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Article

Dependencies of the Adhesion Forces between TiO Nanoparticles on Size and Ambient Humidity 2

Jens Laube, Michael Dörmann, Hans-Joachim Schmid, Lutz Mädler, and Lucio Colombi Ciacchi J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.7b05655 • Publication Date (Web): 21 Jun 2017 Downloaded from http://pubs.acs.org on June 27, 2017

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Dependencies of the Adhesion Forces between TiO2 Nanoparticles on Size and Ambient Humidity Jens Laube,† Michael Dörmann,‡ Hans-Joachim Schmid,¶ Lutz Mädler,§,k and Lucio Colombi Ciacchi⇤,†,k †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 ‡Particle Technology Group, Faculty of Mechanical Engineering, University of Paderborn, 330958 Paderborn, Germany ¶Particle Technology Group, Faculty of Mechanical Engineering, University of Paderborn, 33098 Paderborn, Germany §Foundation Institute of Materials Science (IWT) and Faculty of Production Engineering, University of Bremen, 28359 Bremen, Germany kMAPEX Center for Materials and Processes, University of Bremen, 28359 Bremen, Germany E-mail: [email protected] Phone: +49 0421 218 64570

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Abstract We study the variation of the adhesion forces between wet TiO2 nanoparticles as a function of their size and the ambient relative humidity. Combining all-atom molecular dynamics and capillary simulations we demonstrate that the linear scaling of the interparticle forces with the particle diameter, well established for microscopic and macroscopic particles, can be extended down to diameters of a few nm. At this size scale, however, the molecular nature of the water adsorbates dictates the adhesion forces both via solvation effects and influencing parameters of analytical capillary models such as the equilibrium particle-particle separation distance and the water/particle contact angle. Moreover, the water surface tension becomes considerably larger than the macroscopic bulk value due to combined effects of thin-film confinement and tight curvature, in a way that strongly depends on humidity and particle size. Taking these effects into proper account, classical capillary equations can be used to predict the interparticle forces even of the smallest particles considered here (4 nm), although the circular approximation fails to reproduce the distance at which the water meniscus breaks. Finally, the transition between the dominating effects at the nanoscopic scale and conventional capillary theory valid at microscopic size scales can be only rationalized if the presence of roughness asperities on the surface of the large particles is explicitly taken into account.

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Introduction Nanoparticle adhesion under ambient conditions plays a crucial role in a number of industrial processes such as agglomeration, dispersion and functionalization. 1–3 To develop novel process routes, adequately design the respective devices and achieve the desired product properties, precise quantitative prediction of the interparticle forces at the nanoscale and in the presence of humidity is highly important and should be incorporated in comprehensive interaction force models. It is widely accepted that, especially for small particles, the interparticle forces do not simply depend on the particle size, but also on their surface properties such as roughness, hydrophilicity, charge density, and shape. 3–19,44 In addition, geometrical assumptions typically used for macroscopic particles may not hold at the nanometer scale, where the length scales of the interacting forces and the particle dimensions approach the same order of magnitude. 20 Only recently we have found that the adhesion forces between flame-sprayed TiO2 particles with a diameter of 4 nm in humid air are crucially dominated by the nature of the surface-adsorbed water shell and the surface roughness, and that macroscopic adhesion force models have to be modified to account quantitatively for these effects. 21,22 In our previous study we have modified the capillary theory within the circular approximation including explicitly the thickness of the surface-adsorbed water shell, 4,5,17,23,24 and superimposed oscillatory solvation forces induced by the structuring of the water molecules between the particles. 25–31 However, so far we have only predicted the adhesion forces between monodisperse 4 nm particles, while the size distributions in technical applications are generally rather wide, spanning between 2 and 20 nm for average particle sizes of a few nm. 22 Therefore, a generalization of our developed methods is needed to provide a comprehensive interaction force model for the prediction of the interaction forces in polydisperse nanoparticle assemblies. For particles with sizes above 100 nm it is known that the capillary, Van der Waals and electrostatic forces typically scale linearly with the particle radius and that the capillary 3

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forces dominate for particle sizes below 1 µm. 32–36 It remains unclear whether the same linear-scaling law extends to particle sizes between 1 and 100 nm or if other dependencies are valid in this size range. Additionally, it is still unknown at which particle size the transition between our modified theory and the classical macroscopic theory takes place, and according to which mechanism the transition takes place. To address these open questions, in this work we perform a detailed analysis of the interparticle forces at different particle sizes for smooth and rough particles, combining numerical analysis and molecular modeling. In doing that, we also consider the possible influence of surface charge distributions, mainly arising from protonation/deprotonation equilibria with the surrounding humid atmosphere. Our findings will help to understand how the interparticle force dependencies on size and humdity can be described using a unified approach that is valid at all size scales, and be the basis for coarse-grained interaction models.

Methods All-Atom Molecular Dynamics Simulations The TiO2 nanoparticle models are created using the methods developed by Schneider et al. 37,38 and applied in our former works. 21,22 In this approach, nanoparticles of different diameters and geometries are carved from an infinite TiO2 single rutile crystal respecting the 1:2 Ti:O stoichiometry. To obtain rough particles, a spherical region decorated with asperities is cut from the crystal bulk as demonstrated in Fig. 1 and already described before. 21 Furthermore, a macroscopically flat but atomically rough surface slab model is created by cleaving the crystal along the lattice direction and annealing the model. This results in an amorphous outermost surface layer that represents the surface of nanoparticles more realistically than atomically flat crystal facets with low Miller indexes. Subsequent relaxation and charge rescaling of the different models are then performed as described by Schneider et al. 37 The accessible Ti atoms with less than 5 O neighbours are hydrated assuming disso4

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ciative water adsorption, 38 leading to a coverage of approximately 1.5 terminal OH-groups and 1.5 hydrogenated bridging oxygens per nm2 , independently of the particle size. For the representation of humid ambient conditions, the particles are then covered with different amounts of TIP3P 39 water molecules. 21 To investigate the influence of the local charging state of the surfaces on the interparticle forces, the modified multisite complexation model, as implemented for TiO2 by Koeppen et al. 40 is applied to representative particle models. Namely, at the isoelectric point 75% of the 2.4 potential protonation/hydroxylation sites per nm2 are actually protonated or hydroxylated, in equal proportions. The presence of unitary positive and negative charges at these sites is realised by adjustment of the force field charges according to strategy applied in Ref. 41 We do not consider deviations from the isoelectric point resulting in a total surface charge density different than zero, because the net number of charges on particles with diameter smaller than 10 nm is expected to be of the order of unit or less, which is negligible for the purposes of the present work. 42–44 Interparticle forces and water density profiles between two particles at specific separations are computed from average values of all-atom molecular dynamic (MD) simulation trajectories lasting each 0.6 ns after a previous 0.4 ns equilibration in the NVT ensemble at 300 K, with the crystal atoms kept fixed. All simulations are performed using the LAMMPS programm package. 48 a

b

dsp dasp

indentation

8 nm

Figure 1: Schematic illustration of the creation of rough particle models (a). The asperity diameter and indentation depth determine the desired asperity width and height. (b): Cut through a representative rough 8 nm particle without water, exhibiting several asperities with a height of 0.5 nm.

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Capillary Theory - Capillary Simulations To determine the capillary force FCap between spherical nanoparticles, the force-equilibrium at the neck of the connecting liquid meniscus is used:

FCap = where

P Am + 2

L Um

(1)

P is the pressure difference between the liquid phase of the meniscus and the sur-

rounding gas phase,

L

is the liquid/gas surface tension, and Am and Um are the cross-section

and the circumference of the meniscus neck, respectively. The so-called Laplace pressure

P

results from the curvature of the liquid surface, also named the Kelvin radius rK , as described by the Young-Laplace equation P =

L

rK

(2)

.

The Kelvin radius in turn depends on the water vapour partial pressure P/P0 through the Kelvin-equation: rK =

L VM

RT ln

(3)

⇣ ⌘, P P0

with the kinetic gas constant R , the temperature T and the liquid molar volume VM . In the case of a meniscus, the Kelvin radius depends on the curvature radius rm and the crosssection radius lm (cf. Fig. 2) via

rK =



1 rm

1 lm



1

.

(4)

The method presented by Dörmann et al. 20 is used to calculate numerically the capillary forces between two particles, allowing for the determination of the shape of the meniscus between two particles without a priori geometric assumptions. In this approach, the true shape of the meniscus connecting two particles is determined using an iterative procedure, where the meniscus is constructed considering the local curvature and cross-section radius

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from equations (3) and (4) and varying the wetting angle

at the particle surfaces until

a consistent connection point is achieved at both particles (cf. Fig. 2 bottom). After the meniscus shape is determined, the resulting force exerted by the capillary bridge can be computed according to equations (1) and (2).

Figure 2: Schematic illustration of the coordinates and symbols used in this work to describe the particle model incorporating the surface adsorbed water layer of thickness h. Redrawn and modified from Ref. 21

Circular Approximation To facilitate the calculation of the capillary forces, the meniscus curvature parallel and perpendicular to the symmetry axis can be assumed to be circular. From this approximation, for two spheres of equal size the curvature radii rm and lm are linked to the water/solid contact angle ⇥, the filling angle , the particle radius R1 and the particle particle distance

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D (cf Fig. 2) via rm =

2R1 (1 cos( )) + D 2 cos(⇥ + )

(5)

and lm = R1 sin( )

rm [1

(6)

sin(⇥ + )] .

With these constraints, the capillary force between two particles at a certain humidity and distance can be calculated iterating over equations (3), (5) and (6) until the matching partial pressure is hit and then inserting the final values into equations (1) and (2). Alternatively, the shape of the meniscus between two particles can be extracted from the water density profiles obtained in the atomistic simulations, as described in detail in our previous work. 21 Fitting of the thereby available capillary force from equation (1) to the actually measured force in a simulation yields a modified surface tension

L,

which in turn

can then be used together with the meniscus geometry to determine the humidity of this very simulation from the Kelvin-equtation (3).

Surface tension on a slab model The interfacial tension

between two phases can be calculated from the method proposed

by Kirkwood and Buff 49,50 computing the gradient of the pressure (stress) tensor p across the interface: =

Z

zhi zlo



pz (z)

1 {px (z) + py (z)} dz , 2

(7)

with the variable z pointing in orthogonal direction to the surface plane and the integration boundaries zlo and zhi far enough away from the position of the interface. To calculate the liquid/vapour surface tension

L

of water layers with different thicknesses adsorbed at our

TiO2 slab model, the pressure tensor acting on the water molecules is recorded every 0.002 ns in a 50 ns NVT simulation after a previous relaxation. The pressure gradient present at a specific height z is then computed by summing the pressure tensor of all atoms and snapshots in bins with a thickness of 0.1 Å along the z axis (see supporting information, Fig. S1). As 8

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the system under investigation exhibits one solid/liquid and one liquid/gas interface, the influence of the solid/liquid interface on the pressure gradient has to be eliminated from the calculation. To this aim, the pressure gradient values at a sufficiently thick water layer below a z value of 1.65 nm, where the gradient starts to deviate significantly from the bulk value 0, are subtracted from the respective pressure gradient values of all other layers (see supporting information Fig. S1). To assess the inaccuracy of this method at small layer thicknesses due to the variation of the pressure gradient at the solid/liquid interface, the surface tension of the water layers is alternatively computed from the work of adhesion between the two surfaces. In this approach, two slabs covered with the same amount of water are brought into contact and the work needed to separate the slabs is computed from integrating the force-displacement (F-D) curve recorded during the detachment of the slabs. The detachment F-D curve is computed from a steered MD simulation in the NVT ensemble with a timestep of 2 fs, during which one slab is displaced along the z axis at a speed of 0.5 m/s and the forces acting on both slabs are averaged over 104 steps. To eliminate the portion of the crystal forces, the forces between two slabs without water and OH-groups are subtracted from the respective force curves. As the water meniscus between the slabs is metastable close to the breaking distance, the forces between the slabs are additionaly calculated from quasistatic simulations with a duration of 3 ns as described above after placing two slabs at a specified distance. The detachment force curves are then truncated at the distance where no meniscus is formed during the quasi-static simulations thus providing a hysteresys between approaching and detaching surfaces (see Fig. 3).

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Figure 3: Work of adhesion needed to separate two surfaces from the integration of the detachmend F-D curve as illustrated for a slab covered with 20 water molecules per surface nm2 from top left to bottom right respectively. The dark grey area depicts the work considered for two approaching surfaces while the light grey area depicts the additional area considered for two detaching surfaces.

Results & Discussion We begin our study with the analysis of the adhesion forces between TiO2 nanoparticles at a broad range of particle sizes and size combinations. To this aim, we first explicitly compute the forces between pairs of 6 nm, 8 nm, 10 nm and 20 nm particles, between the caps of two 50 nm particles and between an 8 nm particle and a 4 nm particle by using MD simulations as specified in the methods section. This selection of particle sizes covers the whole size distribution of flame-made TiO2 nanoparticle films and refines towards small particles, where we expect the peculiarities of nano-adhesion to be most significant. Additionally, in order to investigate the influence of roughness on the adhesion forces, we selectively compute the interparticle forces between 6 nm, 8 nm and 50 nm particles decorated with several asperities of 0.5 nm height and 0.5 to 1.7 nm radius depending on the particle size (see Fig. 1b). The resulting force-displacement (F-D) curves between smooth particles reveal the characteristic superposition of monotonous capillary and oscillatory solvation forces as described in our previous work, 21 independently of particle size (Fig. 4). However, since the intensity of the solvation forces depends strongly on the local surface geometry in the contact region, it varies 10

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for the different particle sizes and orientations and decreases for larger particles, although never entirely disappearing. An important point to consider at this stage is the possible presence of charges at the particles’ surfaces, although they generally build up in larger amount for hydrophobic rather then hydrophilic surfaces, 45 and especially during mutual friction between different materials as a consequence of tribocharging phenomena. 46 However, also hydrophilic oxide surfaces in humid air have been observed to present net charged sites, primarily due to intrinsic protonation/deprotionation equilibria of the terminal groups and the selective adsorption/desorption of charged water particles from the atmosphere. 44 Here we neglect the presence of unbalanced positive and negative charge distributions, which for particle sizes smaller than 10 nm and under conditions close to the isoelectric point are expected to be of the order of one charged site per particle or less. 42,43,47 We do, however, consider a balanced distribution of protonated and deprotonated surface sites as present at titania/water interfaces. 40 As shown in the Supporting Information (Section S2), the force-distance curves obtained for charged (but nuetral) particles do not show systematic deviations from those obtained for uncharged surfaces. This is primarily due to the fact that charged sites do not significantly alter the water structure within the adsorbed layers, and therefore do not influence significantly neither the solvation nor the capillary forces. For this reason, all forthcoming simulations are performed with neutral particles.

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a

b

6 nm

d

c

10 nm

8 nm

e

20 nm

f

8 nm

4 nm

~ ~

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50 nm

Figure 4: Representative force displacement curves of two 6 nm, 8 nm, 10 nm and 20 nm particles, of an 8 nm particle together with a 4 nm particle and of two caps cut from 50 nm particles (from a to f) all without asperities and at a coverage of 16.2 water molecules per nm2 . The insets display volumetric density isosurfaces of the crystal and water phase of the respective simulated particle models at a surface distance of approximately 1 nm. The strength of the capillary force contributions increases with particle size, as expected, while its range of action does not vary significantly, 20 except perhaps for the 20 nm particles, probably due to the formation of more than one capillary neck at large separation. This makes the capillary force contribution strongly sensitive to asperities especially at larger particle sizes, as can be observed from the F-D curves between rough particles in Fig. 5. The incorporation of asperities with a height of 0.5 nm 21 effectively pins the distance D to relatively large (and positive) values, which leads to a significant reduction of the capillary forces. At the same time, the oscillatory solvation forces are nearly entirely quenched. 13 This effect is strongest for the 50 nm particles. Here, water menisci build nearly exclusively between the asperities (Fig. 5), whose size therefore dominates (i.e. limits) the adhesion force. 3,6,9

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a

8 nm

b

~ ~

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50 nm

Figure 5: Representative force displacement curves of two 8 nm particles (a) and of two caps cut from 50 nm particles (b) both decorated with asperities of 0.5 nm height and covered with 16.2 water molecules per nm2 . The insets display volumetric density isosurfaces of the crystal and water phase of the respective simulated particle models at a surface distance of approximately 1 nm and a cutout of the asperities in contact. As a next step to generalize our modification of the capillary theory 21 for different particle sizes, we compute the capillary forces from the geometries of the adsorbed water shells on the smooth particles, which exhibit contact angles of 0 even for the larger particles. It is important to remember that the particle radius R1 in equations (5) and (6) needs to include the thickness of the surface adsorbed water layer (see Fig. 6). This thickness of the water layer depends on the water coverage and therefore on the humidity, but is not appreciably dependent on the particle size (Fig. 6).

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Figure 6: Variation of the adsorbate layer thickness h (dashed lines, empty symbols) and particle distance D (continuous lines, filled symbols) at the point of maximum adhesion force with changing humidity and particle size calculated from the dimensions of the surface adsorbed water layer on the particles. Following our previous approach, we can best fit the far-range part of the F-D curves computed from MD with capillary theory by treating the water surface tension

L

as a

variable parameter. 21 To estimate the relative humidity (P/P0 ) for the respective water coverage and particle size, the fitted

L

values can then be inserted together with the meniscus

geometries into the Kelvin-equation (4). The obtained values of

L

are close to the bulk

water surface tension at high relative humidities, but increase significantly with decreasing humidity for all particle sizes (Fig. 7), which is in good agreement with the results from Benet et al. 51 and Werner et al. 52 This is the combined result of the water confinement in thin liquid films on one side and of the small curvature radius of these films on the other side. In order to decouple the two effects, we compute the surface tension of water layers adsorbed to a flat TiO2 surface model (infinite curvature radius) presenting the same amount of water molecules per surface area as for the particle models both using the Kirkwood/Buff method and computing the work of adhesion between the two surfaces. The values obtained via the Kirkwood/buff method (black solid line in Fig. 7) converge to the bulk surface tension at 100 % relative humidity and match the computed values for the largest particle sizes (10 14

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or 20 nm) in the region of high humidity. However, they diverge from them for humidities below 60 % due to the strong variations of the solid/liquid interfacial pressure gradient at small liquid layer thicknesses (cf. Fig S3 in the supporting information). Instead, the values obtained from the work of adhesion (grey area in Fig. 7) do follow the model prediction for larger particles in the entire humidity range, although the uncertainty associated with the calculation increases with humidity. This is due to the increasing hysteresis encountered when thick water layers are put in contact or separated, wich is caused by the decreasing probability to form water menisci between the layers at increasing surface separations (cf. Fig S4 in the supporting information). Overall, the data computed with the various methods allow us to conclude that only for particles smaller than about 10 nm does the small curvature radius of the liquid/vapour interface produce a significant increase of

L.

For small particles,

however, this effect is very pronounced. Interestingly, for pairs of particles of different sizes (8 and 4 nm, in our case), the smaller particle seems to dominate, the computed values corresponding roughly to those of two equal 4 nm particles. From a molecular point of view, the increase of surface tension in thin films arises from the limited mobility of the water molecules due to the film structuring. As a consequence, their rearrangement after separation of the liquid phase in two free surfaces is partly prevented, resulting in a higher net energy penalty associated with the surface creation than in the case of thick films. 51,53,54 The effect of curvature is less intuitive and roots on the inherent spacial separation between the Gibbs’ surface plane (i.e. the equimolar dividing surface between liquid and gas) and the surface plane of force equilibrium in the Laplace equation (the surface of tension). This separation is known as the Tolman length 55,56 and is of the order of -0.05 nm for our water model. 57 For spherical droplets, the surface tension increases by an amount equal to twice the ratio between the Tolman length and the particle radius, 58,59 which becomes appreciable at particle radii of the order of a few 1 nm. Alternatively, both effects can be readily rationalized if the surface tension is thought to originate from the presence of surface “capillary waves”. 60 In confined liquids, either because of very thin films 15

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or of small droplet radii, the long-range amplitudes of the capillary waves are suppressed, which leads to a free energy penalty and thus a surface tension increase. The observed trend for water adsorbed on other materials than TiO2 is expected to differ only slightly, through the material-dependent water structuring at the solid/liquid interface. However, the dominant role is played by the liquid/vapor interface. For different liquid adsorbates, the trend will strongly depend on the value of the Tolman length, that can be either negative or positive, in which case a decrease of

L

with decreasing particle radius

could also be obtained.

Figure 7: Development of the surface tension L with changing humidity and particle size calculated from the fitting of the capillary geometries to the MD forces between the particles. The surface tension for a flat surface representing an infinite particle radius is calculated from the Kirkwood-Buff method (KB, solid line) and from the work of adhesion between the two slabs (WA, the grey area highlights the hysteresis between approaching and retracting surfaces). The humidity values for the flat surface are adopted from the respective coverages of the 10 nm particles. The bulk surface tension of the used water model as calculated by the KB-method on a thick water slab is indicated by a horizontal black dashed line. As found previously, 21 the relation between the water coverage set in the simulations and the air humidity given by the Kelvin equation (3) matches very well experimentally measured adsorption isotherms on different TiO2 surfaces (See Fig. 8). We note that the predicted adsorption isotherms also depend slightly on the particle size and the match to the experimental results is better for the larger particles, which is probably due to the deficiency of the Kelvin equation at small curvatures leading to an overestimation of the humidity at 16

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a given curvature. 61

Figure 8: Comparison of the water adsorption isotherms from experimental works 23,24 to the isotherms on different particle sizes in this work calculated from linking the fitted air humidity of the simulations to the respective water coverages. The predicted capillary forces within the circular approximation using the fitted values of surface tension

L

and air humidity, particle radii including the water layer thickness

and contact angles of 0 reproduce very well the far-ranged region of the force displacement curves calculated with MD (blue FCap,M od curves in Fig. 9; see also section S4 in the supporting information for the results of other air humidity values). If we relax the circular approximation and compute the capillary forces according to the iterative “capillary simulations” described in the Methods section, very similar results are obtained (red FCap,Sim curves in Fig. 9). Instead, conventional capillary theory (nominal particle radii without water thickness, nominal surface tension

L

and a water/TiO2 contact angle equal to 10 deg 21 )

completely fails to reproduce the MD simulations even for the largest particle sizes investigated (green FCap,Conv curves Fig. 9). It is important to note that at the point of maximum adhesion force, at values of

between 0.6 and 0.7 nm, both the capillary simulation and

the simpler circular approximation predict roughly the same force values for all investigated particle sizes and air humidities. This is mainly due to the fact that for negative particle distances D (cf. Fig. 6) the shortcomings of the circular approximation are very minor. Only at the larger distances, close to the point where the water meniscus breaks down, is the cir17

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cular approximation less accurate than the capillary simulations in estimating the position of the the meniscus breakpoint. 20,62 As a direct outcome of our analysis, from Fig. 9 it is apparent that the F-D curves normalized by the particle diameter present nearly identical values for all particle sizes. This demonstrates that the capillary force contribution scales linearly with particle size also for diameters below 100 nm, extending the trend predicted by classical models valid for microscopic and macroscopic sizes. 32–36 Furthermore, the point of meniscus breakdown, and thus the range of action of the capillary forces, is nearly invariant with particle size, as already anticipated above. These two statements also hold for mixed particle sizes (8 nm and 4 nm), where the capillary forces can be calculated using the Derjaguin approximation R⇤ = 2 ·

R1 · R2 . R1 + R2

(8)

In contrast, the amplitudes of the oscillatory solvation forces are less pronounced at larger particle sizes in relation to the capillary forces, or occur only for some particular orientations where specific crystal facets face each other. This underlines a decreasing influence of the solvation forces with growing particle size.

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6 nm

8 nm

10 nm

20 nm

8 nm & 4 nm (R*~5.3 nm)

Figure 9: Comparison of the explicitly calculated forces from MD simulations (black crosses) of various particle sizes at different particle orientations and at a water coverage of 16.2 water molecules per nm2 (air humidity of approximately 65-75%, Fig. 8) to the capillary forces calculated from the capillary simulations (red line), the modified capillary theory (blue line) and from the conventional capillary theory (green line) using the averaged capillary quantities from the water density profile. The variation of the maximum adhesion forces with humidity (Fig. 10) follows for all 19

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particle sizes a characteristic behaviour with an early maximum and a subsequent decrease of the force with increasing humidity. This trend is differently pronounced for the different particle sizes but is always observable and reflects the expected change of capillary force with humidity at negative particle separations D (cf. Fig 6), as already stated in our previous work for smooth 4 nm particles. 21 This finding confirms our former assumption that the force-humidity dependency depends primarily on the particles’ surface hydrophilicity and roughness, irrespective of their size. 21

Figure 10: Variation of the maximum adhesion forces with humidity and in bulk water between representative orientations of differently sized nanoparticles. Figure 11 shows the development of the interparticle forces with particle size, presented for the coverage of 16.2 water molecules per nm2 (air humidity of approximately 65-75%, Fig. 8). Reported are the absolute values of the forces in correspondence of the minimum of the F-D curves computed with MD, at separation distances

between 0.6 and 0.7 nm, which

we define here as Fmax . The total adhesion forces from the MD simulations (black crosses in Fig. 11) are significantly higher than the capillary forces calculated from our modified capillary theory within the circular approximation (blue curve in Fig. 11) because of the additional contribution due to solvation. As explained above, this contribution diminishes for larger sizes and is also dependent on the specific orientation of the particles. However, at all sizes, including the largest ones (20 and 50 nm) the explicitly calculated forces are never lower than the analytical capillary force prediction calculated for negative values of 20

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the particle separation D. Conventional capillary theory with a typical D value of 0.0 nm, 36,63 on the other hand, is expected to be valid at large particle radii. We suggest that an intuitive way of reconciling the validity of the conventional theory in the microscopic size scale with our observed behaviour of the adhesion forces is to postulate that the surface roughness dictates the minimum separation distance between (wet) larger particles. For very small particles, negative D values can occur even in the presence of roughness, especially for particle orientations presenting a good degree of surface crystallinity, as a consequence of the interpenetration of the water adsorption layers. For larger particles, roughness asperities would prevent overlap of the water layers, so that D is pinned to zero or positive values. This seems to be confirmed by the results we have obtained for rough particle models (red crosses in Fig. 11). For smaller particles, the total forces are in the range predicted by capillary theory with D values between zero and -0.34 nm, plus a small additional effect of solvation (not as large as in the case of smooth particles). With our rough model of 50 nm particles we have computed a much smaller adhesion force, even smaller than the predicted capillary force with D equal to 0.0 nm. In fact, one needs to take into account that water menisci are formed almost exclusively between the asperities and not homogeneously distributed across the whole particle-particle cross-section. We note that asperities differing in size and curvatures would need to be considered to determine quantitatively their effect on the interparticle forces. Here we limit ourselves to a qualitative consideration of the effect of roughness, considering asperities of the same nominal size. However, the crystal truncation and annealing protocol used to produce the particle model brings in more irregularities, effectively leading to atomic-scale asperities of different kind and size (round, flat, pointy, etc.), as visible in Figure 1b.

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Figure 11: Maximum interparticle forces against the particle size for smooth and rough particles which are decorated with asperities of 0.5 nm height. In comparison the development of the capillary forces at the same particle distance is shown for the modified circular approximation (blue line) considering the surface adsorbed water layer and thus a negative particle distance D = 0.34 nm (cf. Fig. 6) and for the unmodified circular approximation (green line) assuming direct particle contact with D = 0.0 nm. 36,63

Conclusions In the present work we have demonstrated that the linear scaling of interparticle forces with their diameter, which is well-established for particle sizes above 100 nm, 32–36 extends to the size range down to 4 nm and probably even less. In this range, however, overlap and structuring of the water layers between the particles (i.e. solvation) become a very important contribution to the adhesion forces. Only at particle sizes above about 20 nm does solvation become negligible with respect to the dominant capillary contribution. Moreover, for diameters of a few nm capillary theory needs to incorporate explicitly the presence of water adsorption layers through (i) negative values of D; (ii) contact angles of zero degrees; and (iii) water surface tension values dependent on the degree of humidity and on the particle size (see Fig. 7). It is worth to note that we have shown that the circular approximation can be safely used to predict the capillary force contributions at the point of maximum adhesion force, because at small particle distances also the errors of its geometrical assumptions are small. However, due to the increase of these errors at larger particle distances, the circular 22

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approximation underestimates rather severely the actual range of action of capillary forces, as set by the point of meniscus breakdown in the F-D curves. Accurate prediction of this point may be critical when implementing capillary force models in discrete-element simulation routines, where the dynamics and stability of nanoparticle films depend sensitively on the onset-distance of the nonbonding forces. Attention should also be paid to determine experimentally and include in any simulation model the correct charge state of the terminal surface sites. Accumulation of charges on the particles’ surfaces may arise from triboelectric charging, intrinsic protonation/deprotonation propensity of terminal groups, and the selective adsorption or desorption of charged water particles from humid air, in ways that are strongly dependent on the relative humidity. 44 Here we have found that balanced distribution of extra protons and hydroxide ions do not affect significantly the contact forces of TiO2 particles with diameters of the order of 10 nm or less. However, the situation may be different especially for particles that are larger, more hydrophobic, or more strongly basic or acid, such as e.g. silicon oxide. 45,47 Finally, a transition to conventional capillary theory valid at microscopic size scales 33,36,63–65 can be only rationalized if the presence of roughness asperities on the surface of the large particles is explicitly taken into account. As the surface roughness on particles in technical applications has a fractal character, 9,66,67 the influence of roughness on the interparticle forces will grow with increasing particle size and must be determined in-depth to correctly predict the interparticle adhesion forces. 6,7,9,11,14,15,36,63

Acknowledgement We are grateful to Luis MacDowell (Complutense University of Madrid, Spain), Guido Grundmeier (University of Paderborn, Germany) and Samir Salameh (Delft University of Technology, Netherlands) for stimulating discussions. Furthermore we thank Sebastian Potthoff and Magdalena Laurien for their support in part of the simulations. This work has been

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supported by the Deutsche Forschungsgemeinschaft within the Priority Program “Partikel im Kontakt - Mikromechanik, Mikroprozessdynamik und Partikelkollektiv” (SPP 1486) under grants CO 1043/3 and MA3333/3. Computational time has been provided by the HLRN supercomputing centre at Hannover and Berlin, Germany.

Supporting Information Available Kirkwood/Buff method; Surface tension on a flat surface; MD and capillary forces at different water coverages. This material is available free of charge via the Internet at http://pubs. acs.org/.

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