Nanoscale Capillary Interactions in Dynamic Atomic Force Microscopy

Article pubs.acs.org/JPCC

Nanoscale Capillary Interactions in Dynamic Atomic Force Microscopy Victor Barcons,†,‡,⊥ Albert Verdaguer,§ Josep Font,†,‡ Matteo Chiesa,*,‡ and Sergio Santos*,†,‡,⊥ †

Departament de Disseny i Programació de Sistemes Electrònics, UPC - Universitat Politècnica de Catalunya Av. Bases, 61, 08242 Manresa, Spain ‡ Laboratory for Energy and Nanosciences, Masdar Institute of Science and Technology, Abu Dhabi, United Arab Emirates § Centre d′ Investigació en Nanociència i Nanotecnologia (CIN2) (CSIC-ICN), Esfera UAB, Campus de la UAB, Edifici CM-7, 08193-Bellaterra, Catalunya, Spain S Supporting Information *

ABSTRACT: Standard models accounting for capillary interactions typically involve expressions that display a significant decay in force with separation. These forces are commonly investigated in the nanoscale with the atomic force microscope. Here we show that experimental observations are not predicted by these common expressions in dynamic interactions. Since in dynamic atomic force microscopy methods the cantilever is vibrated over the surface, the nanoscopic tip is submitted to nonlinear interactions with the sample in a periodic fashion. That is, the force dependencies involved in dynamic interactions in the nanoscale can be probed. We describe two extreme experimental scenarios in these dynamic interactions and interpret them as single and multiple asperity cases. In both extremes there is a predominantly attractive component of the net force that is relatively independent of distance and that ranges several nanometers above the surface. The distance dependence approximates that of a square well. Experimental data have been acquired for cantilevers of different stiffness and fundamental resonant frequency indicating that the distance dependencies provided here are valid for a relatively large range of frequencies. The reproducibility of our experiments and the accurate prediction of the experimental data that we present imply that future investigations should take the phenomena that we report into account to describe and interpret dynamic capillary interactions.

1. INTRODUCTION Capillary interactions affect surfaces in ambient conditions where nanoscale water layers cover them and water bridges might form.1−5 This capillary interaction is relatively long in range and might affect the flow of granular materials,6 particle cohesion in powders and adhesion of particles to surfaces,7 and the seismic properties of rocks.8 Since a single meniscus or neck can form between single nanoscale asperities,9 the atomic force microscope (AFM) has played a key role over the past decade in the study of the force dependencies involved in neck formation.2−5,10 In the dynamic AFM mode of operation, the cantilever is vibrated at sufficiently close tip−sample proximities that its dynamics are affected by the intermolecular and surface tip−sample forces. The principle behind the technique is that tip−sample forces affect dynamic parameters such as the amplitude of oscillation A, the phase lag Φ between the oscillation amplitude and the drive amplitude, and the mean cantilever deflection def. That is, all of these parameters are perturbed by the tip−sample interaction and vary, relative to the unperturbed values for the free cantilever, according to the particular forces involved in the interaction. In particular, in ambient conditions, © 2012 American Chemical Society

where the surface of the tip and the sample are covered by a water film of nanoscale dimensions, capillary interactions might control the dynamics of the cantilever. Capillary interactions typically involve the formation and rupture of a capillary bridge and, more generally, the intermolecular forces acting between the water layers on the interacting surfaces.1,11,12 The mechanism for capillary bridge formation and rupture can lead to hysteretic behavior. That is, first, as the tip approaches the sample, a capillary bridge forms between the sample's and the tip's surface. Note that here the terms capillary bridge and capillary neck are used interchangeably. Then, on retraction, the capillary bridge ruptures. If the distances of formation and rupture of the capillary bridge do not coincide, hysteresis follows. It is important to note that, in dynamic AFM, the distance dependencies involved in the formation and rupture of the capillary bridge might differ from those in static measurements. Thus, information about capillary interactions and, in particular, information about the dynamics of Received: November 8, 2011 Revised: March 16, 2012 Published: March 16, 2012 7757

dx.doi.org/10.1021/jp2107395 | J. Phys. Chem. C 2012, 116, 7757−7766

The Journal of Physical Chemistry C

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compactly in terms of the chemistry, elasticity, and geometry of the tip−sample system. The expressions read19

capillary bridge formation and rupture can be probed in dynamic AFM methods.1−3,5,10,11,13 Furthermore, due to the high lateral resolution of the dynamic AFM, these dynamic interactions provide information about capillary processes that occur in nanoscale.4,7,10,11,14 Here, the tip−sample interaction is probed dynamically in ambient conditions. We show that the several expressions typically used to model capillary interactions in dynamic AFM are not experimentally validated. In particular, we show that these expressions are either (1) not valid on their own to model nanoscale capillary interactions and the dynamics of bridge formation and rupture or (2) they simply do not stand the empirical test. The former point would raise questions regarding what the dominant attractive forces are in dynamic nanoscale interactions. With the latter point we imply that, if the capillary force is dominant, then the standard expressions do not capture the essence of the dynamic phenomena. The main implication of this investigation is that the dominant force at nanometers of proximity of the surface, while attractive, is relatively independent of separation for the first few nanometers. That is, the distance dependence is, roughly, that of a square well. Second, experimentally, we find that there are two relatively different responses in terms of the dynamics of the cantilever, and we interpret these to be a consequence of the absence or presence of asperities on the tip. According to our findings, the behavior in the multiple asperity case cannot be reproduced theoretically in a trivial fashion. This is possibly due to the variety of geometrical characteristics and material and chemical properties of the multiple asperities and/or to the stochastic formation and rupture of multiple capillary necks. Despite this complexity, we also discuss general trends in the multiple asperity scenarios. It is remarkable that the behavior that we interpret to be a consequence of multiple asperities is typically observed when the tip radius is relatively large. Conversely, the behavior for the single asperity case, is typically observed when the tip radius is smaller. Finally, we note that the main objective of this work is to arrive to a description of the net tip−sample force that can reproduce experimental observations typically observed in ambient dynamic AFM. Importantly, such dependencies should reproduce experimental behavior especially when the oscillation amplitude A is small. This is because, in these cases, the stored energy in the cantilever is of the same order of magnitude as the tip−sample interaction. In this respect, the results presented here are shown to represent a step in the right direction.

Fa(d) = −

HR

d > a0

6d2

FAD = −4πR γ ≡ −

FDMT(d) =

HR 6a02

4 E* R δ3 3

(1)

d ≤ a0

d ≤ a0

(2)

(3)

where d is the instantaneous tip−sample distance, a0 is an intermolecular distance that implies that matter interpenetration cannot occur, H is the Hamaker constant, R is the effective tip radius here modeled as a sphere throughout, γ is the surface energy, E* is the effective elastic modulus of the contacting bodies, that is, tip and sample, and δ is the indentation during mechanical contact. The suffixes a, AD, and DMT stand for attractive, adhesion, and contact repulsive forces, respectively. The repulsive forces (3) are modeled with the Derjaguin−Muller−Toporov (DMT) model of contact mechanics.19,20 The effective modulus E* is written as a function of the Poisson coefficients of the tip νtip and the sample νsam and their elastic moduli Etip and Esam, respectively, as19,21 E* =

1 1 − ν 2tip E tip

+

1 − ν 2sam Esam

(4)

The Poisson coefficients are 0.3 here throughout. In the above conservative expressions (1−3), chemical information about the tip−sample pair is provided via H and/or γ, information on the elasticity of the tip and the sample is provided via the elastic modulus E*, and the tip−sample geometry is characterized by the effective tip radius R and the distance d between the tip and the sample. The interaction is commonly modeled as occurring between an infinite surface, that is, the sample, and a sphere of radius R. This procedure has also been used here. The indentation can be written in terms of the instantaneous tip−sample distance as δ = a0 − d; indentation occurs when d ≤ a0. The net tip sample force Fts can be written as the sum of the three terms in eqs 1−3. As stated, other than the capillary force, it is well-established that these three forces play an important role in dynamic AFM and, in particular, in amplitude modulation AM AFM.3,13,19,22 If other forces are involved in the interaction, these also add to Fts. That is, in general, if p forces Fi are involved in the tip−sample interaction, i=p the net force can be written as Fts = ∑i=1 Fi where each term Fi might have different tip−sample distance dependencies as in the case of the expressions 1−3. The sum of all of the contributing forces Fts controls the cantilever dynamics, and it is the net dependence of Fts on tip−sample distance d which is of particular interest in this study. The consequences on the dynamics of the addition of the capillary force FCAP to the conservative expressions in eqs 1−3 are discussed in the next section. The distance dependencies of several standard expressions for the (net) tip−sample force Fts are shown in Figure 1a,b. The behavior shown in Figure 1a (top lines in dark blue) is obtained when considering the attractive long-range van der Waals (London dispersion) forces Fa (1), contact adhesion forces FAD (2), which are distance-independent,20 the short-range repulsive forces FDMT20 (3) and a linear relationship (FCAP(d) ∝ d

2. METHODS AND MODEL When accounting for capillary interactions, the common interpretation is given in terms of a force component that dominates at long distances relative to London dispersion and, in general, van der Waals (vdW) forces.3,5,15 The main assumption is that this force behaves in a hysteretic fashion and decays with separation.5,16 One might argue that there are several possibilities in terms of the derivation of the capillary force; nevertheless, the final expressions are consistent with the above traits.2−5,16 Other than the capillary, several conservative forces have been used in the literature to model and interpret the tip−sample interaction. These have shown to provide fundamental insights in terms of cantilever dynamics.3,11,17−19 Three of the most widely used forces are the long-range London dispersion (1), the short-range adhesion (2), and the short-range repulsive (3). These three expressions can be written 7758



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the vdW force to the capillary interaction contracting force. It is worth noting that, while others have reported the lack of convergence when using such discontinuous expressions in numerical simulations,3 no such instabilities have been encountered in the present work (see details on numerical algorithms below). That is, in this work the force has been added discretely to the equation of motion of the cantilever (see below) without the need of algorithms to allow for convergence. The value of d off can be calculated by numerically solving the Laplace−Young equation. This gives3,23 1/3 doff ≈ Vmen −

Vmen = 4πRh2 +

⎛ d ⎞ FCAP(d) = −2πγH ORX ⎜1 − ⎟ 2 doff ⎠ ⎝ d > a0

4 3 πh + 2πr DMT2h 3

(7)

where all of the parameters have been already defined except for rDMT which is the radius of the area of contact when indentation occurs. From the DMT model,24 rDMT = (Rδ)1/2. It is clear that rDMT = 0 when indentation does not occur, that is, δ = 0 or d > a0. Thus Vmen is constant for d > a0 and consists of the displaced water from the tip's and sample's surfaces in the overlapping region where the capillary bridge forms. When d ≤ a0, Vmen increases due to the water displaced in the area where mechanical contact occurs.3 The assumption that Vmen consists of the accumulation of displaced water in regions where the water layers overlap is in accordance with typical approximations used to derive capillary force expressions and, in particular, with expressions using the constant volume Vmen approximation.3,15,16 It should be noted however that eq 5 assumes a constant chemical potential rather than a constant (meniscus) volume Vmen approximation.16 Still, eqs 6 and 7 have been used in our study throughout unless otherwise stated. In any case, it will be shown later that the value of doff is not critical to reproduce experimental data. That is, it will be shown that doff is, to a certain extent and in the above sense, not a critical parameter in this work. Two expressions for the capillary force where the constant volume Vmen approximation is used will be given below,3,15 and one is shown in black in Figure 1a. These expressions have been previously used to discuss capillary interactions in dynamic AFM.1,3,11 As stated, in the constant volume approximation, meniscus growth by condensation and dissolution of water vapor is not considered. That is, the mechanism for the formation of the capillary bridge in the constant volume approximation is related to the coalescence of the water films that are already formed on the surfaces rather than to the condensation of water vapor. The constant volume approximation might be a more realistic situation in dynamic interactions than the constant chemical potential because the required time for water vapor condensation does not play a role.16 Note that in dynamic AFM the formation and rupture of the capillary should occur in the order of μs. In this work, capillary interactions are defined from now on and, in general, as the phenomenon for which the water layers on the tip's and the sample's surfaces meet whether the capillary

in Figure 1a) for the capillary force FCAP (5) commonly written as5

and

(6)

where Vmen is the volume of the meniscus or the volume of water forming the water bridge. Vmen can be calculated from geometrical considerations and by simply assuming that it is only the water already present on the water films covering the tip−sample surfaces that add to Vmen3

Figure 1. Force−distance Fts dependencies for (a) standard models where Fts displays peaks at close proximities and (b) for a conservative system where water might (continuous line) or might not (dotted line) be present. The parameters are: R = 20 nm, γ = 45 mJ (surface energy), γH2O = 72 mJ (surface energy of water), Esam = 10 GPa (elastic modulus of the sample), Etip = 120 GPa (elastic modulus of the tip), and h = 0.6 nm (height of the water layers on the tip and the sample). In b, γ = 45 mJ has been used for d < a0 and γH2O = 72 mJ for d > a0.

provided capillary on

1 2/3 Vmen 5R

(5) 5

where X is the average contact coefficient. We have used X = 1.5−2.5 in our simulations and, in particular, X = 2 in Figure 1a. The coefficient X has already been used in the literature to model capillary interactions in dynamic AFM by giving similar values.5 The expression in eq 5 gives rise to the typical hysteresis in capillary interactions since it acts when d < don on tip approach and when d < doff on tip retraction. In this way, don and doff refer to the distances of formation and rupture of the capillary bridge where doff ≥ don. When doff > don hysteresis occurs and energy is dissipated in the interaction. If don = doff there is no hysteresis, and the interaction is conservative. For d ≤ a0, the value of FCAP is zero5 (see discussion below). Typically doff > don, this being the mechanism leading to the two-valued function and hysteresis, in the distance dependence of the force Fts on d in Figure 1a. In this work, the value of don always lies in the range don = 2h − 3h where h is the height of the adsorbed water layers on the hydrated surfaces, that is, the tip and the sample. A value don = 2h has been assumed in previous studies where instabilities due to capillary condensation and/ or vdW attraction have been neglected.3 If the bridging material is a wetting liquid however, the water films become unstable on approach under the effect of the vdW interactions, and the capillary bridge is thought to form at don ≈ 3h.16 In any case, whether don = 2h, 3h, or values in between these two, the same phenomenon occurs on approach. Briefly, at d = don, a discontinuity in the force occurs from the almost negligible, in comparison, value of 7759



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capillary interaction have been derived in a similar way. For example11,16

bridge is formed or not. The two extremes are illustrated in Figure 2 in the top and middle rows, respectively. In the top

FCAP(d) = −

and

row the capillary bridge is not formed, and the water layers interact due to water overlap only. In the middle row the capillary bridge is formed before the water layers overlap, that is, d = don = 3h, and ruptures at doff > don where doff is calculated from eq 7. These two cases correspond to a single asperity scenario. The multiple asperity case is illustrated in the bottom row in Figure 2, and it is discussed at the end of the next section. The scenario depicted in the middle row in Figure 2 is the most commonly accepted interaction, and most studies are based on such assumptions.3,5,11,13 Quantitatively, this is the phenomena leading to distance dependencies as those shown in Figure 1a, that is, capillary bridge formation and rupture (dark blue and black lines); the tip approach and retraction are shown with the use of continuous and discontinuous lines, respectively. The other distance dependence of Fts on d shown in Figure 1a (bottom black lines) has been plotted by replacing the expression for the capillary force in eq 5 by eqs 8 and 9

and

4πγH OR 2

1+


d h

d > a0

FCAP(d) = −

(8)

4πγH OR 2

a 1+ 0 h

for

2

πRd2 1+ Vmen

d > a0


(10)

where, as for eqs 8 and 9, the force can be assumed to saturate in terms of distance when d ≤ a0, that is, d = a0 for d ≤ a0. However, eq 8 has a dependence on Vmen and thus increases with indentation. Expressions 8−10 are derived using the constant volume approximation.15,16 In summary, the three capillary expressions provided so far, that is, 5, 8, and 10, correspond to scenarios where (1) the capillary bridge forms on approach and ruptures on retraction as shown in Figure 1a and illustrated in the middle row in Figure 2b, (2) the capillary force controls the interactions at the longer distances, and (3) the net force Fts significantly decays with separation from d = a0 to d = doff. In Figure 1a, eq 10 has not been plotted, but note that a general expression FCAP(d) ∝ (1/1 + μdn) is given there (black lines) where both 8 and 10 can be recovered by setting n = 1 or n = 2, respectively; μ is simply a constant. In particular, both expressions, that is, 8 and 10, lead to a similar type of dependence of Fts on d (see below). In Figure 1a the paths correspond to the approach (continuous lines) and retraction (dashed lines) trajectories when the capillary neck forms and ruptures in one cycle. Again, this is the mechanism for hysteretic behavior and capillary dissipation.3,5 A general description of the use of expressions 1−3 and the three expressions for the capillary force already provided, that is, 5, 8, and 10, is given next with the use of Figure 1a. First, from Figure 1a, note that there is a peak (absolute minima) in Fts at d = a0. This peak is mainly due to the vdW (London dispersion force) term in eq 1 and its dependence on d, that is, FCAP(d)∝(1/d2). Then, Fts rapidly increases with distance at just angstroms or fractions of an angstrom past this peak. This behavior gives Fts a convex shape there (circle in Figure 1a) in all cases. As it turns out, in our simulations, none of the force dependencies resembling those in Figure 1a predict the experimental phenomena that we observe in dynamic AFM. This is particularly the case at the smaller cantilever tip−sample separations zc; zc is the unperturbed tip position relative to the solid surface of the sample, and it is a commonly used term when describing cantilever dynamics.25 Since it is at the smallest separations, that is, zc ∼ nm, that the cantilever dynamics are most sensitive to the dependencies of Fts on d, this region is of particular interest and will be the focus of this work in the next section. Note that in ambient AM AFM, the perturbed oscillation amplitude is A ∼ zc,26 implying that small separations typically involve small amplitudes. It should also be noted that most studies directly omit information of this region or state that reproducibility is challenging when driving the cantilever at these very small separations. In general, the tip radius is mentioned as a possible source of irreproducibility.3,18,27 Physically, the relevance of small values of zc is that it is at these separations that the cantilever stored energy Ecant is closest to the tip−sample interaction energies.11 To a first approximation28 Ecant ≈ 1/2kA2, where k is the spring constant. Now, assuming that interaction energies are in the order of 10− 102 eV and considering standard values of k (2−40 N/m), one

Figure 2. Schemes illustrating the single asperity case where (a) water layers might be in contact during an oscillation cycle but the capillary neck does not form and (b) where the capillary neck forms and ruptures during once cycle. (c) Schemes illustrating the scenario where there are multiple asperities and several necks might form at sufficiently large separations. The asperities on the tip might lead to repulsive interactions as these make contact with the sample mechanically.

FCAP(d) = −

4πγH OR

d ≤ a0 (9)

where all of the parameters have already been defined. The above expressions eqs 8 and 9 for the capillary force have also been used in previous studies, together with eqs 1−3, to model capillary interactions in dynamic AFM.3 Other similar expressions for the 7760



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where d* = d − 2h and eq 11 replaces eq 1. Significantly, the maxima in the vdW force is predicted by eq 1 at d = a0 where Fa coincides with the adhesion force FAD (2). Nevertheless, using the above reasoning we write

finds that for A = 1 nm, Ecant = 6−125 eV, and for A = 10 nm Ecant = 0.625−124 keV (for k = 2 and 40 N/m, respectively). Significantly, most studies also emphasize the relevance of an appropriate choice for the magnitude or variation of the capillary force FCAP when sample indentation occurs. For example, when using 5, some5 use thermodynamic arguments to establish that for d < a0 the capillary force should be zero, that is, FCAP = 0 for d < a0. Others3 have made using a constant value for the capillary force during sample indentation, that is, FCAP = constant for d < a0. Still, other studies16 indicate that an increase in FCAP should follow with increasing indentation, that is, FCAP increases with δ. Again, all of these models and variations have been implemented in the present work and solved numerically (see below) to establish whether experimental data are closely reproduced. The main outcome (data not shown) is that none of the models and/or variations, in terms of the behavior of FCAP when indentation occurs, can reproduce the experimental phenomena discussed in this work, that is, the behavior of the amplitude A, phase Φ, and mean deflection def in dynamic AFM when the oscillation amplitudes and/or separations are sufficiently small. Nevertheless, a more counterintuitive Fts/d dependence (see continuous black lines in Figure 1b and Figures 3b,c, 4, and 5), does. We propose to confirm these statements in the rest of this work.

FAD = −

To derive the expressions leading to Figure 1b we have considered what happens to the vdW expression (1) in the presence of water films on the surfaces of the tip and the sample (Figure 3a). Note that (1) was derived by Hamaker in 1937 by summing the atom−atom dispersion interactions for two bodies, i.e. a sphere and an infinite surface. Nevertheless, the pair interactions between the water molecules should be accounted for when there are water films on the surface.1,11 This phenomenon can be described compactly if we write1,11 6d*2

d* > a 0

or

d* ≤ a 0

or

d ≤ 2h + a0

(12)

d2z dt

2

+

mω0 dz + kz = Fts + F0 cos ωt Q dt

(13)

where ω0 is the natural angular frequency, the effective mass of the tip is m = k/(ω0)2, Q is the Q factor, k is the spring constant, and the drive force is written as F0 cos ωt where ω is the drive frequency. The instantaneous position, z, is measured from the equilibrium position of the cantilever zc.1,11,19 The spring model is a good approximation to the real phenomenon in ambient conditions where the Q factor is relatively high,29,30 that is, 102−103, and where higher modes are not excited. The tip−sample distance d from the solid surface of the tip to the solid surface of the sample is related to z by d = zc + z.19 Note that viscous interactions that the cantilever experiences in the medium, that is, air in ambient conditions, is modeled here in the standard way by considering an effective value for the Q factor. The term effective here makes reference to the fact that, while viscosity might arise from the velocity of the cantilever relative to the medium and/or internal cantilever viscosity, Q should account for a net contribution of such phenomena. In the literature reference to this phenomenon is described in terms of a background energy that is dissipated in other than the tip−sample interaction.27,31 Significantly, these contributions can be experimentally accounted for using standard AFM equipment by monitoring the frequency response of the free cantilever and recording the value of Q.25 Here, Q is directly measured experimentally in this way, and similar values as those obtained in experiments are used in the simulations. We have implemented this model in both Matlab and C. While both codes produced identical results, the model implemented in C is computationally superior. For the numerical integration a standard Runge−Kutta algorithm of the fourth order has been used throughout. The results have been compared with fifth and eight order Runge−Kutta algorithms and with the Adams−Bashforth algorithm of the fourth order without

Figure 3. (a) Scheme illustrating the geometry of the tip−sample interaction when there are water molecules, that is, water films, on the surfaces; (b) phase and (c) amplitude curves acquired experimentally (red and blue lines for approach and retraction respectively) on a mica sample. The curves are reproduced with numerical simulations (black dashed lines) when a square conservative well (continuous clack lines in Figure 1b) is used. In the simulation, we obtain a normalized time for the water contact (CTw) of 1 in some regions (blue colored). Experimental parameters: f = f 0 ≈ 300 kHz, k ≈ 40 N/m, R ≈ 5 nm (according to SEM scans, data not shown), Q ≈ 500 and A0 ≈ 1.5 nm. Simulation parameters: f = f 0 = 300 kHz, k = 40 N/m, R = 5 nm, Q = 500, γ = 45 mJ, γH2O = 72 mJ, Esam = 10 GPa, Etip = 120 GPa, A0 ≈ 1.5 nm and h = 0.6 nm. Again, as in Figure 1b, γ = 45 mJ has been used for d < a0 and γH2O = 72 mJ for d > a0.

HR

6a02

In Figure 1b, eqs 3 and 11−12 have been used to obtain the distance dependencies Fts/d shown in Figure 1b. Note that eqs 3 and 11−12 are conservative forces; eq 3 is the DMT repulsive force and acts only when there is sample indentation, that is, d < a0, eq 11 corresponds to the vdW interaction when the water layers on the tip's and the sample's surfaces do not overlap, and eq 12 corresponds to the region where the water layers overlap. An illustration of this process is shown in Figure 2 in the top row. This is the single asperity case when no capillary bridge is formed. It is important to note that the conservative nature of such interaction implies that hysteresis does not occur on tip approach and retraction. The main characteristic of this Fts/d dependence is that a square well appears at the smallest separations (continuous black lines in Figure 1b). That is, the force is independent of distance there. The Fts/d dependence when there are no water films present on the surfaces is also shown in Figure 1b with the use of dashed lines; eqs 1−3 have been used. To test the validity of the above expressions in dynamic AFM, the governing equation of motion (13) has been modeled as a mass on a spring m

Fa(d) = −

HR

d > 2h + a0 (11) 7761



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Figure 4. Force dependencies Fts/d where (a) the force is relatively independent and (f) dependent on distance in the hysteretic region. Simulations b−c and g−h reproduce the experimental outcomes in d−e and i−j, respectively. The samples are quartz (left) and Al (right). Simulation parameters: R = 7 nm, A0 = 25 nm, and others as in Figure 3. Experimental parameters: f = f 0 ≈ 270 kHz, k ≈ 35 N/m, R ≈ 5−10 nm, Q ≈ 500, and A0 ≈ 25 nm.

The experimental work has been carried out on an Asylum Research Cypher and on an Asylum Research MFP-3D-SA AFM when working at the natural frequency of oscillation, that is, ω = ω0 or f = f 0. In the experiments, phase Φ, amplitude A, and deflection def distance curves have been obtained on quartz, mica, aluminum Al, graphite Gr, and silicon Si samples. The experiments have been carried out in ambient conditions. The experiments have also been carried out with different values of cantilever spring constant k (1.2−42 N/m) and the corresponding range in resonant frequency f 0 (70−320 kHz). Some specific cantilever models used in these experiments are the AC240TS (k ≈ 2 and f 0 ≈ 70 kHz) and AC160TS (k ≈ 40 and f 0 ≈ 300 kHz) from Olympus and the Multi75AL (k ≈ 3−5 and f 0 ≈ 75 kHz) from Budget Sensors (see Supporting Information details for a range of experimental curves). In summary, the different types of behavior described in this work (Figures 3−5) have been reasonably reproduced on all of these samples. This indicates that the present study is valid for the whole range of frequencies considered standard in ambient dynamic AFM.25,28,32 In the next section experimental results are compared with the predictions of simulations.

3. RESULTS AND DISCUSSION A single asperity is initially assumed where the capillary bridge does not form. This case is implemented with the use of the conservative forces 3, 11, and 12. These expressions correspond to the scenario where the water layers on the surfaces of the interacting bodies simply overlap and no energy is dissipated via capillary hysteresis. As stated, such a phenomenon is illustrated in the top row of Figure 2. Recall that with single asperity we imply that the tip radius R is sufficiently sharp, that is, R < 10−20 nm, and that the assumptions to derive the

Figure 5. (a) Proposed distance dependency for a type of multiple asperity case. To reproduce b−c the hysteresis or tip trapping on retraction it is the negative slope of the curve at d > a0 that plays a major role. The sample in d−e is mica. Simulation parameters: as in Figure 4 except for R = 20 nm and A0 = 5. Experimental parameters: as in Figure 4 except for R ≈ 20−30 nm and A0 ≈ 5 nm.

significant improvement in the simulations. No problems of convergence have been experienced when implementing tip− sample forces where the onset and the break off is discrete. 7762



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expressions,20,33 both capillary and vdW, are reasonable. The behavior of experimental phase and amplitude curves (red and blue for approach and retraction, respectively) obtained on a mica sample is shown in Figures 3b,c. Note that at close separations zc there is a clear change in the amplitude's slope (Figure 3c), local minima in A, and corresponding variations in phase. Experimentally, we have consistently obtained similar curves on Si, Al, quartz, and mica surfaces (see the Supporting Information for details). This experimental behavior for the phase ϕ and the amplitude A response in Figure 3b and c, respectively, can never be reproduced when using force dependencies such as those shown in Figure 1a, that is, eqs 1−10. In particular, according to our analysis, it is the convex curvature (circled in Figure 1a) of the net tip−sample force Fts at close proximities and its (positive) slope in the nonhysteretic region that inhibit the presence of the phenomena observed in Figure 2. In this study, simulations have been carried out with such models, that is, eqs 1−10, for a range of tip radii, that is, 5 < R < 30 nm, free amplitudes, that is, 1 < A0 < 20 nm, spring constants, that is, 2 < k < 40 N/m, and resonant frequencies, that is, 70 < f 0 < 350 kHz, to confirm that the experimental behavior cannot be reproduced (data not shown). Note that it is with the smallest oscillation amplitudes A and at the smallest separations that remarkable phenomena are observed experimentally, that is, abrupt changes in slope in amplitude and phase. As stated, the dynamics of the cantilever are not commonly considered in the literature under such conditions, and it is in the prediction of such phenomena that the above models particularly fail. On the other hand, by considering only the conservative expressions in 3, 11, and 12 (see distance dependencies in black lines in Figure 1b) the experimental results (dashed black lines in Figures 3b,c) are readily reproduced in the simulations. This outcome confirms the relevance of the interactions between the atoms on the water films. In particular, this data shows that the scenario illustrated in the top row in Figure 2 is plausible in dynamic AFM. That is, the capillary bridge might not form and rupture during an oscillation cycle, and water layers might simply overlap instead. It is important to note, however, that past the local minima in amplitude A, and on tip approach, the tip gets permanently trapped in the water layers according to simulations (Figure 3c). Mathematically, we write the normalized contact time per cycle with the water as CTw. Thus, whether the capillary forms or not, we write CTw = 1 when the tip is in perpetual contact with the water layers as illustrated in blue in Figure 3c. In fact, in the example shown in the Figure 3b,c, there is a small negative step in mean deflection both in the experiments and in the simulations when this occurs (data not shown). The step in mean deflection is positive on retraction at the same position, and both amplitude and phase overlap on approach and retractions. In short, the step in mean deflection coincides with the onset of perpetual water contact CTw = 1 and with the local minima in amplitude (see discussion of Figure 4 below). We now consider what happens when A0 is increased, the repulsive regime is reached, and the capillary neck forms and ruptures during one oscillation cycle (Figure 3b; Figure 3c discussed later). Here, the concepts of don and doff are reintroduced, implying that capillary hysteresis can occur and that the capillary bridge can form before the water layers overlap, that is, don ≥ 2 h. Following the same reasoning as before, we consider two scenarios. Namely, the cases considered are (1) that the magnitude of the net force Fts remains relatively constant in the region don < d < doff (Figure

4a−c) and (2) that the force monotonously decays with increasing separation in that region (Figure 4f−h). The two cases use as the initial motivation the results in Figures 1b and 3b,c. For the first case, that is, if Fts remains relatively constant on retraction when the capillary neck is formed, the situation shown in Figure 4a as follows. This net force dependence on d has been obtained by writing, for the vdW interaction, Fa(d*) = −

HH2OR

capillary off (d > don)

6(d*)2

FAD(H *) = −

H *R

(14)

capillary on(a0 < d < doff )

6a02

(15)

In eq 15 H* implies that the Hamaker needs to be interpolated for that range. We use a linear interpolation H* =

HH2O − H doff − a0

(d − a 0 ) + H

(16)

The constant HH2O is used in eq 14 indicating that the vdW interaction between the tip and the sample might vary from noncontact to mechanical contact due to the presence and absence of water layers under the tip, respectively. That is, in the long range, the vdW interactions are affected by the atom− atom interactions between the water molecules on the tip's and the sample's surfaces. This is indicated by using a value of Hamaker HH2O. Then, when mechanical contact occurs, water molecules are displaced, and the vdW adhesion force originates from the mechanical contact between the solid tip and the solid sample. In this case, the adhesion force is written in terms of the constant H (eq 16). It is clear that these two values should converge from the situation in eq 14 to the situation in eq 16. This is accounted for by eq 15. For the capillary force FCAP, we have used a small variation of eq 10 in Figure 4, as follows FCAP(d) = −

2πγH OR 2

1+ and

πR ad2 Vmen


d > a0

(17)

where Ra = R 0 + d = a0

RCa − R 0 (d − a 0 ) doff − a0

d = doff

to (18)

and Ca = doff/R; R0 = a0. If d ≤ a0, d = a0 in both eqs 17 and 18. Note that due to the choice of R0 and Ca in 18 Ra = d throughout in the region a0 < d < doff. The main motivation for modifying eq 10 is related to the requirement of removing the peaks in Fts observed in Figure 1a near d = a0 (see circle in Figure 1a) to reproduce the experimental observations. As stated, these peaks lead to the appearance of phenomena that is not observed experimentally. The physical interpretation of the expressions used to produce Figure 4a, eqs 14−18, should relate to the actual existence of a plateau in net force Fts in the region a0 < d < don. This interpretation follows from the fact that, as stated, it is only 7763



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constant Coff that allows controlling the decay of the force with separation. The expression reads

when such plateaus are present in the simulations that experimental observations are reproduced (this applies both experimentally and in the simulations). In any case, it should be emphasized that the object of this work is to find expressions for the net Fts/d dependencies that can reproduce the experimental data in the simulations. The results of implementing eqs 2, 3, and 14−18 for the tip− sample forces and solving the equation of motion (13) numerically are shown in Figure 4a−c. Special emphasis is given to the regions of the smallest oscillation amplitudes A and separations zc when interpreting the data. In terms of the amplitude response A, in Figure 4b the situation is relatively similar to that shown in Figure 3c in the blue-colored region and has already been discussed above. Nevertheless some differences can be observed in the simulations. The most obvious difference between Figures 3b,c and 4b−e relates to the fact that, on retraction, the repulsive regime (dashed lines) is reached in the latter. Reaching the repulsive regime with large free amplitudes however is standard behavior in AM AFM34 and can be reproduced with all of the above models according to our results (data not shown). In particular, note that in Figure 3b−c A0 ≈ 1.5 nm whereas in Figure 4 A0 ≈ 25 nm. More significantly, in Figure 4c the phase response is altered by the hysteresis of the capillary bridge in both the attractive (top branch) and the repulsive (bottom branch) regimes. In particular, there is local minima in phase in the attractive branch (green circle in Figure 4c) coinciding with the onset of the negative slope in amplitude and a smooth step in negative mean deflection (see Figure S1 of the Supporting Information for details on mean deflection). Moreover, in this case, in the region coinciding with the negative slope in amplitude, subharmonic excitation follows according to our numerical results. The physical interpretation for subharmonic excitation in the region of negative slope in Figures 4b,c relates to the impossibility of reaching the steady state there. That is, intermittent water impacts occur where, in some oscillations, the cantilever loses stored energy when the capillary bridge ruptures and, as a consequence, the amplitude significantly decays until perpetual water contact occurs. However, during perpetual water contact the oscillation amplitude increases once more until the capillary bridge ruptures and stored energy is lost once more. This situation repeats indefinitely for a given separation under these conditions, and the steady state is never reached. We have recently reported the onset of subharmonic excitation in similar situations.11 It follows that in these cases CTw = 1 is obtained throughout only in the region marked in blue in the inset (see Figure S1), at which point, the capillary bridge never ruptures. We find matching experimental evidence for this behavior in amplitude, phase (Figure 4d,e) and mean deflection (see Figures S1−S7) for all of the samples tested in this work; in Figure 4d,e the sample is quartz. The case for which the force monotonously decays with increasing separation in the region don < d < doff is shown in Figure 4f−h. Here, the tip−sample dependencies are the same as those for Figure 4a−c, that is, eqs 2, 3, and 14−18, except that for don < d < doff the vdW force (15) is allowed to decay monotonously with increasing separation. This is written here by forcing the vdW term to converge at don and by using a

F − FDon Fa(d) = Doff (d − don) + FDon doff − don don < d < doff

(19)

where FDon = −

⎤ R ⎡⎢ HH2O − H ⎥ − + ( d a ) H on 0 ⎥⎦ 6a02 ⎢⎣ doff − a0

FDoff = −Coff

RHH2O 6a02

Coff ≥ 0

(20)

(21)

All of the parameters have already been defined. Note that the use of eqs 19−21 is more general than eq 15. For example, eq 15 can be recovered if Coff = 1 in eq 21; in Figure 4f,i−j Coff = 0.15. The only difference in the Fts/d dependence here is a monotonous decrease in force Fts with separation in the region don < d < doff when the capillary is on (c.f., Figure 4a,f). From the point of view of dissipation, the mean energy dissipated per cycle decreases with decreasing Coff; note that the minimum value of Coff here can be zero. The consequences of a decrease in force in the don < d < doff with increasing separation, according to numerical simulation, are shown in Figure 4g,h. Only the main differences relative to the behavior previously described when discussing Figure 4b,c are discussed to avoid redundancy. First, the phase (Figure 4g) displays a discontinuous step tending to 180° (green circle in Figure 4h). This coincides with the onset of the negative slope in amplitude on approach and the onset of perpetual water contact, that is, CTw = 1. When CTw = 1 there is zero dissipation, and no subharmonics are excited. A smooth step in deflection is also observed (Figure S1 of the Supporting Information). Evidence of the experimental observation of this phenomenon is shown in Figures 4i,j where the sample is Al. The different behavior in phase in these two cases, that is, Figure 4e,j, provides an empirical method to study the force distance dependencies in the hysteretic region in dynamic AFM. It should be noted that one might arbitrarily change the value of doff, that is, don < doff < 2C where C = doff as predicted by eq 6, and according to our simulations, the behavior discussed in Figures 3 and 4 is still reproducible (data not shown). In short, it is the presence of the plateau in force in the region a0 < d < don (Figures 1b and 4a,f) that should always be considered. Furthermore, we experimentally find (Figure S3) that one or the other outcome might occur with a given tip in a sequence of curves. The physical origin of this seemingly stochastic phenomenon might be due to the constant volume approximation not fully holding in dynamic interactions and/or to possible tip water instabilities.4,5,15 For example, a possible mechanism might relate to droplets being left on the surface after the snapping of the capillary bridge on retraction. Then coalescence might or might not occur in the next cycle significantly increasing or decreasing the volume of the meniscus Vmen.16 Finally, we briefly introduce the multiple asperities scenario. This case is illustrated in the bottom row of Figure 3. The motivation behind this analysis is that we have experimentally observed that when the tip is not sufficiently sharp, that is, R > 10−20 nm, the behavior discussed in Figures 3 and 4 cannot be typically reproduced in the experiments. In fact, for a given tip, 7764



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provided FAD is sufficiently large compared to FCAP (eq 17). This is the case shown in Figure 5a. Again we are interested in the regions of small separations zc and small oscillations A in terms of the dynamic response of the cantilever. In summary, the implementation of this model results in numerical simulations that predict that the region with negative slope in amplitude is not present and that there is tip trapping on retraction (Figure 5b,c); the amplitude A is relatively distanceindependent there when the tip is trapped. Importantly, a value of R = 20 nm has been allowed here as compared to the value R = 7 nm used to produced Figure 4a−c and f−h. This is consistent with the above discussion regarding the traits of the multiple asperity scenarios. Our simulations show that, in the region where the tip is trapped, permanent water contact occurs, that is, CTw = 1 (blue in Figure 5b). In Figures 5d,e we provide experimental evidence of this phenomenon where the sample is mica. Again, we have reproduced this behavior in a variety of samples (see the Supporting Information for details).

the behavior might converge from that observed in Figures 3 and 4 to that in Figure 5 as the tip gets blunted. Typical differences observed in these cases include (1) situations where the negative slope amplitude (see Figures 3 and 4) is inhibited, and sometimes not even present, and (2) hysteresis or tip trapping occurs on retraction instead (see Figure 5). The interpretation of this multiple asperities scenario is supported by the large variety of deviations that we find when the behavior is not as described in Figures 1, 3, and 4. In short, it is likely that the variety in geometrical characteristics and material and chemical properties of the multiple asperities (Figure 3c) plays an important role in these cases making an analytical approach complex. Such a variation in tip geometry and number and shape of asperities should be particularly significant for the largest tips, that is, with increasing R, agreeing with our experimental findings. Nevertheless, we can still describe certain phenomena in the multiple asperity scenario where the tip radius is relatively large, that is, R > 10−20 nm. For example, force distance dependencies Fts/d in the hysteretic region don < d < doff should still obey the same rules, in terms of phase, independently of the number or shape of asperities. In Figure 5a we present a distance dependence of Fts on d that (1) is consistent with the scheme in Figure 3c, (2) presents only slight variations from the dependencies in Figure 4a,f, and (3) reproduces major aspects that we observe experimentally. From Figure 5a, note that the decrease in dissipation accounts for multiple but small capillary necks. The multiple neck scenario is illustrated in the schemes in the bottom row of Figure 2. Also note the slope in the attractive conservative region a0 < d < don. There, Fts is relatively constant but has a slight negative slope compared to Figures 4a,f. This slope might account for asperity deformation. In turn, capillary deformation can lead to the onset of the small repulsive interactions depending on the elasticity, size, and number of the asperities (see bottom row in Figure 2). For example, when the capillary is formed on approach, that is, d = don, multiple capillary necks might form increasing the attractive component of the force to the contact value of the capillary contracting force. Nevertheless, this force can be rapidly counteracted by the onset of repulsive forces as the asperities on the tip make contact with the sample's surface. The net result is a slow increase in the net force Fts with decreasing separation until the tip makes solid mechanical contact with the surface, at which point, the repulsive forces in eq 3 start dominating the interaction. To model such interactions, eq 3 is used to model the contact repulsive forces as usual, the capillary force FCAP is modeled with eq 17, and for the vdW interactions we use Fa(d*) = −

HH2OR 6(d*)2

d > don

4. CONCLUSIONS In conclusion, we have shown that the force distance dependencies in nanoscale dynamic capillary interactions might be significantly different to those typically discussed in the literature. In particular, standard expressions fail to predict most of the experimental phenomena that can be observed in a variety of samples in ambient conditions, thus failing to pass the empirical test. This might be due the several reasons, but most significantly, it should be noted that the derivation of expressions related to capillary interactions is typically carried out under the assumption of thermodynamic equilibrium. It is clear that in dynamic AFM, where the oscillation frequency is in the order of kHz, this requirement is not satisfied. Additionally, the fact that dynamic AFM methods can probe the tip−sample interaction by sometimes avoiding the snap into contact, typical of static methods, implies that dynamic methods can provide information where static methods cannot. In this respect, our study shows that realistic force dependencies accounting for dynamic interactions between hydrated surfaces have a relatively flat, that is, distance-independent, region at a few nanometers of separation. We have used the London dispersion forces between the atoms forming the water films on surfaces to interpret this phenomenon. Another significant outcome of this study is that a method has been developed to probe the distance dependence of the hysteretic component of the capillary interactions in the long range. In short, two outcomes have been investigated, namely, the net force might decrease with separation in the hysteretic region in one case and, in the other, it might be relatively distance-independent. These two cases can be distinguished by monitoring the behavior of the phase at small oscillation amplitudes. A multiple asperity scenario has also been discussed, and we have presented a model that could pave the way to interpret these complex interactions. As a final note, we report that the behavior observed in Figures 3 and 4 can be routinely reproduced unless using hydrophobic functionalized tips and/or freshly exfoliated hydrophobic samples (Figure S4 of the Supporting Information). Because of the reproducibility of our study in a variety of samples and the clear match between theoretical predictions and experimentation, we conclude that our discussion of force dependencies is fundamental in ambient conditions. In particular, the study should be relevant for the whole range of frequencies typical of ambient dynamic AFM. Future investigations

(22)

and FAD(H *) = −

H *R 6a02

a0 < d < don

(23)

where H* is interpolated with the use of eq 24 H* =

HH2O − H don − a0

(d − a 0 ) + H

(24)

and eq 2 is used for d < a0. All of the parameters have already been defined. Physically, eqs 23 and 24 lead to a negative slope in Fts in the region a0 < d < don provided HH2O > H and 7765



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(22) Stark, R.; Schitter, G.; Stemmer, A. Phys. Rev. B 2003, 68, 0854011−0854015. (23) Willett, C. D.; Adams, M. J.; Johnson, S. A.; Seville, J. P. K. Langmuir 2000, 16, 9396−9405. (24) Fischer-Cripps, A. C. Nanoindentation, 2nd ed.; Springer: New York, 2004. (25) Garcia, R.; Perez, R. Surf. Sci. Rep. 2002, 47, 197−301. (26) Paulo, A. S.; Garcia, R. Phys. Rev. B 2002, 66, 0414061− 0414064. (27) Martinez, N.; Garcia, R. Nanotechnology 2006, 17, S167−S172. (28) Giessibl, F. J. Rev. Mod. Phys. 2003, 75, 949−983. (29) Rodríguez, T. R.; García, R. Appl. Phys. Lett. 2002, 80, 1646− 1648. (30) Tamayo, J. Appl. Phys. Lett. 1999, 75, 3569−3571. (31) Cleveland, J. P.; Anczykowski, B.; Schmid, A. E.; Elings, V. B. Appl. Phys. Lett. 1998, 72, 2613−2615. (32) Gan, Y. Surf. Sci. Rep. 2009, 64, 99−121. (33) Hamaker, H. C. Physica 1937, 4, 1058−1072. (34) Santos, S.; Gadelrab, K. R.; Souier, T.; Stefancich, M.; Chiesa, M. Nanoscale 2012, 4, 792−800.

should take our study into account to interpret the phenomena involved in dynamic capillary interactions.

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ASSOCIATED CONTENT

S Supporting Information *

Further details on the expressions (section A) and experimental data (section B) used to interpret the effects of capillary interactions in dynamic AFM in the main article. This material is available free of charge via the Internet at http://pubs.acs.org.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Author Contributions ⊥

These authors contributed equally to this work.

Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS Graphic designer Maritsa Kissamitaki has designed and produced all of the artwork. We thank Neus Domingo (Universitat Autònoma de Barcelona) regarding technical support and Alba Santos (Universitat Autònoma de Barcelona) for helpful discussions.

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REFERENCES

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Nanoscale Capillary Interactions in Dynamic Atomic Force Microscopy

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