6.3.1. The modified Multibond (mMB) simulations
In order to demonstrate the ability of an improved algorithm to produce reasonable and experimentally-supported friction results, a MATLAB program based on the MB model was developed, with modifications. This algorithm and the program are referred to as
the modified Multibond (mMB) model. Figure 6.1 illustrates the dynamic system that it simulates numerically. This can be represented with the Langevin equation:
mX¨ = n X i=1 kbond(xi−X)−ηbond ˙ X−v−kcantX−ηcantX˙ +ζp2ηcantkBT cant+ζ p 2ηbondkBT bond (6.8)
where kB is Boltzmann’s constant, is a random Gaussian noise sample with µ = 0 and
σ= 1,nis the number of presently active interactions andζis the noise multiplier parameter which changes the average amplitude of the mechanical noise. For a single interaction i.e.,
n= 1, the dynamics are almost identical to those addressed by Reimann et al. [4, 132], who was the first and only investigator to identify and explore the Fokker-Planck connection. Unlike Reimann’s work however, in the mMB algorithm 1) the noise amplitude is not always set to that described by the fluctuation-dissipation theorem and is a tunable parameter ζ, 2) there can be multiple interactions instead of one, and 3) the interactions are linear springs whereas Reimann employed a sinusoidal potential in line with several PTT studies [33, 131, 22, 124, 133, 5, 4, 132, 6, 23, 122].
As noted, the interactions are represented by linear springs; this differs from the periodic potential employed in PTT studies [33, 131, 22, 124, 133, 5, 4, 132, 6, 23, 122] but is consistent with MBa simulations [16, 15, 17, 125]. The simulations methodology can be modified to incorporate different potentials, including a periodic potential (see SI section 8.3.3). The dampers, ηbond and ηcant, represent dissipative phonon modes in the substrate
and cantilever, respectively, that occur in real experiments.
In the mMB simulation algorithm the fourth-order Runge-Kutta method was used. At each time step, ˙X is given an additional
ζΓµ= 0, σ=p2ηbondkBT
+ζΓµ= 0, σ=p2ηcantkBT
Here, Γ is a normal distribution with mean µ and standard deviation σ. The procedure for determining when an interaction is formed is identical to the original MBa algorithm detailed previously. A bond-forming activation energy of 1.5×10−20J is used to match the previous work [16, 15, 17, 125] and can be tuned as needed. However, for comparison with the FP theoretical framework proposed in this article (which may be improved upon to incorporaten >1 as well as the effect of having a rate of forming the interaction), there is no effect of forming an interaction in the FP theoretical framework as it is only concerned with first passage times and not with what happens after the interaction is broken. The breaking condition occurs when the interaction reaches the critical stretch lengthXc, which
is set to two angstroms. This critical stretch length can be tuned for different materials and interactions.
Though the idea of an attempt frequency only has palpable context in an Arrhenius analysis, which is purposefully being avoided due to its mean-field nature, the effective attempt frequency can arbitrarily but heuristically be taken as 0.16/∆t where ∆t is the simulation time step and the factor of 0.16 is chosen as the tail of the first standard deviation. For the mMB time steps, which range from of 5×10−9to 5×10−10s−1, the effective attempt frequency is on the order of tens or hundreds of MHz, at least four orders of magnitude smaller than the attempt frequencies discussed in most prior PTT analyses.
6.3.2. Fokker-Planck (FP) PDE numerical solutions
The Fokker-Planck equation is relevant when the force from damping is much greater than inertiai.e.,
β ˙ X m ¨ X
(recall thatβrepresents a general damping coefficient). For
most cases involving AFM friction, this is true. Otherwise the speed-dependence of friction would result from inertial effects, which by assumption is not the case.
For the case where n= 1, and
(ηbond+ηcant) ˙X m ¨ X , equation 6.8 reduces to
and the Fokker-Planck equation representing the mMB simulations with these same condi- tions is
∂P(X, t|X0, t0)
∂t =∇ ·
∇D−kbond(vt−X) +ηbondv−kcantX
ηbond+ηcant
P(X, t|X0, t0). (6.11)
together with the moving boundary conditions P(X=vt−Xc, t) = P(X=Xc−vt, t) =
0.
To compare the Fokker-Planck equation to the mMB results, equation 6.11 was solved us- ing MATLAB’s partial-differential equation solverpdepe.mwith the probability at the edges of the spatial domain set to equal zero at all times. The initial normal distribution of mass positions P(X,0|0,0) is given a standard deviation of 0.01 ˚A to approximate a delta func- tion. After each iteration of thepdepe.m solver, the puller position is moved ∆x=v∆t(∆t
is the time step) and the boundary conditionsP(X =vt−Xc, t) =P(X=Xc−vt, t) = 0
are enforced by redefining the spatial domain to only include the region meeting the condi- tion vt−Xc≥X ≥vt+Xc,i.e., an intact interaction. Figure 6.2 shows several snapshots
in time of the Fokker-Planck solver algorithm.
Code Availability Codes for the mMB and FP numerical algorithms can be found on GitHub.com [134, 135].
6.4. Simulation results of the modified Multibond model of nanoscale friction and