP size for NPs at the air−octadecane surface. Both the line tension variation with solvent and NP size can be qualitatively explained using an interface displacement model for the line tension.
1. INTRODUCTION When three bulk phases intersect at, for example, the solid− liquid−vapor contact line this contact line can be characterized by an energy per unit length or line tension τ.1−6 For sufficiently large system sizes, the line tension is expected to be a materials parameter; namely, its value is a constant determined solely by the composition of the three bulk phases. If, however, one of the “bulk” phases is finite, then the line tension is expected to become size dependent for sufficiently small phase size. Finite-size line tension effects for liquid droplets at a solid−vapor interface have been the subject of extensive theoretical discussions;7−10 however, evidence for such an effect has proven elusive. Numerical studies11,12 and an experiment13 may provide indications of a finite-size line tension effect. This subject still requires further clarification. This publication considers the influence of finite-size effects on the line tension for solid spherical colloidal particles at a liquid−vapor surface. As far as the authors are aware, finite-size line tension effects, for this particular situation, have rarely been considered in the past. At the solid−liquid−vapor contact line, the line tension arises via an integration over the surface potential V(l(x)) within the liquid wedge in the vicinity of the contact line. The thickness l(x) of the liquid wedge varies with distance x from the contact line.14 The surface potential V(l) is characterized by an interaction range ξV, for example, for the van der Waals interaction ξV ∼ 100 nm.15 Hence, for macroscopic droplets or particles with size R ≫ ξV, the line tension is expected to be a materials parameter, independent of droplet/particle size. Conversely, for small droplets or particles with R ≪ ξV, the line tension τ is expected to become a function of droplet/particle size R. Indeed, McBride and Law16 found that for “large” spherical particles (hundreds of nanometers in size) the line tension τ for these particles at © XXXX American Chemical Society
the air−liquid surface was a constant, independent of particle size. By contrast, in the current publication, the line tension for nanometer-sized colloidal particles at the air−liquid surface is found to be size dependent. This publication is set out as follows. Section 2 describes a model for determining the line tension of nanoparticles, at the air−liquid surface, from the variation in solution surface tension with bulk nanoparticle concentration. This line tension model improves upon an earlier implementation of these ideas.17 This improved line tension model is then used to determine the following: (i) The line tension variation, as a function of solvent, for 5 nm core diameter Au nanoparticles coated with dodecanethiol ligands at the air−alkane surface where the alkane solvent is varied from n-decane (C10) through to noctadecane (C18). (ii) The line tension variation, as a function of Au core diameter, for Au nanoparticles coated with dodecanethiol ligands at the air−octadecane surface where the Au core diameter is varied from 2.1 to 21.6 nm. Section 3 provides a description of the experimental methods used in this publication. In section 4, the “interface displacement model” for the line tension is numerically analyzed and compared with the experimental line tension results contained in section 2. The interface displacement model qualitatively describes both the variation in line tension with solvent, as well as, the variation in line tension with Au core diameter. A summary and discussion of our results can be found in section 5. Appendices 1 and 2 provide theoretical details for the interface displacement model of the line tension (section 4). Received: October 24, 2017 Revised: December 4, 2017 Published: December 4, 2017 A
DOI: 10.1021/acs.langmuir.7b03700 Langmuir XXXX, XXX, XXX−XXX
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with a solid surface. From eq 4 if either τ → 0 or R → ∞ then θ → θ∞. The number of nanoparticles (NPs) at a surface, relative to the number in the bulk solution, is determined by thermodynamic equilibrium. For thermodynamic equilibrium the chemical potential of a NP at the surface is equal to the chemical potential of a NP in the bulk liquid which implies that21 φs φb = exp((E b − Es)/kT ) 1 − φs 1 − φb (6)
2. NANOPARTICLE SOLUTION SURFACE TENSION Wi et al.17 describe how the line tension of nanoparticles, at the air−liquid surface, can be deduced from the variation in surface tension with bulk nanoparticle concentration. This model is summarized briefly here. A particle at the liquid/vapor surface possesses energy Es, while the same particle in the bulk liquid possesses energy Eb. The energy difference between these two configurations is given by E = Es − E b = (γSV − γSL)2πRh + 2πbτ − πb2γsol
(1)
where γij is the surface energy between phases i and j, with subscript S = solid, L = liquid, and V = vapor. γsol is the surface tension of the neat solvent, τ is the line tension of the threephase solid−liquid−vapor contact line, while R, b, and h are, respectively, the particle radius, lateral radius, and protrusion height defined in Figure 1, inset.
or, from eqs 1 and 5, φs 1 − φs
=
⎡ πb2γ − 2πRhγ cos θ − 2πbτ ⎤ ∞ sol sol ⎥ exp⎢ ⎢⎣ ⎥⎦ 1 − φb kT φb
(7)
where φs (φb) is the NP surface fraction (bulk volume fraction). According to eqs 2−4, τ and h can be rewritten as τ = bγsol(1 − cos θ∞/ 1 − (b /R )2 )
(8)
and h = R(1 −
γ = φγ + (1 − φs)γsol s NP
h = R(1 − cos θ )
(2)
Hence, the contact angle θ that the particles makes with the liquid surface is given by
cos θ =
1−
⎛ b ⎞2 ⎜ ⎟ ⎝R⎠
(3)
The positioning of the particle at the liquid/vapor surface is determined by mechanical equilibrium, namely, dE/dh = 0. Mechanical equilibrium of the particle gives rise to the modified Young’s equation18,19 cos θ =
cos θ∞ τ 1 − bγ
(4)
sol
where Young’s equation
20
cos θ∞ = (γSV − γSL)/γsol
(10)
where γNP represents the surface energy of a NP covered surface, at high coverage (φs ∼ 1). In comparing theory (eqs 7−10) with experiment two parameters, b and γNP, are adjusted in the theory in order to find the best fit to experimental data. Figure 1 examines the influence of these two parameters by comparing this theory with experimental surface tension measurements γ for dodecanethiol ligated Au nanoparticles of radius R = 4.2 nm at the octadecane−air surface as a function of bulk NP volume fraction φb at a temperature of 30 °C. The red squares are experimental data from Wi et al.17 while the black dots represent the best fit curve with b = 2.5363 nm and γNP = 21.4 mN/m. The influence of γNP can be deduced from the upper and lower blue dashed curves which correspond to, respectively, γNP = 21.0 mN/m and γNP = 22.0 mN/m at fixed b = 2.5363 nm. Similarly, the influence of b can be deduced from the upper and lower red solid curves which correspond to, respectively, b = 2.52 nm and b = 2.56 nm at fixed γNP = 21.4 mN/m. Hence, from Figure 1 the lateral radius b determines the initial slope of γ at small φb, whereas, γNP determines the background surface tension value at large φb. Therefore, single particle effects (arising from the single particle energy difference Eb − Es) are prominent at low surface coverage, whereas, multiparticle effects (included in γNP) become important at high surface coverage. Wi et al.17 examined the surface tension behavior upon addition of these 4.2 nm radius dodecanethiol ligated Au NPs to a homologous series of n-alkane solvents from n-decane
By geometry and
(9)
Therefore, the exponential term within eq 7 can be expressed solely in terms of the lateral radius b via use of eqs 8 and 9. In experiments, the surface tension γ is measured as a function of the bulk NP volume fraction φb. In order to compare theory (namely, eq 7 which relates φb to φs) to experiment, a relationship between the surface tension γ and the NP surface volume fraction φs is required. This is achieved via the constitutive equation
Figure 1. Surface tension γ versus bulk NP volume fraction φb at 30 °C for 4.2 nm radius NPs in n-octadecane solvent. Experimental data (red squares).17 Theory, eqs 7−10: best fit with b = 2.5363 nm and γNP = 21.4 mN/m (black dots), b = 2.52/2.56 nm (upper/lower red solid curve) at fixed γNP = 21.4 mN/m, γNP = 21.0/22.0 mN/m (upper/lower blue dashed curve) at fixed b = 2.5363 nm.
b = R sin θ
1 − (b/R )2 )
(5)
has been used in deriving eq 4. The modified Young’s equation (eq 4) describes how the presence of a line tension τ, for a finite sized particle, causes the particle contact angle θ to deviate from its macroscopic value θ∞. Here angle θ∞ is the macroscopic contact angle that a liquid/vapor surface makes B
DOI: 10.1021/acs.langmuir.7b03700 Langmuir XXXX, XXX, XXX−XXX
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Langmuir (C10) through to n-octadecane (C18) at a temperature of 30 °C. Figure 2 provides a comparison between the best fit (solid
Figure 2. Surface tension γ versus bulk NP volume fraction φb at 30 °C for 4.2 nm radius NPs in a selection of n-alkane solvents: C18 (red), C16 (blue), C12 (green), and C10 (black). Experiment (symbols).17 Best fit theory (eqs 7−11) with zero/nonzero Tolman length δ (lines/dashed line).
lines) and experiment (symbols) for selected n-alkane solvents. Theory provides a reasonable description of these surface tension experiments for all of the n-alkane solvents under consideration, however, there are some noticeable deviations between theory and experiment, particularly for the longer nalkane solvents. Could finite-size or, equivalently, high curvature corrections to the surface energy explain these deviations? For small NPs at a liquid surface both the SL and SV surfaces will necessarily exhibit high curvature. Curvature corrections to the surface energy are predicted to scale as ∼δ/R where the Tolman length δ ∼ 10−10 m.22 According to Young’s equation (eq 5), high curvature corrections for the SV and SL surfaces involve replacing γsol cos θ∞ ≡ γSV − γSL by γsol cos θ∞(1−2δ/R) in eqs 7 and 8, where δ is treated as a third adjustable parameter. This three-parameter fit (b = 2.497 nm, γNP = 21.4 mN/m, and δ = 0.396 × 10−10 m; red dashed line Figure 2) fails to provide a better description of the noctadecane experimental data compared with a two-parameter fit (b = 2.5363 nm and γNP = 21.4 mN/m; red solid line). Therefore, in the remainder of this publication, finite-size surface tension corrections will be ignored (i.e., δ = 0). For each n-alkane solvent the best fit b parameter allows a determination of the NP contact angle θ and line tension τ using, respectively, eqs 3 and 8. Figure 3a−c provides plots of, respectively, θ, τ, and γNP as a function of n-alkane solvent. A table of numerical values for these parameters is available in the Supporting Information. Wi et al.17 used an approximate “one-parameter fit” to analyze the surface tension data for dodecanethiol ligated Au NPs adsorbed at the air-alkane surface. In this earlier work b was treated as an adjustable parameter while γNP was fixed with γNP = γcrit (∼20.9 mN/m), the critical surface tension.23 This assumption for γNP is expected to be approximately correct, as any solvents with γsol < γcrit will completely wet the nanoparticles (with θ∞ = 0) and nanoparticles will not adsorb at the air−solvent surface; thus, in this case, γ will not change as the bulk nanoparticle concentration is increased. Therefore, γcrit will serve as a lower bound for γNP, which agrees with the observations in Figure 3c. The “two-parameter” fit, used in Figure 3, provides a better description of experimental data than the “one-parameter” fit used by Wi et al.;17 however, the overall trends for θ and τ are similar in both cases.
Figure 3. (a) Contact angle θ, (b) line tension τ, and (c) γNP versus nalkane solvent for 4.2 nm radius NPs at 30 °C, from eqs 7−10 (red squares). Solid lines in (b): theory (eq 24) with σ = 0.05 nm (red dashed)/0.01 nm (green solid line).
The variation in τ with solvent number (Figure 3b) is somewhat unexpected. For shorter alkane chain lengths, γsol decreases and approaches γcrit and, correspondingly, θ∞ approaches 0. Thus, a decrease in the n-alkane chain length, should correspond to approaching a first-order wetting transition. The line tension τ, on approaching a first-order wetting transition, is expected to change from a negative sign to a positive sign as the distance to the wetting transition is decreased (at least for liquid droplets on a solid surface).14,24,25 This is the opposite trend to that displayed in Figure 3b where n-octadecane C18 (n-decane C10) is expected to be far from (close to) a first-order wetting transition. The line tension trend displayed in Figure 3b is discussed further in section 4. For sufficiently large nanoparticles (radius R > 50 nm) at a liquid−air surface, the nanoparticle line tension will be a materials parameter where this line tension will be independent of nanoparticle size because the particle size (greatly) exceeds the surface potential interaction range. This situation of constant line tension, independent of particle radius, was observed recently by McBride and Law.16 Here, the opposite limit is examined. Small nanoparticles, with dimensions much less than the surface potential range, are expected to possess a line tension whose value is size dependent. The experimental section (section 3) describes our procedure for preparing dodecanethiol ligated Au nanoparticles where the Au core diameter is varied from 2.1 nm through to 21.6 nm by varying the gold to thiol mole ratio. At fixed Au core diameter, the nanoparticle solution surface tension γ (in air) is then measured at a temperature of 30 °C as a function of the bulk nanoparticle volume fraction φb in the solvent n-octadecane. These surface tension γ versus φb experimental results are analyzed using eqs 7 −10 at fixed background surface tension γNP = 21.4 mN/m, namely, the same background value found in Figure 3c for noctadecane. Figure 4a and b shows, respectively, θ and τ versus particle radius C
DOI: 10.1021/acs.langmuir.7b03700 Langmuir XXXX, XXX, XXX−XXX
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Figure 4. (a) Contact angle θ and (b) line tension τ versus NP radius R in n-octadecane solvent at 30 °C, from eqs 7−10 at fixed γNP = 21.4 mN/m. Solid lines in (b): theory (eq 24) with σ = 0.05 nm (red dashed)/0.01 nm (green solid)/0.005 nm (black dashed dotted line).
R = t + RAu
organic phase to a red color which then turned into a black (brown) solution. The reaction mixture was left for 12 h with rapid stirring in air. Upon completion, the organic phase was removed from the vessel and evaporated to 5 mL. Next, 350 mL of 99.5% ethanol was added to this 5 mL of organic phase, and the mixture was kept at −18 °C for 40 h during which precipitation occurred. The black (dark brown) precipitate was then filtered using 5 μm cellulose filter paper and then washed with 300 mL of 99.5% ethanol. The product was dispersed in 30 mL of 99.5% toluene and then reprecipitated using ethanol. Finally the dried product was dispersed in n-octadecane at 30 °C. This nanoparticle solution was used in the surface tension and concentration measurements. 3.3. Surface Tension and Contact Angle Measurements. The surface tension γ was measured as a function of NP volume fraction in n-octadecane solvent at a temperature of (30.0 ± 0.1)°C by analyzing the shape of a pendant NP solution droplet hanging from a glass capillary tip.31 The experimental error in the surface tension, determined from three separate measurements at each concentration, was less than 0.05 mN/m. A long-range microscope, from First Ten Angstroms (model FTA100), was used to measure the contact angle θ∞ of macroscopic nalkane droplets on a n-dodecyl-trichlorosilane coated silicon wafer at a temperature of 30.0 °C.17 The experimental error in the contact angle is estimated to be ±0.5°. 3.4. Particle Radius Measurement. The particle radius was determined using dynamic light scattering using a particle size analyzer ELSZ-0S (Otsuka Electronics Co. Ltd.) at a scattering angle of 165°. Each measurement was averaged over 50 runs. The detectable size range of the instrument is from 0.6 nm to 7 μm. All samples were dispersed in toluene, rather than n-octadecane, to avoid solvent freezing during the measurement. 3.5. Concentration Measurement. The number density N of gold NPs in n-octadecane was determined by measuring the NP absorbance A450 at a wavelength of 450 nm using a Shimadzu UV spectrophotometer UV-1800 (Shimadzu Corp.) where32
(11)
where the ligand length t = 1.7 nm while RAu is the Au core radius. The line tension is a strong function of the nanoparticle radius (Figure 4b) and changes from a negative to a positive value with increasing radius. Tables of b, θ, and τ versus R are provided in the Supporting Information. As a check on these new experimental results, we note that the numerical values for θ and τ at a nanoparticle radius R = 4.2 nm agree closely with the values in Figure 3a and b for the solvent n-octadecane. A model which qualitatively explains the line tension behavior exhibited in Figure 4b is discussed in section 4. Longer n-alkanes (C15−C50) exhibit a surface freezing transition for temperatures between the bulk freezing temperature Tbf and surface freezing temperature Tsf.26−28 In this temperature regime, the first n-alkane monolayer in immediate contact with the air phase becomes frozen, due to surface entropy effects, while the bulk n-alkane phase remains a liquid. For n-octadecane, Tbf = 27 °C and Tsf = 29.5 °C (Figure S1); thus, the data in Figure 4 at 30 °C is above Tsf and it is not necessary to consider surface freezing effects when interpreting the line tension data in Figure 4b.
3. EXPERIMENTAL SECTION 3.1. Materials. Dodecanethiol-coated gold NPs were synthesized from gold(III) chloride, tetraoctylammonium bromide, and sodium borohydride all purchased from Sigma-Aldrich Co. LLC and 1dodecanethiol, purchased from Tokyo Chemical Industry Co. Ltd. nOctadecane was purchased from Tokyo Chemical Industry Co. Ltd. and purified by distillation under reduced pressure (bp 165−167 °C at 7.5 mmHg). The purity of n-octadecane was confirmed via surface tension measurements. 3.2. Synthesis. Dodecanethiol-coated gold NPs of radii R = 2.75− 12.5 nm were synthesized following published protocol.29,30 Gold/ thiol mole ratios of 3.5:1, 4:1, 5:1, 5.5:1, 6:1, and 6.5:1 were used for the reaction. In the following, we describe the 3.5:1 Au/thiol synthesis scheme. The other synthesis schemes are similar. In this synthesis scheme, 8 mL of bright yellow 30 mM HAuCl4(aq) solution was added to an empty glass reaction vessel and rapidly stirred. Then 5.3 mL of a 100 mM N(C8H17)4Br(toluene) solution was added to the reaction vessel. Immediately the solution phase separated into an orange (red) organic phase on top and a clear, orange tinted aqueous phase on the bottom. This mixture was vigorously shaken several times in order to obtain complete removal of color from the aqueous phase. Then 1.7 mL of 43 mM C12H25SH(toluene) solution was added to the organic phase, and 7 mL of a 160 mM NaBH4(aq) solution was added to the reaction mixture with slow stirring. There was an instant color change of the
N=
A450 × 1014 ⎡ ⎛ d 2⎢ − 0.295 + 1.36exp⎜ − ⎝ ⎣
2 ⎞⎤
( d −78.296.8 ) ⎠⎥⎦ ⎟
(12)
Here d = 2R is the diameter of gold NPs measured via dynamic light scattering and N is the number of particles per cubic centimeter.
4. INTERFACE DISPLACEMENT LINE TENSION MODEL In Figure 3b (red squares) the line tension τ for Au NPs of radius R = 4.2 nm, dissolved in various n-alkane solvents at a temperature of 30 °C, is observed to change from a negative to a positive value with increasing n-alkane solvent chain length. In Figure 4b (red squares) the line tension τ changes from a D
DOI: 10.1021/acs.langmuir.7b03700 Langmuir XXXX, XXX, XXX−XXX
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Langmuir negative to a positive value with increasing NP radius R for Au NPs dissolved in n-octadecane solvent at 30 °C. This section considers the interface displacement model for the line tension14 and examines whether or not this model can explain both the sign and the magnitude of the line tension in Figures 3b and 4b. In the interface displacement model, the line tension τ̂[l(x)] is a functional of the film thickness profile l(x), in the vicinity of the contact line, where l(x) is the film thickness at distance x from the contact line. Figure 5a schematically illustrates how
A functional minimization of eq 13 leads to γ0
d2l dV = 2 dl dx
(15)
which has a first integral of 2 1 ⎛ dl ⎞ γ0⎜ ⎟ = V (l) + const 2 ⎝ dx ⎠
(16)
Hence, from eqs 13, 14, and 16 ∞
τ=2
l(x) varies with position x from the contact line (near x ∼ 0) for a liquid droplet on a solid surface. Far from the contact line, in the vapor phase at x → −∞, the solid surface is covered by an adsorbed film of thickness l1. The macroscopic liquid droplet, at x → +∞, can be treated as a liquid wedge of contact angle θ with the solid surface. Hence, the thickness of the liquid film l(x) varies between these two limits as the position x is varied along the solid surface. In the squared gradient approximation the line tension functional takes the form14 ∞
⎡
⎛
⎞2
∫−∞ dx⎢⎣ 12 γ0⎝ ddxl ⎠ ⎜
⎟
∞
C = −2
∫−∞ dx[V (leq(x), s(x); θ∞)]
(18)
and hence ∞
τ=2
⎤ + V (l(x); θ )⎥ + const ⎦
∫−∞ dx[V (leq(x), s(x); θ) − V (leq(x), s(x); θ∞)] (19)
The calculation of τ in eq 19 is complex and a number of approximations will be necessary in order to make this calculation more manageable. The presence of leq(x) in eq 19 implies that there will be a “dip” for attractive potentials (“bump” for repulsive potentials), akin to the behavior exhibited in Figure 5a (for a droplet on a solid surface, red curve) in the vicinity of the contact line. To simplify the calculation, we assume that
(13)
where the first term in the integrand (proportional to (dl/dx)2) arises from the cost in surface energy to distort and increase the liquid surface area of the profile where γ0 is the surface tension. This increase in surface energy is balanced by a decrease in the surface potential energy, V(l(x);θ). This surface potential energy is a function of l(x). For reasons which will become apparent later we have labeled the surface potential energy with a contact angle θ designation as well. In the vicinity of the contact line, where the interaction potential V(l) plays a prominent role, the film profile will deform away from its wedge shape as indicated by the red line in Figure 5a; however, this deformation increases the area of the droplet surface and thus costs energy as determined by the first term which appears in the integrand of eq 13. The value of the line tension functional τ̂[l(x)] varies as the film thickness profile l(x) is varied. At equilibrium, the equilibrium film thickness profile33 leq(x) is that profile which minimizes τ̂[l(x)]. Thus, the line tension τ corresponds to this minimum value τ = τ[̂ leq(x)] = min τ[̂ l(x)]
(17)
where C is a constant whose value will be identified below. This interface displacement model can also be applied to particles at a liquid−vapor surface. In the absence of any line tension effects (τ = 0), a particle would make a contact angle θ∞ with the liquid/vapor surface (Figure 5b, dashed circle). When line tension effects are present the particle now makes a contact angle θ with the liquid/vapor surface (Figure 5b, blue sphere). The surface potential V(l(x), s(x); θ) now depends upon both the liquid thickness l(x) as well as the secant s(x), both of which vary with position x along the liquid/vapor surface. If the particle radius R is much, much greater than the interaction range of V then the dependence on s(x) can be ignored i.e. the solid particle can be treated as a macroscopically large spherical solid surface. van der Waals interactions play a role up to length scales of ∼ 100 nm and, hence, for the particles considered in this publication (diameters ∼ 5−25 nm) the s(x) dependence of V(l(x), s(x); θ) will be important and cannot be neglected. The constant C in eq 17 can now be identified. In the limit when τ = 0, the particle possesses a contact angle of θ∞; therefore, from eq 17
Figure 5. (a) Variation of interfacial thickness profile l(x), in the vicinity of a contact line near x ∼ 0, for a liquid droplet (L) on a solid surface (S) in vapor (V). Far from the contact line, in the vapor phase, l(x) → l1 as x → −∞. In the liquid phase, at x → +∞, the liquid droplet can be treated as a wedge of constant angle θ. (b) Variation of interfacial thickness profile l(x) for a (blue) solid sphere (S) at the liquid/vapor surface. This sphere makes a contact angle θ with the liquid surface where, at position x, a secant through the solid sphere has length s(x). In the absence of a line tension, the sphere would make angle θ∞ with the liquid surface (dashed sphere).
τ[̂ l(x)] =
∫−∞ dx[V (leq(x); θ)] + C
leq(x) ≈ l(x)
(20)
where l(x) represents a purely horizontal liquid/vapor surface up to the contact line. This approximation assumes that the magnitude of the line tension originates primarily from the contact angle change from θ∞ to θ, rather than from l(x) to leq(x). The presence of a ligand coating on a particle also implies that the secant cut through the particle is more complex than the simple s(x) exhibited in Figure 5b. Figure 6 provides a more complete description of the current geometry. In Figure 6, x = 0 coincides with the apex of the particle, xmin corresponds to the three-phase solid−liquid−vapor contact line, while xcore and xmax coincide with, respectively, the terminal points of the Au core and ligand shell. The x integration in eq
(14) E
DOI: 10.1021/acs.langmuir.7b03700 Langmuir XXXX, XXX, XXX−XXX
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Langmuir τAu(θ ) =
ASo0SoAAuLiAu 6πR
∫θ
ϕcore
dϕ cos ϕ
× ([C1 −
sin 2 ϕcore − sin 2 ϕ ]−2
− [C1 +
sin 2 ϕcore − sin 2 ϕ ]−2 )
(26)
and C1 =
19 is therefore limited to points between xmin and xmax where this x integration is subdivided into two regions, namely, (i) xmin ≤ x ≤ xcore and (ii) xcore ≤ x ≤ xmax. In region (i) between bulk air and bulk solvent phase (see the schematic on the right of Figure 6), the integral in eq 19 calculates the interaction energy within a thin solvent film of thickness l with an adjoining ligand layer of thickness L, Au layer of thickness T, and second ligand layer of thickness L. In region (ii) the Au layer is no longer present and the integral in eq 19 calculates the interaction energy within a thin solvent film of thickness l with an adjoining ligand layer of thickness 2L. Equation 19 is readily calculated numerical by transforming into angular coordinates ϕ using
ALiSoLi ≈ ((εLi − εSo)/(εSo − 1))2 ASo0So
and therefore (22)
Regions (i) and (ii) now corresponds to, respectively, ϕmin ≤ ϕ ≤ ϕcore and ϕcore ≤ ϕ ≤ ϕmax where ϕmin = θ ,
ϕcore = sin−1(R c/R ),
and
ϕmax = 90° (23)
An analytic expression for the surface potential V(l(x),s(x);θ) is required in eq 19. In Appendix 1, for the geometry depicted in Figure 6, V(l(x),s(x);θ) is approximated by the nonretarded van der Waals interaction in the “Hamaker” approximation. Appendix 2 provides details on the numerical integration of eq 19. From Appendix 2, the line tension τ = τLi(θ ) + τAu(θ ) − τLi(θ∞) − τAu(θ∞)
(24)
where τLi(θ ) =
ASo0SoALiSoLi 6πR
∫θ
− [C1 + cos ϕ]−2 )
90°
(28)
where εLi and εSo are, respectively, the ligand and solvent (optical) dielectric constants (Supporting Information Table 3). In the vicinity of the contact line, where l → 0, the surface potential V (∼1/l2) is divergent. The cutoff length σ has been introduced into eq 27 to control this divergence. The Au− ligand−Au Hamaker constant AAuLiAu (Supporting Information Table 3), which appears in eq 26, is ∼4 times larger than ASo0So and ∼40 times larger than ALiSoLi; hence, AAuLiAu may significantly impact the line tension τ. Note that in the calculation of τ (eq 24) the ligand contribution τLi (eq 25) is always present, whereas the Au contribution τAu (eq 26) is only present when ϕcore > θ. In particular, for the current experiments, for R = 4.2 nm NPs in Figure 3b, the τAu term does not play a role for n-heptadecane and n-octadecane solvents while, in Figure 4b, with an n-octadecane solvent the τAu term does not play a role for sufficiently small NPs possessing radii R = 4.2, 3.6, and 2.75 nm. The lines in Figures 3b and 4b compare eq 24 with experiments for differing values of the cutoff length scale (σ = 0.05 nm (red dashed), 0.01 nm (green solid), and 0.005 nm (black dashed dotted line)). The theory (eq 24) correctly reproduces the change in sign of the line tension exhibited in Figures 3b and 4b where the sign of τ is determine by the relative magnitudes of θ and θ∞. When θ < θ∞ (θ > θ∞), τ is positive (negative). The line tension magnitude is a sensitive function of the cutoff length σ. In order to qualitatively reproduce the experimental line tension magnitudes a small value for σ is required. A value of σ = 0.05 nm provides a qualitative description of the solvent dependence of the line tension (Figure 3b), whereas, a value of σ = 0.005−0.01 nm provides a qualitative description of the radii dependence of the line tension (Figure 4b). The NPs used in the experiments of Figures 3b17 and 4b (this publication) were prepared using slightly differing experimental procedures; hence, one should not place too much significance upon these slighly differing
(21)
dx = R cos ϕ dϕ
(27)
τLi(θ∞) and τAu(θ∞) are obtained from eqs 25 and 26, respectively, by replacing θ by θ∞. In these expressions Aikj is the Hamaker constant between bodies i and j across medium k. In eqs 25 and 26, the subscripts correspond to So = solvent, 0 = vacuum, Li = ligand, and Au = gold. The Hamaker constant possesses a typical magnitude of A ∼ 5 × 10−20 J while the NP radius R ∼ 5 nm; hence, the line tension magnitude τ ∼ pN, in qualitative agreement with the experiments in Figures 3b and 4b, theoretical expectations,2,34,35 and computer simulations.36,37 The solvent−vacuum−solvent Hamaker constant ASo0So (Supporting Information Table 3) appears in each of the terms in eq 24; thus, this term merely scales the magnitude of τ but does not influence the relative magnitudes of the differing terms. The ligand−solvent−ligand Hamaker constant ALiSoLi, which appears in eq 25, is estimated from38
Figure 6. Contact line occurs at xmin where the particle makes an angle θ with the liquid/vapor surface. At arbitrary angle ϕ, at position x, secant s(x) = 2L(x) + T(x) or, alternatively, s = 2L + T. The Au core terminates at xcore corresponding to angle ϕcore, whereas the NP terminates at xmax corresponding to ϕ = 90o. For xcore < x < xmax, T = 0, and therefore s = 2L.
x = R sin ϕ
σ + cos θ R
dϕ cos ϕ([C1 − cos ϕ]−2 (25) F
DOI: 10.1021/acs.langmuir.7b03700 Langmuir XXXX, XXX, XXX−XXX
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Langmuir
the surface potential V, consists of two contributions τLi(θ) (eq 25) and τAu(θ) (eq 26) from, respectively, the ligand-solvent interaction at all x (xmin ≤ x ≤ xmax) and the Au core−ligand interaction at small x (xmin ≤ x ≤ xcore). A cutoff length σ, in the interaction potential V, is necessary to control the divergence in the integration over this potential. The relative magnitudes of θ and θ∞ (the macroscopic contact angle) determines the sign of τ. Thus, if θ < θ∞ (θ > θ∞) then τ is positive (negative). The magnitude of τ is determine by the cutoff length σ. A cutoff length of order σ ∼ 0.005−0.05 nm is required to explain the experimental magnitude for the line tension where the primary contribution to τ arises from τLi(θ) (Figure 7). This interface displacement model for τ qualitatively explains the experimental line tension observations in Figures 3b and 4b where a slightly differing value for σ is required in these two figures (σ ∼ 0.05 nm and ∼ 0.005−0.01 nm, respectively). These slightly differing values for σ may arise from differing experimental NP preparation methods in these two experiments. The interface displacement model, used in this publication, suffers from two inherent deficiencies, which require further examination in order to ascertain there influence upon the line tension. Specifically, (i) an improved model (better than the nonretarded van der Waals Hamaker approximation39) should be used for the surface potential V and (ii) the equilibrium film thickness profile33 leq(x) should be used in place of l(x) within V (where l(x) corresponds to a flat horizontal liquid−air surface). Even with these improvements, the interface displacement model for the line tension will continue to possess one adjustable parameter, namely, the cutoff length σ which controls the diverging integral at small l within the potential V. The results in this publication are primarily concerned with obtaining an understanding of single NP effects for NPs adsorbed at low surface concentration at the air−liquid surface. The analysis of NP surface tension data additionally provides information about multiparticle effects at high NP surface coverage via the background surface tension γNP (Figure 3c). To date the results in Figure 3c are not understood. Extensions of the experimental and theoretical ideas in this publication are likely to have widespread implications for particle laden surfaces, particle foams, and Ramsden−Pickering emulsions.40,41
magnitudes for σ. Figure 7 demonstrates the dependence of the differing line tension components upon the cutoff length σ.
Figure 7. Variation of line tension components with cutoff length σ for R = 5.9 nm NP in n-octadecane solvent. τ (heavy black line, eq 24), τLi(θ) (solid blue line, eq 25), τLi(θ∞) (dashed blue line), τAu(θ) (solid red line, eq 26), and τAu(θ∞) (dashed red line).
This figure indicates that the line tension difference τLi(θ) − τLi(θ∞), arising from the solvent−ligand interaction, plays the primary role in determining the line tension magnitude whereas the Au-ligand interaction via τAu(θ) − τAu(θ∞) is only of secondary importance. The small magnitude for σ, required to explain experimental data, indicates that the line tension arises primarily from the immediate vicinity of the three-phase contact line. This suggests that a complete understanding of the experimental line tension data in Figures 3b and 4b will only be possible when leq(x) (as opposed to l(x)) is included within the line tension calculation, however, even upon inclusion of leq(x), a cutoff length σ will continue to play a prominent role in any calculation.
5. SUMMARY AND DISCUSSION This publication develops an improved model for determining the line tension of NPs at the liquid−air surface from the variation in NP solution surface tension γ with bulk NP volume fraction φB (eqs 7−10). Two fitting parameters enter this model, namely, the lateral radius b and the background surface tension γNP. b, which arises from a single NP model, controls the surface tension γ at small φB, whereas γNP determines the surface tension behavior at large φB where NP−NP interactions will play a role (Figure 1). The lateral radius b allows a determination of the particle contact angle θ (eq 4) and line tension τ (eq 8). This improved model is used to reanalyze earlier NP solution surface tension data of Wi et al.17 for dodecanethiol ligated Au NPs (radius R = 4.2 nm) at the alkane−air surface at 30 °C (Figures 3) for various n-alkane solvents from n-decane (C10) through to n-octadecane (C18). The line tension τ changes from a negative to a positive value with increasing n-alkane chain length (Figure 3b), while γNP exhibits a complex dependence as a function of n-alkane chain length (Figure 3c). This model has also been used to determine the line tension τ for dodecanethiol ligated NPs in noctadecane solvent at 30 °C as a function of NP radius for radii in the range R = 2.75−12.5 nm. The line tension τ changes from a negative to positive value with increasing radius R (Figure 4b). An interface displacement model for the line tension is developed in section 4 and compared with the line tension data in Figures 3b and 4b. This line tension model (eq 24), analyzed using a nonretarded van der Waals Hamaker approximation for
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APPENDIX 1 In Section 4, the surface potential V(l(x),s(x);θ) is required in order to calculate the line tension τ. According to Figure 6, at position x, this involves finding the surface interaction potential in the solvent layer of thickness l, which possesses an adjacent ligand layer of thickness L, Au layer of thickness T, and second ligand layer of thickness L, between bulk air and bulk solvent phases. For convenience we start with the more general configuration depicted in Figure 8 and follow the notation and analysis of Parsegian and Ninham.42 Their notation makes it simplier to keep track of all terms that are present. Figure 8 shows a seven layer system between two bulk phases, L and R, where the nonretarded van der Waals contribution to the surface energy between surfaces S and S′, across medium m of thickness l, is required. For the situation of specific interest in this publication (Figure 6), the surface layers 1′, 2′, and 3′ are not present, hence, thicknesses a′ = b′ = c′ = 0. In Lifshitz theory, the nonretarded van der Waals surface energy is given by42 G
DOI: 10.1021/acs.langmuir.7b03700 Langmuir XXXX, XXX, XXX−XXX
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APPENDIX 2 From eq 19, in the interface displacement model, the line tension is given by ∞
∫−∞ dx[V (leq(x), s(x); θ) − V (leq(x), s(x); θ∞)]
τ=2 Figure 8. Depiction of a seven-layer system where the surface potential V between surfaces S and S′ in layer m of thickness l between bulk phases L and R is of interest. On the right side of layer m are three layers, 1′, 2′, and 3′ possessing thicknesses a′, b′, and c′, respectively. If a′ = b′ = c′ = 0 then this figure becomes equivalent to Figure 6. kT 8πl 2
V (l ; a , b , c ; T ) =
∞
∑′∫ n=0
∞
r
≈2
dx[V (l(x), s(x); θ ) − V (l(x), s(x); θ∞)]
min
using the approximation that l(x) ≈ leq(x)
+ ln[1 − Δ̅ mL (a , b , c)Δ̅ mR e ]}dx
(39)
where l(x) represents a horizontal liquid/vapor surface. The integral in eq 38 is subdivided into two components
(29)
∫x
To a reasonable approximation, Δ̅ jk = 0 and εi(iξ) − εj(iξ)
xmax
dx =
min
∫x
xcore
dx +
min
∫x
xmax
dx
(40)
core
Considering first xmin ≤ x ≤ xcore
εi(iξ) + εj(iξ)
(30)
2
where εi(iξ) is the dielectric constant of medium i at imaginary frequency iξ. In eq 29 ΔmL (a , b , c) =
xmax
(38)
x{ln[1 − ΔmL (a , b , c)ΔmR e−x] −x
Δij =
∫x
Δm3 + Δ3L (a , b)e
1 + Δm3Δ3L (a , b)e
xcore
dx[V (l(x), s(x); θ ) − V (l(x), s(x); θ∞)]
min
(41)
where from Appendix 1
−(xc / l) −(xc / l)
∫x
V (l , s ; θ) ≡ V (l , L , T ) ⎛ ⎞ A So0So ⎧ 1 1 ⎨ ALiSoLi ⎜ ≈ − 2 2⎟ 12π ⎩ (σ + l + 2L + T ) ⎠ ⎝ (σ + l)
(31)
⎪
⎪
where Δ3L (a , b) =
Δ32 + Δ2L (a)e−(xb / l) 1 + Δ32 Δ2L (a)e
+
−(xb / l)
(42)
x = R sin ϕ
1 + Δ21Δ1L e−(xa / l)
(33)
∫0
∞
dx = R cos ϕ dϕ
ϕmin = θ
ϕcore = sin−1(R c/R )
τ1(θ) = τ11(θ) + τ12(θ) =2
and
b≡T (35)
=
∫0
∫x
xcore
dx[V (l(x), s(x); θ)]
min
A So0So 6πR
∫θ
ϕcore
dϕ cos ϕ{ ALiSoLi ([(σ + l)/R ]−2
− [(σ + l + 2L + T )/R ]−2 ) +
To a good approximation44 3ℏ 4π
(46)
From eqs 41−46
Additionally, for our situation (Figure 6) a = c ≡ L,
(45)
and
⎡Δ Δ32 dξ⎢ m2 3 + ⎣ l (c + l)2
Δ1L = −Δm3 ,
(44)
and the angular integration limits are (from Figure 6)
⎤ Δ1L Δ21 Δ + + 2 2 ⎥ mR (b + c + l) (a + b + c + l) ⎦ (34) Δ21 = −Δ32 ,
(43)
hence,
Following Israelachvili,43 the logarithmic function, within the integrand of eq 29, can be expanded in Δ where only the lowest order terms, quadratic in Δ (i.e., ΔijΔkm), are retained. The surface potential V(l;a,b,c;T) then reduces to an integration over ξ, specifically, ℏ 16π 2
⎪
and σ represents a control parameter which limits the divergence in the integration at small l. As
Δ21 + Δ1L e−(xa / l)
V (l ; a , b , c ; T ) ≈
⎪
(32)
and Δ2L (a) =
⎛ ⎞⎫ 1 1 ⎬ AAuLiAu ⎜ − 2 2⎟ (σ + l + L + T ) ⎠⎭ ⎝ (σ + l + L)
−2
× ([(σ + l + L)/R ]
∞
Δij Δkl dξ ≈
Aiji Aklk
− [(σ + l + L + T )/R ]−2 )}
(47)
(36)
where τ11(θ) (τ12(θ)) represent the first (second) integral appearing in eq 47. By geometry from Figure 6
where Aikj is the Hamaker constant between bodies i and j across medium k. Additionally, Aiji = Ajij
AAuLiAu
(37)
Equations 34−37 will be used in Appendix 2 to calculate the line tension τ.
l = R(cos θ − cos ϕ)
(48)
2L + T = 2R cos ϕ
(49)
and H
DOI: 10.1021/acs.langmuir.7b03700 Langmuir XXXX, XXX, XXX−XXX
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Langmuir T = 2 R c 2 − R2 sin 2 ϕ
constants used in this publication; surface tension versus temperaure for n-octadecane, indicating the temperature regime where surface freezing occurs (PDF)
(50)
therefore R c 2 − R2 sin 2 ϕ
L = R cos ϕ −
■
(51)
τ1(θ) (eq 47) is therefore readily integrated numerically using eqs 48−51. Similarly, τ1(θ∞) is obtained by replacing θ in eq 47 by θ∞. Next considering the region xcore ≤ x ≤ xmax, where T = 0, i.e., the Au core no longer contributes to the integration, from eqs 38−40 τ2(θ ) = 2
∫x
xmax
Corresponding Author
*E-mail:
[email protected]. ORCID
Bruce M. Law: 0000-0002-3877-8497 Notes
The authors declare no competing financial interest.
dx[V (l(x), s(x); θ )]
core
ASo0SoALiSoLi
=
6πR
∫ϕ
90°
dϕ cos ϕ([(σ + l)/R ]−2 (52)
where, for this situation, l = R(cos θ − cos ϕ)
(53)
and L = R cos ϕ
(54)
τLi(θ ) ≡ τ11(θ ) + τ2(θ ) ASo0SoALiSoLi 6πR
∫θ
90°
dϕ cos ϕ([C1 − cos ϕ]−2
− [C1 + cos ϕ]−2 )
(55)
where C1 =
σ + cos θ R
(56)
One can show that τ12(θ ) ≡ τAu(θ )
(57)
where τAu(θ ) =
ASo0SoAAuLiAu 6πR
∫θ
ϕcore
dϕ cos ϕ
× ([C1 −
sin 2 ϕcore − sin 2 ϕ ]−2
− [C1 +
sin 2 ϕcore − sin 2 ϕ ]−2 )
(58)
In this model the line tension is given by τ = τLi(θ ) + τAu(θ ) − τLi(θ∞) − τAu(θ∞)
(59)
which is the equation used to calculate the line tension in Section 4 (eq 24).
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ACKNOWLEDGMENTS
■
REFERENCES
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Hence τ2(θ) and τ2(θ∞) are readily calculated from eqs 52−54. Upon combining τ11(θ) and τ2(θ),
=
■
B.M.L. thanks his hosts for their kind hospitality during his brief visit to Kyushu University. B.M.L. acknowledges partial support for this research from the American Chemical Society Petroleum Research Fund Grant Number 58043-ND5. H.M. appreciates funding for this work from MEXT/JSPS KAKENHI Grant Number JP 17H02943.
core
− [(σ + l + 2L)/R ]−2 )
AUTHOR INFORMATION
ASSOCIATED CONTENT
S Supporting Information *
The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.langmuir.7b03700. Best fit parameters for R = 4.2 nm Au NPs in various nalkane solvents at 30 °C; best fit values for Au NPs of various radii in n-octadecane solvent at 30 °C; Hamaker I
DOI: 10.1021/acs.langmuir.7b03700 Langmuir XXXX, XXX, XXX−XXX
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J
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