Corresponding Author

Christian Amatore(christian.amatore@ens.fr)


The main scope of this work was to elaborate and test a simple mathematical and numerical procedure for reconstructing the probability density distributions f(ρ) characterizing the distribution of electroactive or electrocatalytic nano-components present or deposited on the electrochemically-inert surface of a planar conductor based on the time-dependent chronoamperometric responses of the corresponding electrochemical array. The mathematical and numerical validity of the procedure was established for three types of arrays (one periodical, two involving random dispersions) involving near-spherical nano-components dispersed on a flat surface. Indeed, altogether, these three types represent most 2D-experimental electrochemical nano-arrays used for analytical or electrocatalytic purposes. This reconstruction procedure is easily implementable using most commercial mathematical programs. Albeit the simplicity of its implementation, it allowed recovering probability densities with an excellent precision, even when the available time-range experimentally accessible was too short for its rigorous application, being thus perfectly adequate to most experimental purposes.

Graphical Abstract


electrochemical arrays, chronoamperometry, inverse problem, micro- and nanodisk electrode arrays, density distribution probability, voronoi tessellation

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[1] Murray R W. Nanoelectrochemistry: Metal Nanoparticles, Nanoelectrodes, and Nanopores. Chemical Reviews, 2008, 108(7): 2688-2720.

[2] Tian N, Zhou Z Y, Sun S G, Ding Y, Wang Z L. Synthesis of Tetrahexahedral Platinum Nanocrystals With High-Index Facets and High Electro-Oxidation Activity. Science, 2007, 316(5825): 732-735.

[3] Xiao X, Bard A J. Observing Single Nanoparticle Collisions at an Ultramicroelectrode by Electrocatalytic Amplification. Journal of the American Chemical Society, 2007, 129(31): 9610-9612.

[4] Zhou Y G, Rees N V, Compton R G. The Electrochemical Detection and Characterization of Silver Nanoparticles in Aqueous Solution. Angewandte Chemie International Edition, 2011, 50(18): 4219-4221.

[5] Yoo J J, Kim J, Crooks R M. Direct Electrochemical Detection of Individual Collisions between Magnetic Microbead/Silver Nanoparticle Conjugates and a Magnetized Ultramicroelectrode. Chemical Science, 2015, 6(11): 6665-6671.

[6] Kissling, G P, Miles, D O, Fermin, D J. Electrochemical Charge Transfer Mediated By Metal Nanoparticles and Quantum Dots. Physical Chemistry Chemical Physics, 2011, 13(48): 21175-21185.

[7] Zhao J J, Bradbury C R, Huclova S, Potapova I, Carrara M, Fermin D J. Nanoparticle-Mediated Electron Transfer across Ultrathin Self-Assembled Films. Journal of Physical Chemistry B, 2005, 109(48): 22985-22994.

[8] Costentin C, Saveant J M. Catalysis at the nanoscale may change selectivity. Proceedings of The National Academy of Sciences of The United States of America, 2016, 113(42): 11756-11758.

[9] Gal F, Challier L, Cousin F, Perez H, Noel V, Carrot G. Electrocatalytic (Bio)Nanostructures Based on Polymer-Grafted Platinum Nanoparticles for Analytical Purpose. ACS Applied Materials & Interfaces, 2016, 8(23): 14747-14755.

[10] Multi-Walled Carbon Nanotube Supported Pd Nanocubes with Enhanced Electrocatalytic Activity. Hong W, Bi, P Y, Shang C S, Wang J, Wang E K. Journal of Materials Chemistry A, 2016, 4(12): 4485-4489.

[11] Anderson R M, Yancey D F, Zhang L, Chill S T, Henkelman G, Crooks R M. A Theoretical and Experimental Approach for Correlating Nanoparticle Structure and Electrocatalytic Activity. Accounts of Chemical Research 2015, 48(5): 1351-1357.

[12] Hong W, Wang J, Wang E K. Synthesis of Hollow PdRuCo Nanoparticles with Enhanced Electrocatalytic Activity. RSC Advances, 2015, 5(58): 46935-46940.

[13] Neumann C C M, Laborda E, Tschulik K, Ward K R, Compton R G. Performance of Silver Nanoparticles in the Catalysis of the Oxygen Reduction Reaction in Neutral Media: Efficiency Limitation Due to Hydrogen Peroxide Escape. Nano Research, 2013, 6(7): 511-524.

[14] Li Y Y, Jiang Y X, Chen M H, Liao H G, Huang R, Zhou Z Y, Tian N, Chen S P, Sun S G. Electrochemically Shape-Controlled Synthesis Of Trapezohedral Platinum Nanocrystals with High Electrocatalytic Activity. Chemical Communications, 2012, 48(76): 9531-9533.

[15] Zhou Z Y, Huang Z Z, Chen D J, Wang Q, Tian N, Sun S G. High-Index Faceted Platinum Nanocrystals Supported on Carbon Black as Highly Efficient Catalysts for Ethanol Electrooxidation. Angewandte Chemie International Edition, 2010, 49(2): 411-414.

[16] Zhou Z Y, Tian N, Li J T, Broadwell I, Sun S G. Nanomaterials of High Surface Energy with Exceptional Properties in Catalysis and Energy Storage. Chemical Society Reviews, 2011, 40(7): 4167-4185.

[17] Gara M, Ward K R, Compton R G. Nanomaterial Modified Electrodes: Evaluating Oxygen Reduction Catalysts. Nanoscale, 2013, 5(16): 7304-7311.

[18] Yang H Z, Kumar S, Zou S H. Electroreduction of O2 on Uniform Arrays of Pt Nanoparticles. Journal of Electroanalytical Chemistry, 2013, 688(1): 180-188.

[19] Fontaine O, Laberty-Rober C, Sanchez C. Sol-Gel Route to Zirconia-Pt-Nanoelectrode Arrays 8 nm in Radius: Their Geometrical Impact in Mass Transport. Langmuir, 2012, 28(7): 3650-3657.

[20] Sun P, Mirkin M V. Kinetics of Electron-Transfer Reactions at Nanoelectrodes. Analytical Chemistry, 2006, 78(18): 6526-6534.

[21] Sabatani E, Rubinstein I, Maoz R, Sagiv J. Organized Self-Assembling Monolayers on Electrodes .1. Octadecyl Derivatives on Gold. Journal of Electroanalytical Chemistry, 1987, 219(1-2): 365-371.

[22] Finklea H O, Avery S, Lynch M, Furtsch T. Blocking Oriented Monolayers of Alkyl Mercaptans on Gold Electrodes. Langmuir, 1987, 3(3): 409-413.

[23] Bethell D, Brust M, Schiffrin D J, Kiely C. From Monolayers to Nanostructured Materials: An Organic Chemist's View of Self-Assembly. Journal of Electroanalytical Chemistry, 1996, 409(1-2): 137-143.

[24] Le Poul N, Douziech B, Zeitouny J, Thiabaud G, Colas H, Conan F, Cosquer N, Jabin I, Lagrost C, Hapiot P, Reinaud O, Le Mest Y. Mimicking the Protein Access Channel to a Metal Center: Effect of a Funnel Complex on Dissociative Versus Associative Copper Redox Chemistry. Journal of the American Chemical Society, 2009, 131(49): 17800-17807.

[25] Chazalviel J N, Allongue P. On the Origin of the Efficient Nanoparticle Mediated Electron Transfer across a Self-Assembled Monolayer. Journal of the American Chemical Society, 2011, 133(4): 762-764.

[26] Cancino J, Machado S A S.Microelectrode Array in Mixed Alkanethiol Self-Assembled Monolayers: Electrochemical Studies. Electrochimica Acta, 2012, 72(1): 108-113.

[27] Leroux Y R, Hapiot P. Nanostructured Monolayers on Carbon Substrates Prepared by Electrografting of Protected Aryldiazonium Salts. Chemistry of Materials, 2013, 25(3): 489-495.

[28] Lhenry S, Jalkh J, Leroux Y R, Ruiz J, Ciganda R, Astruc D, Hapiot P. Tunneling Dendrimers. Enhancing Charge Transport through Insulating Layer Using Redox Molecular Objects. Journal of the American Chemical Society, 2014, 136(52): 17950-17953.

[29] Leroux Y R, Hui F, Hapiot P. A protecting-Deprotecting Strategy for Structuring Robust Functional Films Using Aryldiazonium Electroreduction. Journal of Electroanalytical Chemistry, 2013, 688(1): 298-303.

[30] Fu F, Chen S, Kuzume A, Rudnev A, Huang C, Kaliginedi V, Baghernejad M, Hong W, Wandlowski T, Decurtins S, Liu S-X. Exploitation of Desilylation Chemistry in Tailor-Made Functionalization on Diverse Surfaces. Nature Communications, 2015, 6, # 6403.

[31] Amatore C. Electrochemistry at Ultramicroelectrodes. in "Physical Electrochemistry: Principles, Methods and Applications" (I. Rubinstein, Ed.), M. Dekker, New York. 1995. Chap.4. pp.131-208.

[32] Amatore C, Maisonhaute E, Simonneau G. Ohmic Drop Compensation in Cyclic Voltammetry at Scan Rates in the Megavolt per Second Range: Access to Nanometric Diffusion Layers via Transient Electrochemistry. Journal of Electroanalytical Chemistry, 2000, 486(2): 141-155.

[33] Amatore C, Fosset B. Equivalence Between Electrodes of Different Shapes: Between Myth and Reality. Analytical Chemistry, 1996, 68(24): 4377-4388.

[34] Watkins J J, Chen J, White H S, Abruna H D, Maisonhaute E, Amatore C. Zeptomole Voltammetric Detection and Electron-Transfer Rate Measurements Using Platinum Electrodes of Nanometer Dimensions. Analytical Chemistry, 2003, 75(16): 3962-3971.

[35] Sun P, Mirkin M V. Electrochemistry of Individual Molecules in Zeptoliter Volumes. Journal of the American Chemical Society, 2008, 130(26): 8241-8250.

[36] Amatore C, Arbault S, Guille M, Lemaitre F. Electrochemical Monitoring of Single Cell Secretion: Vesicular Exocytosis and Oxidative Stress. Chemical Reviews, 2008, 108(7): 2585–2621.

[37] White R J, White H S. Electrochemistry in Nanometer-Wide Electrochemical Cells. Langmuir, 2008, 24(6): 2850-2855.

[38] Boateng A, Irague F, Brajter-Toth A. Low nM Detection Limits at Porous 13nm Thick Membrane-Coated Nanostructured Microdisk Electrodes. Electroanalysis, 2013, 25(2): 345-355.

[39] Du Y, Li B, Wang E K. "Fitting" Makes "Sensing" Simple: Label-Free Detection Strategies Based on Nucleic Acid Aptamers. Accounts of Chemical Research, 2013, 46(2): 203-213.

[40] Li J, Wang E K. Silver Nanoclusters for Drug Detection in Biological Samples: What is the Future? Bioanalysis, 2014, 6(11): 1421-1423.

[41] Fan D Q, Zhai Q F, Zhou W J, Zhu X Q, Wang E K, Dong S J. A Label-Free Colorimetric Aptasensor for Simple, Sensitive and Selective Detection of Pt (II) Based on Platinum (II)-Oligonucleotide Coordination Induced Gold Nanoparticles Aggregation. Biosensors & Bioelectronics, 2016, 85(1): 771-776.

[42] Li D Y, Li J, Jia X F, Wang E K. Gold Nanoparticles Decorated Carbon Fiber Mat As a Novel Sensing Platform For Sensitive Detection Of Hg(II). Electrochemistry Communications, 2014, 42(1): 30-33.

[43] Jia X F, Dong S J, Wang E K. Engineering the bioelectrochemical interface using functional nanomaterials and Microchip Technique Toward Sensitive and Portable Electrochemical Biosensors. Biosensors & Bioelectronics, 2016, 76(1): 80-90.

[44] Du Y, Guo S J, DongS J, Wang E K. An integrated sensing system for detection of DNA using new parallel-motif DNA triplex system and graphene-mesoporous silica-gold nanoparticle hybrids. Biomaterials, 2011, 32(33): 8584-8592.

[45] Chen A, Tsao M J, Chuang J F, Lin C H.Electrochemical determination of Verapamil with a microchip embedded with gold nanoelectrode ensemble electrodes. Electrochimica Acta, 2013, 89(1): 700-707.

[46] Ongaro M, Ugo P. Sensor Arrays: Arrays of Micro-and Nanoelectrodes, in “Environmental Analysis By Electrochemical Sensors And Biosensors. Vol 1: Fundamentals” (Moretto L, Kalcher K, Eds.). Springer, New York (2014). pp: 583-613.

[47] Dawson K, O'Riordan A. Electroanalysis at the Nanoscale, in “Annual Review of Analytical Chemistry, Vol 7” (Cooks R G, Pemberton J E, Eds.). Annual Reviews, Palo Alto (2014). Pp: 163-181.

[48] Polsky R, Xiao X Y, Wheeler D R, Brozik S M. Multifunctional Electrode Arrays, in “Nanomaterials For Electrochemical Sensing And Biosensing” (Pumera M, Ed.). Pan Stanford Publishing (2013). pp: 89-131.

[49] Virgilio F, Prasciolu M, Ugo P, Tormen M. Development of Electrochemical Biosensors by E-Beam Lithography for Medical Diagnostics. Microelectronic Engineering, 2013, 111(1): 320-324.

[50] Ongaro M, Ugo P. Bioelectroanalysis with Nanoelectrode Ensembles and Arrays. Analytical and Bioanalytical Chemistry, 2013, 405(11): 3715-3729.

[51] Zhang J, Ting B P, Ying J Y. Theoretical Assessment of Binding and Mass-Transport Effects in Electrochemical Affinity Biosensors that Utilize Nanoparticle Labels for Signal Amplification. Chemistry-a European Journal, 2012, 18(47): 15167-15177.

[52] Yang C, Jacobs C B, Nguyen M D, Ganesana M, Zestos A G, Ivanov I N, Puretzky A A, Rouleau C M, Geohegan D B, Venton B J. Carbon Nanotubes Grown on Metal Microelectrodes for the Detection of Dopamine. Analytical Chemistry, 2016, 88(1): 645-652.

[53] Habtamu H B., Ugo P. Miniaturized Enzymatic Biosensor via Biofunctionalization of the Insulator of Nanoelectrode Ensembles. Electroanalysis, 2015, 27(9): 2187-2193.

[54] Shipway A N, Katz E, Willner I. Nanoparticle Arrays on Surfaces for Electronic, Optical, and Sensor Applications. ChemPhysChem, 2000, 1(1): 18-52.

[55] Amatore C, Saveant J M, Tessier D. Charge Transfer at Partially Blocked Surfaces. A Model for the Case of Microscopic Active and Inactive Sites. Journal of Electroanalytical Chemistry, 1983, 147(1-2): 39-51.

[56] O. Sliusarenko O, Oleinick A, Svir I, Amatore C. Validating a Central Approximation in Theories of Regular Electrode Electrochemical Arrays of Various Common Geometries. Electroanalysis, 2015, 27(4): 980-991.

[57] O. Sliusarenko O, Oleinick A, Svir I, Amatore C. Development and Validation of an Analytical Model for Predicting Chronoamperometric Responses of Random Arrays of Micro- and Nanodisk Electrodes. ChemElectroChem, 2015, 2(9): 1279-1291.

[58] Davies T J, Compton R G. The Cyclic and Linear Sweep Voltammetry of Regular and Random Arrays of Microdisc Electrodes: Theory. Journal of Electroanalytical Chemistry, 2005, 585(1): 63-82.

[59] Henstridge M C, Compton R G. Mass Transport to Micro- and nanoelectrodes and their Arrays: A Review. Chemical Record, 2012, 12(1): 63-71.

[60] Zoski C G, Fernandez J L, Imaduwage K, Gunasekara D, Vadari R. Evaluation of the Intrinsic Kinetic Activity of Nanoparticle Ensembles under Steady-State Conditions. Journal of Electroanalytical Chemistry, 2011, 651(1): 80-93.

[61] Masa J, Batchelor-McAuley C, Schuhmann W, Compton R G. Koutecky-Levich Analysis Applied to Nanoparticle Modified Rotating Disk Electrodes: Electrocatalysis or Misinterpretation? Nano Research, 2014, 7(1): 71-78.

[62] Compton R G, Laborda E, Ward K R. Heterogeneous Surfaces, in Understanding Voltammetry: Simulation of Electrode Processes. Imperial College Press. London (2014). pp: 201-227.

[63] Bard A J, Faulkner L R. Electrochemical Methods. J Wiley & Sons. New York (2001). pp: 161-166.

[64] See e.g.: Cai Y, Newby B M Z. Marangoni Flow-Induced Self-Assembly of Hexagonal and Stripelike Nanoparticle Patterns. Journal of the American Chemical Society, 2008, 130(19): 6076–6077.

[65] Fernandez J L, Wijesinghe M, Zoski C G. Theory and Experiments for Voltammetric and SECM Investigations and Application to ORR Electrocatalysis at Nanoelectrode Ensembles of Ultramicroelectrode Dimensions. Analytical Chemistry, 2015, 87(2): 1066-1074.

[67] Leroux Y, Schaming D, Ruhlmann L, Hapiot P. SECM Investigations of Immobilized Porphyrins Films. Langmuir, 2010, 26(18): 14983-14989.

[68] Fang P P, Buriez O, Labbe E, Tian Z Q, Amatore C. Electrochemistry at Gold Nanoparticles Deposited on Dendrimers Assemblies Adsorbed onto Gold and Platinum Surfaces. Journal of Electroanalytical Chemistry, 2011, 659(1): 76-82.

[69] Marques J T, de Almeida R F M, Viana A S. Biomimetic Membrane Rafts Stably Supported on Unmodified Gold. Soft Matter, 2012, 8(6): 2007-2016.

[70] Raya D G, Silien C, Blazquez M, Pineda T, Madueno R. Electrochemical and AFM Study of the 2D-Assembly of Colloidal Gold Nanoparticles on Dithiol SAMs Tuned by Ionic Strength. Journal of Physical Chemistry C, 2014, 118(26): 14617-14628.

[71] Aurenhammer F. Voronoi Diagrams – A Survey of a Fundamental Geometric Data Structure. ACM Computing Surveys, 1991, 23(3):345–405.

[72] Davies T J, Ward-Jones S, Banks C E, del Campo J,Mas R, Munoz F X, Compton R G. Journal of Electroanalytical Chemistry, 2005, 585(1): 51-62.

[73] Belding S R, Dickinson E J F, Compton R G. Journal of Physical Chemistry C, 2009, 113(25): 11149-11156.

[74] Belding S R, Compton R G. Journal of Physical Chemistry C, 2010, 114(18): 8309-8319.

[75] Jarai-Szabo F, Neda Z. On the size distribution of Poisson Voronoi cells. Physica A, 2007, 385(2): 518-526.

[76] Cheng I F, Whiteley L D, Martin C R. Ultramicroelectrode Ensembles - Comparison of Experimental and Theoretical Responses and Evaluation of Electroanalytical Detection Limits. Analytical Chemistry, 1989, 61(7): 762-766.

[77] In order to reach diffusion layers of a few nanometer thickness one needs to use voltammetry in the megavolt per second range. See [32] and the following review: Amatore C, Maisonhaute E. When Voltammetry Reaches Nanoseconds. Analytical Chemistry, 2005, 77(15): 303A-311A.

[78] Chan Y Y, Eng A Y S, Pumera M, Webster R D. Assessments of Surface Coverage after Nanomaterials are Drop Cast onto Electrodes for Electroanalytical Applications. ChemElectroChem, 2015, 2(7): 1003-1009.



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