Assembly of Nanoparticles at Liquid Interfaces: Crowding and Ordering

Feb 24, 2014 - and Emanuela Del Gado*. ,†. †. Department of Civil, Environmental and Geomatic Engineering, and. §. Laboratory for Interfaces, Sof...
0 downloads 0 Views 2MB Size
Download PDF
Article pubs.acs.org/Langmuir

Assembly of Nanoparticles at Liquid Interfaces: Crowding and Ordering Konrad Schwenke,† Lucio Isa,§ and Emanuela Del Gado*,† †

Department of Civil, Environmental and Geomatic Engineering, and §Laboratory for Interfaces, Soft Matter and Assembly, Department of Materials, ETH Zürich, 8093 Zurich, Switzerland S Supporting Information *

ABSTRACT: Experiments with the self-assembly of nanoparticles at liquid interfaces suggest that cooperative and slow dynamical processes due to particle crowding at the interface govern the adsorption and properties of the final assembly. Here we report a numerical approach to studying nonequilibrium adsorption, which elucidates these experimental observations. The analysis of particle rearrangements shows that local ordering processes are directly related to adsorption events at high interface coverage. Interestingly, this feature and the mechanism coupling local ordering to adsorption do not seem to change qualitatively upon increasing particle size polydispersity, although the latter changes the interface microstructure and its final properties. Our results indicate how adsorption kinetics can be used for the fabrication of 2D nanocomposites with controlled microstructure.

■

INTRODUCTION The self-assembly of nanoparticles at liquid interfaces is a promising route to achieving nanoscale design of new composite structures:1−3 suitably functionalized nano-objects can be strongly localized at the interface and organized into a variety of patterns thanks to their interactions and interface mobility.4−7 Adsorption at the interface is driven by the gain in free energy due to the reduction of the direct contact between the two bulk phases, and nanoparticles (NPs) can be used to create nanoporous membranes or nanoscale sensors.8−10 To exploit fully the potential of functionalized NPs11,12 complete control of the assembly process is needed, from the kinetics of particle adsorption to the structure and mechanics of the final nanocomposite interface. This is currently far from being achieved, because of difficulties in following self-assembly at the nanometer scale and concurrent complex processes in the NP− interface and NP−NP (in the bulk and at the interface) interactions.13−15 Theoretical and numerical studies have proven to be very effective in complementing experiments, in particular, in analyzing the hydrodynamics of the interface (and how this may be perturbed by the adsorption of a single solid object). A number of recent studies have also focused, on the atomistic level, on the adsorption of a single nano-object at a liquid interface.16−22 Interparticle interactions and out-ofequilibrium kinetics have hardly been addressed in spite of being crucial for material fabrication and processing, and hence for technological applications.23,24 The adsorption process during the assembly of an NP-laden liquid interface can be monitored by measuring the effective interfacial tension γ of the whole system, which decreases with increasing NP coverage. Although a monotonous decrease toward the final maximal density of particles at the interface is typically expected,1 recent experiments on core−shell nanoparticles with a diameter range from 10 to 30 nm adsorbed at © 2014 American Chemical Society

oil−water interfaces have shown complex adsorption kinetics, with the appearance of a pseudo-plateau in γ followed by a further decrease taking place only at much longer times.25,26 This qualitative change in the adsorption kinetics appears when the bulk concentration c of NPs is increased and cannot be rationalized in terms of single-particle processes or pure hydrodynamics. A deeper understanding of the different basic mechanisms that control the development of such particleladen interfaces is needed to achieve full control of the assembly process and the final obtained properties. Here we combine a new simulation approach with experiments to characterize the adsorption of nanoparticles at a liquid interface, with the aim to unravel the link among particle dynamics within the interface, the adsorption kinetics, and the interface microstructure. The simulations confirm that for sufficiently high adsorption rates a pseudo-plateau in the adsorption kinetics appears at intermediate times, and a second time scale for further relaxation develops. These features can be closely linked to the crowding, and dynamical slowing and ordering of particles at the interface. Defects in the local packing are furthermore found to be crucial to continuing adsorption and constitute preferred adsorption sites. These findings give new indications on how to exploit the adsorption rate to design self-assembly processes by controlling the evolution of local ordering and packing defects. Interestingly, the mechanisms elucidated by this computational study do not significantly vary upon changing the size polydispersity of the NPs, as also confirmed in experiments monitoring the adsorption of core−shell NPs of well-defined size polydispersity at a water−air interface. Received: November 4, 2013 Revised: February 21, 2014 Published: February 24, 2014 3069

dx.doi.org/10.1021/la404254n | Langmuir 2014, 30, 3069−3074

Langmuir

■

Article

lengths are given in units of σ, the particle diameter, and ε is the unit energy. The mass of each NP is m and all times are in units of τ = (mσ2/ε)1/2. In those reduced units, we set the size of the simulation box to L = 70σ. The choice of T allows us to tune further the relaxation time scale τr(ϕ), and in the following section, we use kBT = 0.5ε. The temperature is controlled with a Nose-Hoover thermostat,39 and we used an integration time step δt = 0.005τ for most values of Γ.40 We have performed simulations for monodisperse particle samples and for polydisperse ones, as synthesized NPs always have a finite size polydispersity. For the polydisperse samples we use a Gaussian size distribution with a mean value of ⟨σi⟩ = 1σ and a standard deviation of 0.15, truncated for practical reasons at 0.7σ and 1.3σ. In this case, the interaction energy depends on the diameter of the interacting particles σij via σij = (σi + σj)/2. In the MC, the insertion/deletion probability in this case is determined by the difference in free-energy ΔF = ΔU + ΔE for a randomly chosen particle with ΔE = −ΔE0(σi/σ)2 and ΔU being the repulsion from the particles already present at the interface. All of the data shown here refer to ΔE0 = 200kBT. We consider Γ = 50τ−1, 4τ−1 and 0.02τ−1, and the maximum number Nmax of particles inserted varies between Nmax = 4700 and 6500. All simulations have been averaged over 20 statistically independent samples. We use the sample-to-sample fluctuations to estimate error bars, and if not indicated otherwise, the error bars in the figures are on the order of the symbol size.

MODEL AND NUMERICAL SIMULATIONS For the model and the numerical approach, we consider that the interplay between the adsorption of NPs from the bulk and the dynamics of the particles within the interface can be characterized by two characteristic time scales. The first one is set by the adsorption rate Γ = τa−1, which to a first approximation increases with the NP concentration in the bulk, and the second one is related to the characteristic relaxation time of the particles within the interface τr(ϕ), which increases with increasing interface coverage ϕ. We propose a Monte Carlo (MC) scheme to mimic the adsorption of interacting nanoparticles, where the probability of adding/ deleting one particle to/from the interface is the result of a balance between the interparticle interactions at the interface and the free-energy gain ΔE associated with removing a portion of the liquid interface (and exposing parts of the particle surface to each fluid phase). In a first approximation, ΔE can be taken to be proportional to the square of the particle diameter and is fixed by the chemistry of the NP and the fluids.27 Strong adsorption, relevant to self-assembly at interfaces, corresponds to ΔE ≤ −200kBT.26 Our MC scheme can be seen as a grand canonical Monte Carlo scheme, where ΔE represents the main contribution to the excess part of the chemical potential.28 Each MC cycle consists of N0 = 3000 attempts to adsorb or desorb a particle (N0 can be varied to mimic different particle concentrations in the bulk,26 but here we keep it fixed for simplicity). The dynamics of the particles adsorbed at the interface and interacting via an effective potential is investigated by means of molecular dynamics (MD). We alternate MC cycles with MD runs and use the parameter Γ, the number of MC cycles per elapsed MD time, to mimic the adsorption rate. We thus vary Γ to investigate how the adsorption rate may affect the evolution of the interface coverage. Our approach can be adapted to different experimental systems, with any form of coarse-grained particle interactions and different ΔE values for different particle species. Here we consider that typical liquid interface thicknesses at room temperature are on the order of a few angstroms.29 Therefore, the interface is a sharp boundary between the two phases, and the NP adsorption energy can be, to a first approximation, simply evaluated from the free-energy gain in removing a portion of the interface equal to the NP cross-section.27 We model the interface as a flat, 2D simulation box with periodic boundary conditions and consider that diffusion within the interface is much slower than in the bulk;30,31 therefore, adsorbed particles are added instantaneously to the interface. The effective interactions between NPs are dominated by steric repulsion when the polymer shells are more than 1 nm thick.32 Although very soft potentials have been proposed for this type of NP in the bulk,33,34 it has been shown that when the NPs are adsorbed onto the liquid interface, the polymers grafted on their surfaces get strongly stretched (and stiff), leading to an interaction significantly steeper than the ones in the bulk.14,15 Here we use a repulsive power law potential Uij = ε(σ/rij)12 where rij is the distance between particles i and j. This potential is a widely studied (and therefore convenient and sufficiently general) case with respect to particle dynamics or the melting transition.36,37 In general, we do not expect the specific steepness of the soft sphere potential to change qualitatively the results discussed below, and we have tested them varying the power law to using a Gaussian potential.38 All

■

RESULTS AND DISCUSSION The effective interfacial tension γ of the NP−liquid−interface system measured in experiments can be related to ϕ, the fraction of the interface covered by NPs41−43 For a small portion of the interface with area δA, we assume that the number of adsorbed NPs can be estimated as 4ϕδA/πσ2, where σ is the NP diameter at the interface. Because experimental observations indicate that the effective interactions of the NP at the interface can be fairly short-ranged, we consider that the adsorption strength is much higher than the particle−particle interactions until they are in very close contact. If, on this basis, we neglect the NP interactions at the interface, then the related reduction of the macroscopic surface tension can be directly written in terms of the increase in surface coverage as γ − γ ≃ 4ϕΔE/πσ2, where ΔE is the adsorption energy per particle. Following this argument, γ/γ0 ∝ 1 − const·ϕ. Recent experimental results on ΔE for the type of systems of interest here have allowed us to verify that this relation holds up to fairly high coverages.35,38 From the simulations, we compute the coverage ϕ(t) = ∑iπσi2/4L2, where the sum extends over all particles at the interface at time t, and in Figure 1, we plot 1 − ϕ(t) for both the monodisperse and the polydisperse systems for Γ = 50τ−1. The data display a complex time evolution clearly reminiscent of experimental observations:26 particle adsorption slows down after a first rapid filling of the interface, characterized by a pseudo-plateau in the time evolution, after which adsorption starts again. Upon varying Γ, the simulations show that the slowing down of the adsorption and the plateau-like regime arises when the irreversible filling of the interface takes place so fast that the relaxation time becomes much larger than 1/Γ. We can verify in the simulations that in the plateau regime the relaxation time τ(ϕ) associated with the particle dynamics at the interface has become much larger than τa. 3070

dx.doi.org/10.1021/la404254n | Langmuir 2014, 30, 3069−3074

Langmuir

Article

Figure 1. (Main frame) 1 − ϕ(t) (green △) and ⟨Ψ6⟩ (red ○) as a function of time for monodisperse particles and Γ = 50τ−1. (Inset) As in the main frame for the polydisperse system.

Figure 2. ⟨Ψ6⟩ as a function of the surface coverage ϕ for the monodisperse system with different Γ. The dotted line indicates the melting point as determined in ref 36. (Inset) ⟨Ψ6⟩ as a function of ϕ for polydisperse systems.

This can be immediately evaluated by stopping the adsorption and computing the mean-squared displacement (MSD) ⟨(r(0) − r(t))2⟩ of the particles at a fixed density. In the plateau regime, the MSD indicates particle localization that is typical of the caging effect in glassy dynamics.38,45−47 Upon decreasing Γ, the plateau regime progressively disappears (i.e., sufficiently slow adsorption allows for filling the interface without jamming the particles38). This supports the idea that the plateau-like regime sets in at a surface coverage such that the total repulsion of the particles at the progressively jammed interface overcomes the energy gain as a result of further adsorption. In the monodisperse systems, we expect the NP eventually to order upon increasing the interface density.48 Long-range order in two dimensions is limited by fluctuations,49 and ordering may take place via a hexatic phase, with long-range orientational order developing before quasi-long-range positional order characterizing the solid.50−52 Hence, we investigate more closely the densification process for monodisperse particles by computing the bond orientational order parameter ⟨Ψ6⟩, which quantifies the amount of hexatic order developed at the interface as the adsorption proceeds. The local order parameter is defined as Ψ6k =

1 Nk

To investigate the mechanisms that drive the adsorption beyond the plateau-like regime, we analyze spatial heterogeneities arising during the ordering of the jammed interface. During slow adsorption, large ordered domains are formed and surrounded by defects (e.g., grain boundaries) i.e., very localized regions with a significantly lower degree of hexatic order. Figure 3a shows a snapshot of the interface at the end of

Nk

∑ exp(i6Θjk) j=1

(1)

where the sum is taken over all neighbors of particle k and Θjk is the angle of the bond connecting particles j and k.53 |Ψ6k| is 1 if the particle is in a perfect hexagonal environment. We average over all particles to get ⟨Ψ6⟩ = 1/N∑Nk=1|Ψ6k| as a measure of the degree of orientational order in the system. In Figure 1, ⟨Ψ6⟩ is plotted as a function of time together with 1 − ϕ(t), clearly indicating that the slowing down of the particle adsorption due to the filling of the interface is associated with the rapid growth of orientational order.38,54,55 In Figure 2, we plot ⟨Ψ6⟩ as a function of the surface coverage ϕ for systems with different Γ: the data show that the amount of ordering at the interface at the same surface coverage is indeed controlled by the adsorption rate. The emerging picture is that fast and irreversible particle adsorption is quite similar to supercooling and the higher the adsorption rate, the deeper the supercooling.

Figure 3. (Top) Snapshots of monodisperse (left) and polydisperse (right) systems after the plateau, with the color code indicating Ψ6. (Bottom) Ψ6 distribution within 1.8σ from newly inserted particles (blue □) and from random points (green ○) in the monodisperse (left) and in the polydisperse system (right), after the adsorption plateau and for Γ = 50 τ−1.

the plateau, with the color code indicating the degree of orientational order. The distribution of Ψ6 at the interface at the end of the plateau, shown in Figure 3c (circles), displays in fact a peak at values of Ψ6 close to 1 and extends to low values corresponding to the defects. In Figure 3c, we also plot the distribution of Ψ6 around a new particle about to be inserted at the end of the plateau (squares): the distribution is truncated at 3071

dx.doi.org/10.1021/la404254n | Langmuir 2014, 30, 3069−3074

Langmuir

Article

high Ψ6 showing that further adsorption is taking place mostly in the defect regions, where local rearrangements are still possible. That is, adsorption at long times is driven by the presence of defects developed during the ordering process, and these ultimately allow for higher coverage and further ordering of the interface (Figure 2). Similar results are obtained if one looks at the local area available around random neighboring particles and in the vicinity of a new insertion. From the latter, one obtains that new particles are inserted where locally the area fraction is lower, a consequence of defects in the crystalline structure as evidenced by the Ψ6-analysis.38 Starting from the insight gained for the monodisperse system, it is interesting to recognize that the same mechanisms also come into play in the presence of polydispersity: the insets of Figures 1 and 2 also show that the polydisperse system displays the onset of a significant increase in ⟨Ψ6⟩ at the surface coverages where the slowing down of the adsorption occurs. The absolute value of ⟨Ψ6⟩ attained is of course lower than that in the monodisperse system, but the phenomenon is qualitatively identical. Although polydispersity prevents crystallization, the local packing can be increased through the formation of small ordered domains, reminiscent of the medium-range orientational order found in glassy alloys56 and qualitatively similar to the increase in orientational order in the monodisperse system (Figure 3b). These findings may also account for the observation of discrete changes in the interparticle separation at long times found in recent experiments.14 In Figure 3d, we also observe that adsorption at the end of the plateau regime happens in regions of lower local order, hinting to the same direct link between kinetics, local structure, and locally available space for new insertions, regardless of the size polydispersity.

Figure 4. (Main frame) Reduced interfacial tension γ′ measured for samples of different polydispersities and concentrations: (green △) PDI = 0.15, c = 3.4 μg/cm3 and (red □) PDI = 0.47, c = 2.0 μg/cm3. (Inset) Core size distributions (solid lines are Gaussian fits).

tension and then immersed in the aqueous NP suspension to the same immersion height to measure the pressure due to NP adsorption. Figure 4 shows the reduced effective interfacial tension γ′ = γ/γ0 versus time at the air−water interface measured for the two samples with different polydispersities. The zeroing process lasts a few tens of seconds, and during this time, some NPs already adsorb at the interface, which gives γ′ < 1 at the beginning of the curves. Starting from the low-polydispersity system, we observe that the data show an initial rapid adsorption, followed by a significant slowing down in the uptake of particles, before adsorption continues, in agreement with the numerical results and other experiments. In the experiments, as in the simulations, no spontaneous particle desorption is observed during our observation window and adsorption is continuing, although extremely slowly. This confirms that the adsorption energies correspond to equilibrium coverages that are not reachable over relevant time scales. Increasing the size polydispersity may hinder the formation of ordered domains; moreover, it can also influence size-selection mechanisms that have been suggested to contribute to the slowing down of the adsorption kinetics.26 This might support the idea that the scenario presented above should qualitatively change with the size polydispersity of the NPs, but in agreement with the simulations, in the experiments we also observe a qualitatively identical behavior in the case of the highly polydisperse sample: the only significant differences are on the time scales due to different values of NP bulk concentration.58

■

EXPERIMENTS To further test the scenario developed in the simulations, we have performed new experiments of NPs adsorption at an water−air interface. The NPs have an iron oxide core stabilized by a thick solvated shell of highly monodisperse poly(ethylene glycol) (PEG) of approximately 5 kDa in molecular weight.25,35,57 The core size and polydispersity can be tuned via the synthesis conditions (mainly temperature and time). We have used samples with the same average size but different polydispersities of 0.15 and 0.47, respectively, as measured by quantitative analysis of TEM images of several thousands of particles. Figure 4 (inset) shows the full core size distributions. We follow the adsorption and thus the time evolution of the interface coverage by monitoring the surface pressure of aqueous NP suspensions at a flat air−water interface of fixed area using a Wilhelmy plate balance with a roughened platinum plate (KSV 5000, Finland). As NPs diffuse in the aqueous subphase and adsorb at the interface, the measured surface pressure, defined as the difference between the interfacial tension of pure air−water γ0 = 72 mN/m and the effective interfacial tension in the presence of the NPs γ, increases. The measurements were carried out by filling two identical containers with a cross-section of 9.6 cm2 with 5 mL of pure water and aqueous NP suspensions, respectively. The concentrations of the NP suspensions were tuned to highlight the presence of the intermediate adsorption plateau within similar experimental time windows for the two samples (c = 3.4 μg/cm3 for the PDI = 0.15 sample and c = 2.0 μg/cm3 for the PDI = 0.47 sample). The Wilhelmy plate was first immersed in the pure water container to zero the pure air−water interfacial

■

CONCLUSIONS By combining numerical simulations and experiments, we have shown that a strong adsorption of nanoparticles onto a fluid interface can produce supercooled high-density particle states that can temporarily slow down or arrest the adsorption process. Thanks to particle rearrangements at the interface, the development of local and quasi-long-range order allows for more efficient packing and produces defect regions where further adsorption can take place. This two-stage adsorption process is the same for monodisperse and polydisperse systems, 3072

dx.doi.org/10.1021/la404254n | Langmuir 2014, 30, 3069−3074

Langmuir

Article

(10) Isa, L.; Kumar, K.; Mueller, M.; Grolig, J.; Textor, M.; Reimhult, E. Particle lithography from colloidal self-assembly at liquid-liquid interfaces. ACS Nano 2010, 4, 5665−5670. (11) Fan, J. A.; Wu, C.; Bao, K.; Bao, J.; Bardhan, R.; Halas, N. J.; Manoharan, V. N.; Nordlander, P.; Shvets, G.; Capasso, F. Selfassembled plasmonic nanoparticle clusters. Science 2010, 328, 1135− 1138. (12) Wang, Y.; Wang, Y.; Breed, D. R.; Manoharan, V. N.; Feng, L.; Hollingsworth, A. D.; Weck, M.; Pine, D. J. Colloids with valence and specific directional bonding. Nature 2012, 491, 51−56. (13) Kaz, D. M.; McGorty, R.; Mani, M.; Brenner, M. P.; Manoharan, V. N. Physical ageing of the contact line on colloidal particles at liquid interfaces. Nat. Mater. 2011, 10, 1−5. (14) Isa, L.; Calzolari, D. C. E.; Pontoni, D.; Gillich, T.; Nelson, A.; Zirbs, R.; Sanchez-Ferrer, A.; Mezzenga, R.; Reimhult, E. Coreshell nanoparticle monolayers at planar liquid−liquid interfaces: effects of polymer architecture on the interface microstructure. Soft Matter 2013, 9, 3789−3797. (15) Snijkers, F.; Pasquino, R.; Vermant, J. Hydrodynamic interactions between two equally sized spheres in viscoelastic fluids in shear flow. Langmuir 2013, 29, 5701−5713. (16) Fenwick, N. I. D.; Bresme, F.; Quirke, N. Computer simulation of a Langmuir trough experiment carried out on a nanoparticulate array. J. Chem. Phys. 2001, 114, 7274. (17) Oettel, M.; Dietrich, S. Colloidal interactions at fluid interfaces. Langmuir 2008, 24, 1425−1441. (18) Ranatunga, R. J. K. U.; Kalescky, R. J. B.; Chiu, C.-c.; Nielsen, S. O. Molecular dynamics simulations of surfactant functionalized nanoparticles in the vicinity of an oil/water interface. J. Phys. Chem. C 2010, 114, 12151−12157. (19) Cheung, D. L. Molecular dynamics study of nanoparticle stability at liquid interfaces: effect of nanoparticle-solvent interaction and capillary waves. J. Chem. Phys. 2011, 135, 054704−054712. (20) Aland, S.; Lowengrub, J.; Voigt, A. A continuum model of colloid-stabilized interfaces. Phys. Fluids 2011, 23, 062103−062115. (21) Frijters, S.; Gunther, F.; Harting, J. Effects of nanoparticles and surfactant on droplets in shear flow. Soft Matter 2012, 8, 6542−6556. (22) Luu, X.-C.; Yu, J.; Striolo, A. Nanoparticles adsorbed at the water/oil interface: coverage and composition effects on structure and diffusion. Langmuir 2013, 29, 72217228. (23) Herzig, E. M.; White, K. A.; Schofield, A. B.; Poon, W. C. K.; Clegg, P. S. Bicontinuous emulsions stabilized solely by colloidal particles. Nat. Mater. 2007, 6, 966971. (24) Jordens, S.; Isa, L.; Usov, I.; Mezzenga, R. Non-equilibrium nature of two-dimensional isotropic and nematic coexistence in amyloid fibrils at liquid interfaces. Nat. Commun. 2013, 4, 1917. (25) Isa, L.; Amstad, E.; Textor, M.; Reimhult, E. Self-assembly of iron oxide-poly(ethylene glycol) core-shell nanoparticles at liquidliquid interfaces. Chimia 2010, 64, 145149. (26) Isa, L.; Amstad, E.; Schwenke, K.; Del Gado, E.; Ilg, P.; Kroger, M.; Reimhult, E. Adsorption of core-shell nanoparticles at liquid-liquid interfaces. Soft Matter 2011, 7, 7663−7675. (27) Pieranski, P. Two-dimensional interfacial colloidal crystals. Phys. Rev. Lett. 1980, 45, 569572. (28) The excess chemical potential we use in this GCMC does not correspond to the true chemical potential for the adsorption at the interface because it does not include the contribution arising from the particle−particle interactions, which changes with particle density. Nevertheless numerical simulations using the Widom insertion method indicate that the last contribution is, in a first approximation, reasonably negligible with respect to the adsorption strength. (29) Bresme, F.; Chacn, E.; Tarazona, P.; Tay, K. Intrinsic structure of hydrophobic surfaces: the oil-water interface. Phys. Rev. Lett. 2008, 101, 14. (30) Stocco, A.; Mokhtari, T.; Haseloff, G.; Erbe, A.; Sigel, R. Evanescent-wave dynamic light scattering at an oil-water interface: diffusion of interface-adsorbed colloids. Phys. Rev. E 2011, 83, 111.

where no quasi-long-range order can develop but where liquid patches play the same role as defects in the monodisperse systems. In both cases, the crucial control parameter is the adsorption rate. Our results indicate that a complex interplay between adsorption and self-assembly takes place and that the microstructure of the interface can be accurately designed by controlling the adsorption conditions. In particular, adsorption rates can be tuned by changing surface functionality59 and solvent conditions60 and by applying external fields,61 yielding a powerful and subtle handle to control the structure and mechanics of interfacial monolayers.

■

ASSOCIATED CONTENT

* Supporting Information S

Conversion between the effective interfacial tension measured in experiments and the surface coverage. Additional simulation data for simulations with a Gaussian potential. Comparison of systems with different adsorption rates and different adsorption energies. Analysis of the dynamics of the particles at the interface. Data for the development of the Voronoi area around particles and the orientational and positional correlation length at the interface. This material is available free of charge via the Internet at http://pubs.acs.org.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This work was supported by the Swiss National Science Foundation (grant no. PP00P2_126483/1). L.I. acknowledges Torben Gillich and Esther Amstad for providing the nanoparticles, Nic Spencer for support, and SNSF grants PZ00P2_142532/1 and PP00P2_144646/1 for financial support.

■

REFERENCES

(1) Boker, A.; He, J.; Emrick, T.; Russell, T. P. Self-assembly of nanoparticles at interfaces. Soft Matter 2007, 3, 1231−1248. (2) Grzelczak, M.; Vermant, J.; Furst, E. M.; Liz-Marzan, L. M. Directed self-assembly of nanoparticles. ACS Nano 2010, 4, 35913605. (3) Garbin, V.; Crocker, J. C.; Stebe, K. J. Nanoparticles at fluid interfaces: exploiting capping ligands to control adsorption, stability and dynamics. J. Colloid Interface Sci. 2012, 387, 111. (4) Dai, Q.; Frommer, J.; Berman, D.; Virwani, K.; Davis, B.; Cheng, J. Y.; Nelson, A. High-throughput directed self-assembly of core-shell ferrimagnetic nanoparticle arrays. Langmuir 2013, 29, 74727477. (5) Law, A. D.; Auriol, M.; Smith, D.; Horozov, T. S.; Buzza, D. M. a. Self-assembly of two-dimensional colloidal clusters by tuning the hydrophobicity, composition, and packing geometry. Phys. Rev. Lett. 2013, 110, 138301−138305. (6) Malescio, G.; Pellicane, G. Stripe phases from isotropic repulsive interactions. Nat. Mater. 2003, 2, 97100. (7) Salgado-Blanco, D.; Mendoza, I. C. Self-assembly of anisotropic soft particles in two dimensions. Eur. Phys. J. E 2013, 36, 9852. (8) Yan, F.; Goedel, W. Polymer membranes with two-dimensionally arranged pores derived from monolayers of silica particles. Chem. Mater. 2004, 16, 1622−1626. (9) Jiang, C.; Markutsya, S.; Pikus, Y.; Tsukruk, V. V. Freely suspended nanocomposite membranes as highly sensitive sensors. Nat. Mater. 2004, 3, 721728. 3073

dx.doi.org/10.1021/la404254n | Langmuir 2014, 30, 3069−3074

Langmuir

Article

(31) Wang, D.; Yordanov, S.; Paroor, H. M.; Mukhopadhyay, A.; Li, C. Y.; Butt, H.-J.; Koynov, K. Probing diffusion of single nanoparticles at water-oil interfaces. Small 2011, 7, 3502−3507. (32) Lee, C. K.; Hua, C. C. Nanoparticle interaction potentials constructed by multiscale computation. J. Chem. Phys. 2010, 132, 224904. (33) Everaers, R.; Ejtehadi, M. Interaction potentials for soft and hard ellipsoids. Phys. Rev. E 2003, 67, 041710. (34) Lo Verso, F.; Yelash, L.; Egorov, S. A.; Binder, K. Interactions between polymer brush-coated spherical nanoparticles: the good solvent case. J. Chem. Phys. 2011, 135, 214902. (35) Zell, Z. A.; Isa, L.; Ilg, P.; Leal, L. G.; Squires, T. M. Adsorption energies of poly(ethylene oxide)-based surfactants and nanoparticles on an air-water surface. Langmuir 2014, 30, 110−119. (36) Broughton, J.; Gilmer, G.; Weeks, J. Molecular-dynamics study of melting in two dimensions. Inverse-twelfth-power interaction. Phys. Rev. B 1982, 25, 46514669. (37) Hurley, M.; Harrowell, P. Kinetic structure of a two-dimensional liquid. Phys. Rev. E 1995, 52, 1694. (38) See the Supporting Information for further information. (39) Frenkel, D.; Smit, B. Understanding Molecular Simulation; Academic Press: San Diego, 2002. (40) δt = 0.004τ for the system with the largest Γ. During particle insertion, we use a smaller step (0.005τ ≤ δt ≤ 8 × 10−7τ) to avoid strong temperature fluctuations. (41) Du, K.; Glogowski, E.; Emrick, T.; Russell, T. P.; Dinsmore, A. D. Adsorption energy of nano- and microparticles at liquid-liquid interfaces. Langmuir 2010, 26, 12518−12522. (42) Aveyard, R.; Haydon, D. A. The Principles of Surface Chemistry; Prentice-Hall: Englewood Cliffs, NJ, 1972. (43) The reduced effective surface tension γ/γ0 measured in the experiments does not necessarily (and rigorously) correspond to the thermodynamic quantity defined in ref 42. This would be the case only if, for each time t at which γ(t) is measured, the thermodynamic equilibrium of the whole system (molecules of the two fluids + nanoparticles) was reached. Indeed, this is not the case because the NP adsorption is fast and irreversible. In spite of this, we can still consider that the time resolution of an experiment (i.e., for measuring γ(t)) is significantly larger than the time scale needed by the molecules of water and air at the interface to reach a new local equilibrium state upon uptake of new NPs. Hence, we can assume that each experimental data point corresponds at least to a state of mechanical equilibrium of the interface (if not really thermodynamical) and that the change in γ(t) is still a well-defined quantity. In the simulations, we do not resolve the molecular scale and consider the adsorption to be instantaneous with respect to the time scale of NP reorganization. On this basis, we expect that the relation between the measured γ and the surface coverage is still applicable for our study, and this expectation is supported by the reasonable range of surface coverage obtained when transforming γ to 1 − ϕ for the experimental data, as shown in the Supporting Information (Figure S1). The final coverage (i.e., γ(t ≃ ∞)) could in principle be a better candidate for the thermodynamic effective surface tension, but, as already discussed in ref 44, this is not really the case for a colloid-coated interface where, because of the high energy barriers involved, small changes in area are not necessarily balanced by changes in the interfacial colloid population and true equilibrium measurements (in the thermodynamics sense) would not allow quantifying the change in the effective surface tension γ. Such measurements would be able to detect only the surface change in the bare fluid−fluid interface that does not significantly change upon adsorption/desorption of the NPs. (44) Clegg, P. S.; Herzig, E. M.; Schofield, A. B.; Egelhaaf, S. U.; Horozov, T. S.; Binks, B. P.; Cates, M. E.; Poon, W. C. K. Emulsification of partially miscible liquids using colloidal particles: nonspherical and extended domain structures. Langmuir 2007, 23, 5984−94. (45) Mazoyer, S.; Ebert, F.; Maret, G.; Keim, P. Dynamics of particles and cages in an experimental 2D glass former. Europhys. Lett. 2009, 88, 66004.

(46) Tanaka, H.; Kawasaki, T.; Shintani, H.; Watanabe, K. Criticallike behaviour of glass-forming liquids. Nat. Mater. 2010, 9, 324331. (47) Candelier, R.; Widmer-Cooper, A.; Kummerfeld, J.; Dauchot, O.; Biroli, G.; Harrowell, P.; Reichman, D. Spatiotemporal hierarchy of relaxation events, dynamical heterogeneities, and structural reorganization in a supercooled liquid. Phys. Rev. Lett. 2010, 105, 135702. (48) Alder, B.; Wainwright, T. Phase transition in elastic disks. Phys. Rev. 1962, 127, 359360. (49) Peierls, R. Remarks on transition temperatures. Helv. Phys. Acta 1934, 7, 183. (50) Kosterlitz, J.; Thouless, D. Ordering, metastability and phase transitions in two-dimensional systems. J. Phys.: Condens. Matter 1973, 6, 1181−1203. (51) Halperin, B.; Nelson, D. Theory of two-dimensional melting. Phys. Rev. Lett. 1978, 41, 121124. (52) Bernard, E.; Krauth, W. Two-step melting in two dimensions: first-order liquid-hexatic transition. Phys. Rev. Lett. 2011, 107, 1−4. (53) Nelson, D. Defects and Geometry in Condensed Matter Physics; Cambridge University Press: Cambridge, U.K., 2002. (54) We also observe an increase in positional order at the interface, although the correlation length is always smaller than that associated with Ψ6. (55) Engel, M.; Anderson, J. A.; Glotzer, S. C.; Isobe, M.; Bernard, E. P.; Krauth, W. Hard-disk equation of state: first-order liquid-hexatic transition in two dimensions with three simulation methods. Phys. Rev. E 2013, 87, 042134. (56) Tanaka, H. Bond orientational order in liquids: towards a unified description of water-like anomalies, liquid-liquid transition, glass transition, and crystallization: bond orientational order in liquids. Eur. Phys. J. E 2012, 35, 113. (57) Gillich, T.; Acikgoez, C.; Isa, L.; Schlueter, A. D.; Spencer, N. D.; Textor, M. PEG-stabilized core-shell nanoparticles: impact of linear versus dendritic polymer shell architecture on colloidal properties and the reversibility of temperature-induced aggregation. ACS Nano 2013, 7, 316−329. (58) In the experiments, the interplay among particle size distribution, bulk concentration, and adsorption kinetics is delicate, and here we have adjusted the bulk concentrations to yield adsorption curves with similar time scales in the experimental window to allow for a more direct comparison. (59) Duan, H.; Wang, D.; Kurth, D.; Mohwald, H. Directing selfassembly of nanoparticles at water/oil interfaces. Angew. Chem., Int. Ed. 2004, 43, 5639−5642. (60) Reincke, F.; Hickey, S.; Kegel, W.; Vanmaekelbergh, D. Spontaneous assembly of a monolayer of charged gold nanocrystals at the water/oil interface. Angew. Chem., Int. Ed. 2004, 43, 458−462. (61) Turek, V. A.; Cecchini, M. P.; Paget, J.; Kucernak, A. R.; Kornyshev, A. A.; Edel, J. B. Plasmonic ruler at the liquid-liquid interface. ACS Nano 2012, 6, 7789−7799.

3074

dx.doi.org/10.1021/la404254n | Langmuir 2014, 30, 3069−3074