Molecular Dynamics Simulation of Nanoforces between Substrates

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Molecular Dynamics Simulation of Nanoforces between Substrates Mediated by Liquid Bridges: Controlling Separation and Force Fluctuations Gerson E. Valenzuela, Roberto E. Rozas, and Pedro G. Toledo J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.7b10447 • Publication Date (Web): 31 Oct 2017 Downloaded from http://pubs.acs.org on November 2, 2017

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The Journal of Physical Chemistry

Molecular Dynamics Simulation of Nanoforces between Substrates Mediated by Liquid Bridges: Controlling Separation and Force Fluctuations Gerson E. Valenzuela,∗,† Roberto E. Rozas,∗,‡ and Pedro G. Toledo∗,¶ †Chemical Engineering Department, Universidad de La Frontera, Av. Francisco Salazar 01145, Temuco, Chile. ‡Department of Physics, University of B´ıo-B´ıo, Av. Collao 1202, PO Box 5-C, Concepción, Chile. ¶Chemical Engineering Department and Surface Analysis Laboratory, University of Concepción, PO Box 160-C, Concepción, Chile. E-mail: [email protected]; [email protected]; [email protected]

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Abstract Molecular dynamics simulations allow determination of the force between a tip and a sample mediated by a fluid or a nanoscale capillary bridge at any separation distance in approach-retract cycles. However, without further consideration, the procedure leads to force-distance curves without control of tip-to-sample distance and tip oscillations. Here we show that the solution of a macroscopic model describing the dynamics of the tip-sample interaction can precisely guide the choice of optimum parameters, for instance tip elastic constant and approach speed, for controlled simulations. The method is applied to the interaction between two substrates in two cases of intervening fluid: a water bridge and a Lennard-Jones bridge. The method can be very useful in the determination of force-distance curves in more complex systems.

Introduction The interaction forces between nanoparticles or between colloidal particles almost exclusively determine the surface properties of these particles and therefore the result of every process in which they are involved, whether industrial or environmental. Such forces are routinely measured for example by atomic force microscopy (AFM), 1 also with the surface force apparatus (SFA), 2 between two solid surfaces mediated by a fluid as a function of the distance between the surfaces. The force measured between two surfaces of particular geometry, typically a spherical tip or probe and a plane in AFM and two crossed cylinders in SFA, can be easily converted to interaction energy between flat surfaces through the Derjaguin approximation. 3 In AFM the tip or probe is a colloidal sphere attached to a cantilever. AFM allows continuous measurement of cantilever deflection vs. position as tip and sample plane approach, commonly named extension, or separate, commonly named retraction. The force acting between tip and sample at any given separation is determined from the deflection of the cantilever. Force-distance curves are measured in conventional static contact-mode and also while operating in dynamic modes, amplitude modulated (tapping) and frequency 2


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modulated (non-contact). A known challenge in the measurement of force-distance curves is the condensation of water which eventually forms a liquid bridge between tip and sample; the bridge strongly draws the tip towards the sample with a force, the capillary force, which dominates over nanoscales contacts, atomic bonding, van der Waals forces, Casimir forces, and electrostatic forces. 4–19 The force exerted by a capillary bridge is of great interest, but there is a problem in separating it from the measured total interaction force. Also, capillary forces largely influence AFM operations if utilized to determine friction forces 10,20,21 and when employed as a litho tool in dip-pen nanolithography. 22–24 The great interest in force-distance curves and in particular its components is evident from the extensive literature on laboratory experiments 4–19 and also from the growing literature on computational simulation. 25–38 AFM and related techniques such as friction force microscopy, chemical force microscopy and dip-pen nanolithography are amenable to simulation since both force and displacement measured are of the order of magnitude covered by the simulations. The most used simulation techniques in the determination of forcedistance curves include molecular dynamics (MD), 32–35 Monte Carlo (MC), 25–31,37 classical continuum 26,35,36,38–43 and hybrid methods. 44,45 These techniques are also useful for testing interaction potentials. Recently, we used molecular dynamics simulation of a nanoscale capillary water bridge amidst two planar substrates to determine the resulting force between the substrates with no arbitrariness on the geometry and location of the free surface of the bridge. 46 A major difficulty both in the experimental measurement of force-distance curves and in their simulation is the a priori lack of knowledge of some material parameters, such as tip and corresponding elasticity, or operating parameters, such as tip-to-sample approach speed, which makes difficult to control tip-to-sample distance and tip oscillation. The experience is very valuable in the choice of these parameters. Alternatively, the more expensive method of choosing them by trial and error is used.

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Here we show that the solution of a macroscopic model describing the dynamics of the tip-sample interaction can precisely guide the choice of optimum parameters for controlled simulation of the interaction. The method is applied to the simulation by molecular dynamics of the interaction between two substrates in two cases of intervening fluid: a water bridge and a Lennard-Jones bridge. The first application shows the ability of the method to control system parameters to refine a force-distance curve and the second application shows the ability of the method to unravel the complex force due to a layered liquid bridge.

Simulation system The system simulated with MD is shown in Fig. 1, wherein the nomenclature on the left

Figure 1: Scheme used to calculate the force due to a liquid water bridge between two flat substrates. S1 is the sample which is fixed, S2 is the mobile probe, C is the liquid water bridge, and SP is an array of parallel springs with sites (atoms) at both ends. Atoms a are attached to S2 and sites b are either fixed or displaced depending on the experiment.

refers to the simulation itself, while that on the right to the AFM technique which we intend to imitate. The scheme of Fig. 1 was first used to determine the force exerted by a water bridge between two substrates, of great interest in many areas, 46 here we show that is also useful to determine the force due to a non-aqueous bridge and to evaluate the effect on the force when the liquid making the bridge is structured in several layers. Substrates S1 and S2 are made of atoms that interact according to the Lennard Jones potential, hereafter LJ. 4


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Substrate S1 is the sample whose position is fixed, and S2 is part of the tip which is mobile. C is a concave water bridge between S1 and S2 composed of water molecules that interact for instance according to the TIP4P/2005 potential. SP is a parallel combination of harmonic springs with LJ atoms a at the end touching with S2 and sites b at the other end. The springs share the same value of elastic constant and obey Hooke’s law in the z coordinate. Atoms a and S2 represent the tip or probe. Depending on the experiment sites b may remain fixed or may be moved to a predetermined position. The force (atomic/electrostatic/capillary) between the tip and the sample that is sensed from the stretching of SP (deflection of a cantilever in AFM) is given by FS =

Pn

i=1

k∆zi where k is the elastic constant of each

spring making SP, ∆zi is the stretching of the i-spring and n is the total number of springs. Each element in Fig. 1 is prepared separately by means of conventional MD simulation and then all the elements are placed in a simulation box, as shown in Fig. 1, until they reach equilibrium.

Simulation method Extension force-distance curves were obtained by alternating two simulation steps, as shown in Fig 2. In Step 1 both the sites b and sample were fixed while the tip-sample distance (D) attained a stable value. Mean values and fluctuations of the bridge capillary force F and the distance D were calculated from their instantaneous values during the equilibrium simulation step. In the simulation of Step 2, the spring constant k was increased and the sites b were approached to the fixed sample, at constant speed vb , up to a distance d. After this, the sites b were fixed and the system was let to reach a new equilibrium. The procedure of Step 1 was used to calculate a new pair F -D.

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Figure 2: Schematic representation of the method used to generate an extension F -D curve. In the simulation of Step 1, sites b are fixed and the bridge is in equilibrium. In the simulation of Step 2, the spring constant k is increased and sites b are approached to the sample at constant speed vb . This non-equilibrium step ends once a predefined distance d is attained, after which, sites b are again fixed and a new equilibrium is reached. In each Step 1 the tip oscillates about equilibrium distances D1 and D2 . Figure adapted from Valenzuela et al.. 46

Application to a water bridge Here, we describe the application of the simulation procedure to a capillary bridge of liquid water between two planar substrates. First, the various elements of the system were prepared separately and then assembled into a simulation box and equilibrated. Tip and sample preparation Atoms in the sample S1 were initially arranged as an fcc crystal with an initial density of −3 140000 mol/m3 (0.0843 ˚ A ) in a rectangular simulation box applying periodic boundaries

and then relaxed in an N P T simulation at 1 bar and 300 K. After equilibration, the density −3 of the substrate was 116600 mol/m3 (0.0702 ˚ A ). Atoms of S1 follow the LJ 12-6 interaction

potential, with parameters σ = 0.2471 nm and ǫ/kB = 8053.8. The tip S2 was a replica of the equilibrated configuration of the sample S1. S2 followed the same LJ interactions as S1. The springs SP were harmonics and had LJ sites at their ends. They were disposed in a hexagonal array with a sectional area xy equal to that of S2. The spacing between sites a and b was 2.5 nm which was the length of the springs when unperturbed. In a new

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simulation, the array of springs SP was placed close to S2 such that the separation distance between sites a of SP and tip S2 were slightly larger than the cut-off radius. Then, the sites b of SP were moved at -5 m/s (-0.05 ˚ A/ps), in the z coordinate, until the sites a and the atoms of the tip stick each other. The spring constant k at this step was large enough (30 mN/m) so that the springs maintain their initial length of 2.5 nm. Water preparation In a separate simulation, 500 water molecules in a cubic simulation box with an initial density ˚−3 ) were relaxed in an N P T simulation run at 300 K and 1 bar. 55000 mol/m3 (0.0331 A Periodic boundary conditions were applied. Water interaction potential was TIP4P/2005. 47 System preparation In a final simulation all the items were placed in the order shown in Fig. 1. In the new simulation box, periodic boundaries were applied only in the longitudinal directions x and y. Initially, water molecules formed a cube after the periodic boundaries of the preparation step were removed, this cube was placed between substrates S1 and S2 with an initial separation water-substrate of 0.3 nm (about the size of a water molecule) as Fig. 3 shows. In an N V T simulation both the substrates at fixed position and the water molecules were simulated until a capillary water bridge was formed and its total energy reached a stationary value. The shape of the bridge depended on the interaction parameters between water and S1 and water and S2. Cross parameters used here were given elsewhere. 46 Force-distance curve measurement The prepared capillary bridge was stretched in the z direction by moving the spring sites b at 5 m/s (0.05 ˚ A/ps) until the distance between substrates was 4 nm. After this, the substrates were held fixed and the water bridge was equilibrated until its total energy stabilized. In this simulation a sufficiently high value was chosen for the force constant of the spring (k = 30

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Figure 3: Initial system. Side view. mN/m) so that the bridge was stretched and not the springs. Finally, the spring constant was decreased to 1 mN/m (S1 and b remain fixed) and consequently the capillary bridge shrank and the springs stretched. The simulation of Step 1 (see Fig. 2) led to equilibrium at which the corresponding F -D pair was calculated. Before applying Step 2 to obtain a new data point for the extension, the value of k had to be chosen not too large to prevent a jump backward nor too small to prevent a jump forward, a common decision in AFM. In a preliminary approach, we used a trial and error procedure for both values of k and vb . For the predefined distance d we used 0.25 nm. The resulting extension force-distance curve is shown in Fig. 4 which highlights large error bars associated with the lack of a criterion to select appropriate values of k. The simulation result in Fig. 4 resembles the experimental force-distance curve 48 even if the systems are not identical, what matters is that the trend is captured correctly. In the following section, we show how a mass point description of the AFM scheme in Fig. 1 can precisely guide the correct choices of k and vb to produce precise and reproducible force-distance curves. The simulation method described here was implemented in CUDA-C language running on NVIDIA graphic processing units.

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Figure 4: Preliminary force-distance curve for a liquid water bridge between two planar substrates. The dotted line is a polynomial fit to the simulation data.

Macroscopic model A macroscopic model is proposed to describe the dynamics of the tip-sample interaction in Fig. 1. The model, illustrated in Fig. 5, does not have any molecular detail only the relevant elements of the system. The springs are replaced by a single effective spring with constant

Figure 5: Simplified description of the tip-sample interaction in Fig. 1. keff and equilibrium length l, the tip is described as a point S2 with total mass m, the sample as a fixed point S1 and the bridge as a nonlinear spring. Distance between S1 and S2 is D and that between S1 and sites b is L. The force exerted by the effective spring and by the liquid bridge on the tip S2 are denoted FS and F respectively. The force between S1 and S2 is zero since this interaction is turned off during the simulation. Additionally, a dissipative force −γeff D˙ might be introduced to describe the damping of the system. Newton’s equation describes the position of the tip with respect to the sample

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¨ = FS + F − γeff D. ˙ mD

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(1)

According to Fig. 5 the elongation of the springs is L − D − l, therefore the elastic force is FS = keff (L − D − l). When sites b are moved at velocity vb then L(t) = L0 + vb t. The nonlinear force F is approached by its first order expansion around the equilibrium tip-sample distance De , i.e. F (D) = F (De ) + F ′ (De )(D − De ). After defining x = D − De and replacing the expressions for the forces F and FS , the Eq. (1) reads as follows

m¨ x + γeff x˙ + [keff − F ′ (De )]x = keff [L − De − l] + F (De ).

(2)

In the following sections, we examine the solutions of this equation when the dissipative term is very small or nonexistent, appropriate for the system shown in Fig. 1. Such a problem has a simple analytical solution that allows the choice of optimal values to parameters such as the elastic constant of the tip and the velocity of the tip. We consider the cases when vb = 0 (Step 1) and vb 6= 0 (Step 2).

Step 1 (vb = 0) When sites b remain at rest the system reaches a stationary state where the tip oscillates around its equilibrium position De . When L is constant, the right-hand side of Eq. (2) is zero, following the equilibrium condition

FS (De ) = −F (De ),

(3)

(∂t2 + ω 2 )x = 0.

(4)

and the Eq. (2) simplifies to

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When keff > F ′ (De ) the solution is a stationary wave

x(t) = A cos(ωt + φ).

(5)

where

ω = A =

q

[keff − F ′ (De )]/m

q

x20 + (v0 /ω)2

φ = − arctan

v0 /ω x0

(6) (7)

!

(8)

are the frequency, the amplitude and the phase respectively. The amplitude is thus determined by the initial position x0 and velocity of the tip v0 , it is lower when the velocity v0 is low and the frequency ω is high. In practice, the frequency depends on the choice of keff . A low frequency increases the amplitude of the oscillation of D and thus o FS .

Step 2 (vb 6= 0) The distance between the tip and the sample is modified by changing the value of the spring + . During this step sites b are moved at constant velocity vb , therefore replacing constant to keff

L = L0 + vb t into the force balance, Eq. (2), leads to the equation that describes the motion of the tip [∂t2 + (ω + )2 ]x = C0 + C1 t

(9)

where x = D − De+ . De+ and ω + are the new tip-sample equilibrium distance and tip frequency, respectively, according to the modification of the effective spring constant. The + (L0 −De+ −l)]/m constants on the left-hand side of the Eq. (9) are given by C0 = [F (De+ )+keff + vb /m. Eq. (9) is equivalent to ∂t2 [∂t2 + (ω + )2 ]x = 0 which solution is and C1 = keff

x(t) = c0 + c1 t + c2 cos(ω + t) + c3 sin(ω + t) 11


(10)


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i.e. the sum of linear and oscillatory contributions. The constants of the linear contribution are obtained by replacing the solution Eq. 10 into Eq. 9, they are c0 = C0 /(ω + )2 and c1 = C1 /(ω + )2 . The constants of the oscillatory part are determined by the initial conditions + + + + of Step 2, x(0) = x+ 0 and v(0) = v0 , they are c2 = x0 − c0 and c3 = (v0 − c1 )/ω . Eq. 10

can be written as x(t) = c0 + c1 t + A+ cos(ω + t + φ+ )

(11)

with amplitude, frequency and phase

A+ =

c22 + c23

s

+ − F ′ (De+ ) keff m c3 = − arctan c2

ω+ = φ+

q

(12) (13) (14)

respectively. During Step 2 the tip velocity is given by v(t) = c1 − A+ ω + sin(ω + t + φ+ ) and is bounded by c1 ± A+ ω + . As Step 2 is applied to reach a new equilibrium it is important to predict the effect of keff and vb over the fluctuation of x and v, when sites b are stopped (a new Step 1). Replacing the values of c2 and c3 into Eq.(12) leads to A+ =

q

+ 2 2 + 2 (x+ 0 − c0 ) + (v0 − c1 ) /(ω )

(15)

+ it is simple to note that it has a minimum when x+ 0 = c0 and v0 = c1 , condition that defines

the following set of equations + ∆L keff = x+ 0 + keff − F ′ (De+ ) + vb keff = v0+ + keff − F ′ (De+ )

(16) (17)

where ∆L is the increase in L from one equilibrium position to another. Equations (16) and (17) suggest the following. On the one hand, a very stiff spring 12


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(keff >> F ′ ) facilitates the control of ∆D (x+ 0 ) through the choice of ∆L (which depends on the choice of vb ), but increases the oscillation of the tip reducing the sensitivity in the measurement of the force. The increase of the frequency, in turn, can affect the dynamics of the fluid in contact with the sample S2, indirectly affecting the measurement of the force. On the other hand, a very soft spring (keff → F ′ ) would facilitate the transfer of the tipto-sample approach speed vb to the tip requiring low speeds to avoid large oscillations. For a controlled operation these observations suggest the use of soft springs, taking care that keff > F ′ and that vb is sufficiently low.

Procedure for a refined F (D) curve According to the previous results a procedure for the refinement of an available force-distance curve controlling frequency ω and gap ∆D, is 1. Estimate an approximate curve F (D) as in Fig. 4 (see Section ). 2. Set a value of De , the equilibrium distance between the sample and the tip. 3. Set the effective spring constant equal to keff = mω 2 + F ′ (De ) in order to get the given wave frequency ω. F ′ is taken from the approximate curve F (D). keff is related to the spring constant of the simulation through k = keff /n, where n is the number of springs in the SP. 4. Run the first Step 1. 5. Calculate De+ = De + ∆D where ∆D is the gap between two data points in the forcedistance curve. + . Also set a value of vb to move sites b to a new 6. Run Step 2. Set a new value of keff

position L respect to the sample at which the distance S1-S2 is the equilibrium distance De+ . The latter is necessarily iterative in vb .

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7. Run the new Step 1 and return to point 5 above. Repeat the loop until the entire force vs. distance curve is complete. The simulation method described above is also helpful to explore the force-distance relationship in small regions around given equilibrium distances De . For this, it is enough to change the value of k (and thus the value of keff ) keeping all other parameters fixed. Next we illustrate the use of the procedure to refine force vs. distance curves by setting values of vb .

Refinement of the curve F (D) Figure 6 shows D vs t data for some of the points of the preliminary force-distance curve in Fig. 4. D fluctuates around a mean value in each case.

As an example, the transition

Figure 6: D versus t data for the F -D points s1 to s5 in Fig. 4. The spring constant keff = nk is shown for each data point.

from simulation s1 to s2 in Fig. 5 is shown in Fig. 7 with a red dotted line. According to 14


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Figure 7: D versus t for the transition from s1 to s2 . The forcing speed in the simulation of Step 2 is vb = −5 m/s. The dashed line indicates the value of De predicted for s2 .

the equilibrium condition, Eq. (3), for a given liquid bridge and springs, the equilibrium distance between the tip and the sample De in Step 1 fluctuates around a value that depends exclusively on the equilibrium separation L between sites b and the sample. The dashed line in Fig. 7 shows the value of De (2.14 nm) at which we expected to arrive in s2. The forcing speed used in the transition was vb = −5 m/s. Clearly, the tip is oscillating around a smaller value of D. The continuum solution can help to find the conditions for the tip to oscillate around the predefined value of De . But first we show the numerical solution of the continuum model for the transition s1 to s2 in Fig. 8. The solution clearly reproduces the simulation results in Fig. 7 but the interesting question is whether the continuum model can help to find a value of the forcing speed vb that causes the tip to oscillate around the predefined value of De . The velocity vb = −5 m/s used in Fig. 7 turned out to be very high and thus the tip reached a value of L very far from the equilibrium value corresponding to De . The procedure was repeated at lower velocities until the equilibrium value of L corresponding to De was reached. Figure 9 shows the continuum solution for several values of vb and clearly the solution for vb = −1 m/s corresponds to a wave oscillating around De = 2.14 nm. Then we repeated the simulations for the transition from s1 to s2 using the forcing speed vb = −1 m/s in Step

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Figure 8: Same as Fig. 7 but from the numerical solution of the continuum model. Parameters used D(0) = 2.6 nm and D′ (0) = 0, for F (D) the polynomial fit in Fig. 4 is used, other parameters keff = 0.9 N/m, L0 = 5.35 nm, l = 2.5 nm, m = 2.137 · 10−22 kg and vb = −5 m/s.

2. The result is shown in Fig. 10, the tip oscillates around De = 2.14 nm as expected. These simple examples predict the usefulness of the continuum solution in the measurement and prediction of AFM force curves between surfaces separated a few nanometers.

Figure 9: D versus t for the transition s1 to s2 with forcing speeds vb from -1 to -5 m/s used in the solution of the continuum model.

Force-distance curves The application of the procedure in Section to the determination of the force versus distance curve is illustrated in three different cases, with each case showing different aspects of the methodology. The cases are: (1) a water bridge between two substrates, (2) a Lennard-Jones bridge between two substrates, and (3) two substrates without a bridge. To focus only on 16


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Figure 10: D versus t for the transition s1 to s2 for two forcing speeds in the simulation of Step 2, that is, vb = −1 m/s and -5 m/s. the force exerted by the bridge, in the first two cases the interaction between the substrates is turned off. Two sizes of substrates and number of springs are used. System I: S1 and S2 are formed by 1800 atoms of Cu (MCu = 63.5 g/mol) and SP by 900 springs (Ma = 16 g/mol). Lateral dimensions of the substrates are Lx = Ly = 5.89 nm. Mass of the tip is m = 2.14 · 10−22 kg. System II: S1 and S2 are formed by 4232 atoms of Cu and SP by 576 springs. Lateral dimensions of the substrates are Lx = Ly = 8.85 nm. Mass of the tip is m = 4.62 · 10−22 kg.

Water bridge As a first application, we reconstruct the force versus distance curve of Fig. 4, we seek to have control over the spacing of the data and also over the associated error bars. The simulation time of each data point is defined as 2 ns during which 50 tip oscillations are expected, so the period is 40 ps and the frequency is ∼ 160 GHz. The spacing between each pair of contiguous data points is defined as ∼ 0.2 nm. For System I, the effective elastic constant of the springs is keff = 5.267 + F ′ [N/m] (from Eq. 6). Once the parameters are fixed, the liquid water bridge is stretched as much as possible without breaking it, at the resulting separation distance the slope F ′ is evaluated from Fig. 4, if the stretch is close to the bridge rupture then F ′ = 0 is a good guess, finally the elastic constant k is calculated as keff /n

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and the simulation is started. Applying Step 1 leads to the first point of the new force vs. distance curve. Step 2 is then applied with a tip velocity vb such that the predefined spacing is reached as close as possible. The repetition of the sequence of Steps 1 and 2 leads to the force versus distance curve shown in Fig. 11. The large fluctuations in the force observed in this figure are due to the high values of the elastic constant that result from the parameters chosen for System I. Although in System I it is possible to control the spacing of the data points and the fluctuations of the force, it is necessary to reduce the fluctuations associated with the force, for this we simulate System II. Parameters are: simulation time of each data point is 4 ns, period and corresponding tip oscillation frequency are 200 ps and 31 GHz respectively, and spacing between contiguous data points is ∼ 0.2 nm. The effective elastic constant of the springs for System II follows keff = 0.455 + F ′ [N / m]. The repetition of the sequence of Steps 1 and 2 leads to the force versus distance curve shown in Fig. 12, clearly the data point spacing and the force fluctuation are controlled. It is important to note that the force versus distance curves in Figs. 4, 11 and 12 are the same, except that the latter is better controlled through a better choice of the parameters used. This simple exercise shows two aspects that may be of interest, from the point of view of simulation there is a method that can be very useful in the determination of more complex force versus distance curves and from the practical point of view there are relations based on first principles to guide the best selection of parameters for the measurement of force curves.

Lennard-Jones bridge The simulation is complicated when the force curve shows jumps or oscillatory behavior due to underlying non-monotonic forces. In these cases, it may happen that after the simulation of a given data point with a tip with given elastic constant, the slope F ′ changes so sharply that it alters the oscillation frequency of the tip beyond the permitted limits (see Eq. 6). An example of a non-monotonous force curve is that exerted by a liquid argon bridge between two flat surfaces. Consider a bridge of 500 molecules of argon (LJ) that is simulated at 80 18


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Figure 11: Force-distance curve for a water bridge between two substrates at 300 K. System I is used, for which keff changes from 4.77 to 7.02 N/m.

Figure 12: Force-distance curve for a water bridge between two substrates at 300 K. System II is used, for which keff changes from 0.57 to 1.61 N/m.

K according to the configuration of System II. The interaction energy between argon and substrates is set to provide a contact angle of 50o , measured through the wetting phase, from a simulation of a drop of argon over the same substrate. Successive application of Steps 1 and 2 repeatedly leads to the force versus distance curve (red dots) shown in Fig. 13. The slope of the curve changes significantly between consecutive red points at separation distances less than 2 nm, with a minimum force at 1.7 nm which may suggest that the attraction of the substrates due to the argon bridge is maximum at that distance. However, it is known that liquid argon is organized in layers under confinement (see for example 35,49 ). In fact, density profiles of liquid argon confined at D = 0.9, 1.3, 1.7 and 2 nm show three, four, five and

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Figure 13: Red dots represent the force-distance curve for a liquid argon bridge between two substrates at 80 K. Black symbols represent the force when the elastic constant of the tip is slightly changed. Capillary bridge configurations A2 and B2 are somewhat more compressed than configurations A1 and B1 respectively. System II is used for tip and sample.

six layers respectively. The question that arises is whether the curve changes if the elastic constant of the tip changes once the system is dominated by layering. Here we try to answer by exploring the behavior of the force in the vicinity of some of the red points. We find that a better way to do that is to fix the sites b in the position leading to a red dot and then simulate again after introducing differential changes in the value of the elastic constant of the tip. In Fig. 13, black symbols reveal forces that are larger (smaller) than those of the 20


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red dots due to a differential increase (decrease) of the elastic constant (not greater than 20% between consecutive data points). According to the forces obtained, the three-layer configuration is very stable and exhibits an attractive force (point A1) larger than the force at D = 0.9 nm (red dot). The three-layer configuration even exhibits repulsion when it is compressed (point A2). The situation is similar when the configuration of liquid argon is in four, five and six layers. Density profiles for A1, A2 and B1, B2 suggest that the structure of the layers changes in some complex way between the extreme points. The full F -D curve (black and red symbols) in Fig. 13 is qualitatively very similar to the curves measured recently by Cheng and Robbins for a LJ fluid between parallels plates. 50 The force due to a water bridge with layered structure, for D < 1 nm, has been also detected by this method. 46 It should be clear that the black symbols in Fig. 13 are difficult to obtain because they require differential changes in the elastic constant and advancements and setbacks of the tip, in cases, less than a molecular diameter. Once this is possible experimentally, feedback with simulation would allow the study of systems that under confinement adopt complex configurations.

Conclusion Molecular dynamics allowed determination of the force (F ) between two substrates, tip and sample, mediated by a fluid at any separation distance (D) and during approach and retract cycles. Without further considerations, however, the results are force-distance curves with uncontrolled distance between the interacting surfaces and uncontrolled oscillations of the substrate acting as tip. In this work we found that the closed-form solution of a point-mass model describing the dynamics of the tip-sample interaction can precisely guide the choice of optimum measurement parameters both physical, for example tip elastic constant, and operational, for example tip-to-sample approach speed. The solution relates the resonance frequency of the point-mass system to the tip stiffness (keff ) and mass (m), and the local

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gradient of the force (F ′ (De )) through the equation ω 2 = (keff − F (De ))/m where De is the equilibrium tip-sample distance, and thus, for instance, given ω and a rough value of F ′ an optimum value of keff is anticipated. Also, for optimal control of the distance between substrates at which the force is to be determined, during approach, the tip speed must be kept low so that oscillation of the tip is small and controlled. The simulation method was applied to the interaction between two substrates in two cases of intervening fluid: a water bridge and a Lennard-Jones bridge. In the first case the method allowed a precise control of the spacing between data points and especially of the oscillation of the tip, in the second case the method showed that a correct choice of the elastic constant of the tip can reveal force curves and bridge underlying molecular structure that would not be possible with a non-optimal tip. In the latter application the method applied to a symmetric fluid (LJ) confined between two substrates clearly shows that when the liquid is layered a single value of the elastic constant reveals only a portion of the force-distance curve and that the determination of the complete curve requires different values of the elastic constant. Feedback between experiments and simulations, using the method presented here, should help to understand complex systems in a better way. Determination of complete force-distance curves between similar and dissimilar substrates with different intervening fluids is direct with the method in this paper.

Acknowledgement We thank Centro CRHIAM Project Conicyt/Fondap-15130015 for financial support. GEV thanks Centro CRHIAM for a postdoctoral fellowship position and CEMCC UFRO for computational support.

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Molecular Dynamics Simulation of Nanoforces between Substrates

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