Electrical Conductivity in Granular Media and Branly's Coherer: A

Electrical conductivity in granular media and Branly’s coherer: A simple experiment Eric Falcon, Bernard Castaing

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Eric Falcon, Bernard Castaing. Electrical conductivity in granular media and Branly’s coherer: A simple experiment. 2004. hal-00002394v1

HAL Id: hal-00002394 https://hal.archives-ouvertes.fr/hal-00002394v1 Preprint submitted on 29 Jul 2004 (v1), last revised 16 Nov 2004 (v2)

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The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. ccsd-00002394, version 1 - 29 Jul 2004 yee yREVT by Typeset ens-lyon.fr/eric.falcon/ ∗ spoue nisvcnt.Dsoee n19 yE. by 1890 in Discovered vicinity. action wave its this electromagnetic Branly, in an as produced ini- soon severa is The as by magnitude falls of [1]. irreversibly orders oxi- constraint resistance powder a slightly high under a tially powder in metallic appears dized which instability conduction lcrmgei ae seant eetypbihdin published of recently influence note Education the Physics a as (see well waves as media, electromagnetic granular as such non-homogeneous of media properties al- point, thermal starting also and properties historical electrical paper an the from This local explore granular). ones to the students a collective lows on of the disorder chain of focus typical a instead (e.g., to with grains) [8] between experiment (contacts beads model a metallic of of means being 6]. by each 5, duction [3, [4]; percolation “A-fritting” of as ef- process global Joule labelled a a also with by combined 7] microcontacts of [6, welding fect electrostati or local molecular [5], by effect grains forces tunnel of attraction modified [4], [3], oxide tion - grains metal on the layers through breakdown oxide electrical the invoked demonstrations; of been clear Several have any scale understood. contact without well the at not processes still possible are media metal- granular in phenomena lic transport electrical these 1900, near temporal [2]. occur ii) conditions that certain resistance voltage, under of threshold relaxations slow a and an fluctuations elec- from exceeds an transition as source observed con- a state trical i) conductive directly a as; to is such state source insulating powder the electrical with an nected when notably ena, lcrclcnutvt ngaua ei n rnyscoh Branly’s and media granular in conductivity Electrical mi drs:Ei.acneslo.r URL: [email protected]; address: Email h oee ffc r“rnyeet sa electrical an is effect” “Branly or effect coherer The eew e u oudrtn hstasto fcon- of transition this understand to out set we Here transmission radio wireless first the for used Although ASnmes 54.n 52.d 45.70.-n meta 05.20.Dd, of 05.45.Jn, made numbers: PACS detector wave radio a - transmission. effect radio illustrate coherer also Branly’s measurements microco of The the thermometer. of determination mi resistive imp implicit where a an grains critical obtain between to low microcontacts us transiti the allows a This of vicinity observed. At the is in state conductive media. a granular to insulating metallic of properties eso o ipelbrtr xeietcnb sdt exhi to used be can experiment laboratory simple a how show We .INTRODUCTION I. E 9) fe re eiwo h history the of review brief a After [9]). X tdistance at ∼ eiodco ea junc- metal - semiconductor 1 aoaor ePhysique, de Laboratoire eae oohrphenom- other to relates M 62 6 lédIai,6 0 yn France Lyon, 007 69 d’Italie, allée 46, 5672, UMR rcFalcon Eric http://perso. Dtd uy2,2004) 29, July (Dated: 1, ∗ cl oml u´rer eLyon, Supérieure de Normale Ecole ´ n enr Castaing Bernard and c l ann h odr h ale ihvleo h powder the of con- value tube restored. high the was earlier shaking resistance the Upon powder, the circuit. taining the closing open- and successively of by ing conductivity action powder the the considerable of a under increase observed waves. and powders forces, hertzian metallic electromotive on various the of experiments of behaviour performed discovery Calzecchi-Onesti [10], the T. later the years 1884, after convinced, 20 In not time some was long published Society were a Royal results the his conductiv- time, and in a the change at At sudden sparks its electric ity. by to important indicated sensitive the was as discovered tube distance voltaic a have such a to that with fact appears series Hughes in designed placed granules cell. first waves) metallic was acoustic with it detect since filled to microphone tube car- called two a was in with (which grooves E. in or con- D. resting blocks, loose rod 1879, bon carbon a In a for of phenomenon formed capacitor). tact similar former current a (the discharge observed jar Hughes a Leyden of a passage metal of the of from mixture a resulting of filings conductivity electrical the in- in permanent crease the 1835 Rosenschöld in af observed loop Munk away. wire meters a few across a arcing gap, induce small could a or- containing MHz) the 100 of that of waves der electromagnetic noticed frequency He (high waves. sparks and electromagnetic generation space of free by propagation demonstrated performed clearly Hertz experiments H. electromagnetism, of theory ial,orcnlsosaegvni e.IV. Sec. in respectively. given D are III conclusions and our C Finally, III Sec. conduction in qual- the mechanism a of transition by interpretation followed quantitative easily B, and III present Sec. be itative we in Then, can results laboratory. experimental A which physics III the standard Sec. here a in used in introduce built we setup II, experimental Sec. in the effect coherer the of hsdsoeywsatcptdb aypol;P S. P. people; many by anticipated was discovery This Maxwell’s of publication the after time short a 1887, In ncmsfo neetotemlcoupling electro-thermal an from comes on tc eprtr,wihi nlgu to analogous is which temperature, ntact 0-erodpolm h explanation the problem, 100-year-old a I RE ITRCLREVIEW HISTORICAL BRIEF A II. rwligocr.Teaprtsused apparatus The occurs. crowelding sdvlae rniinfo an from transition a voltage, osed lig sdfrtefis wireless first the for used fillings l i eti lcrcltransport electrical certain bit 1 rr ipeexperiment simple A erer: 2

The action of nearby electromagnetic waves on metal- ments along the same lines with 2 balls in contact [13]. lic powders was observed and extensively studied by E. However, in 1906, the invention by Lee de Forest of the Branly in 1890 [1]. When metallic filings are loosely ar- triode, the first vacuum tube known as audion, progres- ranged between 2 electrodes in a glass or ebonite tube, it sively supplanted the coherer as a receiver, and Branly’s has a very high initial resistance of many megohms due effect sank into oblivion without being fully elucidated. to an oxide layer likely present on the particle surfaces. Several decades later in the beginning of the 1960s, a When an electric spark was made at a distance, the resis- group in Lille (France) became interested again in this old tance was suddenly reduced to several ohms. This con- problem. They suggested a molecular relaxation arising ductive state remained until the tube was tapped restor- from the removal of applied electrostatic field that held ing the resistance to its earlier high value. Since the the particles in contact [5]. In the 1970s, numerous works electron was not known at this time (discovered in 1895- dealt with the conductivity of granular materials for bat- 96), Branly called his device a “radio conductor” to re- teries, but not with the transition of electrical conduction call that “the powder conductivity was settle under the [14]. In 1975, a group in Grenoble (France) suggested a influence of the electric radiations from the spark”; the mechanism of electrical breakdown of the oxide layer on meaning of the prefix “radio” being at this time “radi- the grain surfaces, and were interested in the associated ant” or “radiation”. He performed other experiments 1/f resistance noise [3]. In 1997, the transition of con- with various powders, lightly or tighly compressed pow- duction was observed by Vandembroucq et al. by direct ders, and also found the same effect occurred in the case visualization (with an infrared camera) of the conduc- of two metallic beads in contact, or in the case of two tion paths when a very high voltage (more than 500 V) slightly oxidized steel or copper wires laid across each is applied to a monolayer of aluminium beads [6]. More other with light pressure [11]. This loose or “imperfect” recently, a group in Liège (Belgium) has revisited the contact was found to be extremely sensitive to a distant action at a distance of sparks [7]. For additionnal infor- electric spark. mation about the coherer history, the reader is invited to This discovery caused a considerable stir as soon as see Refs. [15, 16]. O. Lodge in 1894 repeated and extended Hertz’s experiments by changing the wire loop by Branly’s tube, a much more sensistive detector. Lodge improved it into a III. ELECTRICAL CONDUCTIVITY OF A reliable, reproducible detector, and automated it by as- CHAIN OF METALLIC BEADS sociating a mechanical shock to the tube. Lodge called it a “coherer” from the Latin cohaerere synonymous with Understanding the transition of electrical conduction stick together. He said that the fillings were “cohered” through granular materials is a complicated many body under the action of the electromagnetic wave and needed problem which depends on a large number of parameters: to be “decohered” by a shock. Later, E. Branly and global properties concerning the grain assembly (i.e. sta- Lodge focused their fundamental research on mechanisms tistical distribution of shape, size and pressure) and local of powder conductivity, and not on practical applications properties at the contact scale of two grains (i.e. degree such as wireless communications. However, based on the of oxidization, surface state, roughness). Among the phe- coherer, and using it as a wave detector, the first wireless nomena proposed to explain the coherer effect, it is easy telegraphy communications were transmitted in 1895 by to show that some have only a secondary contribution. G. Marconi, and independently by A. S. Popov. Popov For instance, since coherer effect has been observed by also used the coherer as a lightning recorder to detect the Branly with a single contact between 2 grains [11], perco- atmospheric electrical dicharges at a distance. lation can not be the predominant mechanism. Moreover, Under the action of the voltages that are induce by when two beads in contact are connected in series with electromagnetic waves, T. Tommasina advanced in 1898 a battery, a coherer effect is observed at high enough the opinion that the metallic grains were welded together. imposed voltage [13], in a similar way as the action at According to Lodge, as a consequence of the electric field distance of a spark or an electromagnetic wave. In this of the wave, the filings became dipoles and attracted paper, we deliberately reduce the number of parameters, each other by electrostatic forces, inducing small mo- without loss of generality, by focusing on the electrical tions of grains to stick together, forming thus conductive transport within a chain of metallic beads directly con- chains [12]. A shock should be enough to break these nected to an electrical source. fragile chains, and to restore the resistance to its origi- nal value. Branly never believed this hypothesis, and to demonstrate that motion was not necessary, he immersed A. Experimental setup the particles in paraffin, wax, resin, or used a column of six hard steel balls or disks, few centimeters in diame- The experimental setup is sketched in Fig. 1. It con- ter. Since the coherer effect persisted, he thought that sists of a chain of 50 identical AISI 420 stainless-steel the properties of the dielectric between the grains played beads, each 8 mm in diameter, and 0.1µm in roughness. an important role, and was responsible for the “radio- The beads are surrounded by an insulating framework of conduction”. In 1900, Guthe et al. performed experi- polyvinylchloride (PVC). A static force F 500 N is ap- ≤ 3

Computer U, I, R when the current applied to the chain is reversed (arrows - + I 4 and 5). When repeating this symmetrical loop up to Force F diﬀerent values of Imax (solid and dashed arrows), and Displacement x for various applied forces F , one can show that the back

Stepper motor 1 N trajectories depend only on Imax, and follow the same

Force Force sensor 0 x back trajectory when expressed in the variables U as a F < 500 N PVC function of R0bI (see inset of Fig. 2). The values of R0b are determined by the slopes of the U–I back trajectories FIG. 1: Schematics of experimental setup. at low decreasing current (see Fig. 2). plied to the chain of beads by means of a stepper motor, 2 6 U and is measured with a static force sensor. The number 0 F = 56 N, I = 1 ; - 1 A of motor steps is measured with a counter to determine max F = 100 N, I = 0.5 ; - 1 A 4 max F =220 N, I = 1 ; - 1 A 3 the total deformation of the chain, x, necessary to reach max 3 this speciﬁc force. During a typical experiment, we chose 1 −6 2 to supply the current (10 I 1 A) and to simulta- R ~ kΩ - MΩ ≤ ≤ o R ~ few Ω neously measure the voltage U, and thus the resistance ob

R = U/I. Similar results have been found when repeat- 0 6 ing experiments with imposing the voltage and measuring 5 5

Voltage (V) 4 4 I and R. The bead number N between both electrodes -2 is varied from 1 to 41 by moving the “electrode beads” 3

5 Voltage (V) 2 within the chain. Note that the lowest resistance of the 4 -4 1 0 whole chain (few Ω) is always found to be much higher U 0 1 2 3 max 4 I * R (I ) (V ) than the electrode or the stainless steel bulk material. -U 0b max -6 0 -1 - 0.5 0 0.5 1 Current I (A) B. Experimental results FIG. 2: Symmetrical characteristics U–I of a chain of N = 13 −Imax ≤ I ≤ The mechanical contact of the bead chain is measured beads, for various current cycles in the range +Imax, and for various forces F . Inset shows the reversible in very good agreement with the non linear Hertz law R0b Umax ≡ R0b ∗Imax ≃ . 3/2 back trajectories rescaled by . 3 5 V. (given by linear elasticity), that is F x . This leads (See text for details). to an estimation of the typical ranges∝ of deformation between two beads from 2 to 20 µm, and of the apparent contact radius, A, from 40 to 200 µm, when F ranges This decrease of the resistance by several orders of from 10 to 500 N. magnitude (from R0 to R0b) has similar properties as The electrical behavior is much more amazing! Since that of the coherer effect with powders [1] or with a single no particular precautions has been taken for the beads, contact [11, 13]. Note that after each cycle in the current, the bead-bead contact is not metallic and an insulating the applied force is reduced to zero, and we roll the beads film (oxide and/or contaminant), few nanometer thick, along the chain axis to have new and fresh contact be- is probably present. When the applied current to the tween beads for the next cycle. With this methodology, chain increases, we observe a transition from an insulat- the resistance drop (the coherer effect or Branly effect) ing to a conductive state as shown in Fig. 2. At low and the saturation voltage are always observed and are applied current and fixed force, the voltage-current U–I very reproducible. characteristic is reversible and ohmic (arrow 1) of high This saturation voltage U0 is independent of the applied force, but depends on the number of beads, N, be- and constant resistance, R0. This resistance at low cur- 4 7 tween the electrodes. When varying N from 1 to 41, rent (R0 10 10 Ω) depends in a complex way on the ≃ − the saturation voltage per contact U U0/(N +1) is applied force and on the contaminant film properties (re- 0/c ≡ sistivity, thickness) at the contact location. The values of found constant and on the order of 0.4 V per contact. However, this saturation voltage slightly depends on the R0 are determined by the slopes of the U–I trajectories bead materials (U 0.4 V for stainless steel beads; at low increasing current. As I is increased enough, the 0/c ≃ resistance strongly decreases and reaches a constant bias U0/c 0.2 V for bronze beads; 0.3 V for brass beads), but is≃ of the same order of magnitude [8]. U0 (arrow 2). As soon as this saturation voltage U0 is reached, the resistance is irreversible upon decreasing the current (arrow 3): this back trajectory is reversible and C. Qualitative interpretation the resistance reached at low decreasing current, R0b (of the order of 1 - 10 Ω), depends on the maximum current previously applied, Imax, such as a memory effect. The Assume a mechanical contact between two metallic non linear reversible back trajectory is also symmetrical, spheres covered by a thin contaminant film ( few nm). ∼ 4

The interface generally consists of a dilute set of micro- Metal contacts due to the roughness of the bead surface at a Contaminant/oxide film (few nm) I specific scale [4]. The mean radius, a, of these micro- Metal contacts is of the order of magnitude of the bead rough- 2A ~ 100 µm ness 0.1 µm, which is much smaller than the apparent 2ai 2af Hertz∼ contact radius A 100 µm. Figure 3 schematically ∼ shows the building of the electrical contact by transfor- Electrical contact area metallic ( ) mation of this poorly conductive film. At low applied quasi-metallic ( ) current, the high value of the contact resistance (kΩ – + Mechanical contact area MΩ) probably comes from a complex conduction path + + Apparent contact area found by the electrons through the film within a very small size ( 0.1 µm) of each microcontact (see lightly R ~ kΩ - MΩ R ~ Ω ≪ Diameter grey zones in Fig. 3). The electron flow then damages 2a 2af the film, and leads to a “conductive channel”: the crowd- Increasing current ing of the current lines within these microcontacts gen- 2ai erates a thermal gradient in their vicinity, if significant Joule heat is produced. The mean radius of microcontacts then strongly increases by several orders of mag- FIG. 3: Schematic view of the electrical contact building nitude (e.g., from ai 0.1 µm to af 10 µm), and through microcontacts by transformation of the poorly con- thus enhances their conduction≪ (see Fig.∼ 3). This corre- ductive contaminant/oxide film. At low current I, the electri- sponds to a nonlinear behaviour (arrow 1 until 2 in Fig. cal contact is mostly driven by a complex conduction mechanism through this film via conductive channels (of areas in- 2). At high enough current, this electro-thermal process I I can reach the local welding of the microcontacts (arrow creasing with ). At high enough , an electro-thermal coupling generates a welding of the microcontacts leading to ef- 2 in Fig. 2); the film is thus “piercing” in a few areas ficient conductive metallic bridges (of constant areas). where purely metallic contacts (few Ω) are created (see black zones in Fig. 3). [Note that the current-conductive channels (bridges) are rather a mixture of metal with where L = π2k2/(3e2) = 2.45 10−8 V2/K2 is the the film material rather than a pure metal. It is proba- Lorentz constant, k the Boltzmann’s× constant, and e the ble that the coherer action results in only one bridge – electron charge. Stemming from the thermal equilibrium the contact resistance is lowered so much that punctur- and Ohm’s law, this relationship shows that the max- ing at other points is prevented]. The U–I characteristic imum temperature Tm reached at the contact is inde- is then reversible when decreasing and then increasing I pendent of the contact geometry and of the materials in (arrow 3 in Fig. 2). The reason is that the microcontacts contact! This is a consequence that both the electrical have been welded, and therefore their final size af does resistivity, ρ(T ), and thermal conductivity, λ(T ), are re- not vary any more with I < Imax. The U–I back tra- lated to the conduction electrons, leading thus that their jectory then depends only on the temperature reached respective dependance on the temperature, T , follows the in the metallic bridge through its parameters (electrical Wiedemann-Franz law [4] and thermal conductivities), and no longer on its size as previously. λρ = LT , (2) A voltage near 0.4 V across a contact thus leads from Eq. (1) to a contact temperature near 1050oC for a bulk o D. Quantitative interpretation temperature T0 = 20 C. This means that U 0.3 0.4 V leads to contact temperatures that exceed the≃ softening− Let us now check quantitatively the interpretation in or/and the melting point of most conductive materials. Sect. III C. Assume a microcontact between two clean Efficient conductive metallic bridges are therefore created metallic conductors (thermally insulated at uniform tem- by microwelding. Beyond the quantitative agreement perature T0, with no contaminant or tarnish film on their with the experimental saturation voltage U0/c (see Sect. surfaces). Such a clean microcontact is generally called IIIB and Fig. 2), Eq. (1) also explains why U0/c is the a “spot”. If an electrical current flowing through this relevant parameter in the experiments in Sect. III B, and spot is enough to produce Joule heating (assumed totally not the magnitude of the current. In addition, when U drained off by thermal conduction in the conductors), approaches U0/c on Fig. 2, the local heating of microcon- then a steady-state temperature distribution is quickly tacts is enough, from Eq. (1), to soft them. Then, their reached ( µs) in the contact vicinity. The maximum contact areas increase thus leading to a decrease of local ∼ temperature reached, Tm, is located at the contact, and resistances, and thus stabilizing the voltage, the contact is linked to the applied voltage, U, by [8] temperatures and the contact areas, since the current is fixed. The phenomenon is therefore self-regulated in U 2 voltage and temperature. T 2 T 2 = , (1) m − 0 4L Our quantitative model only describes the electrical 5 behavior of a welded contact (i.e. as soon as the satura- or on the metal used for the contact! However, in case tion voltage is reached, saying at I = Imax). It describes of alloys in contact, Eq. (7) depends on α the tempera- the reversible U–I back trajectory (when this contact ture coefficient of electrical resistivity of the alloy. This is cooled by decreasing the current, then eventually re- additional parameter is related to the defects in the ma- heated by increasing I). Here, the contact area is as- terial. The normalized U–I back trajectory (i.e., U as a sumed constant since it has been welded, and I < Imax. function of IR0b of the inset of Fig. 2) are compared in Let us recall the principles of the derivation for the an- Fig. 4 with the theoretical solutions [Eq. (7)] for the pure alytical expression of the nonlinear U–I back trajectory metals, and for a stainless steel alloy. A very good agree- [8]. ment is shown between the experimental results and the The thermal equilibrium means that the heat flux, electro-thermal theory, notably for the alloy case. Qual- λ(T )grad T , across the isothermal surfaces T is due to itatively, the alloy solution has a better curvature than the electrical power, ϕ(T )I, where ϕ(T ) is the potential the pure metal one. The agreement is quantitatively ex- −1 between the two isothermals on both sides of the contact. cellent (see Fig. 4) when choosing α = 4T0 instead of −1 Let us introduce the “cold” contact resistance R0b pre- 3.46T0 (the α value for AISI 304 stainless steel), since sented to a current low enough not to cause any apprecia- the α value for the bead material (AISI 420 stainless ble rise in the temperature at the contact [the conductor steel) is unknown, but should be close. During this ex- bulk being at the room temperature T0, with an electri- perimental back trajectory, the equilibrium temperature, cal resistivity ρ0 ρ(T0)]. The derivation of R0b involves Tm, on a microcontact is also deduced from Eq. (6) with the same equipotential≡ surfaces during a virtual change no adjustable parameter (see inset of Fig. 4). Therefore, ⋆ between the “cold” state (denoted with a star) ϕ , and when the saturation voltage is reached (U0 = 5.8 V), o the real “hot” state ϕ(T ). This means that the current Tm is close to 1050 C which is enough to soften or to density is invariant along an equipotential (which is also melt the microcontacts between the N = 13 beads of the ⋆ an isothermal) grad(ϕ )/ρ0 = grad(ϕ)/ρ(T ). With the chain. Our implicit measurement of the temperature is thermal equilibrium, this gives equivalent to the use of a resistive thermometer.

⋆ dϕ dϕ λ(T ) 6 = = dT . (3) U ρ0 ρ(T ) ϕ(T ) 0 Using the Ohm’s law and integrating Eq. (3) between the 5 isothermal surfaces T0 and Tm leads to [17] 4 IR Tm λ(T ) 0b =4 dT . (4) ρ0 ϕ(T ) T0 3 6 Z U

U (V) 0 The temperature dependance for the thermal λ(T ) and 5 4 electrical ρ(T ) conductivities for the material in contact 2

3 U (V) are given by Eq. (2) and 2 1 1 T (oC) m ρ(T ) ρ0[1 + α(T T0)] , (5) 0 ≡ − 0 300 600 900 1200 0 α being the temperature coeﬃcient of electrical resistiv- 0 1 2 3 U 4 I * R (V) max ity. Equations (2) and (5) then allow to explicit both 0b λ(T ) and ϕ(T ) in Eq. (4). The current IR0b through the contacts only depends on the temperature Tm (i.e. on FIG. 4: Comparison between experimental U–I back trajecto- U) ries of Fig. 2 (symbols), and theoretical curves from Eq. (7) for −1 the case of an alloy with stainless steel properties [α = 4T0 −1 − . T0 · · · α T0 −.− U 2 ( ) or 3 46 ( )], and for a pure metal [ = ( )]. 2 Inset shows the theoretical maximum temperature, Tm [Eq. Tm = T0 + 2 , (6) s 4L(N + 1) (6)], reached in one contact when the chain of N = 13 stainless steel beads is submitted to a voltage U. and ﬁnally reads [8]

√L θ0 cos θ IR0b = 2(N + 1) −1 dθ , α 0 [1 + (αT0) ]cos θ0 + cos θ Z (7) IV. CONCLUSIONS where θ0 arccos(T0/Tm). Note that only R0b depends on the contact≡ geometry, and its value is easily reached We have reported the observation of the electrical experimentally (see Sect. III B) transport within a chain of oxidized metallic beads un- −1 Here again, for pure metals (α = T0), Eq. (7) does der applied static force. A transition from an insulating not depend explicitly on the geometry of the contact, to a conductive state is observed as the applied current 6 is increased. The U–I characteristics are nonlinear, hys- termination of the microcontact temperature all through teretic, and saturate to a low voltage per contact ( 0.4 this reverse trajectory, with no adjustable parameter, as V). Electrical phenomena in granular materials related≃ an analog of a measurement with a resistive thermometer. to this conduction transition such as the “Branly effect” Moreover, one could attempt to assert that this process were previously interpreted in many different ways but may be easily witnessed through direct visualization (mi- without a clear demonstration. Here, with this simple croscope, infrared camera). But for that, it is clear that teaching experiment, we have shown that this transi- very powerful electrical source must be applied, far in ex- tion, triggered by the saturation voltage, comes from an cess of that necessary to produce true coherer phenomena electro-thermal coupling in the vicinity of the microcon- (see for instance Ref. [6]). tacts between each bead. The current flowing through these spots generates local heating which leads to an increase of their contact areas, and thus enhances their conduction. This current-induced temperature rise (up Acknowledgments to 1050oC) results in the microwelding of contacts (even for so low voltage as 0.4 V). Based on this self-regulated temperature mechanism, an analytical expression for the We thank D. Bouraya for the realization of the ex- nonlinear U–I back trajectory is derived, and is found in perimental setup, and Madame M. Tournon-Branly, the very good agreement with the data. It also allows the de- granddaughter of E. Branly, for discussions.

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