The effect of mutual angular misalignment in the quantized sliding of solid lubricants

Transcription

1 Facoltà di Scienze e Tecnologie Laurea Triennale in Fisica The effect of mutual angular misalignment in the quantized sliding of solid lubricants Relatore: Prof. Nicola Manini Correlatore: Prof. Rosario Capozza Alessandro Culatti Matricola n A.A. 2014/2015 Codice PACS: p

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3 The effect of mutual angular misalignment in the quantized sliding of solid lubricants Alessandro Culatti Dipartimento di Fisica, Università degli Studi di Milano, Via Celoria 16, Milano, Italia 11/12/2015 Abstract Recent studies showed that under specific circumstances the motion of a lubricant solid layer between two sliding crystalline surfaces exhibits a quantized sliding motion where the velocity of the center of mass of the lubricant is a fixed fraction of the externally fixed velocity of the top solid surface. This quantized movement, due to the forward dragging of solitons, or mismatch density waves, remains unaffected under wide changes of the parameters of the system. The studies made on this topic considered one-, two- or three-dimensional simulations, but they focused mostly on aligned layers. This work uses computer simulations to investigate the same quantized sliding motion simulation of mutually misaligned layers of atoms. Advisor: Prof. Nicola Manini Co advisor: Prof. Rosario Capozza 3

4 Contents 1 Introduction 5 2 The model The interaction The geometry Results Overall rotation Discussion and conclusion 15 Bibliography 19 4

5 1 Introduction Previous work [1] [2] using a one-dimensional model showed the existence of quantized sliding motion of a lubricant layer between two layers of solids. These two solids are modeled by a static and a moving sinusoidal potential, with two different spacings, and the lubricant atoms interact with each others via harmonics potentials. These works [1] [2] showed that the velocity ratio between the velocity of the top sinusoidal potential and the lubricant center of mass remains constant for large changes of the system parameters, and depends on the commensurability ratio between the top and bottom spacings. Subsequent work [3] generalized these findings to 1+1-dimensions. More realistic three-dimensional models [4] showed the existence of similar velocity plateaus. These models usually take rigid triangular lattice for the two sliding solids. The initial condition for the lubricant layer is a triangular lattice as well, but it is not rigid and the atoms composing the layer can move one with respect to another. These models usually use the Lennard-Jones interaction as two body potential for the interaction between the particles. These models were mostly used to investigate aligned lattices. Recent works about mutual rotations [5] [6] investigated if misalignment is compatible with quantized motion, and on the dependence on the angle of mutual rotation of several quantities, such as the velocity ratio or the friction coefficient. In this work we investigate the existence of the velocity plateau in systems where the three layers are mutually rotated. We will consider three different natural orientation of the external velocity: the basis vector of the three lattices. We will firstly consider the top layer velocity aligned with one of the bottom layer basis vector, and we will then analyze the alignment with the vector of the top layer and of the initial conditions of the lubricant layer. 2 The model 2.1 The interaction We model the lubricant by a layer of point-like classical particles. These particles are confined by two solid surfaces each represented by a triangular lattice of atoms. The reciprocal position of the particles composing the top and the bottom layers are fixed, while the particles of the lubricant can move freely according to the classical equations of motion. We take the two-body Lennard-Jones (LJ) interaction as model for the interaction between the particles, using the expression: 5

6 [ (σ ) 12 ( σ ) ] 6 φ LJ (r) = ε 2, (1) r r which has a minimum of depth ε at distance σ. This interaction has an unlimited range of action, but for computational reasons we need to introduce a cutoff radius R c and neglect all interaction between particles which are more distant than R c. A simple cut to the LJ would create problems because when a particle crosses the cutoff radios its energy would jump. In order to solve this problem we shift the LJ potential by φ LJ (R c ), thus eliminating this discontinuity: { φlj (r) φ LJ (R c ) r R c φ(r) =. (2) 0 r > R c We adopt a cutoff radius R c = 2.5 σ, so that: [ ( ) 12 ( ) ] 6 σ σ φ LJ (R c ) = ε 2 = ε. (3) A lubricant particle moves according to the equations: m r i = N l j=1,j i r c r c r i φ ll ( r i r j ) j=1 N b j=1 r i φ lb ( r i r bj ) N t φ lt ( r i r tj ) + f r damping i, i where r i, r b and r t are the positions of the lubricant, bottom and top particles, N l, N b and N t are the numbers of lubricant, bottom and top particles and φ ll, φ lb and φ lt are the truncated LJ lubricant-lubricant, lubricant-bottom and lubricant-top interactions, each characterized by its own parameters. We adopt a frame of reference where the bottom layer is not moving. In this frame of reference we force the top layer to move at a fixed velocity in the ˆx direction, while it can move on the ŷ and ẑ axes due to the forces between its own atoms and the lubricant particles. The force required to maintain the top layer velocity compensate the friction force exerted by the lubricant layer at each instant of time: N l N t F frict = φ lt ( r i r tj ). (5) r txj i=1 j=1 This force times the velocity integrated in time gives the work it makes per unit time, i.e. the power dissipated by friction forces: W frict = T 0 (4) T F frict v ext dt = v ext F frict dt = T v ext Ffrict. (6) 6 0

7 On the ŷ and ẑ axes the top layer evolves according to appropriate equation of motion: N l N t N t mÿ top i = φ lt ( r i r tj ), (7) N t m z top i i=1 y t j=1 j N l N t = i=1 z t j=1 j φ lt ( r i r tj ). (8) These equations are the same for all the atoms in the top layer, so we have only one solution, as all top atoms move rigidly as a whole. To remove the heat produced by friction we add a damping force to the equation of motion of the lubricant particles. In a real experiment the atoms would dissipate heat by creating phonons that propagate into both the top and the bottom sliders. As this mechanism is absent in our rigid-layer model, to represent it we use a symmetric damping: f damping j = η r j η( r j r t ), (9) where we initially used the standard underdamped value η = 0.05, but we observed that this small dissipations allows wide fluctuations in the lubricant velocity for certain values of the external driving velocity. In order to reduce this effect we increased the value to η = 0.1. The lubricant, bottom and top atoms are generally three different kinds of atoms, characterized by three different lattice spacing and whose Lennard- Jones interactions have different equilibrium distance. The σ pp describing the interaction between lubricants particles is taken 0.83% larger than the equilibrium lattice spacing, to take into account the non-nearest neighbor interactions. This gives us σ pp = 0.855a b. For σ tp and σ bp we initially took the values given by the Lorentz-Berthelot mixing rule: σ ij = σ ii + σ jj 2, (10) where for σ tp and σ bp one should assume lengths of the order of a t and a b respectively. However with these values a significant fraction of the lubricant atoms would be pulled outside the layer, thus breaking the layered model entirely. We then increased them and used the values σ tp = 3 and σ bp = 1.1. Even if the atoms are different, for simplicity and to avoid proliferations of parameters, we assume the same interaction energy ε for all the Lennard-Jones potentials. We use this interaction energy as the model energy unit. We also use the lattice spacing of the bottom layer as length unit and the mass of the particles as mass unit. These three quantities provide a system of natural units 7

8 Physical quantity Natural units Typical value length a b 0.2 nm mass m 50 a.m.u. = kg energy ε 1 ev = J time m 1 2 ε 1 2 a b s = 140 fs velocity ε 1 2 m m/s viscous dissipation coefficient ε 1 2 a 1 b m kg/s Table 1: Natural units and typical values for several physical quantities. of this system; all the results are implicitly expressed in terms of these units. To obtain the actual value of the simulated result one should then multiply it by the typical value of the corresponding natural unit, listed in Table The geometry Both the bottom and the top layers are rigid triangular lattices. The initial condition for the lubricant lattice is a triangular lattice as well, see Figure 1, but this layer is not rigid and during the simulation the atoms can move one with respect to another. The primitive vectors of the bottom and lubricant lattices are: a b1 = a b (1, 0) ( 1 a b2 = a b, ) 3 2 2, (11) a l1 = a l (cos θ, sin θ) ( ( ) ( )) a l2 = a l cos θ + π 3, sin θ + π 3 where θ is the angle of rotation between the lattices. The fourth equation can be written as: ( 1 ( a l2 = a l cos θ ) 3 sin θ, 1 ( sin θ + 3 cos θ) ). (12) 2 2 The number of atoms a computer is capable of handling is much smaller than the number of atoms in a realistic experiment. Simulations using a small number of particles would then be determined by edge effects. To mitigate this problem we introduce lateral (2D) periodic boundary conditions in our model. This means that each particle in the simulated cell represents an infinite array of replicated particles in the other cells, with position shifted by k 1 a 1 + k 2 a 2, where a 1 and 8

9 Figure 1: A part of the supercell used in the simulations: bottom layer (red), lubricant layer (light blue) and top layer (dark blue). a 2 are the primitive vectors of the periodic supercell and k 1 and k 2 are integer numbers. In our simulations the three lattices are mutually rotated. We must select the appropriate values of the lattice spacing ratios ρ b = a b /a p and ρ t = a t /a p and mutual rotation angles to obtain a correctly periodic supercell. Focus on the bottom and the lubricant layers, for the moment. We have taken the bottom lattice spacing a b as unit distance, and we take one of the primitive vector of the bottom layer aligned along the ˆx axes. We consider a condition where the lubricant layer has a smaller lattice spacing a l and its primitive vectors are rotated with respect to the vectors of the bottom layer. To generate an overall periodicity these primitive vectors must satisfy: n 1 a b1 + n 2 a b2 = m 1 a l1 + m 2 a l2, (13) where n 1, n 2, m 1 and m 2 are appropriate integer numbers. The supercell must then of course be selected to be compatible with the common vector of Eq.( 13).We can now use the components of the vectors in Eq.( 13) to form a linear system for the variables x = cos θ and y = sin θ: { (2m1 + m 2 )x m 2 3y = (2n1 + n 2 )ρ m 2 3x + (2m1 + m 2 )y = n 2 3ρ. (14) 9

10 The solutions of this system are: x = m 1(2n 1 + n 2 ) + m 2 (n 1 + 2n 2 ) ρ m m 1 m 2 + m y = 3 m 1n 2 m 2 n 1 m m 1 m 2 + m 2 2 (15) ρ 2, (16) We can then calculate ρ by imposing the condition x 2 + y 2 = 1. We find: m m 1 m 2 + m 2 2 ρ =, (17) n n 1 n 2 + n 2 2 and we can use tan θ = y/x to obtain: ( ) 3 m 1 n 2 m 2 n 1 θ = arctan. (18) m 1 (2n 1 + n 2 ) + m 2 (n 1 + 2n 2 ) Equations ( 17) and ( 18) are simpler if we take n 1 = n 2 = n: ρ = 1 m m 1 m 2 + m 2 2, (19) n 3 θ = arctan ( 3 3 m 1 m 2 m 1 + m 2 ). (20) In our simulation we take n 1 = n 2 = 20, m 1 = 20 and m 2 = 27. This gives us a misalignment angle θ = rad = and lattice-spacing ratio ρ = To increase the probability of observing solitons dragging, we aligned the top layer with the Moiré pattern generated by the bottom and the lubricant layers. The rotation angle ψ of this pattern, according to [7], satisfies: cos θ = sin2 ψ ρ + cos ψ 1 sin2 ψ ρ 2. (21) This gives us ψ = rad = 30. The top lattice spacing, according to [6], is: a l a t = 1 + ρ 2 2ρ 1 cos θ. (22) This gives us a t =

11 v top y v cm x / v ext v cm y / v ext time Figure 2: Periodic time dependence in the typical transient. Centermass velocity of the (a) lubricant, ˆx component, (b) lubricant, ŷ component and (c) top layer, ŷ component. In this simulation the ˆx component of the top layer velocity is v ext = and η = Dashed lines: the time-average values. 3 Results In our simulations the lubricant center of mass usually starts moving with an apparently chaotic behavior for several hundreds of time units. After this initial transient it may reach a steady state where the center of mass velocity fluctuates periodically. Figure 2 shows an example of this transient. Figure 3 shows the fluctuations of the velocities in the periodic state. Alternatively the sliding system can end up in a non periodic steady state where the lubricant velocity show no recognizable pattern. In such situation the top layer acquires a nonzero lateral drift velocity, see Figure 4. Figure 5 shows a few successive snapshots of a periodic steady motion. The purpose of these simulations was to identify a quantized velocity plateau. Within this plateau the ratio between the average velocity of the lubricant center of mass and the velocity of top layer remains constant. To discover a range of velocities where this constant ratio may occur, we start off with an external velocity v ext = 0.01 and let the system reach a steady state. We then increased and decreased the top layer velocity in small steps and we let the system reach the new steady state. 11

12 v top y v cm x / v ext v cm y /v ext time Figure 3: Periodic time dependence in the steady state. Center-mass velocity of the (a) lubricant, ˆx component, (b) lubricant, ŷ component and (c) top layer, ŷ component. In this simulation the ˆx component of the top layer velocity is v ext = and η = Dashed lines: the time-average values. While increasing v ext we find that the lubricant reaches a periodic steady state only as long as the external velocity is smaller than a certain velocity v c. This critical velocity is v c = for the simulations with damping η = 0.05 and v c = for η = 0.1. For velocities larger than v c, the steady state is non periodic and the ratio between the average ˆx component velocity of the lubricant center of mass and the top layer velocity deviates from the ratio of the quantized velocity plateau. In this plateau state the average ŷ component velocity of top layer vanishes. In contrast, in a non-periodic state the transverse top velocity is nonzero and fluctuates non periodically, similarly to the lubricant center of mass. We also explored the lower boundary of the velocity plateau. We find that the velocity ratio remains constant only for a small range of decreased top layer velocity. Indeed, for v ext = (η = 0.1) or v ext = 0.06 (η = 0.05) we could never identify a periodic steady state, see Figure 6. Also here, in the non periodic state the top layer acquires a nonzero transverse velocity component. The resulting small velocity quantized plateau is shown in Figure 6. We investigate whether a frozen transverse top layer motion could be beneficial to the quantized velocity plateau. We then fix to zero the top layer velocity 12

13 v cm x / v ext v top y v cm y / v ext time Figure 4: Non periodic time dependence in the steady state. Centermass velocity of the (a) lubricant, x component, (b) lubricant, y component and (c) top layer, y component. In this simulation the x component of the top layer velocity is vext = and η = 0.1. Figure 5: Subsequent snapshots of the quantized motion at vext = 0.01 and η = 0.05, obtained after the initial transient, at a time interval t = 40. Note the horizontal advancement of the top layer, and the dragging of the underlying soliton pattern. The white lubricant atom helps detecting lubricant motion. 13

14 η = 0.1 η = v cm / v ext v ext Figure 6: The velocity plateau. along ŷ. We repeat the simulations with η = 0.01 obtaining the results summarized in Figure 7. We find no lower limit to the velocity plateau down to velocities as small as v ext = Overall rotation All the simulations done so far were made with the external velocity aligned with one of the primitive vectors of the bottom layer. We repeat these simulations with different alignments of the layers with the external velocity. This is equivalent to dragging the top layer along different directions. Specifically we consider the external dragging velocity aligned parallel to two other natural directions: one primitive vector of the lubricant or alternatively of the top lattice. With these alignments follow the same protocol, starting with an external velocity v ext = In these simulations we find, after a transient, a steady state similar to those we had with the original alignment. In the rotated layers we find a similar velocity ratio as the one we obtained previously. We then increase and decrease the external velocity to identify the velocity plateau boundaries: we find a periodic motion for a few values of the top layer velocity. Figures 8 and 9 show the velocity ratios we evaluated for these two alignments. With these alignments the plateau where the velocity ratio remains constant is extremely small. Like in the bottom aligned simulations we fix the top layer transverse velocity: Figure 10 and Figure 11 show the results of these calculations. As a results 14

15 0.165 v cm / v ext v ext Figure 7: The velocity plateau, when the ŷ-component of the top layer velocity is fixed to zero. In these simulations η = 0.1. of fixing the top layer lateral motion, we obtain again broad plateaus. 4 Discussion and conclusion The main results of the present thesis are: The quantized plateau state does indeed exist for mutually rotated layers. The dynamic of the quantized state is consistently periodic. This state involves a constant ratio of the longitudinal lubricant average velocity components to the imposed top speed. The lubricant transverse velocity is generally nonzero. In contrast, the top layer velocity has a null average transverse component in the quantized sliding state. The termination of the quantized sliding state is often associated to the top layer acquiring a lateral component of its motion. By constraining the top layer to move in a straight line (no transverse motion) the quantized plateau state is significantly extended. 15

16 v cm / v ext v ext Figure 8: The velocity ratios in the simulations where the external velocity is aligned with a basis vector of the lubricant lattice. Due to time limits, we have not found yet the end of the velocity plateau for increasing velocity. More calculation to investigate the edge of those plateaus are currently being run. 16

17 v cm / v ext v ext Figure 9: The velocity ratios in the simulations where the external velocity is aligned with a basis vector of the top lattice v cm / v ext v ext Figure 10: The velocity ratios in the simulations where the external velocity is aligned with a basis vector of the lubricant lattice. 17

18 v cm / v ext v ext Figure 11: The velocity ratios in the simulations where the external velocity is aligned with a basis vector of the top lattice. 18

19 References [1] A. Vanossi, N. Manini, G. Divitini, G. E. Santoro, and E. Tosatti, Phys. Rev. Lett. 97, (2006). [2] A. Vanossi, N. Manini, F. Caruso, G. E. Santoro, and E. Tosatti, Phys. Rev. Lett. 99, (2007). [3] I. E. Castelli, N. Manini, R. Capozza, A. Vanossi, G. E. Santoro and E. Tosatti, J. Phys.: Condens. Matter 20, (2008). [4] A. Vigentini, B. Van Hattem, E. Diato, P. Ponzellini, T. Meledina, A. Vanossi, G. Santoro, E. Tosatti and N. Manini, Phys. Rev. B 89, (2014). [5] D. Mandelli, A. Vanossi, M. Invernizzi, S. Paronuzzi, N. Manini, and E. Tosatti, Phys. Rev. B 92, (2015). [6] D. Mandelli, A. Vanossi, N. Manini, and E. Tosatti, Phys. Rev. Lett. 114, (2015). [7] F.Grey and J. Bohr, Europhys. Lett. 18, 717 (1992). 19