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Electrochemical Simulations Package, ESPESPElectrochemicalSimulationsPackage(ESP v. 2.4)(C) June 1994 - February 1998,Carlo Nervi( )Note: ESP 2.4 is the last version. This manual (ASCII format) can be found inside the zip file. Download ESP. Read the file readme.esp for last news. You can visit also our mainelectrochemical WWW pages. Links2GoElectrochemistrySummary: Introduction. General Considerations. Input Mechanism from Keyboard. Meanings of Experimental Parameters. Edit Mechanism. Best Fitting. Analysis Tool. References. Notes.INTRODUCTIONESP [1] is a program to perform general electrochemical simulations andBest Fitting of experimental data. Some ideas (i.e. expanding space gridand Runge-Kutta integration) are inspired by Gosser's simulator CVSIM[2], who applied them into his simulator. Usually electrochemicalsimulators, i.e. like CVSIM [2], implement an "analog" waveform; ESPimplement a "digital" one. It means that CV is performed by ESP asStaircase Voltammetry (SCV) rather than Cyclic Voltammetry (CV). Manyelectrochemists mistake analog for staircase CV. It is important torecognize that SCV and analog response are different. To my knowledgethis is the first implementation of a digital-ramp simulator. With thelarge diffusion of digital potentiostat, like EG&G 273, SCV plays animportant role with respect to analog potentiostat. Digital potentialwaveform allow the easy design of new techniques, as well as a lessdependence of the shape of E/i plots from double layer capacitance, dueto the exponential decay of the charging current. In the remaining partof this manual, CV term is used instead of the proper SCV. Availabletechniques in ESP are Cyclic Voltammetry (CV), Square Wave Voltammetry(SWV), Cronoamperometry (CA) and Sampled DC Polarography (SDC). Thislatter technique can be simulated either by solid electrode (constantarea) or by dropping mercury electrode. The former can be supposed to bethe simulation of a vibrating solid electrode, whereas the latter issimulated adopting the concept of a flat electrode moving towards thebulk of the solution which surface area (a sphere) increase by time [3].ESP can simulate virtually any electrochemical mechanism, build as acombination of: a maximum of 20 species a maximum of 10 chemical reactions a maximum of 10 redox coupleESP perform diffusion by fast implementation (in C) of expanding spacegrid algorithm, to minimize computational time. Spatial grid double insize every fourth grid-increment [4]. Homogeneous chemical complicationsare solved by means of the Runge-Kutta integration of the fourth-order[5]. Best-Fitting routines for non-linear optimization are based on theSimplex technique [6]. COOL algorithm [7] is used; the experimentalcurrent i(exp) is expressed as a linear function of the dimensionlesscurrent function, i(sim):i(exp) = slope * i(sim) + intcpHere "slope" and "intcp" are constants that come from the linearregression between experimental and simulated current. Neither slope norintcp depend by simulation parameters. If the ScaleFlag (SF) parameteris set to 0 then the output is that one of a pure simulation, e.g.slope=1 and intcp=0.Simulation input can be done by keyboard or by file. There are two kindof informations that ESP need to perform simulation: 1) a collection ofexperimental parameters (like scan rate, initial and final potentials,etc.), and 2) the mechanism you want to simulate. Both this informationsare stored in the mechanism file, so that there is no necessity tore-type by keyboard every things. There are different files used by ESP,and every extension identify the kind of file. File organization follows: *.mec is an ASCII Mechanism file, where the mechanism is stored. This file is used for both direct simulation and Best Fitting. It can be written by any text editor or by the build-in mechanism editor. Its structure is self-explaining; see supplied examples. *.prn is an ASCII file. The first line must be: /* ASCII E i t */ or any permutation of {E, i, t}, i.e.: /* ASCII i E t */ (Capsare significative!). You can omit t (and type '-' instead of't'), but never omit E or i. Moreover, if t is omitted, the '-'must be the last, i.e.: /* ASCII E i - */ Each subsequent lineis a curve point; each point is made by two (t omitted) or three(t not omitted) values, and the order of the values must be inagreement with the order choosen in the first line. i.e.:/* ASCII E i - */0.010 1.234E-70.020 1.567E-7 ...END The last line of *.prn must be "END". The same file format is used by the output simulation file (P option). The main aim of *.prn file is to provide an alternative input/output file format which can be used by non-EG&G users. *.sim is the simulation output file. It has approximately the same structure of M270 binary file, so who use the EG&G software can read it as a normal experimental file. Reading *.sim by EG&G software a warning could appear: don't worry, you can continue. The best way to avoid trouble with *.sim files is to load them, skip the edventually warning message and immediately overwrite files without any further manipulations. This because I have no official information about the structure of M270 files: I just approximately analysed those files and tryed to reproduce them. *.sim can also be optimized like M270 file. *.fit is the Best Fit settings file (in binary). Store the whole VAR_OPT struct, wich include the experimental current as well as the value of parameters to be Fitted. It is possible to save *.fit an re-start the Best Fitting where you stopped it. *.??? is a M270 experimental file and can be Best Fitted. Each file which doesn't fit in above seen extensions, is supposed to be an M270 experimental file.You can run ESP as "ESP" alone (and you will be prompted to choosevarious possibilities) or you can type "ESP namefile" where the namefileextension reflect the kind of calculation you want. (i.e. ESP *.mec meanssimulation of the selected mechanism, ESP m270_file, ESP *.sim, ESP*.prn, ESP *.fit means Best Fitting).GENERAL CONSIDERATIONSSymbolsMeaningUnitsCConcentrationmM (millimolar, 10-3 M)DDiffusion coefficientcm2/s(cm * cm / s)EPotentialVKeHeterogeneous rate constantcm/sKf, KbHomogeneous rate constant(forward and backward)1/s for first-order rate constants,1/(mM*s) for second-order rate consantsARElectrode areacm2iCurrent outputAmperetTimes (seconds)INPUT MECHANISM FROM KEYBOARDThe prompt "How many species ->" request the total number of speciesthat are electrochemically active or are involved in the mechanism (i.e.a non-electro active fragments produced by an homogeneous irreversiblereaction should not be considered if such a fragment doesn't produce anyfurther electro active species). Each species is associated with aninteger, in order of entry. Refer to this number identification. Foreach species you should enter the corresponding initial concentration;the diffusion coefficient is taken 1E-05 cm2/s by default,but you can modify it later. The concentrations are millimolar.In entering the redox reactions, look at the following example:Enter 1 redox as: [i]: [Ox] [Red] [n e-] [E] [Ke] [alpha][1]: 1 2 1 -.567 .12 .5It means that the redox No. 1 ([1]) is entered: the Ox species is 1, theRed one is 2, reduction processes involves 1 electron, at the formalpotential of E=-0.567 V, with an heterogeneous rate constant of Ke=0.12cm/s, and transfer coefficient alpha=0.5 i.e.: Ox + 1e <--> Red is: 1 + 1e <--> 2Every input must be separated by at least one space.As a guideline, let's say that log(Ke * sqrt(ST/D))>0.2 producenernstian behavior (D=diffusion coefficient); e.g. for ST=25 ms, a=0.5,and D=1E-5 cm2/s, Ke should be greater than 0.2 cm/s.You cannot specify less than 1 redox processes. If a EE mechanism shouldbe entered, i.e. (0/1-) (1-/2-) processes, it is VERY IMPORTANT to enterthe two redox consecutively, for instance: 1 + 1e <--> 2 2 + 1e <--> 3 example:Enter 2 redox as: [i]: [Ox] [Red] [n e-] [E] [Ke] [alpha][1]: 1 2 1 -.300 .25 .5[2]: 2 3 1 -.600 .25 .5This input is an EE mechanism having a separation between the tworeduction potentials of 300 mV. If you don't enter consecutively the tworedox, the EE mechanism is not recognized and unpredictable result isobtained. Higher number of consecutive electron processes are recognized(and calculated correctly) only if you follow the same approach. It meansthat: the [Ox] specie of the first redox (i.e. No 1) can appear only into the first redox. Never in next redox processes. the [Red] specie of the first redox (i.e. No 2) must appear as [Ox] specie into the second redox. the [Red] specie of the second redox (i.e. No 3) must appear as [Ox] specie into the third redox.ecc... ecc...There are no restrictions in the case of independend redox processes.You can set the number of homogeneous chemical reactionsequal to 0. Each chemical reaction is described as: "a + b <--> c + d".For example:Enter 1 reactions (a+b <--> c+d) as [i]: [a] [b] [c] [d] [Kf] [Kb][1]: 1 2 3 0 10.123 0 is: 1 + 2 ---> 3 (Kf=10.123)means a=1, b=2, c=3, d=0, Kf=10.123, Kb=0. Species number 0 means nospecies (i.e. is not an electroactive specie and doesn't participatefurthermore to the overall mechanism). If 0 is specified you must put itas last product or reactant. If there is only one reactant b must be 0 (acan't be 0). If there are no products, both c and d must be 0; if thereis one product d must be 0 (and not c). The homogeneous rate constantmust be positive. They are in unity of 1/s, when first order rateconstant is entered, and unity of 1/(mM*s) when second order rateconstant is entered.MEANINGS OF EXPERIMENTAL PARAMETERSThis section is mainly devoted to non EG&G users, to explain variousexperimental parameters, and their relations. NI, AP, IR (and RU plus DL)are introduced by ESP, and are described here. Both potential and timevalues can be a number or PASS.CO: in this field you can store any comment (67 characters maximum).CP: Condition PotentialCT: Condition Time. The potential CP is applied for the time CT before to start electrochemical experiment.ET: Equilibration Time. The potential IP is hold for ET seconds just before to start experiment.SR: Scan Rate. SR=SI/ST; by this equation ST is calculated (except SDC).FR: Frequency. ST=1/FRSI: Scan Increment. The height of the step: absolute value of the potential increment (in mV) between the actual and the next value of applied potential.ST: Step Time. The duration of the step (in s). Also drop time.TP: Time per Points. Has the same meaning of ST, but in ChronoAmperometry represents the sampling time.AM: Acquisition Mode. AM select the point where, on the potential step wide ST s, the current must be sampled. The Step Time ST is formally divided into 4 equal parts. For CV technique AM, can be: AM=1 the current is sampled at the first quarter of SI AM=2 the current is sampled at the second quarter of SI AM=3 the current is sampled at the third quarter of SI AM=4 the current is sampled at the fourth quarter of SI AM=Ramp the current is a mean of all four points sampled AM=All the current is sampled at all four quarter (Warning: this quadruple the number of points) In other techniques AM doesn't have any meaning.PH: Pulse Height. Used in SWV: the potential in first half of ST is increased by PH, whereas in the second half is decreased by PH, so that the range of potential within the step is wide 2*PH mV.NP: Number of Points. This parameter is automatically calculated by ESP, and can't be modifyed. A maximum of 4000 points are allowed.NI: Number of Subdivision. Is the No. of parts in which the step wide ST is divided for computational purposes. Larger values of NI increase the accuracy. ESP suggest (and set) NI at the beginning of mechanism edit and every time you modify another parameter. To select NI you should set NI and immediately run simulation.AP: Approximation of chemistry. Can be Fast or Exact. Usually it is near the electrode that the effect of associated chemistry must be evaluated, and as the distance from the electrode increase, effects are less pronounced. ESP evaluate the homogeneous rate in the first box of the grid; Fast method check this value and in the case of a low value, computation inside all other boxes is skipped. This speed up calculations and usually is an acceptable approximation. Exact method simply perform all calculations in every boxes (useful only in particual mechanisms). Normally Fast approximation is enough; Fast may fail when homogeneous chemistry reactions are significative only away from the electrode.WE: Working Electrode. Actually three different kind of electrodes are supplied: Solid, HMDE and DME. Solid and HMDE are completely equivalent. DME is implemented as a flat electrode moving towards the bulk of the solution which surface area increase by time. Surface area of DME depends by ST (e.g. drop time) and MF (Mercury Flow).MF: Mercury Flow. MF and ST define the area of the spheric DME.AR: Electrode Area. The dimensional current is proportional to AR.IP: Initial Potential. Applied potential start from IP, goes to V1 (if not equal to PASS), to V2 (if not equal to PASS) and finally to FP.V1: Vertex 1. Optional potential at which sweep rate is switched.V2: Vertex 2. The second optional switching point.VD: Vertex Delay. The holding time at vertex potentials.E1: Potential 1. Is the first potential applied to CA experiments.T1: Time 1. E1 is applied for T1 seconds (CA).E2: Potential 2. Second potential of CA experiments.T2: Time 2. E2 is applied for T2 seconds (CA).FP: Final Potential.NC: Number of Cycles. How many times the IP-V1-V2-FP cycle must be repeated.SC: Store Cycles. Last cycle is always stored. SC select the other cycle that can be stored.IR: IR method. Can be None, RU (Uncompensated Resistamce), or DL (Double Layer and uncompensated resistance).RU: Uncompensated Resistance. Appear only if IR=RU or IR=DL. Must be always greater than 0.DL: Double Layer Capacitance. Appear only if IR=DL. Both DL and RU must be always greater than 0.TE: Temperature.EDIT MECHANISMThe first line of the screen, in YELLOW on the right, shows the name ofthe mechanism file you are editing. The subsequent lines display for eachspecies (in LIGHTRED) by first the concentration and by second thediffusion coefficient. After there are the redox couples. On the right ofthe redox progressive number (in LIGHTMAGENTA), the redox itself isdisplayed. Then there are the chemical reactions with their number (inLIGHTBLUE). If the parameter SF is set to 1 then the output of thesimulation during best-fit is scaled. If SF=0 the output is not scaled.The green writings "Range to Optimize" show all selectedranges (max 10) of experimental file to be Best-Fitted. The "low" and"high" limit are the starting and ending points of the range. To modifyinteractively these limits just type "r n", where n is the number of therange you want to modify. Further instructions are given in thesubsequent text window.The parameters that can be optimized are Kf and Kb for homogeneouschemical reactions; E, Ke, a (alpha), and n (No. of electron, althoughtit is better do not optimize n) for electrochemical reactions; D(diffusion coefficient) and C (concentration) for each species. If oneparameter is selected for optimization, his color will be LIGHTGREEN. Tomodify one parameter (or to deselect the optimization) just type its name(one of the {kf|kb|e|ke|a|n|d|c}), the number of species, redox orchemistry you want to change, and the new value of the parameter. If youwant to optimize one parameter, you should type an "o " at the beginningof the line. Examples:e 2 -.5 Change the E value of the second redox to -0.5 Vkf 1 10. Change Kf of the first chemical reaction to 10.0o e 2 -.5 Optimize the E of redox No. 2, and change it to -0.5 Vex Exit without save mechanism to disk file.sa Save to file *.mec and exitBut the mechanism file include also the experimental electrochemicalparameters as SR (Scan Rate), IP (Initial Potential), etc... ESP iswritten by an electrochemist (me) that use the potentiostat EG&G M273.Although everybody can use ESP, the electrochemical parameters(experimental) are a subset of those of M270 software. The M273 is adigital potentiostat. Each wave form (E vs t) is digital, and theparameter SI (Scan Increment) is the digital step. Each step (wide SI mV)is divided by NI sub-intervals. NI depends from how fast chemicalcomplications are. Higher are the rate constants, higher NI is. NI mustbe multiple of 4 (condition automatically setted by ESP). This becauseeach step, in M273 potentiostat, is divided into 4 sub-intervals. HigherNI is, higher the accuracy of the calculation is, but higher thecomputational time is. You can manually set NI to get high accuracy. Youcan choose the value of NI you want (i.e. you could want to get veryaccurate simulations by setting NI=512, or, before to do finalsimulation, you could want to see how it looks, so fast -but inaccurate-simulations are performed by NI=8). However, as ESP starts and whenever aparameter is modifed (or by setting NI to any values lower than 4), ESPsuggest an optimum value of NI. The best value of NI (and to get valuableresults), is rather a complex function of SI, AM, and kind of techniquechoosen, I suggest you adopt the ESP suggestion, but try manually byyourself the effect of NI over results. To choose another value of NI youmust set NI and immediately run the simulation.BEWARE: the total Number of Points (NP) CANNOT exceed 4000 (as in M270).If you have more than 4000 points a message appear and ESP refuse to exitfrom edit mode until you have less than 4000 points (i.e. by changingSI). The NC (Number of Cycles) select how many cycles will be performed.The last cycle is always collected; the SC (Store Cycle) select the othercycle stored. NC=3, SC=1 means 3 cycles; the first and the third cyclesare collected. Only two cycles can be stored. If NC>SC, a double numberof point is required. BEWARE that increasing NC you greatly increase thetotal time of the experiment; this require a larger space-grid. Actuallythere is a limit of 100 space-increments. If this limit is reached theprogram will ask whether you want to continue or to stop. Also CT, VD andET (Condition Time and Equilibration Time) could greatly increase bothspace-increments and computational time.Another feature introduced starting from ESP 2.1 is the possibility toroughly evaluate the effect of Uncompensated Resistance and Double LayerCapacity. The IR parameter can be None (no uncompensated resistance), RU(only uncompensated resistance) or DL (double layer capacity anduncompensated resistance). If you specify IR=RU, you must supply apositive value for the RU parameter (which appear only if IR is differentfrom None). Having IR=DL a further positive value must be provided to theDL parameter (the double layer capacity).However care should be taken in using this option: no cycles areperformed to reach a self-consistent output current. If you are runningBest Fit, RU/DL can be safely used: the potential correction is done byusing the experimental current, so no cycles are required and routineworks well. Instead, in pure simulations the last computed current at theprevious sub-interval is used to calculate the ohmic drop, so thesimulated curve is not 100% quantitative; however it is useful to have anidea of the shape of current/potential wave. High values of NI producesbest results; again, convergency is achieved by increasing NI.BEST-FITTINGIn Best-Fitting mode experimental curve is green, whereas simulated onesare yellows. It is very important to input parameters not very far fromthe real value. To choose and have an idea of the parameters (i.e. thehomogeneous rate constant and reduction potentials) it is better to dosome simulation before to start Best Fit. The best way to pick bestinitial values is to type "ESP experimental_filename" and begin BestFitting with no parameters to optimize. Just leave the edit window bytyping SA (save *.mec file). This perform a Single Fit and shows thesimulated curve (yellow) together the experimental one (green). At theend of each Single Fit you can press the ESC key to exit from ESP, orother keys to re-enter into mechanism edit mode, so you can changeparameters manually step by step, to look by your own eyes the differencein the next Single Fit run. If in the edit window you type EX (exit)you'll exit from ESP. When you select a parameter to optimize, the BestFit over all experimental file begins, according selected ranges too.Ranges are useful when you want to have good agreement only for someparts of the electrochemical shape (i.e. peaks in CV) excluding parts ofexperimental data affected by large noise (i.e. adsorption, impurities,etc.). If full Best Fit is choosen, Log file optionally will report theBest Fit proceeding. Into Log file are stored every optimized parameters.ANALYSIS TOOLThis is a special feature of ESP, mainly useful for theoretical study.There are few check of the validity of input, and impredictable resultscould be obtained when the rules here outlined are not strictly followed.Many times in electrochemical studies peoples look only potential andcurrent peaks and so on, simply because those points are simplest tolocate. And most of times it is interesting to know how those pointsshift varying some parameters, like SR. Well, analysis tool of ESP allowyou to automatically change some parameters and pick up selected pointsof electrochemical responses. The parameters that can be automagicallychanged are: SI, NI, PH, SR, FR, ST, TE, E, Ke, a, Kf, Kb, KtE, KlE, KtC,KlC, where last four new parameters have the following meaning:KtE = log(sqrt(ST) * Ke/(Dox^((1-a)/2) * Dred^(a/2)))KlE is defined in the same way except that in lieu of ST you should read(ST/(SI*nF/RT)). These parameters redefine Ke (ST is kept to the samevalue, e.g. SR is unchanged, so that comparison of dimensionless currentis possible). The use of KtE resemble the Osteryoung approach, whereasKlE resemble the Matsuda and Ayabe definition of dimensionless parameterLambda.KtC = log(ST * (Kf + Kb))KlC is defined in the same way except that in lieu of ST you should read(ST/(SI*nF/RT)), as in the previous case. These parameters redefine bothKf and Kb, however the equilibrium constant Kf/Kb is unchanged.Each simulation calculated by Analysis tool is saved with a file namelike l0000001.sim. The file number is increased according to the"logout" file. Output is both on screen and into the file "logout"(WARNING: the file is overwritten). Potentials are in mV. The potentialscans (see each techniques) are divided into equals nredox intervals(where nredox is the number of redox of the current mechanism). Insideeach intervals only one peak is searched, as the largest value ofcurrent. To obtain the usual measures of forward (cathodic) peakpotential and current, a second-order polynome is fitted to the threelargest currents, and the peak potential and current are chosen as theposition and amplitude of the maximum of the parabola. The anodic peakis defined in the same way. Depending on the selected techniques, thefollowing applies: CV: only a full cycle is considered. IP must be equal to FP and V1 must be different from PASS; V2 must be equal to PASS; V1 should be more negative than IP. Scans should be IP-V1 (forward) and V1-FP (reverse). Output will contain, for each redox processes, Epc (cathodic peak potential), ipc (cathodic peak current), Epa (anodic peak potential), ipa (anodic peak current), E1/2 (mean of Epa and Epc), DEp (Epa-Epc).SWV: the only allowed scan is IP-FP, with FP more negative than IP (V1=V2=PASS). Output consists of Esu (summit potential), isu (summit current), W1/2 (wide of the peak at the half of height), St (peak area in microcoulomb).SDC: the only allowed scan is IP-FP, as in SWV. Output consists of E1/2 (the half wave potential), il (the current limit), slope (the slope of logarithm analysis used to calculate E1/2). il is taken as the highest current value.CA: CA is not supported by the Analysis tool.To select the analysis tool, instead of normal run, you should manuallyedit mechanism file by your favorite text editor. The magic keyword"Analysis" should be added at the end of the file. Subsequently the nameof the parameter you want to vary, the initial, final and step valuesshould be entered. Examples:AnalysisNI 4 512 4SI 1 20 1Ke 2 0.01 1 0.01This will vary NI from 4 to 512 by the increment of 4 (the minimum!!please take care of NI values: NI must be a multiple of 4!!!). Then SIstart from 1 and will be 20 with step of 1 mV. KE (heterogeneousconstant) of redox number 2 start from 0.01 and will be 1 with step of0.01 cm/s. This means 128x20x100= 256000 simulations! As you can see thenumber of simulations quickly grow by increasing number of parameters...KtE, KlE, KtC, KlC, E (thermodynamic reduction potential), Ke(heterogeneous rate constants), a (alpha), Kf (homogeneous forward raterate constant) and Kb (homogeneous backward rate constant) require thenumber of selected redox or chemistry.An example that illustrate this feature is furnished by the e_an.mec andec_an.mec files. By typing "esp e_an.mec" the file "logout" is created,which shows the effect of SR (Scan Rate) onto potential and current peaksfor a quasi-reversible mechanism. A more complex example is included inec_an.mec, where the automatic change of SR shows the effect of SR whenthe mechanism include an homogeneous coupled chemical reaction.References: Source code of ESP is available upon request. It will be delivered by e-mail only. See notes in README.1ST for details and copyright. D.K.Gosser, F.Zhang, Talanta, 38 (1991) 715. Z.Galus, Fundamentals of Electrochemical Analysis., Ellis, 1994. R.Seeber, S.Stefani, Anal.Chem., 53 (1981) 1011. (a) M.F.Nielsen, K.Almadal, O.Hammerich, V.D.Parker, Acta Chem.Scand., A41 (1987) 423. (b) A.C.Norris, Computational Chemistry, Wiley, 1981. (a) J.A.Nelder, R.Mead, Comput.J., 7 (1965) 308. (b) P.B.Ryan, R.L.Barr, H.D.Todd, Anal.Chem., 52 (1980) 1460. J.O'Dea, J.Osteryoung, T.Lane, J.Phys.Chem., 90 (1986) 2761.NotesAbout Manual:Since my native language is italian, please don't hate me for manylanguage mistake that belong to this manual; just smile and go on ;-)For basics about electrochemistry, see for instance:A.J.Bard, L.R.Faulkner, Electrochemical Methods. Fundamentals andApplications, Wiley, New York, 1980.Z.Galus, Fundamentals of Electrochemical Analysis. 2nd (revised)edition., Ellis Horwood and Polish Scientific Publishers, New York andWarsaw, 1994.D.Britz, Digital Simulations in Electrochemistry, Second, revised andextended edition, Springer-Verlag, Berlin, 1988.Author, Bugs and Suggestions:Any suggestions/bugs reports both about ESP and manual are welcome:Dr. Carlo Nervi,Dipartimento di Chimica IFM,via P. Giuria 7, 10125 Torino, ITALY.Voice: +39 11 6707508Fax : +39 11 6707855e-mail: nervi@ch.unito.itWWW: http://lem.ch.unito.it/Acknowledgments:I wish to thank Dr. Serge V. Kukharenko (Moscow) for helpful discussionsand Dr. M. Ravera (Torino) for beta-testing.Hardware requirements: PC IBM or compatible with 386+387, 486, 586 CPU,color VGA graphics. The Borland Turbo C version of ESP (esp-tc.exe) canbe run on 8086 PC without math co-processor; in any cases, mathcoprocessor is highly recommended. |
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