Modelling of frictional and adhesive contacts at silicon/polymer interface for nanotechnological applications

Model

According to molecular mechanical theory proposed in detail by Tabor and Kragielski,5^–7 the friction force between two interacting surfaces consists of two components: deformational and adhesive one. The first (mechanical) component is related to the necessity of mutual deformations of the mating surfaces surface layer's asperities. The adhesive component is, however, caused by the molecular bonds between two contacting surfaces.

In the process of friction between two surfaces, particularly polymeric ones, the adhesive component of the total friction force is significant and cannot be neglected. Decreasing the surface roughness leads to the bigger value of the adhesive component, and in contrary the increased roughness effects that the mechanical component is becoming more important. The exact magnitudes of both components depend on several factors, mostly external load, surface roughness and waviness, mechanical properties, adhesive properties, etc.6,7

An important issue in the description of the tribological behaviour of the frictional contacts is the determination of the real contact area A_r, which is a sum of the elementary areas of the asperities of the surfaces being in contact. The real contact area is the zone where direct force interaction occurs and hence friction force and wear are highly dependent on its actual value.7 On the other hand, the real contact area depends mainly on the surface roughness and is a result of the surface asperities deformation and penetration under the load.

Simulation input data

Most of the input data for modelling must be obtained from the real samples provided and identified during additional testing. For each sample there is a need to use an atomic force microscopy (AFM) to identify the topography of the surface on nanoscale, as well as other parameters like Young modulus E_a (can be obtained from the nanoindentation), Poisson's ratio ν and the surface free energy γ (estimated with the use of the dedicated wettability tester). These parameters (properties) of the contacting materials/components are most important in frictional interactions in the contact area.

There are also other important parameters, e.g. intermolecular distance ϵ (assumed to be ∼2 nm) and Winkler's layer thickness h (assumed as 2000 nm). These parameters of the sample/countersample interface and two simulation experiment variables, external load N and standstill time t_p, have been taken into consideration in the model of contact and consecutively varied in the following simulations.

For polymers their time dependent, rheological behaviour must be additionally taken into account and consequently additional model parameters are determined: Young modulus E_b and E_c and retardance times t_ret1 and t_ret2. The methodology of the experimental tests to identify these parameters are described in Ref. 8.

The Voigt–Kelvin model of the second degree was chosen (Fig. 1), as it can describe viscoelastic polymer behaviour in the time domain providing reasonable accuracy. According to the model, all three values of Young modulus and two retardance time values can be recalculated into one time dependent Young modulus of the material The time dependent elastic modulus E(t) is calculated as

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Figure 1

Voigt–Kelvin model applied for description of rheological behaviour of used polymeric material

Modelling procedure

At the first stage, sample and countersample topographies are put virtually together (Fig. 2). Contact area is under constant, external load N and contacting surfaces are separated by the distance d. Topography height z_i is identified by AFM as the distance to the reference plane situated at the lowest point of the topography image.

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Figure 2

Model of contact with Winkler's layers

Tribological interaction modelling requires additional assumption to simplify the model. Elastic deformation of the surface is represented as the deformation of many, individual Winkler's springs with the elementary surface area S_i and height h, referenced later as Winkler layer height. No deformation is expected below the Winkler's layer.

In order to simplify the simulation process, initial contact interface of the both, rough samples is changed into one rough sample interfacing flat, rigid surface (Fig. 3). Owing to this, the model input data must be recalculated. The surface deformation process is characterised by parameters related to physical mechanical behaviour of the samples. The numbers in the bottom indexes indicate first and second sample (countersample) respectively In the areas, where local surface topography is greater than the distance between surfaces d, the mechanical deformation occurs, whereas in sample-to-plane proximity areas surface deformation can be observed as the result of the activity of adhesive forces.

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Figure 3

Model of rough surface with reference plane

Conforming to the model similarity rule, the elementary force P_wi opposed to elastic deformation of the surface can be described as where Δz_i = d−z_i and k = (1+ν)(1−2ν)/(1−ν²).

The exact description of interacting surfaces separated by small nanometric size gap cannot be described properly without the intermolecular forces which should be taken into account. The molecular force F(l_i) is a derivate of potential energy described with the Lenard–Jones potential and can be calculated with the use of the following equation In the assumed contact the above equation describes the molecular interaction of the surfaces on the elementary area S_i which depends directly on the measured topography discretisation step (AFM topography measurement single step).

The force equilibrium requires that all forces sum up to zero We can calculate the surface deformation Δz_i and then the topography parameter describing the deformation z Finally, adding together the non-contact molecular pulling forces F_s (adhesion forces) and the contact deformation forces P over all topography points and for certain surfaces distance will give us The adhesion and mechanical forces for given samples are not equal for any distance d not being equilibrium distance d₀. Therefore, equilibrium equation is as follows The external force N is experiment parameter, therefore we can calculate the equilibrium distance between the surfaces d₀. Knowing equilibrium distance d₀, the real contact area can be determined as a sum of all elementary areas where we can observe mechanical contact where n are the points with the distance z_i′<d₀.

The relative real contact area A_r can be also specified in per cent as a ratio of topography real contact points A_r to the nominal (contour) area of contact A_n

Calculation of friction forces

As the result of the previous equations, for every topography contact point, the adhesion and mechanical forces (adhesive and mechanical components of the friction force) can be calculated for every equilibrium distance.

Based on the knowledge which surfaces points are in the contact and which are under non-contact interactions, two above mentioned components (contact and non-contact) of the friction force can be therefore well identified.

When two surfaces in the contact, bonded by adhesion forces and under the influence of lateral force are sheared, every singular Winkler's layer (in the points of interaction) is sheared as well.

The lateral force work W_iad can be calculated as The energy needed to break the contact can be determined as van der Waals forces acting over the distance from l_ia to ∞, where l_ia is the slid between two surfaces (l_ia = d−z′) The energy E_n is smaller than the lateral force work W_iad, therefore assuming that friction process does not generate any wear products, lateral force needed to destroy adhesive bonding Q_i can be calculated as where Δx is the elementary step at the AFM surface topography measurements.

Adding together Q_iad forces for all non-contact interacting points gives finally non-contact friction force component T_ad In order to find contact component of the friction force, the breaking situation for every contact point must be analysed. Knowing contact pressure P_i for every singular area S_i, the distance l_im can be calculated, where molecular force F(l_i) equals to this contact pressure The work of the lateral force W_im must equalise molecular interaction energy of both contacting materials molecules for the change of their distance from l_im to ϵ and cause adhesion contact to be broken for l_ia = ϵ, i.e. the molecules distance changes from ϵ to ∞.

It can be written as and therefore Finally the lateral force needed to rapture the mechanical bonding is Adding together Q_im forces for all contact interacting points gives finally the contact (mechanical) friction force component T_m The total friction force needed to initiate the movement of the contacting surfaces on the calculated area can be found, therefore

Macroscale contact results

The proposed model and simulation algorithm can be used for both macroscale polymers as well as here in particular for nanotechnological contacts, e.g. to estimate the frictional/adhesive interactions in the mould/imprinted material interface in NIL fabrication processes. The polymeric components from the first group of macroscale sliding contacts usually indicate much higher surface roughness. When they are put into contact, initial, real surface of contact is relatively small (<20%) and increases highly with the increased contact pressure (Fig. 4).

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Figure 4

Real area of contact dependence on contact pressure (external load) for macroscale samples. Simulation size is 1×1 mm. Standstill time t_p = 10 s. Both samples identical (material: polyamide 6; E_a = 1800 MPa, ν = 0·35, γ = 72 mJ m⁻², E_b = 1470 MPa, E_c = 1850 MPa, t_ret1 = 9·9 s, t_ret2 = 65 s, R_a = 9·3 nm)

Friction force depends evidently on the contact pressure but the relationship between contact pressure and relative area of contact is not linear. Relatively strong dependence is observed for small loads and much lesser for high loads (Fig. 5).

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Figure 5

Friction force dependence on contact pressure (external load) for macroscale samples. Simulation area size is 1×1 mm (area of contact). Standstill time t_p = 10 s. Both samples identical (material: polyamide 6; E_a = 1800 MPa, ν = 0·35, γ = 72 mJ m⁻², E_b = 1470 MPa, E_c = 1850 MPa, t_ret1 = 9·9 s, t_ret2 = 65 s, R_a = 9·3 nm)

The significant friction force decrease can be observed with the increase of the contact pressure (Fig. 6).

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Figure 6

Friction coefficient dependence on contact pressure for macroscale samples. Simulation area size: 1×1 mm. Standstill time t_p = 10 s. Both samples identical (material: polyamide 6; E_a = 1800 MPa, ν = 0·35, γ = 72 mJ m⁻², E_b = 1470 MPa, E_c = 1850 MPa, t_ret1 = 9·9 s, t_ret2 = 65 s, R_a = 9·3 nm)

Nanoscale contact results

The sliding contacts used directly in NIL fabrication process (silicon mould/polymeric thin film interface) show totally different behaviour. The high flatness of the samples causes very fine fit of the couple at initial contact and therefore huge superiority of the adhesion friction component over the mechanical one. In such a case, pulling forces generated in the interface are usually much higher than external load applied, making it negligible. The typical friction coefficient values calculated according the rules for the macrocontacts can easily exceeds the value of 10. Friction force dependence on contact pressure cannot be seen and can be considered as constant (Fig. 7).

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Figure 7

Friction force dependence on contact pressure (external load). Simulation area size: 1×1 mm. Standstill time t_p = 10 s. Samples: silicon mould (E_a = 169000 MPa, ν = 0·27, γ = 1200 mJ m⁻², R_a = 3·2 nm) versus PMMA imprint (E_a = 550 MPa, ν = 0·4, γ = 40 mJ m⁻², E_b = 1470 MPa, E_c = 1850 MPa, t_ret1 = 9·9 s, t_ret2 = 65 s, R_a = 3·2 nm)

The real area of contact is close to 100% at the initial contact and does not change much with the increasing contact pressure (external load) (Fig. 8).

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Figure 8

Real area of contact dependence on contact pressure (load). Simulation area size: 1×1 mm. Standstill time t_p = 10 s. Samples: silicon mould (E_a = 169000 MPa, ν = 0·27, γ = 1200 mJ m⁻², R_a = 3·2 nm) versus PMMA imprint (E_a = 550 MPa, ν = 0·4, γ = 40 mJ m⁻², E_b = 1470 MPa, E_c = 1850 MPa, t_ret1 = 9·9 s, t_ret2 = 65 s, R_a = 3·2 nm)

There is also very small difference at all in friction for the samples with very small roughness unless it hits a certain value (3 nm for R_a parameter in the presented case) (Fig. 9). At that point the friction force is reduced significantly (>50% in the analysed case). There is also an increase in the friction force expected when the roughness increases further, but it was not possible to observe this phenomenon with studied sample materials.

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Figure 9

Total friction force change (narrow bars) as a function of the sample roughness (wide bars); both samples of the contacting components used in NIL process come from the same process and have the same roughness R_a. Simulation area size: 1×1 mm. Standstill time t_p = 10 s. Samples of each contact: silicon mould (E_a = 169000 MPa, ν = 0·27, γ = 1200 mJ m⁻²) versus PMMA imprint (E_a = 550 MPa, ν = 0·4, γ = 40 mJ m⁻², E_b = 1470 MPa, E_c = 1850 MPa, t_ret1 = 9·9 s, t_ret2 = 65 s)