Theory of metal/metal multi-contact interfaces: implications of the coupling between the electrical and thermal transfer processes

Transcription

1 Theory of metal/metal multi-contact interfaces: implications of the coupling between the electrical and thermal transfer processes N. Foy 1,, R. Bouzerar 1, P. Dassonvalle, V. Bourny,3, J. Fortin,3, A. Rabdane 1 and B. Joncheere 1 1: Laboratoire de Physique des ystèmes Complexes, EA 4663, Université de Picardie J. Verne : EIEE, Ecole d ingénieurs, 14 Quai de la omme, Amiens 3: Laboratoire des Technologies Innovantes, EA 3899 Université de Picardie J. Verne International Conference on Mesoscopic and Condensed Matter Physics, Boston, UA June -4, 015 1

2 Outline What is an MCI? (Multi-Contact Interface) > Academic situations (experimental issues) and industrial applications > Complexity of MCIs Theory of static metallic MCIs > Approximations > One spot model: solutions of the electro-thermal problem and consequences (critical current ) > Two spots model: Matrix formalism Conclusion: Extension to many spots interfaces > Experimental evidence for the critical current > Constriction vs tunneling, Branly effect

3 AFM image of a steel bead surface 10 X7 mm What is an MCI? - Random surfaces (roughness) exhibit asperities with randomly distributed heights and diameters. (O. Durand-Drouhin/V. Bourny/R. Bouzerar) mm 100mm mm Mechanical spots (typical size mm) - The mechanical contact between hard solids under moderate loads consist of an MCI [1]: A set of contacting asperities (spot size typically a few mm) with density controlled by the compression force (typically a few 100 mm apart). For higher loading conditions, molecular contact (nanometric scale) should be considered. uch discrete interfaces are involved in many processes and applications: tribological, electrical and /or thermal transfer, sliding contact, - Complexity arising from the strong coupling between different processes with different length/time scales. 3

4 Compression F Complexity of MCIs under mechanical sollicitations: electrical and thermal transport features spots Bul 1 Tunneling through the oxide barrier - pot induced constriction effect of current/heat streamlines, resulting in electrical/thermal singular resistances located at the contact zones (Holm s constriction resistance []). Force dependent resistances Bul Current streamlines constriction effect spots Capacity (oxide film) tunneling - Electro-thermal coupling due to interdependent thermal and electrical transfer (Wiedemann-Franz law [3]): temperature dependent constriction resistance (Joule heating) and interface temperature controlled by the resistance. - Competition with electron tunneling through the interface: reduction of the overall electrical/thermal resistances. 4

5 Approximations - Neglecting shear stress component within the interface: normal load conditions (no creep of the contact area [4]) see opposite - Compression force treated as a control parameter: mechanical conditions influence thermal/ electrical transfer with no feedbac (separation of the mechanical degrees of freedom) see figure below - Low spot density (low compression force): independent spots condition and focus on constriction effect with a posteriori evaluation of the tunneling contribution U (Volt) 0,6 0,4 0, Theory of static metallic MCIs Voltage creep effect for a contact between two steel beads (V. Bourny/R. Bouzerar) F=4N Two steal beads 0, log 10 t Dependence of the linear resistance upon the compression force at increasing/decreasing current (V. Bourny/R. Bouzerar) 5

6 One spot model a Metallic medium (g,) (g,) - ingle spot idealized to a dis () with radius a, the upper and lower metallic pieces (electrical/thermal conductivity g/) corresponding to domains D+/D- - The current flowing through () being set to I, the potential and temperature distributions in D+/D- are solutions to the equations (resistance of interface R Σ ) div( g V ) I ( x) g T V R I ( x) Theory of static metallic MCIs Upper domain D+ lower domain D- V+, T+ V-, T- a where the distribution ( x) 0, x 0, x 3, D : ( x) d x 1 D Characteristic function of surface 6

7 Complete these equations with the Wiedemann-Franz law, T L V K 8 / g, / (Lorentz const.) Theory of static metallic MCIs Introducing the vector field built up from the electrical and thermal field (electrothermal coupling), we get: LI div e ( x) / T e V T RI L e L ( x) T In the limit of small amplitudes of the field e enough temperature, these equations simplify to: LI V T ( x), x D RI T T ( x) - smallness of L and high Charge localized on the spot (ingoing and outgoing current seen as opposit charges) Q LI 0 T Power heat source located at the spot (temp. T) 7

8 olutions to the simplified equations (T a : ambient temp.) IT ( x ') V x L d x 3 ( ) ' 4 D x x' Theory of static metallic MCIs R I T ( x ') 3 ( ) a ' 4 D x x' T x T d x - potential discontinuous on the spot (singular voltage) resulting in a localized resistance, but continuity of temperature: R ( V V ) / I LT / a x0 T T x T I I ( 0) a /(1 ( / C) ) - Critical current (new feature!) of a single spot reads: Holm s constriction resistance recovered! I a / L C a Fe Cu Al Pb teel 1 mm.4 A 11.4 A 6.9 A 1.0 A 0.76 A - pot resistance increasing with current (thermal effect): the spot acts as a localized phonon source with power driven by the current. - Infinite resistance at I c manifesting a transition to an insulating state. - Non linear V-I characteristic: LT V I I I I a a ( ) /(1 ( / C ) ) 8

9 Two spots model - traightforward extension of the constriction model to two spots L V ( I T ( x) I T ( x)) L T I T I T x x a a 1 1 ( ( ) ( )) I I I ; I / I R / R Approach to the problem through a matrix formalism (X) - Condition for thermal equilibrium of the -spots interface: eigenvalue problem for matrix B T T eq a 4 e. L T eq B e - ame dependence of the equilibrium temperature with total current I as for one spot but a critical current: I C ( d a L(4 d a a ) 1 a a 1 ) -Matrix associated with two spots Theory of static metallic MCIs a1 d - admits as an eigenvector, - Dilute limit a I / a I / a d 1 1 B I T T 1 I / a d I / a 1 1 e 1, 1 C 1 1 d a a I L (, ) a 9

10 - We built up a model of metal/metal MCIs accounting for the electrothermal coupling within such interfaces dominated by the constriction effect. - A worable -spots model (simplest one) evidences the influence of the spots distribution on the potential and temperature fields. Evidence for a specific matrix formalism for computation of the MCI properties such as the equilibrium temperature of the interface. - Prediction of a transition from metal to insulating state driven by the current (thermal effect localized at the MCI) through a critical current depending on spot radius and thermal conductivity of the material. Conclusion: extension to many spots interface I/V ( -1 ) 0,16 0,15 0,14 0,13 0,1 experiment 0,11 linear fitting, I/V=a+b.I a=0,1594 ± 3E-5 ( -1 ) b=-,75336 ± 0,0064 (A -1.V -1 ) 0,10 0,000 0,005 0,010 0,015 I (A ) Evidence for a critical current: Plot I/V vs I validating the prediction of the model (V. Bourny/R. Bouzerar) - Prediction of a non linear I-V characteristic of the interface: validated by experimental data (see figure) - The matrix formalism allows an extension to many spots situations and especially disordered interfaces (more realistic) consisting of randomly distributed spots: natural connection with random matrices theory. - Influence of the shear component of the stress field within the interface to be incorporated: possible correlation between friction forces and the electrical properties of the MCI. - Derivation of the continuum limit of the model (relevant for high compression forces) to explore the transition to the smooth Hertz contact (real contact area close to the apparent area). 10

11 - The starting model focuses on constriction effects: it gives an incomplete account for the usually observed I-V characteristic (see figure below) and especially the hysteretic behaviour (Branly effect) [5]. - The upward branch (increasing current/decreasing resistance) requires a supplementary process : electron tunneling through the interface is a liely candidate. - Competition between the compelling processes tunneling [6] and constriction might lead to new effects: interface electrical instabilities, - MCI endowed with these modified properties acts as an effective medium: tunneling and electron scattering by mechanical spots (phonon sources) resulting in the renormalization of the thermal/electrical properties Typical I-V characteristic for two steel beads under compression (V. Bourny/R. Bouzerar): Branly effect V (V) 1,0 0,8 0,6 0,4 0, R up =70 Conclusion: extension to many spots interface R down =6,4 experimentel data linear fitting N-L fitting V=R down.i+b.i 3 R down =6.415±0.005() b=158.35±0.73(a - ) 0,0 0,00 0,0 0,04 0,06 0,08 0,10 I (A) 11

12 References [1] T. Baumberger, C. Caroli, olid friction from stic-slip down to pinning and aging, Advances in Physics, vol.55, n 3-4. (006) [] R. Holm, Electric contacts: Theory and applications, 4 th edition, pringer-verlag Germany (000) [3] N.W. Aschroft, N.D. Mermin, olid tate Physics, aunders College Publishing, 1976 [4] A. Teaya, Ph.D. Thesis University of Picardie, France (01) [5]. Dorbolo, M. Ausloos, N. Vandwalle, Applied Physics Letters Vol.81 (00); E. Falcon, B. castaing, Bulletin de la FP 148, 9 (005) [6] A. Teaya, R. Bouzerar, V.Bourny, I. Teaya, World Journal of Engineering 8 () (011) 1

13 What is an MCI? Detailed Content > Academic situations (experimental issues) and industrial applications > Complexity of MCIs: How to handle multiple couplings? Mechanical spots, electrical and thermal properties Theory of static metallic MCIs > Approximations - Characteristic scales separation - Prevailing processes (constriction, tunneling) - High and low spot density limits and compression forces range > One spot model: solutions of the electro-thermal problem and consequences (critical current, ) > Two spots model: Matrix formalism. Condition for thermal equilibrium of the interface. Conclusion: Extension to many spots interfaces > Experimental evidence for the critical current > Constriction vs tunneling. Branly effect 13