Contact Forces between TiO2 Nanoparticles Governed by an Interplay

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Contact forces between TiO2 nanoparticles governed by an interplay of adsorbed water layers and roughness Jens Laube, Samir Salameh, Michael Kappl, Lutz Mädler, and Lucio Colombi Ciacchi Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.5b02989 • Publication Date (Web): 28 Sep 2015 Downloaded from http://pubs.acs.org on October 5, 2015

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Contact forces between TiO2 nanoparticles governed by an interplay of adsorbed water layers and roughness Jens Laube,† Samir Salameh,‡ Michael Kappl,¶ Lutz Mädler,‡,§ and Lucio Colombi Ciacchi∗,†,§ Hybrid Materials Interfaces Group, Faculty of Production Engineering, Bremen Center for Computational Materials Science and Center for Environmental Research and Sustainable Technology (UFT), University of Bremen, 28359 Bremen, Germany, Foundation Institute of Materials Science (IWT) and Faculty of Production Engineering, University of Bremen, 28359 Bremen, Germany, and Max Planck Institute for Polymer Research, 55128 Mainz, Germany E-mail: [email protected]

∗

To whom correspondence should be addressed Hybrid Materials Interfaces Group, Faculty of Production Engineering, Bremen Center for Computational Materials Science and Center for Environmental Research and Sustainable Technology (UFT), University of Bremen, 28359 Bremen, Germany ‡ Foundation Institute of Materials Science (IWT) and Faculty of Production Engineering, University of Bremen, 28359 Bremen, Germany ¶ Max Planck Institute for Polymer Research, 55128 Mainz, Germany § MAPEX Center for Materials and Processes, University of Bremen, 28359 Bremen, Germany †

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Abstract Interparticle forces govern the mechanical behavior of granular matter and direct the hierarchical assembling of nanoparticles into supramolecular structures. Understanding how these forces change under different ambient conditions would directly benefit industrial-scale nanoparticle process units such as filtering or fluidization. Here we rationalize and quantify the contributions of dispersion, capillary and solvation forces between hydrophilic TiO2 nanoparticles with sub-10 nm diameter and show that the humidity dependence of the interparticle forces is governed by a delicate interplay between the structure of adsorbed water layers and the surface roughness. All-atom molecular dynamics modelling supported by force-spectroscopy experiments reveal an unexpected decrease of the contact forces at increasing humidity for nearly spherical particles, while the forces between rough particles are insensitive to strong humidity changes. Our results also frame the limits of applicability of discrete solvation and continuum capillary theories in a regime where interparticle forces are dominated by the molecular nature of surface adsorbates.

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Controlling the agglomeration and dispersion behavior of oxide nanoparticles is key to technical applications such as catalysis, sensorics or hybrid materials synthesis. 1,2 Especially in large-scale industrial process units such as filtering and fluidization, the contact forces between individual nanoparticles directly influence the process efficiency and thus, indirectly, the production costs. Often nanoparticle assemblies are processed or employed under normal ambient conditions, which implicates the presence of a layer of adsorbed water molecules on the nanoparticles’ surfaces in contact with humid air. For this reason, numerous experimental and theoretical studies have attempted to rationalize the dependencies of interparticle contact forces on the surrounding humidity. 3–13 As a result, it has become clear that such dependencies rest on a non-trivial combination of particle size, shape, surface hydrophilicity and roughness. 14 However, predictive models that account for controlled changes of the interparticle forces are mostly limited to each single contribution and are often applied to model systems with little transferability to real material/adsorbate combinations. Of particular interest is the case of nanoscopic particles, for which the thickness of the water layer reaches the same order of magnitude as the particle size 15,16 and the discrete molecular nature of the adsorbed layer becomes perceptible. It has been recently shown that continuum capillary theory may be applied to rationalize contact forces in humid air if the adsorbed water layer is explicitly taken into account. 5,8 Moreover, the molecular ordering at solid/liquid interfaces 17,18 has been long known to result in structure-dependent, oscillatory forces between material surfaces separated by a liquid solvent, 19,20 as observed for instance by means of AFM force spectroscopy 21–23 or molecular dynamics simulations. 9,13,24 The oscillation amplitude and frequency can be quantified in terms of a product (interference) of the solvent density profiles over isolated surfaces when they come into close contact, 25 and are dependent on the degree of solvent affinity of the material surface (on its degree of hydrophilicity, in the case of water). 26–29 A rationalization of the dependence of dispersion, solvation and capillary interparticle forces on ambient humidity is often difficult and not intuitive. Especially challenging is to

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predict the relative changes between these contributions across the transition between dry and wet conditions. 4,30–32 For micrometer-sized particles, classical capillary effects dominate and are modulated by both the overall particle shape and surface roughness. 6,10,33 However, it remains an open question whether the same conclusions can be transferred to particles with sizes of a few nm. 34,35

Results and Discussion Molecular dynamic simulations at different water coverages In order to investigate the issue outlined above, we perform an extensive study of the water-mediated adhesion between TiO2 nanoparticles using all-atom molecular dynamics modelling, as already introduced in ref. 12 We first concentrate on fairly smooth, spherical rutile particles with a diameter of d = 4 nm terminated by 3 nm−2 OH groups (Fig. 1). The surface/water interactions are modelled via a recently developed force field optimized for TiO2 /water interfaces. 36–38 The computed force-displacement (F-D) curves between two particles whose surfaces are separated by a distance δ (see scheme in Fig. 1) at different water coverages are shown in Fig. 2 and in the Supporting Information (Fig. S1 and S2). They display an oscillatory behaviour at short particle-particle distances, superimposed on a typical long-range capillary behaviour. 12 In agreement with common knowledge, 39,40 the slope of the capillary contribution to the F-D curves decreases and the length of the capillary interaction increases with increasing water coverage. Obviously, the capillary contribution vanishes when the particles are immersed in bulk water or in the absence of adsorbed water. The F-D oscillations display nearly constant amplitudes at all water coverages (including in bulk liquid), and are only absent when no adsorbed water molecules are present on the surface, i.e. when the interparticle attraction force is only due to van-der-Waals interactions.

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Figure 2: Force-displacement curves calculated for 4 nm TiO2 particles facing each other along the h110i direction (see Fig. 1) with different amounts of adsorbed water molecules. The curves are shifted along the y axis with respect to their respective zero-force lines (dashed) and divided by the particle diameter d.

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Solvation forces The oscillatory behaviour of the force-distance curves at δ . 1 nm commonly arises between interacting surfaces in an ordered molecular solvent. 29,41 The solvent structuring around a single TiO2 nanoparticle in bulk water can be expressed in terms of the radial distribution function g(r) and the radial-angular distribution function g(r, α) of the water O atoms in surface proximity. The distance r is defined as the difference between the length of the radial position vector of each water oxygen atom from the center of mass (COM) of the particle and the length of the position vector of the nearest titanium atom of the particle surface (Fig. 1), whereas α is the angle between the water molecular dipole and the r direction. The superimposed g(r) and g(r, α) (Fig. 3(a)) highlight a tightly bound first hydration layer (up to r ∼0.25 nm) with the O atoms of all molecules pointing towards the surface, and a second hydration double layer (r between 0.25 and 0.5 nm) including molecules with opposite orientations. Integration of g(r) over the single density peaks reveal, on average, 5.2 molecules per nm2 of particle surface in the first layer (which we will refer to as ‘chemisorbed’ water) and 12.2 molecules per nm2 in the second layer (which includes ‘physisorbed’ water).

In the case of particles in humid air separated by a distance δ, analogous structuring of the adsorbed water molecules in the interparticle region is quantified by means of the unidirectional distribution functions gδ (z) and gδ (z, α), computed in a cylindrical volume with an arbitrarily small cross-section A around the z axis connecting the COM of the two particles. As shown in Fig. 3(b), for a separation corresponding to the first potential energy minimum (δ = 0.33 nm, see Fig. 2) a single water double-layer is present between the two particles. Increasing δ leads first to stretching of this double-layer (attractive forces up to a maximum at δ = 0.37 nm and later to insertion of a new double layer, which is under compression (repulsive forces) up to the next potential energy minimum at δ = 0.56 nm (Fig. 3(c)). The evolution of gδ (z) and gδ (z, α) for increasing separations and for different orientations of the interacting particles is reported in detail in the Supporting Information 7


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a

b

d

c

Figure 3: Analysis of the solvation contributions to the interparticle forces. (a) Superimposed g(r) (red curve) and g(r, α) (colour plot, with red representing regions higher than in the bulk, on a logarithmic scale) for a single TiO2 particle in bulk water. (b, c) Same as in (a) for g(z) and g(z, α) between particles in humid air at δ = 0.33 nm and 0.56 nm (A = π(1.4)2 nm2 ). The black lines are theoretical predictions from Eq. 2 using the g(z) in bulk water over an isolated particle in the same orientation. (d) Solvation forces between two particles in bulk water calculated via Eq. 2 (A = π(0.25)2 nm2 ) compared to the forces calculated explicitly along several mutual orientations (black symbols). The latter are shifted to match the position of the maximum attractive force at δ = 0.62 nm and divided by the particle diameter d.

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(Figs. S3,S4,S5). This qualitative description of the solvation contribution of the interparticle force can be put in more rigorous terms by noting that gδ (z) corresponds to the product of the unidirectional distribution functions of the solvent around each single particle (δ → ∞) assuming a linear superposition of the potentials of mean force, 25

gδ (z) = g∞ (z) · g∞ (δ − z) ,

(1)

and that the solvation pressure p between the two particle surfaces can be obtained by integration of the particle-water force function f (z) weighted by the density excess in the interparticle region: p(δ) =

Z

∞

[gδ (z) − g∞ (z)] · ρb · f (z) dz .

(2)

0

Here, ρb is the solvent bulk density, and f (z) and g(z) are both averaged over several particle orientations as explained in detail in the Supporting Information (Section S3). The gδ (z) functions predicted by Eq. 1 agree very well with the corresponding distribution functions calculated explicitly in the MD simulations, irrespective of the chosen water coverage and for all considered orientations. Examples are shown in Fig. 3(b,c) (black lines) and in Supporting Figs. S3,S4,S5. This allows us to apply Eq. 2 and compare the predicted solvation-like contribution of the interparticle forces Fsolv (δ) = p(δ) · A with the results of the explicit MD calculations, which is shown for the case of particles in bulk water interacting along several orientations in Fig. 3(d). The overall agreement between the predicted (blue line) and the explicitly calculated (black symbols) solvation forces is excellent both regarding the position and the intensity of the force oscillations, if an optimal value of the cross-section A is chosen (here its diameter is 0.5 nm, which is considerably smaller than the diameter of the capillary meniscus). However, notable differences related to the specific interaction orientations are visible and point towards an important role played by the local surface structure in governing the interaction 9


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force. This role will emerge more evidently later in this paper when we consider rough particles, and will be a key to understand the dependence of the maximum interaction force on the relative humidity. However, at this stage, it suffices to observe that flatter surfaces (e.g. (110)) result in deeper attractive forces, while more corrugated surfaces (e.g. (001)) result in less pronounced oscillations than the ensemble average (predicted blue curve), as one may intuitively expect.

Capillary forces For particles in humid air, i.e. with a limited amount of adsorbed water molecules, the solvation-like oscillatory force contribution described above is superimposed to a long-range contribution due to the formation of a capillary water neck (meniscus) between the particles (see Fig. 2). While the former contribution is insensitive of variation of water coverage, except for the extreme case of completely dry particles, the latter is strongly humiditydependent. 40,42 The reason for this is the explicit dependence of the meniscus’ geometry (expressed via the curvature radius rm and the cross-section radius lm , see Fig. 1) on the water vapour partial pressure P/P0 through the Kelvin equation P 1 γL · VM 1 , = exp − · − P0 RT rm l m

(3)

where R is the ideal gas-constant, T the temperature, VM the molar volume of water and γL its surface tension. In the circular approximation, the radii of the meniscus, in turn, are linked to the water/solid contact angle Θ and the filling angle β defining the contact point between solid and liquid (Fig. 1) via

rm =

2R1 (1 − cos(β)) + D 2 cos(Θ + β)

(4)

and lm = R1 sin(β) − rm [1 − sin(Θ + β)] , 10


(5)

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where R1 is the particle’s radius and D is the particle-particle separation distance. For a given solid/solvent pair (i.e. given γL and Θ), the shape of the meniscus (rm , lm and β) uniquely defines an attractive force between two particles of equal radius R1 , which can be calculated within the limits of the circular approximation 33,39 through

F = πγL R1 sin(β) 2 sin(Θ + β) + R1 · sin(β)

1 1 − rm l m

.

(6)

When we apply the equations above to our system for a relative humidity of 80 % (see below) using a contact angle Θ = 10◦ (representative of highly hydrophilic TiO2 particles), a surface tension γL = 52.3 mJ/m2 (for bulk TIP3P water, 43 which is known to differ from the experimental value of 72 mJ/m2 ) and a nominal particle radius of 2 nm, we obtain force-displacement curves that miss by far the force values calculated in our MD simulations (Fig. 4(b), green curve, labelled FCap,Conv ). The reason for this striking discrepancy is the large influence of adsorbed water molecules forming an adsorbate layer with non-negligible thickness h (about 0.5 nm). In order to take this effect properly into account, we obtain directly all parameters necessary to compute the capillary force according to eq. 6 via a geometrical analysis of our atomistic particle/water/particle models at each particle-particle separation distance in the MD trajectories. To do so, we first perform a radial averaging of the water density around the z coordinate connecting the COM of the two particles (taking into account all different mutual orientations) and plot the averaged iso-density lines at different fixed values (Fig.4(a), top panel). We then use the equimolar dividing surface between liquid and vapour (at the density value of 0.5 g/cm3 ) to fit one circle to the region of the interconnecting meniscus and two circles to the regions of the surrounding particles in line with the analysis of Ko et al. 44 (Fig.4(a), bottom panel). This leads to variable values of β, rm and lm , to nearly constant values of R1 for all separation distances of a specific water coverage, and to a contact angle Θ ∼ 0◦ for all separation distances and water coverages, which is very reasonable given that

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a

b

c

d

Figure 4: Analysis of the capillary contributions to the interparticle forces. (a) Top: isovalue lines of the radially-averaged water density between two particles with 16.2 adsorbed water molecules per nm2 , at a distance δ = 0.6 nm, considering different mutual orientations. Bottom: Geometrical fit of the meniscus geometry to the iso-density line at 0.5 g/cm3 . (b) Comparison between the F-D curves computed explicitly in MD simulations (black crosses, shifted as in Fig. 3) and the theoretically predicted capillary force contributions with nominal geometries (green) and geometrical fits of the MD trajectories without (red) and with (blue) constraints on particle radius and humidity. The magenta curve is the total sum of steric, dispersion, solvation and capillary force contributions. (c) Isotherm relating P/P0 to the adsorbed water coverage, as obtained from our capillary force analysis (green squares) in comparison with previous experiments 15,16 (black symbols) and an own TGA measurement (red circle) on FSP-produced particles. (d) Predicted variations of the effective interparticle separation D and the adsorbed water layer thickness h (see Fig. 1) with P/P0 . The error bars in (c) and (d) denote the standard deviation relative to all particle orientations and separations.

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the water molecules within the meniscus are in contact with the water molecules adsorbed on the particle surface (See also Supporting Section S4, Figs. S7,S8,S9). We are now able to compute the capillary contribution to the interparticle force via eq. 6. In doing so, the water surface tension γL is treated as a free parameter and optimized to best-fit the results of the MD simulations in the long-range regions of the force-displacement curves for all considered mutual orientations between the particles (Fig. 4(b), red curve, labelled FCap,Geom ) and all water coverages (Fig. S7,S8,S9). The resulting γL values correctly converge to the surface tension of bulk TIP3P water at high coverages and increase steadily with decreasing coverage (Supporting Table S1). This agrees with previous results, 4,45,46 and stems from the increased structuring and decreased mobility of water molecules in strongly confined, thin liquid layers. The capillary contribution calculated in this way agrees remarkably well with the results of the MD simulations. Fundamental for the obtained agreement is not only the contact angle Θ = 0◦ , but especially the value of the radius R1 , which is considerably larger than the nominal particle radius and now includes the adsorbed water layer, consistently with previous works. 4,5,47 This implies negative values of the separation distance D at the point of maximum attractive force between the particles (Fig. 4(d)), which is an important factor influencing the humidity-dependency of the capillary force contribution (see below). We note that forcing the system to obey the circular approximation leads to very small variations of the fitted R1 values and to slightly more evident variations of the humidity P/P0 , which we obtain from the Kelvin equation 3 over the whole range of particle separations (see Supporting Fig. S10). Choosing the distance-averaged values of humidity, particle radius and surface tension together with a contact angle Θ = 0◦ and solving iteratively Eqs. 3–6 leads to predicted F-D curves that deviate from the ones obtained through local geometrical fitting only at very high separation distances (Fig. 4(b), blue curve, labelled FCap,M od ). However, in the region of contact equilibrium and of maximum attractive force (δ between 0.3 and 0.7 nm), the agreement is very good, thus allowing us to predict the capillary force contribution analytically at each humidity value, once a relationship between

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the humidity and the thickness of the adsorbed water layer (necessary to obtain correct R1 and γL values) is established 4,5 (Fig. 4(d)). Starting from our MD trajectories at different water coverages, after fitting the capillary neck geometry, optimizing the value of γL , applying the Kelvin equation and averaging over all separation distances at each fixed coverage, we obtain isotherms of water adsorption on the hydrophilic particles which are in fairly good agreement with available experimental measurements on TiO2 single crystals and particles (Fig. 4(c)).

Humidity dependence Having rationalized both the structural (solvation) and the capillary contributions to the interparticle forces, we are now able to understand how the latter varies with humidity (the former being, as stated above, roughly constant for all adsorbed water coverages). As a first observation, we note that a key parameter determining whether the capillary attraction either increases or decreases with humidity is the separation distance D between the particles at the point of maximum attractive force. This is visible when plotting force-humidity curves after solving analytically Eq. 3–6 at constant D (Fig. 5(a), grey lines). For D values greater than 0.2 nm, which corresponds to the classical picture of capillary theory where the adsorbate thickness can be neglected because of much larger particle radii, the attractive forces increase. For smaller D values, which are a consequence of adsorbate layer thicknesses only about ten times smaller than the particle diameter, the forces decrease. However, it has to be taken into account that D is itself a function of humidity, and moves from positive values for completely dry particles to negative values for high water coverages (Fig. 4(d)). It is therefore crucial to quantify the actual capillary contribution at the point of maximum attractive force for each different water coverage. Following our previous work, 12 we can safely assume that measured interparticle forces (e.g. by means of AFM force spectroscopy) are the maximum attractive forces corresponding to the minima of the F-D curves before the onset of steric repulsion, located at a separation 14


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δ = 0.37 nm or 0.62 nm, depending on the specific orientation (See Fig. 4(b)). We call Fmax the force at this point. Regarding the explicitly calculated forces in MD simulations, the capillary contributions can be obtained by direct subtraction of the maximum attractive forces in bulk water, Fbulk , from the corresponding Fmax values for each considered particle orientation. The differences Fmax − Fbulk are plotted as a function of humidity in Fig. 5(a) over the parametrized curves at constant D and together with the theoretically predicted FCap,M od values at δ = 0.37 nm and 0.62 nm, which are independent of the actual particle orientation. a

b

roughness: 0.5nm no roughness

c

d

Figure 5: Variation of the capillary force contribution with humidity for smooth (a) and rough (b) particle models. The values predicted by our modified capillary model (blue symbols) are compared with explicitly calculated maximum attractive forces along different orientations after subtraction of the corresponding forces in bulk water (black crosses), and plotted over parametrized force-humidity curves at constant particle-particle separation D (grey lines). (c) Model of two rough particles in capillary contact showing the isovalue water density surface at about 0.5 g/cm3 . (d) Contact forces between FSP-produced TiO2 nanoparticles measured by AFM force spectroscopy at different relative humidities (box plot showing median, quartiles, extremes and outliers of the force distributions). 15


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In the obtained plot, at humidities between 10 and 30 % the calculated forces initially increase. This is consistent with initially positive values of D for very thin adsorbed water layers, which shift towards smaller and eventually negative values with increasing water adsorption, leading to a force increase (since the curves at constant D shift to larger forces as D decreases). The forces then decrease with a slope that does not depend on the orientation (apart from a slight scatter due to phase space sampling issues in the simulations). Indeed, at higher humidities the decrease of force dictated by negative D values cannot be compensated by the force increase that results from thicker adsorbed layers. In summary, the analysis above suggests that the force dependency on humidity is strongly affected by the way the adsorbate layer thickness h (on which both γL and D directly depend) varies for increasing P/P0 values (see Fig. 4(d)).

Particles with surface roughness It has to be noted that the shape of commonly produced nanoparticles deviates from ideal spheres. In this case, the onset of steric repulsion and thus the separation distance at maximum contact force are dominated by the local asperities. 6,48 Therefore, since the adsorbate layer thickness is comparable to the surface roughness, even large changes of the humidity (and thus of the water coverage) have little influence on D. In fact, through explicit calculation of force-displacement curves between rough particle models (Fig. 5(c) and Supporting Fig. S11,S12) we obtain slightly positive and roughly constant D values (see Supporting Section S5 and Table S2). Consequently, we also compute maximum attractive forces which do not vary with humidty over the whole range of investigated water coverages (Fig. 5(b)). Note, however, that even for rough particles we can distinguish a solvation and a capillary contribution to the interparticle forces (Supporting Figs. S11,S12,S13,S14). In order to frame our theoretical prediction in an experimental context, we have measured the interparticle forces within FSP-deposited TiO2 films of nanoparticles with an average radius of 12 nm using our previously developed AFM force-spectroscopy technique. 12 A 16


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thermogravimetric analysis confirms that a layer of adsorbed water is present under normal ambient conditions (50% of relative humidity), with a number of water molecules (14.1 nm−2 ) very close to our theoretical prediction and previous literature results (Fig. 4(c)). In spite of the adsorbed water, however, the measured forces (Fig. 5(d)) present nearly constant median values and distribution widths at all measured humidities, from vacuum up to 90 % RH. This finding agrees with our simulations of rough particles and shows that the irregular particle shape is key to rationalize the humidity-dependence of FSP-produced TiO2 nanoparticles.

Conclusions In conclusion, we have developed a comprehensive approach for the combined experimental measurement and atomistic calculation of interparticle forces at the nanometer scale. We have demonstrated that classical theories can still predict the dependence of the interparticle forces on their surrounding environment (here, the air humidity) in terms of dispersion, solvation and capillary contributions, if the crucial effects of adsorbate layer formation, structuring and nanoscopic surface roughness are correctly taken into account. An unbiased, quantitative prediction of these contributions relies on the knowledge of three fundamental properties, namely (i) the radial distribution function of the adsorbate molecules around a single particle; (ii) the isotherm of adsorption of the adsorbates linking their vapour pressure to the condensation layer thickness; and (iii) the variation of the surface tension of thin liquid films with their thickness and possibly with their curvature, if particles of different sizes are considered. These properties can be all obtained from accurate atomistic simulations (as performed in this work), and at least the first two from experimental measurements that are directly accessible for nanoparticulate materials. Knowledge of other, less accessible properties such as the solvent/solid contact angle are neither required, nor play an important role for particles about 10 times larger than the adsorbate layer thickness. We stress that our analysis is formulated for a realistic solid/vapour couple (here, TiO2 / water), thus capturing effects absent in model systems such as ideally spherical particles 17


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in van-der-Waals solvents. Nevertheless, we have been able to extract conclusions of great generality, which can be potentially applied to a variety of different systems and provide a comprehensive adhesion contact model between nanoparticles that can be readily used in Discrete Particle Modelling approaches. In particular, we have revealed the governing role played by the dependence of the effective particle-particle separation distance D on the adsorbate’s vapour pressure. We therefore suggest that direct tuning of interparticle forces can be accomplished by acting on any property that influences such dependence, most notably the material’s affinity for different adsorbate molecules or the surface roughness. This could be achieved e.g. by surface functionalization, particle doping, or deliberate insertion of other type of (surface) defects. Comparing our results to the adhesion forces between silica nanoparticles as investigated atomistically by Leroch et al. 3 , we find an only slightly different trend of the capillary forces with humidity and less pronounced force oscillations at close particle contact. These variations are due to a differently shaped adsorption isotherm and a weaker structuring of water molecules at the SiO2 surface. Future work shall be devoted to investigate the influence of the aggregation pathway of freely rotating particles 13,49 and to elucidate the scaling laws which relate the particle size with the above-mentioned effects, in order to frame and understand the transition between the nanoscopic particle-particle interaction behaviour presented here and the macroscopic behaviour which is common knowledge in the field of colloidal materials.

Methods Molecular Dynamics Simulations Smooth nanoparticles with a diameter of 4 nm are created by carving a sphere off a rutile single crystal respecting the TiO2 stoichiometry. Rough particles including asperities are created by carving the union region of a 4 nm sphere together with 1 nm spheres placed with their geometrical centers aligned on the particle surface (see scheme in Supporting Fig. S11), 18


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thus producing a 0.5 nm roughness. For the sake of simplicity the water coverage is calculated assuming a surface of 50 nm2 for both particle models, which corresponds to a sphere with a diameter of 4nm (since the coverage is not a direct parameter in the equations used to predict the forces analytically, the resulting approximation for the case of rough particles is negigible for the purposes of this work). The particle models are fully relaxed, and the Ti atoms with less than 5 O neighbours are hydrated assuming dissociative water adsorption, 37 leading to a coverage of 3.0 nm−2 and 4.5 nm−2 terminal OH-groups for smooth and rough particles, respectively. Interparticle forces are calculated from classical Molecular Dynamics (MD) simulation runs (each lasting 1 ns) at fixed distances, in the NVT ensemble at 300K, using the LAMMPS programme package. 50 A detailed description of the MD simulation details and parameters can be found in. 12 The interactions between the oxide atoms of the particles are described by a Matsui/Akaogi force field, as described in Ref. 36 TIP3P 51 water molecules interact with the oxide atoms via the force field developed in Refs. 37,38 All figures were produced using the open-source Python software with the matplotlib package.

Force measurements TiO2 nanoparticle aggregates are synthesized by Flame Spray Pyrolysis (FSP) and in-situ deposited on mica sheets (Science Service) as highly porous films based on our developed standard protocol. 12 The precursors are mixed with a molarity of 0.5 M. The precursor feed rate and the gas flow rates are adjusted to 5 ml/min and 5 l/min, respectively. Force measurements are performed on the nanoparticle films with an atomic force microscope (Nanowizard 3 from JPK Instruments) with Si3 N4 cantilevers (DNP-S from Bruker Nano GmbH) using a force mapping over 10x10 µm2 with 8x8 force measurements. The force mapping is performed using a fixed setpoint load of 2.5 nN and a retraction speed of 2 µm/s. The spring constant of each single cantilever (nominal stiffness 0.12 nN/nm) is determined using the thermal noise method. 52 The measurements are performed in an environmental cell (JPK) at specified humidities. The humidity in the cell is controlled from 0 to 90% 19


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(at 20◦ C) using a humidity generator (Surface Measurement Systems). Before starting a force measurement, the system is equilibrated for a few minutes. 6 The measurements are started at low humidity and carried on increasing it towards higher values. To exclude hysteresis effects or an increasing force due to multiple particles collected on the cantilever tip, 12 a final measurement set is recorded in reversed humidity order. AFM measurements in vacuum are performed using the same AFM-protocol as above in an Enviroscope AFM (Veeco Instruments) with a non-coated silicon cantilever of nominal stiffness 0.20 nN/nm. The porous film of TiO2 nanoparticle aggregates is placed in vacuum (9mbar), heated up to 200◦ C for 2 hours to remove all physisorbed water, and cooled down before measurements.

Acknowledgement We are grateful to Uri Sivan (Technion, Haifa, Israel), Guido Grundmeier (University of Paderborn) and Wolfgang Peukert (University of Erlangen-Nürnberg) for stimulating discussions. This work has been supported by the Deutsche Forschungsgemeinschaft within the Priority Program “Partikel im Kontakt - Mikromechanik, Mikroprozessdynamik und Partikelkollektiv” (SPP 1486) under grants CO 1043/3, MA 3333/3 and KA 1724/1. Computational time has been provided by the HLRN supercomputing centre at Hannover and Berlin, Germany.

Supporting Information Available Force-displacement curves between smooth particles; Water structure between smooth particles in contact; Analytical calculation of solvation forces; Water density profiles and capillary forces between smooth particles; Details on rough particle models. This material is available free of charge via the Internet at http://pubs.acs.org/.

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References (1) Min, Y.; Akbulut, M.; Kristiansen, K.; Golan, Y.; Israelachvili, J. The role of interparticle and external forces in nanoparticle assembly. Nature Mater. 2008, 7, 527–538. (2) Majmudar, T. S.; Behringer, R. P. Contact force measurements and stress-induced anisotropy in granular materials. Nature 2005, 435, 1079–1082. (3) Leroch, S.; Wendland, M. Influence of Capillary Bridge Formation onto the Silica Nanoparticle Interaction Studied by Grand Canonical Monte Carlo Simulations. Langmuir 2013, 29, 12410–12420. (4) Asay, D. B.; Kim, S. H. Effects of adsorbed water layer structure on adhesion force of silicon oxide nanoasperity contact in humid ambient. J. Chem. Phys. 2006, 124, 174712. (5) Hsiao, E.; Marino, M. J.; Kim, S. H. Effects of gas adsorption isotherm and liquid contact angle on capillary force for sphere-on-flat and cone-on-flat geometries. J. Chem. Phys. 2010, 352, 549–557. (6) Farshchi-Tabrizi, M.; Kappl, M.; Cheng, Y.; Gutmann, J.; Butt, H.-J. On the Adhesion between Fine Particles and Nanocontacts: An Atomic Force Microscope Study. Langmuir 2006, 22, 2171–2184, PMID: 16489804. (7) He, M.; Szuchmacher Blum, A.; Aston, D. E.; Buenviaje, C.; Overney, R. M.; Luginbühl, R. Critical phenomena of water bridges in nanoasperity contacts. J. Chem. Phys. 2001, 114, 1355–1360. (8) Jang, J.; Schatz, G. C. Lattice Gas Monte Carlo Simulation of Capillary Forces in Atomic Force Microscopy. J. Adhes. Sci. Technol. 2010, 24, 2429–2451. (9) Kim, H.; Smit, B.; Jang, J. Monte Carlo Study on the Water Meniscus Condensation

21


Langmuir

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

and Capillary Force in Atomic Force Microscopy. J. Phys. Chem. C 2012, 116, 21923– 21931. (10) Xiao, X.; Qian, L. Investigation of Humidity-Dependent Capillary Force. Langmuir 2000, 16, 8153–8158. (11) Xu, L.; Lio, A.; Hu, J.; Ogletree, D. F.; Salmeron, M. Wetting and Capillary Phenomena of Water on Mica. J. Phys. Chem. B 1998, 102, 540–548. (12) Salameh, S.; Schneider, J.; Laube, J.; Alessandrini, A.; Facci, P.; Seo, J. W.; Ciacchi, L. C.; Mädler, L. Adhesion Mechanisms of the Contact Interface of TiO2 Nanoparticles in Films and Aggregates. Langmuir 2012, 28, 11457–11464. (13) Qin, Y.; Fichthorn, K. Solvation forces between colloidal nanoparticles: Directed alignment. Phys. Rev. E 2006, 73, 020401. (14) Köber, M.; Sahagún, E.; García-Mochales, P.; Briones, F.; Luna, M.; Sáenz, J. J. Nanogeometry Matters: Unexpected Decrease of Capillary Adhesion Forces with Increasing Relative Humidity. Small 2010, 6, 2725–2730. (15) Mamontov, E.; Vlcek, L.; Wesolowski, D. J.; Cummings, P. T.; Wang, W.; Anovitz, L. M.; Rosenqvist, J.; Brown, C. M.; Garcia Sakai, V. Dynamics and Structure of Hydration Water on Rutile and Cassiterite Nanopowders Studied by Quasielastic Neutron Scattering and Molecular Dynamics Simulations. J. Phys. Chem. C 2007, 111, 4328–4341. (16) Ketteler, G.; Yamamoto, S.; Bluhm, H.; Andersson, K.; Starr, D.; Ogletree, D.; Ogasawara, H.; Nilsson, A.; Salmeron, M. The Nature of Water Nucleation Sites on TiO2(110) Surfaces Revealed by Ambient Pressure X-ray Photoelectron Spectroscopy. J. Phys. Chem. C 2007, 111, 8278–8282.

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Page 22 of 27

Page 23 of 27

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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(17) Zobel, M.; Neder, R. B.; Kimber, S. A. J. Universal solvent restructuring induced by colloidal nanoparticles. Science 2015, 347, 292–294. (18) Heuberger, M. Density Fluctuations Under Confinement: When Is a Fluid Not a Fluid? Science 2001, 292, 905–908. (19) Horn, R. G.; Israelachvili, J. Direct measurement of structural forces between two surfaces in a nonpolar liquid. J. Chem. Phys. 1981, 75, 1400. (20) Israelachvili, J. N.; Pashley, R. M. Molecular layering of water at surfaces and origin of repulsive hydration forces. Nature 1983, 306, 249–250. (21) Jarvis, S. P.; Uchihashi, T.; Ishida, T.; Tokumoto, H.; Nakayama, Y. Local Solvation Shell Measurement in Water Using a Carbon Nanotube Probe. J. Phys. Chem. B 2000, 104, 6091–6094. (22) Li, T.-D.; Gao, J.; Szoszkiewicz, R.; Landman, U.; Riedo, E. Structured and viscous water in subnanometer gaps. Phys. Rev. B 2007, 75 . (23) Fukuma, T.; Ueda, Y.; Yoshioka, S.; Asakawa, H. Atomic-Scale Distribution of Water Molecules at the Mica-Water Interface Visualized by Three-Dimensional Scanning Force Microscopy. Phys. Rev. Lett. 2010, 104 . (24) Argyris, D.; Ashby, P. D.; Striolo, A. Structure and Orientation of Interfacial Water Determine Atomic Force Microscopy Results: Insights from Molecular Dynamics Simulations. ACS Nano 2011, 5, 2215–2223. (25) Snook, I. K.; van Megen, W. Calculation of solvation forces between solid particles immersed in a simple liquid. J. Chem. Soc., Faraday Trans. 2 1981, 77, 181–190. (26) Grimson, M. J.; Rickayzen, G.; Richmond, P. Short range solvation forces in fluids. Mol. Phys. 1980, 39, 61–73.

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(27) Das, S. K.; Sharma, M. M.; Schechter, R. S. Solvation Force in Confined Molecular Fluids Using Molecular Dynamics Simulation. J. Phys. Chem. 1996, 100, 7122–7129. (28) Trokhymchuk, A.; Henderson, D.; Wasan, D. T. A Molecular Theory of the Hydration Force in an Electrolyte Solution. J. Colloid Interface Sci. 1999, 210, 320–331. (29) Israelachvili, J. N. Intermolecular and Surface Forces, 3rd ed.; Elsevier, 2011; pp 341– 380. (30) Yaqoob, M. A.; Rooij, M. B.; Schipper, D. J. On the Transition from Bulk to Ordered Form of Water: A Theoretical Model to Calculate Adhesion Force Due to Capillary and van der Waals Interaction. Tribol. Lett. 2013, 49, 491–499. (31) Binggeli, M.; Mate, C. M. Influence of capillary condensation of water on nanotribology studied by force microscopy. Appl. Phys. Lett. 1994, 65, 415. (32) Yong, C. W.; Smith, W.; Kendall, K. Surface contact studies of NaCl and TiO2: molecular dynamics simulation studies. J. Mater. Chem. 2002, 12, 2807âĂŞ2815. (33) Pakarinen, O. H.; Foster, A. S.; Paajanen, M.; Kalinainen, T.; Katainen, J.; Makkonen, I.; Lahtinen, J.; Nieminen, R. M. Towards an accurate description of the capillary force in nanoparticle-surface interactions. Modell. Simul. Mater. Sci. Eng. 2005, 13, 1175–1186. (34) Fradin, C.; Braslau, A.; Luzet, D.; Smilgies, D.; Alba, M.; Boudet, N.; Mecke, K.; Daillant, J. Nature 2000, 403, 871–874. (35) Style, R. W.; Hyland, C.; Boltyanskiy, R.; Wettlaufer, J. S.; Dufresne, E. R. Surface tension and contact with soft elastic solids. Nature Comm. 2013, 4 . (36) Schneider, J.; Ciacchi, L. C. First principles and classical modeling of the oxidized titanium (0001) surface. Surf. Sci. 2010, 604, 1105–1115.

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Page 24 of 27

Page 25 of 27

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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(37) Schneider, J.; Ciacchi, L. C. A Classical Potential to Model the Adsorption of Biological Molecules on Oxidized Titanium Surfaces. J. Chem. Theory Comput. 2011, 7, 473–484. (38) Schneider, J.; Colombi Ciacchi, L. Specific Material Recognition by Small Peptides Mediated by the Interfacial Solvent Structure. J. Am. Chem. Soc. 2012, 134, 2407– 2413. (39) Dörmann, M.; Schmid, H.-J. Simulation of Capillary Bridges between Nanoscale Particles. Langmuir 2014, 30, 1055–1062. (40) Pietsch, W.; Rumpf, H. Haftkraft, Kapillardruck, Flüssigkeitsvolumen und Grenzwinkel einer Flüssigkeitsbrücke zwischen zwei Kugeln. Chem. Ing. Tech. 1967, 39, 885–893. (41) Israelachvili, J. Solvation forces and liquid structure, as probed by direct force measurements. Acc. Chem. Res. 1987, 20, 415–421. (42) Butt, H.-J.; Kappl, M. Surface and Interfacial Forces; Wiley-VCH, 2010. (43) Vega, C.; de Miguel, E. Surface tension of the most popular models of water by using the test-area simulation method. J. Chem. Phys. 2007, 126, 154707. (44) Ko, J.-A.; Choi, H.-J.; Ha, M.-Y.; Hong, S.-D.; Yoon, H.-S. A Study on the Behavior of Water Droplet Confined between an Atomic Force Microscope Tip and Rough Surfaces. Langmuir 2010, 26, 9728–9735. (45) Benet, J.; Palanco, J. G.; Sanz, E.; MacDowell, L. G. Disjoining Pressure, Healing Distance, and Film Height Dependent Surface Tension of Thin Wetting Films. J. Phys. Chem. C 2014, 118, 22079–22089. (46) Werner, A.; Müller, M.; Schmid, F.; Binder, K. Effect of long-range forces on the interfacial profiles in thin binary polymer films. J. Chem. Phys. 1999, 110, 1221.

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(47) Zakerin, M.; Kappl, M.; Backus, E. H. G.; Butt, H.-J.; Schönfeld, F. Capillary forces between rigid spheres and elastic supports: the role of Young’s modulus and equilibrium vapor adsorption. Soft Matter 2013, 9, 4534. (48) You, S.; Wan, M. P. Mathematical Models for the van der Waals Force and Capillary Force between a Rough Particle and Surface. Langmuir 2013, 29, 9104–9117. (49) Raju, M.; van Duin, A. C. T.; Fichthorn, K. A. Mechanisms of Oriented Attachment of TiO2 Nanocrystals in Vacuum and Humid Environments: Reactive Molecular Dynamics. Nano Lett. 2014, 14, 1836–1842. (50) Plimpton, S. Fast Parallel Algorithms for Short-Range Molecular Dynamics. J. Comput. Phys. 1995, 117, 1–19. (51) Jorgensen, W. L.; Chandrasekhar, J.; Madura, J. D.; Impey, R. W.; Klein, M. L. Comparison of simple potential functions for simulating liquid water. J. Chem. Phys. 1983, 79, 926. (52) Hutter, J. L.; Bechhoefer, J. Calibration of atomic-force microscope tips. Rev. Sci. Instrum. 1993, 64, 1868–1873.

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Graphical TOC Entry

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Contact Forces between TiO2 Nanoparticles Governed by an Interplay

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