摘要:

Gas Kinetics Group and Division de Chimie-Physique de la Societe Francaise de Chimie 9th International Symposium on Gas Kinetics To be held in Bordeaux, France on 20-25 July 1986 Further information from Dr R. Lasclaux, Lab. Photophys. Photochim. MolBculaire, Universite de Bordeaux I, 33405 Talence Cedex, France Poiymer Physics Group Biologically Engineered Polymers To be held at Churchill College, Cambridge on 21-23 July 1986 Further information from Dr M. J. Miles, AFRC,Food Research Institute, Colney Lane, Norwich NR4 7UA Polymer Physics Group with the British Rheological Society Deformation in Solid Polymers To be held at the University of Leeds on 9-1 1 September 1986 Further information from Dr J. V. Champion, Department of Physics, City of London Polytechnic, 31 Jewry Street, London EC3N 2EY ~~_____________ ~~~~ Carbon Group Carbon Fibres- P ro pe rt i es and A p p I i cat i o ns To be held at the University of Salford on 1 5 1 7 September 1986 Further information from The Meetings Officer, The Institute of Physics, 47 Belgrave Square, London SW1 X 8QX ~ ~~~~~~~~ ~ Division with the Surface Reactivity and Catalysis Group-Autumn Meeting Promotion in Heterogeneous Catalysis To be held at the University of Bath on 23-25 September 1986 Further information from: Professor F.S. Stone, School of Chemistry, University of Bath, Bath BA2 7AY (viii)Gas Kinetics Group and Division de Chimie-Physique de la Societe Francaise de Chimie 9th International Symposium on Gas Kinetics To be held in Bordeaux, France on 20-25 July 1986 Further information from Dr R.Lasclaux, Lab. Photophys. Photochim. MolBculaire, Universite de Bordeaux I, 33405 Talence Cedex, France Poiymer Physics Group Biologically Engineered Polymers To be held at Churchill College, Cambridge on 21-23 July 1986 Further information from Dr M. J. Miles, AFRC,Food Research Institute, Colney Lane, Norwich NR4 7UA Polymer Physics Group with the British Rheological Society Deformation in Solid Polymers To be held at the University of Leeds on 9-1 1 September 1986 Further information from Dr J. V. Champion, Department of Physics, City of London Polytechnic, 31 Jewry Street, London EC3N 2EY ~~_____________ ~~~~ Carbon Group Carbon Fibres- P ro pe rt i es and A p p I i cat i o ns To be held at the University of Salford on 1 5 1 7 September 1986 Further information from The Meetings Officer, The Institute of Physics, 47 Belgrave Square, London SW1 X 8QX ~ ~~~~~~~~ ~ Division with the Surface Reactivity and Catalysis Group-Autumn Meeting Promotion in Heterogeneous Catalysis To be held at the University of Bath on 23-25 September 1986 Further information from: Professor F. S. Stone, School of Chemistry, University of Bath, Bath BA2 7AY (viii)

ISSN:0300-9599    

DOI:10.1039/F198581FX029

出版商:RSC    

年代:1985

数据来源: RSC

摘要:

xxxij AUTHOR INDEX Singh, Km. S., 751 Sircar, S., 1527, 1541 Slade, R. C. T., 847 Smith, I. G., 1095 Snelling, C. M., 1761 Sobczyk, L., 311 Siiderberg, D., 17 15 Solar, S., 1101 Solar, W., 1101 Soma, M., 485 Somorjai, G. A., 1263 Somsen, G., 1015 Sorek, Y., 233 Souto, F. A., 2647 Spencer, S., 2357 Spichiger-Ulmann, M., 7 13 Spoto, G., 1283 Spotswood, T. M., 1623 Srivastava, R. D., 913 Stachurski, J., 1447, 2813 Staricco, E. H., 1303 Stock, T., 2257 Stockhausen, M., 397 Stokes, R. H., 1459 Stone, F. S., 1255 Strachan, A. N, 1761 Strohbusch, F., 2021 Stuckless, J. T., 597 Su, Z., 2293 Subrahmanyam, V. S., 1655 Sugimoto, N., 1441, 2959 Suminaka, M., 2287 Suprynowicz, Z., 553 Sutcliffe, L. H., 679, 1467, 1215 Suzanne, J., 2339 Suzuki, H., 3117 Swallow, A. J., 1225 Symons, M.C. R., 433, 565, 727, 2131, 2775, 1095, 1963, 242 1 Takagi, Y., 1901 Takahashi, Y., 3 117 Takeshita, H., 2805 Tamilarasan, R., 2763 Tamura, K., 2287 Tanaka, T., 1513 Taniewska-Osinska, S., 695, Tascon, J. M. D., 939, 2399 Taylor, M. J., 1863 Taylor, N., 2357 Tejuca, L. G., 939, 2399, 1203 Teller, R. G., 1693 Tempere, J-F., 1357 Teramoto, M., 2941 Theocharis, C . R., 857 Thomas, J. K., 735 Tielen, M., 2889, 3049 Tindwa, R. M., 545 Tissier, C., 3081 Toi, K., 2835 Tokuda, T., 2835 Torrez-Mujica, T., 343 Townsend, R. P., 1071, 173 1, Trasatti, S., 2995 Treiner, C., 2513 Trenwith, A. B., 745 Trifiro, F., 1003 Troncoso, G., 1631, 1637 Tseung, A. C. C., 1883 Tuck, J. J., 833 Turner, J. E., 1263 Uemoto, M., 2333 Uma, K., 2733 Valencia, E., 1631. 1637 Valigi, M., 813 Vallmark, T., 1389 Van Oort, M.J. M., 3059 Varma, M. K., 751 Vattis, D., 2043 Vecli, A., 433 Veseli, V., 2095 Vink, H., 1677, 1725 Vliers. D. P., 2009 Vukovid, Z., 1275 3081, 1913 3127 Waghorne, W. E., 2703 Ward, A. J., 2975 Watanabe, H., 1569 Waugh, K. C., 3073 Weckstrorn, K., 2947 Weinberg, N. N., 875 Weingartner, H., 1031 Wells, C. F.. 801, 1057, 1401, White, M. A., 3059 Williams, J. O., 271 1 Williams, P. A., 2635 Williams, P. B., 3067 Williams, R. T., 847 Wojcik, D., 1037 Wood, G. L., 265 Wood, R. M., 273 Woolf, L. A., 769, 2821 Wright, C. J., 2067 Wright, J. P., 1471 Wright, T. H., 1819 Wurie, A. T., 2605 Yadav, G. D., 161 Yadava, R. D., 751 Yamaguchi, M., 1513 Yamaguti, K., 1237 Yamasaki, S., 267 Yamashita, H., 2485 Yamatera, H., 127 Yelon, W., 1693 Yoshida, S., 1513, 2485 Yoshikawa, M., 2485 Zambonin, P.G.. 621 zdanov, S. P., 2541 Zecchina, A., 1283 Zelano, V., 2365 Zhan, R. Y., 2083 Zhao, Z., 185 Zhulin, V. M., 875 Zilnyk, A., 679, 1215 Zulauf, M., 2947 Zundel, G., 1425, 2375 1985. 2145, 2475, 3091xxxij AUTHOR INDEX Singh, Km. S., 751 Sircar, S., 1527, 1541 Slade, R. C. T., 847 Smith, I. G., 1095 Snelling, C. M., 1761 Sobczyk, L., 311 Siiderberg, D., 17 15 Solar, S., 1101 Solar, W., 1101 Soma, M., 485 Somorjai, G. A., 1263 Somsen, G., 1015 Sorek, Y., 233 Souto, F. A., 2647 Spencer, S., 2357 Spichiger-Ulmann, M., 7 13 Spoto, G., 1283 Spotswood, T. M., 1623 Srivastava, R. D., 913 Stachurski, J., 1447, 2813 Staricco, E. H., 1303 Stock, T., 2257 Stockhausen, M., 397 Stokes, R. H., 1459 Stone, F. S., 1255 Strachan, A.N, 1761 Strohbusch, F., 2021 Stuckless, J. T., 597 Su, Z., 2293 Subrahmanyam, V. S., 1655 Sugimoto, N., 1441, 2959 Suminaka, M., 2287 Suprynowicz, Z., 553 Sutcliffe, L. H., 679, 1467, 1215 Suzanne, J., 2339 Suzuki, H., 3117 Swallow, A. J., 1225 Symons, M. C. R., 433, 565, 727, 2131, 2775, 1095, 1963, 242 1 Takagi, Y., 1901 Takahashi, Y., 3 117 Takeshita, H., 2805 Tamilarasan, R., 2763 Tamura, K., 2287 Tanaka, T., 1513 Taniewska-Osinska, S., 695, Tascon, J. M. D., 939, 2399 Taylor, M. J., 1863 Taylor, N., 2357 Tejuca, L. G., 939, 2399, 1203 Teller, R. G., 1693 Tempere, J-F., 1357 Teramoto, M., 2941 Theocharis, C . R., 857 Thomas, J. K., 735 Tielen, M., 2889, 3049 Tindwa, R. M., 545 Tissier, C., 3081 Toi, K., 2835 Tokuda, T., 2835 Torrez-Mujica, T., 343 Townsend, R.P., 1071, 173 1, Trasatti, S., 2995 Treiner, C., 2513 Trenwith, A. B., 745 Trifiro, F., 1003 Troncoso, G., 1631, 1637 Tseung, A. C. C., 1883 Tuck, J. J., 833 Turner, J. E., 1263 Uemoto, M., 2333 Uma, K., 2733 Valencia, E., 1631. 1637 Valigi, M., 813 Vallmark, T., 1389 Van Oort, M. J. M., 3059 Varma, M. K., 751 Vattis, D., 2043 Vecli, A., 433 Veseli, V., 2095 Vink, H., 1677, 1725 Vliers. D. P., 2009 Vukovid, Z., 1275 3081, 1913 3127 Waghorne, W. E., 2703 Ward, A. J., 2975 Watanabe, H., 1569 Waugh, K. C., 3073 Weckstrorn, K., 2947 Weinberg, N. N., 875 Weingartner, H., 1031 Wells, C. F.. 801, 1057, 1401, White, M. A., 3059 Williams, J. O., 271 1 Williams, P. A., 2635 Williams, P. B., 3067 Williams, R. T., 847 Wojcik, D., 1037 Wood, G. L., 265 Wood, R. M., 273 Woolf, L. A., 769, 2821 Wright, C. J., 2067 Wright, J. P., 1471 Wright, T. H., 1819 Wurie, A. T., 2605 Yadav, G. D., 161 Yadava, R. D., 751 Yamaguchi, M., 1513 Yamaguti, K., 1237 Yamasaki, S., 267 Yamashita, H., 2485 Yamatera, H., 127 Yelon, W., 1693 Yoshida, S., 1513, 2485 Yoshikawa, M., 2485 Zambonin, P. G.. 621 zdanov, S. P., 2541 Zecchina, A., 1283 Zelano, V., 2365 Zhan, R. Y., 2083 Zhao, Z., 185 Zhulin, V. M., 875 Zilnyk, A., 679, 1215 Zulauf, M., 2947 Zundel, G., 1425, 2375 1985. 2145, 2475, 3091

ISSN:0300-9599    

DOI:10.1039/F198581BX031

出版商:RSC    

年代:1985

数据来源: RSC

摘要:

JOURNAL OF THE CHEMICAL SOCIETY FARADAY TRANSACTIONS, PARTS I AND I 1 The Journal of the Chemical Society is published in six sections, of which five are termed Transactions; these are distinguished by their subject matter, as follows: Dalton Transactions (Inorganic Chemistry). All aspects of the chemistry of inorganic and organometallic compounds; including bioinorganic chemistry and solid-state inorganic chemistry; of their structures, properties, and reactions, including kinetics and mechanisms; new or improved experimental techniques and syntheses. Faraakay Transactions I (Physical Chemistry). Radiation chemistry, gas-phase kinetics, electrochemistry (other than preparative), surface and interfacial chemistry, heterogeneous catalysis, physical properties of polymers and their solutions, and kinetics of polymerization, etc.Faraday Transactions II (Chemical Physics). Theoretical chemistry, especially valence and quantum theory, statistical mechanics, intermolecular forces, relaxation phenomena, spectroscopic studies (including i.r., e.s.r., n.m.r., and kinetic spec- troscopy, etc.) leading to assignments of quantum states, and fundamental theory. Studies of impurities in solid systems. Perkin Transactions I (Organic Chemistry). All aspects of synthetic and natural product organic, organometallic and bio-organic chemistry, including aliphatic, alicyclic, and aromatic systems (carbocyclic and heterocyclic). Perkin Transactions II (Physical Organic Chemistry). Kinetic and mechanistic studies of organic, organometallic and bio-organic reactions.The description and application of physicochemical, spectroscopic, and theoretical procedures to organic chemistry, including structure-activity relationships. Physical aspects of bio-organic chemistry and of organic compounds, including polymers and biopolymers. Authors are requested to indicate, at the time they submit a typescript, the journal for which it is intended. Should this seem unsuitable, the Editor will inform the author. The sixth section of the Journal of the Chemical Society is Chemical Communications, which is intended as a forum for preliminary accounts of original and significant work, in any area of chemistry that is likely to prove of wide general appeal or exceptional specialist interest. Such preliminary reports should be followed up eventually by full papers in other journals (e.g.the five Transactions) providing detailed accounts of the work. NOTES I t has always been the policy of the Faraday Transactions that brevity should not be a factor influencing acceptability for publication. In addition however to full papers both sections carry at the end of each issue a section headed ‘Notes’, which are short self-contained accounts of experimental observations, results, or theory that will not require enlargement into ‘full’ papers. The Notes section is not used for preliminary communications. The layout of a Note is the same as that of a paper. Short summaries are required. The procedure for submission, administration, refereeing, editing and publication of Notes is the same as for full papers.However, Notes are published more quickly than papers since their brevity facilitates processing at all stages. The Editors endeavour to meet authors’ wishes as to whether an article is a full paper or a Note, but since there is no sharp dividing line between the one and the other, either in terms of length or character of content, the right is retained to transfer overlong Notes to the full papers section. As‘a guide a Note should not exceed I500 words or word-equivalents. (9NOMENCLATURE AND SYMBOLISM Units and Symbols. The Symbols Committee of The Royal Society, of which The Royal Society of Chemistry is a participating member, has produced a set of recommendations in a pamphlet ‘Quantities, Units, and Symbols’ (1975) (copies of this pamphlet and further details can be obtained from the Manager, Journals, The Royal Society of Chemistry, Burlington House, London W 1 V OBN).These recommendations are applied by The Royal Society of Chemistry in all its publications. Their basis is the ‘Systeme International d’Unites’ (SI). A more detailed treatment of units and symbols with specific application to chemistry is given in the WPAC Manual of Symbols and Terminology for Physicochemical Quantities and Units (Pergamon, Oxford, 1979). Nomenclature. For many years the Society has actively encouraged the use of standard IUPAC nomenclature and symbolism in its publications as an aid to the accurate and unambiguous communication of chemical information between authors and readers. In order to encourage authors to use IUPAC nomenclature rules when drafting papers, attention is drawn to the following publications in which both the rules themselves and guidance on their use are given: Nomenclature of Organic Chemistry, Sections A , B, C, D, E, F, and H (Pergamon, Oxford, 1979 edn). Nomenclature of Inorganic Chemistry (Butterworths, London, 197 1, now published by Pergamon).Biochemical Nomenclature and Related Documents (The Biochemical Society, London, 1978). A complete listing of all IUPAC nomenclature publications appears in the January issues of J. Chem. SOC., Faraday Transactions. It is recommended that where there are no IUPAC rules for the naming of particular compounds or authors find difficulty in applying the existing rules, they should seek the advice of the Society’s editorial staff.(ii)THE FARADAY DIVISION OF THE ROYAL SOCIETY OF CHEMISTRY SYMPOSIUM NO. 20 Phase Transitions in Adsorbed Layers University of Oxford, 17-1 8 December 1985 Organising Committee: Professor J. S. Rowlinson (Chairman) Dr E. Dickinson Dr R. Evans Mrs Y. A. Fish Dr N. Parsonage Dr D. A. Young The aim of the meeting is to discuss phase transitions at gas/liquid, liquid/liquid and solid/fluid interfaces, and in other systems of constrained geometry or dimensionality less than three. Emphasis will be placed on molecularly simple systems, whereby liquid crystal interfaces and chemisorption phenomena are excluded. The preliminary programme may be obtained from: Mrs Y. A. Fish, The Royal Society of Chemistry, Burlington House, London W1V OBN THE FARADAY DIVISION OF THE ROYAL SOCIETY OF CHEMISTRY GENERAL DISCUSSION NO.81 Lipid Vesicles and Membranes Loughborough University of Technology, 15-1 7 April 1986 Organising Committee: Professor D. A. Haydon (Chairman) Professor D. Chapman Mrs Y. A. Fish Dr M. J. Jaycock Dr I. G. Lyle Professor R. H. Ottewill Dr A. L. Smith Dr D. A. Young The aim of the meeting is to discuss the physical chemistry of lipid membranes and their interactions, in particular theoretical and spectroscopic studies, polymerised membranes, thermodynamics of bilayers and Iiposomes, mechanical properties, encapsulation and interaction forces between bilayers leading to fusion but excluding preparation and characterisation methodology. The preliminary programme may be obtained from: Mrs Y.A. Fish, The Royal Society of Chemistry, Burlington House, London W1V OBN Full papers for publication in the Discussion Volume will be required by December 1985. (iii)THE FARADAY DIVISION OF THE ROYAL SOCIETY OF CHEMISTRY GENERAL DISCUSSION NO. 82 Dynamics of Molecular P hotof rag mentat ion University of Bristol, 15-1 7 September 1986 Organising Committee : Professor R. N. Dixon (Chairman) Dr G. G. Balint-Kurti Dr M. S. Child Professor R. Donovan Professor J. P. Simons The discussion will focus on the interaction of radiation with small molecules, molecular ions and complexes leading directly or indirectly to their dissociation. Emphasis will be given to contributions which trace the detailed dynamics of the photodissociation process. The aim will be to bring together theory and experiment and thereby stimulate important future work.Contributions for consideration by the Organising Committee are invited and abstracts of submitted as soon as possible, and abstracts of about 300 words by 30 September 1985, to: Professor R. N. Dixon, Department of Theoretical Chemistry, University of Bristol, Cantock's Close, Bristol BS8 1TS ~~ ~_______ ~ THE FARADAY DIVISION OF THE ROYAL SOCIETY OF CHEMISTRY SYMPOSIUM NO. 21 Interaction-induced Spectra in Dense Fluids and Disordered Solids University of Cambridge, 1&11 December 1986 Organising Committee : Professor A. D. Buckingham (Chairman) Dr R. M. Lynden-Bell Dr P. A. Madden Professor E. W. J. Mitchell Dr J. Yarwood Dr D. A. Young Mrs Y. A. Fish Whilst interaction-induced spectra have been studied in the gas phase for many years, their importance in the spectroscopy of condensed matter has been appreciated only relatively recently.At present a considerable number of studies of induced spectra are taking place in what are (nominally) widely separated fields of study. It is highly desirable to bring these communities together so that common issues can be identified and the progress of one field appreciated in another. Contributions for consideration by the Organising Committee are invited and abstracts of about 300 words should be sent by 25 October 1985 to: Professor A. D. Buckingham, University Chemical Laboratory, Lensfield Road, Cambridge CB2 1 EW30TH INTERNATIONAL CONGRESS OF PURE AND APPLIED CHEMISTRY Advances in Physical and Theoretical Chemistry Manchester, S13 September 1985 The Faraday Division is mounting the following symposia as part of the 30th IUPAC Congress: A. B.C. D. Reaction Dynamics in the Gas Phase and in Solution This symposium will examine the ways in which modern techniques allow detailed study 01 the dynamical motion of molecules which are undergoing chemical reaction or energy exchange. Micellar Systems The symposium will discuss various aspects of micellization, including size and shape factors, micellization in biological systems, chemical reactions in micellar systems, micelle structure and solubilitation. Emphasis will also be given to modern techniques of examining micellar systems, including small-angle neutron scattering, neutron spin echo, photocorrelation spectroscopy, NM R and use of fluorescent probes.Surface Science of Solids The symposium will centre on recent advances in the study of kinetics and dynamics at surfaces and of phase transitions in adsorbate layers on single crystal surfaces. Both experimental and theoretical aspects will be reviewed with an emphasis on metal single crystal surfaces. New Electrochemical Sensors (in collaboration with the Electroanalytical Group of the Analytical Division) The symposium will cover such topics as the fundamentals of the subject, new gas sensors based on membrane electrodes and on ceramic oxides, the development of new ion- Iselective electrodes and the synthesis of new guest-host carriers, the development of CHEMFETS and other integrated devices together with the theory of the operation of such devices, and finally the development of biosensors including for instance enzyme electrodes, direct' electron transfer to biological molecules and new potentiometric techniques for protein analysis.The full programme and application form may be obtained from: Dr J. F. Gibson, 30th IUPAC Congress, Royal Society of Chemistry, Burlington House, London W1V OBNI THE FARADAY DIVISION OF THE ROYAL SOCIETY OF CHEMISTRY GENERAL DISCUSSION NO. 83 Brownian Motion University of Cambridge, 7-9 April 1987 Organising Committee Dr M. La1 (Chairman) Dr R. Ball Dr E. Dickinson Dr J. S. Higgins Dr P. N. Pusey Or D. A. Young Mrs Y. A. Fish The aim of the meeting is to discuss new developments in the experimental and theoretical studies of Brownian motion of colloidal particles and macromolecules, with particular emphasis on the dynamics of aggregate formation and breakdown, computer simulation and many- body hydrodynamic interactions.Contributions for consideration by the Organising Committee are invited and abstracts of about 300 words should be sent by 15 June 1986 to: Dr M. Lal, Unilever Research, Port Sunlight Laboratory, Bebington, Wirral L63 3JW Full papers for publication in the Discussion volume will be required by December 1986 JOURNAL OF CHEMICAL RESEARCH Papers dealing with physical chemistry/chemical physics which have appeared recently in J.Chem.Research, The Royal Society of Chemistry’s synopsis+microform journal, include the following: Quantum-mechanical Studies of Catalysis. Part 1.A Model for Nucleophilic Attack on Carbonyl, catalysed by Non-functional Cationic Surfactants Amiram Goldblum and Jehoshua Katzhendler (1 985, Issue 3) Cyclopropane Parameters for Molecular Mechanics Pekto M . lvanov (1 985, Issue 3) Inorganic Analogue of the Ethyl Radical Jehan A. Baban, Vernon P. J. Marti, and Brian P. Roberts (1 985, Issue 3) The Iron-Vanadium-Oxygen System at 11 23, 1273, and 1373 K. Part 1. Phase Equilibria Larbi Marhabi, Marie-Chantal Trinel-Dufour and Pierre Perrot (1 985, Issue 3) Solvent Effects on the Rotational Barriers of the N,N- Dimethylamides of 2- and 3- Furoic and 2- and 3-Thenoic Acids Gaetano Alberghina, Francesco Agatino Bottino, Salvatore Fisichella, and Caterina Arnone (1 985, Issue 4) A Partial Determination of the Stability Fields of Ferrierite and Zeolites ZSM-5, ZSM-48, and Nu-10 in the K,O-AI,O,-SiO,-NH, [CH,],NH, System Barrie M.Lowe (1 985, Issue 6) Stereochemical Applications of Potential Energy Calculations. Part 4. Revised Electron Spin Resonance Studies of the Ammonia-Boryl Radical (H,N + BH;); an Abraham Araya and The Level of Prochirality : the Analogy between Substitutional and Distortional Amitai E. Halevi (1 985, Issue 6) DesymmetrizationFARADAY DIVISION INFORMAL AND GROUP MEETINGS Polymer Physics Group Biennial Conference : Physical Aspects of Polymer Science To be held at the University of Reading on 11-1 3 September 1985 Further information from Professor Bassett, J. J. Thompson Physical Chemistry Laboratory, University of Reading, Whiteknights, Reading RG6 2AF Statistical Mechanics and Thermodynamics Group Multicomponent Mixtures To be held at the University of East Anglia on 16-1 8 September 1985 Further information from: Dr M.J. Grimson, Food Research Institute, Colney Lane, Norwich NR4 7UA Carbon Group Strength and Structure in Carbons and Graphites To be held at the University of Liverpool on 16-1 8 September 1985 Further information from The Meetings Officer, The Institute of Physics, 47 Belgrave Square, London SWlX 8QX Division with the Institute of Physics Seventh National Quantum Electronics Conference To be held at the Abbey Hotel, Great Malvern on 16-20 September 1985 Further information from: Dr E. Jakeman, Treasurer QE7, RSRE, St Andrews Road, Great Malvern WR14 3PS Surface Reactivity and Catalysis Group with the Catalysis Section of the KNCV Mechanism and Structure in Heterogeneous Catalysis To be held at Noordwijkerhout, The Netherlands on 18-20 September 1985 Further information from: Dr R.Joyner, BP Research Centre, Chertsey Road, Sunbury on Thames TW16 7LN Industrial Physical Chemistry Group A Molecular Approach to Lubrication and Wear To be held at Girton College, Cambridge on 23-25 September 1985 Further information from Mr M. P. Dare-Edwards, Shell Research Ltd, Thornton Research Centre, Chester CH1 3SH Neutron Scattering Group jointly with the Materials Testing Group of the Institute of Physics Industrial Uses of Particle Beams To be held at the Institute of Physics, London on 26 September 1985 further information from The Meetings Officer, The Institute of Physics, 47 Belgrave Square, London SWlX 8QX - Colloid and Interface Science Group with the Colloid and Surface Science Group of the SCI lnterf acial Rheology To be held at Imperial College, London on 16 December 1985 Further information from Dr R.Aveyard, Department of Chemistry, The University, Hull HU6 7RX High Resolution Spectroscopy Group and Theoretical Chemistry Group Title to be Announced To be held at the University of York on 16-18 December 1985 Further information may be obtained from: Dr J. M. Hollas, Department of Chemistry, University of Reading, White'knights, Reading RG6 2AD Neutron Scattering Group Time-resolved Scattering and Transition Kinetics To be held at Imperial College, London on 17 December 1985 Further information may be obtained from: Dr J.S. Higgins, Department of Chemical Engineering, Imperial College London SW7 2BY (vii)Molecular Beams Group with CCP6 Molecular Scattering-Theory and Experiment To be held at the University of Sussex on 19-21 March 1986 Further information from Dr A. Stace, School of Molecular Sciences, University of Sussex, Falmer, Brighton BN1 9QJ Division-Annual Congress Structure and Reactivity of Gas Phase Ions To be held at the University of Warwick on %11 April 1986 Further information from: Professor K. R. Jennings, Department of Molecular Sciences, University of Warwick, Coventry CV4 7AL ~ ~~ ~ ~~ ~ ~ Polymer Physics Group with the Statistical Mechanics and Thermodynamics Group Macromolecular Flexibility and Behaviour in Solution To be held at the University of Bristol on 16-1 8 April 1986 Further information from The Meetings Officer, The Institute of Physics, 47 Belgrave Square, London SW1 X 8QX Industrial Physical Chemistry Group Physical Chemistry of Water Soluble Polymers To be held at Girton College, Cambridge on 1-3 July 1986 Further information from Or 1. D. Robb, Unilever Research Laboratory, Port Sunlight, Bebington, Wirra L63 3JW ~ ~~~ Polymer Physics Group Biologically Engineered Polymers To be held at Churchill College, Cambridge on 21-23 July 1986 Further information from Dr M. J. Miles, AFRC Food Research Institute, Colney Lane, Norwich NR4 7UA Carbon Group Carbon Fibres-Properties and Applications To be held at the University of Salford on 15-1 7 September 1986 Further information from The Meetings Officer, The Institute of Physics, 47 Belgrave Square, London SWl X 8QX Division with the Surface Reactivity and Catalysis Group-Autumn Meeting Promotion in Heterogeneous Catalysis To be held at the University of Bath on 23-25 September 1986 Further information from: Professor F. S. Stone, School of Chemistry, University of Bath, Bath BA2 7AY (viii)

ISSN:0300-9599    

DOI:10.1039/F198581FP065

出版商:RSC    

年代:1985

数据来源: RSC

摘要:

J . Chem. Soc., Faraday Trans. I , 1985,81, 1761-1766 Dehydration of Sodium Carbonate Monohydrate BY MATTHEW C. BALL,* CHRISTINE M. SNELLING AND ALEC N. STRACHAN Department of Chemistry, University of Technology, Loughborough, Leicestershire LE11 3TU Received 21st October, 1982 The kinetics and mechanism of the thermal decomposition of solid sodium carbonate monohydrate have been studied between 336 and 400 K at water-vapour pressures up to 20 kN mP2. Thedehydration follows Avrami-Erofeyev kinetics (n = 2) with a limiting activation energy of 71.5 kJ mo1-I and a frequency factor of 2.2 x lo7 s-l. The reaction rate is reduced by increasing the water-vapour pressure and this has been accounted for in terms of competition from the rehydration reaction. Sodium carbonate monohydrate is commercially important as the precursor of dense sodium carbonate (heavy ash), which is widely used in the chemical and allied industries.The phase diagram for the system Na,CO,-NaHC0,-H,O has been studied and the general stability of the sodium carbonate hydrates outlined., The monohydrate is stable in the presence of saturated solution only above 308 K, with the decahydrate stable at room temperature. In air, however, the decahydrate loses water to form the monohydrate, which is sufficiently stable to have been suggested as an acidimetric standard., The general stability of the monohydrate has been studied by vapour-pressure meas~rernent.~~ The crystal structure has been determined.8 The unit cell is orthorhombic with a = 10.72 A, b = 6.649 A and c = 5.249 A, and has sodium ions coordinated with 4 or 5 oxygen atoms from carbonate ions and 1 or 2 oxygen atoms from water molecules.The water molecules lie in strings parallel to b. Little work has been done on the kinetics of the dehydration. Rigaud and Ripotot studied the decomposition of pellets pressed at 30000 p.s.i.t to 86% of the theoretical density.’ An activation energy of 54 kJ mol-l was obtained, which is close to the enthalpy of dehydration. Differences in behaviour were noted above and below 342 K and it was suggested that diffusion became important above this temperature and at high gas flow rates. Contracting-sphere kinetics were used to calculate the Arrhenius parameters. Dubil studied8 the dehydration using a fluidised-bed apparatus with large samples (1 50 g) and temperatures between 358 and 378 K.Low temperatures and high water-vapour content in the air stream reduced the rate of dehydration. The activation energy was found to vary with water-vapour content from 50 kJ at 0% water to ca. 1000 kJ at 100% water. EXPERIMENTAL Sodium carbonate monohydrate was supplied by ICI Mond Division and had an X-ray diffraction pattern in excellent agreement with that publi~hed.~ The weight loss on heating at 383 K was in good agreement with that required for loss of water (theory 14.52% ; observed 1 p.s.i. x 6895 Pa. 17611762 DEHYDRATION OF SODIUM CARBONATE MONOHYDRATE 14.60%). The dehydration studies were carried out using a C.I. Microforce balance (mark 2C) attached to a potentiometric recorder. Sample weights of 40 mg+ 10% were used throughout, with an expected weight loss of 5.8 mg, allowing a recorder sensitivity of 10 mg f.s.d.to be used. Constant temperatures were maintained by refluxing solvents; nitrogen (99.9% ) was passed over the sample at a constant flow rate of 10cm3min-l. This gas flow was kept at constant water-vapour pressure by passing through drying columns or salt solutions of known relative humidity.1° SEM examination of starting material, partially decomposed material and product was made, as was optical examination of the decomposition using a hot stage and a transmission microscope. Attempts to measure the specific surface area of the starting material by nitrogen adsorption failed because the surface area was very low. Estimates of the likely value were made from SEM photographs; measurement of the surface area of the decomposed material gave no problems.RESULTS The experimental plots of weight loss against time were all S-shaped with relatively short initiation periods, and reduced-time plots based on the time for 50% reaction1' showed that the best fit between experimental data and theoretical curves was in terms of the Avrami-Erofeyev equation with n = 2, i.e. [-In (1 -a)$ = kt. Typical reduced-time plots are shown in fig. 1 together with theoretical curves for several models. Rate constants were calculated from the slopes of graphs of [-In (1 -a)]; against t ; some examples of these are shown in fig. 2. An Arrhenius plot of these rate constants (Ink against 1/T) is shown in fig. 3, and it can be seen that increasing water-vapour pressure has a considerable effect on the rate of dehydration.Plates 1 and 2 show SEM photographs of starting material and product; porosity in the anhydrous material is obvious. The surface area increased from an estimated 0.02 m2 g-l, which suggests little microporosity (no macroporosity is visible in plate I), to 0.63 m2 g-l in the dehydrated material. Plate 3 shows the development of decomposition nuclei with time in a cleaved fragment. The rate of growth is constant. Plate 4 shows electron micrographs of a single nucleus, from which it can be seen that these are approximately circular [plate 4(a)] but very shallow [plate 4(b)], i.e. essentially two-dimensional. DISCUSSION REACTION MECHANISM The present results suggest that the Avrami-Erofeyev equation, with n = 2, gives the best agreement between experimental data and theoretical curves within the range of models considered.These cover nucleation and growth of product phase, boundary control and diffusion control as the rate-controlling step. When n = 2 there are two possible interpretations, uiz. that the number of nuclei increases with time and each nucleus is one-dimensional (i.e. when n = 2, y = 1, S = 1) or that the number of nuclei remains constant and the nuclei are two-dimensional (n = 2, y = 0, 6 = 2). Examination of plate 3 shows that all of the dehydration nuclei are produced at small circular defects (possibly occluded solution) present in the original crystal, i.e. the number of nuclei is constant, and each grows linearly with time.These observations suggest that the second interpretation is the most likely, i.e. y = 0,S = 2. ARRHENIUS PARAMETERS The straight line shown in fig. 3 for the runs carried out in dry nitrogen gives an activation energy of 71.5 kJ mol-' and a frequency factor of 2.2 x lo7 s-l. The rate constants for the decomposition in dry nitrogen can therefore be described by theJ . Chem. SOC., Faraday Trans. 1, Vol. 81, part 8 Plates 1 and 2 Plate 1. Scanning electron micrograph of monohydrate (bar = 25 x m). Plate 2. Scanning electron micrograph of product (bar = 25 x lop6 m). M. C. BALL, C. M. SNELLING AND A. N. STRACHAN (Facing p . 1762)J. Chem. SOC., Faraday Trans. 1, Vol. 81, part 8 M. C. BALL, C. M. SNELLING AND A. N. STRACHAN Plate 3J .Chem. SOC., Faraday Trans. 1, Vol. 81, part 8 Plate 3. Optical micrographs showing growth of nuclei with time (7' = 355 K, magnification = x 125): (a) 0, (6) 120, ( c ) 360 and ( d ) 600 s. Plate 3 M. C. BALL, C. M. SNELLING AND A. N. STRACHANJ . Chem. SOC., Faraday Trans. 1, Vol. 81, part 8 Plate 4 Plate 4. Scanning electron micrograph of dehydration nucleus (bar = 1 x m): (a) tilt = 0" and (b) tilt = 50". M. C. BALL. C. M. SNELLING AND A. N. STRACHANM. C. BALL, C. M. SNELLING AND A. N. STRACHAN 1 .o 0 . 8 n v 0 . 6 m & c .- Y 2 0 . 4 a 0.2 1763 0 0.4 0.8 1.2 1.6 2.0 reduced time Fig. 1. Reduced-time plots for monohydrate decomposition: 0, 336 K/0.06 Nm-2; x , 379 K / l . l kN m-2; a, 335 K/20 kN rn-? and A, 397 K/20 kN mP2. (a) [-ln(1 --a)]$ = kt, (b) [-In (1 - = kt, (c) 1 - (1 - a)i = kt and ( d ) a2 = kt.2 .o 1 . 6 0 10 20 30 LO t/min Fig. 2. Fit of experimental data to [ -ln(l -a)]i = kt: a, 336 K/0.06 N m-2; 375 K/0.06 N m-2; T, 357 K/0.61 kN m-2; ., 355 K/20 kN rnP2 and 355 K/3.3 1 kN m-2. x , A,1764 DEHYDRATION OF SODIUM CARBONATE MONOHYDRATE 1 0 3 KIT 2.5 2.6 2.7 2 . 8 2 . 9 3.0 1 1 1 A Fig. 3. Arrhenius plot for experimental data: A, 0.06; 0, 0.39; x , 0.61; 0, 1.1; V, 2.6; A, 3.3; 0, 15.7 and m, 20 kN m-*. equation k = 2.2 x lo7 exp (- 71 500/RT) s-l. Such a low frequency factor, compared with 1013 s-l for an isolated gaseous molecule, suggests a lcss of entropy in the formation of the transition state and probably indicates a surface reaction.12 This is consistent with the reaction occurring at the reactant/product interface.The activation energy is greater than the enthalpy of decomp~sitionl~ (57.7 kJ mol-l), which suggests that the reaction is kinetically simple. EFFECT OF WATER-VAPOUR PRESSURE The obvious effect of increasing water-vapour pressure is to reduce the reaction rate in a non-linear manner, as can be seen in fig. 3. There are two possible reasons for the reduction in rate: (a) the reactant is decomposing, but owing to surface sorption of water on an increased surface area the apparent rate of dehydration is effectively reduced, or (b) the reactant does not decompose at low temperatures and high humidities, and the reduced rate is a consequence of normal kinetic equilibrium between a forward and an enhanced back reaction. The first possibility was effectively disproved by holding a sample of monohydrate at 339 K over saturated sodium carbonate solution ( p / p o = 0.90) for several days and periodically examining crystals of reactant in the SEM.These conditions produce no decomposition in the kinetic studies. No evidence for decomposition such as increased porosity or cracking was observed. A very slow rate of dehydration, following the sorption example, is only possible if increased surface area becomes available through decomposition. It is likely therefore that the effect of water vapour is to slow down the apparent rate of decomposition by increasing the rate of the backward, rehydration, reaction as follows. Consider a small area of the reactant/product interface and let the vapour pressure of water at the interface be p(H20).Then the rate of loss of water is given by k, A, where k, is the number of moles of monohydrate decomposed per second per unit area. The backward rate (whether reaction or sorption) will depend on the number of water molecules colliding, and will therefore be proportional to p(H,O) and A. It is given by k,Ap(H,O), where kr is the number of moles of water taken up per second per unit area. Hence the net rate of loss of water is given by dm/dt = k, A - k, Ap(H,O)M. C. BALL, C. M. SNELLING AND A. N. STRACHAN 1765 Table 1. Calculated versus measured low-temperature rate constants p(H,O)/N rn-, T/K k,,,,,/10-4 s-l k,,,,/10-4 s-l ratio 10-4 387 613 1053 2 560 3 306 15 732 19 865 336 357 375 337 358 379 339 357 379 336 358 379 336 357 376 338 355 379 355 376 356 374 1.84 7.41 1.56 5.89 1.97 5.09 1.42 7.02 1.41 7.13 1.37 5.75 1.57 1.35 24.4 25.0 23.7 21.6 24.3 26.2 24.2 17.2 1.69 7.34 1.73 8.07 1.97 7.41 1.45 7.8 1 1.11 6.84 1.21 5.46 I .89 1.08 23.4 30.6 30.5 30.3 24.7 29.5 19.8 15.4 0.92 0.99 0.96 1.11 1.35 1.22 1 .oo 1.46 1.28 1.02 1.1 1 1.40 0.79 0.96 1.02 0.90 0.95 1.12 1.21 0.82 0.80 0.90 This composite rate constant can be substituted in the normal derivation of the Avrami-Erofeyev equation to produce the following : [-In (1 - a)]i = B(kd - k, p(H,O)] t (1) where B is the product of a series of constants and also subsumes A.The important point is that using a composite rate constant does not alter the Avrami-Erofeyev derivation, and it is therefore possible to have this mechanism obeyed across the range of experimental variables studied.The difference term in eqn (1) is the experimental rate constant, made up of the forward and backward rates. The Arrhenius parameters for the forward reaction have been determined in dry nitrogen, and the activation energy for the backward, rehydration reaction is the difference between that for the forward reaction and the overall enthalpy change, i.e. (71.5-57.7) kJ = 13.8 kJ mol-l. The rate can therefore be calculated: kcalc = [2.2 x lo7 exp (- 71 500/RT)] - [ A , exp (- 13 800/RT)p(H,O)]. (2) Table 1 gives a comparison of calculated and measured rate constants, using a value of A, = 3.1 x s-l throughout. The ratio of these constants lies between 1.46 and 0.80 with an average of 1.06, which suggests that the effect of water vapour on the dehydration rate is reasonably described by eqn (2), allowing for the inaccuracies in the experimental rates.The value of A, used, 3.1 x lo-" s-l, is obviously composite and made up of a true reaction frequency, probably in the range lo8 to 10l2 s-l, and1766 DEHYDRATION OF SODIUM CARBONATE MONOHYDRATE an efficiency term relating the external pressure of water vapour to the number of collisions occurring at the interface. This efficiency term is obviously very low. HIGH-TEMPERATURE RESULTS It is apparent from fig. 3 that at temperatures > ca. 385 K reaction rates are lower than predicted from those at lower temperatures. Such a lowering in the overall rate of decomposition, in terms of the previous discussion, suggests that the backward rate is increased or the forward rate is reduced.Fig. 3 shows that the runs in dry nitrogen also have lower rates, so that the latter possibility is correct. The inflexion temperature is just above that at which the vapour pressure of the monohydrate reaches atm~spheric.~ It seems likely that under such conditions [p(H,O) > 1-01 the removal of water by the carrier gas is altered so that the 'dry-nitrogen' regime cannot be maintained. If the internal pressure of water vapour is high then it is also likely that the backward, rehydration, rate would be accelerated. The reduced overall rate above 385 K is therefore probably due to both increased backward and reduced forward rates but not to any change in reaction mechanism. COMPARISON WITH OTHER RESULTS As mentioned previously, little work has been done by others on this decomposition.It is difficult to compare the present results with those of Rigaud and Ripotot' because of the large differences in both experimental and theoretical approaches. However Dubil's fluidised-bed study,8 which found that the activation energy varied with partial pressure of water vapour, is confirmed by the present work. The curves of ln(rate) against 1/T found at higher humidities in the present work, if they are assumed to be straight lines, would produce high activation energies. The authors gratefully acknowledge support for this programme of research from ICI Mond Division Ltd and particularly thank Dr N. Rolfe for his interest in the work. R. W. Purcell, in The Modern Inorganic Chemicals Industry, ed. R. Thompson (The Chemical Society, London, 1977), p. 128. Kirk-Othmer, Encyclopaedia of Chemical Technology (Interscience, New York, 2nd edn, 1969), vol. 18, p. 466. W. L. W. Ludekens and T. Thirunamachandran, Chem. Ind. (London), 1954, 1265. R. M. Caven and H. J. S. Sand, J. Chem. SOC., 191 1,99, 1359. C. G. Waterfield, R. G. Linford, B. B. Goalby, T. R. Bates, C. A. Elyard and L. A. K. Staveley, Trans. Faraday Soc., 1968, 64, 868. J. P. Harper, Z. Kristallogr., 1936, 95, 266. M. Rigaud and J. Ripotat, L'lngenieur, 1969, February, p. 16. E. S. Dubil, Zh. Prikl. Khim., 1973, 46, 1906. A.S.T.M. Index, card no. 8-448. lo J. F. Young, J. Appl. Chem., 1967, 17, 241. l 1 J. H. Sharp, G. W. Brindley and B. N. N. Achar, J. Am. Ceram. SOC., 1966,49, 379. l2 H. F. Cordes, J . Phys. Chem., 1968,72, 2185. l3 J. H. Mellor, Treatise on Inorganic and Theoretical Chemistry (Longmans, London, 1961), vol. 2, suppl. 2, p. 1099. (PAPER 21 1797)

ISSN:0300-9599    

DOI:10.1039/F19858101761

出版商:RSC    

年代:1985

数据来源: RSC

摘要:

J. Chem. SOC., Faraahy Trans. 1, 1985,81, 1767-1787 A Critique of the Adsorption Isotherms used in Electrochemical Processes involving the Adsorption of Organic Compounds BY PANAGHIOTIS NIKITAS Laboratory of Physical Chemistry, University of Thessaloniki, Thessaloniki, Greece Received 24th July, 1984 The main adsorption isotherms used in electrochemical processes involving the adsorption of organic compounds have been analysed both theoretically and experimentally in order to clarify the models, the limits of their applicability and the validity of the results obtained. The isotherms are divided into two categories : (a) isotherms based on continuous-solvent models and (b) isotherms based on discrete-solvent models. The isotherms of category (a) have, in general, low applicability, while their models do not give a realistic description of the adsorbed phase.However, the discrete-solvent models are closer to the physical reality of the electrode/ solution interface, although only the generalized Flory-Huggins isotherms seem to provide both high applicability and believable conclusions concerning the particle-particle interactions of the adsorbed phase. Quantitative analysis of experimental adsorption data from electrolyte solutions or from any phase requires evaluation of the proper adsorption isotherm. This describes the relationship between the concentrations of the adsorbate in the adsorption layer and in the bulk of the phase from which the adsorption is taking place. Moreover, an adsorption isotherm may give information about the intermolecular interactions occurring in the adsorption layer.However, it is necessary to know what model is being used for the adsorption isotherm and to be sure about its applicability to the adsorption system under investigation. The purpose of this paper is to test, both theoretically and experimentally, the main adsorption isotherms used in electrochemical processes involving the adsorption of organic compounds in order to determine the models, the limits of their applicability and consequently the validity of the results for interactions of the adsorbed particles. ISOTHERMS USED IN ELECTROCHEMICAL PROCESSES The main adsorption isotherms used for the interpretation of experimental data on the adsorption of organic compounds on homogeneous electrode surfaces are as follows.Frumkin’s exp (- 2a8) 8 1-8 pc = - where /? = (1 / c s ) exp (- AGe/RT), c and c, are the concentrations of the adsorbate and solvent, respectively, in the bulk of the solution, 8 is the surface coverage, a an interaction parameter due to interfacial particle-particle interactions and AG* is the standard free energy of adsorption. 17671768 ADSORPTION ISOTHERMS FOR ELECTROCHEMICAL PROCESSES The Hill-de Boer isotherm4v5 exp (Go) exp ( - 2aO). (2) 0 1-0 pc = - The isotherm from the free-volume theory6* Parsons isotherm8 pc = - exp (x) exp ( - 2aO). 1-0 (i-ey The isotherm with virial coefficients9~ lo pc = 0 exp (- 2aO). The Blomgren-Bockris isotherm for adsorption of organic ions1’ (4) where p > 0 and q > 0 are constants depending on the specific characteristics of the adsorption system under investigation.The Conway-Barradas isotherm for the adsorption of polar compounds12 exp (p, O3I2 - q63) pc = - 1-8 0 wherep, > 0. Bennes’ isotherm13 X pc = - exp {Bo[( 1 - x ) ~ - rx2]} (1 -X)r which can be written in the generalized form (7) where x = O/[r-(r- l)tq (10) ln(f’/fl;) = B , ( 2 ~ - 1 ) ~ - ~ { [ ( 1 - x ) 2 - r x ~ ] ( 2 ~ - 1 ) + 2 i x ( l - x ) ( l - x + r x ) ) n i - 0 (1 1) andf, andf, are the activity coefficients of the adsorbate and the solvent, respectively, at the interface, Bi (i = 0, 1, . . ., n) are constants due to particle-particle interactions at the interface and the parameter r provides the number of solvent molecules displaced from the surface solution by the adsorption of one molecule of adsorbate and is approximately determined from the ratio of the molar volumes U, and us of the adsorbate and solvent respectively, i.e.r = uA/us. The Flory-Huggins isotherm14-16 V pc = exp [a( 1 - 20)] exp(r- 1) (1 -0).P. NIKITAS 1769 which can be written in the generalized forms pc = e 4% exp(r- 1) (1 -8).f; pc = e (1 -7 B)'-lfA exp[-(B+l)] (1-0)' fk where In (fA/fi) is given by eqn (1 1) and Eqn (14) holds for the adsorption of rigid molecules, whereas eqn (1 5 ) holds for the adsorption of flexible molecules. The parameters b; and q which appear in these isotherms are constants depending on the geometrical characteristics of the adsorbed molecules and the lattice of the adsorption layerls and the parameter r is the size ratio of the adsorbate and solvent.Assuming a unimolecular adsorption layer, r in eqn (12) and (1 3) is given by the ratio SA/Ss, where S, and Ss are the areas covered by an adsorbate and solvent molecule, respectively, on the saturated surface, whereas in the case of eqn (14) and (15) for the size parameter r we have to use the integer closest to the ratio SA/Ss. THEORETICAL ANALYSIS OF THE ISOTHERMS An adsorption isotherm can be written in the general form wheref(8) is the configurational term due to the entropy of mixing of the adsorbed particles and g(Bo, . . . , B,, 8) takes into account the particle-particle interactions at the adsorbed phase. Obviously an isotherm must be based on the proper treatment of these two terms. However, their exact determination is difficult if not impossible.At the electrode/electrolyte interface the adsorbate molecules are in competition with the solvent molecules. At every position the solvent molecules may have m x e than one possible orientation whose possibility depends on the electric field of the interface. Therefore accurate determination of the configurations of the adsorbed molecules is impossible and consequently exact determination of the entropy term f(8) is also impossible. The same is true for the energetic term g(Bo, B,, . . ., B,, 8), since at every configuration different particle-particle interactions correspond. Moreover, at the interface there are solvated ions, which are also in competition with the solvent and the adsorbate molecules and which are expected to affect the configurational as well as the energetic term of eqn (18).Thus the complexity of the adsorption layer formed on an electrode makes it necessary to proceed to essential simplifications, the most drastic of which concern the behaviour of solvent at the interface. On the basis of the assumptions made for the solvent, the isotherms fall into two categories: (a) isotherms based on continuous- solvent models and (b) isotherms based on discrete-solvent models.1770 ADSORPTION ISOTHERMS FOR ELECTROCHEMICAL PROCESSES ISOTHERMS BASED ON CONTINUOUS-SOLVENT MODELS This category includes the Hill-de Boer, free-volume, virial, Parsons, Blomgren- Bockris and Conway-Barradas isotherms. These isotherms are based on the assumption that the solvent at the interface behaves as a continuous and inert medium.Therefore the presence of solvent at the interface is not taken into consideration. However, it is clear that this simplification leads to models giving unreal interfaces, as described schematically above. This is why these isotherms fail to describe the adsorption of organic compounds on electrode surfaces satisfactorily. Only in the case of ionic adsorption, where the properties of the adsorbate differ considerably from those of the solvent, is the use of continuous-solvent models justified, but even in this case only for determination of the energetic term,” g(B,, B,, . . ., B,, 8). Of the isotherms of this category, the Hill-de Boer, free-volume, virial and Parsons isotherms describe the mobile adsorption of a substance on an energetically homog- eneous surface, i.e.the adsorption layer behaves as a two-dimensional liquid or a compressed gas. For eqn ( 6 ) and (7) the situation is not so clear, and on the basis of statistical thermodynamics they may be obtained as follows. If we consider that the interface has a two-dimensional crystalline-lattice structure and that there are no lateral interactions between the adsorbed molecules (a langmuirian model), then the chemical potential of an adsorbed molecule is given by4,18,19 p(adsorbed) kT = In (1 - 0) +j$-ln 4in where qin is the internal partition function of an adsorbed molecule and U is the adsorbate-adsorbent interaction energy. In order to take into consideration the particle-particle interactions we have to introduce into eqn (19) an activity coefficient (f ).We thus have ,u( adsorbed) U = In (L) +- - In qin + In$ kT 1-8 kT If we then assume that the particle-particle interactions do not affect the distribution of the adsorbed molecules over the lattice sites of the interface, In f is given by where wij is the interaction energy between the pair (i-j). For the determination of an analytical form of the term In f it is necessary to use a model of the interface. If we use the crystalline-lattice model adopted above and restrict it to the nearest neighbours, we obtain In f = z0wA,/2kT (22) where z is the coordination number of the lattice. in the bulk of the solution gives the Frumkin isotherm, eqn ( l ) , with Equating the chemical potential of the adsorbed molecules to that of the molecules a = - ZW,,/ZkT.( 2 3 ) However, if we use the hexagonal-lattice model” for the determination of lnf, we obtain eqn ( 6 ) and (7). Indeed, according to this model, the adsorbed molecules form at the interface a non-static hexagonal lattice. Every change of the surface coverage 8 results in a change of the intermolecular distances between the adsorbed moleculesP. NIKITAS 1771 and consequently a change in the lattice dimensions. If in this model ro is the distance between two adsorbed molecules, then we have et 3hva2 WAA = --- Er, 4 ~ r : for the case where the adsorbed molecules are spherical organic ions with charge e, and they interact with Coulombic and dispersion forces, and we have when the adsorbed particles are spherical polar molecules with dipole moment p.In eqn (24) and (25) E is the dielectric constant of the interface, a is the polarizability of an adsorbed particle, v is the corresponding characteristic frequency and h is Planck's constant. Now the energetic terms of eqn (6) and (7) result directly from eqn (24) and (25) if we take into consideration the relation ri = 22/'3 nR2/38 (26) where R is the radius of the adsorbed particles. Note that different models have been used for the determination of the configurational term and the intermolecular interactions. However, this is not crucial. The most simplistic approximation is restriction of the intermolecular interactions between the molecules of the adsorbate for determination of the energetic term. This approximation may be considered as satisfactory only for coulombic interactions.The dipole-dipole interactions and particularly the dispersion forces between solvent-solvent and solvent-adsorbate molecules are comparable with those between adsorbate-adsorbate molecules. Therefore eqn (24) and particularly eqn (25) do not represent, even approximately, the potential of the particle-particle interactions of a real interface. ISOTHERMS BASED ON DISCRETE-SOLVENT MODELS This category includes isotherms based on models in which the discrete presence of solvent at the interface is recognized and the adsorption layer is considered to behave as a binary mixture of solvent and adsorbate molecules. For this reason the theories of binary mixtures play an important role in the development of such isotherms. Of these, lattice theoriesls*lg are the easiest to apply to adsorption from solutions, which results in all the known isotherms of this category being based on a lattice model.We will now examine the configurational and energetic terms resulting from lattice models of the adsorption layer. First we consider the case where the interface behaves as a strictly regular so1utionlsV l9 composed of adsorbed molecules of solvent S and adsorbate A, i.e. we assume that the adsorbed molecules are distributed over the points (sites) of a quasi-crystalline lattice with a mean coordination number z. All the molecules which can occupy a lattice site are approximately spheres of the same size. If we also assume that the adsorbed molecules are randomly distributed over the lattice sites of the interface and neglect non-nearest-neighbour interactions, then the chemical potential of an adsorbed molecule is given bylR pi( ad sorbed) k T ZW ZWii = Inxi+(l --lnqin,i+- k T 2k T1772 ADSORPTION ISOTHERMS FOR ELECTROCHEMICAL PROCESSES where N is the total number of the adsorbed molecules, Ni is the number of the adsorbed molecules of type i (= A or S), xi is their mole fraction, qin, is their internal partition function, in which we have formally included the adsorbate-adsorbent interactions, and M! is given by w = wAs-(~AA+wss)/2 (28) where wij is the interaction energy between nearest-neighbour pairs (i-j).Adsorption from solution is considered to take place according to the reaction A (in solution) + rS (adsorbed) +A (adsorbed) + rS (in solution).Therefore at equilibrium we have pA (adsorbed) - rps (adsorbed) = pA (in solution) - rps (in solution). (29) If the chemical potentials of the substances in the bulk of the solution are expressed as a function of the mole fraction and the unsymmetrical system is taken as the reference state,20921 then at the region where Henry’s law is valid we have pA (in solution) = p 2 + k T In (c/cs) (30) and ps (in solution) = pg (3 1) where p p is the chemical potential of the ith component in its standard state. isotherm, eqn (8), with xA = x and Now if we introduce eqn (27), (30) and (31) into eqn (29) we obtain the Bennes B, = zw/kT = z[wAS-(WA,+ ws,)/2]/kT. (32) According to the model, this isotherm is strictly valid for r = 1, where it is reduced to the Frumkin isotherm e I -e PC = - exp [B,( 1 - 2e)] (33) or for values of r very close to unity.However, for solutions of small molecules the effect of the difference in the molecular size of the two components does not effect the results obtained using the model of strictly regular solutions.22 Therefore eqn (8) can also be used in adsorption systems with values of r differing appreciably from unity. However, in this case the interaction parameter B, is not strictly given by eqn (32), since the difference in the sizes of the adsorbed molecules is expected to contribute to the value of B,. If the adsorbed molecules of the solvent and the adsorbate are of different molecular size, then the athermal lattice model leads to the following configurational termsl8 A r-i when the adsorbed molecules are rigid and occupy r lattice sites, (35) (36) for flexible molecules andP.NIKITAS 1773 when Flory's approximationl69 l8 ( z + 00) is used with eqn (34) and (35). A and B are obtained from eqn (16) and (17). It can easily be proved that the configurational term, eqn (36), holds irrespective of whether we have an interfacial solution of r-mer molecules of adsorbate with monomer solvent molecules or a solution of rA-mer adsorbate molecules with r,-mer solvent molecules, provided that r is equal to rA/rS. Indeed, if we denote by pi the number of ways in which the r,-mer as a whole can be placed in the lattice after one of its end elements has been placed, then the ratio a of the probability that a group of TATs sites, which can be occupied by rA molecules of solvent and r, molecules of adsorbate, can be wholly occupied by rA-rners to the probability that the group be completely occupied by r,-mers is given by with the assumption that the probability for an r,-mer (i = A or S) to occupy ri sites on the group of rA r, sites is independent of the presence of other r,-mers on this group.The free energy of mixing AA may be calculated from1sv23 AA = (%+$) ( Jo'lna do-8 Jol h a do) which leads to -=NAln8+N,ln(l-8) AA kT (39) which is identical to the equation for a mixture of r-mers with monomers. Therefore the chemical potentials of the adsorbed molecules and consequently the resulting adsorption isotherm will be identical to that for the adsorption of r-mers from a solution of monomers. In the multisite models particleparticle interactions may be introduced into the adsorption isotherm as follows.It is known from the work of Guggenheimls on chain molecules that for a random distribution the total number of first neighbouring contacts i, j (Nij) is given by However, these equations also hold for molecules containing closed rings. So the free energy of mixing18 where the subscript a denotes the value of the athermal solution and w is defined by the fact that the contribution of each contact of elements of two kinds to the configurational potential energy is (wAA + wss + w)/z, can be applied to all cases irrespective of whether we have chain or ring molecules. From eqn (42) we can calculate the chemical potentials of the adsorbed molecules, and applying the equilibrium condition eqn (29) we obtain the general isotherml61774 ADSORPTION ISOTHERMS FOR ELECTROCHEMICAL PROCESSES where B” = qw/kT.Using Flory’s approximation, eqn (40) is reduced to the simple Flory-Huggins isotherm,ls eqn (12). Finally, eqn (9) and (13b(15) are obtained if we introduce the particle-particle interaction energy on a macroscopic level by expanding the excess free energy of the adsorption layer (ge) to a power series:24 The details of this treatment are given in ref. (1 6). isotherms in this category. On the basis of the above analysis the following results are obtained for the THE FRUMKIN ISOTHERM The Frumkin isotherm is valid when the adsorbed molecules have equal molecular sizes and the particle-particle interactions are weak giving a random distribution of molecules among the sites.Only under these circumstances can eqn (32) be used to obtain reliable information about the particle-particle interactions at the adsorption layer. However, this isotherm has been used in almost all studies of the adsorption of organic compounds from solutions because of its wide applicability. In the case of unassociated solvents, e.g. aprotic solvents and possibly methanol at the mercury/ solution interface,25 there is no explanation on the molecular level for this behaviour. In associated solvents, and especially in the case of water, the wide applicability of the Frumkin isotherm is justified by assuming that solvent molecules at the interface are not independent but are linked in small c1usters.26-28 Dhar et al.15 have rejected this idea on the basis of theories of the structure of water and also following observations that water molecules do not adjust themselves into displaceable groups equivalent to the area of the adsorbate for various sizes of the latter.We also note that the main experimental finding supporting the view of clusters is that eqn (1 2) is applicable only for values of r close to unity. However, this may show the weakness of this isotherm, caused by the approximations involved, in describing experimental systems with r % 1. The model for the Flory-Huggins isotherm results from Frumkin’s model with the ‘improved’ assumption that the adsorbate molecules occupy r simultaneous sites at the interface. However, in lattice models generally, where the main approximations are physical and where their success is due, to some extent, to the elimination of the errors introduced by the approximations, it is not necessarily true that a better physical approximation gives more reliable results.This is also the case for Frumkin and Flory-Huggins isotherms. We believe that maintenance of the Bragg- Williams approximation18 in the Flory-Huggins isotherm, i.e. in a system of particles with different molecular sizes, is the reason why deviations from the experimental data occur when values of r other than unity are used. Moreover, the existence of clusters may cause a problem in evaluating the size parameter r in which case, we cannot justify the use of r = 1 in all experimental systems. Therefore, the success of Frumkin’s isotherm in describing experimental systems must be attributed to the elimination of the errors introduced by the configurational and energetic terms of the isotherm when we pass to systems with r % 1.Consequently, in thiscase, eqn (32) should not be used to obtain information about the particle-particle interactions of the interface. Frumkin’s isotherm is also valid for localized pure gas adsorption on to an energetically homogeneous surface. In this case, the interaction parameter a (or B,) of the isotherm is given by eqn (23). However, from the analysis of the continuous-P. NIKITAS 1775 solvent models it is obvious that, at least for the adsorption of organic compounds from electrolyte solutions, these models are not acceptable for an interpretation of the experimental data at a molecular level.THE ISOTHERMS OF EQN (8) AND (9) Bennes’ isotherm is valid when the adsorbed molecules have sizes that are not essentially different and they are randomly distributed over the surface. Therefore, Bennes’ isotherm is expected to be a better approximation than Frumkin’s isotherm in cases where the adsorbed particles have differences in molecular sizes. As was seen, this isotherm is also applicable in systems with values of r quite different from unity under the limitation that the interaction parameter B, is not given by eqn (32). Eqn (9) is expected to be applicable to all experimental systems since it contains more than two adjustable parameters. In systems with r close to unity the values of Bi are determined from the strength of the particle-particle interactions. However, in cases where r is not unity, the values of Bi are affected by differences in the molecular sizes of the adsorbed particles.This must be taken into account if we wish to obtain reliable information about the intermolecular interactions from the values of the Bi parameters. ISOTHERMS BASED ON FLORY-HUGGINS STATISTICS In these isotherms we have an improved approximation of the entropic term. This term is treated better in eqn (14) and (1 9, although their applicability is limited to experimental systems where the size parameter r takes integer values. The main problem with these isotherms is elaboration of the particle-particle interactions at a molecular level. It has been shown that the Bragg-Williams approximation does not give satisfactory results.16 The same is true of the quasi- chemical approximation.l6 Only the introduction of particle-particle interactions at a macroscopic level, eqn (1 I), resolves the problem of the applicability of these isotherms. Regarding the interaction parameters Bi, which appear in the energetic term of these isotherms, note that they are not expected to be influenced from the differences in the sizes of the adsorbed molecules, because the configurational term of these isotherms is valid for all values of r. Recently, Sangaranarayan and Rangarajan28 reported a new adsorption isotherm based on Flory-Huggins statistics, and they proved that the configurational term of the isotherms of eqn (8) and (9) is a limiting case of the corresponding term of their isotherm.Actually the isotherm of Sangaranarayan and Rangarajan is a generalization of the isotherm defined from eqn (1 5 ) and (43) of the present work, including an extra term from a more detailed treatment of the field-dipole interactions. Therefore the configurational term of the Sangaranarayan-Rangarajan isotherm is identical to that of eqn (15) and it reduces to the corresponding entropic term of eqn (8) and (9) only when z = 2, i.e. in the one-dimensional approximation. However, it is obvious that this approximation is physically meaningless, and therefore we cannot say that Bennes’ isotherm is a particular case of some model based on Flory-Huggins statistics. EXPERIMENTAL ANALYSIS OF THE ISOTHERMS Three sources of experimental data were used for analysis of the isotherms under investigation. The experimental data for the adsorption of acetonitrile (ACN), succinonitrile (SN) and propionitrile (PN) on Hg from aqueous solutions of NaF are taken from ref.(29)-(3 l), the experimental data for the adsorption of methyldiphenylADSORPTION ISOTHERMS FOR 9 ’ 1 4 ELECTROCHEMICAL PROCESSES 15 c-, l o 5 c3 Q I 5 I I I I I I 0.2 0.4 0.6 0.5 1 .o e 0 Fig. 1. Tests of Parsons’ (l), free-volume (2), Hill4e Boer (3), Frumkin’s (4) and virial (5) isotherms for the adsorption of ACN and PN on Hg from 0.25 mol dmP3 NaF aqueous solutions at uM = - 4 and - 5 pC cmW2, respectively. Points are experimental data plotted according to eqn (45). Plots of the Frumkin and virial isotherms are shifted - 1.0 and -2.0, respectively, along the y axis.15 c-, t3 5 10 P 5 0.5 1 .o e 20 c-, l 5 5 t3 4 I 10 O D 1 1 0.5 1.0 e Fig. 2. Tests of the same isotherms as in fig. 1 for the adsorption of MDPO and DDPO on Hg from 0.1 mol dm-3 LiCl methanolic solutions at potentials indicated by the following symbols: 0, Emax; 0, -1.0; 0 , -0.8; A, -0.7; x , -0.5 V (us SCE). Points are experimental data plotted according to eqn (45). Plots of the Frumkin and virial isotherms are shifted - 1.0 and -2.0, respectively, in the case of MDPO, and -0.5 and - 1.0, respectively, in the case of DDPO, along the y axis.P. NIKITAS 1777 I 0.5 1 .o 8 2 0 15 L4 u Q 5 ’ 10 5 0.5 1 .o 8 Fig. 3. Tests of the same isotherms as in fig. 1 for the adsorption of TPP and TPO on Hg from 0.1 mol dm-3 LiCl methanolic solutions.Symbols as in fig. 2 plus: +, -0.6 and A, -0.4 V (us SCE). Plots of the Frumkin and virial isotherms are shifted -0.5 and - 1.0, respectively, along the y axis. phosphine oxide (MDPO), ethyldiphenyl phosphine oxide (EDPO), dodecyldiphenyl phosphine oxide (DDPO), tri(n)octyl phosphine oxide (TOPO), triphenyl phosphine oxide (TPO), triphenyl phosphine (TPP), triphenyl arsine (TPAs), triphenyl antimony (TPSb) and triphenyl bismuth (TPBi) at the mercury/methanolic solution of LiCl interface are taken from ref. (32)-(35), and the high precision experimental data for the adsorption of butan-2-01 on Hg from aqueous solutions of Na,SO, are taken from ref. (36) and (37). For all these systems, 0 against c data are available at least at the adsorption maximum. The procedure used for testing the applicability of the isotherms has been described elsewhere.ls The experimental values of the free energy of adsorption were first determined directly from 8 against c data using AGexp = - RT In UO)/c] (45) (46) (47) and the values obtained were compared with the corresponding theoretical values from - RT In [B/g(B,, * * * 9 B,, @I.AGcalc = In the case of butan-2-01, we used AGexP = -RT In UO)/(aA/ab)] where aA and a, are the bulk activities of butan-2-01 and water, respectively, instead of eqn (45). Some of these comparisons are given in fig. 1-5. The values of the parameters b; used in these calculations are given in table 1, whereas the values of B and Bi, in eqn (46), were determined by a least-squares fit of AGexP against 0 and are given inADSORPTION ISOTHERMS FOR ELECTROCHEMICAL PROCESSES 0.5 1 .o e 1 I 0.5 1 .o e Fig.4. Tests of eqn (9), (14) and (13) in plots (l), (2) and (3), respectively, for the adsorption of MDPO and EDPO on Hg from 0.1 mol dmP3 LiCl methanolic solutions at potentials indicated by the symbols of fig. 2 and 3. Points are experimental data plotted according to eqn (49, solid lines are calculated from eqn (46) using four Bi terms taken from table 3 and broken lines are calculated using only the B, term taken from table 3. 0.5 1 .o e 0.5 1 .o e Fig. 5. Tests of the same isotherms as in fig. 4 for the adsorption of TPP and TPO on Hg. Symbols as in fig. 4. tables 2-4. The values of 0, the root-mean-square deviation between calculated and experimental values of AG, are also shown.These values can be considered as a measure of the applicability of an isotherm. In table 4 the data for butan-2-01 refer only to the region where butan-2-01 lies flat on the mercury surface. We do not include data obtained using eqn (9), (13) and (14) with n 2 1 since they were not used forP. NIKITAS 1779 Table 1. Values of b; r - - 2 0.167 3 0.333 0.50 - 4 0.167 0.333 0.50 further analysis. Also, as expected, the applicability of these isotherms increases following the increase of n. Generally for values of n 3 4 a satisfactory description of the experimental data is observed (a < 0.2 in all cases). From fig. 1-5 and from the values of a we obtain the following results on the applicability of the isotherms. ISOTHERMS BASED ON CONTINUOUS-SOLVENT MODELS In this category the Hill-de Boer, free-volume, virial and Parsons isotherms generally give deviations from the experimental data.Only in the case of ACN, where we have experimental isotherms with values of 8 < 0.5, is there satisfactory agreement, but in all cases where the isotherms correspond to a wide range of 8 values the plots of AG against 8 are non-linear. The virial isotherm shows greater applicability if we take into consideration the linearity of the plots of AG against 8 or the values of a. However, these data do not enable us to accept the validity of the model of this isotherm. For this to be done it is necessary that the values of the second virial coefficient obtained from the isotherm are positive and almost twice the value of the area S, covered by an adsorbate molecule at saturation.The values of the second virial coefficient (B,) are given in table 5. These values are calculated from the interaction parameter a of eqn (5) by means of the relation B, = -a& (48) where Ts is the limiting value of the adsorbate surface excess. Note that only rarely does B, approach its expected value. The Conway-Barradas isotherm is the most applicable if we take into account the values of 0. However, as with the virial isotherm, this does not mean that the model of this isotherm is valid. Indeed, according to the model of the Conway-Barradas isotherm the parametersp, and qshould be positive, but this is not so for almost one-half of the systems studied, and even in the systems where p,, q > 0 they are not always in agreement with the expected values.Thus taking into consideration the structure and the orientation of the adsorbed molecules we expect p, and q to have the same values for TPP, TPAs, TPSb and TPBi, while p1 should increase in the order DDPO < TPO < EDPO < MDPO. However, the values ofp, and q show considerable deviations from the expected behaviour, showing that the model of the Conway- Barradas isotherm is invalid. ISOTHERMS BASED ON DISCRETE-SOLVENT MODELS From the isotherms of this category, the simple Flory-Huggins isotherm, eqn (1 2), has the lowest applicability, Bennes’ isotherm, eqn (8), gives a better description of the experimental data, while Frumkin’s isotherm, eqn (I), and obviously the isotherms of eqn (9) and (1 3)-( 15), which have more than two adjustable parameters, have the1780 ADSORPTION ISOTHERMS FOR ELECTROCHEMICAL PROCESSES Table 2.Results of statistical analysis of the electrosorption data for adsorption on a mercury electrode using Frumkin’s isotherm and isotherms based on continuous-solvent models isotherm Hill- Conway- parameter Frumkin de Boer free volume virial Parsons Barradas -AGe/RT a, PI 4 d - AGe/RT a, P1 4 d -AGe/RT a, PI 4 d -AGe/RT a, PI 4 d - AGe/RT a, PI 4 d -AGe/RT a, PI 4 0 -AGe/RT a, PI 4 d -AGe/RT a, PI 4 d -AGe/RT a? PI 4 U 5.08 0.95 0.06 3.61 0.06 0.05 4.09 1.54 0.15 10.78 - - - - 0.07 - 0.17 10.99 - 0.03 - 0.09 12.61 0.03 0.10 13.02 0.02 0.1 1 6.61 - - - 0.39 - 0.08 7.1 1 -0.1 1 - 4.34 3.54 0.80 3.46 1.16 0.08 2.84 5.42 1.21 1.73 13.13 10.60 - - - - - 0.99 16.48 12.91 - - 36.71 41.07 18.38 - 57.55 52.12 18.12 5.80 1.94 0.48 4.19 4.76 - - - - SN 4.12 7.87 1.62 3.88 3.90 0.12 2.03 11.20 2.57 - ACN - PN - TPP - 7.26 28.66 21.55 - TPAs - 12.99 35.42 26.26 - TPSb - 87.63 85.56 37.22 - TPBi - 130.68 108.19 36.62 5.53 5.93 0.98 1.66 11.52 - MDPO - EDPO - 5.24 0.08 0.10 3.66 - -0.66 - 0.06 4.32 0.35 0.28 11.61 - - 1.92 - 0.51 11.85 - 1.89 - 0.54 15.27 -3.01 0.52 16.70 - - 3.56 - 0.45 6.85 - 1.39 0.12 7.59 - 1.45 - - 0.37 20.0 - 8.53 4.84 4.56 0.38 - - 8.40 39.88 18.29 - - 375.29 505.29 9 100 - 565.83 - 724.38 9 100 - 3005.90 - 2366.74 B 100 -4433.95 3109.91 9 100 2.07 14.38 4.40 - - - - 30.03 54.93 - 5.23 - 2.22 - 0.35 0.06 3.61 -0.34 -0.51 0.05 4.19 - 5.54 - 3.06 0.18 10.85 0.72 0.59 0.16 10.98 0.00 -0.05 0.09 12.69 0.22 0.20 0.10 13.30 0.8 1 0.60 0.10 6.6 1 1.26 0.70 0.08 7.07 0.15 - 0.06 0.1 1 1.81 3.76 0.23 33.23 0.11P. NIKITAS 1781 Table 2.(continued) isotherm Hill- Conway- parameter Frumkin de Boer free volume virial Parsons Barradas 7.65 0.06 0.10 11.37 - 0.75 - 0.17 12.28 0.23 0.10 - 2.70 11.78 5.43 -4.80 14.76 6.98 2.99 13.61 9.2 1 - TPO - 13.17 25.84 11.07 - DDPO - 22.35 33.20 14.16 - TOPO - 6.26 29.35 18.87 - 8.73 - 1.80 - 0.28 12.93 - 2.89 0.22 - 13.06 - 1.56 - 0.5 1 - 253.90 266.27 9 100 - -510.67 453.21 9 100 - - 323.94 436.40 9 100 7.83 0.65 0.71 0.1 1 11.85 4.26 2.58 0.13 12.31 - 0.60 -0.18 0.10 highest applicability. However, the applicability of an isotherm does not mean that the model of the isotherm and the conclusions obtained from it are valid.In order to reach certain conclusions about the validity of these isotherms we have analysed the physical meaning of the interaction-parameter values for TPP, TPAs, TPSb, TPBi, MDPO, EDPO, DDPO, TPO and TOPO. In the case of Frumkin’s isotherm the interaction parameter a has almost the same value for TPP, TPAs, TPSb, TPBi, and TPO, while it decreases in the order TOPO > TPO > EDPO > MDPO > DDPO. The constancy of a may be justified as follows. TPP, TPAs, TPSb, TPBi and TPO are adsorbed with their three phenyl groups in contact to the mercury surface. Hence (a) the attractive London forces between the molecules have the same intensity, because they are limited to being between the phenyl groups, and (b) the dipoles of TPP, TPAs, TPSb and TPBi, because they have the same orientation at the interface, interact repulsively.However, they contribute insignificantly to the value of wAA, since the dipole moments of these molecules are very low and also because the distances between the adsorbed dipoles are significant. For the same reasons the contribution of the dipole-dipole interactions between molecules of TPP, TPAs, TPSb or TPBi and a solvent molecule to the potential wAS is expected to be small. Therefore the particle-particle interactions at the interface do not differentiate, within experimental error, between TPP, TPAs, TPSb and TPBi. For TPO the dipole-dipole interactions are larger and result in a small increase in the value of wAA. However, in the negative region, where TPO has its maximum adsorption, an increase in the value of wAS is expected because of dipole-dipole interactions.These changes of wAA and wAS should be small, they are in the same direction and become neutralized to such an extent that a does not change within experimental error. The decrease in a in the order TOPO > TPO > EDPO > MDPO > DDPO can be justified as follows.33 With the only exception being the value of a for DDPO, the order seems reasonable, since the decrease in the attractive1782 ADSORPTION ISOTHERMS FOR ELECTROCHEMICAL PROCESSES Table 3. Results of statistical analysis of the electrosorption data for adsorption on a mercury electrode using isotherms based on discrete-solvent models isotherm ~~ parameter eqn (8) eqn (9) eqn (12) eqn (1 3) eqn (1 3) eqn (14) eqn (14) 4.49 5.99 2.03a - - - 0.19 2.89 3.71 1.05 0.04 3.90 5.45 2.loa - - - - 0.40 5.51 10.57 1.84 - - - 0.82 5.60 10.83 1.77 - - - 0.55 5.82 12.13 1.97 - - - 0.82 4.49 5.57 1.03 - 0.86 - 0.68 -0.70 0.03 2.89 3.56 0.77 0.04 3.90 5.22 1.23 -0.14 - 0.82 -0.61 - 1.13 0.1 1 5.5 1 10.69 1.67 0.30 0.50 0.40 0.15 5.60 10.93 1.67 0.28 0.44 0.2 1 0.10 5.82 12.72 1.70 0.26 0.32 0.45 0.12 SN 2.80 5.79 2.80 - - - 0.29 1.95 3.46 0.82 0.05 2.28 5.59 3.07 ACN - PN - - - 0.2 1 3.98 10.77 5.46 TPP - - - 1.80 3.89 11.00 5.50 TPAs - - - 1.63 3.96 10.72 9.08 TPSb - - - 2.80 5.99 2.34a - - - 0.18 1.95 3.67 0.94 0.05 2.28 5.51 2.26 - - - - 0.3 1 3.98 10.66 2.44 - - - 0.60 3.89 10.94 2.39 - - - 0.4 1 3.96 12.53 2.46 - - - 1.64 0.60 2.80 5.94 2.07 -0.19 0.10 - 0.26 0.03 1.95 3.55 0.72 -0.13 0.04 2.28 5.48 1.91 -0.33 - 0.03 -0.71 0.08 3.98 10.69 2.3 1 0.67 0.43 0.17 3.89 10.94 2.3 1 -0.10 0.62 0.23 0.10 3.96 12.86 2.26 - 0.02 0.25 0.66 0.13 - 0.09 3.0 5.98 2.09 - - - 0.14 2.0 3.65 0.89 0.05 2.0 5.54 2.03 - - - - 0.26 4.0 10.63 2.22 - - - 0.62 4.0 10.90 2.17 - - - 0.41 4.0 12.37 2.31 - - - 0.60 3.0 5.92 1.85 0.03 0.03 2.0 3.55 0.70 -0.11 0.04 2.0 5.52 1.77 -0.21 0.02 - 0.63 0.07 4.0 10.69 2.06 0.05 0.58 0.49 0.15 4.0 10.94 2.06 0.04 0.53 0.27 0.09 4.0 12.85 2.01 0.10 0.3 1 0.49 0.13 -0.16 - 0.27P.NIKITAS 1783 Table 3. (continued) isotherm parameter eqn (8) eqn (9) eqn (12) eqn (13) eqn (1 3) eqn (14) eqn (14) 5.90 12.40 1.98 - - - 0.84 4.35 6.09 1 .oo - - - 0.09 4.75 6.87 1.33 - - - 0.14 5.45 7.57 1.64 - - - 0.34 8.96 10.33 1.73 - - - 0.44 10.90 12.46 1.94 - - - 0.3 1 5.90 13.37 1.64 0.34 0.18 0.64 0.13 4.35 5.94 0.72 - 0.3 1 -0.19 0.1 1 0.07 4.75 6.95 1.50 0.28 0.19 - 0.08 0.10 5.45 7.72 1.71 0.3 1 0.38 0.18 0.08 8.96 10.64 1.86 0.40 0.27 0.25 0.09 10.90 12.42 1.99 0.03 0.33 - 0.08 0.08 TPBi 4.15 9.62 11.15 - - - 1.53 MDPO 3.06 5.87 1.72 - - - 0.22 3.36 6.68 3.18 EDPO - - - 0.5 1 3.78 7.10 5.50 TPO - - - 1.06 DDPO 6.77 8.38 11.92 - - - 2.12 8.67 12.55 14.30 TOP0 - - - 3.70 4.15 12.94 2.49 - - - 0.62 3.06 6.18 1.50 - - - 0.08 3.36 6.96 1.87 - - - 0.16 3.78 7.76 2.20 - - - 0.26 6.77 11.41 3 .06" - - - 0.59 8.67 13.16 3.89" - - - 2.25 4.15 13.62 2.20 0.10 0.06 0.84 0.15 3.06 5.90 0.98 - 0.61 - 0.4 1 -0.16 0.07 3.36 6.98 1.98 0.08 0.42 - 0.02 0.10 3.78 7.74 2.3 1 - 0.0 1 0.54 0.18 0.09 6.77 10.80 3.26 0.79 0.09 8.67 12.39 3.84 1.14 0.26 -0.36 - 0.09 - 1.14 - 1.05 4.0 4.0 13.68 13.63 2.33 1.82 - 0.3 1 - -0.10 0.93 0.65 0.15 - 3.0 3.0 6.1 1 5.96 1.03 0.73 - -0.35 - 0.38 - 0.07 - - 0.08 0.07 3.0 3.0 6.90 6.99 1.31 1.48 - 0.29 0.37 0.12 0.15 0.10 - - 4.0 4.0 7.67 7.73 2.06 2.1 1 0.1 1 0.47 0.21 0.27 0.08 - - - a Values greater than the critical value of Bo.1784 ADSORPTION ISOTHERMS FOR ELECTROCHEMICAL PROCESSES Table 4.Results of statistical analysis of the electrosorption data for butan-2-01 on a mercury electrode at different electrode charges electrode charge/& cm-2 -7.0 -6.0 -5.0 -4.25 -4.0 -3.25 -2.2 -AGe/RT a 0 -AGe/RT a 0 -AGe/RT a 0 -AGe/RT a 0 -AG@/RT a d - AGe/RT P1 4 d -AGe/RT BO d - AGe/RT a d - AGe/RT BO 0 - AGe/RT BO d 3.02 0.28 0.09 2.44 2.12 0.12 2.44 5.58 0.33 3.19 0.14 1.19 9.99 1.42 2.84 -0.65 - 2.99 - 3.84 0.0 1 3.26 1.64 0.14 Frumkin’s isotherm 3.1 1 3.15 3.05 0.49 0.72 0.98 0.09 0.05 0.05 Hill-de Boer isotherm 3.1 1 4.87 7.52 0.35 0.96 2.15 2.00 0.84 - 1.27 free-volume isotherm 1.40 - 1.07 -5.37 7.44 10.91 16.14 0.83 2.06 4.47 3.37 3.59 3.67 0.18 0.20 0.23 virial isotherm -0.58 -0.58 -0.54 Parsons’ isotherm 17.49 37.39 83.01 3.93 12.55 38.13 Conway-Barradas isotherm 2.96 3.14 3.22 -3.75 - 19.21 -58.13 -3.21 -2.83 -2.45 -2.97 -1.68 -0.69 0.02 0.03 0.06 Bennes’ isotherm with r = 5 3.55 3.82 3.98 1.67 1.68 1.75 0.18 0.20 0.20 2.97 1.10 0.08 - 2.56 9.05 2.99 - 7.96 19.17 6.15 3.68 0.23 -0.53 -90.17 119.42 62.62 3.27 - 2.09 -0.12 0.07 4.02 1.81 0.20 Flory-Huggins isotherm with r = 2.79 2.86 3.24 3.60 3.84 3.84 1.93 2.41 3.03 3.71 3.71 0.01 0.07 0.22 0.44 0.44 isotherm of eqn (13) with n = 0 and r = 2.79 3.30 3.58 3.84 4.00 4.03 1.81 1.85 1.91 2.05 2.13 0.15 0.19 0.18 0.17 0.17 3.23 3.54 3.81 3.97 4.0 1 1.53 1.61 1.71 1.87 1.95 0.13 0.16 0.15 0.14 0.15 isotherm of eqn (14) with n = 0 and r = 3 2.73 2.43 1.45 1.81 0.20 0.44 -8.31 -22.19 15.29 30.69 6.79 22.48 - 19.16 -47.22 32.04 62.37 15.14 45.14 3.69 3.70 0.25 0.25 -0.48 -0.46 - 3 16.07 - 1684.57 370.30 1828.62 280.14 B 100 3.36 3.57 - 1.25 0.55 1.25 3.45 0.14 0.30 4.06 3.91 2.05 2.47 0.3 1 0.90 4.12 4.89 0.97 4.07 2.43 0.30 4.05 2.26 0.3 1 4.24 5.88 1.63 4.0 1 2.86 0.76 3.98 2.70 0.80P.NIKITAS 1785 Table 5. Values of the second virial coefficient for the adsorption systems studied B2 S A adsorbate /nm2 molecule-' /nm2 molecule-' /nm2 molecule-' SN ACN PN TPP TPAs TPSb TPBi MDPO EDPO TPO DDPO TOPO butan-2-01 0.03 0.16 -0.10 1.42 1.36 2.22 2.75 0.79 0.90 1.27 3.63 2.5 1 0.22-0.16 0.0 1 0.08 - 0.05 0.7 1 0.68 1.1 1 1.37 0.40 0.45 0.63 1.82 1.26 0.1 1-0.08 0.35 0.24 0.28 0.74 0.72 0.74 0.77 0.57 0.62 0.70 1.25 1.61 0.34 London interactions and the increase in the repulsive interactions due to the -0 dipole between the adsorbed molecules are in the same order.However, we expect a of TOPO to have a greater positive value. As far as DDPO is concerned, its a value should be near to that of TOPO and between those of TOPO and TPO. The weakness of the Frumkin isotherm in giving these values of a for TOPO and DDPO supports our view that the differences in sizes of the adsorbed molecules contribute to the value of the interaction parameter a of this isotherm.In the case of Bennes' isotherm, the interaction parameter B, decreases in the order TOPO > DDPO > TPO > EDPO > MDPO, i.e. the change of B, values follows the expected order. Though B, does not have the same value for TPP, TPAs, TPSb and TPBi, the scatter of its values, which may be due to experimental errors, extends from the Bo value of TPO to that of TOPO. This shows that the values of Bo for TOPO and DDPO are also small, obviously because Bennes' isotherm does not take fully into account the size of the adsorbed molecules. The contribution of the differences in sizes of the adsorbed molecules to the values of the interaction parameters becomes clear in the case of eqn (9) and (13).These isotherms have more than one interaction parameter and a direct comparison between them does not lead to any conclusion. However, the values of Bi may be correlated with the interfacial behaviour of the adsorbate if the excess free energy of the adsorbed phase is calculated using eqn (44), which gives the excess free energy with respect to the free energy of an ideal, in the case of eqn (9), and an athermal, in the case of eqn (1 3), interfacial solution. Fig. 6 shows ge curves calculated from the values of Bi of eqn (9) and (1 3). The positive values of ge show that both the adsorbate and solvent molecules tend to form clusters at the interface. This means that there are relatively strong attractive interactions among the adsorbate molecules.Hence, the observed increase of ge values in the order MDPO < EDPO < TPO < DDPO < TOPO is reasonable. However, the values of ge appear to be depressed for DDPO and TOPO when calculated using eqn (9). This can be interpreted as follows. The values of the excess free energy of athermal binary solutions with respect to an ideal one are negative and in absolute values they increase following the increase of r.18 Consequently, in the case of eqn (9) and for the substances studied, the size of the adsorbed molecules 59 FAR 11786 ADSORPTION ISOTHERMS FOR ELECTROCHEMICAL PROCESSES 1.0 0.0 0.5 1.0 0.0 0.5 1 .o X A Fig. 6. Change in the excess free energy ge of adsorption of (1) MDPO, (2) EDPO, (3) TPO, (4) DDPO and (5) TOP0 as a function of their surface molar fraction.ge is calculated from eqn (44) using four Bi terms of eqn (9) (A) and eqn (13) (B). and the intermolecular interactions contribute in a competitive manner to the value of ge. However, in the case of eqn (13), because the size factor is taken more into account, the free energy ge is due mainly to the particle-particle interactions at the interface, and thus eqn (1 3) is preferred if we want a molecular interpretation of the adsorption data. The same is expected to hold for eqn (14) and (1 5) since the size factor is again treated more correctly. CONCLUSIONS The theoretical and experimental analysis of the isotherms leads to the following conclusions. The Hill-de Boer, free-volume, Parsons and virial isotherms based on continuous- solvent models are inadequate to describe the behaviour of an adsorbed layer, on both a molecular and a macroscopic level.Their models are far from the reality of the adsorbed phase and their applicability to the experimental systems studied is small. The Conway-Barradas isotherm of the same category has a higher applicability although the validity of its model is also small, since the conclusions about the interactions obtained from it are usually erroneous. The Frumkin, Bennes, Flory-Huggins and generalized isotherms, eqn (9) and (1 3)-( 15), all based on discrete-solvent models, are closer to the physical behaviour of the adsorbed layer. Of these isotherms, the simple Flory-Huggins isotherm presents remarkable deviations from the experimental data, Bennes’ isotherm gives a satisfactory description of the adsorption isotherms, while eqn (9) and (1 3)-( 15) and Frumkin’s isotherm have the highest applicability. Irrespective of the applicability of the various isotherms of this category, the experimental and theoretical data we have analysed show that eqn (13) is preferred if we wish to obtain reliable information aboutP.NIKITAS 1787 particle-particle interactions at the interface. The validity of eqn (14) and (1 5 ) is also expected to be high in experimental systems which fulfill the requirements of their models. A. N. Frumkin, Z. Phys., 1926, 35, 792. E. Guggenheim, Trans. Faraday Soc., 1945,41, 150. B. B. Damaskin, 0. Petrii and V. Batrakov, Adsorption of Organic Compounds on Electrodes (Plenum Press, New York, 1971).R. Aveyard and D. A. Haydon, An Introduction to the Principles of Surface Chemistry (Cambridge University Press, Cambridge, 1973). T. L. Hill, J. Chem. Phys., 1952, 20, 141. S. K. Rangarajan, J. Electroanal. Chem., 1973, 45, 283. R. Parsons, J. Electroanal. Chem., 1964, 7, 136. R. Parsons, Trans. Faraday Soc., 1955,51, 1518. lo R. Parsons, Trans. Faraday Soc., 1959, 55, 999. I ' E. Blomgren and J. O'M. Bockris, J. Phys. Chem., 1959,63, 1475. l 2 B. E. Conway and R. G. Barradas, Electrochim. Acta, 1961, 5, 319. l 3 R. Bennes, J. Electroanal. Chem., 1979, 105, 85. l4 J. Lawrence and R. Parsons, J. Phys. Chem., 1969, 73, 3577. l 5 H. P. Dhar, B. E. Conway and K. M. Joshi, Electrochim. Acta, 1973, 18, 789. P. Nikitas, J. Chem. Soc., Faraday Trans. I , 1984, 80, 3315. S. Levine, J. Mingins and G. M. Bell, J. Electroanal. Chem., 1967, 13, 280. 'I S. Sarangapani and V. K. Venkatesan, Electrochim. Acta, 1979, 24, 975. l8 E. A. Guggenheim, Mixtures (Oxford University Press, London, 1952). l9 T. L. Hill, An Introduction to Statistical Thermodynamics (Addison-Wesley, London, 1962). 2o A. Sanfeld, in Physical Chemistry: an Advanced Treatise, ed. H. Eyring, D. Henderson and W. Jost 21 P. Nikitas, J. Electroanal. Chem., 1984, 170, 333. 22 A. Miinster, Statistical Thermodynamics (Academic Press, New York, 1974), vol. 2. 23 E. A. Guggenheim and M. L. McGlashan, Proc. R. SOC. London, Ser. A , 1950, 203,435. 24 M. L. McGlashan, J. Chem. Educ., 1963, 10, 516. 25 W. R. Fawcett, J. Phys. Chem., 1978, 82, 1385. 26 R. Parsons, J. Electroanal. Chem., 1964, 8, 93. 27 R. Parsons, J. Electroanal. Chem., 1975, 59, 229. 28 M. V. Sangaranarayanan and S. K. Rangarajan, J. Electroanal. Chem., 1984, 176, 29. 29 A. De Battisti and S. Trasatti, J. Electroanal. Chem., 1973, 48, 213. 30 A. De Battisti, V. Faggiano and S. Trasatti, J. Electroanal. Chem., 1976, 73, 327. 31 B. AM-El-Nabey and S. Trasatti, J. Chem. Soc., Faraday Trans. 1, 1975, 71, 1230. 32 P. Nikitas, A. Anastopoulos and D. Jannakoudakis, J. Electroanal. Chem., 1983, 145, 407. 33 P. Nikitas, A. Pappa-Louisi and D. Jannakoudakis, J. Electroanal. Chem., 1984, 162, 175. 34 A. Pappa-Louisi, P. Nikitas and D. Jannakoudakis, Electrochim. Acta, 1984, 29, 515. 35 P. Nikitas, A. Pappa-Louisi and D. Jannakoudakis, J. Electroanal. Chem., 1985, 184, 109. 36 H. Nakadomari, D. M. Mohilner and P. R. Mohilner, J. Phys. Chem., 1976, 80, 1761. 37 D. M. Mohilner, H. Nakadomari and P. R. Mohilner, J. Phys. Chem., 1977, 81, 244. (Academic Press, New York, 1971), vol. I. (PAPER 4/ 1290) 59-2

ISSN:0300-9599    

DOI:10.1039/F19858101767

出版商:RSC    

年代:1985

数据来源: RSC

摘要:

J. Chem. SOC., Faraday Trans. I , 1985,81, 1789-1796 Solid-Solid Phase Transitions in K,Cr,O, at Pressures below 0.4 GPa from Differential Thermal Analysis under Hydrostatic Pressure BY LEWIS H. COHEN* Department of Earth Sciences, University of California, Riverside, California 92521, U.S.A. AND WILLIAM KLEMENT JR* School of Engineering and Applied Science, University of California, Los Angeles, California 90024, U.S.A. Received 10th September, 1984 Differential thermal analyses of reagent-grade K,Cr,O,, under Ar pressure up to 0.4 GPa, yield accurate temperatures for two reversible solid-solid transitions and some data for melting. Onset temperatures for the transition of the low-temperature polymorph (low) to the high-temperature polymorph (high) increase linearly with pressure from 255 "C at 0.1 MPa, with slope 0.28, pK Pa-' ; for the high + low transition onset temperatures also increase linearly with pressure, but from 237 "C at 0.1 MPa and with slope 0.30, pK Pa-'.A new phase, y, was discovered, with the low-high-y triple point located near 0.13, GPa and 279 "C. The y-high transition was followed to melting, near 0.41 GPa and 457 "C, and also metastably below the triple point with low; hystereses are 5 4 K for the y-high transition. Relative transition entropies were deduced from areas under the peaks of the curves of differential temperature against time; near the low-high-y triple point the y-high transition entropy appears slightly greater than that for low-high. The low-y transition was not detected by these techniques and a quench from the y field to ambient conditions did not yield an X-ray powder diffraction pattern distinctly different from low.Variation of volume with temperature at 0.1 MPa is deduced from the critically evaluated 0.1 MPa data plus the constraints imposed by the hgh-pressure data. In 1833 Mitscherlichl reported a solid-solid transition in K2Cr20, at ambient pressure. Numerous subsequent investigations at 0.1 MPa still have not produced good agreement on the characteristics of the phase change(s) involved. It seems accepted2-* that crystals grown from aqueous solution contain 2 0.02 % water present in cavities; this water is released if heated below the transition(s) for lengthy times and more easily if the crystals are powdered. Most investigators who have used crystals grown from aqueous solution have observed a phase transition on the initial heating which could not be reproduced on subsequent heating cycles.This initial, 'ephemeral' transition has attracted more experimental investigation than the reproducible, rapidly reversible solid-solid transition which persists subsequent to the initial heating. The extensive studies of Clark et aL5 on the solid-solid transitions effectively discredited many previous data and interpretations but also presented hypotheses, such as that on transition hysteresis, which have encouraged the present, precise investigation by differential thermal analysis (d. t.a.) under hydrostatic pressures. The only previous high-pressure investigations was exploratory in nature and its conclusions are discredited by the present work in the overlapping pressure range.17891790 PHASE TRANSITIONS IN K,Cr,O, The present investigation shows that the initial ephemeral transition is of as little importance at high pressures as it is at 0.1 MPa. The rapidly reversible 0.1 MPa solid-solid transition is followed with increasing pressure until another rapidly reversible solid-solid transition is encountered. This discovery establishes the existence of another polymorph of K,Cr,O,. The variations with pressure of the hystereses in these two transitions are accurately established and thus delimit the equilibrium phase relations. Transition entropies are obtained from the d.t.a. experiments; thermo- dynamic data, such as the temperature variation of volume at 0.1 MPa, are also deduced.EXPERIMENTS AND RESULTS Unpowdered Mallinckrodt analytical-reagent-grade K,Cr,O, was loaded directly into Pt capsules7 which had been sealed only at one end so that water could escape from the sample. The portion of the capsule containing the sample was carefully deformed to produce a nearly planar surface to which the chromel-alumel measuring junction was lashed and silver paint was used to enhance thermal contact. D.t.a. experiments were carried out as b e f ~ r e , ~ with Ar pressure known accurately to f 0.7 MPa and temperature believed accurate to within 2 K; the effects of pressure on thermocouple e.m. f. were considered negligible. Heating and cooling rates through the transitions ranged from 0.1, to 1.1 K s - ~ , ~ but no variation in transition temperature with rate was discernible.Most cycles were made with rates of ca. 0.4-0.6 K s-l. In the first run, with 9.8 mg of sample in 115 mg of Pt, the initial heating at 15, MPa yielded a step-like differential temperature signal with onset 274.5 "C. Further heatings did not reproduce this signal or any other at nearby temperatures and pressures. Signals observed thereafter upon cycling the temperature were well defined, and their onsets, taken for the transition temperatures, were reproducible to _+ 1 K. Shown in fig. 1 are the temperatures obtained for the low -+ high transition on heating and the high + low transition on cooling. Upon increasing the pressure, signals from another transition were first encountered near 0.15 GPa; the onsets were reproducible to & 1 K and were especially abrupt on cooling.A signal corresponding to the low + high transition was observed only on the first heating (fig. 1). A hitherto unknown phase transition had been encountered, and the new phase is herein designated y. The y + high and high+y transition temperatures then determined (fig. 1) vary differently with pressure than do the high-low transition temperatures. Also there is considerably less hysteresis for the y-high transition than for the low-high transition. The small hysteresis for the y-high transition also provides an upper limit for any thermal lag which might be envisioned for the very small sample under the relatively fast heating/cooling rates. Near 0.406 GPa and 457 "C a transition was encountered on heating which yielded an enormous d.t.a.signal; this was interpreted as melting and, because the capsule had not been sealed, temperature was quickly reduced. There was no attempt at confirmation because of the unsealed capsule. Subsequently, only poor signals for the solid-solid transitions were observed at lesser pressures and so the run was terminated. In the second run, with 10.6 mg of sample in 113 mg of Pt, the initial heating at 23., MPa yielded a step-like signal as before, with onset 275 OC, which was not reproducible. Low + high and high -+ low signals were observed with increasing pressure; the corresponding transition temperatures are shown in fig. 1. Near 0.12, GPa, d.t.a. signals were observed which correspond to y + high and low + high on heating and to high + y and high -+ low on cooling; onsets for the high -+ y signalL.H. COHEN AND W. KLEMENT JR v - vv - LOW 3791 400 V i2 300[Li A overlapped the high + low peaks and thus could not be determined with the usual precision. Near 0.14, GPa a d.t.a. signal corresponding to the low + high transition was observed only on the first heating (fig. 1). y + high and high+? transition temperatures were then determined upon increasing the pressure. Attempts were made to find signals corresponding to the y -+ low transition. First, the sample near 0.15, GPa was cooled from below the high + y transition to CQ. I70 "C. Secondly, pressure on the sample near 240 "C was reduced from 0.174 to 0. I 15 GPa fairly steadily over ca. 6 min. Thirdly, pressure on the sample near 275 "C was reduced from 0.186 to 0.129 GPa over ca.6 min. No potential signal was observed in any of these excursions. In the latter two cases, however, the y + high signal was observed on heating. So as to try to quench y and retain it metastably at 0.1 MPa, the sample was cooled as rapidly as possible from ca. 295 "C near 0.33 GPa towards ambient temperature and then the pressure was released rapidly. This quenched specimen was then examined by X-ray powder diffraction with monochromatized Cu K, radiation, mechanical working of the sample being mini- mized. The seven largest peaks in the interval 28 = 18-28" were similar in location, if not in relative intensities, to those in published patterns [fig. 3 of ref. (9)] for stock K,Cr,O,, as-received and well powdered.X-ray diffraction on the unpowdered stock material used here similarly agreed with the published patterns. Since careful X-ray diffraction work5 notes the ' unpredictable X-ray intensity variations' for K,Cr,O, cycled through the transformations at 0.1 MPa, the present work, with only milligram samples of quenched specimen, finds no useful distinctions among the diffraction patterns for the quenched specimen and the present or others' starting material. Areas under d.t.a. peaks were obtained by counting squares on the recorder traces after drawing plausible 'base lines'. The validity of this analysis10 depends on the peak areas being proportional to transition enthalpies independent of heating/cooling rate,1792 3 v) U 2 : .- ; A 2 2 a a ) - E - a U ." .- c - 2 - e I - U - PHASE TRANSITIONS IN K,Cr,O, I 1 ~ ~ ~ , ~ 1 1 , 1 1 1 1 , 1 1 1 1 , 1 1 , , , 1 1 1 , - - iZ' I I; - - - 7' - xi !/y7! +\ 7 - \! h\ - - - - F 3; pressure or temperature.' Relative transition entropies ' were obtained by dividing each peak area by the onset temperature. The data from the two runs were combined by weighting per sample mass and are shown in fig. 2 with the spread shown by error bars or symbol size. No systematic discrepancies between heating and cooling data are evident. Near the triple point the transition entropy for y + high appears greater than that for low -, high, which will require that the transition entropy for y -+ low 2 0. The low-y transition must be first-order because of the previously noted metastability. Linear regressions of transition entropies per unit mass (in the arbitrary units indicated in fig.2) as a function of pressure yield the following intercepts and slopes (per GPa): for low-high: 2.2, and - 7.8 for the seven data from multiple cycles; 2.3, and -8.6 for all data, with the square of the linear-correlation coefficient r2 = 0.89 for both; for high-y: 0.9, and 5.3 for the twenty data from multiple cycles, with r2 = 0.81 ; 0.9, and 5.4 for all data, with r2 = 0.86. For lack of statistically significant difference in these fits to the two classes of data for each transition, only one line for each is shown in fig. 2. Regression analysis of the pressure dependence of the transition temperature yields the following results: low + high: T/"C = 254., + 0.28, p/MPa high + low: T/"C = 23%, + 0.30, p/MPa with r2 = 0.998 for both.These agree well with the 0.1 MPa data,5 uiz. low -+ high near 528 K and high -+ low near 508 K. Over the pressure interval 0.12435 GPa, hystereses of 5 4 K, unaffected by pressure, are discerned. Least-squares fits of high-y transition temperatures yield, for 14 data from multiple cycles on heating for both runs combined: T/"C = 171.,+0.87, x lO3p-O.40, x 103p2 (p in GPa)L. H. COHEN AND W. KLEMENT JR 1793 with a standard error of 0.57 K; the 6 cooling data points of the first run yield a curve of similar initial slope (within 5 % ) and initial curvature (within IS%), with 0.14 K standard error. Extrapolation to 0.406 GPa yields the temperature for the onset of melting (the y-high-liquid triple point) in good agreement with the value of 457 "C found experimentally. DISCUSSION The several modern investigations of the solid-solid transition(s) at 0.1 MPa show too little agreement; the following critical evaluation is made so as to obtain preferred thermodynamic values necessary to correlate the high-pressure data.Calorimetry1' yields the preferred ASt/R = 0.347 f 9 (using their peak temperatures to convert enthalpy); their onset of 257 f 1 "C for low + high differs only slightly from the present (extrapolated) 0.1 MPa value. Others9 find values which can be considered corroborative. The decrease in ASJR with pressure (fig. 2) does not appear to dominate the low-high hysteresis variation as found in red-yellow Hg12.10 Rather, the dominant effect seems that of decreasing hystereses often accompanying increasing transition temperatures. l2 Since it is always assumed that the equilibrium phase boundary lies within a hysteresis interval, the y-high equilibrium boundary is well constrained within the hysteresis interval of < 4 K.However, where is the high-low equilibrium boundary and, concomitantly, the high-low-y triple point? It may be a relevant clue that only the first heatings yielded low + high signals (near 0.14 and 0.15 GPa), with only the y-high transitions observed thereafter. If this were interpreted as implying metastability of low with respect to y there, then the triple point would be 5 0.140 GPa. The y-high equilibrium boundary intersects the high + low transition line near 0.13, GPa, with y-high readily reversible (metastably) to lower pressures.Thus, from this reasoning the low-high equilibrium boundary is very close to (but above) the high +low transition line near a triple point with coordinates 0.13, GPa, 279 "C. The Clausius-Clapeyron equation, d TJdp = A &/ASt, constrains the volume and entropy changes across reversible transitions. For the equilibrium low-high transition, A&/R x 0.35 requires A z 0.86 f 0.02 cm3 mol-l, which is roughly compatible with the ca. 1 % volume change reported from di1at0metry.l~ Using the estimate for the initial variation in low-high transition entropy with pressure (fig. 2): dAS, - = -r$)p+%r2) = -(10.5+0.8) x lop3 cm3 mol-1 K-l. dP dP aT p Linear extrapolation of the calorimetric data" for c,/T against temperature, in the transition region, leads to bounds on (dTJdp) (aASt/aT), of (1.9f0.6) x Unfortunately the 0.1 MPa volumetric data are incomplete, but previous data can be critically assessed and, along with the present results, can be incorporated in a self-consistent analysis.For low at 25 "C the preferred volume is from X-ray diffraction work1, which has been corroborated.16 The thermal expansion for low,l7 however, is suspect and that for high is lacking. The average volumetric thermal expansion for low over 0-100 "C, obtained from various sources,17 is 13., x cm3 mol-l K-l. A possible problem with this value is that it was derived from macroscopic measurements, and there is a discrepancy at 25 "C between the volume of low from preferred macroscopic measurements1* and cm3 mol-l K-l.Thus @A &/t3T)p = (12.4 f 1.4) x cm3 mol-l K-l.1794 PHASE TRANSITIONS IN K,Cr,O, 118 110 108 0 I 00 200 300 400 T I T Fig. 3. Volumes as a function of temperature of different K2Cr,0, phases at 0.1 MPa. Arrows show ranges containing high-low transition temperature and melting temperature. Circles indicate volumes for high, low and monoclinic, from X-ray diffraction data (see text). the preferred datum from X-ray diffraction; presumably this discrepancy is due to porosity. As yet unindexed X-ray diffraction lines at 310 and 380 OC13 may be important; they cannot be indexed by analogy with the accepted high structure' and may not even be characteristic of pure K,Cr,O,. Vesnin and Khripin13 state that the X-ray diffraction pattern of a sample which had been melted was significantly different from the original, and it is believed4 that melting sometimes causes decomposition.A self-consistent analysis for the variation of volume with temperature at 0.1 MPa (fig. 3) can be made by assuming volumes vary linearly with temperature and selecting (aV/aT), x 17 x lop3 cm3 mol-l K-l for low and x 30 x cm3 mol-l K-l for high, which are compatible with the A 6 and @A K/a T)P selected above. The volume of high near the melting temperature is consequently estimated as 118.0 cm3 mol-l. Good agreementlg9 2o for the densities of the liquid yield the volume near the melting temperature as 128.7, cm3 mol-l. For AS,/R = 6.4,,' the predicted initial melting slope is ca. 0.2 pK Pa-'. This is compatible with the observed melting near 0.41 GPa, assuming that the curvature, -d2Tm/dp2, of the melting curve is positive, as is usual. For the initial, ephemeral transition observed on heating, this work presents the only high-pressure data so far obtained and is consistent with the view that it is related to water contained in crystals grown from aqueous solution. The enthalpies or entropies reported for this transition are not in good agreement. The best calorirnetryl1 suggests ASJR x 0.35, similar to the low-high transition, and presumably the heating was slow enough so that 'most' entrapped water was released below the temperatureL.H. COHEN AND W. KLEMENT JR 1795 ranges of the transitions. Another in~estigationl~ yields ca. 0.38 despite large (1 5 - 3 g) sample(s) and high (4-10 K min-I) heating rate(s).Others'. @ ? 22 agree less, with one' obviously based on apparatus calibration too far removed from the K2Cr,07 transition temperatures. On the other hand, the hypothesis5 of entrapped water would seem to require extra enthalpy in the initial, ephemeral transition corresponding to the vaporization of the water. Indeed, a finding that d.t.a. peaks for 'dried' material have about twice the enthalpy for the ephemeral transition as for the subsequent reversible transitions5 is approximately corroborated9 for as-received material. It is doubtful that the Clausius-Clapeyron equation applies to the ephemeral transition because it is not reversed, but were it applicable the small slope (fig. 1) combined with the volume change14 of 5.2% would require an enormous entropy change (which is never observed).The initial, ephemeral transition seems not to have any thermodynamic description nor any bearing on K,Cr,O, phase equilibria. Another hypothesis5 is that water vapour initially in the crystal exerts 'internal pressure' such that low is stabilized relative to high in the initial heating. As deduced from the present results for the trajectory of the high-low phase boundary, ca. 60 MPa would be necessary to increase the low + high transition temperature ca. 15 K, and this might exceed the strength of the crystal. The d.t.a. results5 for different particle sizes and for the physical fracturing on cooling cannot be correlated easily with the present results. Since only small samples are used here, thermal gradients within the sample would not seem to be a problem, as they may have been with the massive (ca.1 g) samples5 used in thermosonimetry. The fits (fig. 2) to the transition-entropy data suggest that the entropy change in the y -+ low transition is positive near the low-high-y triple point, and possibly of magnitude ASJR z 0.06. Using the slopes for the low-high and y-high boundaries the volume change for low -+ y near the triple point is estimated as - l., cm3 mol-1 and thus the slope for the low-y boundary there is 2., pK Pa-'. Given the foregoing estimate for the volume difference between low and y near the low-high-y triple point and the fact'? 23 that the volume difference near 25 "C at 0.1 MPa (fig. 3) between low and the well known metastable monoclinic form is not too different, it is attractive to suggest that y is this monoclinic polymorph.It is not impossible that y is the stable phase at or below room temperature at 0.1 MPa. One consequence might be that, were this monoclinic polymorph heated rapidly enough at 0.1 MPa, there would be a transition to high near 171 "C. So far, the only information about heating the monoclinic polymorph at 0.1 MPa is that it reverts to low at ca. 120 23 or 5 170 0C.24 The present results (fig. 1) for the phase relations of K,Cr,O, differ drastically from the previous high-pressure investigations with the simple squeezer because that apparatus is well known for its inaccuracy in pressure, certainly in the range < 0.4 GPa. Since melting temperatures in the previous investigations are so much lower than those encountered or deduced in the present work, the other phase boundaries in that investigation are also suspect. U.Klement and G-M. Schwab, 2. Kristallogr., 1960, 114, 170. F. W. Schwab and E. Wichers, J. Res. Natl Bur. Stand., 1944, 33, 121. J. Knoeck and H. Diehl, Talanta, 1969, 16, 181. T. Yoshimori and N. Sakaguchi, Talanta, 1975,22, 233. G. M. Clark, M. Tonks and M. Tweed, J . Thermal Anal., 1977, 12, 23. C. W. F. T. Pistorius, 2. Phys. Chem. (Neue Folge), 1962,35, 109. L. H. Cohen and W. Klement, J. Chem. Eng. Data, 1974, 19, 210. L. H. Cohen and W. Klement, Aust. J . Chem., 1981, 34, 1845. M. Natarajan and E. A. Secco, Can. J . Chem., 1979, 57, 2703. lo W. Klement and L. H. Cohen, 1984, J . Chem. Soc., Faraday Trans. 1, 1984,80, 1831.1796 PHASE TRANSITIONS IN K,Cr,O, l1 M. M. Popov and G. L. Gal’chenko, J. Gen. Chem. USSR, 1951,21, 2489. I2 L. H. Cohen and W. Klement, Philos. Mag., Part A, 1979, 39, 399. l3 Yu. I. Vesnin and L. A. Khripin, Russ. J. Znorg. Chem., 1966, 11, 1 188. l4 J. Jaffray and A. Labary, C. R. Acad. Sci., 1956, 242, 1421. l5 J. K. Brandon and I. D. Brown, Can. J. Chem., 1967,46,933. l6 G. Brunton, Muter. Res. Bull., 1973, 8, 271. Gmelins Handbuch der Anorganische Chemie (Verlag Chemie, Weinheim, 1962), vol. 8, part B, pp. 548 and 550. le W. H. Hartford, Znd. Eng. Chem., 1949, 41, 1993. J. M. Jaeger, 2. Anorg. Allg. Chem., 1917, 101, 1. 2O J. P. Frame, E. Rhodes and A. R. Ubbelohde, Trans. Faraday SOC., 1959, 55, 2039. 21 F. D. Rossini, D. D. Wagman, W. H. Evans, S. Lmine and I. Jaffe, Natl Bur. Stand. (U.S.) Circ., 22 J. L. Holm, Therm. Anal., 1980, 6, 105. 23 P. L. Stedehouder and P. Terpstra, Physica, 1930, 10, 113. 24 M. A. Duffour, C. R. Acad. Sci., 1913, 156, 1022. 1952,500. (PAPER 4/ 1557)

ISSN:0300-9599    

DOI:10.1039/F19858101789

出版商:RSC    

年代:1985

数据来源: RSC

摘要:

J. Chem. SOC., Faraahy Trans. I, 1985,81, 1797-1817 Double-layer Interactions of Unequal Spheres Part 1 .-The Effect of Electrostatic Attraction with Particles of Like Sign of Potential BY EYTAN BAROUCH* AND EGON MATIJEVIC* Departments of Mathematics and Chemistry and Institute of Colloid and Surface Science, Clarkson University, Potsdam, New York 13676, U.S.A. Received 1 1 th September, 1984 The Poisson-Boltzmann equation for two unequal spheres with fixed surface potentials is analysed. When both potentials are of the same sign but of different magnitude it is shown that the electrostatic energy decreases owing to the fact that a portion of the surfaces may be attractive even though the rest of the surfaces may be repulsive. This effect depends on the separation, as well as on the difference in the potentials and the sizes of the two spheres. I.INTRODUCTION The double-layer theory for a system of unequal spheres is more complicated than for the classical case of two parallel plates. The reason for the difficulties lies in the symmetry properties of the former system. While two parallel plates represent essentially a one-dimensional problem, two spheres of different sizes have only one axis of symmetry, connecting their centres as illustrated in fig. 1. The potential, 4, in the intervening medium between two unequal spheres with radii a, and a, of fixed surface potentials 41 and d2 and separation H, is usually described by the Poisson- Boltzmann (PB) equation, which for a symmetrical electrolyte reads 8mez ez4 v24 = - sinh - E kT where n is the number of positive (or negative) ions per cm3, e is the electronic charge, z is the valency of either type of ion, E is the dielectric constant in solution, k is Boltzmann’s constant and T is the absolute temperature.If one conveniently defines the dimensionless potential, w, by ry = ez4/kT (1.2) u2 = 8nne2z2/kT~ (1.3) V2w = K , sinh v / . (1.4) The above equation cannot be solved analytically for a two-dimensional system. One can either treat the non-linear boundary-value problem numerically or devise a reasonable approximation that can be systematically improved by an iteration scheme. Hoskin and Levine1*2 utilized the former approach for systems consisting of equal and Debye’s inverse length, K, as then the PB eqn takes the form 17971798 DOUBLE-LAYER INTERACTIONS Fig.1. Two spheres with radii a, and a, and separation of their surfaces H,. spheres. In the cited works equipotential surfaces were computed for various typical cases in which the symmetry of the systems was evident. Furthermore, the repulsive free energy and the force of repulsion between two spherical particles was given in a series as well as in a tabular form., It is the purpose of this study to use the analytical method in solving the two-dimensional PB equation, which is then applicable to systems of either equal or unequal spheres. The employed method consists of an initial approximation determined from scale analysis and an iteration scheme that improves it systematically. In the case of equal spheres a comparison with Hoskin’s results can then be made.The first serious attempt to tackle the question of interactions of unequal spheres was carried out by Hogg, Healy and Fuerstenau3 (HHF). They developed an approximate expression for the double-layer energy utilizing the procedures which Derjaguin applied to equal sphere^.^ The equation derived by HHF can be written in dimensionless form as Further progress was reported by various author^.^-^ Although eqn (1.5) is appealing because of its simplicity, the calculated energies often exceed values suggested by experiments.1°-15 It was suspected that one of the reasons for these discrepancies was the treatment of the symmetry of the system. To remedy this difficulty an expression was derived based on the non-linear PB equation in its two-dimensional form.18 This approach is sufficiently systematic such that higher-order corrections can be computed once the basic scale balance is understood.The developed equation for the double-layer energy reads (for the case tyl w2 > 0): E = (yy ( - nal ~ 2 9 6” y1 [(a, K ) ~ - R2]-i [cosh v / , - 4(R)]iR dR - na, ~ 2 ; luz V ~ [ ( ~ , I C ) ~ - R2]-1 [cosh w2 - 4 (R)]i R dR) (1.6) where q is the unit charge and 4(R) is determined fromE. BAROUCH AND E. MATIJEVIC 1799 where Aj = cosh lyj, for j = 1 , 2, H , = IC-, h, R = Icr and 2 = ( z - a'), z and r being the radial coordinates. Eqn (1.7) is valid for reasonably large separation h or for large enough radial coordinate R, for which the potential contains an extremal point. However, it was shown by Parsegian and Gingell17 that in the linearized parallel-plates case the system may become attractive at a sufficiently close separation.We are therefore facing the possibility that for smaller values of R the surfaces are attractive, while for larger R they are repulsive for the same fixed surface potentials. Thus a system of spheres could have a lower total interaction energy owing to the above reasoning.'* In this paper it is demonstrated that such a reduction in the total energy can indeed take place. For the sake of completeness, sections 2 and 3 summarize the classical aspects of the problem4 in terms of notation to be utilized throughout this work. 2. FORMULATION OF THE PROBLEM Consider the simplest PB geometry, i.e. parallel plates in the one-dimensional case : ly" = ~2 si nh v/ (2.1) Y/(O) = w1, w(L) = v / 2 7 w 1 # w 2 and in its linearized form: ly" = u2ly.The induced surface charge densities are given by (in suitable units) and the two surfaces are attractive [repulsive] when p(O)p(L) < 0 [p(O)p(L) > 01. The solution of eqn (2.2) with IC = 1 is [ly, sinh (L - x) + ly2 sinh x] (2.4) 1 ly=- sinh L and the induced surface charges are 1 sinh L p(L) = -- [ - ~ 1 + v / 2 cash L]. (2.5a) (2.5 b) We distinguish two cases: Case (i): lyl < 0, ty2 > 0, dtyldx > 0, 0 d x 6 L. For this case, v/(x) is a monotonic increasing function, p(0) > 0, p(L) < 0 and the system is always attractive. Case (ii): ly, > 0, ly, > lyl. For this case the product p(O)p(L) takes the form v / , > 0,1800 DOUBLE-LAYER 1N"ERACTIONS Eqn (2.6) implies the unexpected fact that the system is attractive for small separations, yet repulsive for the large ones.For p(0) = 0 the system has no electrostatic force, which takes place at a critical value L, given by (2.7) Consequently the system is attractive (repulsive) for L < L, ( L > L,). One might question the applicability of these considerations and the possible implications to less simplified systems. Note that a change of repulsion to attraction cannot occur when lyl = ly,. Rewriting eqn (2.6) for this case yields CoshL, = ly2/ly1 > 1. p(O)p(L) = v/4 (1 + COSh2 L - 2 cash L) = y/q(coshL-1)2 > 0 (2.8) which shows that the system is always repulsive and L, vanishes. In contrast one may expect that the more dissimilar the particles the more complicated will be the pattern of forces and interaction energies.This idea forms one of the basic points in this paper. 3. PARALLEL PLATES Consider eqn (2.1), with parallel-plates boundary conditions and K = 1. The first immediate integral is given by ( v / ' ) ~ = 2 cosh y/ + C (3.1) where C is a constant. The sign of the derivative at x = 0 determines the positivity or negativity of the surface charge density; hence one must consider two cases: Case (i): ly is monotonic. Since y1 < ly,, ly is a monotonic increasing function, i.e. dly/dx > 0 for 0 < x ,< L. In other words we choose the positive square root for 0 < x < L to yield explicitly dly/dx = +(2 cosh v / + 0;. (3.2) One more integration results in where C is obtained from This system is attractive and ~ ( x ) can be calculated by inversion of eqn (3.3), which Case (ii): ly is not monotonic.In this case the derivative of ly must vanish at an is a straightforward problem of elliptic integrals of the first kind. interior point. Let this point be x = Z, namely Note that in case of lyl = ly2, I = L / 2 for any L. Eqn (3.1) yields 0 = 2 C O S ~ ~ ( l ) + C.E. BAROUCH AND E. MATIJEVIC 1801 Thus, for 0 < x < 1 the solution v/ is decreasing monotonically, while for 1 < x < L it is increasing. This system is repulsive. Explicitly, dx - dW = - [2 cosh v / + c]: (0 < x < l) We integrate eqn (3.7a) and (3.7b) to obtain (3.7a) (3.7 b) (3.8a) (3.8 b) A simple manipulation gives which is the well known transcendental equation for ry(l) already shown as eqn (1.3) when Z 2 - Z , = L.If W , = y2, the symmetric case 1 = L/2 is immediate. To establish the existence of the critical length L, for ry, # ry2 one takes ~ ( 0 ) = ry, and dy//dxl,,, = 0, namely 0=2~0shry,+C. (3.10) Since C is known, L, is given by dW 2;(~0sh t// -cash t//$ ' As in the linear case, the system is attractive (repulsive) for L < L, ( L > L,). (3.11) 4. CURVED SURFACES Curved surfaces are clearly more common than parallel plates, and focusing our attention on two spheres we also retain the parameter K . Since the system of two spheres is cylindrically symmetric with the axis of symmetry connecting the two centres of the spheres, we reuse the scaling introduced earlieP and approximate the sphere surfaces by parabolae.This consideration raises the issue that different parts of the same surfaces could act (attract or repel) differently whenever the action is separation- dependent. Here it is demonstrated that this effect is true, resulting in smaller energies that lower the adhesion energy gaps. Consider the PB equation in cylindrical coordinates : a2 l a a az2 rar ar -ry+--r-W = ic2 sinhy/ (4.1) and boundary conditions ry = ry, when z2+r2 = a; ry = ry2 when ( z - a, -a2 - Ho)2 + r2 = af.1802 DOUBLE-LAYER INTERACTIONS Note that the angular-dependent term disappears owing to the symmetry around the main axis. Consider the scaling R = icr, Z = ic2 ( z - a l ) , Ho = IC-, h. (4.3) Eqn (4.2) then takes the form Ic-2R2+(al+iC-2Z)2 = a: (IC-~Z - a, - K-2h)2 + U - ~ R ~ = a: (4.4) and neglecting terms of order I C - ~ one obtains R2+2a1Z x 0 (on sphere 1) (4.5 a) R2 - 2a, (2- h) x 0 (4.5 b) (on sphere 2) and the scaled PB equation takes the form a 2 l a a K,-Y+--R-I,U = sinhw 322 RaR aR which is eqn (1.11) of ref.(16). (4.6), namely At this juncture it is convenient to consider the exact problem associated with eqn (4.7) In other words we are considering the two particles as parabolae, bearing in mind that this is an exact mathematical problem whose results are closely related to the two-sphere system. We also wish to consider the linear case [with boundary conditions (4.7)] as it is much simpler, yet contains a fair amount of useful information. Since both eqn (4.6) and (4.8) have on their left-hand sides a term of O(K,) and a term of 0(1), it is tempting to drop the radial term and expect that the radial dependence of w(2,R) is obtained from the boundary conditions.To check the validity of this argument, eqn (4.8) is most useful. Consider the approximate equation a 2 K2 w o = w o with eqn (4.7) as the boundary condition. The solution yo is given by (4.9) . (4.10) sinh {.' [ h +: R2 (i +-!-)I}E. BAROUCH AND E. MATIJEVIC 1803 On the surfaces Z = - R2/2a,, 2 = h + R2/2a,, t,uo = w, and yo = y,, respectively. In other words wo = O( l), K~ a2 wo/aZ2 = O( 1). Next, the radial term R-l(a/aR) R(a/aR) yo is of interest and can be written as = O(max [(rca,)-l, (~a,)-~]} = O( 1). (4.11) Eqn (4.1 1) explicitly demonstrates what is meant by stating a, and a, as being ‘fairly large’.This argument justifies the omission of the radial term, and it remains valid for the non-linear case [eqn (4.6)], although the proof is more cumbersome. Therefore we first consider the approximate scaled problem given by (4.12) a 2 az2 IC, - w (2, R) = sinh w (2, R) with boundary conditions (4.7), and immediate first integral 1 2 - ,K ( z;), = cosh r// (Z, R) - 4 (R). (4.13) Once #(R) is determined from the boundary conditions, y(Z, R) can be obtained Two cases need to be distinguished : by one more integration and classical inversion. Case (i): 1 R2 < 2 6 h+-R2 dw 1 - f O for -- dZ 2a1 2a2 (this is the case for w, > w1 and ty, w2 < 0, and for yl w, > 0 if Z < 2,). Since the derivative does not vanish, it does not change sign and the square root is taken positive. Explicitly, one obtains and q51 (R) is determined from the transcendental functional equation Under these conditions the two surfaces always attract each other.Case (ii) : 9 = o for some z = Zo(R). d Z In this case and R2 - < O for -- d z d Z,(R) dw dZ 2a1 dw 1 d Z 2% - > O for Zo(R)<Zdh+-R2. (4.14) (4.15)1804 DOUBLE-LAYER INTERACTIONS Thus the square root is obtained as R2 < z < zo(R) -- dv - - 2 9 ~ - ~ [ c o s h ~ / ( Z , R ) - # ~ ( R ) ] i for -- d Z 2 6 R2 9 = 2 k 1 [cosh v(Z, R) - 42 (R)]4 for Zo < Z < h +- . (4.16) d Z 2a2 Integrating eqn (4.16) formally yields Z,(R) = j (-2irc-l)dZ' (4.17a) = Jh+(1/Za2)R* (24~-l) dz' (4.17 b) 0 = cash v[Zo (R), R] -42 (R). (4.17~) The remaining task is to determine #(R).Combining eqn (4.17a-c) one obtains the functional equation for b2(R) : d v d v v[Z,(R)I jvl rcosh v-42 (R)li -(l/2a,)R2 ~ ~ ~ Z , , ) [cash v-42 (R)l' Z o ( R ) (4.18) To determine the critical function Z,(R) one may utilize eqn (4.13), the left-hand cash v1 - 4, (R) = 0. (4.19a) For the critical case, where v/ is monotonic, integration of eqn (4.13) using eqn (4.19 a) yields with Ai = cosh vi, i = 1, 2. Eqn (4.18) is identical to eqn (1.7). side of which vanishes at 2 = - R2/2a1, namely Since Zc is on surface 2, it satisfies the equation In other words, eqn (4.19b) determines a critical separation h from the equation (4.20) (4.19 b) radial coordinate R, for a given (4.21) The interpretation of this relation is rather surprising and counter-intuitive : the portions of the surfaces that are sufficiently close to each other (such as R < R,) are attractive, while the portions of the surfaces that are further apart (such as R > R,) are repulsive. In other words, the repulsive double-layer forces are decreased by the attractive forces.Such a conclusion does not follow from the original double-layer theory. This fact most likely accounts for the discrepancy between calculated and measured interaction energies for unlike SphereslO or plate-sphere systems,13-15 when approximate theories were applied. In these cases previously computed energies were too high, while the computed energies here are indeed systematically lower.E. BAROUCH AND E. MATIJEVIC 1805 5. INTERACTION ENERGY The attention is now turned to the analytical evaluation of eqn (1.6), using the ideas developed in sections 2-4.The computation of E is deceiving, since it looks straightforward yet contains several hidden difficulties. The main complication is embedded in the fact that the integrand is a function of R and #(R), with 4 ( R ) determined from a transcendental equation. The latter could have two different forms for the same separation and different radial coordinate R. A second complication is the necessity to subtract the self-energy, Eself, which should be done prior to evaluation of the energy to avoid the pitfalls of subtraction of two large numbers. In other words eqn (1.6) is viewed as an implicit form of the energy E and is derived below in an explicit form. In eqn (1.7), valid for separations above critical, #(R) + 1, yields h -+ co.The self-energy can now be written as Eself = P ( -al v1 JoKul [(a, K ) , - R2]-1 (Al - 1): R dR -a, y / , I""' [(a, K ) , - R2]-:(A2 - 1): R dR) (5.1) 0 with the constant P given by P = n&2:(kT/q)2 and the interaction energy is obtained by subtracting eqn (5.1) from (1.6) as E = P( -al ty, joKu' [ ( ~ , K ) ~ - R ~ ] - : { [ A ~ - ~ ( R ) ] ~ - ( A ~ - l);}RdR 0 KUZ --u2y2[ [ ( ~ , K ) ~ - R * ] - : { [ A , - ~ ( R ) ] : - ( A , - l):]RdR At this juncture one must distinguish two cases: (i)K-l h = K H ~ > I,, ( i i ) c l h = icHO c I,. Case (i). The system is always repulsive and 4(R) is always determined from eqn (4.18); 4 descends monotonically from its value at R = 0 to 1 and it varies in the range A, > A, > b ( R ) > 1.This monotonicity suggests a change of integration variable. Case (ii). The system can be either repulsive or attractive, which represents a more complicated case. Given K-' h = icHo, 4 (R) is determined again from eqn (4.18) for R > R,, but from for R < R,. In particular, the range of #(R) is bounded by A1 yet it does not have a lower bound, and in principle approaches - co as Ho,R -, 0. At R,, 4 = A1 as illustrated by eqn (4.21). The integration by parts has to be done with care. In the evaluation of E one must again consider two cases. Case (i): KH, > I,. We define the functions 6+ - (4) by1806 DOUBLE-LAYER INTERACTIONS and the constants (bh, b1 and 42 by KHO = d+(4h) ( 5 . 6 ~ ) (5.6 b) Integration by parts of eqn (5.3) yields Since #(I?) is monotonic over the entire range of integration, we change the integration variable and integrate over 4 to obtain which is the desired result.Note that the transcendental equation is solved only for three constants and not for the entire integrand. Case (ii): KH, < I,. For this case 4 h is determined from K H o = d-($h) while q$ are obtained if (5.9) (5.10) The condition (5.10) is sufficiently rare and presents no major difficulty, so it is not considered here. However, it is included in the numerical work and will be discussed in the next paper in this series.19 In formula (5.7) we change the variables and obtainE. BAROUCH AND E. MATIJEVIC 1807 A2-4 - 2 (&+ --!--)-' [a+ (4) - KH,] A2-4 )]} . (5.13) Note that both eqn (5.8) and (5.13), despite their cumbersome appearance, are considerably simpler than eqn (1.6), since they are explicit integrals with explicit integrands.Both a+(#) and a_(& are incomplete elliptic integrals of the first kind, and their computation (if done correctly) is very fast.20 6. SPHERE-PLATE LIMIT The case of the sphere-plate limit (a, finite, a, -+ GO) is of special interest because it applies to problems of particle adhesion, which has been extensively studied experimentally. Case (i): KH, > I,. We rewrite eqn (5.3) with P defined by eqn (5.2) as As before, one must distinguish two cases. E = P ( - a , ~ , E , - a 2 y / , E 2 ) . (6.1) The energy E2 can be read from eqn (5.8) as 1 $2 ( ~ a , ) 2 - 2 ( ~ a , ) [a, (4) - KH,] t ) (6.2) Ica2 (4h- l) - E2 = [(A2 - dh)t + (A2 - l)'] +? j4h " ( A2-4 with 4 h and d2 determined by eqn (5.6a) and (5.66).take the limit under the integral sign. We define El by However, E, needs to be computed explicitly. Since the limit is singular, one cannot In this region 4(R) = 1 +f(R), f(R) > 0 and given approximately by eqn (3.10) of ref. (1 1). Thus [A, - 4 (R)]i - (A, - 1)i x - (A, - l ) - i f ( R ) . (6.4)1808 DOUBLE-LAYER INTERACTIONS Changing the integration variable R = Kaly, dR = Ka,dy yields a,E, x- Ka? Jol yf(lca1 Y ) dY. 2 (A, - 1): (1 -y2)): Let the constants C, and B be defined by C, = &exp(-rcHo) (6.6) and B = [(A, + 1)i- 241 [(A2 + 1): - 241 [(A, + 1); + 2;]- [(A2 + 1): + 2iI-l. (6.7) It is clear that eqn (6.5)-(6.7) are valid for KH, > 1, while f(Ka, y ) is approximated by f h a , u) = 2 ({ 1 + c, exp [ - t (Ka,)2y2/Ka21>2 { 1 - C, exp [ -4 (~a,)~y~/lca,]}-~ - 1).(6.8) Note that C, < 1 since B < 1 and exp ( -KH,) < 1 or Substitution of eqn (6.9) in (6.5) gives (6.10) The leading asymptotic term of eqn (6.10) yields (6.1 1) Zdz exp (- 4Ga2 Z2/Ka2) z -~ (A, - 1): a, El z - Finally, the sphere-plate limit for large separation and KH, > I, is given explicitly as Correction terms to eqn (6.12) can be obtained by keeping more terms infand more terms in the asymptotic series obtained from eqn (6.10). Note that with a, fixed, as a2 -+ co, E takes the form Eqn (6.12) and (6.13) display the absence of interchange symmetry of the system. The interaction energy of the plate-sphere system with a sphere of radius a [a = a, in eqn (6.12) or (6.13)] depends on whether the larger potential is on the plate or on the sphere.This effect is absent at HHFl where the limit is trivial!Case (ii) : I, > H,. a, E, = a, JoRc E. BAROUCH AND E. MATIJEVIC 1809 We again take the case a, + co, a, and R, finite, and compute RdR [&a,), - R2]: {[A,- 4 (R)lt - (A1 - 119 Kul RdR {[Al-Q(R)]i-(Al- 1)i> = 11+12. (6.14) [@a,), - R2]i In the first integral R, is finite and the limit can be taken under the integral sign lim Il(al) = joRc RdR{[A,-Q(R)]l-(A,-l)i}. (6.15) The second integral I, is very similar to the previous case. The function Q(R) is bounded in this region by A1 3 Q (R) 2 1. Since Q (R) x A, where R x R,, the statement Q (R) x 1 +f(R) is not as meaningful, unless care is exercised. One integration by parts and change of variables R = ica, y, dR = Kal dy, yield to obtain U l - t c o with Q(y) given implicitly by and I, given by d+(d) = K H o +B(~a1)2Y21(~a,) (6.17) I, = -ica;(A,-l)i [ 1- (K:)2]i-$ica:J:, - (1 -~[d+(Q)-icHo]):(Al-Q)-~dQ. ( K d 2 (6.18) The expression of the square root under the integral yields (6.19) a 1 = $(A1- ~ ) : R ; I C - ~ + ~ [~+(Q)-KH,](A,-Q)-~~Q J A l or Expressions I, from eqn (6.20) and Il from eqn (6.15) are the leading terms in the asymptotic series of El for large a,; both are finite and their sum is the exact limit, which is non-isotropic.Note that eqn (6.15) can be integrated by parts directly. 7. ITERATION SCHEME The solution of the linearized PB equation with boundary conditions (4.7) is given explicitly by eqn (4.10).The non-linear case [once Q (R) is obtained] is given explicitly by eqn (4.12) with the same boundary conditions. In this chapter an iteration method is devised which allows, in principle, computation of the potential for the exact two-sphere problem and hence the interaction energy.,OV 211810 DOUBLE-LAYER INTERACTIONS Let Yo(x,y,z) be given by Y(2,R) from eqn (4.12). Consider the systematic iterations AYn+l = tc2 sinhYn. (7.1) First one introduces bispherical coordinate systems22 p, q and 4 by a sinq sin# a sinhp (7.2) a sin q cos # X = z = cosh p -cos q’ = cosh p -cos q’ cash P - cos q with - GO < p < GO, 0 < q < 7c and 0 < # < 2n; the interfocal length a is given by 1 2d a = - [(d - a2, - - 4a: a$ (7.3) where d = al+az+Ho.(7-4) The surface p = pj is a sphere of radius alcosec pj( whose centre is at z = a cothpj, x = y = 0. The main idea of the iteration scheme (7.1) and its implementation is dependent on obtaining a good initial approximation vo(r). Then the correction t,ul(r) can be found as a solution of the Poisson equation with exact spherical boundary conditions (v/ = v/i for p = pi, i = 1’2). The major tool used in obtaining yl(r) 3 yl(p, q, 9) is the Green’s function method. In particular one needs the Green’s function of the Laplace equation with homogeneous boundary conditions, i.e. G = 0 for p = pl, ,u = ,u2. Let E, = 1, E, = 2 for rn > 0. After a fair amount of labour one obtains the explicit homogeneous Green’s function in terms of the intrafocal parameter and the bispherical coordinates as G = a-l [(cosh p - cos q) (cosh po - cos qo)]d where f l are the Legendre functions.The system at hand is independent of 4, so only the term with m = 0 contributes. When A is expressed in bispherical coordinates, the Green’s function of interest satisfies the formal equation AG 01, Po, q, 70) = - 47d 01 -Po) wl - ‘lo) (7-6) and is given explicitly by G = a-’ [(cosh p - cos q) (cosh po - cos qO)]i Z (P, (cos q) P, (cos qo) 00 n-0 x (exp [ - (n + h) IP -Poll + 1 -exp [2 (n +t) IPZ -P111>-’ x {exp [- (n +a> 01 + P O - 2Pd +exp [(n +h) 01 +PO - 2Pl)l - 2 cash [(n +a> 01 - Po)l>) where Pn (cosq) is the nth Legendre polynomial. Next one needs to solve the Laplace equation AY, = 0 (7.7)E. BAROUCH AND E.MATIJEVIC 181 1 with boundary conditions The solution of eqn (7.8) with (7.9) is obtained as Note that eqn (7.10) yields the limit of a sphere of potential v, and a grounded plane [given by Morse and Feshbach as formula 10.3.71 of ref. (22), p. 12991 if v1 --+ 0, w 2 +vo, Pl +*o and P2 +Po- Finally, the iterate v/,(p, q) is expressed as L J p , J O where J(po, 'lo) is the Jacobian given by a3 sinq, J@O' 'lo) = (cosh po - cos (7.12) and the 'charge density' po is given by Po @o, 'lo) = K2 sinh [ w o @09 'l0)l (7.13) with w0 (po, 'lo) being the approximate solution in bispherical coordinates. Since Yl (p, q) is independent of the actual iteration (solution of the Laplace equation), as well as of Green's function G and the Jacobian J, eqn (7.1 1) takes the iterative form VjoL9'l) = with thejth iterate 'charge density' given by Pj @o, yo) = u2 sinh [ w j OLo, ~011- (7.15) This consideration, in principle, completes the exact solution of the PB equation for two unequal spheres. We have estimated that in most practical cases the correction to the interaction energy of the first iterate is a decrease of lCrl5%.However, a detailed numerical and analytical study is in its final stages and is beyond the scope of this paper. From the iteration scheme (7.14) it seems that the iterate vj is a power series in ic2, which is of course absurd. To avoid this possible confusion a simple scaling is introduced : KX = X, uy = Y, KZ = 2, Kai = Ai, ua = A , ud = D. This scaling is consistent with all results obtained in this paper, and it does not change the coordinates p, q.For illustration purpose we rewrite eqn (4.10) in terms of these variables as1812 DOUBLE-LAYER INTERACTIONS Table 1. Parameters used in calculating the interaction energies shown in fig. 2-9 parameter P1 P2 N1 N2 a1 /m a2lm K/m-' V/I/V V/2/v 4 A2 AH/10-20 J critical separation /A 2.2 x 10-7 1.7 x 1.04 x lo8 0.030 0.053 1.752 3.949 1 .o 99.0 2.2 x 10-7 3.6 x 10-7 3 . 2 9 ~ lo8 0.025 0.043 1 SO52 2.7333 1 .o 31.4 ~ ~~~~ 1.0 x 10-7 1.0 x 10-5 7.35x 107 0.040 0.050 2.4558 3.5313 1.4 77.1 1.0 x 10-7 1.0 x 10-5 2.33 x lo8 0.040 0.050 2.4558 3.5313 1.4 24.4 The same yo in terms of bispherical coordinate system is given by I}-' yo (p, q) = {sinh [ icHo +: (i+-$ (A2 sin2 q) (coshp - sin q)-, 1 1 x [ y, sinh ( icH0 +% A2 sin2 q (cosh p - sin q)-, - A[sinh p(cosh p - cos ~ ) - l - (7.17) .P o o l o , ?lo) = sinh [wo@o, ro)I (7.18) and eqn (7.18), (7.12), (7.10) and (7.7) in eqn (7.11) to obtain the first iterate. The convergence is slower but considerably easier to handle since no inversion of the elliptic functions at hand is involved. )I 1 + y, sinh - A2 sin2 q(cosh p - sin q ) - , + A[sinh p(cosh p - cos q)- - 11 (2Al In principle eqn (7.17) can be substituted in 8. EXAMPLES The numerical computation of the electrostatic energy E as a function of separation is a complicated task. This is not surprising, since eqn (5.8) and (5.13) contain many numerical pitfalls. Detailed discussion of the involved analysis, code development and a systematic study of the results will be presented in the second paper of this series.19 Here a few examples are offered to illustrate cases that are based on certain experimentally studied systems.Table 1 summarizes the parameters used in the calculations. Systems designated as P refer to two unlike spheres, whereas those marked N are essentially representations of the plate-sphere configuration. The overall Hamaker constants, A,, for cases P are based on a mixed aqueous dispersion of spherical titania (a,) and haematite (a,) particle^,,^ and for cases N for haematite (a,) and steel (a,) ~ysterns.,~ The values of w1 and w, correspond to electrokinetic potentials determined at appropriate pH values. Fig. 2 and 3 give the electrostatic interaction energies as a function of distance for cases P1 and P2 calculated using the theory developed in this paper (BM, solid lines)E.BAROUCH AND E. MATIJEVIC Lo& 1813 -LO I 1 I 1 I 0 50 100 150 2 00 A separation/ A i0 Fig. 2. Electrostatic energy E (in kT units) plotted against separation for case P1 (see table 1): -, BM model; ---, HHF model. 200 h ly 100 2 . A c y o e $ -100 m 4-3 VI +d u -200 '. '. /- - I 1 I I 20 LO 60 80 100 SeparationlA Fig. 3. Electrostatic energy E (in kT units) plotted against separation for case P2 (see table 1): -, BM model; ---, HHF model. and the expression by Hogg, Healy and Fuerstenau3 (HHF, dashed line). The HHF values are consistently higher over all particle separations. The electrostatic interaction energies start to decrease below a certain distance; however, this effect is much more pronounced when calculated using eqn (5.8) and (5.13) (BM).Indeed, in the latter case, the electrostatic interactions even display net attraction at reasonable separations. This effect is not observed using the HHF approximation for the examples given in fig. 2 and 3. The distance at which the net electrostatic energy vanishes is dependent on the critical separation, as can be seen by comparing fig. 2 and 3 with the values in table 1. Fig. 4, which relates to case PI, gives the ratio EBM/EHHF over a large range of separations, both above and below critical. At ca. 12 A this ratio turns negative exhibiting a dramatic change, since EBM becomes attractive while EHHF is still net repulsive ! Finally, fig. 5 compares total energies, ETOT, for case P2, in which the van der Waals energy, calculated for the system titania-haematite, is added to the electrostatic1814 DOUBLE-LAYER INTERACTIONS 1 0 LL X 2 z -1 -2 ( 50 100 150 200 250 separation1A Fig. 4.Ratio EBM/EHHF for case P1 (see table 1). I -400 I I I I 0 10 20 30 40 separationlli Fig. 5. Total interaction energy EToT (in kT units) plotted against separation for case P2 (see table 1): -, BM model; ---, HHF model. energy. It is obvious that EToT becomes negative much earlier when calculations are based on eqn (5.8) and (5.13) rather than using the HHF model. Fig. 6-9 are analogous plots to those shown in fig. 2-5 for the cases N1 and N2 (table 1). Similar trends are observed for systems of large difference in particle radii (equivalent to the plate-sphere configuration), as was seen for two spheres of comparable size.In the former case no net attraction is to be expected until the particles are very close (ca. 2 separation). However, the effect of partial attraction, which decreases the repulsion energy, is quite obvious. Fig. 8 displays the ratio E B , / E H H p - for the N1 case over a broad range of interparticle distances, which shows that the discrepancy increases as the particles approach each other. It is at small separations that various phenomena (heteroco- agulation, adhesion etc.) are most sensitive to the value of the interaction energy. The examples given in fig. 4 and 8 explain why many experimental results disagreed with calculations based on HHF approximations. In fig. 9 the total interaction energy is plotted for the system steel-haematite, which can be used to assess the adhesion effects of rust on iron.The following final comments are in order. (i) The analysis of this paper is devotedE. BAROUCH AND E. MATIJEVIC 1815 300 3 I I I I I I 0 25 50 75 I00 125 150 separation/A Fig. 6. Electrostatic energy E (in kT units) plotted against separation for case N1 (see table 1): -, BM model; ---, HHF model. 300 -1001 I I I I 0 20 40 60 80 1 separation/A 0 Fig. 7. Electrostatic energy E (in kT units) plotted against separation for case N2 (see table 1); -, BM model; ---, HHF model. ' . O ~ 0.21 I I 1 1 I 1 0 25 50 75 100 125 150 17 separation/ A Fig. 8. Ratio EBM/EHHF for case N1 (see table 1).1816 DOUBLE-LAYER INTERACTIONS 300 I -100 1 I I I I I I 0 20 GO 60 80 100 120 1 separation/ A 0 Fig.9. Total interaction energy EToT (in kT units) plotted against separation for case N2 (see table 1); -, BM model; ------ , HHF model. to particle interactions where potentials are of the same sign. If potentials are opposite in sign, the system is always attractive and it will be dealt with in a separate publication. (ii) It is clear that the simple expression of HHF offers satisfactory results in some regions. It is not possible to ascertain a priori for which system the HHF formulation will give acceptable answers. The regions of agreement and disagreement between the BM and HHF models will be discussed elsewhere. This task is by no means simple since the functions involved contain two independent variables. Furthermore, certain features are not taken into account in the HHF analysis; in particular, the inherent coupling of the geometry and the potentials of the system is weaker because the geometrical term is factorized out.In view of the simplicity of the HHF expression it is useful to establish the region of its applicability. (iii) Claims have been made25 that E obtained from exact solutions of the PB equation and E H H F are in good agreement. These statements refer to the trivial case of equal spheres and equal and opposite potentials. In both cases the critical separation is zero. No attempt was made to make the same comparison for unlike spheres of different potentials. Thus these authors had no justification to generalize their conclusions. (iv) In view of the complexity of the numerical computations, we are willing to carry out analysis of a certain number of different systems of interest to workers in the field.We thank Mr T. H. Wright (Clarkson University) for his assistance in solving computational problems and many stimulating discussions. Further thanks are expressed to Dr N. Kallay (University of Zagreb) and to Mr S . Kulkarni (Clarkson University) for useful exchanges. Prof. E. Ruckenstein (SUNY, Buffalo) offered interesting comments with respect to the problems of this paper. This work was supported by N.S.F. Grant no. CPE-8111612. N. E. Hoskin, Philos. Trans. R. SOC. London, Ser. A , 1956, 248,433. * N. E. Hoskin and S. Levine, Philos. Trans. R. Soc. London, Ser. A, 1956,248,449. R. Hogg, T. W. Healy and D. W. Fuerstenau, Trans. Faraday Soc., 1966, 62, 1638. B. V. Derjaguin, Discuss. Faraday SOC., 1954, 18, 85; Kolloid Z., 1934, 69, 155; Physicochim. Acta USSR, 1939,10, 333. G. M. Bell, S. Levine and L. N. McCartney, J . Colloid Interface Sci., 1970, 41, 542. G. M. Bell and G. C. Peterson, J. Colloid interface Sci., 1972, 41, 542. D. C. Pneve and E. Ruckenstein, J. Theor. Biol., 1976,56, 205. D. C. Prieve and E. Ruckenstein, J, Colloid Interface Sci., 1978, 63, 317.E. BAROUCH AND E. MATIJEVIC 1817 A. B. Glendinning and W. B. Russel, J. Colloid Interface Sci., 1983. 93, 95. lo J. K. Marshall and J. A. Kitchener, J. Colloid Interface Sci., 1966, :,2, 342. l 1 M. T. Boughey, R. M. Duckworth, A. Lips and A. L. Smith, J. Chtm. SOC., Faraday Trans. I , 1978, l2 H. Sasaki, E. MatijeviC and E. Barouch, J. Colloid Interface Sci., 1980, 76, 319. l 3 R. J. Kuo and E. Matijevid, J. Colloid Interface Sci., 1980, 78, 407 l4 E. MatijeviC, R. J. Kuo and H. Kolny, J. Colloid Interface Sci., 19: Il, 80, 94. l5 F. K. Hansen and E. MatijeviC, J. Chem. SOC., Faraday Trans. I , 1 ?80, 76, 1240. l6 E. Barouch, E. Matijevic, T. A. Ring and J. M. Finlan, J. Colloid lnterface Sci., 1978, 67, 1 ; 1979, l7 V. A. Parsegian and D. Gingell, Biophys. J., 1972, 12, 1192. 74,2200. 70, 400. E. Barouch, Application of Field Theory to Statistical Mechanics, ed. L. Garido (Springer Verlag, Berlin, 1985), p. 231. 19 E. Barouch, E. Matijevid and T. H. Wright, J. Chem. SOC., Farada, 7 Trans. I , 1985,81, 1819. 2o S. Kulkami, M S c . Thesis (Clarkson University, 1984). 22 P. M. Morse and H. Feshbach, Methods of Theoretical Physics (M :Graw-Hill, New York, 1953). 23 P. Gherardi and E. MatijeviC, J. Colloid Inter$ace Sci., submitted f )r publication. 24 N. Kallay and E. MatijeviC, J. Colloid Interface Sci., 1981, 83, 289 25 B. K. C. Chan and D. Y. C. Chan, J. Colloid Interface Sci., 1983, ! I, 28 1. E. Barouch and S. Kulkami, work in preparation. (PAPER 4/ 1572) 60 FAR 1

ISSN:0300-9599    

DOI:10.1039/F19858101797

出版商:RSC    

年代:1985

数据来源: RSC

摘要:

J. Chem. Soc., Faraday Trans. 1, 1985,81, 1819-1832 Double-layer Interactions of Unlike Spheres Part 2.-Numerical Analysis of Electrostatic Interaction Energy BY EYTAN BAROUCH,* EGON MATIJEVIC* AND THOMAS H. WRIGHT Departments of Mathematics and Chemistry and the Institute of Colloid and Surface Science, Clarkson University, Potsdam, New York 13676, U.S.A. Received 22nd November, 1984 A comprehensive numerical analysis of the electostatic interaction energy of a two-surface system has been performed. A partial attraction between two spheres differing in size and the magnitude of the same sign potential is established. As a consequence conditions which lead to either heterocoagulation or selective coagulation in a binary mixture as a function of ionic strength are identified. The analysis is applicable to the plate/sphere configuration, which allows for interpretation of particle-adhesion phenomena.In our preceding paper' the electrostatic interaction energy for two unequal spheres was derived analytically. The potential used in the derivation was a solution of the Poisson-Boltzmann equation for two dissimilar spheres, obtained by a scale balancing that allowed for the two-dimensionality of the system, since there is only one axis of symmetry. This approach refines the earlier methods where the interaction energy was essentially obtained from a two-plate system multiplied by a geometrical factor. Indeed, the dependence of the interaction energy on surface potentials and particle sizes is considerably more complicated than indicated by previous results.This work deals with the numerical analysis of the expressions derived in ref. (1). It is shown that one of the major computational problems is accurate evaluation of incomplete elliptic integrals of the first kind, which is a direct consequence of the non-linearity of the Poisson-Boltzmann equation. After a number of difficulties have been overcome a series of programs is developed that allow computations of the electrostatic interaction energies for any set of conditions of physical interest. The input parameters are particle radii (in m), surface potentials (in V), Debye's K (in m-l) and particle separation (in A). A number of cases are offered in which the effects of varying particle size, surface potential and K values have been investigated.In addition to some predictable trends, it was noted that under certain conditions the calculations indicated phenomena that are not present in the classical theory. One such observation is the absence of interchange symmetry, i.e. switching potentials on two particles of different radii does not yield the same electrostatic interaction energy. This effect increases as the size of the particles becomes more divergent, yet it is totally absent when the classical model is employed. At large separations (KH > ca. 10) the electrostatic energies as calculated in this work, EBM, and by the model of Hogg et aZ.,2 E H H F , yield comparable results. However, for two particles that differ in size and in the magnitude of the potential of the same sign, the discrepancy increases as the separation decreases.At shorter 1819 60-21820 DOUBLE-LAYER INTERACTIONS OF UNLIKE SPHERES distances EBM changes sign, which means that particles attract each other electro- statically; at the same separation EHHF still indicates strong repulsion. On changing the ionic strength ( K ) another consequence of the partial attraction becomes evident. Under certain conditions, mixed systems at high ionic strength could be more stable than the same systems at lower ionic strengths, which is the opposite of what is expected for particles of identical potential. This effect is so pronounced that in some cases an increase in K may change an attractive to a repulsive system. As a practical consequence of this finding it may be possible to distinguish between heterocoagulation and selective coagulation.At a given higher ionic strength homofloc forms but not heterofloc; at a lower ionic strength the opposite may be true. FORMULATION The energy expression, E, as derived earlier' is given by c -[ 1 1 The upper and lower limits are solutions of the transcendental equations = KX (3 a> where 6+(4) - are given by eqn (5.5) of ref. (1) and 4 = [A, 4,, 421 for KX = W0, K H ~ + C,, K H ~ + ( ~ a ~ ) ~ C,] (3 b)E. BAROUCH, E. MATIJEVIC AND T. H. WRIGHT 1821 respectively, with It is here that a selection process begins. The purpose is to determine which equation must be solved, and it is explicitly dependent on the given separation for which the energy is computed. The selection starts by comparing K H with the dimensionless critical separation I, given by Note that in eqn (5) A2 > A, > 1.Therefore eqn ( 5 ) can be rewritten as a particular case of the formula3 with the condition and the definitions 1: dt [ ( t - a ) ( t - b) ( t - c)]-; = gF(a, k ) y > a > b > c F(a,k) = dO[l -k2 sin2 81-i Joa (7) sin a = [( y - a)/( y - b)]k (9 c) Clearly, eqn (8) is a representation of the incomplete elliptic integral of the first kind. Using y = A2, a = Al, b = 1 , c = - 1 and eqn (6)-(9), I, takes the form I, = 2i(A1 + l)-;F(4,, k,) k, = [2/(A., + l)];. (10) (1 1 ) (12) In other words, the selection process begins with the need to compute the elliptic integral of the first kind, Ic. Next one compares I, with KH,. If KH, > I, the system is totally repulsive and we call it case (1).If rcHo < I, the system is partially (or totally) attractive and we call it case (2). with sin = [(A, - A1)/(A2 - l)]: CASE (1): REPULSIVE SYSTEM One must obtain the integration limits &,, 4, and 42, which are determined from the equation1822 DOUBLE-LAYER INTERACTIONS OF UNLIKE SPHERES respectively. Eqn (13) contains two elliptic integrals with A2 > A, > 4 > 1 and definitions (144 (14W k, = [2/(#+ 01: (144 g = 2(4+ 1)-1. ( 1 4 4 (15) Note that k , = k(4), aj = q(4) and eqn (1 5) needs to be solved three times. Furthermore, the need to have a fast and accurate routine that computes F(a, k) is now clear, since it is embedded in every computation at a low-level integration. A discussion of this routine is presented below.As long as $bh is sufficiently far from the singularity of the first integral in eqn (l), a standard 512 Gauss-Legendre quadrature in double precision is fairly straightforward. It is employed as the external (high-level) integration, while the function 6+(q5) is given by the left-hand side of eqn (13). The need for the elliptic integral routine is again clear. The transcendental eqn (15) is solved by a modified regula-falsi (m.r.f.) r o ~ t i n e . ~ A solution is sought in the interval (1 + A,). If a solution is bracketed, a fast m.r.f. obtains a solution after a few iterations; if not, the solution is assumed in the interval (1, 1 + 10-14). The interval gets bisected until bracketing of the solution takes place, then the routine loops back to the m.r.f.part to speed up convergence. If ten bisections do not bracket the solution, the routine returns the value 1 as its solution. This routine is called TRNSND. sin a, = [(A, --&/(A, - l)]: sin a, = [(A2 -4)/(A2 - l)]: Using eqn (8), one may rewrite eqn (1 3) as 2-;g[F(a,, k,) + F(a17 k,)] = AZ. CASE (2): PARTIALLY (OR TOTALLY) ATTRACTIVE SYSTEM requires a great deal of attention, since there are many logical selections necessary. This case is considerably more complicated analytically as well as numerically and First one computes the lower limit of integration by solving the equation The potential is a monotonically increasing convex function; in other words, it is bracketed by a constant and a straight line. Since a natural bracketing of the solution 4 takes place, only regula-falsi is needed and it is included in the routine MONTON.The limiting case lcH0 -+ 0 is allowed when (bh -+ - 00. Thus #h is not bounded from below and the expression of 6-(4) as a difference of two elliptic integrals is dependent on the value of 4 h . This expression may be obtained as ' - (4) = gLF(y2, k, - F(yl 7 k)l (17) where g, y,, y2 and k are dependent on the value of 4. (a) 1 < 4 < A,. 6 - (4) = 2 -; ( J4A' dY [(Y - 4) (Y - 1) (Y + 1114 - J#A' dv [(Y - 4) (Y - 1 ) (Y + 111-9 = k1[F(%7 kl) - F(a17 kl)l (18) with k,, a, and a, given by eqn (14).E. BAROUCH, E. MATIJEVIC AND T. H. WRIGHT For this case one uses eqn (6) with a = 1, and c = - 1 to obtain 1823 (b) - 1 < 4 < 1 . The functions k,, k,, /3, /I1, A2, Al, g, and g3 are given explicitly by g, = 2q1 -&-i g , = 1 k3 = [( - 1 - 4 / ( 1 - 4)li k2 = [(4+ 1)/(2)1i sin A2 = [(A, - 1)/(A2 - 4)]4 sin A1 = [(A, - l)/(Al -4)y sin B2 = [(A, - 1)/(A2 + 1)]i sin 8, = [(A, - l)/(A, + l)];.The boundary cases 4 = 1 and 4 = - 1 are elementary functions and are given below. (d) $4 = 1. A2 &(1) = 2-iJA1 dy[(y+ l)]-i(y- l)-l = log [tanh(ly2/4) coth (ly1/4)]. (25) (e) 4 = - 1 . &(- 1) = 2-;J:dy[(y-- l)]-;(y+ l)-l = 2{arctan [exp ( ly2/2)] - arctan [exp ( ly1/2)]}. (26) A relation which is extremely useful for both accuracy and speed of computation is given as a summation formula of two incomplete elliptic integrals with the same modulus k by3 with z = tan (4/2) = [sin 8( 1 - k2 sin2 8); sin p( 1 - k2 sin2 8);]/(cos 8+cos 8). (29) A particular case of eqn (27)-(29) is distinguished when z = 1 and 4 = n/2, which takes place when cot /I = k’ tan 8 (30) where k’ is the complementary modulus and eqn (27) takes the explicit form F(8, k ) + F(B, k ) = K(k) (31)1824 DOUBLE-LAYER INTERACTIONS OF UNLIKE SPHERES with K(k) being the complete elliptic integral of the first kind.Eqn (30) and (31) are crucial to one's ability to write an accurate elliptic integration routine (called ELINT).~ These considerations form the necessary ingredient needed to produce the program that computes $bh. In most cases of interest one may obtain 41 and d2 by the use of TRNSND discussed above. However, one must make the following comparisons: (32 a) or (32b) for i = 1, 2. If eqn (32a) is true, 4i is indeed obtained from TRNSND.In certain cases where eqn (32b) holds, the corresponding q5i is determined from MONTON together with its associated complications. The actual computation of the energy E in eqn (2) is straightforward for the third and fourth integrals, since they are evaluated far from the branch point at A, = 4. The first integral is divided into two integrals, one of which is in (& 4,) and the other in (+,,A,). The latter is combined with the second integral and the resulting subtraction is done analytically. A change of variable is then performed to eliminate the common singularity and the resulting integral is evaluated numerically. These complications do not occur if 4, is obtained from MONTON, since the singularity at A, is never reached. Again, all the high-level integrations are done using a 512 Gauss-Legendre quadrature scheme.uHo + Cz(uai)2 > I, uH, + C2(uai)2 < I, ELINT ROUTINE The routine that computes the incomplete elliptic integral F(4, k ) is the backbone of the entire numerical process. This routine is exceedingly fast as well as accurate to 26 digits. It is computed by a combination of two Landen transformation^.^ (i) Descending Landen transformation : kn+, = [ 1 - (1 - k:)i]/[ 1 + ( 1 - k:)i] sin 4n+l = (1 + k i ) sin 4n cos q5n( 1 - k2, sin2 t$n)-i (33) (34) with k, = k and with sin 4, = sin 4. The evaluation of F(+,, k,) is then given by Eqn (35) is very useful in computing the complete integral K(k) since 4, =n/2 and 4n = n/2 for all n. The convergence is quadratic to zero and K(k) is given by n K(k) = an lim n ( I + k j ) .n+oo j - 1 However, it is not at all easy to compute F(4, k ) accurately, since the need to compute 4n (after the correct quadrant is located) involves the computation of inverse trigonometric functions. (ii) Ascending Landen transformation : (37)E. BAROUCH, E. MATIJEVIC AND T. H. WRIGHT 1825 sin 4j+l = (1 + k j ) sin 4j/( 1 + kf sin2 4 j ) (38) (39) ~ ( 4 7 k ) = n-cc lim ( ~ ( 4 n y k n ) 3-0 i”I (1 +k;)-l). In this case the convergence of the iterated kj is also quadratic but its limiting value is 1, which allows one to compute F ( 4 , l ) explicitly as the elementary integral The use of the ascending transformation eliminates the problem of computing inverse trigonometric functions. However, a different computational problem arises.The transformation pushes k -, 1 and can push sin4 close to 1, which may introduce a spurious singularity with the obvious disastrous consequences. This problem is dealt with by using eqn (30) and (31). A logical selector computes F(4,k) for the smaller phase of eqn (31). If the smaller phase is the required one, a fast ascending transform is used. Otherwise the complete integral is computed using the descending transform, the second incomplete integral is computed using the ascending transform and the result is subtracted from K(k). POTENTIALS OF OPPOSITE SIGN The interaction energy of two spheres with potentials of opposite sign can be essentially written in terms of eqn (1) with two modifications. It is assumed that ‘yl < 0 and ry, > 0 without any loss of generality.In eqn (l), ‘yl is substituted by I ‘yl I, but 6, (4) is exchanged with another function d2(#), given explicitly as The constants dh, b1 and 42 are determined from eqn (3) with d2(4) substituted for 6,. The solutions dh, b1 and b2 have 1 as their upper bound and are not bounded below. Their solver routine is similar to TRNSND with a modified regula-falsi speeding up the convergence. In most cases studied numerically, the results obtained by this procedure were within 80% of the results obtained by the HHF formula.2 We conclude that for potentials of opposite sign the latter approximation is practical and gives fairly reasonable results. NUMERICAL RESULTS AND DISCUSSION Systematic computations were carried out to illustrate the dependence of the electrostatic energy on a variety of parameters including the particle sizes and the potentials of two spheres, as well as the ionic strength.A coding system is offered to identify different cases according to their physical characteristics, as follows : (a) IC values code A B C K/rn-l 2 x lo7 5 x lo7 8 x l o 7 I (b) surface potentials on two spheres code X Y Z ryl,ry,/mV 45,75 30,75 30,60 E H F J x 108 1 . 5 ~ lo8 2 x lo8 3 x lo8 U V W 5,75 25,65 35,551826 DOUBLE-LAYER INTERACTIONS OF UNLIKE SPHERES 0 h -200 2 a, - 4 0 0 E ..y h \ 0 m .d * -600 0, a, -800 U 0 - -1000 0 100 200 300 4 00 -400 h ..y P 1 -500 E: a, 0 m .d U c Y 2 -600 0 b) - -700 1 sepa ra t ion/A /- .- I I I I I 50 60 70 80 90 100 separation1 A Fig. 1. (a) Electrostatic interaction energy calculated using eqn (1) and (2) (EBM) between two spheres with radii a, = 1 x m, potentials tyl = 15 mV and tyz = 75 mV and K = 2 x lo7 m-l (AU5, -), and also for a, and a2 interchanged, keeping all other conditions the same (AUSI, ---).The calculated values using the expression of Hogg et al.2 (EH,,)arealsogiven(EHHF,. * * * .). (b)Sectionofthesamecurves(E,,)onanexpandedscale. m and a2 = 1 x (c) radii of two spheres code 1 2 3 4 5 a,/m 1 x 10+ 5 x 1 x 1 x 1 x lo-' a2/m 1 x10+ 1 x 2 x 1 x lo+ 1 x For example, a system designated EX31 refers to two spheres of radii a, = 2 x ryl = 45 mV, ry, = 75 mV and IC = 1 x lo8 m-l. The letter I added to the code identifies a system in which the two radii are interchanged while the potentials are not. The computed electrostatic energies using expressions derived by BM' [eqn (1) and (2)] and by HHF2 [eqn ( 1 .9 , ref. (l)] are designated by E,, and EHHF, respectively. Fig. 1 (a) gives the electrostatic energy as a function of separation for cases AU5 and AU5I in order to show that an interchange of particle radii, while keeping the potentials the same, leads to different results. The absence of symmetry is demonstrated if the BM model is used, yet no such effect exists in the HHF formulation. Fig. l(6) illustrates more clearly the magnitude of this effect. Note that dissymmetry was m, a, = 1 xE. BAROUCH, E. MATIJEVIC AND T. H. WRIGHT 500 - .............. ..* ....... .............. - I 1827 Fig. 2. EBM ...................... c1 -20 -- 2 + W - - 4 0 - -60. I I 1 I 100 150 200 250 500 0 50 separation/ A Fig.3. EBM (-) and EHHF ( - - . . a ) for a, = 1 x m, a, = 1 x m, w1 = 45 mV, y, = 75 mV and K = 1 x m-l (EXl). 300 4 separation/ A y, = 75 mV and K = 1 x lo8 m-l (EX3). Fig. 4. E,, (-) and EHHF ( * * * .) for a, = 1 x lo-' m, a2 = 2 x lo-' m, y, = 45 mV,1828 DOUBLE-LAYER INTERACTIONS OF UNLIKE SPHERES / i I ! -40. I I I I I 0 50 100 150 200 250 300 separation/ A Fig. 5. EBM for tp, = 15 mV, tp2 = 75 mV, K = 1 x lo8 m-l, a, = 1 x m and a2 = 1 x lo-* m (EUlI, -), a, = 1 x lov6 and a2 = 5 x 10-8m (EU21, ---) and a, = 2 x lO-’m and a, = 1 x 10-7 ( E U ~ I , -.-.-). observed in all cases studied ; with increasing difference in particle sizes the absence of symmetry becomes more pronounced. Fig. 2-4 show the dependence of the electrostatic energy on distance for three different combinations of particle sizes and two different sets of corresponding potentials, all at a constant value of K .In fig. 2-4 the solid and dotted lines represent EBM and E H H F values, respectively. In all cases the BM expressions give lower energies than the HHF approximation over the entire range of separations. Note that the reversal of repulsion to net attraction for two spheres approaching each other is observed in all three examples when the BM model is used. Under certain conditions (such as shown in fig. 2) the HHF expression displays the same effect but at a much closer separation. The attraction is commonly detected when the BM treatment as applied, yet rarely so when calculations are based on the HHF model.The discrepancy between the EBM and E,,, values always increases with decreasing distance between the two spheres. This effect is due to the partial attraction contained in the BM model. At large separations the ratio E,M/EH,F approaches a constant, the magnitude of which depends on the physical parameters of a given system. The deviation from unity is attributed to the non-linearity of the system. The fact that E H H F can change sign is not due to partial attraction but is caused by the logarithmic singularity when Kho -+ 0, which is present explicitly in the HHF expression. The latter is essentially an integral of the energy of the two-plate model multiplied by a geometric factor [a, a,/(a, +a,)]. The effect of varying the radii of the two spheres is illustrated in fig.5 , which gives EBM as a function of separation for three different pairs of particles with fixed surface potentials. It is evident that in the case of a large difference in particle size (which can be interpreted as a sphere/plate model) the electrostatic repulsive energy is considerably lower than for spheres of less disparate sizes. At closer separations, where attraction prevails, the absolute value of the energy decreases as the difference in particle size increases. Note that for a fixed pair of surface potentials but different particle sizes there is a distance at which the interaction energy is identical. The interaction-energy curves as a function of separation for two particles of fixed sizes but varying potentials are illustrated in fig.6. The higher potential ( y 2 ) is kept constant at 75 mV while the lower potentials (u/J are 15, 30 and 45 mV. WithI50 & 100 4 x . P 50 aJ U cj .- + z o z b, -50 c-l 0 - -1 00 E. BAROUCH, E. MATIJEVIC AND T. H. WRIGHT . . I I I I I 0 50 100 150 2 separation1 A m, K = 1 x lo8 m-l, v / , = 45 Fig. 6. EBM for a, = 5 x lop8 m, a, = 1 x 1829 10 mV and wz = 75 mV . - ~ ~~ (EX2, -), ty, = 30 mV and ty, = 75 mV (EX2, ---) and tyl = 15 mV and tyz = 75 mV (EU2, -.-.-). Fig. 7. EBM for a, = 1 x m-l, a, = 2 x lo-' m-l, ty, = 15 mV, K = 2 x lo7 m-l (AU3, -), K = 5 x lo7 m-l (BU3, ---), K = 1 x lo8 K = 3 x lo8 (JU3, -..-.. ). yz=75mV and (EU3, -.-.-) and decreasing values of y1 the interaction energies become smaller and the critical distance larger. This effect might appear counter-intuitive if one were to consider the difference in the potentials only. However, the dominant parameter is the critical separation, which accounts for this observation.Fig. 7 shows the effect of varying ionic strength (expressed as K) on the electrostatic energy of two unlike spheres of fixed sizes and surface potentials. Any two systems in fig. 7 exhibit identical electrostatic interaction energy at a specific separation, d. As a consequence, at separations d the energy of the system with the lower ionic strength (smaller K) is higher, whereas at separations -c d the reverse is observed. Indeed, electrostatic attraction can be charged to repulsion by an increase of the ionic strength because of partial attraction between two unlike spheres.The calculated effect shows that under certain conditions the repulsion energy of a dispersion of unlike particles of different potentials increases with the ionic strength (K), which is contrary1830 DOUBLE-LAYER INTERACTIONS OF UNLIKE SPHERES Fig. 8. Total interaction energy as a function of separation for the system Q, = 1 x m, a, = 2 x m, y1 = 45 mV, y = 75 mV and K = 5 x lo7 m-l (BX3, -), K = 8 x lo7 m-l (CX3, - - * * . -), K = 1.5 x lo8 m-l (HX3, ---) and K = 3 x lo8 m-l (JX3, --.-.-). Hamaker constant used for the calculation of the van der Waals energy: 1 x J. D potential, $/nV Fig. 9. Critical separation, I,,, as a function of the lower potential (wl) for w, - yl = 50 (-), 30 (---) and 10 mV (- - - -). Q, = 2 x lo-' m, Q, = 1 x m and K = 1 x lo8 m-l.to the behaviour in homocoagulation. The computations have shown that in the latter case no reversal of the normal influence of electrolytes is to be found. This finding explains the differences in the stability behaviour of the same mixed systems at various ionic strengths; it demonstrates the reasons why in some cases a mixed aggregate is formed (heterocoagulation), while in some other cases selective coagulation dominates. Specifically, the same system of particles may yield heteroflocs at lower electrolyte contents but produce homoflocs at higher concentrations of the same electrolyte. The total interaction energy, including the van der Waals contribution for systems of fixed particle sizes and surface potentials but varying K values, is illustrated in fig.8. The crossing of curves described in fig. 7 is still obvious, which again implies that a mixed dispersion can, in principle, be more stable at higher electrolyte concentrations, provided that the partial attraction is present. The dependence of the critical separation on the lower potential (y1) for threeE. BAROUCH, E. MATIJEVIC AND T. H. WRIGHT 4 1831 0 I, 2 4 6 8 separat ion/KH, Fig, 10. 4h as a function of separation. a, = 2 x lo-’ m, a, = 1 x lo-’ m, ry, = 45 mV, ryz = 75 mV and IC = 1 x lo8 m-l. fixed-potential differences (10, 30 and 50 mV) is shown in fig. 9. These calculations were carried out for constant K , a, and a, values. With increasing difference (w, - w,) the critical distance becomes larger. In fig. 10 the solution of the transcendental equations that compute the limits of integration as a function of K a is given.For K a c I, eqn (16) and for rca > I , eqn (13) are used. The cusp at I, is shown with the value 4 = A,, as K a -+ 0 4 --+ - co. As rca increases 4 asymptotically -+ 1. CONCLUSIONS For systems of unlike particles with potentials opposite in sign the chemical approximation based on a two-plate model (HHF) yields acceptable results. If potentials on such particles are of the same sign and similar in magnitude the same model is again applicable. However, as the potential difference increases the critical separation becomes larger and the effects of partial attraction are more pronounced. Consequently, the deviations from the simple model (HHF) are enhanced. Similar trends are observed with the change in particle sizes.The results of the calculations presented in this work illustrate unequivically the extent of variations between EBM and E H H , . In some cases the ratio E , , / E H H , is very small, at close separations it is negative and at large separations it approaches a constant value asymptotically. These findings are in direct contradiction to the statements by Chan and White,6 who claimed a maximum deviation of ca. 15%. Note that Hoskin and Levine’v * performed the calculations for equal spheres and equal potentials, both numerically and analytically, using finite difference techniques and series expansion. Computations based on expressions derived in this paper reduced to the case of equal spheres are in good agreement with the results of Hoskin and Levine. This work was supported by N.S.F. grant no. CPE-8111612. Note added in proof: We have compared our results with those obtained using Derjaguin approximation as applied to the non-linear equation between two parallel plates. We also too into consideration the electrostatic attraction when the two plates are closer than the critica1832 DOUBLE-LAYER INTERACTIONS OF UNLIKE SPHERES distance. We find a significant variation in values at the maximum of the interaction-energy curve. The detailed analysis of these findings is the subject of another paper that is presently being prepared in collaboration with Dr A. V. Parsegian. E. Barouch and E. Matijevid, J. Chem. SOC., Faraday Trans. I , 1985, 81, 1797. P. F. Byrd and M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Scientists (Springer Verlag, Berlin, 1976). Handbook of Mathematical Functions, ed. M. Abramowitz and I. A. Stegun (National Bureau of Standards, Washington D.C., 1972). E. Barouch and E. Doberkat, A Note on the Numerical Evaluation of an Incomplete Elliptic Integral, to be published. P. Y . C. Chan and L. White, J . Colloid Interface Sci., 1980, 74, 303. N. E. Hoskins, Philos. Trans. R. SOC. London, Ser. A , 1956, 248, 433. N. E. Hoskin and S. Levine, Philos. Trans. R. SOC. London, Ser. A , 1956, 248, 449. * R. Hogg, T. W. Healy and D. W. Fuerstenau, Trans. Faraday SOC., 1966, 62, 1638. (PAPER 4/ 1994)

ISSN:0300-9599    

DOI:10.1039/F19858101819

出版商:RSC    

年代:1985

数据来源: RSC

摘要:

J. Chem. SOC., Faraday Trans. I, 1985,81, 1833-1840 Constant-composition Study of the Kinetics of the Dissolution of Strontium Fluoride in Aqueous Solution BY SALEM M. HAMZA~ AND GEORGE H. NANCOLLAS* Chemistry Department, State University of New York at Buffalo, Buffalo, New York 14214, U.S.A. Received 14th September, 1984 A highly reproducible constant-composition technique has been used to study the rates of dissolution of strontium fluoride crystals over a range of undersaturation. At 25 "C and at higher driving forces (relative undersaturation 0.06-0.25) the process appears to follow normal diffusion-controlled dissolution, while at very low undersaturation (0.008-0.0 15) a surface- controlled reaction predominates. The two regions of undersaturation show markedly different dependences upon the addition of organic phosphonate inhibitors, the action of which can be interpreted in terms of a Langmuir-type adsorption equation. The dissolution and crystallization of fluoride salts of the alkaline-earth metals is of importance in view of their applications in spectroscopy, electronics, lasers and glass manufacture.The salts are also involved in water-treatment processes in which control of crystallization and dissolution reactions is of great importance. The influence of foreign anions and cations which may influence the rates of dissolution either through adsorption or by lattice substitution is also of considerable importance. The involvement of fluoride ion provides an opportunity for using the constant- composition method developed in our laboratory,'* together with the specific fluoride ion electrode.The method enables dissolution rates to be measured under conditions of constant solution composition for extended periods, even at very low undersatu- ration levels, with a precision unattainable by conventional dissolution experiments. The study of a strontium salt was prompted by an earlier observation that these systems may be especially sensitive to the influence of inhibitor^.^ EXPERIMENTAL Undersaturated solutions of strontium fluoride were prepared from both Ultrapure (Alfa Chemicals) and Reagent Grade (J. T. Baker Chemicals) and triply distilled deionized water. Solutions were filtered through 0.22 pm Millipore filters which had been prewashed in order to remove any residual wetting agents or surfactants.Solutions were analysed (k 0.1 % ) by passing aliquots through an ion-exchange resin (Dowex 50) in the hydrogen form and titrating the eluted acid with standardized sodium hydroxide solutions. All fluoride solutions were prepared and stored in polyethylene or polypropylene bottles in order to prevent fluoride attack on glass surfaces. Seed crystals were prepared by the rapid mixing (in ca. 4 min) of 1 dm3 of potassium fluoride (0.780 mol dm-3) and strontium nitrate (0.390 mol dm-3) solutions. The crystals were washed with distilled water and allowed to age for at least six months at 25 "C. Their composition was confirmed by X-ray powder diffraction (Philips XRG 3000 X-ray diffraction photometer, Ni filter and Cu Ka radiation). The specific surface area of the crystals -f On leave from Menoufia University, Egypt.18331834 S . M. HAMZA AND G. H. NANCOLLAS Table 1. Dissolution of strontium fluoride crystalsa expt T,r seed rate no. /lo-, mol dm-, 102a /mg / lop6 mol min-l m-2 ~~ 24-30 31 32 33 34 35 36 37 38 39 40 41 42 43 131 1.359 1.106 1.180 1.250 1.310 1.360 1.387 1.453 1.457 1.460 1.464 1.468 1.471 1.474 1.457 8.0 25.0 20.0 15.0 11.2 8.0 6.0 1 S O 1.25 1 .oo 0.75 0.50 0.30 0.10 1.25 4.52- 1 4.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 10.0 4.3 k0.3 12.6 f0.5 9.1 rf: 0.3 7.5 k0.15 5.7 +O. 1 4.14 k 0.1 3.56 & 0.07 2.9 1 rf: 0.05 2.13 f 0.06 1.45 f. 0.04 0.73 f 0.02 0.42 f 0.0 1 0.24 f 0.0 1 0.13 f. 0.01 2.10 f 0.07 a &,: TF = 1 :2, [KNO,] = 0.15 mol dm-3, 25 "C, stirring speed 200 r.p.m., with 300 r.p.m.for expt (35) and (38). was measured (+_ 1 %) by single-point B.E.T. nitrogen adsorption using 30% nitrogen + helium gas mixtures (Quantasorb, Quantachrome, Greenvale, New York). Scanning electron micro- graphs (IS1 model I1 scanning electron microscope) and transmission electron micrographs (Leitz) showed the seed materials to consist of small aggregates of particles with cube-like morphology. Dissolution experiments were made in a nitrogen atmosphere using a double-walled polyethylene-lined cell of 300 cm3 capacity. The cell contents were maintained at 25 f 0.1 "C by circulating thermostatted water through the outer jacket. Stirring was effected using a variable-speed magnetic stirrer with Teflon stirring bar (1.1 x 3.5 cm).Undersaturated solutions of strontium fluoride were prepared by mixing strontium nitrate and potassium fluoride solutions in the range of relative undersaturation 0.001 < a < 0.25, where a is given by 0 = {([Sr2+],[F-]$- ([Sr2+] [F-]2)g}/([Sr2+]0 [F-]:): where [Sr2+], [F-] and [Sr2+]0,[F-]0 are the undersaturation and saturation molar concentrations of free lattice ions, respectively. The ionic strength was maintained at 0.15 mol dm-, during the dissolution experiments by the addition of potassium nitrate (Ultrapure) solution. Following the addition of seed crystals to the undersaturated solutions, the undersaturation was maintained constant by the potentiometrically controlled addition of this background electrolyte from a piston-driven burette.The rate of titrant addition was controlled by the e.m.f. of a specific fluoride ion electrode (Orion model 94-09) using a Metrohm pH-stat (Combititrator 3D, Brinkmann Instrument Co.). The thermal electrolytic silver, silver chloride reference electrode immersed in a 4 mol dm-, potassium chloride solution saturated with silver chloride was maintained at 25 f 0.1 "C and was separated from the cell by an additional liquid junction containing 0.15 mol dm-3 potassium nitrate in order to eliminate errors in e.m.f. caused by leakage of salt-bridge solution. The tip of the reference electrode was also constructed of Teflon to avoid fluoride-glass interactions. In order to verify constant composition, aliquots of reaction mixture were withdrawn at known times, filtered (0.22 pm Millipore) and analysed for strontium ion by atomic absorption (Perkin-Elmer model 503).The solid phases collected during the experiments were also investigated by scanning electron microscopy, specific-surface-area analysis and X-ray powder diffraction.DISSOLUTION OF STRONTIUM FLUORIDE 1835 40 60 ‘O t/min Fig. 1. Plots of amount of strontium fluoride dissolved against time: expt (32) 0, (37) 0, (91) A, (125) 0 and (104) a. RESULTS AND DISCUSSION The concentrations of free-ion species in the solutions were calculated from mass-balance and electroneutrality expressions as described previously,2 using the thermodynamic equilibrium constants, K , for the various associated species : H F e H + + F - K = 6.61 x ref. (4) HF + F- HF, K = 0.295, ref.(4) SrF+ + Sr2+ + F- K = 0.147, ref. ( 5 ) SrOH+ + Sr2+ + OH- K = 0.150, ref. (6) H,O f H+ +OH- Kw = 1.002 x lo-’*, ref. (7). Activity coefficients were calculated from the extended form of the Debye-Hiickel equation proposed by Davies.8 The results of the dissolution experiments are summarized in table 1, in which T,, and TF are the molar concentrations of strontium and fluoride, respectively. Typical plots of the amount of strontium fluoride dissolved, calculated from the titrant addition, as a function of time are shown in fig. 1. Assuming, as a first approximation, simple spherical or cubic particles, corrections were made for changes in surface area during the dissolution reactions by introducing a factor (wi/wt)i, where wi and wt are the masses of solid phase present initially and at time t , respectively.It can be seen in table 1 that the rates of dissolution were proportional to the mass of seed crystals used to initiate the reactions, and a plot of the rate of dissolution according to rate = d&,/dt = kan (2)1836 12- N I E .5 0 E E z 4 i! 4 I - P, Y S. M. HAMZA AND G . H. NANCOLLAS - - I I 1 1 15 20 25 lo lo2 (3 Fig. 2. Plots of rate of dissolution of strontium fluoride against Q for expt (31)-(36) at high undersaturation. 18 105 0 2 I2 6 Fig. 3. Plots of rate of dissolution of strontium fluoride against o2 for expt (37)-(41) at low undersaturation. shown in fig. 2 confirms a first-order dependence upon undersaturation (n = 1) in eqn (2) at relatively large driving forces. A calculated rate of diffusion of 3 x mol min-' mV2 under the conditions for expt (34)9 is in satisfactory agreement with the measured value, 5.7 x mol min-l m-2.In contrast, at low undersatur- ations (fig. 3) it can be seen that the reaction rates follow a parabolic rate law with n = 2 in eqn (2). A similar striking change in the mechanism of dissolution at lower undersaturations has also recently been observed for calcium oxalate monohydrate.10 The suggestion of a predominantly surface-controlled process at low undersaturations may also be supported by the observed independence of the experimental rate of1837 DISSOLUTION OF STRONTIUM FLUORIDE Table 2. Effect of additives on the rates of dissolution of strontium fluoride crystalsa expt. T,, additive rate / 1 O-' mol dm-3 / 1 O-s mol min-' m-2 no.mol dm-3 10% 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 1.387 1.387 1.387 1.387 1.387 1.387 1.387 1.387 1.387 1.310 1.310 1.310 1.310 1.310 1.310 1.310 1.310 1.310 1.310 1.3 10 1.180 1.387 1.387 1.387 1.387 1.387 1.387 1.387 1.387 1.387 1.387 1.310 1.310 1.310 1.310 1.310 1.310 1.310 1.310 1.310 1.310 1.180 1.180 1.180 1.180 1.180 1.180 1.180 1.180 1.180 1.180 1.46 1.46 1.46 6.00 6.00 6.00 6.00 6.00 6.00 6.00 6.00 6.00 11.2 11.2 11.2 11.2 11.2 11.2 11.2 11.2 11.2 11.2 11.2 20.0 6.00 6.00 6.00 6.00 6.00 6.00 6.00 6.00 6.0 6.0 11.2 11.2 11.2 11.2 11.2 11.2 11.2 11.2 11.2 11.2 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 20.0 1.25 1.25 1.25 HEDP 5.0 HEDP 10 HEDP 15 HEDP 20 HEDP 30 HEDP 40 HEDP 50 HEDP 75 HEDP 100 HEDP 5.0 HEDP 10 HEDP 15 HEDP 20 HEDP 30 HEDP 40 HEDP 50 HEDP 75 HEDP 100 HEDP 125 HEDP 150 HEDP 5 ENTMP 5 ENTMP 10 ENTMP 15 ENTMP 20 ENTMP 25 ENTMP 30 ENTMP 40 ENTMP 60 ENTMP 75 ENTMP 100 ENTMP 5 ENTMP 10 ENTMP 15 ENTMP 20 ENTMP 25 ENTMP 30 ENTMP 40 ENTMP 50 ENTMP 75 ENTMP 100 ENTMP 5 ENTMP 10 ENTMP 15 ENTMP 20 ENTMP 30 ENTMP 40 ENTMP 50 ENTMP 60 ENTMP 75 ENTMP 100 ENTMP 5 HEDP 5 - - 2.87 f 0.2 2.28 f 0.07 1.78 f 0.05 1.52 & 0.04 1.37 f O .1 1.13k0.1 1.40f0.3 1.54 f 0.04 1.96 f 0.03 3.91 k0.2 3.08 k 0.2 2.53 f 0.15 2.05 f 0.2 1.57 f 0.05 1.38 f 0.03 1.72 & 0.03 2.15 f0.2 2.39 k 0.06 2.61 f0.14 2.53 k0.15 4.31 f0.17 2.53 +_O. 12 1.83 f 0.04 1.58 f 0.02 I .42 f 0.08 1.30 k 0.05 1 .OO & 0.03 0.85 & 0.02 0.9 1 f 0.04 1.27 k 0.05 1.26 f 0.04 3.60 f 0.1 2.60 & 0.04 2.15 f 0.07 1.60 & 0.05 1.48 f 0.03 1.38 f 0.05 1.02 f 0.02 1.12 k 0.02 1.44 f 0.04 1 S O f 0.05 4.41 f 0.20 3.70 t 0.1 5 3.01 f0.16 2.45 f 0.14 1.95 10.10 1.78 &O.10 1.60 & 0.07 1.90 & 0.05 2.15 f 0.13 2.50 0.10 2.10 f 0.07 0.60 f 0.0 1 0.74 & 0.0 1 a Gr: TF = 1 : 2, [KNO,] = 0.15 mol dm-3, 10 mg seed, 25 "C, stirring speed 200 r.p.m., with 300 r.p.m. for expt (1 3 1).1838 S. M. HAMZA AND G . H. NANCOLLAS 20 40 60 80 100 [ENTMP]/lO-' mol dm-3 Fig. 4. Plots of rate of dissolution against [ENTMP] for expt (121k(130) A, ( I 1 lH119) 0, (101)-(110) 0, (133) 0 and (132) 0. I 1 I 1 I 50 I00 I50 200 [ ENTMP] - l / lo4 dm3 mol-' Fig. 5.Plots of R,/(R,-R,) against [ENTMPI-l. Expt (101)--(107) 0, (lllH116) 0 and (121)-(127) A. dissolution of changes in fluid dynamics, as shown in table 1 [compare expt (38) and (1 3 I)]. However, this evidence may be inconclusive for such small particles for which changes in stirring rate may have little influence on the fluid shear forces at the crystal surfaces. The particles will tend to move with the fluid flow. Thus at higher undersaturation, at which diffusion control is proposed [table 1 , expt (30) and (35)],DISSOLUTION OF STRONTIUM FLUORIDE 1839 a change in fluid dynamics also has no observable effect on the measured rate of dissolution. The crystallization rates of strontium fluoridell and divalent metal ion salts in generallo* 12-14 are greatly inhibited by added substances.In the present work, the rate of dissolution of strontium fluoride was studied in the presence of hydroxyethylidene 1,l -diphosphonic acid (HEDP) and ethylenediaminetetra(methy1ene phosphonic acid) (ENTMP). Table 2 summarizes the data at two undersaturations for each additive; each experiment was made in duplicate or triplicate and the uncertainties in the rates are also given. Typical plots of the extent of dissolution as a function of time are shown in fig. 1. Initial rates of dissolution, plotted against [ENTMP] in fig. 4, suggest that the influence of additives on the dissolution kinetics of strontium fluoride may be interpreted in terms of at least two factors: (i) specific adsorption on the crystal surface and (ii) sequestration of strontium ion in the solution.In general, specific inhibition of the rate of dissolution may be expected to be induced at much lower concentrations of additive molecules than for the simple complexation of lattice cations. It can be seen that concentrations as low as 5 x lop7 mol dm-3 for each additive markedly reduce the dissolution rate. As the concentration of additive molecules increases, the active dissolution sites on the crystal surfaces are blocked through adsorption and the rate of crystal dissolution decreases. Strontium ion chelation by the phosphonate additives will become more pronounced as the concentration of the latter increases. This is clearly seen in fig. 4, in which the rate of dissolution increases at high concentration of additive molecules, reflecting the increase in undersaturation caused by sequestration of strontium ions. On the assumption that the decreased dissolution rate reflects the adsorption of phosphonate ions at active dissolution sites, the influence of both inhibitors can be interpreted in terms of a Langmuir-type eq~ati0n.l~ The reduction in dissolution rate then reflects the increased crystal surface area covered by the adsorbed inhibitor molecules.If R, and Ri are the rates of dissolution in the absence and presence of inhibitors, respectively, the Langmuir equation requires a linear relationship between the reciprocal of the relative reduction in rate, R,/(R, - Ri), and the reciprocal of the inhibitor concentration.16 Fig. 5 confirms the applicability of this interpretation for both phosphonates under diffusion-controlled dissolution.Moreover, the degree of inhibition increases with increasing undersaturation. Thus for ENTMP, the adsorption affinity (rate constant for adsorption/rate constant for desorption, calculated from the slopes of the linear plots in fig. 5 ) has values of (8.9f0.1) x lo5, (1 1.7fO.l) x lo5 and (1 8.9 Ifi 0.1) x lo5 dm3 mol-l at relative undersaturations, o, of 0.06,O. 11 and 0.20, respectively. For HEDP the corresponding values are (5.1 f 0.5) x lo5, (8.2 f0.3) x lo5 and (1 7.3 0.1) x lo5 dm3 mol-l. In contrast to this behaviour, in the region of surface control at lower o values the effectiveness of the inhibitors is increased as anticipated. Thus for ENTMP at 5 x lo-’ mol dm-3, a 7 1.3 % reduction in the rate of dissolution at 0 = 0.013 may be compared with a 19.4% decrease at o = 0.06 under diffusion control. We acknowledge financial support from the U.S.National Science Foundation, grant no. CPE83 13383. P. G. Koutsoukos, Z. Amjad, M. B. Tomson and G. H. Nancollas, J. .4m. Chem. Soc., 1980, 102, 1553. L. J. Shyu and G. H. Nancollas, Croat. Chem. Acta, 1980, 53, 281. :’ J. R. Campbell and G. H. Nancollas, J . Phys. Chem., 1969, 73, 1735. A. J. Ellis, J . Chem. Soc., 1963, 4300.1840 S. M. HAMZA AND G. H. NANCOLLAS 5 6 7 8 9 10 11 12 13 14 15 16 R. E. Connick and M. S. Tsao, J . Am. Chem. Soc., 1954,76, 531 1. F. G. Gimbell and C. B. Monk, Trans. Faraday Soc., 1954, 50, 965. H. S. Harned and W. J. Hamer, J . Am. Chem. Soc., 1933, 55, 2194. C. W. Davies, Ion Association (Butterworths, London, 1962). A . E. Nielsen and J. Christoffersen, in Biological Mineralization and Demineralization, ed. G. H. Nancollas (Springer-Verlag, Berlin, 1982), p. 37. D. J. White and G. H. Nancollas, J . Cryst. Growth, 1982, 57, 267. G. H. Nancollas, Adv. Colloid Interface Sci., 1979, 10, 215. J. Christoffersen, M. R. Christoffersen, S. B. Christensen and G. H. Nancollas, J . Cryst. Growth, 1983, 62, 254. G. H. Nancollas, R. A. Bochner, E. Liolios, L. J. Shyu, Y. Yoshikawa, J. P. Barone and D. Svrjek, Am. Inst. Chem. Eng., Symp. Ser., 1982, 215, 26. F. C. Collins and J. P. Leineweber, J . Phys. Chem., 1956, 60, 389. R. A. Bochner, A. Abdul-Rahman and G. H. Nancollas, J . Chem. Soc., Faraday Trans. I , 1984,80, 217. P. G. Koutsoukos, Z. Amjad and G. H. Nancollas, J . Colloid Interface Sci., 1981, 83, 599. (PAPER 4/ 1592)

ISSN:0300-9599    

DOI:10.1039/F19858101833

出版商:RSC    

年代:1985

数据来源: RSC

摘要:

J . Chem. SOC., Faraday Trans. 1, 1985,81, 1841-1862 Determination of the Adsorption Behaviour of ' Overpotential-deposited ' Hydrogen-atom Species in the Cathodic Hydrogen-evolution Reaction by Analysis of Potential-relaxation Transients BY BRIAN E. CONWAY* AND LIJUN B A I ~ Chemistry Department, University of Ottawa, 365 Nicholas Street, Ottawa, Canada KIN 9B4 Received 17th September, 1984 Despite its major importance in electrocatalysis and mechanistic aspects of electrode processes, the experimental determination of the adsorption behaviour of kinetically involved intermediates, e.g. H in H, evolution, in electrode reactions proceeding at appreciable Faradaic currents has hitherto remained little developed. A new method of analysis of the kinetics of the decay of overpotential on open-circuit, following interruption of a polarizing current, has been applied to the study of the adsorption behaviour of the 'overpotential-deposited' (0.p.d.) H generated as an intermediate in the cathodic evolution of H, at appreciable currents.Results are described for Ni and electrodeposited Ni-Mo-Cd composite cathode materials. The method enables, for the first time, the H coverage to be reliably determined as a function of potential, at appreciable overpotentials, and the corresponding adsorption pseudocapacitance behaviour to be directly and accurately evaluated for the 0.p.d. species in a continuous Faradaic reaction. The results show that H coverage at the surfaces of these metals attains limiting values at overpotentials of - 0.15 to -0.25 V, so that the Faradaic process of H, evolution proceeds, at these metals, on a surface incompletely but appreciably covered by H atoms.By application of a steady-state analysis to a discharge and electrochemical-desorption mechanism this behaviour is explained theoretically and the ratios of rate constants required to account for the observed H-adsorption behaviour are evaluated. The Tafel slopes for the H,-evolution reaction at the metals studied are discussed in terms of the observed H-adsorption behaviour and steps in the reaction mechanism. One of the principal problems in the kinetics of multistep electrode processes is the experimental evaluation of the adsorption behaviour and potential dependence of coverage by the electroactive adsorbed intermediates involved. Despite a number of theoretical treatment~l-~ of this question in relation to interpretation of Tafel (dq/d In i) or ' Lefat '4 (d In i/dq) slopes and to evaluation of electrocatalysis at various electrode material^,^^ no reliable method yet exists for the evaluation of the potential dependence of coverage by electroactive intermediates or the corresponding adsorption p~eudocapacitance,~ C,, in the ' overpotential region' where appreciable continuous Faradaic currents pass, except by the a.c.impedance method over limited ranges of potential. In contrast, the phenomenon of so-called ' underpotential deposition ' (u.p.d.) of strongly bound adatoms at electrode surfaces has been widely studied, both experirnentally*-ll and theoreti~ally.~? I2-l4 Such species arise by Faradaic deposition, usually in sub-monolayer arrays, at potentials corresponding to lower free energies than those for deposition of the same element in the bulk state at unit activity.In most t Permanent address: Department of Chemistry, University of WuHan, WuHan, China. 18411842 ADSORPTION BEHAVIOUR OF H IN H, EVOLUTION cases it is believed that these strongly bound u.p.d. species are not the kinetically involved intermediates in the corresponding overall continuous Faradaic reactions, e.g. in H,, Cl, or 0, evolution and bulk metal electrocrystallization processes. This is certainly the case for u.p.d. H in the hydrogen-evolution reaction at Pt. Experimentally, u.p.d. is easy to study since the small current densities associated with deposition or desorption ofmonolayer arrays 'by definition' suffer no interference from any large currents of parallel Faradaic processes associated with the corresponding bulk reaction. In order to understand the kinetics of multistep Faradaic reactions of the electrocatalytic type, e.g.the cathodic H,-evolution reaction (h.e.r.), it is necessaryl9 l5 to have information on the adsorption behaviour of the ' overpotential deposited ' (0.p.d.) intermediate species that are deposited (and removed) in the steady state of the Faradaic reaction proceeding at appreciable net current densities, i. In the case of the h.e.r., the intermediate species are, of course, H atoms usually adsorbed at the cathode metal interface.l? 2 * l6 The potential dependence of the fractional coverage, 8, of electrodeposited adspecies is conveniently characterized by the experimentally determinable pseudo~apacitance~~ defined by C4 = qld8/dq, where q1 is the charge required for deposition of a monolayer.If C4 is accessible over a range of q, then 8, or changes of 8, asfTq) can be evaluated by integration of (C4/ql) dq. Direct studies of chemisorbed H in the h.e.r. under conditions where appreciable net cathodic currents, i, are passing at corresponding overpotentials, q , related to i by the Tafel equation, q = (RT/aF) In (i/i,), where i, is the exchange current density (at q = 0) and a is the transfer coefficient, are difficult if not wholly unsatisfactory. For example, cyclic voltammetry,17 which provides results of very high sensitivity and accuracyg+ l7 for u.p.d.species, is inapplicable over potential ranges where the Tafel relation for appreciable continuous currents applies, giving potential-dependent background currents for the main Faradaic reaction that are orders of magnitude larger than those corresponding to changes of coverage by the chemisorbed intermediate species. Similarly, attempts to use the charging-curve method1* lead to either uncertainlg or physically meaningless results,20 e.g. for adsorption of H in the h.e.r. A.c. impedance studies on H adsorbed at Pt have been performed by several Russian and German w o r k e r ~ ~ l - ~ ~ and many C.V. studies have also been 17* 23 however, the results of these methods apply, at Pt, to the strongly bound H, which is not necessarily involved as a kinetic intermediate in the h.e.r.Usually C.V. is not applicable to the study of H at base metals such as Ni, Fe etc. because of overlap1g with metal oxidation processes. Also, impedance measurements are unsatisfactory when there is continuous bubble formation, e.g. of H,, at appreciable current densities, > ca. lo3 A crn-,. In this paper we show how the electrochemical adsorption behaviour of the kinetically involved adsorbed H intermediate in the h.e.r. at Ni and some Ni-Mo-Cd electrocatalyst preparations can be studied quantitatively by analysis of the kinetics of relaxation of overpotential (potential decay) upon interruption of current at various polarization potentials. Note that experimental evaluation of the potential dependence of coverage of electroactive intermediates, e.g.H in the h.e.r., is an essential requirement for understanding the origin of Tafel slopes, b, lower than ca. RT/0.5 F, which arise because the Lefat slope (6-l = d In i/dq) is a linear combination of F/RT times the potential dependence of the natural logarithm of the coverage 8 and of F/RT times the barrier symmetry factor /? for the charge-transfer process. Hitherto this has been treated almost exclusively in theoretical ways.'* 2- 3 9 l5 Low Tafel slope behaviour (as well as favourable i, values) is one of the essential requirements for good polarization performance of electrocatalytic electrodes, so that low overpotentials canB. E. CONWAY AND L. BAI 1843 be maintained at practically significant high current densities (2 100 mA cm-,), as is realized with RuO,/Ti electrodes for C1, evolution or Ni-Mo-type preparations for the h.e.r.BASIS OF THE METHOD EQUATIONS FOR POTENTIAL-DECAY BEHAVIOUR On open circuit, following interruption of a continuous steady-state current at (1) where c d l is the double-layer capacitance of the electrode interface (normally 6 Cd except in the limits 0 + 0 or 1) and i is the experimentally measured steady-state current density at potential q.7 Using recently available digital instruments such as the Nicolet digital oscilloscope (or other digital data-acquisition systems), q(t) data can be acquired with high accuracy over a range of several decades (5 or 6 in the present work) of time and the data can be processed reliably in a coupled computer to enable eqn (1) to be used for interpretation of the data or to be integrated to give related q against log? inf~rmation.~~? 2 5 9 27 density i, the kinetics of relaxation q(t) in time t are given by24-27 -(Gal + C4) dq/dt = i, exp (aqF/RT) = i Three operations with eqn ( I ) can usefully be made.27 (a) Integration to give an q(t) against logt relation of the form [cf.ref. (24) and (231 - q( t)/ b = In ( t + z) + In ( i,/ Cb) (2) where z is an integration constant equal to bC/i, b is the Tafel slope, RT/aF, from (1) and C is the overall capacitance, Cdl+C,, which, to give eqn (2), must be independent ofq.26 For such conditions, the potential decay slope 6, = -dq/d In (t+ z) is equal to the Tafel slope b. If C4 is appreciable, it is usually potential de~endent.~ Previously, two limiting cases of potential-dependent C, [ = K exp ( f qF/RT)] were also ~onsidered,~~ which enables eqn (2) still to be integrated explicitly giving values of 6, that are either greater or smaller than 6, depending on conditions of coverage by the electroactive intermediate, corresponding to the + or the - sign in the exp ( qF/RT) factor in C,.The use of the integrated form of eqn ( I ) , eqn (2), follows the analysis given previously by Conway and Bourgault2* and by Morley and W e t m ~ r e , ~ ~ based on the treatment of potential decay first given by Butler and Armstrong.26 (b) For the work described here, the approach is to evaluate C4 directly from eqn (3 a) = i/( -dq/dt) (3 b) (1) as C, + Cdl = i, exp [aqF/RT]/( - dy/dt) without any limiting approximations. This constitutes the new procedure described in this paper.It depends on being able to evaluate dq/dt accurately; this is achieved quite precisely by computer differentiation of digitally acquired q(t) information from the potential relaxation following interruption of current. Thus, with eqn (3), C, + C,, can be evaluated as a function of potential over the whole range of q through which the relaxing potential q(t) falls. t We recognize that eqn ( l ) , with Cd, and Cd added in a simple parallel relationship, is an oversimplification when the rate constant for the H-deposition step is comparable or small compared with that for the desorption step. In that case a fuller analysis of the relaxation characteristics of the equivalent circuit must be made, as for an impedance analysis.This will be published elsewhere.1844 ADSORPTION BEHAVIOUR OF H IN H, EVOLUTION (c) Alternatively, In (c, + c d ] ) can be evaluated as ln(C,+C,,) = -In(-dq/dt)+lni,,+q(t)/h (4 4 (4 b) and its variation with q plotted. Eqn (3) or (4) give information about the total electrode interface capacitance, C,+ c d l . Only the term C, is of interest in the electrode kinetics, i.e. when C, 9 c d ] . As we shall see, the significant results of the above treatment are readily recognized: (i) when C, B c d 1 and is potential dependent and (ii) when C = C, + C,, + constant, equal to the value of Cdl. = In i- In ( - dq/dt) MECHANISMS OF RELAXATION OF THE POTENTIAL WITH TIME When C = c d l , potential relaxation takes place [cJ ref. (26)] by self-discharge of the double-layer capacitance through continuing passage of electronic charge across it at a rate determined by the potential-dependent Faradaic reaction resistance as characterized by the charge-transfer kinetics.When C, 9 c d l and the electrode surface is appreciably covered by the reaction intermediate, e.g. H, the self-discharge process must proceed by mixed anodic and cathodic reactions, viz. for the h.e.r. in alkaline solution OH-+ MHadS -+ M + H,O+e (anodic) ( 5 ) MH,,, + H,O +e -, H, + OH- (cathodic) (6) since the charge for H removal at appreciable coverages is of the order of 25 times greater than the charge required for changing the potential difference across c d ] over the range of q(t) during decay. When C, B c d l it is presumed, in the usual way [cf ref. (1) and (2)], that a desorption step [e.g.reaction (6)] is rate controlling in the overall reaction so that the partial reaction ( 5 ) is almost in equilibrium. On open-circuit d e ~ a y , ~ - , ~ it is reasonable to assume that the same conditions must obtain, i.e. with reaction (6) continuing to be rate controlling so that the same values of i,, and b apply in eqn (1) as in the corresponding Tafel equation for the steady-state process. The equivalent circuits involved were discussed by Tilak and Conway in ref. (26). The possibility should be mentioned here, following the comment of a referee, that eqn (1) should contain a term (1 -Obubbles) to allow for ‘coverage’ of the electrode by bubbles. However, numerous papers on the log (current) against overpotential characteristic of electrodes up to quite high current densities show that this effect is not manifested nor is ‘bubble overvoltage’ significant.Thus, for potential decay, e.g. at Hg, both the Tafel relation in log i and the decay relation in log t are linear over wide ranges of i and t ; also, the correct double-layer capacitance is retrieved from analysis of the result^.^^^ 26 For high i at active electrodes, bubbles seem only to increase the effective solution resistance. EXPERIMENTAL METHOD In an early part of the work electrodes were cathodically polarized at various controlled current densities, i, and the corresponding overpotentials were registered by a digital milli- voltmeter. ‘Steady-state’ values of q were recorded 30 s after each change of i.In the later development of the work, corresponding to most of the results presented here, a slow potential sweep was applied to the electrode by means of a potentiodyne system ( 1 mV s-l for Ni and 0.24 mV s-l for the Ni-Mo electrodeposited composite electrodes, see below) and the currentsB. E. CONWAY AND L. BAI 1845 and corresponding potentials were recorded digitally. Tests were made to show that the above sweep rates gave polarization results that were almost independent of any further diminution of sweep rate. Following characterization of the Tafel relation by the above procedure, currents were then interrupted by a vacuum Hg relay and the potential against time relaxation transients recorded digitally by means of a Nicolet digital oscilloscope.In some experiments two oscilloscopes were operated in tandem to cover a total of 5 or 6 decades of time change, at the same time maintaining an adequate sampling density in the time data. The digitally acquired q(t) data were then processed in a PDP 11 /34 computer to obtain C asflq) using eqn ( 3 0 ) or (36), or In C asf(q) using eqn (4a) or (4b), after computing dq/dt. The q(t) data acquired and stored by the Nicolet oscilloscope had sufficient points (at least 8000) for each decay curve to enable the decay rate, dq/dt, to be accurately evaluated over the range of potentials where appreciable Faradaic currents are initially passing, and thus were able to give reliable quantitative information on the electrochemical behaviour of the ' overpotential- deposited' H species in the h.e.r.studied in this work through evaluation of Cb and coverage The method is applied to a comparative study of the h.e.r. in alkaline solutions at Ni-wire electrodes and at special electrocatalytic preparations of electrodeposited Ni and Mo with traces of Cd. 0,. ELECTRODE MATERIALS High-purity Ni wires (Johnson Matthey and Mallory Co.) were degreased in acetone and sealed in glass bulbs in H, using the special method we have employed for many years.,* Ni (80% )-Mo (19% )-Cd (1 % ) composite electrodes were prepared by electrolytic codeposition of Ni and Mo with Cd on to an Fe plate according to the method described p r e v i o ~ s l y , ~ ~ based on a patented procedure30 for the preparation of high-activity electrocatalytic materials for H, evolution.These materials are of considerable interest as they give low polarizations, i.e. ca. 0.1 V at 0.1-0.2 A cm-, for the h.e.r., particularly at elevated temperatures, i.e. 353-363 K. This is largely due to the low Naperian Tafel slope, uiz. 0.044-0.060 V, that prevails up to high current densities. Composition of the Ni-Mo-Cd electrodes was checked by X-ray emission analysis in a s.e.m. For comparison, the behaviour at an 80% Ni, 20% Mo bulk alloy, thermally prepared, was also examined. REFERENCE ELECTRODE Potentials of the test electrode were referred to that of a reversible Pt/H, electrode in the same solution. The potential of this reference electrode was communicated to the digital oscilloscopes through a Tektronix type 0 amplifier in a cathode follower mode capable of handling frequencies up to the 10 MHz range, equivalent to fast decay transients.These hydro- gen overpotentials are shown in all the graphs as positive quantities. SOLUTIONS Aqueous NaOH solutions (1 .O and 0.2 mol dm-3) were made up from AnalaR grade NaOH recrystallized in an N, atmosphere from pyr~distilled~~ water and dissolved in the requisite volume of such water. In some experiments the solutions were pre-electroly~ed.~~ REPRODUCIBILITY All polarization and potential-decay results were excellently reproducible if attention was given to controlling the duration of prior cathodic polarization or the potential sweep rate and thus the times at which the current and potential measurements were taken. This is a necessary procedure in work with Ni and Ni-based electrodes [cf.ref. (29) and (30)]. BEHAVIOUR AT VARIOUS TEMPERATURES Experiments were conducted over a range of temperatures from 274 to 353 K. MORPHOLOGY The Ni-Mo-Cd electrocoated electrodes have a nodular cauliflower appearance under the s.e.m. at moderate magnifications [plate 1 (a)] and some finer-scale structure is observable at higher magnifications [plate 1 (b)]. The structure is evidently porous as Kr desorption B.E.T. measurements gave a real/apparent area ratio of ca. 450.1846 ADSORPTION BEHAVIOUR OF H IN H, EVOLUTION LEACHING OF MO The possibility of leaching of Mo from the Ni-Mo electroplated alloy electrodes must be considered. In another project related to this one, experiments were conducted in aqueous NaOH and 5.5 mol dm-3 NaCl up to 363 K in order to evaluate the extent of leaching of Mo.Mo was determined by atomic absorption spectroscopy using a range of standard solutions of molybdate. Only after leaching at 363 K for 2 h in the presence of hypochlorite was any Mo detectable in pH 13 NaCl solution. At pH 4.5, however, significant and detectable amounts of both Ni and Mo appear in solution after 0.25 h. At the ordinary temperatures of most of the present decay experiments (298 K) and at pH > 13, and over the short duration (ca. 20 s) of a decay transient, the extent of Mo leaching would be quite insignificant, also bearing in mind the complete absence of C1- and of OC1- ions in the present work and the ‘cathodic protection’ that exists over most of the potential ranges involved in the decay experiments.RESULTS AND DISCUSSION LOG (CURRENT DENSITY) AGAINST POTENTIAL RELATIONSHIPS We first refer to the Tafel relations for the Ni, bulk Ni-Mo and electroplated Ni-Mo-Cd electrodes in order to characterize the polarization behaviour and to evaluate C4 as f ( q ) through eqn (1) or (4). Fig. 1 (a) and (6) show plots of q against log i for Ni wire, bulk Ni-Mo alloy and the Ni (80% )-Mo (19% )-Cd (1 % ) composite electrodes at several temperatures. The Ni-Mo-Cd electrodes exhibit two-slope Tafel relations characteristic of a change of mechanism in a consecutive two-step pathway and/or of a change of adsorption behaviour of the H intermediate with potential. A potential for transition between one region to another is seen at x - 0.075 to - 0.1 V.At Ni wire, usually a single-valued Tafel slope is observed, as found previo~sly,~~ at a given temperature but sometimes with a little curvature. POTENTIAL-DECAY BEHAVIOUR Ni-Mo-Cd ELECTRODES The digitally acquired potential-relaxation data are shown in fig. 2(a) directly as Alogt) for four initial polarization current densities. Fig. 2(b) shows the same data plotted asf[log (t + z)] with the integration constant evaluated empirically [cf. ref. (24)] to give a linear relation in log (t + z) for short times when z > t at the early stage of potential decay. Note that an estimate of C can also be made from z using the definition of z given below eqn (2). The plots of q against log(t+z) are not quite coincident for all initial currents as they are if only a Cdl determines the capacitance26 or when 0,-evolution behaviour at Ni - 0 - OH electrodes is This suggests that electrode prehistory determines, in part, the potential-decay behaviour, e.g.due to some sorption of H as Ni hydride [cf ref. (29)]. Fig. 3 shows plots of q against log(-dq/dt) at 298 K for the data processed according to eqn (4). In contrast to the q against log t plots shown in fig. 2(a), the plots of q against log (-dq/dt) are coincident in the cases of curves 3-6 of fig. 3 at their upper ends where q > 0.10 V. Note that in these plots there is no arbitrariness involved, as there is unavoidably in plots such as those in fig. 2 (b) because of empirical evaluation of z. This difficulty is avoided by treating the data according to eqn (4).We see from fig. 3 that plots of q against log (-dq/dt) also exhibit two principal linear regions with different slopes, like the Tafel relations shown in fig. I , to which they correspond. Fig. 4 shows similar plots for i = 0.15,0.20 or 0.30 A (apparent) cm-, for four temperatures. Like the Tafel plots of fig. 2, there is evidently a progressive change of the H-electrosorption behaviour with temperature: the lower slope region (30 mV) becomes progressively extended as the temperature is increased to 341 K.J . Chern. SOC., Faraday Trans. I , Vol. 81, part 8 Plate 1 Plate 1. S.e.m. pictures of electrocoated Ni (80%)-Mo (19%)--Cd (1%) electrodes at two magnifications: (a) 10 pm scale and (b) 1 pm scale. B. E. CONWAY AND L. BAI (Facing p . 1846)B. E.CONWAY AND L. BAI 1847 0 . L 0.3 $0.2 0 * I 0 0 Fig. 1. (a) Comparative Tafel plots for the h.e.r. on Ni-wire and bulk Ni-Mo alloy electrodes at 298 K in 0.2 mol dm-3 aqueous NaOH and on an electrocoated Ni-Mo-Cd electrode at 298 and 353 K (these plots are corrected for the back-reaction current near the reversible potential by the function shown). (b) Tafel plots, corrected for back-reaction current, for the Ni-M*Cd electrocoated electrode: ( 1 ) 274, (2) 298, (3) 308, (4) 325, (5) 338 and (6) 353 K. Ni-WIRE ELECTRODES At Ni-wire electrodes, two regions of the decay kinetics are again observed [fig. 5 (a)] but the Tafel plot (fig. 1) is a single line, as is usually found. The corresponding plots of q against log ( t + T) for Ni wire are shown in fig. 5 (b) for four initial values of i, while fig.6 shows the corresponding plots of q against d log (-dq/dt), giving the capacitance behaviour [eqn (4)]. As with the Ni-Mo-Cd electrodes, two regions of the log ( - dq/dt) behaviour are observed, corresponding to the 1 18 and 61 mV slopes,1848 - 1 0.25 0.20 $ 0.15 0.10 0.05 0 ADSORPTION BEHAVIOUR OF H IN H, EVOLUTION log ( t t 7)/ms 1 3 L I 5 0.25 3.20 5.15 > 3.10 2 0.05 0 -1 1 3 5 Fig. 2. Potential-decay plots for the h.e.r. at Ni-Mo-Cd-plated electrodes at 298 K in 1 .O mol dm-3 aqueous NaOH : (a) q against log t and (6) q against log (t + z) for interruption of polarization at four apparent current density values: (1) 0.10, (2) 0.15, (3) 0.20 and (4) 0.30 A cm-2. log t/ms I - 4 - 3 - 2 - 1 0 1 log ( G d T l d t ) Fig.3. Plots of q against log (-dq/dt) for the h.e.r. at Ni-Mo-Cd-plated electrodes for six initial apparent current densities: (1) 0.020, (2) 0.060, (3) 0.10, (4) 0.15, (5) 0.20 and (6) 0.30 A cm-2; T = 298 K. Curves 3, 4, 5 and 6 represent the same data as in fig. 2 (curves 1, 2, 3 and 4, respectively).B. E. CONWAY AND L. BAI 1849 0 0 > \ E 0 - 4 - 3 -2 -1 0 1 log (-dq/df) Fig. 4. As in fig. 3 but for four temperatures: (1) 278 K (0.15 A cm-2), (2) 298 K (0.15 A cm-2), (3) 319 K (0.20 A cmP2) and (4) 341 K (0.30 A cm-2). log ( t + T)/ms log rims Fig. 5. Potential-decay plots for the h.e.r. at pure Ni-wire electrodes at 298 K in 0.2 mol dm-3 aqueous NaOH : (a) 7 against log t and (6) 7 against log ( t + z) after interruption of polarization at four current densities: (1) 0 .5 5 ~ (2) 1 . 4 5 ~ (3) 4.3 x and (4) 15.3 x lop3 A cm-2. 61 FAR 11850 0 *4 2 0.2 0 ADSORPTION BEHAVIOUR OF H IN H, EVOLUTION I -3 3 Fig. 6. Plots of q against log(-dqldt) for the h.e.r. at pure Ni-wire electrodes at 298 K in 0.2 mol dm-3 aqueous NaOH at four initial current densities: (1) 0.55 x (3) 4.3 x and (4) 15.3 x A cm-2. (2) 1.45 x respectively, in fig. 6. Also, with the Ni-wire electrode, the populations of points corresponding to the potential decays become almost coincident for the four initial values of i when t % z. This means that the decay behaviour itself in log ( t + z) must, in this case, also be almost coincident, as is in fact seen from fig. 5(b). PSEUDOCAPACITANCE BEHAVIOUR OF ADSORBED O.P.D. H AT Ni-wIm AND Ni-Mo-Cd-COATED ELECTRODES (i) C, FOR ADSORBED H ~ s f ( q ) From the plots of log (-dq/dt) against q it is possible to calculate the derivative d log C/dq using eqn (4a).Since, experimentally, log (-dq/dt) appears limitingly linear in q over two regions, d log C/dq is obtained simply as the sum of minus the reciprocal slope, d-l, of those linear regions and the Lefat slope, b-l. Results are given in table 1 . Since d log C4/dq is approximately constant over an appreciable range of potential, C4 is evidently almost an exponential function of q over that range. This is illustrated in fig. 9, to be discussed below, where the plots of log C against q are shown for the three types of electrode materials. However, the results in table 1 also show that the arguments of the exponents are not simply qF/RT but are rather arbitrary numbers.We now show how a more complete evaluation of the capacitance behaviour can be made directly with eqn (1) in the forms eqn (3 b) or (4b), using the q against log i and the q(t) behaviour differentiated to give dq/dt. Fig. 7 shows C asf(q) at Ni wire at 276, 298 and 318 K, while fig. 8 shows similar plots for an Ni-Mo-Cd electrode at 274, 296 and 353 K. Fig. 9 shows plots of logC against q for the three electrode materials for the defined conditions. In these plots C (= Cdl + C,) or log C have been calculated using eqn (3b) or (4b), respectively, using the dq/dt values and the experimental i(q) values over the whole range of the decay experiments. Evidently, from fig. 7 and 8 there is a continuous variation of Ch with q, exhibiting a maximum,B.E. CONWAY AND L. BAI 1851 Table 1. Values of d log C+/dq for the two linear regions of the plots in fig. 3 and 6 (T = 298 K) electrode b (upper) b (lower) d (upper) d (lower) upper lower Ni wire 0.12 0.12 0.1 1 0.060 ca. 0 - 8.0 N i - M d d 0.125 0.044 0.065 0.030 -7.4 -10.6 Fig. 7. Plots of capacitance C against overpotential q for the h.e.r. at pure Ni-wire in 0.2 mol dm-3 aqueous NaOH at three temperatures : 276, 298 and 3 18 K. as in fact is expected the~retically.~* l4 However, in the plots of log C in fig. 9 it appears that the ascending regions of these plots exhibit two slopes corresponding possibly to two populations of the H intermediate involved in the h.e.r. at Ni or Ni-based electrodes, depending on q and corresponding i values, with experimentally distin- guishable exponential dependences on potential.For q more negative than ca. 0.15-0.25 V, C tends to reach a constant value (fig. 7, larger-scale plots show this more clearly), which we presume is the double-layer capacitance for the electrodes under the prevailing experimental conditions. However, at smaller q, C rapidly increases and maxima are clearly seen for Ni-wire electrode (fig. 7) and the Ni-Mo-Cd electrode (fig. 8). Since C increases in the direction of decreasing negative potential and the adsorbed H is deposited in a cathodic reaction, the constant C above ca. 0.15-0.25 V must correspond to the double-layer capacitance at a surface fully or partially covered by H (see below). The values of this limiting double-layer capacitance are 60-80 pF (apparent) cm-2 for Ni wire.Similar values are found for c d l at the Ni-M&d electrode, allowing for the real/apparent area factor of 450 x measured by the Kr B.E.T. desorption experiment referred to earlier. From the potential dependence of C-C,,, we can calculate the coverage of the ' overpotential-deposited ' H as flq), shown later in fig. 1 1. 61-21852 ADSORPTION BEHAVIOUR OF H IN H, EVOLUTION 800 I 6 00 ?7/V Fig. 8. Plots of capacitance C against overpotential 7 for the h.e.r. of the Ni-Mo-Cd-plated electrodes in 0.2 mol dm-3 aqueous NaOH at three temperatures: 274, 296 and 353 K. 0 . 5 n N s 2 0 % M c( - 0 . 5 - 1 .o - 1 . 5 I 3*0 \ I. :/?: N i - Mo-Cd p l a t e d , 278 K /- f .o h N E 1 .o I 0 VIV Fig.9. Plots of log C against overpotential 7 for the three types of electrode materials.B. E. CONWAY AND L. BAI 1853 POSSIBLE SURFACE OXIDATION PROCESS Before proceeding further, we recognize the possibility that, at the electrodes studied, on decline of potential towards the reversible potential for the h.e.r., a corrosion process could set in, producing a surface oxide film: Ni + 2H20 -+ Ni(OH), + H,. It is conceivable that formation of such a film could give rise to the observed C, near the reversible potential. We consider this possibility in the light of the following information and experiments. First, in terms of the Pourbaix diagram35 for Ni in alkaline solution, oxidation can begin only at a small positive potential, ca. 0.1415 V, relative to the H, reversible potential. Secondly, cyclic voltammograms (fig.10) for both Ni and Ni-Mo-Cd electrodes show that surface oxidation processes do not commence until potentials are attained that are significantly more positive than those corresponding to the range of potentials where C, is appreciable (fig. 7-9). The onset of surface oxidation occurs, in fact, at potentials more positive than thermodynamically expected,36 owing to irreversibility in the process, familiar in oxide-film f o r m a t i ~ n . ~ ~ l7 Also, the formation charge involved is very much greater than that corresponding to integration of the Ctb against q profiles derived here for Ni. The development of oxide films, even at sub-monolayer levels, can be readily detected by means of ellipsometry. Work is in progress to adapt the chronoellipsometric method for the study of the surface of smooth Ni electrodes during a potential decay transient towards the H, reversible potential.It is anticipated that the results of this experiment will demonstrate clearly if any oxide film is generated by corrosion during the course of the transient. HYDROGEN-ATOM COVERAGE From the potential-decay behaviour, information on H coverage is revealed through the pseudocapacitance C, (= q1 de,/dq). For an electrochemical adsorption isotherm of the Frumkin form [see ref. (1 4)] : the corresponding pseudocapacitance (for reversible behaviour) is14 where g is an interaction ~arameter.~? l4 ~btainable,~ uiz. Only in the limiting case of g = 0 (no interactions) is an explicit function of potential For other conditions numerical evaluation of C4(q) is required.14 The observed potential ranges of appreciable H pseudocapacitance are - 0.02 to -0.3 V EH at Ni-wire electrodes and -0.01 to -0.25 V EH at the Ni-Mo-Cd electrodes.The corresponding fractional coverages by H, or changes in coverage, are evaluated by integrating C4(q)/q1 with respect to q, giving OH asflq), as shown in fig. 11.1854 ADSORPTION BEHAVIOUR OF H IN H, EVOLUTION 2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 rll V 2 - 1 - 4 E 0 - 2 - 1 - - 2 - - 3 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 OlV Fig. 10. Cyclic-voltammetry plot of i against q over various potential ranges for (a) pure Ni wire and (b) Ni-Mo-Cd-plated electrode in 0.2 mol dm-3 aqueous NaOH at 298 K with a sweep rate of 18.9 mV s-l in N,-bubbled solution.Before each cycle the potential was held at q = -0.09 V for 180 s. The H deposition charges corresponding to the above potential ranges for Ni and Ni-MMd electrodes at 298 K are 75.6 5 pC cm-2 and 22.8 _+ 2 mC cm-2, re- spectively, and are temperature dependent. These values are substantially less than that corresponding to a monolayer of H, ca. 257 pC cm-2 for the (1 11) plane at Ni or 210 pC cm-2 for Pt,8 allowing for a real/apparent area of ca. 450 at the electrocoated material and ca. 2 at Ni wire. These data suggest that a steady-state coverage by H of less than a monolayer arises at the Ni or Ni-Mo-Cd electrodes as the overpotentialB. E. CONWAY AND L. BAI 1855 0.8 o.7[ 0 0.05 0.1 0.15 0.2 0.25 0.30 VIV Fig. 11. Plot of coverage of H(8) against overpotential q for the h.e.r.at Ni-Mo-Cd-plated electrodes at (1) 353 and (2) 298 K and at pure Ni-wire electrodes at (3) 298 and (4) 276 K. Coverage, 0, derived from the C data shown in fig. 7 and 8 assuming real/apparent area ratio is ca. 450 for the composite electrodes and ca. 2 for Ni wire, taking 8, = 1 corresponding to ca. 257 pC cm-2. increases. This, as well as the observed Tafel slopes, shows that recombination of chemisorbed H cannot be the rate-controlling step in the h.e.r. at the Ni electrode surfaces. In the case of the step M + H,O + e e MH,,, +OH- (10) coupled with H desorption and H,-molecule formation by MH,,,+H,O+e M+H,+OH- (6) we can show, as follows, that limiting steady-state H coverages of < 1 are expected.The relevant rate equations for reactions (6) and (10) are: V10 = kl,,(l - O H ) eXp (-pqF/RT) 1(1 -OH), Say (11) U-lo = k-lo COH -exp [( 1 -B) qF/RT] me,, say (12) Us = k, 8H exp (- yqF/RT) n8H, say (13) where p and y are symmetry factors for charge transfer and 1, m and n are defined according to eqn (1 1H13). The steady-state condition for H coverage is then i.e. l(1 -OH)-m8,-n8, = 0 (15) or 8 = l/(l+m+n) (16)1856 ADSORPTION BEHAVIOUR OF H rn H, EVOLUTION noting that I, m and n are exponential functions of q as eqn (1 1 )-( 13). Corresponding to eqn (1 6), the steady-state pseudocapacitance q1 d8H/dq is Fq1 lm C# = - RT ( I + m + n), taking = y = 0.5. Inserting the potential-dependent factors 8, = 1 /[ 1 + Ky,’ COH -exp (qF/RT) + k6/kio]. (18) At appreciable cathodic overpotentialst it is seen that OH reaches limiting values for the reactions (6) and (10) given by i.e.8, is determined by the extent to which the discharge step, producing chemisorbed H, and the reverse are in ‘equilibrium’,$ thus, for k, .4 klo, 8, -+ 1 limitingly and reaction (10) is in almost complete equilibrium. Alternatively, when k,/klO > 1, i.e. when the h.e.r. is ‘discharge controlled’, there is a varying degree of quasi-equilibrium in reaction (10) and, with increasing overpotential, limiting fractional coverages by H, 8H of < 1 can arise, as shown in fig. 12(a). The limiting values of 8, as a function of k,/k,o are shown in fig. 13. The corresponding C, maxima are then smaller [fig. 12(6)] than for the quasi-equilibrium case.l3, l 4 Note that as k,/klo goes from 6 1 to % 1 the kinetics of reactions (10) and (6) change from being ‘desorption controlled’ [through reaction (6)] to being ‘discharge controlled’ [through reaction (lo)], respec- tively, with 8H then $ 1.This rather special feature of the behaviour of the dual-reaction sequence of reactions (6) and (10) arises because both the desorption and the adsorption (discharge) steps have the same dependence on potential, if /? = y, and is a diagnostic aspect of this mechanism. In the general case, corresponding to eqn (7) where effects of significant interactions are included, the steady-state condition gives 8H then has limiting values at appreciable cathodic overpotentials dependent on g and on the rate constant ratio k,/kl,. Correspondingly, the H pseudocapacitance also depends on these two parameters.As may be expected, for g > 0 a much wider spread of values of the ratio k,/klO is required$ to cause to vary from 1 to smaller values than is the case for Langmuir adsorption (g = O ) . 7 9 3 7 RELATION TO THE EXPERIMENTAL BEHAVIOUR It is clear that the experimental coverage and H pseudocapacitance behaviour shown in fig. 7 and 8, which correspond to attainment of a limiting coverage 8, substantially < 1 at both types of electrode, is rationalized by the above treatment of reactions (6) and (10) in the steady-state. From the experimental data, we see from fig. 1 1 that k,/klO calculated from using eqn (19) (Langmuir case) would be t At low overpotentials, eH is influenced by K;: and by q, so that the general eqn (18) must be used # When this is the case, the Cd differs1* from its ‘reversible’ value given by eqn (9).0 In the case where g may be significantly > 0, it follows from eqn (20) that the ratio k,/k,, required for evaluation of OH asf(q) under such conditions. for a given value of OH, lim to be attained is smaller.B. E. CONWAY AND L. BAI 1857 2500 2000 ’? 1500 6 Lii 1000 LL a 1 500 0 Fig. 12. Calculated plots of (a) 8, against overpotential for three k,/k,, values, using eqn (16), and (b) C+ against overpotential using eqn (17), taking coH- = 1, q1 = 257 pC cm-2, T = 298 K, k,,, = 1, kLl0 = 30 with (1) k6 = 0.1, (2) k, = 1 and (3) k, = 10 (arbitrary units for k since only I 1 1 1 1 I 0 10 20 30 k6lk 10 Fig. 13. Calculated plots of limiting coverage OH, Ilm for reactions (10) and (6) against ka/k,o, using eqn (1 9).4.7 for Ni wire and ca. 1.4 for the Ni-Mo-Cd electrode material at T = 298 K. These results also show that it is not correct to assume, as has often been done previously, that when the proton-discharge, H-adsorption step reaction (10) is rate controlling, the H coverage is very small. Evidently, at the various Ni-based electrodes studied here, the rate constants of the ‘discharge’ and ‘desorption’ steps are comparable so1858 ADSORPTION BEHAVIOUR OF H IN H, EVOLUTION that it is not possible to define unambiguously the ‘rate-controlling’ step for the be havi our actually observed . From the present results at Ni and Ni-Mo-Cd electrodes, we can conclude that, with increasing q, the h.e.r.at these metals proceeds on a surface incompletely but still appreciably covered by H, the limiting fractional coverage being determined in the steady state by the ratio of the rate constants of the H-discharge and the H-desorption steps. Integration of the observable C, against q profiles gives experi- mentally determined values for these limiting coverages and hence the ratio of the rate constants. We see from fig. 7, 8 and 11 that the limiting coverages by H are quite temperature dependent. This is easily rationalized in terms of the above treatment since OH, lim, being determined by the ratio k6/klo, will normally be temperature dependent as k, and klo will be expected to be associated with different activation energies. For the mechanism involving reactions ( 6 ) and (lo), it is expected that b will remain constant with a value of ca.0.12 V when OH is near 1 or is otherwise constant. In the Langmuir case [eqn (7), g = 01, b will in fact remain constant down to ca. -50 mV from the standard potential (0, = 0.5) for H adsorption through reaction (10). Below this potential, 8, will start to decrease in a potential-dependent manner and the Tafel slope will become less steep. Correspondingly, C, increases approximately exponentially with -qF/RT as I q I decreases (fig. 9). Thus we see that the slopes of the log(t+z) and log(-dqldt) plots in relation to dq/d logi are consistent and correspond to the h.e.r. proceeding with a limiting constant OH at high q and a potential-dependent OH at q < ca. 0.15 V for the Ni-Mo-Cd electrodes.Simulated Tafel plots were calculated for reactions (6) and (10) for 3 values of k,/k,,, with reaction (6) or (10) rate controlling. The results are shown in fig. 14. A transition from almost single-slope behaviour (b = 0.1 18 V) to double-slope behaviour (b = 0.044-0.1 18 V) arises as k,/k,O is decreased from 10 to 0.1, corresponding to the calculated coverage against potential relation of fig. 12. This change of slope behaviour corresponds, of course, to a change of rate-controlling step and to the different coverage conditions that obtain as k,/klO varies. From the data of fig. 11 derived from the experimental decay behaviour, the self-consistency of the results can be demonstrated by calculating the corresponding expected Tafel relations, taking account of the variation of OH with q and using exp (qFI2RT) as the charge-transfer factor.The results of such calculations are shown in fig. 15, from which it is seen that the double-slope forms of the experimental Tafel plots [fig. 1 (a) and (b)] are well reproduced and thus can be clearly attributed to the dependence of OH on q . The mechanism of the h.e.r. at Ni in alkaline solution and the extent of H coverage during steady evolution of H, have hitherto been controversial 33 The present results now give a clear conclusion regarding this matter: at potentials more negative than 0.15-0.25 V E,, H, evolution at Ni-wire or Ni-Mo-Cd electrodes from alkaline solutions proceeds on a surface that is appreciably covered, but to a limitingly constant value of < 1, by the electroactive intermediate adsorbed H, giving a Tafel slope 0.125 V, while nearer the reversible potential, as coverage by H decreases, a potential-dependent pseudocapacitance is developed with a characteristic’ * l 4 maximum giving a Tafel slope of 0.04-0.06 at the Ni-Mo-Cd electrode.At Ni wire, the limiting OH is smaller than at Ni-Mo-Cd and its potential dependence is insufficient to give rise to a well defined lower-slope region at this metal; however, near the reversible potential, after the back reaction current, which complicates linearity of the Tafel relation, is subtracted out [fig. l ( a ) and (b)], the Tafel plot for the h.e.r. at Ni wire [fig. 1 (a)] does show some change of slope at low values of q. We see that the new method of analysis of potential decay leads, for the first time,B. E.CONWAY AND L. BAI 1859 log ( i / i o ) Fig. 14. Numerical simulation of Tafel plots for h.e.r. according to reactions (6) and (10) using the relation i / i o = 8, exp (qF/2RT) with same values of parameters as for plots in fig. 12 (values of k, shown on curves). log (i/A cm-2) Fig. 15. Tafel relations derived from the experimentally determined 8, asfTq) from fig. 1 1 : (1) Ni, 298 K, i, = 1 x A cm-2, (2) Ni-Mo-Cd-plated, 298 K, i, = 1 x A (apparent) cm-2 and (3) as in (2) but for 353 K. to rather complete experimental knowledge of the coverage and pseudocapacitance behaviour of the electroactive H intermediate in the h.e.r. proceeding at appreciable currents at Ni, a Ni-Mo bulk alloy and electrocoated Ni-Mo-Cd cathodes, as well as the condition of the kinetics of the steps in the overall reaction pathway in terms of the ratios of the rate constants involved.1860 ADSORPTION BEHAVIOUR OF H IN H, EVOLUTION COMPARISON OF POLARIZATION BEHAVIOUR OF Ni-Mo-Cd AND Ni ELECTRODES The principal difference in polarization behaviour is the existence of a low-slope region at Ni-Mo-Cd that is not observed at Ni-wire electrodes (fig.1) but the high-current-density behaviour is similar, allowing for the ratio of the electrochemical real/apparent area factors of ca. 220. This also corresponds to the capacitance behaviour (fig. 7-9) where the double-layer capacitance is found to be ca. 250 times larger at the coated electrodes than at Ni. However, an additional difference that may be connected with the appearance of the low-slope Tafel region at Ni-Mo-Cd is the trend to a relatively high capacitance between = - 0.10 and 0 V (fig.8) corresponding to the second linear region of the log C relation in fig. 9. It is possible that within 0.1 V of the H, reversible potential, the h.e.r. at Ni-Mo electrodeposits or Raneys proceeds through mediation of a decomposing hydride, as Conway et ~ 1 . ~ ~ suggested, and as is supported by cyclic-voltammetry results at these materials. It appears that the coated electrode does not differ just in real area; its method of preparation and composition, which may introduce a hydride phase through codeposition of H,299 387 39 has a substantial effect on the type of polarization behaviour observed (fig.1) and on the potential dependence of the adsorption of H (compare the slopes of the lines in fig. 2,3 and 9). In the case of Ni-Mo, it should also be noted29 that a bulk, thermally prepared alloy does not behave in the same way as an electrocoated material of the same nominal composition (see fig. 1). Further work on comparative aspects of electroplated Ni-Mo and bulk Ni-Mo alloys is in progress and will be reported elsewhere. These observations clearly have important consequences for the possibility3* of preparing by electrocoating and other techniques various high-area electrode materials that differ electrocatalytically from the corresponding bulk substances other than because of the large real area. ACTIVATION ENERGIES The apparent activation energies3s for the two Tafel regions exhibited by the h.e.r.at Ni-Mo-Cd electrodes (fig. 1) were obtained from the derivative d In &/d(l/T) of electrochemical Arrhenius plots. As follows from fig. 1, the activation energy for the low-slope region (17 kJ mol-l) is much lower than that for the higher-slope sections (32 kJ mol-l). This is the basis of the practically important situation that low Tafel slopes increasingly characterize the polarization behaviour of Ni-Mo-Cd electrode preparations as the temperature is raised (fig. 1). The AH%(q = 0) for the h.e.r. at Ni-wire electrodes is ca. 35 kJ mol-l, and the same value characterizes the Tafel relation from to 2 x 10-l A ern-,. The activation energy at q = 0 for Ni-Mo-Cd cathodes (32 kJ mol-l) is quite similar to that at Ni wire (35 kJ mol-l) derived from the Tafel relations at high q, which suggests that over this region the reaction conditions are the same at these two materials, i.e.the h.e.r. proceeds on a surface whose coverage by H is incomplete but limitingly constant with further increase of q. At low q, for Ni-Mo-Cd electrocoated electrodes the activation energy (17 kJ mol-l) is lower, which corresponds to the h.e.r. proceeding on a surface where OH is potential dependent and has not yet attained its constant limiting value. Under these conditions OH, at a given q and rate constant k,, can also be temperature dependent, leading to a different value of AH% from that at high q. Alternatively, the AH% for the region of low q may correspond [cf. ref. (29)] to a different mechanism of H desorption involving a surface hydride.B.E. CONWAY AND L. BAI 1861 CONCLUSIONS (i) The method of analysis of digitally recorded potential decay transients, coupled with kinetic information provided by the Tafel polarization behaviour, allows a complete and accurate evaluation to be made of the potential dependence of H coverage under conditions where appreciable continuous Faradaic curves are passing ; such information was not previously obtainable in an unambiguous way, e.g. from charging curves or cyclic-voltammetry measurements. (ii) The experimental behaviour shows that the h.e.r. at Ni and Ni-Mo-Cd materials proceeds on a surface partially covered to a limiting extent by the adsorbed intermediate, H. Experimentally determinable limiting coverages are attained at ca.- 0.15 to - 0.25 V E , and are temperature dependent. (iii) The situation in (ii) allows determination of the ratio of the rate constants for the rate-controlling desorption step to that for the discharge-chemisorption step. (iv) Good polarization performance of the Ni-Mo-Cd-electrocoated cathodes arises, in part, from the low-slope region that persists increasingly to higher current densities as the temperature is raised. (v) The low-slope region is associated with a region of high pseudocapacitance. A facile desorptive mechanism in H, evolution, involving this H species, which may be in the form of a surface-phase hydride, is indicated. (vi) The behaviour of the Ni-Med-coated electrodes differs in some important ways from metallic Ni and bulk Ni-Mo alloy, so the good performance of the first electrode material is not due simply to the large real-to-apparent areas that can be realized with coating procedures.(vii) Applications of the method to the study of intermediates in other Faradaic reactions, e.g. anodic 0,-, C1,-and F,-evolution processes, can be made and are in progress in this laboratory. We thank the Natural Sciences and Engineering Research Council of Canada for support of this work. 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ISSN:0300-9599    

DOI:10.1039/F19858101841

出版商:RSC    

年代:1985

数据来源: RSC