Electrical conduction and Joule effect in one-dimensional chains of ...

Electrical conduction in chains of beadsFig. 6 Variation of voltage U as a function of current I . Chain of 5stainless beads (diameter 11 mm). Comparison between experimentaldata (×) and analytical model (−) with the adjusted parameters (Eq. 6).Decreasing current: α ↓ =+1.1×10 −3 and β ↓ = −6.9×10 −4 ;increasingcurrent α ↑ =+2.1 × 10 −3 and β ↑ =+7.0 × 10 −4of beads has a slightly higher resistance to the flow of electriccharge R lc↑ ≃ 2.1 (part DE, Figs. 2 and 6) thanforthe weak decreasing currents R lc↓ ≃ 2.0 (part CD, Fig. 6).This difference is mainly due to the thermal delays associatedto the relatively fast evolution of the electric current. Indeed,for a given small current in the decreasing phase (part CD),the bulk of the beads presents a lower temperature and thus alower measured voltage compared to those of the increasingphase of current (part DE). Thus the chain of beads has anelectromechanical behavior of hysteretic type for which thevoltage–current characteristics present a loop (Fig. 6). Thelarger is the loop, the greater is the thermal energy releasedin the chain and consequently the more the beads warm up.These phenomena are more pronounced for cycles of higherelectric currents, as shown later in Fig. 7.4 Modeling of the hysteretic behavior CD-DEIn the following, the measured hysteretic current–voltagecharacteristics (parts CD and DE) are compared with thoseresulting from an analytical model [7,8,12–14,24].The two beads in contact are considered as two homogeneousand isotropic metal conductors with large dimensionscompared to the contact radius. The electric currentgoes into and leaves by surface ends S 1 and S 2 (Fig. 3).The heat flux only runs out through these same surfaces.For side surfaces, the losses by conduction, convection orradiation are neglected. It should be noted that the thermalconductivities of the air (λ air ≃ 0.026 W m −1 K −1 )andthePVC tubes (λ PVC ≃ 0.16 W m −1 K −1 )aremuchsmallerthan thermal conductivity within the stainless steel (λ steel ≃15 W m −1 K −1 ).Equipotential surfaces in the whole bead are also isothermalsurfaces. The two networks of equipotential andFig. 7 Voltage U,currentI and power P = UI for a chain of 3 stainlessbeads (diameter 11 mm) submitted to four successive cycles ofcurrent. ↗ and ↖:increasingcurrents;↙ and ↘:decreasingcurrentsisotherm present values that depend on the boundary conditionsimposed on the potential and the temperature. Moreover,the equipotential network is independent of the heatflux. Taking into account the previous considerations, phenomenaare considered in a steady-state, and the equationof energy balance leads to the Kohlrausch relation. It providesthe following expression for the potential difference ubetween the contact surface and the surface at temperatureT :u 2 (T, T c ) = 2∫ T cTρ(θ)λ(θ) dθ, (3)where T c , λ and ρ are respectively the temperature of thecontact surface, the thermal conductivity and the electricalresistivity.The conservation equation of the charged particle flow1 −−→ρ(T 0 ) grad[vlc (T 0 )]=ρ(T 1 −−→) grad[u(T )] allows to obtain thepotential difference v lc (between the contact surface and surfaceS 0 )forweakelectricalcurrentswithnoheatflux:∫ T cλ(θ)v lc (T 0 , T c ) = ρ(T 0 ) dθ. (4)U(θ)T 0The dependencies of the electrical resistivity and thermalconductivity on the temperature are considered linear andrespectively of the form:ρ(T ) = ρ r {1 + α(T − T r )}[ m],λ(T ) = λ r {1 + β(T − T r )} [Wm −1 K −1 ], (5)where ρ r ≃ 0.810 −6 mandλ r ≃ 15 W m −1 K −1 arereference values of the electrical resistivity and thermal conductivityat temperature T r ≃ 293 K [22,25]. The values ofcoefficients α and β can be obtained by minimization of thedifference between the voltage–current (U −I ) model (Eq. 6)and the experimental data. It is necessary because of the lack123

P. Béquin, V. Tournatof existing data on the dependence of electrical resistivityand thermal conductivity on temperature for stainless steels[22,25,26], and in particular for AISI 440C (code used bythe American Iron and Steel Institute) or 1.4125 (Europeanmaterial number).For small electric currents, the voltage V lc (between surfacesS 1 and S 2 )andthecurrentI are observed to be linearlydependent such as V lc = R lc I ,whereR lc is determined bythe slope of the U–I plot at low current (Fig. 2).The substitution of the first two terms of the Taylor expansionof ρλ ≃ ρ r λ r {1 + (α + β)(T − T r )} into Eqs. 3 and 4,leads to the potential difference between electrodes given byHolm [7]:V lc (T 0 ) = R lc I = ρ 0 2(n + 1) √ ρ r λ rρ r (α + β) 3 2{ []W(T 0 )× α arctan1 + (α + β)(T 0 − T r )}+ β W(T 0 )√where W(T 0 ) = α+β U(T0 )2(n+1) √ and ρρ r λ 0 = ρ r {1 + α(T 0 − T r )}.rBy introducing the maximum values of the electric currentI MAX and of the voltage U MAX ,Eq.6 can be rewritten as[]W(Tα arctan0 )1+(α+β)(T 0 −T r )+ β W(T 0 )I = I MAX [] , (7)α arctan+ β W MAXW MAX1+(α+β)(T 0 −T r )√ α+β UMAX2(n+1) √ ρ r λ r.where W MAX =Equations 6 and 7 give the relation between the current Iand the potential difference U between n beads, with α and βtwo parameters to be adjusted. For weak currents I in Eq. 6,the linear dependence between current and voltage is foundby imposing |β| ≪ α. Forhighercurrents,thefollowingsimplifications can be used:– the temperature T r associated to the reference variablesis equal to the temperature of the beads T 0 when theycarry a weak current I ;– the material at the contact follows the Wiedemann–Franz’s law which states that the product of the thermalconductivity by the electrical resistivity of a metalis proportional to the temperature: ρ r λ r = LT 0 ,whereL = 2.45 × 10 −8 V 2 /K 2 is the Lorenz number;– consequence of the preceding assumption, the quantityα + β must be close to 1/T 0 .Under these assumptions, the relation between the current Iand the voltage U can be written in the simplified form [7,8]:I = V lc(T 0 )= 2(n + 1)T √0 LR lc R lc× {αT 0 arctan [U(T 0 )] + (1 − αT 0 ) U(T 0 )} . (8)(6)where U(T 0 ) = U(T 0)2(n+1)T 0√L.4.1 DiscussionTo identify the limitations of the previous models (Eqs. 6 and8), experimental data (U, I ) obtained in a chain ofthree beads (diameter 11 mm) carrying a current with cyclesof increasing and decreasing values, are now presented.Figure 7 shows how voltage U and power P vary withcurrent I .From the preceding paragraph it can be seen that both Eqs.6 and 8, providingarelationbetweenthecurrentI and thevoltage U,presentasimilarform:I = A arctan(BU)+CU.Parameters A, B and C can be adjusted to obtain the best fitwith the experimental data, as shown in Figs. 6 and 8.4.1.1 Simplified modelsBy identifying parameters A, B and C to the correspondingterms of the simplified relation between current and voltage(Eq. 8), R lc , α and T 0 can be determined. Table 1 givesthe evolution of the latter quantities for successive cycles ofelectric current. Values of R lc are identical to those determinedby the slope of the U–I plot at low current (see valuesof R lc listed in Table 3). The values of α are positive anddecrease slowly with the current I MAX . This effect is discussedlater in this article. The values (α ≃ 2 × 10 −3 [K −1 ])are close to the values (α ≃ 6 × 10 −3 [K −1 ])giveninRef.[12]forvarioussolid/solidcontactsbetweenmetals(Al,Ni,Fe) and (α ≃ 1.6 × 10 −3 [K −1 ])giveninRef.[22] forastainless steel AISI 420. Finally, an important feature of thesimplified model is the prediction of temperatures T 0 of thebulk of the beads with anomalously high values comparedto the expected values around 300 K. This discrepancy maybe attributed to the use of Wiedemann–Franz’s law whichhas been found to be inadequate for some materials (iron andnickel for instance [8]).However, it is interesting to note that an equivalent formof this simplified model is used by Falcon et al. [13,14](Eq.(8) in Ref. [14]) and it correctly models the electro-thermalphenomena in the chain of stainless steel beads (martensiticstainless steel AISI 420). In this model [13,14], the coefficientα is initially deduced from the relation α −1 = 3.46 T 0(for an austenitic stainless steel AISI 304—see Ref. [26] from[14]) and is then replaced by the relation α −1 = 4 T 0 in orderto obtain a better agreement between predictions and experimentaldata for martensitic stainless steel beads AISI 420used in the experimental system.Figure 8 compares the experimental data (obtained in ourchain of 3 martensitic stainless steel beads AISI 440C) andthe model [13,14] with α −1 = 1.9 T 0 .Thislastexpressionwith factor 1.9 isfoundtoprovidethebestagreement123

Electrical conduction in chains of beadsTable 2 Quantities corresponding to the experimental characteristicsU–I of Figs. 7 and 8—chain of 3 stainless beads (diameter 11 mm)1 2 3 4I MAX (A) 0.5 1 1.5 2α ↓ ∗ 10 −3 (K −1 ) 1.8 1.7 1.4 1.2α ↑ ∗ 10 −3 (K −1 ) 1.8 1.7 1.4 0.9T 0↓ (K) 295 315 365 440T 0↑ (K) 295 315 365 600Fig. 8 Voltage U as a function of current I for a chain of 3 stainlessbeads (diameter 11 mm) submitted to four successive cycles of current.Experimental data (multisymbol), analytical model (solid lines) correspondingto Eqs. 6 and 8. Parameters of the fit are given in Table 3.Analyticalmodel from Falcon et al. (Eq. (8) in Ref. [14]) with α −1 = 1.9 T 0(Table 2 gives the evolution of T 0 ): (dashed lines with dotted) increasingcurrents and (dashed lines) decreasing currentsbetween model and experiments for low electric currentsI MAX = 0.5 A.Fortheothercyclesofelectriccurrent,thetemperature T 0 is adjusted to give the best agreement betweenmodel and experiments. The results are gathered in Table 2.The values of α are sensibly smaller than those from modelEq. 8 and decrease a little faster with I MAX .Thismodelpredictsalso anomalously high temperatures T 0 .Figure8 showsthat with increasing I MAX ,themodel[13,14] deviatessignificantlyfrom the experimental cycles. Although the experimentalU–I characteristics can be well fitted by the modelsEqs. 6 and 8 (see Fig. 8), there exist disagreements on theobtained parameters from the both preceding models (seeparameter values in Table 1 and especially the temperatureT 0 ). This shows the failure of these models to describe correctlythe electro-thermo-mechanical behavior of a chain ofTable 1 Quantities corresponding to the experimental characteristicsU–I of Figs. 7 and 8—chain of 3 stainless beads (diameter 11 mm)1 2 3 4I MAX (A) 0.5 1 1.5 2R lc↓ () 1.09 0.58 0.43 0.35R lc↑ () 1.08 0.58 0.44 0.41α ↓ ∗ 10 −3 (K −1 ) 2.4 2.3 2.0 2.0α ↑ ∗ 10 −3 (K −1 ) 2.4 2.2 1.9 1.5T 0↓ (K) 543 601 702 968T 0↑ (K) 513 544 632 1071The parameters R lc , α (related to electrical resistivity) and T 0 are providedby the simplified model Eq. 8 using a nonlinear minimizationroutine. Quantities for increasing current are R lc↑ , α ↑ and T 0↑ ; quantitiesfor decreasing current are R lc↓ , α↓ and T 0↓The model of Falcon et al. [13,14] withα −1 = 1.9 T 0 is used. Quantitiesfor increasing current are α ↑ and T 0↑ ; quantities for decreasingcurrent are α↓ and T 0↓beads carrying various currents and submitted to a weak force∼ 1N.4.1.2 Extended modelUnder these conditions, the ‘extended’ model (Eq. 6) describedpreviously must be considered. Parameters A, B andCarenowidentifiedtothecorrespondingtermsoftherelationbetween current and voltage (Eq. 6). Using the resistanceR lc evaluated through the slope of the U–I plot at low currentand assuming α(T 0 − T r ) ≪ 1 (a realistic assumption),quantities α and β can be estimated. Table 3 gives the evolutionsof these quantities for cycles of electric current (seeFig. 7).For all the cycles of electric current, parameters α arepositive and β negative. The obtained values are close toα ≃ 6×10 −3 K −1 and 2×10 −4 ≤ β ≤ 12×10 −4 K −1 givenby reference [12] forvarioussolid/solidcontactsbetweenmetals (Al, Ni, Fe) and α ≃ 1.6 × 10 −3 K −1 given inTable 3 Quantities corresponding to the characteristics U–I of Fig. 7—chain of 3 stainless beads (diameter 11 mm)1 2 3 4I MAX (A) 0.5 1 1.5 2U MAX (V) 0.93 0.98 1.05 1.1R lc↓ () 1.09 0.58 0.43 0.36R lc↑ () 1.09 0.58 0.44 0.42α ↓ ∗ 10 −3 (K −1 ) 2.2 1.9 1.4 1.0α ↑ ∗ 10 −3 (K −1 ) 2.3 2.0 1.5 0.7β ↓ ∗ 10 −4 (K −1 ) −5.2 −5.0 −4.2 −4.8β ↑ ∗ 10 −4 (K −1 ) −4.3 −3.6 −2.7 −2.5R lc is obtained from the slope of the U–I curve at low current. Parametersα (connected to the electrical resistivity) and β (connected to thethermal conductivity) are provided by Eq. 6 using a nonlinear minimizationroutine. Quantities for increasing current are R lc↑ , α ↑ and β ↑ ;quantities for decreasing current are R lc↓ , α ↓ and β ↓123

P. Béquin, V. TournatRef. [22] forastainlesssteelAISI420.Thus,anincreasein temperature when large currents pass through the contactregion between beads leads to an increase in the electricalresistivity ρ and to a decrease in the thermal conductivityλ within the contacting material. Consequently a risein temperature increases resistivity—and Joule effect—anddecreases thermal conductivity—which reduces the abilityof the material to dissipate heat by thermal conduction. Inthe contact region between beads it is sufficient to cause asignificant heat accumulation. Moreover, it can be observedthat the saturation voltage U MAX increases with the maximumcurrent I MAX .Consequently,thetemperatureatthecontact surface increases and tends gradually towards themelting temperature of the contacting material. A softeningof the asperities is expected [13], and can lead to alocal micro-sintering. As a result, the electrical resistivityis found to decrease with current I MAX (α decreases) andthe thermal conductivity increases with current I MAX (|β|decreases).For a cycle of electric current, coefficients α ↑ (increasingcurrent) and α ↓ (decreasing current) present roughly equalvalues; in contrast coefficients β ↑ and β ↓ of thermal conductivityhave different values (|β ↓ | > |β ↑ |)thatmeasurementuncertainty cannot explain alone. In model Eq. 6, thedependencies of electrical resistivity and thermal conductivityon temperature are considered linear (Eq. 5) overthewhole range of temperature (293–1800 K). On the basis ofthe presented results, one improvement could be to seek moreelaborated relations for ρ(T ) and λ(T ). Thedependenciescan be expressed as a power series of temperature T withmore terms than used in models [12,27]. Furthermore, it isinteresting to note that a positive value of β ↑ for the increasingcurrents is deduced from Fig. 6. A thorough analysis ofthe experimental curve shows a point of inflection close toI = 0.44 A. This change of behavior could be ascribed toan initial softening of the contact surface when the currentgrows, and could partly explain the difference with the valuesin Table 3. The role of the latent heat in the possiblemicro-melting of the asperities is a highly complex phenomenonto model at this level of understanding. It has notbeen taken into account in the presented models: all the estimatedtemperatures remain significantly lower than the meltingpoint. Only softening of the contacts and micro-sinteringhave been invoked here to interpret the results. The studyof the micro-structural changes of the contact areas by othermeans could be profitable to guide the future improvementsof the models.In conclusion, these measurements show the diversity ofelectrical and thermal phenomena within the chains of beads.The simplified models based on the Wiedemann–Franz’slaw fail to describe the electro-thermo-mechanical phenomenain a chain of beads carrying currents under a small appliedcompression force ∼1 N.5ElectromagneticeffectsonachainofbeadsThe linear and nonlinear acoustic properties of granular materialsare strongly sensitive to the mechanical properties at thelevel of the contacts between beads [4]. For such studies, itcould be extremely convenient to be able to modify contactmechanical properties without any modification of the geometryof the medium. A controlled electromagnetic wave hasthe ability to produce these mechanical transformations withoutaffecting the remainder of the granular structure. In thiscontext, and as a complement to the previous part, this sectionreports an analysis of the electrical and thermal behaviors ofachainofbeadssubjectedtoastrongelectromagneticpulsecreated by a spark.This study can be viewed as a continuation of the workof Dorbolo et al. [21] whichwasdealingwiththeinfluenceof an electromagnetic wave on the electrical resistance ofthe contacts between two beads. In [21], the electromagneticperturbation is found to affect preferentially the contacts havinga resistance higher than a threshold value which dependsmainly on both the mechanical characteristics of the chain ofbeads and of the distance between the latter and the spark (theelectromagnetic source). Here, we carry out measurementswith a near field electromagnetic excitation. They confirmpartly and extend the observations of Ref. [21] madeinthefar field.5.1 Experimental set-upThe experimental set-up is presented in Fig. 9. Theelectromagneticwave is produced by a discharge across a spark gapas in Refs. [17,28].Experiments were performed using a “three-electrode system”in which the storage capacitor is kept charged at a voltage(from 0 to 20 kV) lower than the breakdown voltage ofthe gap in air at atmospheric conditions. A third electrodeis placed between the main electrodes to provide an initialionization of the air necessary to cause the spark breakdownmechanism.5.2 ResultsFigure 10 shows the voltage–current characteristics for acycle of current (0–1 A). During this cycle, 5 sparks are created(distance between the chain of beads and the electrodesr ≃ 120 mm) causing a voltage drop between the electrodesof the chain of beads. These voltage drops can be connectedto strong variations of the contact impedances. It is observedthat for increasing dc currents I ,theelectricbehaviorofthechain of beads is less and less disturbed by the electromagneticpulses.The electromagnetic radiation created by these sparksinduces an electric current I sp in the chain of beads which is123

Electrical conduction in chains of beadsFig. 11 Diameter of the contact d versus current I deduced from theellipsoidal model for the contact surface. Solid curve: model I U = d ρ —Eq. 2. Chain of 5 stainless beads (diameter 11 mm). During the cycleof current, 5 sparks were created; r ≃ 120 mm distance between thechain of bead and the sparkFig. 9 Schematic diagram of the experimental set-up to study the influenceof an electromagnetic wave pulse on the electromechanical behaviorof a chain of beads. A “three-electrode system” is used for the generationof the electromagnetic pulse. V T and V HT are the trigger voltageand the capacitor voltage. The gap distance is 10 mm and r is the distancebetween the axis of the discharge and the axis of the chain ofbeadsFig. 12 Induced current I sp versus distance r between the chain ofbeads and the spark. Values estimated at I ≃ 0.5 Aduringacycleofcurrent (0–0.5 A) on a chain of 5 stainless beads (diameter 11 mm).Dashed line corresponds to I sp ∝ 1/r 2.1Fig. 10 Voltage U as a function of current I . Chain of 5 stainless beads(diameter 11 mm). During the cycle of current, 5 sparks are producedand are indicated by the arrows. The distance between the chain ofbeads and the spark is r ≃ 120 mmadded in a quadratic way to the dc current I due to the fact thatthey are incoherent current sources. Using the model fromEq. 2 giving the diameter of the contact as a function of theadmittance I U = d ρ ,thecontactdiameterisplottedasafunctionof the current in Fig. 11.Thecurrentpulseinducedbytheelectromagnetic pulse contributes to the sudden rise of thecontact diameter between beads. After this sudden increase,the diameter remains constant as long as the dc current Ican flow without modifying the area of contact by softening[9–11,21].The value of the impulse current I sp can be estimated byidentification to the value of the dc current I necessary toobtain the same diameter of contact. With this method, theinduced current I sp for the first spark (data corresponding toFigs. 10 and 11)isestimatedtobearound0.35 A. In this currentcycle, for dc currents higher than 0.9 A, the presence ofinduced impulse currents does not further modify the contactgeometry between the beads. As was previously observed in[21], the Fig. 11 shows that the contacts having a small diameterand thus a high electrical resistance (U I = ρ d —Eq. 2)arestrongly affected by the electromagnetic perturbation.Figure 12 gives the current induced by the electromagneticpulse of a spark placed at a distance r from a chain of 5 beads.The values of this current are estimated using the methoddescribed previously. This figure shows that the induced currentdecreases monotonically as distance r increases. A firstanalysis at the distances r higher than 50 mm (distances123

P. Béquin, V. Tournatroughly equal or larger than the length of the chain of beads:5 × 11 mm) shows that the current induced by the spark isproportional to 1/r 2.1 (obtained through a linear regressionon log[I sp ]). These results are consistent with those obtainedin [21] whereaninducedcurrentproportionalto1/r m withm ≃ 1.2 fordistancesr varying between 0.1 and 2.2 mis observed. Note that the spatial dependence of monochromaticelectromagnetic wave amplitude is respectively proportionalto 1/r 2 in the near field and proportional to 1/r inthe far field.At this level, the lack of information on the electromagneticfield (amplitude, frequency spectrum and directivity)does not allow the development of a realistic model to becompared to experimental results. These particular electromagneticbehaviors as a function of distance (near field/farfield) will have to be taken into account in the models andexperiments dedicated to the electromechanical behavior ofagranularmediumwithlargedimensions.6ConclusionContinued interest in granular materials has stimulated newexperiments and models to characterize these complex structures.In this article the electrical and thermal behaviors of achain of beads carrying a dc and an impulse electric currenthave been investigated. This work provides theoretical andexperimental continuations to the studies of Refs. [13,14,21]mostly carried out in the context of the Branly effect.The passage of an electric current in a chain of beadsaffects the nature of the contact between beads by causingaheatingandasofteningofthemetalneartheinterface.These changes are consequent to the heat which is generatedin the constricted region of the current flow and lead to anincrease in the contact surface. We developed specifically anexperimental setup suited to the measurement of the electriccurrent and voltage for a chain of metallic beads. The analysisof the voltage–current characteristics of such a chainindicates that the diameter of the contact surface varies linearlywith electric current I .Subjectedtoacycliccurrent,the chain of beads presents an electromechanical behaviorof hysteretic type. The larger are the electric current and thenumber of beads the larger is the hysteretic effect (the loop)and the greater is the heat released in the chain.Steady states models derived on the assumption of equilibriumbetween Joule heating in the contact area andthermal dissipation by conduction within the material areproposed. The simplified models of Eq. 8 and the modeldeveloped in [13,14], all based on the Wiedemann–Franz’slaw (W–F) (which states that the product of the thermal conductivityand the electrical resistivity of a metal is proportionalto the temperature), are observed to be inadequate todescribe electromechanical conduction in chains of beadswith cyclic currents under a weak static compression force∼1N.AnextendedmodelEq.6 is then developed without therestrictions of the W–F’s law. The comparisons with experimentaldata provide the variations of parameters α and βcorresponding respectively to the linear evolution of the electricalresistivity and to the thermal conductivity (Eq. 5). Theseresults could benefit in the future from temperature measurementson the beads and from the use of various materials andbead diameters.The effect of an electromagnetic pulse producing a pulseof electric current through a chain of beads is finally studied.The electromagnetic wave pulse is created by an electric discharge(electric spark) placed near the chain of beads. Theexperimental results show that the current can cause appreciablechanges in the contact surface when the magnitude ofthe induced current is sufficient but also when the dccurrent remain weak. Evolution of the induced current withthe chain/spark distance in near field (100 mm) in Ref.[21] wherethe1/r dependence is observed.The control of the contact surface between beads usingasuitableelectromagneticwavecouldfindapplicationsinparticular in the study of the fundamental processes involvedin the acoustic propagation through granular media [29–32].Acknowledgments We would like to thank James Blondeau, StéphaneLebon and Eric Egon for their assistance in the experiments andtheir technical advice. We particularly thank Jean-Marie Genet for hisexperimental work.References1. Duran, J.: Sands, Powders and Grains. Springer-Verlag, NewYork (1999)2. Kakalios, J.: Granular physics or nonlinear dynamics in a sandbox.Am. J. Phys. 73(1), 8–22 (2005)3. Nesterenko, V.F.: Dynamics of Heterogeneous Materials. Springer-Verlag, New York (2001)4. 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