influence of surface topology on the electrical response of many bead assemblies

Influence of surface topology on the electrical response of many bead assemblies A Tekaya, R Bouzerar, and V Bourny Citation: AIP Advances 2, 032108 (2012); doi: 10.1063/1.4738952 View online: http://dx.doi.org/10.1063/1.4738952 View Table of Contents: http://aip.scitation.org/toc/adv/2/3 Published by the American Institute of Physics AIP ADVANCES 2, 032108 (2012) Influence of surface topology on the electrical response of many bead assemblies A Tekaya,1,2 R Bouzerar,1 and V Bourny2 Physique des Syst`emes Complexes - Universit´e de Picardie - UFR Sciences, 33 Rue Saint –Leu 80039 Amiens, France Laboratoire des Technologies Innovantes (LTI) EA3899, 02100 St-Quentin, France (Received 21 February 2012; accepted July 2012; published online 17 July 2012) We propose an interpretation of the voltage creep observed in metallic grain assemblies based on the sensitivity of the electrical properties to the surface topology of the beads From the real area of contact between adjacent beads, directly measured through the bulk resistance of a single bead, we derive a simple relation between the voltage creep rate and the creep rate of the interface friction coefficient, regarded as an aging process expressing asperity creep The likely influence of the Branly effect on the aging process is briefly discussed Copyright 2012 Author(s) This article is distributed under a Creative Commons Attribution 3.0 Unported License [http://dx.doi.org/10.1063/1.4738952] A granular material is a complex medium made of a large number of deformable particles where the prevailing features are the contact interactions between grains and/or the grain deformations The complexity of such assemblies relies not only on the number of particles but also on the topology and geometry of the contact network This network is responsible for the rich variety of behaviors of granular media such as its flowing properties, e.g avalanches, manifesting the complex (strongly dissipative!) dynamics of the network or its electrical properties influenced by both individual contacts and some ‘organized’ states subtended by any percolative structure Granular media are thus influenced by the contact network, itself being controlled by their simplest component, the contact between adjacent grains The interplay between individual effects (isolated contacts) and collective ones (network dependent) is of central importance To separate these effects, the study of an individual contact is required This fundamental step provides information about the nature of the interface between two grains as well as the key to understand the influence of the single contact on the collective properties of complex assemblies That interface consists mainly of the surface chemical composition and the complex surface geometry Due to air exposure of metallic beads, the surface is covered with a thin oxide layer responsible for the high electrical resistance (at low currents) usually reported Upon increasing current, this high initial resistance is decreased down to several orders of magnitude: this is the Branly effect.1 The most spectacular signature of that irreversible process consists in well known voltage hysteresis loops2 exhibiting, at high enough currents, voltage saturation effects.3 Several mechanisms of the resistance reduction have been pointed up among which an electrical breakdown scenario,4 electron tunneling through the oxide layers and the voids between grains5 or even a current-induced welding of the beads in very tiny contact zones.3, The latter mechanism is suggested by the mechanical contact between the rough surfaces asperities of neighboring grains resulting in many contact points Such a multiple contacts interface is involved not only in the electrical features of the contact but controls many features such as the friction force between grains or mechanical ones as the shear strength of the interface.6 Though electrical and roughness related properties are of different nature, we evidence a basic connection between them arising from the sensitivity of electrical measurements to both the surface topology of the beads and the nanojunctions due to the mechanical contact of rough surfaces More precisely, the voltage creep effect is shown to be driven by the multicontact interface aging 2158-3226/2012/2(3)/032108/8 2, 032108-1 C Author(s) 2012 032108-2 Tekaya, Bouzerar, and Bourny AIP Advances 2, 032108 (2012) FIG Current-Voltage characteristics obtained on a system of 16000 beads with a compression force F = 41 N The different colors correspond to the successive cycles The initial state revealed by the first loop exhibits a very high resistance, up to 17 k , and a plateau at high current, suppressed after a few loops The low-current resistances (ohmic) after several loops are strongly decreased (lowered down to 50 after a few loops) Notice the difference between the backward characteristics (low dispersion) and the upward characteristics (higher dispersion) Our electrical studies consist mainly of two experimental configurations dedicated to I-V characteristics and voltage relaxation under mechanical compression To clarify likely influence of the dimensionality, geometry or even complexity on the observed processes, these studies were carried out on many types of bead assemblies such as 1D channels from two up to a few beads and a 3D disordered assembly composed of a single layer (400 beads / layer) up to 45 layers corresponding to a maximal number of 18000 beads (diameter mm) For each configuration we used copper electrodes of appropriate size In the last case, the electrodes are copper discs with diameter 40 mm The I-V characteristics were determined through a classical method: the upward characteristic is picked up by increasing the current from zero up to a maximal value, the voltage being recorded at each step without delay The current is then decreased through the same steps allowing to pick up the voltages in the same conditions Prior to any measurement cycle, the assemblies were shaken Owing to that procedure, the chemical surface states are swept out The unavoidable variability of the surface states induces a chemical disorder arising mainly from the nanometric oxide layer with random composition This results in a fluctuating component of the contact resistance Though we are not interested here in the absolute values of the resistance but rather on global features of the I-V characteristics, we have checked this effect on a separate setup recording the electrical resistance between two mobile electrodes in contact with a single steel bead: their varying relative position reproduces the effect of the chemical disorder as a randomly fluctuating resistance In spite of the ‘chemical’ noise, the global shape of the I-V characteristics and their main features are not affected Anyway, that noise can be cancelled out by averaging over a great number of I-V characteristics (averaging over chemical disorder) The absence of delay (no waiting time) is a precaution imposed by the voltage creep effect evidenced through the study of the time evolution of the voltage across our bead assemblies The electrical relaxation was studied at different currents applied through a simple protocol consisting in applying a constant current, recording the time varying voltage and shaking the assembly prior to the next step Typical voltage – current loops obtained on 3D assemblies are presented on Fig The most striking features of the upward characteristic regard the highly resistive initial state of the 3D assembly with a resistance about 17 k reflecting the large number of beads, and the voltage ‘peak’ above which the voltage decreases towards a ‘plateau’ This negative resistance behavior is observed only for the first measurement cycles and low compression forces: at high enough compressive 032108-3 Tekaya, Bouzerar, and Bourny AIP Advances 2, 032108 (2012) FIG Selection of a series of voltage relaxation curves obtained on a 4000 bead system under a compression force of 100 N and input current I = A The slope of these curves dV/dLnt is reproducible while the initial state is slightly variable, reflecting the randomly varying surface state forces or after a few measurement cycles the hysteresis loop adopts a more conventional shape The suppression of the negative resistance regime under mechanical compression indicates a likely change of the contact network topology and a suppression of the bead-bead interface resistance The non-symmetrical voltage dispersion in the upward and backward characteristics is likely due to Joule heating Indeed, the voltage across the assembly depends on both the injected current and the temperature distribution within the bead contact network manifesting as a random resistor network The temperature field then fluctuates accordingly, exhibiting a higher dispersion at increasing current (increasing temperature) and a reduced one when current is decreased The relaxation dynamics of our assemblies is evidenced on Fig for an assembly of 4000 steel beads distributed over 10 layers under a compressive force F=100 N These curves were obtained for different measurement cycles and exhibit no significative difference It consists in a voltage creep observed in all types of assemblies, whatever their complexity More precisely the time evolution of the voltage across the sample obeys a well defined logarithmic law:7 V (t) V0 (1 + B Ln(1 + t/τ )), (1) where V0 is the initial voltage, τ a characteristic time scale of the phenomenon involved in the creep process and the relaxation rate B at long time : B ≈ d V /V0 d Lnt, (2) being systematically negative and depending on the applied current The initial voltage is proportional to the applied current with an initial resistance about R0 ≈ 3.37 That initial state is reproducible with an average voltage 0.4 V/ layer (voltage across one single interface) These values yield a relaxation rate B ≈−0.023 at I=1A Surprisingly, these values obtained on the large 3D bead assemblies are close to those obtained in two bead assemblies, especially at high currents The current dependence of the relaxation rate is presented on Fig 3(a) and Fig 3(b) On Fig 3(a) we plotted |B| against the applied current: it decreases from 0.15 at low currents down to 0.02 at higher values This curve evidences a current-driven ‘transition’ between a fast creep regime below 26 mA (inflexion point of the curve 3a) and a slow regime above 26 mA associated with a constant relaxation rate (see Fig 3(b)) The closeness of the relaxation rates observed on simple and complex 032108-4 Tekaya, Bouzerar, and Bourny AIP Advances 2, 032108 (2012) FIG (a) Evolution of the relaxation rate dV/V0dLnt against the applied current for a single bead-bead interface After a decrease between and 40 mA, it saturates at a constant value around 0,03 (b) The curve is the derivative of the relaxation rate B with respect to I It evidences the position of the inflexion point at 26 mA assemblies indicates the individual nature of B This point is confirmed on Fig establishing the additive character of the slope dV/dLnt, proportional to the number of beads The different points on the graph correspond to similar experiments realized at different currents We notice a dispersion increasing with the number of beads That dispersion is connected to the greater complexity of the medium: for larger assemblies, the current flows through a larger number of beads with different surface states This random exploration of a larger set of surface geometries results in a wide dispersion That dispersion enhancement by the increasing complexity manifests the sensitivity of the relaxation rate to the surface state of the beads Finally, the dependence of B upon the bead number constrains and simplifies the interpretation of the relaxation rate, to be viewed as a single interface feature The physical interpretation of our results requires the knowledge of the electrical resistance of a bead For the simplest situation of two beads, the total resistance combines their proper resistance and the contact resistances due to bead-bead interface and bead-electrode interface But, for many bead assemblies, in agreement with our observations, the single bead resistance and the interface one should be multiplied by the number of layers The bead-bead interface can be neglected at high enough compression forces This effect proceeds directly from the electron tunneling through the 032108-5 Tekaya, Bouzerar, and Bourny AIP Advances 2, 032108 (2012) FIG Relaxation rates plotted against the beads number The multiple points correspond to different experiments testing the reproducibility (for different currents) This curve evidences the extensive character of the relaxation rate, suggesting the individual nature (that is connected to individual contacts) of the processes controlling the relaxation oxide layer controlled by the specific ohmic conductance5 G L ≈ K q G (E F )2 |M|2 , (3) (where the squared matrix element of the tunneling hamiltonian |M|2 ∝ 2π e−kd ) scaling as the squared electron density of states at the Fermi level and proportional to a characteristic exponential factor summing up the influence of the tunnel barrier thickness d as well as its height hidden in the parameter κ At high compression the interface width d vanishes, leading to a very small resistance, strongly reduced by the high Fermi level density of states in metals But even if tunneling is absent, the Holm’s constriction resistance8 R H ∼ 1/2nγ a due to the contact between asperities (number of contacts n, conductivity γ ) with transverse radius a accounts for that reduction through a significant increase of both a and n under compression Indeed, the Holm’s contact resistance in our compression range is typically about a few 10−3 for steel spheres,9 that is much smaller than the bead resistance (a few Ohms) The total conductance of a single two bead interface combines all these contributions G int = G L + G H + G B where the Holm’s conductance G H = 1/R H and GB accounts for the contribution of the Branly effect.1, The relative importance of these contributions depend on both the force and current ranges At low enough force and current, the Branly effect is inactive and the interface is dominated by tunnelling and nanojunctions (contact between asperities) Oppositely, at high compression and current the proliferation of contacts and their likely merging under thermal effects of current favor the Branly effect For a two bead system, the total resistance reads Rt = 2R + 1/G int where R is the bulk resistance of a single bead The transition to the many bead system can be achieved through a global equivalent electrical network The two bead system provides then the ‘elementary cell’ from which that network is built up Though such networks might be very complex, an interesting approach to their handling was proposed by Renouf et al.10 Nevertheless a global description of the contact network is not relevant in the situation dealt with in that paper since, as was pointed out earlier, the voltage relaxation manifests properties associated with a single interface It is clear that whatever the current and compression range, the bulk resistance and the contact resistance 1/G int are sensitive to the interface processes More precisely, R should manifest individual interface processes through the real area of contact The knowledge of the bulk resistance of a single bead is required for understanding the electrical behavior of both a two bead system or a larger contact network Computing the electrical resistance of a spherical bead between two contact zones (being either electrodes or neighboring beads) of fixed areas is not a trivial problem In an earlier work, that resistance was found11 to be : R 2Vb /γ , (4) 032108-6 Tekaya, Bouzerar, and Bourny AIP Advances 2, 032108 (2012) where Vb is the bead volume for contacts located at the poles The area has a complex microscopic origin Many studies of solid frictional interfaces and their rheology12 point out the prevailing influence of surface roughness on the effective area of contact.13 The geometry and the dynamics of the contact between rough hard surfaces (e.g steel beads) are controlled by two characteristic length scales: the first is a mesoscopic scale giving the size of microcontacts, the second lying in the nanometric range controls the frictional dissipation due to molecular reorganizations Moreover, these studies evidence an aging process of the interface consisting of a creep growth of the contact area:14 (t) (1 + m Ln(1 + t/τ )), (5) where the rate m ≈ d / d Lnt is systematically a few 10−2 at room temperature The logarithmic growth of the real area of contact is thought of as the expression of plastic creep within the contact interface.12 Indeed, the average pressure pav W/ felt by the microcontacts due to the normal load W is about the yield stress σ Y of the contacting materials, that is plastic deformation is expected It should be noticed that, in the plastic regime, the friction coefficient reads μ = σ S / pav , σ S being the shear strength: The friction coefficient thus undergoes a similar creep with a rate dμ/d Lnt ≈ mσ S /σY which appears to be of the same order of magnitude as m.15 These considerations and equation (4) suggest that the interface aging should result in a resistance ‘creep’ (or equivalently a voltage creep at fixed current) : R(t) = V (t)/I R0 (1 + B Ln(1 + t/τ )), (6) with an initial resistance R0 2Vb /γ 02 and a negative relaxation rate B −2m According to that interpretation, the observed decreasing voltage manifests the increase of the real contact area that is the plastic relaxation of the microcontact interface We are led to an ‘electrical’ estimation of the parameter m which is around 0.07 at low currents and decreases down to a constant value about 0.01 at higher currents These values agree with the usually reported values of m.12 From the resistivity of our steel beads 1/γ ≈ 1.710−8 m, their radius Rb ≈ mm and taking into account the statistical distribution of the contacts for a many bead system, we can assess the initial area ≈ 6.3710−8 m which can be viewed as the area of a disc of radius a ≈ 250 μm This estimate of the real area of contact agrees with data of Bowden and Tabor6 extracted from both contact resistance and yield pressure measurements (indentation experiments) and reporting a linear increase of the contact area with load : B−T (F) ≈ 8, 36.10−10 F[m ], (7) that is B−T (100N ) ≈ 8, 36.10−8 m To reinforce our interpretation, one can assess the yield stress of the beads from σY ≈ W/ For a normal load W = 100 N we then get σY ≈ 1.4109 Pa which agrees with the values reported for steel16 lying in the range − 7.109 Pa From studies of steel-steel contact reporting12, 15 friction creep rates about dμ/d Lnt 3.10−2 we also get the shear strength to yield stress ratio about σ S /σY ≈ 0.4 − Our interpretation raises two fundamental questions First, the current–driven transition should be a thermal effect due to Joule’s heating: the relaxation of the contact area might be thermally activated Second, the interface ageing effect should be affected by the Branly effect, that is the nucleation of melted microcontacts17 acting as shortcuts According to a simple model proposed by Falcon et al.,3 due to the voltage U = R0 I between adjacent beads the contact zone temperature is heated to TJ = T + U /4L where the Lorentz constant L 2.45.10−8 V /K Assuming now an Arrhenius’ law 1/m ∝ e−E/k B T in the absence of current we are lead to the voltage dependence m m exp(E/k B TJ ), (8) The corresponding profiles presented on Fig for three arbitrarily chosen activation energies account qualitatively for our observations especially in the √ low current sector It is worth noticing that this model evidences a natural voltage scale Uc = L T 93 mV or equivalently a current scale Ic 27 m A close to the relaxation regime change threshold (see Fig 3) This model fits our data only for current below 30 mA with an activation energy E ≈ 0.11 eV The average decreasing rate of m is about dm/d I ≈ −3.5A−1 For larger values we underestimate the parameter m This 032108-7 Tekaya, Bouzerar, and Bourny AIP Advances 2, 032108 (2012) FIG Plot of the parameter m vs the voltage U/T (in the room temperature units) expected in the case of a thermally activated model of the voltage relaxation process As U∼I, the observed decrease at low currents is reproduced with a ‘flattening’ at high currents evoking a trend to saturation FIG Comparison of the observed influence of the compression force on the relaxation rate in shaken and unshaken bead systems In the unshaken case, the slope are about dm/dI = −0.18 A−1 to be compared to −1.5A−1 in the shaken case discrepancy can be explained by the likely surface melting of the beads induced by the heating and the subsequent metallic bridges formation involved in the Branly effect The influence of the Branly effect on the voltage relaxation is evidenced on Fig where we compare two typical experimental situations: shaken (shaking the system prior to relaxation experiment) and unshaken two bead system Shaking the beads amounts to cutting the microcontacts while a high compression force favors their nucleation The relaxation rates are clearly lower for the unshaken system where the effects of microcontacts are reinforced CONCLUSIONS This study affords us to have a certain voltage creep interpretation The relation between the voltage creep rate and the interface friction coefficient one is made, as well as the accession to the elementary contact area thanks to the new electrical resistance expression Our estimations agree with experimental results and a possible connection between electrical resistance and tribology could 032108-8 Tekaya, Bouzerar, and Bourny AIP Advances 2, 032108 (2012) be made with the proviso that experimental study coupling electrical and tribological measurements should be carried out ACKNOWLEDGMENTS This study has profited by the financial support provided by the FEDER and the Region of Picardie E Branly, “Variations de conductivit´e sous diverses influences e´ lectriques,” C R Acad Sci Paris 111, 785 (1890) Falcon, B Castaing, and M Creysels, “Transport e´ lectrique non lin´eaire dans les milieux granulaires,” in proceedings of Rencontre du Non Lin´eaire, 97 (2004) E Falcon and B Castaing, “Electrical conductivity in granular media and Branly’s coherer,” Am J Phys 73(4), 302 (2005) S Dorboloo, M Ausloos, and N Vandevalle, “Hysteretic behavior in metallic granular matter,” Appl Phys Lett 81(5), 936 (2002) A Tekaya, R Bouzerar, V Bourny, and I Tekaya, “Electron tunneling in metallic beads systems and slow electric relaxation,” World Journal of Engineering 8(2), 171 (2011) F P Bowden and D Tabor, The friction and lubrication of solids (Clarendon Press Oxford, 1954) E Falcon, B Castaing, and M Creysels, “Propri´ et´es e´ lectriques de la mati`ere granulaire : Bruit et intermittence,” Bulletin de la S.F.P 149 (2005) R Holm, Electric contacts Theory and applications, 4th edition (Springer Verlag, Germany, 2000) and references therein F P Bowden and D Tabor, Proc Roy Soc A 169, 391 (1939) 10 S Descartes, M Renouf, N Fillot, B Gautier, A Descamps, Y Berthier, and Ph Demanche, “A new mechanical–electrical approach to the wheel-rail contact,” Wear 265, 1408 (2008) 11 K Bourbatache, M Guessasma, E Bellenger, V Bourny, and A Tekaya, “Discrete modelling of electrical transfer in multi-contact systems,” Granular Matter 14, (2012) 12 T Baumberger and C Caroli, “Solid friction from stick-slip down to inning and aging,” Advances in Physics 55(3–4), 279 (2006) 13 J H Dieterich and B D Kilgore, Pure App Geophys 143, 283 (1994) 14 B N J Persson, Sliding friction Physical principles and applications, 2nd Edition (Springer Verlag, Germany, 2000), 363, see also ref 12 15 S J Dokos, J Appl Mech 13, 148 (1946) 16 F A McClintock and A S Argon, Mechanical behavior of Materials (Addison-Wesley, 1966) 17 E Falcon and B Castaing, “Propri´ et´es e´ lectriques de la mati`ere granulaire: l’effet Branly continu,” Bulletin de la SFP 148, (2005) E  ... are the contact interactions between grains and/or the grain deformations The complexity of such assemblies relies not only on the number of particles but also on the topology and geometry of the. .. dispersion That dispersion enhancement by the increasing complexity manifests the sensitivity of the relaxation rate to the surface state of the beads Finally, the dependence of B upon the bead. .. well as the key to understand the influence of the single contact on the collective properties of complex assemblies That interface consists mainly of the surface chemical composition and the complex