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A New “Quasi-Dynamic” Method for Determining the Hamaker Constant of Solids Using an Atomic Force Microscope Sean G. Fronczak, Jiannan Dong, Christopher A. Browne, Elizabeth C. Krenek, Elias I. Franses, Stephen P. Beaudoin, and David S. Corti* Davidson School of Chemical Engineering, Purdue University, 480 Stadium Mall Drive, West Lafayette, Indiana 47907-2100, United States S Supporting Information *

ABSTRACT: In order to minimize the effects of surface roughness and deformation, a new method for estimating the Hamaker constant, A, of solids using the approach-to-contact regime of an atomic force microscope (AFM) is presented. First, a previous “jump-into-contact” quasi-static method for determining A from AFM measurements is analyzed and then extended to include various AFM tip-surface force models of interest. Then, to test the efficacy of the “jump-into-contact” method, a dynamic model of the AFM tip motion is developed. For finite AFM cantilever-surface approach speeds, a true “jump” point, or limit of stability, is found not to appear, and the quasi-static model fails to represent the dynamic tip behavior at close tip-surface separations. Hence, a new “quasi-dynamic” method for estimating A is proposed that uses the dynamically well-defined deflection at which the tip and surface first come into contact, dc, instead of the dynamically ill-defined “jump” point. With the new method, an apparent Hamaker constant, Aapp, is calculated from dc and a corresponding quasi-static-based equation. Since Aapp depends on the cantilever’s approach speed, vc, and the AFM’s sampling resolution, δ, a double extrapolation procedure is used to determine Aapp in the quasi-static (vc → 0) and continuous sampling (δ → 0) limits, thereby recovering the “true” value of A. The accuracy of the new method is validated using simulated AFM data. To enable the experimental implementation of this method, a new dimensionless parameter τ is introduced to guide cantilever selection and the AFM operating conditions. The value of τ quantifies how close a given cantilever is to its quasi-static limit for a chosen cantilever-surface approach speed. For sufficiently small values of τ (i.e., a cantilever that effectively behaves “quasistatically”), simulated data indicate that Aapp will be within ∼3% or less of the inputted value of the Hamaker constant. This implies that Hamaker constants can be reliably estimated using a single measurement taken with an appropriately chosen cantilever and a slow, yet practical, approach speed (with no extrapolation required). This result is confirmed by the very good agreement found between the experimental AFM results obtained using this new method and previously reported predictions of A for amorphous silica, polystyrene, and α-Al2O3 substrates obtained using the Lifshitz method.

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INTRODUCTION The Hamaker constant, A, quantifies the effects of composition on the van der Waals (vdW) interactions among particles and between particles and surfaces.1,2 Accurate values of A are required for understanding, for example, colloidal stability,3 interfacial adhesion,4 and nanoparticle self-assembly.5 The Hamaker constant can be calculated from the quantum electrodynamic Lifshitz theory,6 which requires detailed data of the complex dielectric constants of the interacting materials over a wide range (in principle, an infinite range) of frequencies. Application of the Lifshitz theory becomes less precise, however, when optical data are known over a limited frequency range.2 Furthermore, such a limited data set is generally available only for a small number of materials.7 Thus, developing alternative accurate experimental methods to determine A reliably is important. For example, Hamaker constants can be determined using a surface force apparatus (SFA).8 With an SFA, an interfero© XXXX American Chemical Society

metric technique is used to measure accurately the separation distance between two surfaces. The attractive force between the surfaces is measured as a function of separation distance, which enables an estimate of A to be obtained. The SFA requires, however, the availability of molecularly smooth cylinders.9 An atomic force microscope (AFM) can also be used to determine the attractive force, Fts, between an AFM tip and a surface of interest as a function of the separation distance, z, between them. With an AFM, a cantilever is moved into and out of contact with the surface, and its deflection, d, is measured, with the magnitude of d being related to the competition between Fts and the restoring force of the cantilever, Fc. An estimate of A can be obtained from this resulting deflection curve. Almost all solid materials can be investigated with an AFM.10 Received: November 9, 2016 Published: December 30, 2016 A

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Figure 1. Qualitative AFM tip deflection curve and snapshots of the cantilever’s approach-to-contact. (A) As the piezo height increases, or as the cantilever approaches the surface (moving to the right on the x-axis), the tip deflects downward by an amount d (0). The tip−surface separation distance is z (>0), where z = h + d = h − |d|, and |d| ≤ h. (B) The critical, or jump, point where the tip first becomes unstable. The critical values of h, d, and z are denoted as h*, dj, and z*, respectively. (C) The first contact point just after the tip becomes unstable and immediately jumps to the surface. This is the point when z first equals zero, and now the corresponding tip deflection is denoted as dc and is equal to −h*. The “jump-into-contact” distance, Δd, is defined as the difference of the deflections just at and just after the jump point: Δd = dj − dc = z*. (D) The tip finally snaps out of contact with the surface as the piezo height decreases or as the cantilever moves away from the surface.

analysis applicable to other tip−surface interactions. Since the effects of the cantilever tip’s inertia are neglected in the quasistatic model, we also introduce a dynamic model of the tip’s motion to test the quasi-static assumption. By comparing the dynamic and quasi-static models’ predictions for the tip’s behavior, the limitations of the quasi-static assumption become evident. Consequently, we propose a new method for estimating A that is not quasi-static, but rather “quasi-dynamic”, in which the dynamic nature of the tip is considered explicitly. First, the deflection at which the tip and surface first come into contact, dc, is determined from the dynamic AFM data. Then, an apparent Hamaker constant, Aapp, is calculated from a corresponding quasi-static-based equation. This apparent Hamaker constant depends, however, on the cantilever-surface approach speed and the sampling resolution of the AFM. A double extrapolation procedure is therefore used to obtain the value of Aapp in the limits of zero approach speed and infinite AFM resolution, thereby recovering the “true” value of A. An analysis of simulated deflection curves indicates that this new method is capable of yielding very accurate estimates of A. Nevertheless, the experimental time to determine dc for a range of approach speeds and sampling resolutions is large. Thus, we also present guidelines for the practical implementation of the new method. Invoking various scaling arguments and noting the instrumental noise inherent to an AFM measurement, we show for certain AFM operating conditions that the double extrapolation procedure can be avoided, and reliable estimates of A can still be obtained from only a single AFM data set. Finally, an experimental proof-of-concept of the method demonstrates that the obtained values of A for amorphous silica, polystyrene, and α-Al2O3 are in excellent agreement with those predicted from the Lifshitz theory.3,19−21

A qualitative AFM deflection curve is shown in Figure 1 for a system in which only attractive vdW forces are present. To begin, the piezoelectric controller in the AFM apparatus raises the surface toward the cantilever. Eventually, the tip and surface interact, and the cantilever is deflected (in the negative direction) toward the surface. When the tip is finally close enough to the surface, |Fts| exceeds |Fc| and the tip “jumps” into contact with the surface. An apparent discontinuity, labeled as Δd, therefore appears in the deflection curve. Once in contact, the cantilever may be pressed further toward the surface, resulting in a positive tip deflection. The piezoelectric controller then retracts the surface away from the cantilever. Only when |Fc| exceeds |Fts| does the tip finally snap out of contact with the surface. AFM tip deflection data within the “pull-off” regime (Figure 1: D) have been used to estimate a value of A.11,12 Just before the tip snaps out of contact, Fc is assumed to be equal and opposite to Fts. If Fts is solely due to the tip−surface vdW interaction, A can then be estimated from Fc, by assuming the cantilever is a Hookean spring. This “pull-off” method is, however, quite sensitive to the surface roughness, contact separation distance, and surface deformation,13−16 which are aspects of the contact regime that are difficult to quantify precisely. Therefore, methods for determining A that rely on the approach regime of the AFM deflection curve (Figure 1: A,B), which should be affected less by the surface roughness and deformation, have also been developed.17,18 Improving these approach-to-contact methods is the main goal of this paper. The prevailing description of the behavior of the AFM tip in the approach regime is based on a “quasi-static” model, in which for sufficientlty large separation distances |Fc| is always assumed to be equal to |Fts|. We revisit the quasi-static model for a simple tip−surface force profile and then provide a general B

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QUASI-STATIC MODEL The quasi-static model of the AFM tip’s behavior in the approach regime assumes that for sufficiently large z mechanical equilibrium is achieved,17,18,22,23 or Fnet = Fc(d) + Fts(z) = 0

A11 =

(1)

d=

1 Fts(z , A , R t , ...) kc

(6)

1=

1 ∂Fts kc ∂d

(7)

(2)

where the spring constant of the cantilever, kc, can be determined experimentally.24 To determine Fts, one must first describe the shapes of the interacting bodies. The tip−surface attractive force is then evaluated as a function of z and A using a volume element integration between the two bodies.1 Three common geometries, and the resulting force profiles, for describing an AFM tip interacting with a flat surface are presented in Part A of the Supporting Information. Scanning electron microscope (SEM) images of many cantilever tips suggest that a truncated pyramid with a spherical cap often accurately captures a tip’s geometry.25 A simpler description is to represent the apex of the tip as a sphere of radius Rt interacting with a flat surface. For this model, the force−distance dependence when Rt ≫ z is given by1 Fts(A , z) = −

where Fts is a function of z, A, and various geometric tip parameters.25,28

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DYNAMIC SIMULATIONS OF THE TIP MOTION The quasi-static model seems to provide a way to determine A from the apparently measurable jump-into-contact distance Δd. The quasi-static model, however, is based on the assumption that the tip is always in mechanical equilibrium up until the critical point.17,18,22 But since the piezoelectric controller moves the surface at a finite speed, an exact force balance at each z cannot be strictly achieved during an AFM experiment, particularly near the (unstable) critical point (see Figure S4). Moreover, the quasi-static “jump-into-contact” method implicitly assumes that upon reaching the critical point the tip immediately jumps onto the surface with an (unphysical) infinite speed. To investigate how closely the quasi-static analysis describes the AFM tip dynamics, we introduce a timedependent model of the tip deflection. The motion of an AFM tip is modeled as a ball−spring system without friction or damping.18 One side of the spring, with spring constant kc, is attached to a platform that is located at a varying height h from a smooth surface, and the other side is attached to the ball. The displacement of the ball from the platform, denoted as d, represents the deflection of the AFM cantilever tip. The ball representing the tip is considered to be a “point particle” with mass m (with a “resonance frequency”, ω0, equal to kc/m ) and interacts with the surface via a force vs distance profile, Fts(z). For convenience, damping is not accounted for, which does not alter the main conclusions of our analysis. The cantilever approaches the surface with a constant speed of vc, and so h decreases linearly with time t from an initial height h0

AR t 6z 2

(3)

where A is the Hamaker constant (i.e., A12) between the flat substrate “1” and the tip material “2” (if in a vacuum). If the intervening medium were a material denoted by “3”, then the corresponding Hamaker constant would be labeled as A132. Describing the approach portion of an AFM deflection curve with eq 1 seems to provide a simple method to estimate A.25 But at some (small) critical separation distance, z* (Figure 1: B), the tip appears to become unstable and then immediately “jumps” into contact with the surface.22 This instability first occurs at a value of z beyond which |Fts| > |Fc|, and thus eq 1 can no longer be satisfied. With the tip rapidly jumping into contact as soon as the instability is reached, the resolution of the deflection data around this jump point greatly affects the accuracy with which eq 1 fits the deflection data. 26 Furthermore, the precise tip−surface separation distance z is unknown prior to contact in an AFM experiment and is a fitted parameter when using eq 1. To overcome some of these issues, an alternative method that uses information exclusively from the jump portion of the approach regime has been developed. The change in the cantilever deflection across this jump is called the “jump-intocontact” distance and is denoted as Δd = dj − dc (Figure 1). Using eq 3, A12 and Δd are related by18 3k (Δd)3 A12 = c Rt

(5)

where A11 and A22 are the two materials’ self-Hamaker constants. The derivation of eq 4 and its description of an unstable equilibrium point are provided in the Supporting Information (Parts B and C). In general, the critical conditions, z*, dj, and dc, as well as Δd, are obtained for any physically meaningful Fts by simultaneously solving the following two equations (see eqs S18 and S19 in the Supporting Information)

where Fnet is the net force on the tip and d ( 0.979 s are shown. The line with filled dots represents the quasi-static case; the solid line, which overlaps in part with the line of dots, represents the dynamic case; the dashed line BD indicates the quasi-static “jump-into-contact” distance. The dots highlight points along the quasi-static curve and are used for better visualization. Adapted from ref 23.

as a function of time t. The quasi-static curve overlaps with the dynamic curve, except for the very end of the tip movement. For this cantilever speed, the quasi-static assumption is quite accurate for most of the tip motion, except when h becomes very small. Then, there is a significant deviation between the dynamic and quasi-static deflections. As expected, the quasi-static curve reaches a limit before |d| = h, with corresponding values of dstat and h of −0.71 and 2.12 nm, respectively. If the quasi-static model were exact, the “jump-into-contact” would occur at a well-defined critical point, in which Δd = dstat + h = 1.41 nm. Instead, the tip described by the dynamic model moves continuously until it touches the surface, where ddyn = −1.94 and h = 1.94 nm. Hence, there is no well-defined “jump-into-contact” point for the physically relevant dynamic case (i.e., the cantilever’s approach speed is nonzero). In other words, if one were able to track precisely the location of the tip throughout the entire approach to the surface, one would never witness a “jump-into-contact” event. The sudden “jump” observed in an experiment only arises D

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Figure 3. Representative AFM cantilever deflection data obtained using a Bruker Corp. MultiMode PicoForce AFM and a silicon nitride tip against amorphous silica at two different resolutions along a ∼ 1 μm ramp (h0) with an approach speed of 1000 nm/s. The left figure (a) was generated with the AFM preset resolution, δ ≅ 0.2 nm, and the right figure (b) was generated at the instrument’s finest sampling resolution, δ ≅ 0.015 nm. The apparent “jump” point, circled in (a), disappears as more data points are obtained prior to contact.

Figure 4. (a) Calculated cantilever tip deflection as a function cantilever height based on the dynamic model (eqs 8 and 9), for approach speeds ranging from 1000 to 1 nm/s. The parameter values for A, Rt, tc, h0, and kc were the same as those used in Figure 2. The deflection at first contact with the surface, dc, has been labeled for both the 250 and 500 nm/s approach speed curves. (b) Effect of the sampling resolution, δ, on the apparent Hamaker constant, Aapp, obtained from the direct quasi-static interpretation (eq 4) of the simulated dynamic AFM data for different cantilever approach speeds (and normalized with respect to the inputted A used to simulate the deflection curves). The lines in (b) are guides to the eye.

Aapp as δ → 0 for any finite approach speed again leads to the conclusion that a true “jump point” does not in fact exist.

not equal to the value of A used in the dynamic model, these “apparent” Hamaker constants were labeled Aapp. In Figure 4b, values of Aapp/A are plotted as a function of the sampling resolution for different cantilever approach speeds. The Aapp calculated from eq 4 and the “jump-into-contact” distance are found to depend strongly on vc and δ and do not tend toward a unique value. As δ decreases, all the curves in Figure 4b converge toward a zero value of Aapp, which is a nonphysical result (there should be no dependence of the vdW force on the AFM resolution). As the resolution of the AFM becomes finer (δ decreases), the “jump point” appears closer to the point of first contact between the tip and surface. Consequently, as δ decreases, smaller values of Δd are used in eq 4, resulting in the smaller values of Aapp. The vanishing of

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NEW “QUASI-DYNAMIC” HAMAKER CONSTANT ESTIMATION METHOD The “jump” point and the “jump-into-contact” distance can only be uniquely identified in the quasi-static limit (i.e., vc → 0). In contrast, the deflection at first contact between the tip and surface, dc, is a well-defined location along any AFM deflection curve and is uniquely and reproducibly measurable with an AFM apparatus. The more precisely is the trajectory of the tip tracked, the more precisely is dc determined. The value of dc will therefore reach a nonzero value as δ → 0, in contrast to Δd approaching zero in this same limit. In addition, dc should approach its quasi-static limiting value as vc → 0. Consequently, E

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Figure 5. Testing of the double extrapolation to the zero sampling resolution (a) and quasi-static limits (b) for recovering the inputted value of A (i.e., Aapp/A = 1). In (a), the values of Aapp for each vc are extrapolated to the zero sampling resolution limit (circled in red) to obtain Aapp(vc, δ = 0). In (b), the values of Aapp(δ = 0) are extrapolated to the quasi-static limit to obtain Aapp(vc = 0, δ = 0). Dynamic model parameter values are the same as used in Figure 2. The lines are guides to the eye.

the effect of vc and δ on dc should be greatly reduced compared with what is observed for Δd. A quasi-static relation that connects A to dc should provide an alternative and ultimately accurate method for estimating the Hamaker constant. In the quasi-static limit, dc is equal and opposite to the undeflected height of the cantilever at the critical point, h* (Figure 1). Hence, using eq 3, and noting that h* = −dc at the critical point, the relationship between A and dc is found to be (see eq S14) A12 = −

A11 =

3 8 kc(dc) 9 Rt

2 6 (A12 )2 64 kc (dc) = A 22 81 R t 2A 22

tc, h0, and kc as for Figure 2. The resulting values of Aapp/A are plotted as a function of δ for various approach speeds in Figure 5a. For all vc > 0, Aapp/A < 1 and is a nearly linear function of δ and approaches a nonzero value as δ approaches zero. As seen in Figure 4a, the magnitude of dc in the quasi-static limit is always greater than that for the dynamic case, a trend that has also been observed in AFM experiments (Part E in the Supporting Information). Thus, as follows from eq 12, Aapp(vc, δ) ≤ A. Finally, the value of Aapp in the zero sampling resolution limit, Aapp(vc, δ = 0) tends toward the inputted value of A as vc decreases. Hence, as expected, Aapp(vc = 0, δ = 0) ≈ A. To test the effectiveness of the double extrapolation procedure, we considered only the data in Figure 5a for sampling resolutions of 0.015 nm or greater. From this limited data set, linear extrapolations for each vc were performed in order to obtain Aapp(vc, δ = 0). These estimated continuous sampling limits were then plotted as a function of vc (Figure 5b), and a second extrapolation of Aapp(vc, δ = 0) to the quasistatic limit, vc → 0, was performed. Based on a best fit cubic polynomial, a value of Aapp/A = 0.999 was obtained. In this regard, similar accuracies were obtained using lower- and higher-order polynomials and a linear extrapolation based on the last three data points. Thus, eq 12, with its use of dc, provides an approach-to-contact method for estimating A with a high degree of accuracy.

(12)

(13)

both of which now include only experimentally measurable parameters. For other tip−surface force profiles, eqs 6 and 7 are again solved simultaneously, along with h* = −dc, to obtain a connection between A12, or A11, and dc. With this new method, dc is determined from a given dynamic deflection curve, and then eq 12 or 13 is used to calculate an apparent Hamaker constant, Aapp. Although the new method makes use of the quasi-static relations in eq 12 or 13, the method is not strictly quasi-static. Rather, the method is “quasi-dynamic”, in that we acknowledge explicitly that this apparent Hamaker constant is not the “true” value of A and will depend on the approach speed and the sampling resolution of the dynamic data, or Aapp(vc, δ). To extract the value of A that would be obtained when the quasi-static model is strictly valid, a double extrapolation procedure must be employed. First, for a given vc, the value of Aapp in the infinite resolution limit is obtained, or Aapp(vc, δ = 0). Then, to recover the quasi-static limit, the values of Aapp(vc, δ = 0) are extrapolated to zero approach speed to obtain Aapp(vc = 0, δ = 0), which should in principle be equal to A. To test this new method, simulated dynamic data for the sphere−tip model (eq 3) were generated at various approach speeds and sampling resolutions using the same values of A, Rt,

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OPTIMIZATION OF THE EXPERIMENTAL PROCEDURE Experimental implementation of this new method seems to require that several AFM experiments be conducted, employing a range of approach speeds and sampling resolutions in order to confidently extrapolate to their ideal limits. Unfortunately, the experimental time to determine dc, particularly at slow approach speeds, where each reported value of dc is an average of several repeated contacts at different locations on the surface, is large. Fortunately, the results presented in Figure 5 suggest that high accuracy may be achievable with just a few experiments or perhaps a single carefully chosen experiment F

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∼2.7% lower than the inputted value of A, while Aapp for the stiffer cantilever F is only ∼1.3% lower (both of which are well within the corresponding expected error of dc from the AFM measurements29). Hence, when E and F are used at these slow approach speeds, no statistically meaningful improvement in the accuracy of the estimated value of A would be obtained upon extrapolating to the quasi-static limit (vc → 0). In other words, cantilevers E and F have effectively reached their quasistatic limits for vc ≤ 200 nm/s (i.e., Aapp/A ∼ 1). As seen in Figure 6a, each cantilever effectively behaves quasi-statically (say, when Aapp/A ≥ 0.97) below some critical value of vc, which depends upon ω0 and kc. To provide guidance for selecting approach speeds at which a given cantilever behaves as if it were at its corresponding quasi-static limit, we introduce scaling arguments for understanding the effects of ω0 and kc on the dynamic behavior of the cantilever tip. To begin, we note that the maximum deflection of the tip is dc. While dc depends upon vc, the largest value of dc will be approximately equal to (except for very large approach speeds) its value in the quasi-static limit, or dc,qs. Thus, for a given vc, the time scale, tc, for the cantilever to move a distance |dc,qs| is equal to

using a slow approach speed and a sufficiently fine sampling resolution. In Figure 5a, for the low speed of vc = 1 nm/s, Aapp is within 1% of A at the instrument’s finest resolution (δ = 0.015 nm). Given eq 12, this difference in Aapp corresponds to a value of dc that is within 0.33% of the extrapolated value of dc (at δ → 0). Decreasing this small deviation of dc even further would be difficult, as it falls well within the expected errors of dc inherent to a typical AFM experiment, arising from, for example, thermal noise.29 Hence, as long as the instrument’s resolution is sufficiently fine, the experimentally obtained values of dc, as well as the calculated values of Aapp, should have no statistical dependence on δ. AFM experiments confirm this expectation, as no statistically significant effect of δ is observed for resolutions around δ = 0.015 nm (Part F in the Supporting Information). Consequently, for a fine enough resolution, only the dependence of Aapp on vc needs to be considered, and just a single extrapolation to the quasi-static limit (vc → 0) is required. Furthermore, Figure 5 also indicates that Aapp is very close to A at suitably low approach speeds. In other words, if the inertial effects of the cantilever’s properties were minimal, such that the dynamic tip behavior was sufficiently close to that of a quasi-static cantilever, then A could be reliably estimated from only one optimally chosen AFM experiment (taken at an appropriately small vc and δ). To investigate further this possibility for actual AFM measurements, we again turn to the dynamic model (eqs 8 and 9) to analyze the dependence of Aapp on vc for each of the six silicon nitride cantilevers provided in Bruker’s commercially available MSCT cantilever tip pack (a common cantilever tip assemblage). The nominal values of the resonance frequencies, ω0, and spring constants of these cantilevers are provided in Table 1.

tc =

ω0 [kHz]

kc [N/m]

A B C D E F

22 15 7 15 38 125

0.07 0.02 0.01 0.03 0.1 0.6

vc

(14)

While the above is relevant to the movement of the entire cantilever, the time scale that is relevant to the motion of the cantilever tip is tresp =

1 ω0

(15)

which characterizes the inertial response of the tip, or how rapidly it responds to changes in the net force acting on it. As the cantilever moves toward the surface, the attractive force on the tip will change. A rapidly responding tip, or a cantilever with a small value of tresp, will quickly adjust to this change, such that the tip will quickly return to a deflection for which the right-hand side of eq 8 is, on average, close to zero. In this case, the tip deflection will always be near to its corresponding quasi-static value. But if the inertial response of the tip is too slow, or tresp is too large, the tip will not be able to return to its quasi-static deflection before the location of the cantilever, and hence the attractive force, has substantially changed again. The dynamic behavior of the tip should be effectively quasi-static when tresp vc = ≪1 τ≡ tc ω0|dc,qs| (16)

Table 1. Resonance Frequencies, ω0, and Spring Constants, kc (as reported by Bruker Corp.), of the Six Prefabricated Cantilevers, A−F, in a Standard Commercially Available MSCT Assemblage cantilever

|dc,qs|

The tip−surface interaction was again modeled using eq 3, with Rt = 100 nm and A = 1.1 × 10−19 J. For each pair of kc and ω0 in Table 1, deflection curves were generated with the dynamic model and then “filtered” as before using δ = 0.015 nm. The value of dc was obtained for each deflection curve and then inputted into eq 12 to determine Aapp. The resulting values of Aapp for each cantilever are plotted as a function of vc in Figure 6a. As expected, Aapp for each cantilever tends toward the value of A inputted into the dynamic model as the quasi-static limit is approached (vc → 0) In addition, the results for cantilevers E and F, which have the highest resonance frequencies and spring constants in Table 1, suggest that only one very slow (vc ≤ 200 nm/s) experiment for these stiff cantilevers might be needed to generate a reliable estimate of A. Using the slow, yet practical, approach speed of 200 nm/s, Aapp for cantilever E is only

The above is consistent with the quasi-static limit being reached for vc → 0 or ω0 → ∞ (i.e., m → 0). Equation 16 also indicates that quasi-static behavior is approached for large values of |dc,qs|, which itself depends upon the spring constant of the cantilever. Using the sphere-plate result of eq 3 for |dc,qs|, the ratio of characteristic time scales, τ, becomes

τ=

1/3 vc ⎛ 8 kc ⎞ ⎜ ⎟ ω0 ⎝ 9 AR t ⎠

(17)

Equation 17 fully explains the trends exhibited by Figure 6a. For a given approach speed (and fixed values of A and Rt), G

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Figure 6. (a) Effect of the approach speed, vc, on the apparent Hamaker constant, Aapp, for the cantilevers A−F provided in Table 1 obtained using the dynamic model (eqs 8 and 9). The value of the Hamaker constant inputted into the dynamic model is denoted as A. The sampling resolution is δ = 0.015 nm, which is the finest achievable resolution for the AFM used in this study. The lines are guides to the eye. (b) Dynamics of the cantilever is captured by τ as demonstrated by the behavior of all the studied cantilevers collapsing onto one universal curve. (Each curve extends from zero to a maximum value of τ corresponding to the cantilever approaching at 1000 nm/s. For example, curve A ranges 0 < τ < 0.008.) Cantilevers operated at approach speeds corresponding to τ ≤ 1.2 × 10−3 will have a predicted theoretical accuracy of ≥97% from only one experimental trial (i.e., effectively behaving “quasi-statically”).

those cantilevers with smaller values of kc1/3/ω0 will be closer to the quasi-static limit or will have values of Aapp/A closer to unity. According to Table 1, the ranking of cantilevers from highest to lowest values of kc1/3/ω0 is (in appropriate units) C (=31), D (=21), A (=19), B (=18), E (=12), and F (=6.7). This ordering is fully consistent with the ranking of cantilevers from lowest to highest values of Aapp/A in Figure 6a and also explains the similar values found for D, A, and B. While B and D have the same resonance frequency, B has the smaller value of kc, which corresponds to the larger value of |dc,qs|. Hence, B is closer to the quasi-static limit than is D. Cantilever A has an even larger value of kc, which increases the inertial effects. But that increase is partially offset by A’s larger value of ω0, with its value of τ falling in between that of D and B. So while eq 17 suggests that flexible (small kc) cantilevers are closer to their quasi-static limits, a stiffer cantilever (large kc) can still be used if the resonance frequency is large enough. The stiffest cantilevers, E and F, are closest to the quasi-static limit since they have, by far, the largest values of ω0. The dimensionless parameter τ provides a quantitative indicator of the quasi-static behavior of a given cantilever and can be used to determine which approach speeds should yield an accurate estimate of A. To demonstrate this capability, Aapp/A from Figure 6a is plotted as a function of τ in Figure 6b. When plotted against τ, instead of vc, all the separate curves in Figure 6a collapse onto one universal curve. If the quasi-static limit is defined to be reached, at least statistically speaking, when Aapp/A ≥ 0.97, then a single experiment should yield an estimate of A that is as accurate as is feasible for τ ≲ 1.2 × 10−3. Thus, in general, the tip should be well-described by its quasistatic behavior for the following approach speeds

vc ≲ 0.001ω0|dc,qs|

F, respectively, indicating that these cantilevers are effectively behaving quasi-statically. The predicted scaling behavior in Figure 6b can be tested using AFM experimental results. Cantilevers C, D, E, and F were selected, and values of dc for each cantilever were determined for δ ≈ 0.015 nm and approach speeds of 50, 200, 500, and 1000 nm/s. The substrate chosen for these experiments was amorphous silica as obtained from Bruker Corporation. (Other details of the experimental procedure are provided in the next section. The resulting data are provided in Part G of the Supporting Information.) Since the sphere-plate geometry is not necessarily relevant to the actual cantilevers, we use the general expression for τ in eq 16, along with the replacement of |dc,qs| by |dc,qs*| (the value at the extrapolated limit). In Figure 7, values of (dc/dc,qs*)3 are plotted as a function of τ. (Note that if the tip and surface were well-described by the sphere-plate geometry, then (dc/dc,qs*)3 would be equivalent to Aapp/A.) Similar to what is observed in Figure 6b, the results for each cantilever appear to fall onto a single curve. In addition, the values of τ at approach speeds of 50 and 200 nm/s for both E and F are always less than 1.2 × 10−4, which correspond to values of (dc/dc,qs*)3 very close to 1. The expected values of Aapp/A should therefore also be close to 1 as a result of the nearly quasi-static behavior of each tip. Hence, an accurate estimate of A should be obtainable from these cantilevers when operated at one of these slow approach speeds. Finally, as predicted by eq 17 and Figure 6b, the accuracy of the estimated Hamaker constant should also depend on the magnitude of A, though the effect of A is relatively weak (since τ scales as A−1/3). The expected errors in Aapp for a realistic range of Hamaker constants (50−300 zJ) are provided in Part H of the Supporting Information, which are found to be less than or equal to 3%. Again, considering the errors inherent to an AFM measurement, this degree of error is negligible. In summary, by properly selecting the cantilever’s properties, highly accurate estimates of A should be obtainable from AFM data generated at one practical cantilever approach speed.

(18)

As noted earlier, cantilevers E and F yield estimates of A with less than 3% theoretical error for vc = 200 nm/s. At this approach speed, τ equals 1.05 × 10−3 and 5.83 × 10−4 for E and H

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of contact with the surface 200 times at each of eight distinct locations on the surface. These 1600 individual measurements were then averaged to produce the measured value of dc, with error bars corresponding to one standard deviation from the average. For each experiment, the cantilever spring constant was determined using the thermal tune method before and after data collection.24 The spring constant was determined to be 0.150 ± 0.004 N/m. To ensure the absence of capillary bridging effects, the AFM measurements were conducted in a humidity-controlled environment with a relative humidity below 10%. An Amstat 1U400 Staticmaster ionizer was placed in close proximity to the substrate and cantilever to mitigate electrostatic charge buildup on the surfaces. The new method requires that dc be determined, which in principle can be obtained by taking the difference between the cantilever deflection at first contact and the deflection at the start of an experiment when the tip−surface separation distance is large. But due to optical interference and electronic background effects of the instrument’s data collection,31 the photodetector registers an apparent steady increase in the tip deflection, as well as an apparent small periodic fluctuation of the tip position, while the cantilever approaches the surface. For similar reasons, the initial registered tip deflection is nonzero and changes with each contact experiment. These effects are typically ignored when determining A from the pull-off methods but must instead be removed from our data analysis, since the measured deflections are of the same order as the instrument noise (i.e., 1−10 nm). Thus, for each deflection curve, a linear regression was used to represent the effects of this instrument noise, as shown in Figure 8. The tip’s initially undeflected state at contact (α) is then estimated using the deflection that falls on this linear regression and is located directly above the estimated first tip−surface contact point (β). Point β is identified as the first point (located around where the tip deflection rapidly changes with a change in piezo height) at

Figure 7. Experimentally observed values of (dc/dc,qs*)3 plotted as a function of τ. The error bars for both τ and (dc/dc,qs*)3 correspond to the propagated errors that result from measuring the tip deflections, approach speeds, and resonance frequencies with an absolute error of ca. 2−3%. For clarity, only one set of error bars has been included, but the errors for each cantilever are included in the Supporting Information (Part G). The lines through the points are guides to the eye.

While eq 17 indicates that the chosen cantilever should have the largest value of ω0/kc1/3, other experimental concerns not yet accounted for have to be considered when selecting a cantilever. For example, while cantilever F has the lowest values of τ at all approach speeds, its large spring constant yields values of dc that are less than 3 nm (Figure S6). Consequently, only when F is in close proximity to the surface will its tip be measurably deflected. But at these close separation distances, even small (and unavoidable) surface asperities are expected to have a relatively large effect on the experimental results, leading to large relative variations in the measured values of dc. Thus, cantilever E was instead selected for the proof-of-concept experiments presented in the following section. While being flexible enough to minimize the effects of surface roughness (with a larger value of |dc|), E is still optimal (i.e., low values of τ). Using this cantilever and an approach speed of 200 nm/s, the Hamaker constant is theoretically predicted (from Figure 6) to be estimated with an error of only ∼2−3% (i.e., τ ≲ 1.2 × 10−3).

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EXPERIMENTAL PROOF OF CONCEPT Experimental Procedures, Analysis, and Materials. Cantilever tip deflections were again measured using a Bruker Corporation MultiMode PicoForce AFM with a NanoScope V controller. The surfaces chosen for the experimental validation of the new method were fused silica, polystyrene, and α-Al2O3. The first two were prefabricated and used as produced by Bruker Corporation. The third substrate was fabricated in the Parsons Lab (North Carolina State University) as a model smooth surface using atomic layer deposition to coat alumina onto a flat silicon plate. The thickness of this deposited layer was ∼40 nm and was previously shown to be thick enough to guarantee vdW behavior of bulk alumina.30 The cantileversurface approach speed was set at 200 nm/s, and the specific cantilever tip used was the Bruker MSCT cantilever E in Table 1. For each experiment, the cantilever was brought into and out

Figure 8. Measured cantilever tip deflection as a function of piezo height as the piezo-controller raises the substrate toward the cantilever (cantilever E in Table 1). The apparent steady drift in the registered deflection resulting from the AFM’s collection protocol are canceled by fitting the noncontact portion of the deflection curve to a linear regression (red line). The deflection at first contact, dc, is estimated by finding the difference between the registered deflections at points α and β. The sampling resolution is δ ≈ 0.015 nm. I

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Figure 9. (a) SEM image of the tip of cantilever E taken at 20 000× magnification and (b) the AFM cantilever tip modeled as a truncated pyramid with a spherical cap.

which the slopes of the deflection curve on either side of that point transitioned from a negative to a positive value. (After β, the direct contact portion of the deflection curve is presumed to begin.) The difference between the registered deflections at points α and β is the estimated value of dc for each experiment. The choice of the sphere-plate geometry to represent the tip−surface interaction is convenient for theoretical discussions of the new method. Since the quasi-static analysis using eq 3 gives rise to an analytically solvable cubic equation, the simple connections between A and dc in eqs 12 and 13 can be easily obtained. But, as seen in the SEM image (Figure 9a), the tip is more closely represented by a truncated pyramid with a spherical cap. For this cantilever shape, the tip−surface interaction force is25

Table 2. Measured Values of dc and Estimated Self-Hamaker Constants (Aii) Obtained Using the New Method and the Lifshitz Theory for Three Different Substrate Materials

4 tan ϕ ⎛ R t cos ϕ + tan ϕ(z + λ) ⎞⎤ ⎟⎥ ⎜ π ⎝ (z + λ)2 ⎠⎥⎦

Lifshitz

estimation

approximation

material

|dc| [nm]

[10−20 J]

[10−20 J]

polystyrene silica α-Al2O3 silicon nitride

5.22 ± 0.26 4.98 ± 0.12 5.67 ± 0.14 −

7.9 ± 1.4 6.5 ± 0.8 13.2 ± 2.1 −

7.1,20 7.921 6.5,19 6.63 14.5,3 15.219 18.019

close to its true quasi-static value (which should be the case if vc is low enough), the results in Table 2 correspond to values of τ of (1.01 ± 0.06) × 10−3 (polystyrene), (1.06 ± 0.04) × 10−3 (silica), and (9.28 ± 0.34) × 10−4 (α-Al2O3). Thus, the experimental measurements were indeed performed at an approach speed that is within the effective quasi-static limit of the chosen cantilever (i.e., τ < 1.2 × 10−3). Note that the focus of the new method is on the experimental determination of dc, a parameter that is independent of the geometric model used to describe the cantilever tip. Only after dc is measured is a geometric tip model needed to “convert”, via eqs 6 and 7, the obtained value of dc into an estimated value of A. Hence, the accuracy of the new method will be mainly limited by the accuracy of the AFM measurements themselves. Errors will, of course, arise from the inability to measure precisely the geometric dimensions of the cantilever tip. But the main sources of error in the dc measurements should stem from, for example, surface roughness, the limited resolution of the instrument, and the noise inherent to an AFM experiment.29 These effects decrease one’s ability to reliably track and predict the true motion of the cantilever during its approach.26 Even if dc could be estimated with small relative error, the relative error of Aapp would nonetheless be much larger. At least for the sphere-plate geometry, eq 13 indicates that A11 varies as dc6. Hence, the relative errors in the estimates of A11 will be 6 times that of dc (not including the relative errors for the other terms appearing in eq 13). The error bars for dc in Table 2 are reasonable, being ±5.0% or ±2.6 Å for polystyrene, ±2.4% or ±1.2 Å for silica, and ±2.5% or ±1.4 Å for α-Al2O3. A significant reduction in these relative errors is difficult to achieve, as this would require accuracies of less than 1 Å. A more flexible cantilever may lead to a decrease in the relative error, since |dc| increases as kc

2 (R t cos ϕ)2 A ⎡ λ (3R tz + R tλ − zλ) Ft(A , z) = − ⎢ + 6 ⎢⎣ (z + λ)3 z 2(z + λ)3

+

new method measured

(20)

where the geometric parameters λ, ϕ, and Rt are defined in Figure 9b. These parameters were estimated from Figure 9a and are as follows: λ = 5 nm, Rt = 75 nm, and ϕ = 60°. In order to use eq 20, the quasi-static connection between A and dc must be obtained via eqs 6 and 7. Numerical methods were used to evaluate these equations to determine the value of Aapp that was consistent with the given value of dc estimated from the AFM measurements. Method Validation and Discussion. Since Hamaker constants are typically reported in terms of a material’s selfattraction (Aii), the geometric mean approximation27 and the value of A22 for silicon nitride predicted by the Lifshitz theory19 were employed to determine A11 from each measured tip− surface interaction (A12). The average values of dc and the resulting values of A11 for amorphous silica, polystyrene, and αAl2O3 are shown in Table 2. The values of A11 estimated using the Lifshitz method3,19−21 are also provided in the table. The agreement between the Lifshitz predictions and the experimental results obtained with the new method at a single approach speed and sampling resolution is excellent. Equation 16 or 18 can be used to check the reliability of the experimental results. The single chosen approach speed should have been low enough such that the effective quasi-static limit was reached for the selected cantilever, with a value of τ much less than unity. Assuming the measured value of |dc| is very J

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decreases, thereby minimizing the effects of surface roughness. This is counteracted by the fact that thermal noise typically becomes more important for flexible cantilevers. A stiffer cantilever would lessen the impact of thermal noise, but this leads to a decrease in |dc|, which may then increase the impact of surface roughness. Error bars of dc on the order of ∼2−3% are perhaps expected, which seem to correspond to the current feasible limit of the AFM. Hence, the expected error bars in A11 obtained with the new method will be ∼12−18%.

AUTHOR INFORMATION

Corresponding Author

*(D.S.C.) E-mail [email protected]. Present Address

J.D.: Nalco Water, 6216 W. 66th Place, Chicago, IL 60638. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This material is based upon work supported by the U.S. Department of Homeland Security, Science and Technology Directorate, Office of University Programs, under Grant Award 2013-ST-061-ED0001. The views and conclusions contained in this document are those of the authors and should not be interpreted as necessarily representing the official policies, either expressed or implied, of the U.S. Department of Homeland Security.

CONCLUSIONS The determination of the Hamaker constant of a solid material using the approach-to-contact regime of an AFM deflection curve is usually based on the following steps: (1) the dynamic AFM tip deflection under the influence of the attractive force between the tip and the solid for a given cantilever approach speed is obtained and (2) a quasi-static model is used to fit the dynamic data, enabling one to infer the value of the Hamaker constant. In this paper, we evaluated the validity of this quasistatic method and developed an improved method, in which the motion of the tip and the finite resolution of the AFM are explicitly considered in the estimation of the Hamaker constant. A dynamic model of the tip motion revealed that for small tip−surface separation distances the tip motion deviates from the predictions of the quasi-static model, except at the limit of zero cantilever approach speeds. The dynamic analysis indicated that a “jump-into-contact” distance, which can be defined only in the quasi-static limit, cannot therefore be uniquely determined from the inherently dynamic AFM data. The Hamaker constants determined using the “jump-intocontact” method were determined to be highly sensitive to vc and δ. Thus, a new “quasi-dynamic” method was developed for properly interpreting the dynamic data. In this new method, the dynamically well-defined deflection at first contact between the tip and surface, dc, is determined from the dynamic data. A double extrapolation method is employed to generate Aapp(vc = 0, δ = 0), which was shown to very closely estimate the true value of A. To minimize the experimental effort needed for the new quasi-dynamic method, scaling arguments were developed that provided a quantitative indicator used to determine which cantilever-surface approach speeds correspond to the effective quasi-static limit for a given cantilever. Hence, for an appropriately chosen cantilever, Hamaker constants can be reliably determined from AFM data collected at only one slow, but practical, approach speed, as was demonstrated experimentally using three previously studied materials. The resulting estimates of the Hamaker constants were found to be in agreement with the corresponding predictions of the Lifshitz theory. The new method therefore provides a valid and practical approach for estimating the Hamaker constants of solid materials as accurately as feasible with AFM measurements.

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ABBREVIATIONS vdW, van der Waals; SFA, surface force apparatus; AFM, atomic force microscope; SEM, scanning electron microscope. REFERENCES

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.langmuir.6b04063. Figures S1−S10, Table S1, and eqs S1−S24 (PDF) K

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