\r\n\tIn the book the theory and practice of microwave heating are discussed. The intended scope covers the results of recent research related to the generation, transmission and reception of microwave energy, its application in the field of organic and inorganic chemistry, physics of plasma processes, industrial microwave drying and sintering, as well as in medicine for therapeutic effects on internal organs and tissues of the human body and microbiology. Both theoretical and experimental studies are anticipated.
\r\n
\r\n\tThe book aims to be of interest not only for specialists in the field of theory and practice of microwave heating but also for readers of non-specialists in the field of microwave technology and those who want to study in general terms the problem of interaction of the electromagnetic field with objects of living and nonliving nature.
",isbn:"978-1-83968-227-8",printIsbn:"978-1-83968-226-1",pdfIsbn:"978-1-83968-228-5",doi:null,price:0,priceEur:0,priceUsd:0,slug:null,numberOfPages:0,isOpenForSubmission:!1,hash:"8f6a41e4f5ce0e9c48628516d7c92050",bookSignature:"Prof. Gennadiy Churyumov",publishedDate:null,coverURL:"https://cdn.intechopen.com/books/images_new/10089.jpg",keywords:"Electromagnetic Wave, Microwave Energy Application, Electromagnetic Energy Generation, Intelligent Microwave Heating, Microwave Organic Chemistry, Microwave Reactor, Microwave Discharge, Microwave Plasma, Microwave Drying System, Tissue Microwave Heating, Measurement Automation, Industrial Microwave Process",numberOfDownloads:224,numberOfWosCitations:0,numberOfCrossrefCitations:0,numberOfDimensionsCitations:0,numberOfTotalCitations:0,isAvailableForWebshopOrdering:!0,dateEndFirstStepPublish:"July 3rd 2020",dateEndSecondStepPublish:"July 24th 2020",dateEndThirdStepPublish:"September 22nd 2020",dateEndFourthStepPublish:"December 11th 2020",dateEndFifthStepPublish:"February 9th 2021",remainingDaysToSecondStep:"7 months",secondStepPassed:!0,currentStepOfPublishingProcess:5,editedByType:null,kuFlag:!1,biosketch:"Prof. Gennadiy I. Churyumov is a professor at two universities: Kharkiv National University of Radio Electronics, and Harbin Institute of Technology and a senior IEEE member.",coeditorOneBiosketch:null,coeditorTwoBiosketch:null,coeditorThreeBiosketch:null,coeditorFourBiosketch:null,coeditorFiveBiosketch:null,editors:[{id:"216155",title:"Prof.",name:"Gennadiy",middleName:null,surname:"Churyumov",slug:"gennadiy-churyumov",fullName:"Gennadiy Churyumov",profilePictureURL:"https://mts.intechopen.com/storage/users/216155/images/system/216155.jfif",biography:"Gennadiy I. Churyumov (M’96–SM’00) received the Dipl.-Ing. degree in Electronics Engineering and his Ph.D. degree from the Kharkiv Institute of Radio Electronics, Kharkiv, Ukraine, in 1974 and 1981, respectively, as well as the D.Sc. degree from the Institute of Radio Physics and Electronics, National Academy of Sciences of Ukraine, Kharkiv, Ukraine, in 1997. \n\nHe is a professor at two universities: Kharkiv National University of Radio Electronics, and Harbin Institute of Technology. \n\nHe is currently the Head of a Microwave & Optoelectronics Lab at the Department of Electronics Engineering at the Kharkiv National University of Radio Electronics. \n\nHis general research interests lie in the area of 2-D and 3-D computer modeling of electron-wave processes in vacuum tubes (magnetrons and TWTs), simulation techniques of electromagnetic problems and nonlinear phenomena, as well as high-power microwaves, including electromagnetic compatibility and survivability. \n\nHis current activity concentrates on the practical aspects of the application of microwave technologies.",institutionString:"Kharkiv National University of Radio Electronics (NURE)",position:null,outsideEditionCount:0,totalCites:0,totalAuthoredChapters:"2",totalChapterViews:"0",totalEditedBooks:"0",institution:null}],coeditorOne:null,coeditorTwo:null,coeditorThree:null,coeditorFour:null,coeditorFive:null,topics:[{id:"24",title:"Technology",slug:"technology"}],chapters:[{id:"74623",title:"Influence of the Microwaves on the Sol-Gel Syntheses and on the Properties of the Resulting Oxide Nanostructures",slug:"influence-of-the-microwaves-on-the-sol-gel-syntheses-and-on-the-properties-of-the-resulting-oxide-na",totalDownloads:94,totalCrossrefCites:0,authors:[null]},{id:"75284",title:"Microwave-Assisted Extraction of Bioactive Compounds (Review)",slug:"microwave-assisted-extraction-of-bioactive-compounds-review",totalDownloads:12,totalCrossrefCites:0,authors:[null]},{id:"75087",title:"Experimental Investigation on the Effect of Microwave Heating on Rock Cracking and Their Mechanical Properties",slug:"experimental-investigation-on-the-effect-of-microwave-heating-on-rock-cracking-and-their-mechanical-",totalDownloads:28,totalCrossrefCites:0,authors:[null]},{id:"74338",title:"Microwave Synthesized Functional Dyes",slug:"microwave-synthesized-functional-dyes",totalDownloads:21,totalCrossrefCites:0,authors:[null]},{id:"74744",title:"Doping of Semiconductors at Nanoscale with Microwave Heating (Overview)",slug:"doping-of-semiconductors-at-nanoscale-with-microwave-heating-overview",totalDownloads:45,totalCrossrefCites:0,authors:[null]},{id:"74664",title:"Microwave-Assisted Solid Extraction from Natural Matrices",slug:"microwave-assisted-solid-extraction-from-natural-matrices",totalDownloads:25,totalCrossrefCites:0,authors:[null]}],productType:{id:"1",title:"Edited Volume",chapterContentType:"chapter",authoredCaption:"Edited by"},personalPublishingAssistant:{id:"252211",firstName:"Sara",lastName:"Debeuc",middleName:null,title:"Ms.",imageUrl:"https://mts.intechopen.com/storage/users/252211/images/7239_n.png",email:"sara.d@intechopen.com",biography:"As an Author Service Manager my responsibilities include monitoring and facilitating all publishing activities for authors and editors. From chapter submission and review, to approval and revision, copyediting and design, until final publication, I work closely with authors and editors to ensure a simple and easy publishing process. I maintain constant and effective communication with authors, editors and reviewers, which allows for a level of personal support that enables contributors to fully commit and concentrate on the chapters they are writing, editing, or reviewing. I assist authors in the preparation of their full chapter submissions and track important deadlines and ensure they are met. I help to coordinate internal processes such as linguistic review, and monitor the technical aspects of the process. As an ASM I am also involved in the acquisition of editors. 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\n\t\t\t
1. Introduction
\n\t\t\t
We perform a critical analysis of some quantum mechanical models such as the hydrodynamic model (Madelung’s model), de Broglie’s theory of double solution etc., specifying both mathematical and physical inconsistencies that occur in their construction.
separation of the physical motion of objects in wave and particle components depending on the scale of resolution (differentiable as waves and non-differentiable as particles) - see paragraphs 5-7;
solidar motion of the wave and particle (wave-particle duality) - see paragraph 8, the mechanisms of duality (in phase wave-particle coherence, paragraphs 9 and 10 and wave-particle incoherence, see paragraph 11);
the particle as a clock, its incorporation into the wave and the implications of such a process - see paragraphs 12 and 13;
Lorentz-type mechanisms of wave-particle duality - see paragraph 14.
where \n\t\t\t\t\t\n\t\t\t\t\t\tℏ\n\t\t\t\t\t\n\t\t\t\t is the reduced Planck’s constant,\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tm\n\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t the rest mass of the test particle, U the external scalar field and \n\t\t\t\t\t\n\t\t\t\t\t\tΨ\n\t\t\t\t\t\n\t\t\t\t the wave-function associated to the physical system. This differential equation is linear and complex.
\n\t\t\t
Starting from this equation, Madelung (Halbwacs, F., 1960; Madelung R., 1927) constructed the following model. One separates real and imaginary parts by choosing \n\t\t\t\t\t\n\t\t\t\t\t\tΨ\n\t\t\t\t\t\n\t\t\t\t of the form:
is called the quantum potential. Equation (5) corresponds to the momentum conservation law and equation (6) to the conservation law of the probability’s density field (quantum hydrodynamics equations).
\n\t\t\t
We have the following:
\n\t\t\t
any micro-particle is in constant interaction with an environment called „subquantic medium” through the quantum potential Q,
the „subquantic medium” is identified with a nonrelativistic quantum fluid described by the equations of quantum hydrodynamics.
\n\t\t\t
In other words, the propagation of the Ψ field from wave mechanics is replaced by a fictitious fluid flow having the density \n\t\t\t\t\t\n\t\t\t\t\t\tρ\n\t\t\t\t\t\n\t\t\t\t and the speed\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tv\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t, the fluid being in a field of forces\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t∇\n\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\tU\n\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\tQ\n\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t. Moreover, the following model of particle states (Bohm D. & Hiley B.J., 1993; Dϋrr D. et al,1992; Holland P.R., 1993; Albert D.Z., 1994; Berndl K. et al, 1993; Berndl K. et al, 1994; Bell J.S., 1987; Dϋrr D. et al, 1993): Madelung type fluid in „interaction” with its own „shell” (there is no space limitation of the fluid, though of the particle).
\n\t\t
\n\t\t
\n\t\t\t
3. DeBroglie’s theory of double solution. The need for introducing the model of Bohm and Vigier
\n\t\t\t
One of the key observations that de Broglie left in the development of quantum mechanics, is the difference between the relativistic transformation of the frequency of a wave and that of a clock’s frequency (de Broglie L., 1956; de Broglie L., 1957; de Broglie L., 1959; de Broglie L., 1963; de Broglie L., 1964; de Broglie L., 1980). It is well known that, if \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tυ\n\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t is the frequency of a clock in its own framework, the frequency confered by an observer who sees it passing with the speed \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tv\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\tβ\n\t\t\t\t\t\t\tc\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t is
This is what is called the phenomenon of “slowing down of horologes”. This phenomenon takes place due to the relative motion of horologes. On the contrary, if a wave within a certain reference system is a stationary one, with frequency \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tυ\n\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t and is noticed in a reference system animated with speed\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tv\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\tβ\n\t\t\t\t\t\t\tc\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t, as compared with the first one, it will appear as a progressive wave that propagates in the sense of the relative motion, with frequency
and if we admit that within the appropriate system of the corpuscle the associated wave is a stationary one, with frequency\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tυ\n\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t, all the fundamental relations of undulatory mechanics and in particular\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tλ\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\th\n\t\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t, in which p is the impulse of the corpuscle, are immediately obtained from the previous relations.
\n\t\t\t
Since de Broglie considers that the corpuscle is constantly located in the wave, he notices the following consequence: the motion of the corpuscle has such a nature that it ensures the permanent concordance between the phase of the surrounding wave and the internal phase of the corpuscle considered as a small horologe. This relation can be immediately verified in the simple case of a corpuscle in uniform motion, accompanied by a monochromatic plain wave. Thus, when the wave has the general form
in which A and Φ are real, the phase concordance between the corpuscle and its wave requires that the speed of the corpuscle in each point of its trajectory be given by the relation
Nevertheless it was not enough to superpose the corpuscle with the wave, imposing it to be guided by the propagation of the wave: the corpuscle had to be represented as being incorporated in the wave, i.e. as being a part of the structure of the wave. De Broglie was thus directed to what he himself called the theory of “double solution”. This theory admits that the real wave is not a homogeneous one, that it has a very small area of high concentration of the field that represents the corpuscle and that, besides this very small area, the wave appreciably coincides with the homogeneous wave as formulated by the usual undulatory mechanics.
\n\t\t\t
The phenomenon of guiding the particle by the surrounding undulatory field results from the fact that the equations of the field are not linear ones and that this lack of linearity, that almost exclusively shows itself in the corpuscular area, solidarizes the motion of the particle with the propagation of the surrounding wave (de Broglie L., 1963; de Broglie L., 1964; de Broglie L., 1980).
\n\t\t\t
Nevertheless there is a consequence of “guidance” upon which we should insist. Even if a particle is not submitted to any external field, if the wave that surrounds it is not an appreciably plain and monochromatic one (therefore if this wave has to be represented through a superposition of monochromatic plain waves) the motion that the guidance formula imposes is not rectilinear and uniform. The corpuscle is subjected by the surrounding wave, to a force that curves its trajectory: this “quantum force” equals the gradient with the changed sign of the quantum potential Q given by (7). Therefore, the uniform motion of the wave has to be superposed with a “Brownian” motion having random character that is specific to the corpuscle.
\n\t\t\t
Under the influence of Q, the corpuscle, instead of uniformly following one of the trajectories that are defined by the guidance law, constantly jumps from one of these trajectories to another, thus passing in a very short period of time, a considerably big number of sections within these trajectories and, while the wave remains isolated in a finite area of the space, this zigzag trajectory hurries to explore completely all this region. In this manner, one can justify that the probability of the particle to be present in a volume element \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\td\n\t\t\t\t\t\t\tτ\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t of the physical space is equal to\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t|\n\t\t\t\t\t\t\t\t\t\tΨ\n\t\t\t\t\t\t\t\t\t\t|\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\td\n\t\t\t\t\t\t\tτ\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t. This is what Bohm and Vigier did in their statement: therefore they showed that the probability of repartition in \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t|\n\t\t\t\t\t\t\t\t\t\tΨ\n\t\t\t\t\t\t\t\t\t\t|\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t must take place very quickly. The success of this demonstration must be correlated with the characteristics if “Markov’s chains.”(Bohm, D., 1952; Bohm D. & Hiley B.J., 1993; Bohm D., 1952;., Bohm D 1953).
the frequency of the corpuscle that is assimilated to a small horologe must be identified with the frequency of the associated progressive wave;
the coherence of the inner phase of the corpuscle-horologe with the phase of the associated wave;
the corpuscle must be “incorporated” into the progressive associated wave through the “singularity” state. Thus, the motion of the corpuscle “solidarizes” with the propagation of the associated progressive wave. Nevertheless, once we admit these statements, de Broglie’s theory does not answer a series of problems, such as, for example:
What are the consequences of this “solidarity”? And we could continue …. Moreover, Madelung’s theory (Halbwacs, F., 1960; Madelung R., 1927) brings new problems. How can we built a pattern of a corpuscle (framework + Madelung liquid) endlessly extended in space?
\n\t\t\t
Here are some of the “drawbacks” of the patterns in paragraphs 2 and 3 which we shall analyze and remove by means of introducing the fractal approximation of the motion.
\n\t\t
\n\t\t
\n\t\t\t
5. The motion equation of the physical object in the fractal approximation of motion. The Eulerian separation of motion on resolution scales
where \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tV\n\t\t\t\t\t\t\t\t^\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t is the complex speed field
Now we can apply the principle of scale covariance by substituting the standard time derivate (d/dt) with the complex operator\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\td\n\t\t\t\t\t\t\t\t^\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t/\n\t\t\t\t\t\t\td\n\t\t\t\t\t\t\tt\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t. Accordingly, the equation of fractal space-time geodesics (the motion equation in second order approximation, where second order derivates are used) in a covariant form:
This means that the sum of the local acceleration\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t∂\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tV\n\t\t\t\t\t\t\t\t\t\t^\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t/\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t∂\n\t\t\t\t\t\t\t\t\tt\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t, convection \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tV\n\t\t\t\t\t\t\t\t^\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t⋅\n\t\t\t\t\t\t\t∇\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tV\n\t\t\t\t\t\t\t\t^\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t and ”dissipation” \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t∇\n\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tV\n\t\t\t\t\t\t\t\t^\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\treciprocally compensate in any point of the arbitrarily fractal chosen trajectory of a physical object.
\n\t\t\t
Formally, (10) is a Navier-Stokes type equation, with an imaginary viscosity coefficient,
This coefficient depends on two temporary scales, as well as on a length scale. The existence of a pure imaginary structured coefficient specifies the fact that “the environment” has rheological features (viscoelastic and hysteretic ones (Chioroiu, V. et al, 2005; Ferry, D. K. & Goodnick, S. M., 2001; Imry, Y., 2002)).
Equation (14a) corresponds to the law of the impulse conservation at differentiable scale (the undulatory component), while (14b) corresponds to the same law, but at a non differentiable scale (corpuscular component). As we will later show, in the case of irotational movements (14) it will be assimilated to the law of mass conservation.
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6. Rotational motions and flow regimes of a fractal fluid
\n\t\t\t
For rotational motions, \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t∇\n\t\t\t\t\t\t\t×\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tV\n\t\t\t\t\t\t\t\t^\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t≠\n\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\trelation (10) with (9) through separating the real part from the imaginary one, i.e. through separating the motions at a differential scale (undulatory characteristic) and non differential one (corpuscular characteristic), results (Harabagiu A. et al, 2010)\n\t\t\t
We can now characterize the flow regimes of the fractal fluid at different scales, using some classes of Reynolds numbers. At a differential scale we have
In previous relations V, L, D, are the specific parameters, while U, l, D are the parameters of the non differential scale. The parameters V, U are specific speeds, L, l specific lengths and D is a viscosity coefficient. Moreover, the common “element” for R(D-N), R(N-N), R(TDN-D) and R(TND-D) is the ”viscosity” which, through (20) is imposed by the resolution scale.
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Equations (15) are simplified in the case of the stationary motion for small Reynolds numbers. Thus, equation (15) for small R (D-N) becomes
7. Irotational motions of a fractal fluid. The incorporation of the associate wave corpuscle through the solidarity of movements and generation of Schrodinger equation
The Schrodinger “geodesics” can be obtained as a particular case of equation (36), based on the following hypothesis (conditions of solidarity of the motion, incorporating the associated wave corpuscle):
\n\t\t\t
the motions of the micro-particles take place on fractal curves with the fractal dimension DF=2, i.e. the Peano curves (Nottalle, L., 1993; Nottale, L., 2004);
with m0 the rest mass of the microparticle, c the speed of light in vacuum and \n\t\t\t\t\t\n\t\t\t\t\t\tℏ\n\t\t\t\t\t\n\t\t\t\t the reduced Planck constant. The parameters (38) should not be understood as “structures” of the standard space-time, but as standards of scale space-time; iv) function F(t) from (36) is null. Under these circumstances, (36) is reduced to the standard form of Schrodinger’s equation (Ţiţeica, S., 1984; Peres, A., 1993)
In such a context, the scale potential of the complex speeds plays the role of the wave function.
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8. Extended hydrodynamic model of scale relativity and incorporation of associated wave corpuscle through fractal potential. The correspondence with Madelung model
\n\t\t\t
Substituting the complex speed (9) with the restriction (27) and separating the real part with the imaginary one, we obtain the set of differential equations (Harabagiu A. et al, 2010)
with \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tρ\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t an amplitude and S a phase, then (34) under the form
or, still, through integration with \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tρ\n\t\t\t\t\t\t\t≠\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\n\t\t\t
Equation (40) corresponds to the impulse conservation law at differential scale (the classical one), while the impulse conservation law at non differential scale is expressed through (45) with\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tT\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\tt\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t≡\n\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t, as a probability density conservation law
with Q given by (41) or (44) forms the set of equations of scale relativity extended hydrodynamics in fractal dimension DF. We mention that in references (Nottalle, L., 1993; Chaline, J. et al, 2009; Chaline, J. et al, 2000; Nottale, L., 2004) the model has been extended only for DF=2. The fractal potential (41) or (44) is induced by the non differentiability of space-time.
\n\t\t\t
In an external scalar field U, the system of equations (46) modifies as follows
Now the quantum mechanics in hydrodynamic formula (Madelung’s model (Halbwacs, F., 1960)) is obtained as a particular case of relations (47), using the following hypothesis:
the motion of the micro-particles takes place on Peano curves with DF=2;
\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\td\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t±\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\t\tξ\n\t\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\tare the Markov-Wiener variables (Nottalle, L., 1993; Chaline, J. et al, 2009; Chaline, J. et al, 2000; Nottale, L., 2004);
the time space scale is a Compton one. Then, (38) have the expressions
Let us choose the null power density in (51b). Then there is no impulse transport at differential scale between corpuscle and wave. Moreover, for \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tρ\n\t\t\t\t\t\t\t≠\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\n\t\t\t
In other words, the fluid becomes coherent (the fluid particles have the same phase). Such a state is specific for quantum fluids (Ciuti C. & Camsotto I., 2005; Benoit Deveand, 2007), such as superconductors, superfluids, etc. (Felsager, B., 1981; Poole, C. P. et al, 1995). Under such circumstances, the phase of the corpuscle considered as a small horologe equals the phase of the associated wave (coherence in corpuscle-wave phase).
\n\t\t\t
At non-differential scale, equation (51), with restriction (52) takes the form
field A is expelled from the structure, its penetration depth being
the one-dimensional geodesics of the space are described through function
the dominant of the corpuscular characteristic is accomplished by means of “self-expulsion” mechanism of the fractal field from its own structure that it generates (that is the corpuscle), the penetration depth being\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tΛ\n\t\t\t\t\t\t\t\t\t¯\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t. The identification
there is an impulse transfer between the corpuscle and the wave on the fractal component of the speed field, so that all the attributes of the differential speed could be transferred on the fractal speed.
\n\t\t\t
All the above results indicate that wave-particle duality is an intrinsic property of space and not of the particle.
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10. Wave-corpuscle duality through flowing stationary regimes of a coherent fractal fluid in phase. The potential well
For movements on Peano curves (DF=2) at Compton scale \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\tm\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tD\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\tℏ\n\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t(83) with substitutions
while for \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tn\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t∞\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\tit implies a Ruelle-Takens’ type criterion of evolution towards chaos (Ruelle D. & Takens, F., 1971\n\t\t\t\t; Ruelle, D., 1975). Therefore, the wave-corpuscle duality is accomplished through the flowing regimes of a fractal fluid that is coherent in phase. Thus, the laminar flow (small n) induces a dominant ondulatory characteristic, while the turbulent flow (big n) induces a dominant corpuscular characteristic.
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11. Wave-corpuscle duality through non-stationary regimes of an incoherent fractal fluid
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In the one dimensional case the equations of hydrodynamics (46) take the form
both differential scale speed V and non-differential one U are not homogeneous in x and t. Under the action of fractal force F, the corpuscle is assimilated to the wave, is a part of its structure, so that it joins the movement of the corpuscle with the propagation of the associated progressive wave;
the timing of the movements at the two scales, V=U implies the space-time homographic dependence
the uniform movement V=c is obtained for null fractal force F=0 and fractal speed U=0, using condition x=ct. The fractal forces in the semi space. \n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\t\t∞\n\t\t\t\t\t\t\t\t\t≤\n\t\t\t\t\t\t\t\t\tx\n\t\t\t\t\t\t\t\t\t≤\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tx\n\t\t\t\t\t\t\t\t\t\t¯\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\tand \n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tx\n\t\t\t\t\t\t\t\t\t\t¯\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t≤\n\t\t\t\t\t\t\t\t\tx\n\t\t\t\t\t\t\t\t\t≤\n\t\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\t\t∞\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t are reciprocally compensated.
This means that the corpuscle in “free” motion simultaneously polarizes the “environment” of the wave behind \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tx\n\t\t\t\t\t\t\t≤\n\t\t\t\t\t\t\tc\n\t\t\t\t\t\t\tt\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t and in front of\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tx\n\t\t\t\t\t\t\t≥\n\t\t\t\t\t\t\tc\n\t\t\t\t\t\t\tt\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t, in such a manner that the resulting force has a symmetrical distribution as compared with the plane that contains the position of the noticeable object \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tx\n\t\t\t\t\t\t\t\t¯\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\tc\n\t\t\t\t\t\t\tt\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t at any time moment t. Under such circumstances, the physical object uniformly moves (the corpuscle is located in the field of the associated wave).
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12. The corpuscle as a horologe and its incorporation in the associated wave. Consequences
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According to de Broglie’s theory, the corpuscle must be associated to a horologe having the frequency equal to that of the associated progressive wave. Mathematically we can describe such an oscillator through the differential equation
where ω defines the natural frequency of the oscillator as it is dictated by the environment (the wave), and the point above the symbol referes to the differential as compared with time. The most general solution of equation (94) generally depends not on two arbitrary constants, as it is usually considered, but on three: the initial relevant coordinate, the initial speed and the phase of the harmonic oscillatory within the ensemble that structurally represents the environment (the isolated oscillator is an abstraction !). Such a solution gives the relevant co-ordinate
where \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\th\n\t\t\t\t\t\t\t¯\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t refers to the complex conjugate of h and \n\t\t\t\t\t\n\t\t\t\t\t\tΦ\n\t\t\t\t\t\n\t\t\t\t is an initial phase specific to the individual movement of the oscillator. Such a notation allows us to solve a problem that we could name “the oscillators with the same frequency”, such as Planck’s resonators’ ensemble-the basis of the quantum theory arguments in their old shape. That is, given an ensemble of oscillators having the same frequency in a space region, which is the relation between them?
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The mathematical answer to this problem can be obtained if we note that what we want here is to find a mean to pass from a triplet of numbers –the initial conditions- of an oscillator towards the same triplet of another oscillator with the same frequency. This process (passing) implies a simple transitive continuous group with three parameters that can be built using a certain definition of the frequency. We start from the idea that the ratio of two fundamental solutions of equation (94) is a solution of Schwarts’ non linear equation (Agop, M. & Mazilu, N., 1989; Agop, M. & Mazilu, N., 2010; Mihăileanu, N., 1972)
This equation proves to be a veritable definition of frequency as a general characteristic of an ensemble of oscillators that can be scanned through a continuous group of three parameters. Indeed equation (96) is invariant to the change of the dependent variable
which can be verified through direct calculation. Thus, \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tτ\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\tt\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\tcharacterizes another oscillator with the same frequency which allows us to say that, starting from a standard oscillator we can scan the whole ensemble of oscillators of the same frequency when we let loose the three ratios a: b: c: d in equation (97). We can make a more precise correspondence between a homographic change and an oscillator, by means of associating to each oscillator a personal \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tτ\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\tt\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\tthrough equation
Let us notice that \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tτ\n\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tτ\n\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\tcan be freely used one instead the other, which leads to the next group of changes for the initial conditions
Now, through a Matzner-Misner type principle one can obtain Ernst’s principle of generating the symmetrical axial metrics (Ernst, F.J., 1968; Ernst, F.J., 1971)
where γ = det γαβ with γαβ the metrics of the “environment”.
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Therefore, the incorporation of the corpuscle in the wave, considering that it functions as a horologe with the same frequency as that of the associated progressive wave, implies gravitation through Einstein’s vacuum equations (equivalent to Ernst’s principle (106d)). On the contrary, when the frequencies do not coincide, there is an induction of Stoler’s group from the theory of coherent states (the parameter of the change is the very ratio of frequencies when creation and annihilation operators refer to a harmonic oscillator (Agop, M. & Mazilu, N., 1989)).\n\t\t\t
\n\t\t\t
Let us note that the homographic changes (99) generalize the result (92). Moreover, if \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\ta\n\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\tb\n\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\tc\n\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\td \n\t\t\t\t\t\t\tє\n\t\t\t\t\t\t\tℤ\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\tthen the Ernst type equations describe supergravitation N=1 (Green, M.B. et al, 1998).
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13. Informational energy through the fractal potential of complex scalar speed field. The generation of forces
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The informational energy of a distribution is defined through the known relation (Mazilu N. & Agop M., 1994),
where \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tρ\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\tx\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t is the density of distributions, and we note by x, on the whole, the random variables of the problem, dx being the elementary measure of their field.
\n\t\t\t
This functional represents a measure of the uncertainty degree, when defining the probabilities, i.e. it is positive, it increases when uncertainty also incresases taken in the sense of expanding distribution and it is additive for sources that are independent as compared to uncertainity. If we admit the maximum of informational energy in the inference against probabilities, having at our disposal only a partial piece of information this is equivalent to frankly admitting the fact that we cannot know more. Through this, the distributions that we obtain must be at least displaced, as compared to the real ones, because there is no restrictive hypothesis regarding the lacking information. In other words, such a distribution can be accomplished in the highest number of possible modalities. The partial piece of information we have at our disposal, is given, in most cases, in the form of a f(x) function or of more functions.
are now constraints the variation of the functional (107) has to subject to, in order to offer the distribution density corresponding to the maximum of informational energy. In this concrete case, Lagrange’s non determined multipliers method directly leads to the well known exponential distribution
the distribution density \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tρ\n\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\tr\n\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t through the maximum of informational energy implies the expression
Taking into account that the fractal term, \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\tU\n\t\t\t\t\t\t\t⋅\n\t\t\t\t\t\t\t∇\n\t\t\t\t\t\t\tU\n\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\tD\n\t\t\t\t\t\t\tΔ\n\t\t\t\t\t\t\tU\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\tintervenes as a pressure (for details see the kinetic significance of fractal potential Q (Bohm, D., 1952)) then we can admit the relation
In particular, if \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t∇\n\t\t\t\t\t\t\tU\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\tg\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t is a gravitational accelaration (125) becomes
In particular, for \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tm\n\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\tD\n\t\t\t\t\t\t\tΩ\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t with \n\t\t\t\t\t\n\t\t\t\t\t\tΩ\n\t\t\t\t\t\n\t\t\t\t the wave pulsation (for movements on Peano curves with
at Compton scale\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\tℏ\n\t\t\t\t\t\t\tΩ\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t) the previous relation becomes
For an incompressible fractal fluid, the balance equations of the “impulse” -see (126), of the energy -see (129) and ”mass” – see (46) with \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tρ\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\tc\n\t\t\t\t\t\t\to\n\t\t\t\t\t\t\tn\n\t\t\t\t\t\t\ts\n\t\t\t\t\t\t\tt\n\t\t\t\t\t\t\t.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t and \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t∇\n\t\t\t\t\t\t\t⋅\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tU\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t become
Let us take into account the following simplyfing hypothesis:
\n\t\t\t
constant density, \n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tρ\n\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t\tρ\n\t\t\t\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\tc\n\t\t\t\t\t\t\t\t\to\n\t\t\t\t\t\t\t\t\tn\n\t\t\t\t\t\t\t\t\ts\n\t\t\t\t\t\t\t\t\tt\n\t\t\t\t\t\t\t\t\t.\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\texcepting the balance equation of the impulse where density is disturbed according to relation
In order to study the dynamics of system (134), our description closely follows the approach in (Bârzu, A. et al, 2003).
\n\t\t\t
The convection in the fractal fluid takes place when the ascending force that results from energy “dilatation” overcomes the viscous forces. Then we can define the Rayleigh number
where \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tβ\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\tΔ\n\t\t\t\t\t\t\tε\n\t\t\t\t\t\t\t/\n\t\t\t\t\t\t\td\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t is the energy gradient between the superior and inferior frontiers of fluid layer. In the case of convection, Rayleigh’s number plays the role of control parameter and takes place for
In general, R is controlled through the gradient β of the energy.
\n\t\t\t
As reference state, let us choose the stationary rest state\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tV\n\t\t\t\t\t\t\t\t\t¯\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t, for which equations (134a-c) take the form
where \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tz\n\t\t\t\t\t\t\t^\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t represents the versor of vertical direction. We take into account that pressure and ε vary only in vertical direction due to the considered symmetry. For ε the conditions on the frontier are
Integrating equation (139b) with these conditions on the frontier, it will follow that in the reference rest state, the profile of ε on vertical direction is linear.
The features of the system in this state do not depend on coefficient D that appears in balance equations.
\n\t\t\t
We study now the stability of the reference state using the method of small perturbations (Bârzu, A. et al, 2003). The perturbed state is characterized by
One can notice that the perturbations are time and position functions. Substituting (143) in equations (134) and taking into account (141) and (142) the following equations for perturbations (in linear approximation) are obtained:
We introduce adimensional variables \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tr\n\t\t\t\t\t\t\t\t\t¯\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\'\n\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\tt\n\t\t\t\t\t\t\t\'\n\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\tθ\n\t\t\t\t\t\t\t\'\n\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\tδ\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tV\n\t\t\t\t\t\t\t\t\t¯\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\'\n\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\tδ\n\t\t\t\t\t\t\tp\n\t\t\t\t\t\t\t\'\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t through the changes
For R>RC, the reference state becomes unstable, and the convection “patterns” appear. We consider them as being parallel therefore the speed vector will be always perpendicular to their axis. We assume the patterns parallel to the y axis, i.e., the speed component along this direction is zero.
where \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tψ\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tx\n\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\ty\n\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\tz\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t defines Lagrange’s current function. The speed field must satisfy the conditions on frontiers (the inferior and superior surfaces)
Using Lagrange’s function, \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tψ\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tx\n\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\ty\n\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\tz\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\tthe limit conditions (148) and (149) become
Therefore, the phase volume exponentially diminishes in time, as the system tends towards the atractor. For any value of the control parameter r, the system (156) admits as a fixed point the origin
Since parameters b and r are positive ones, it follows that the first eigenvalue \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tω\n\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\tb\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t is negative for any values of the parameters. The other two eigenvalues ω2 and ω3 satisfy the relations
According to (160), if \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t0\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\tr\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t the sum of the two eigenvalues is negative and the product is positive. Therefore, all the eigenvalues are negative and the origin is a stable node. For r > 1, according to (160), the origin becomes unstable and two new fixed points appear in a fork bifurcation. These points are noted with \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tC\n\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t and \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tC\n\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t which corresponds to patterns
Let us study their stability. Replacing the values that correspond to the branch (\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tC\n\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t) in (158), the characteristic equation becomes
If the fixed points (161) will bear a Hopf bifurcation, for a value of control parameter\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tr\n\t\t\t\t\t\t\t\tH\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t, there will be two complex conjugated purely imaginary eigenvalues. Replacing \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tω\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\ti\n\t\t\t\t\t\t\tβ\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t in (162) we obtain
From equation (164a) it follows that\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tβ\n\t\t\t\t\t\t\t\t2\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\tb\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\t\tr\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t. Replacing this value in equation (164), Hopf bifurcation takes place in
Considering that \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tr\n\t\t\t\t\t\t\t\tH\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\tthe condition for b results
For this value of the control parameter, the two fixed points \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tC\n\t\t\t\t\t\t\t\t+\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\tand \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\tC\n\t\t\t\t\t\t\t\t−\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\tlose their stability in a subcritical Hopf bifurcation. Beyond the bifurcation point all the periodical orbits are unstable and the system has a chaotic behavior. Figures 2a-c to 8a-c show the trajectories, the time evolutions, the phase portraits and the Fourier transform for the different values of the parameters. It follows that when the value of the parameter r increases, there is a complicated succession of chaotic regimes with certain periodicity windows. The limit cycle appears through a reverse subarmonic cascade and loses stability through intermittent transition towards a new chaotic window.
\n\t\t\t
Figure 2.
a) Trajectory b) time evolution c) phase pattern for r=80, b=0.15
\n\t\t\t
Figure 3.
a) Trajectory b) time evolution c) phase pattern for r=100, b=0.19
\n\t\t\t
Figure 4.
a) Trajectory b) time evolution c) phase pattern for r=100, b=0.06
\n\t\t\t
Figure 5.
a) Time evolution b) phase portrait c) the Fourier transform for r=416, b=0.067
\n\t\t\t
Figure 6.
a) Time evolution b) phase portrait c) the Fourier transform for r=403, b=0.067
\n\t\t\t
Figure 7.
a) Time evolution b) phase portrait c) the Fourier transform for r=401, b=0.067
\n\t\t\t
Figure 8.
a) Time evolution b) phase portrait c) the Fourier transform for r=380, b=0.067
\n\t\t\t
In Fig.9 we present the map of the Lyapunov exponent with the value \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tσ\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t (the co- ordinates of the light points represent the pairs of values \n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tx\n\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\ty\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t(\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tb\n\t\t\t\t\t\t\t\t\t,\n\t\t\t\t\t\t\t\t\tr\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t)\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t for which the probability of entering in a chaotic regime is very high.
\n\t\t\t
Figure 9.
The Lyapunov exponent map for value \n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\t\tσ\n\t\t\t\t\t\t\t\t\t=\n\t\t\t\t\t\t\t\t\t1\n\t\t\t\t\t\t\t\t\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t of the Lorenz system
\n\t\t\t
The previous analysis states the following:
\n\t\t\t
a model of a physical object can be imagined. This model is built from a Madelung type fluid limited by two carcases that are submitted to an energy “gradient”, from the inferior carcase towards the superior one;
for small energy gradients, i.e. R<RC the reference state is a stable one. The ascending force resulting from energy ”dilatation” is much smaller than the dissipative one.
for energy gradients that impose restriction R>RC the reference state becomes unstable through the generation of convective type “rolls”. The ascensional force is bigger than the dissipative one;
the increase of energy gradient destroys the convective type ”patterns” and induces turbulence;
this behavior of fractal fluid can correspond to a Lorenz type “mechanism”: limit cycles the convective type “rolls”, intermitences (“jumps” between limit cycles) with the “destroy” of the convective type “rolls”, chaos with “turbulence” of the convective type state etc.;
the stability of solutions corresponds to the dominant undulatory feature, the wave-corpuscle duality can be correlated with the Lorenz type mechanism: self-organization of the structure through the generation of convective type “rolls” implies the wave-corpuscle transition, while the “jumps” among limit cycles, i.e. the intermittences induce a critical state that corresponds to chaos transition, thus ensuring the dominance of corpuscular effect.
\n\t\t
\n\t\t
\n\t\t\t
15. Conclusions
\n\t\t\t
Finally we can display the conclusions of this chapter as follows:
\n\t\t\t
a critical analisys of the hydrodinamic model of Madelung and of the double solution theory of de Broglie’s theory of double solution was performed – departing from here, we built a fractal approximation of motion;
we got the equation of motion of the physical object in the fractal approximation and the Eulerian case was studied;
the flowing regimes of a rotational fractal fluid were studied;
we studied the irotational regime of a fractal fluid and the incorporation of the particle into the associated wave by generating a Schrödinger equation;
the extended hydrodinamic model of scale relativity was built and the role of the fractal potential in the process of incorporation of the particle into the wave, specified;
we indicated the mechanisms of wave–particle duality by their in phase coherences;
we studied the wave-particle duality by stationary flow regimes of a fractal fluid which is coherent in phase, and by non-stationary flow regimes of an incoherent fractal fluid by means of a „polarization” type mechanism;
considering the particle as a singularity in the wave, we showed that its incorporation into the associated wave resulted in Einstein’s equations in vacuum - contrary, its non-incorporation led to the second quantification;
we established a relation between the informational energy and the fractal potential of the complex speed field - it resulted that the generation of forces implies the maximum of the information energy principle;
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we showed that a particle model in a fractal approximation of motion induced a Lorenz type
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mechanism.
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\n\t\n',keywords:null,chapterPDFUrl:"https://cdn.intechopen.com/pdfs/29592.pdf",chapterXML:"https://mts.intechopen.com/source/xml/29592.xml",downloadPdfUrl:"/chapter/pdf-download/29592",previewPdfUrl:"/chapter/pdf-preview/29592",totalDownloads:3124,totalViews:168,totalCrossrefCites:0,totalDimensionsCites:1,hasAltmetrics:0,dateSubmitted:"April 29th 2011",dateReviewed:"September 28th 2011",datePrePublished:null,datePublished:"February 24th 2012",dateFinished:null,readingETA:"0",abstract:null,reviewType:"peer-reviewed",bibtexUrl:"/chapter/bibtex/29592",risUrl:"/chapter/ris/29592",book:{slug:"theoretical-concepts-of-quantum-mechanics"},signatures:"Călin Gh. Buzea, Maricel Agop and Carmen Nejneru",authors:[{id:"24020",title:"Dr.",name:"Maricel",middleName:null,surname:"Agop",fullName:"Maricel Agop",slug:"maricel-agop",email:"m.agop@yahoo.com",position:null,institution:null},{id:"99400",title:"Dr.",name:"Calin Gheorghe",middleName:null,surname:"Buzea",fullName:"Calin Gheorghe Buzea",slug:"calin-gheorghe-buzea",email:"calinb2003@yahoo.com",position:null,institution:{name:"National Institute of Research and Development for Technical Physics",institutionURL:null,country:{name:"Romania"}}}],sections:[{id:"sec_1",title:"1. Introduction",level:"1"},{id:"sec_2",title:"2. Hydrodynamic model of quantum mechanics (Madelung’s model)",level:"1"},{id:"sec_3",title:"3. DeBroglie’s theory of double solution. The need for introducing the model of Bohm and Vigier",level:"1"},{id:"sec_4",title:"4. Comments",level:"1"},{id:"sec_5",title:"5. The motion equation of the physical object in the fractal approximation of motion. The Eulerian separation of motion on resolution scales",level:"1"},{id:"sec_6",title:"6. Rotational motions and flow regimes of a fractal fluid",level:"1"},{id:"sec_7",title:"7. Irotational motions of a fractal fluid. The incorporation of the associate wave corpuscle through the solidarity of movements and generation of Schrodinger equation",level:"1"},{id:"sec_8",title:"8. Extended hydrodynamic model of scale relativity and incorporation of associated wave corpuscle through fractal potential. The correspondence with Madelung model",level:"1"},{id:"sec_9",title:"9. “Mechanisms” of duality through coherence in corpuscle-wave phase",level:"1"},{id:"sec_10",title:"10. Wave-corpuscle duality through flowing stationary regimes of a coherent fractal fluid in phase. The potential well",level:"1"},{id:"sec_11",title:"11. Wave-corpuscle duality through non-stationary regimes of an incoherent fractal fluid",level:"1"},{id:"sec_12",title:"12. The corpuscle as a horologe and its incorporation in the associated wave. Consequences",level:"1"},{id:"sec_13",title:"13. Informational energy through the fractal potential of complex scalar speed field. The generation of forces",level:"1"},{id:"sec_14",title:"14. Lorenz type mechanism of wave-corpuscle duality in non stationary systems",level:"1"},{id:"sec_15",title:"15. Conclusions",level:"1"}],chapterReferences:[{id:"B1",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tAgop\n\t\t\t\t\t\t\tM.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tMazilu\n\t\t\t\t\t\t\tN.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t1989\n\t\t\t\t\tFundamente ale fizicii moderne, Ed. Junimea, Iasi\n\t\t\t\t\n\t\t\t'},{id:"B2",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tAgop\n\t\t\t\t\t\t\tM.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tChicoş\n\t\t\t\t\t\t\tL.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tNica\n\t\t\t\t\t\t\tP.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tHarabagiu\n\t\t\t\t\t\t\tA.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t2008\n\t\t\t\t\tEuler’s fluids and non-differentiable space-time, Far East Journal of Dynamical systems\n\t\t\t\t\t10\n\t\t\t\t\t1\n\t\t\t\t\t93\n\t\t\t\t\t106 .\n\t\t\t'},{id:"B3",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tAgop\n\t\t\t\t\t\t\tM.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tHarabagiu\n\t\t\t\t\t\t\tA.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tNica\n\t\t\t\t\t\t\tP.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t2008\n\t\t\t\t\tWave-Particle duality through a hydrodynamic modul of the fractal space time theory,\n\t\t\t\t\tActa physica Polonica A, 113\n\t\t\t\t\t6\n\t\t\t\t\t1557\n\t\t\t\t\t1574\n\t\t\t\t\n\t\t\t'},{id:"B4",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tAgop\n\t\t\t\t\t\t\tM.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tColotin\n\t\t\t\t\t\t\tM.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tPăun\n\t\t\t\t\t\t\tV.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t2009\n\t\t\t\t\tHaoticitate, fractalitate şi câmpuri, Elemente de teorie a fractalilor I. Gottlieb şi C. Mociuţchi, paginile Editura ArsLonga Iaşi\n\t\t\t\t\t12\n\t\t\t\t\t46 , \n\t\t\t'},{id:"B5",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tAgop\n\t\t\t\t\t\t\tM.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tMazilu\n\t\t\t\t\t\t\tN.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t2010\n\t\t\t\t\tLa răscrucea teoriilor. Între Newton şi Einstein- Universul Barbilian, Ed. Ars Longa, Iaşi\n\t\t\t\t\n\t\t\t'},{id:"B6",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tAlbert\n\t\t\t\t\t\t\tD. Z.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t1994\n\t\t\t\t\tBohm’s alternative to quantum mechanics.\n\t\t\t\t\tScientific American, 270\n\t\t\t\t\t32\n\t\t\t\t\t39\n\t\t\t\t\n\t\t\t'},{id:"B7",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tBarbilian\n\t\t\t\t\t\t\tD.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t1935\n\t\t\t\t\tApolare und Uberpolare Simplexe, Mathematica (Cluj), (retipărit în opera matematică vol I)\n\t\t\t\t\t11\n\t\t\t\t\t1\n\t\t\t\t\t24 , \n\t\t\t'},{id:"B8",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tBarbilian\n\t\t\t\t\t\t\tD.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t1935\n\t\t\t\t\tDie von Einer Quantika Induzierte Riemannsche Metrik, Comptes rendus de l’Academie Roumaine de Sciences, (retipărit în Opera Matematica vol. I)\n\t\t\t\t\t2\n\t\t\t\t\t198\n\t\t\t\t\n\t\t\t'},{id:"B9",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tBarbilian\n\t\t\t\t\t\t\tD.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t1938\n\t\t\t\t\tRiemannsche Raum Cubischer Binarformen, Comptes rendus de l’Academie Roumanie des Sciences, (retipărit în Opera Matematica vol. I)\n\t\t\t\t\t2\n\t\t\t\t\t345\n\t\t\t\t\n\t\t\t'},{id:"B10",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tBarbilian\n\t\t\t\t\t\t\tD.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t1971\n\t\t\t\t\tAlgebră elementară, în Opera didactică vol II, Ed. Tehnică Bucureşti\n\t\t\t\t\n\t\t\t'},{id:"B11",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tBarnsley\n\t\t\t\t\t\t\tM.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t1988\n\t\t\t\t\tFractals Everywhere. Deterministic Fractal Geometry, Boston\n\t\t\t\t\n\t\t\t'},{id:"B12",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tBârzu\n\t\t\t\t\t\t\tA.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tBourceanu\n\t\t\t\t\t\t\tG.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tOnel\n\t\t\t\t\t\t\tL.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t2003\n\t\t\t\t\tDinamica neliniară, Editura Matrix-Rom, Bucureşti\n\t\t\t\t\n\t\t\t'},{id:"B13",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tBell\n\t\t\t\t\t\t\tJ.S.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t1987\n\t\t\t\t\tSpeakable and unspeakable in quantum mechanics. Cambridge University Press, Cambridge\n\t\t\t\t\n\t\t\t'},{id:"B14",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tBenoit\n\t\t\t\t\t\t\tDeveand. Ed\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t2007\n\t\t\t\t\tPhysics of Semiconductor microcavities from fundamentals to nano-scale decretes, Wiley-VCH Verlag GmbH Weinheim Germany\n\t\t\t\t\n\t\t\t'},{id:"B15",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tBerge\n\t\t\t\t\t\t\tP. .\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tPomeau\n\t\t\t\t\t\t\tY.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tVidal\n\t\t\t\t\t\t\tCh.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t1984\n\t\t\t\t\t L’Ordre dans le chaos, Hermann\n\t\t\t\t\n\t\t\t'},{id:"B16",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tBittner\n\t\t\t\t\t\t\tE. R.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t2000\n\t\t\t\t\tQuantum Tunneling dynamics using hydrodynamic trajectories, Journal of Chimical- Physics, 112, 9703 \n\t\t\t\t\t9710 \n\t\t\t\t\n\t\t\t'},{id:"B17",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tBohm\n\t\t\t\t\t\t\tD.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t1952\n\t\t\t\t\tA Suggested Interpretation of Quantum Theory in Terms of „Hidden” \n\t\t\t\t\tVariables I, Phys. Rev.\n\t\t\t\t\t85\n\t\t\t\t\t166\n\t\t\t\t\n\t\t\t'},{id:"B18",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tBohm\n\t\t\t\t\t\t\tD.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tHiley\n\t\t\t\t\t\t\tB. J.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t1993\n\t\t\t\t\tThe Undivided Universe: An Ontological Interpretation of Quantum Theory.\n\t\t\t\t\tRoutledge and Kegan Paul, London.\n\t\t\t'},{id:"B19",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tBohm\n\t\t\t\t\t\t\tD.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t1952\n\t\t\t\t\tA suggested interpretation of quantum theory in terms of „hidden variables”: Part II.\n\t\t\t\t\tPhysical Review, 85\n\t\t\t\t\t180\n\t\t\t\t\t193\n\t\t\t\t\n\t\t\t'},{id:"",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tde Broglie\n\t\t\t\t\t\t\tL.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t1956\n\t\t\t\t\tUn tentative d’interprétation causale et non linéaire de la Mécanique ondulatoire: la theorie de la double solution,\n\t\t\t\t\tGauthier-Villars, Paris\n\t\t\t\t\n\t\t\t'},{id:"B21",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tErnst\n\t\t\t\t\t\t\tF. J.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t1968\n\t\t\t\t\tNew formulation of the Axially Symmetric Gravitational Fielf Problem II\n\t\t\t\t\tPhys Rev.\n\t\t\t\t\t168\n\t\t\t\t\t1415\n\t\t\t\t\n\t\t\t'},{id:"B22",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tde Broglie\n\t\t\t\t\t\t\tL.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t1957\n\t\t\t\t\tLa theoree de la Mesure on Mécanique ondulatoire, Gauthier-Villars, Paris\n\t\t\t\t\n\t\t\t'},{id:"B23",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tde Broglie\n\t\t\t\t\t\t\tL.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t1959\n\t\t\t\t\tL’interprétation de la Mécanique ondulatoire, J. Phys. Rad\n\t\t\t\t\t20, 963\n\t\t\t\t\n\t\t\t'},{id:"B24",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tde Broglie\n\t\t\t\t\t\t\tL.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t1963\n\t\t\t\t\tÉtude critique des bases de l’interprétation actuelle de la Mécanique ondulatoire, Gauthier-Villars, Paris\n\t\t\t\t\n\t\t\t'},{id:"B25",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tde Broglie\n\t\t\t\t\t\t\tL.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t1964\n\t\t\t\t\tLa Thermodynamique de la particule isolée (Thermodynamique cachée des particules), Gauthier-Villars, Paris\n\t\t\t\t\n\t\t\t'},{id:"B26",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tde Broglie\n\t\t\t\t\t\t\tL.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t1980\n\t\t\t\t\tCertitudinile şi incertitudinile ştiinţei, Editura Politică, Bucureşti\n\t\t\t\t\n\t\t\t'},{id:"B27",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tBudei\n\t\t\t\t\t\t\tL.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t2000\n\t\t\t\t\tModele cu fractali. Aplicaţii în arhitectura mediului, Editura Univ. ”Gh. Asachi”, Iaşi\n\t\t\t\t\n\t\t\t'},{id:"B28",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tChaline\n\t\t\t\t\t\t\tJ.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tNottale\n\t\t\t\t\t\t\tL. .\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tGrou\n\t\t\t\t\t\t\tP.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t2000\n\t\t\t\t\tLes arbres d’evolution: Univers S, Vie, Societes, Edition Hachette\n\t\t\t\t\n\t\t\t'},{id:"B29",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tChaline\n\t\t\t\t\t\t\tJ. .\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tNottale\n\t\t\t\t\t\t\tL. .\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tGrou\n\t\t\t\t\t\t\tP.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t2009\n\t\t\t\t\tDes fleurs pour Scrödinger: La relativite d’echelle et ses applications\n\t\t\t\t\tEditure Ellipses Marketing\n\t\t\t\t\n\t\t\t'},{id:"B30",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tChioroiu\n\t\t\t\t\t\t\tV. .\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tMunteanu\n\t\t\t\t\t\t\tL. .\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tŞtiucă\n\t\t\t\t\t\t\tP. .\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tDonescu\n\t\t\t\t\t\t\tŞ.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t2005\n\t\t\t\t\tIntroducere în nanomecanică, Editura Academiei Române, Bucureşti\n\t\t\t\t\n\t\t\t'},{id:"B31",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tCiuti\n\t\t\t\t\t\t\tC. .\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tCamsotto\n\t\t\t\t\t\t\tI.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t2005\n\t\t\t\t\tQuantum fluids effects and parametric instabilities in microcavities\n\t\t\t\t\tPhysica Status Solidi B, 242, 11, 2224\n\t\t\t\t\n\t\t\t'},{id:"B32",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tErnst\n\t\t\t\t\t\t\tF. J.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t1968\n\t\t\t\t\tNew formulation of the Axially Symemetric Gravitational Fielf Problem I,\n\t\t\t\t\tPhys Rev. 167\n\t\t\t\t\t1175\n\t\t\t\t\n\t\t\t'},{id:"B33",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tErnst\n\t\t\t\t\t\t\tF. J.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t1971\n\t\t\t\t\tExterior Algebraic Derivation of Einstein Field Equation Employing a Generalized Basis, J. Math. Phys., 12, 2395\n\t\t\t\t\n\t\t\t'},{id:"B34",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tFeder\n\t\t\t\t\t\t\tJ.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tAharony\n\t\t\t\t\t\t\tA. .\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tEds\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t1990\n\t\t\t\t\tFractals in Physics\n\t\t\t\t\tNorth- Holland, Amsterdam\n\t\t\t\t\n\t\t\t'},{id:"B35",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tFelsager\n\t\t\t\t\t\t\tB.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t1981\n\t\t\t\t\tGeometry, Particles and fields. Odense Univ. Press\n\t\t\t\t\n\t\t\t'},{id:"B36",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tFerry\n\t\t\t\t\t\t\tD. K.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tGoodnick\n\t\t\t\t\t\t\tS. M.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t2001\n\t\t\t\t\tTransport in Nanostructures, Cambridge University Press\n\t\t\t\t\n\t\t\t'},{id:"B37",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tGouyet\n\t\t\t\t\t\t\tJF.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t1992\n\t\t\t\t\tPhysique et Structures Fractals, Masson Paris\n\t\t\t\t\n\t\t\t'},{id:"B38",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tGreen\n\t\t\t\t\t\t\tM. B.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tSchwarz\n\t\t\t\t\t\t\tJ. H.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tWitten\n\t\t\t\t\t\t\tE.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t1998\n\t\t\t\t\tSuperstring Theory vol I, II. Cambridge University, Press, Cambridge\n\t\t\t\t\n\t\t\t'},{id:"B39",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tGrössing\n\t\t\t\t\t\t\tG.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t2008\n\t\t\t\t\tDiffusion waves in sub-quantum thermodynamics: Resolution of Einstein’s\n\t\t\t\t\t“Particle-in-a-box” objection, (in press) http://arxiv.org/abs/0806.4462\n\t\t\t\t\n\t\t\t'},{id:"B40",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tHalbwacs\n\t\t\t\t\t\t\tF.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t1960\n\t\t\t\t\tTheorie relativiste des fluids a spin, Gauthier-Villars, Paris\n\t\t\t\t\n\t\t\t'},{id:"B41",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tHarabagiu\n\t\t\t\t\t\t\tA.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tAgop\n\t\t\t\t\t\t\tM.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t2005\n\t\t\t\t\tHydrodyamic model of scale relativity theory,\n\t\t\t\t\tBuletinul Institutului Politehnic Iaşi, tomul LI (LV) Fasc. 3-4, 77-82, secţia Matematică, Mecanică teoretică, Fizică\n\t\t\t\t\n\t\t\t'},{id:"B42",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tHarabagiu\n\t\t\t\t\t\t\tA.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tNiculescu\n\t\t\t\t\t\t\tO.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tColotin\n\t\t\t\t\t\t\tM.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tBibere\n\t\t\t\t\t\t\tT. D.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tGottlieb\n\t\t\t\t\t\t\tI.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tAgop\n\t\t\t\t\t\t\tM.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t2009\n\t\t\t\t\tParticle in a box by means of fractal hydrodynamic model,\n\t\t\t\t\tRomanian Reports in Physics, 61\n\t\t\t\t\t3\n\t\t\t\t\t395\n\t\t\t\t\t400\n\t\t\t\t\n\t\t\t'},{id:"B43",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tHarabagiu\n\t\t\t\t\t\t\tA. .\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tMagop\n\t\t\t\t\t\t\tD. .\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tAgop\n\t\t\t\t\t\t\tM.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t2010\n\t\t\t\t\tFractalitate şi mecanică cuantică, Editura Ars Longa Iaşi\n\t\t\t\t\n\t\t\t'},{id:"B44",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tHeck\n\t\t\t\t\t\t\tA.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tPerdang\n\t\t\t\t\t\t\tJ. M. .\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tEds\n\t\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t1991\n\t\t\t\t\tApplying Fractals in Astronomy; Springer Verlag\n\t\t\t\t\n\t\t\t'},{id:"B45",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tHolland\n\t\t\t\t\t\t\tP.R.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t1993\n\t\t\t\t\tThe Quantum Theory of Motion.\n\t\t\t\t\tCambridge University Press, Cambridge\n\t\t\t\t\n\t\t\t'},{id:"B46",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tImry\n\t\t\t\t\t\t\tY.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t2002\n\t\t\t\t\tIntroduction to Mesoscopic Physics, Oxford University Press, Oxford\n\t\t\t\t\n\t\t\t'},{id:"B47",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tMadelung\n\t\t\t\t\t\t\tR.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t1927 Zs f. Phys.40, 322\n\t\t\t'},{id:"B48",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tMandelis\n\t\t\t\t\t\t\tA.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tNicolaides\n\t\t\t\t\t\t\tL.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tChen\n\t\t\t\t\t\t\tY.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t2001\n\t\t\t\t\tStructure and the reflectionless/refractionless nature of parabolic diffusion-wave fields\n\t\t\t\t\tPhys. Rev. Lett\n\t\t\t\t\t87020801\n\t\t\t'},{id:"B49",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tMandelis\n\t\t\t\t\t\t\tA.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t2000\n\t\t\t\t\tDiffusion waves and their uses, Phys. Today\n\t\t\t\t\t53, 29 \n\t\t\t\t\n\t\t\t'},{id:"B50",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tMazilu\n\t\t\t\t\t\t\tN.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tAgop\n\t\t\t\t\t\t\tM.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t1994\n\t\t\t\t\tFizica procesului de măsură, Ed. Ştefan Procopiu, Iaşi\n\t\t\t\t\n\t\t\t'},{id:"B51",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tLe Mehante\n\t\t\t\t\t\t\tA.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t1990\n\t\t\t\t\tLes Geometries Fractales, Hermes, Paris\n\t\t\t\t\n\t\t\t'},{id:"B52",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tMihăileanu\n\t\t\t\t\t\t\tN.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t1972\n\t\t\t\t\tGeometrie analitică, proiectivă şi diferenţială, Complemente Editura Didactică şi Pedagogică, Bucureşti\n\t\t\t\t\n\t\t\t'},{id:"B53",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tMunceleanu\n\t\t\t\t\t\t\tC. V.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tMagop\n\t\t\t\t\t\t\tD.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tMarin\n\t\t\t\t\t\t\tC. .\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tAgop\n\t\t\t\t\t\t\tM.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t2010\n\t\t\t\t\tModele fractale în fizica polimerilor, Ars Longa\n\t\t\t\t\n\t\t\t'},{id:"B54",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tEl Naschie\n\t\t\t\t\t\t\t. MS\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tRössler\n\t\t\t\t\t\t\tO. E. .\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tPrigogine\n\t\t\t\t\t\t\tI. .\n\t\t\t\t\t\tEds\n\t\t\t\t\t1995\n\t\t\t\t\tQuantum mechanics, diffusion and chaot icfractals, Oxford: Elsevier\n\t\t\t\t\n\t\t\t'},{id:"B55",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tNelson\n\t\t\t\t\t\t\tE.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t1985\n\t\t\t\t\tQuantum Fluctuations, Princeton University Press, Princeton, New York\n\t\t\t\t\n\t\t\t'},{id:"B56",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tNottale\n\t\t\t\t\t\t\tL.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tSchneider\n\t\t\t\t\t\t\tJ.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t1984\n\t\t\t\t\tFractals and non-standard analysis;\n\t\t\t\t\tJ. Math. Phys.\n\t\t\t\t\t25\n\t\t\t\t\t12\n\t\t\t\t\t96\n\t\t\t\t\t300\n\t\t\t\t\n\t\t\t'},{id:"B57",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tNottale\n\t\t\t\t\t\t\tL.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t1989\n\t\t\t\t\tFractals and the quantum theory of space-time\n\t\t\t\t\tInt. J. Mod. Phys. A\n\t\t\t\t\t4\n\t\t\t\t\t50\n\t\t\t\t\n\t\t\t'},{id:"B58",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tNottalle\n\t\t\t\t\t\t\tL.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t1993\n\t\t\t\t\t Fractal Space-Time and Microphysics. Towards a Theory of Scale Relativity, World Scientific\n\t\t\t\t\n\t\t\t'},{id:"B59",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tNottale\n\t\t\t\t\t\t\tL.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t1996\n\t\t\t\t\tScale relativity and fractal space-time: Applications to quantum physics, cosmology and chaotic systems. Chaos Solitons & Fractals, 7\n\t\t\t\t\t877 \n\t\t\t\t\t938 \n\t\t\t\t\n\t\t\t'},{id:"B60",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tNottale\n\t\t\t\t\t\t\tL.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t2004\n\t\t\t\t\tThe theory of scale relativity: Nondifferentiable geometry, Fractal space-time and Quantum Mechanics, Computing Anticipatory systems: CASYS’ 03-33 Sixth International Confference, AIP Confference Proceedings\n\t\t\t\t\t718\n\t\t\t\t\t68\n\t\t\t\t\t95\n\t\t\t\t\n\t\t\t'},{id:"B61",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tPeres\n\t\t\t\t\t\t\tA.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t1993\n\t\t\t\t\tQuantum teory: Concepts and methods,\n\t\t\t\t\tKlauwer Acad. Publ., Boston\n\t\t\t\t\n\t\t\t'},{id:"B62",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tPoole\n\t\t\t\t\t\t\tC. P.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tFarach\n\t\t\t\t\t\t\tK. A.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tCreswick\n\t\t\t\t\t\t\tR.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t1995\n\t\t\t\t\tSuperconductivity, San Diego, Academic Press\n\t\t\t\t\n\t\t\t'},{id:"B63",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tPopescu\n\t\t\t\t\t\t\tS.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t2004\n\t\t\t\t\tProbleme actuale ale fizicii sistemelor autoorganizate, Editura Tehnopress, Iasi\n\t\t\t\t\n\t\t\t'},{id:"B64",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tRuelle\n\t\t\t\t\t\t\tD.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tTakens\n\t\t\t\t\t\t\tF.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t1971\n\t\t\t\t\tOn the Nature of Turbulence, Commun. Math. Phys., 20, 167, 23, 343\n\t\t\t\t\n\t\t\t'},{id:"B65",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tRuelle\n\t\t\t\t\t\t\tD.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t1975\n\t\t\t\t\tStrange Attractors, The mathematical Intelligencer\n\t\t\t\t\t2, 126\n\t\t\t\t\n\t\t\t'},{id:"B66",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tSakurai\n\t\t\t\t\t\t\tJ.J.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tSan\n\t\t\t\t\t\t\tFu Taun\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t1994\n\t\t\t\t\tModern Qunatum Mechanics, Addison-Wesley, Reading, MA\n\t\t\t\t\n\t\t\t'},{id:"B67",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tŢiţeica\n\t\t\t\t\t\t\tS.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t1984\n\t\t\t\t\tMecanică cuantică, Editura Academiei, Bucureşti\n\t\t\t\t\n\t\t\t'},{id:"B68",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tWeibel\n\t\t\t\t\t\t\tP.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tOrd\n\t\t\t\t\t\t\tG.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tRössler\n\t\t\t\t\t\t\tO. E. .\n\t\t\t\t\t\tEds\n\t\t\t\t\t2005\n\t\t\t\t\tSpace time physics and fractality,\n\t\t\t\t\tFestschroft in honer of Mohamad El Naschie Vienna, New York: Springer\n\t\t\t\t\n\t\t\t'},{id:"B69",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tBerndl\n\t\t\t\t\t\t\tK.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tDϋrr\n\t\t\t\t\t\t\tD.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tGoldstein\n\t\t\t\t\t\t\tS.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tPeruzzi\n\t\t\t\t\t\t\tG.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tZanchi\n\t\t\t\t\t\t\tN.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t1993\n\t\t\t\t\tExistance of Trajectories for Bohmian Mechanics.\n\t\t\t\t\tInternational Journal of Theoretical Physics, 32\n\t\t\t\t\t2245\n\t\t\t\t\t2251\n\t\t\t\t\n\t\t\t'},{id:"B70",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tBerndl\n\t\t\t\t\t\t\tK.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tDϋrr\n\t\t\t\t\t\t\tD.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tGoldstein\n\t\t\t\t\t\t\tS.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tZanchi\n\t\t\t\t\t\t\tN.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t1994\n\t\t\t\t\tSelfadjointness and the Existance of Deterministic Trajectories in Quantum Theory, In On three Levels: Micro-, Meso-, and Macroscopic Approches in Physics, (NATO ASI Series B: Physics, Plenum, New-York)\n\t\t\t\t\t324\n\t\t\t\t\t429\n\t\t\t\t\t434\n\t\t\t\t\n\t\t\t'},{id:"B71",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tDϋrr\n\t\t\t\t\t\t\tD. .\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tGoldstein\n\t\t\t\t\t\t\tS. .\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tZanghi\n\t\t\t\t\t\t\tN.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t1992\n\t\t\t\t\tQuantum Mechanics, Randomness and deterministic Reality. Physics Letters A, 172\n\t\t\t\t\t6\n\t\t\t\t\t12\n\t\t\t\t\n\t\t\t'},{id:"B72",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tDϋrr\n\t\t\t\t\t\t\tD.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tGoldstein\n\t\t\t\t\t\t\tS.\n\t\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tZanchi\n\t\t\t\t\t\t\tN.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t1993\n\t\t\t\t\tA Global Equilibrium as the Foundation of Quantum Randomness, Foundations of Physics, 23\n\t\t\t\t\t712\n\t\t\t\t\t738\n\t\t\t\t\n\t\t\t'},{id:"B73",body:'\n\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\t\n\t\t\t\t\t\t\tBohm\n\t\t\t\t\t\t\tD.\n\t\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t1953\n\t\t\t\t\tProof that probability density approaches in causal interpretation of quantum theory\n\t\t\t\t\tPhysical Review\n\t\t\t\t\t89\n\t\t\t\t\t458\n\t\t\t\t\t466\n\t\t\t\t\n\t\t\t'}],footnotes:[],contributors:[{corresp:null,contributorFullName:"Gh. Buzea Călin",address:null,affiliation:'
National Institute of Research and Development for Technical Physics, Romania
Materials and Engineering Science, Technical “Gh. Asachi” University, Romania
'}],corrections:null},book:{id:"1613",title:"Theoretical Concepts of Quantum Mechanics",subtitle:null,fullTitle:"Theoretical Concepts of Quantum Mechanics",slug:"theoretical-concepts-of-quantum-mechanics",publishedDate:"February 24th 2012",bookSignature:"Mohammad Reza Pahlavani",coverURL:"https://cdn.intechopen.com/books/images_new/1613.jpg",licenceType:"CC BY 3.0",editedByType:"Edited by",editors:[{id:"101263",title:"Prof.",name:"Mohammad Reza",middleName:null,surname:"Pahlavani",slug:"mohammad-reza-pahlavani",fullName:"Mohammad Reza Pahlavani"}],productType:{id:"1",title:"Edited Volume",chapterContentType:"chapter",authoredCaption:"Edited by"},chapters:[{id:"29575",title:"Complementarity in Quantum Mechanics and Classical Statistical Mechanics",slug:"complementarity-in-quantum-mechanics-and-classical-statistical-mechanics",totalDownloads:6458,totalCrossrefCites:0,signatures:"Luisberis Velazquez Abad and Sergio Curilef Huichalaf",authors:[{id:"92857",title:"Dr.",name:"Luisberis",middleName:null,surname:"Velazquez",fullName:"Luisberis Velazquez",slug:"luisberis-velazquez"},{id:"125424",title:"Prof.",name:"Sergio",middleName:null,surname:"Curilef",fullName:"Sergio Curilef",slug:"sergio-curilef"}]},{id:"29576",title:"The Physical Nature of Wave/Particle Duality",slug:"the-physical-nature-of-wave-particle-duality",totalDownloads:6348,totalCrossrefCites:0,signatures:"Marcello Cini",authors:[{id:"95127",title:"Prof.",name:"Marcello",middleName:null,surname:"Cini",fullName:"Marcello Cini",slug:"marcello-cini"}]},{id:"29577",title:"The Bicomplex Heisenberg Uncertainty Principle",slug:"the-bicomplex-heisenberg-uncertainty-principle",totalDownloads:3875,totalCrossrefCites:0,signatures:"Raphaël Gervais Lavoie and Dominic Rochon",authors:[{id:"98444",title:"Prof.",name:"Dominic",middleName:null,surname:"Rochon",fullName:"Dominic Rochon",slug:"dominic-rochon"},{id:"98454",title:"MSc.",name:"Raphaël",middleName:null,surname:"Gervais Lavoie",fullName:"Raphaël Gervais Lavoie",slug:"raphael-gervais-lavoie"}]},{id:"29578",title:"Correspondence, Time, Energy, Uncertainty, Tunnelling, and Collapse of Probability Densities",slug:"correspondence-time-energy-uncertainty-tunnelling-and-collapse-of-probability-densities",totalDownloads:3309,totalCrossrefCites:0,signatures:"Gabino Torres–Vega",authors:[{id:"93519",title:"Dr.",name:"Gabino",middleName:null,surname:"Torres-Vega",fullName:"Gabino Torres-Vega",slug:"gabino-torres-vega"}]},{id:"29579",title:"Anisotropic Kepler Problem and Critical Level Statistics",slug:"anisotropic-kepler-problem-and-critical-level-statistics",totalDownloads:3568,totalCrossrefCites:1,signatures:"Kazuhiro Kubo and Tokuzo Shimada",authors:[{id:"97094",title:"Prof.",name:"Tokuzo",middleName:null,surname:"Shimada",fullName:"Tokuzo Shimada",slug:"tokuzo-shimada"},{id:"102420",title:"Dr.",name:"Kazuhiro",middleName:null,surname:"Kubo",fullName:"Kazuhiro Kubo",slug:"kazuhiro-kubo"}]},{id:"29580",title:"Theory of Elementary Particles Based on Newtonian Mechanics",slug:"theory-of-elementary-particles-based-on-newtonian-mechanics",totalDownloads:3792,totalCrossrefCites:0,signatures:"Nikolai A. 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1. Introduction
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Voltammetry is an electrochemical technique for current-voltage curves, from which electrode reactions at electrode-solution interfaces can be interpreted. Since current-voltage curves, called voltammograms, include sensitive properties of solution compositions and electrode materials, their analysis provides not only chemical structures and reaction mechanisms on a scientific basis but also electrochemical manufacture on an industrial basis. The voltammograms vary largely with measurement time except for steady-state measurements, and so it is important to pay attention to time variables. Voltage is a controlling variable in conventional voltammetry, and the current is a measured one detected as a function of applied voltage at a given time.
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The equipment for voltammetry is composed of electrodes, solution, and electric instruments for voltage control. Electrodes and electric instruments are keys of voltammetry. Three kinds of electrodes are desired to be prepared: a working electrode, a counter one, and a reference one. The three will be addressed below.
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Let us consider a simple experiment in which two electrodes are inserted into a salt-included aqueous solution. When a constant current is applied to the two electrodes, reaction 2H+ + 2e− → H2 may occur at one electrode, and reaction 2OH− → H2O2 + 2e− occurs at the other. The current is the time variation of the electric charge, and hence it is a kind of reaction rate at the electrode. Since the applied current is a sum of the two reaction rates, one being in the positive direction and the other being in the negative, it cannot be attributed to either reaction rate. A technique of attributing the reactions is to use an electrode with such large area that an uninteresting reaction rate may not become a rate-determining step. This electrode is called a counter electrode. The current density at the counter electrode does not specifically represent any reaction rate. In contrast, the current density at the electrode with a small area stands for the interesting reaction rate. This electrode is called a working electrode. It is the potential difference, i.e., voltage, at the working electrode and in the solution that brings about the electrode reaction. However, the potential in the solution cannot be controlled with the working electrode or the counter one. The control can be made by mounting another electrode, called a reference electrode, which keeps the voltage between an electrode and a solution to be constant. However, the constant value cannot be measured because of the difference in phases. A conventionally employed reference electrode is silver-silver chloride (Ag-AgCl) in high concentrated KCl aqueous solution.
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An electric instrument of operating the three electrodes is a potentiostat. It has three electric terminals: one being a voltage follower for the reference electrode without current, the second being a current feeder at the counter electrode, and the third being at the working electrode through which the current is converted to a voltage for monitoring. A controlled voltage is applied between the working electrode and the reference one. These functionalities can readily be attained with combinations of operational amplifiers. A drawback of usage of operational amplifiers is a delay of responses, which restricts current responses to the order of milliseconds or 10 kHz frequency.
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Voltammetry includes various types—linear sweep, cyclic, square wave, stripping, alternating current (AC), pulse, steady-state microelectrode, and hydrodynamic voltammetry—depending on a mode of the potential control. The most frequently used technique is cyclic voltammetry (CV) on a time scale of seconds. In contrast, currently used voltammetry at time as short as milliseconds is AC voltammetry. We describe here the theory and tips for practical use of mainly the two types of voltammetry.
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2. Theory
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The theory of voltammetry is to obtain expressions for voltammograms on a given time scale or for those at a given voltage. First of all, it is necessary to specify rate-determining steps of voltammograms. There are three types of rate-determining steps under the conventional conditions: diffusion of redox species in solution near an electrode, adsorption on an electrode, and charging processes at the double layer (DL). Electric field-driven mass transport, called electric migration, belongs to rare experimental conditions, and hence it is excluded in this review. When a redox species in solution is consumed or generated at an electrode, it is supplied to or departed from the electrode by diffusion unless solution is stirred. When it is accumulated on the electrode, the change in the accumulated charge by the redox reaction provides the current. Whenever electrode voltage is varied with the time, the charging or discharging of the DL capacitor causes current. Therefore, the three steps are frequently involved in electrochemical measurements.
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A mass transport problem on voltammetry is briefly described here. The redox species is assumed to be transported by one-directional (x) diffusion owing to heterogeneous electrode reactions. Then, the flux is given by f = −D(∂c/∂x), where c and D are the concentration and the diffusion coefficient of the redox species, respectively. Redox species in solution causes some kinds of chemical reaction through chemical reaction rates, h(c, t). Then the reaction rate is the sum of the diffusional flux and the chemical reaction rate, ∂c/∂t = −∂f/∂x − h(c, t). Here the equation for h = 0 is called an equation of continuum. Eliminating f with the above equation on the assumption of a constant value of D yields ∂c/∂t = D(∂2c/∂x2) − h(c, t). This is an equation for diffusion-chemical kinetics. The expression at h = 0 is the diffusion equation. A boundary condition with electrochemical significance is the control of c at the electrode surface with a given electrode potential. If the redox reaction occurs in equilibrium with the one-electron transfer at the electrode, the Nernst equation for the concentrations of the oxidized species, co, and the reduced one, cr, holds.
The other conditions are concentrations in the bulk (x → ∝) and the initial conditions.
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2.1. Diffusion-controlled current
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If the mass transport is controlled only by x-directional diffusion, cr and co are given by the diffusion equations, ∂c/∂t = D(∂2c/∂t2) for c = cr or co. An electrochemically significant quantity is not concentration in any x and t, but a relation between the surface concentrations and the current (the flux at x = 0). On the assumption of Do = Dr = D, of the initial and boundary conditions, (cr)t = 0 = c*, (co)t = 0 = 0, and (cr)x = ∞ = c*, (co)x = ∞ = 0, a solution of the initial-boundary problem is given by [1].
where j is the current density. The common value of the diffusion coefficients yields co + cr = c* for any x and t. Inserting this relation and Eq. (3) into the Nernst equation, (co)x = 0 = c*/[1 + exp[−F(E − Eo)/RT]], we obtain the integral equation for j as a function of t or E.
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2.1.1. Linear sweep voltammetry by diffusion
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When the voltage is linearly swept with the time at a given voltage scan rate, v, from the initial potential Ein, Eq. (3) through the combination with the Nernst equation becomes
The above Abel’s integral equation can be solved by Laplace transformation. When the time variation is altered to the voltage variation through E = Ein + vt, the current density is expressed as
where ζ = (E − Eo)F/RT and ζi = (Ein − Eo)F/RT. Evaluation of the integral has to resort to numerical computation. Current at any voltage should be proportional to v1/2, as can be seen in Eq. (5). The voltammogram for v > 0 rises up from Eo, takes a peak, and then deceases gradually with the voltage. The decrease in the current is obviously ascribed to relaxation by diffusion. The peak current density is expressed by
at Ep = Eo + 0.029 V at 25°C, where 0.446 comes from the numerical calculation of the integral of Eq. (5).
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Practical voltage-scan voltammetry is not simply linear sweep but cyclic voltammetry (CV), at which applied voltage is reversed at a given voltage in the opposite direction. The theoretical evaluation of the voltammogram should be at first represented in the integral form with the time variation and then express the time as the voltage. One of the features of the diffusion-controlled cyclic voltammograms is the difference between the anodic peak potential and the cathodic one, ΔEp (in Figure 1), of which value is 59 mV at 25°C.
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Figure 1.
Voltammograms calculated from Eq. (5) for v = (a) 180, (b) 80 and (c) 20 mV s−1.
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2.1.2. AC voltammetry by diffusion
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AC voltammetry can be performed when the time variation of voltage is given by E = Edc + V0eiωt, where ω is the frequency of applied AC voltage, i is the imaginary unit, V0 is its voltage amplitude, and Edc is the DC voltage. A conventional value of V0 is 10 mV. When this voltage form is inserted into Eq. (3) together with the Nernst equation, the AC component of the current density is represented by [2].
A voltammogram (j vs. Edc) at a given frequency takes a bell shape, which is expressed by sech2{(Edc − Eo)/RT}. The functional form of sech2 is shown in Figure 2. The peak current appears at Edc = Eo.
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Figure 2.
Voltammogram calculated from Eq. (10).
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The AC-impedance technique often deals with the real impedance, Z1, = 1/2Y1 and the imaginary one, Z2 = −1/2Y1, where Y1 is the real admittance given by
Here Y2 is the imaginary admittance, equal to Y1. Since Z1 = −Z2, the Nyquist plot, i.e., −Z2 vs. Z1, is a line with the slope of unity. The term 1 + i in Eq. (7) has come from (Dω)1/2, originating from (Diω)1/2. Therefore, it can be attributed to diffusion. In other words, diffusion produces the capacitive component as a delay.
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2.2. Adsorption-controlled current
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When the redox species with reaction R = O + e− is adsorbed on the electrode and has no influence from the redox species in the solution, the sum of the surface concentrations of R and O is a constant, Γ*. Then the surface concentration of the oxidized species, Γo, is given by the Nernst equation:
The time derivative of the redox charge corresponds to the current density, j = d(FΓo)/dt. Application of the condition of voltage sweep, E = Ein + vt, to Eq. (9) yields.
The voltammogram takes a bell shape (Figure 2), of which peak is at E = Eo, similar to the AC voltammogram. The current at any voltage is proportional to v. Since the negative-going scan of the voltage provides negative current values, the cyclic voltammogram should be symmetric with respect to the I = 0 axis. The peak current is expressed as jp = F2Γ*v/4RT. The width of the wave at jp/2 is 90 mV at 25°C.
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2.3. Capacitive current
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Since a phase has its own free energy, contact of two phases provides a step-like gap of the free energy, of which gradient brings about infinite magnitude of force. In order to relax the infinity, local free energy varies from one phase to the other as smoothly as possible at the interface. The large variation of the energy is compensated with spontaneously generated space variations of voltage, i.e., the electric field, which works as an electric capacitor. The capacitance at solution-electrode interface causes orientation of dipoles and nonuniform distribution of ionic concentration, of which layer is called an electric double layer (DL).
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When the time variation of the voltage is applied to the DL capacitance, Cd, the definitions of the capacitance (q = CdV) and the current lead
where Cd generally depends on the time. This dependence is significant for understanding experimentally observed capacitive currents.
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2.3.1. Capacitance by AC impedance
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The DL capacitance has exhibited the frequency dispersion expressed by Cd = (Cd) 1Hzf −λ, called the constant phase element [3, 4, 5] or power law [6, 7], where λ is close to 0.1. Inserting this expression and V = V0eiωt into Eq. (11) yields
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\n\nI\n=\n\n\ni\n+\nλ\n\n\n\nωC\nd\n\nV\n\nE12
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This is a simple sum of the real part of the current and the imaginary one, indicating that the equivalent circuit should be a parallel combination of a capacitive component and a resistive one, both depending on frequency. Since the ratio, −Z2/Z1, for Eq. (12) is 1/λ, the Nyquist plots have slopes less than 10 rather than infinity.
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2.3.2. Capacitive current by CV
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If the capacitive charge is independent of the time, the capacitive current should be I = d(CV)/dt = C(E − Eo)/v. Therefore, it takes a horizontal positive (v > 0) and a negative line (v < 0), as shown in Figure 3 (dashed lines). When the time dependence of C, i.e., Cd = (Cd)0t−λ, is applied to Eq. (11), for the forward and the backward scans, respectively, we have
Capacitive voltammograms by CV at v= 0.5 V s−1 for (dashed lines) the ideal capacitance and for Eq. (13) (solid curves) at λ = 0.2.
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The variation of CV computed from Eq. (13) (Figure 3, solid curves) is similar to our conventionally observed capacitive waves.
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3. Tips of voltammetric analysis
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Voltammograms can identify an objective species by comparing a peak potential with a table of redox potentials and furthermore determine its concentration from the peak current. Their results are, however, sometimes inconsistent with data by methods other than electrochemical techniques if one falls in some pitfalls of analytical methods of electrochemistry. For example, a peak potential is influenced by a reference electrode and solution resistance relevant to methods. Peak currents are varied complicatedly with mass transport modes as well as associated chemical reactions. Since the theory on voltammetry covers only some restricted experimental conditions, it can rarely interpret the experimental data successfully. This review is devoted to some voltammetric tips which can lead experimenters to reasonable interpretation.
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3.1. Understanding outline of voltammograms
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It is rare to observe a reversible voltammogram in which both oxidation and reduction waves appear in a symmetric form with respect to the potential axis at a similar peak potential, as in Figure 1. Frequently observed voltammograms are irreversible, i.e., either a cathodic or an anodic wave appears; a value of a cathodic peak current is quite different from the anodic one in magnitude; a cathodic peak potential is far from the anodic one. These complications are ascribed to chemical reactions and/or phase transformation after the charge-transfer reaction. A typical example is deposition of metal ions on an electrode. The complications can be interpreted by altering scan rates and reverse potentials.
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A wave at a backward scan is mostly attributed to electrode reactions generated by experimenters rather than to species latently present in the solution. That is, it is artificial. It is caused either by the reaction of the wave at the forward scan or the reaction of the rising-up current just before the reverse potential. A source of the backward wave can be found by changing the reverse potentials.
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Some voltammograms have more than two peaks at one-directional scan. The appearance of the two can be interpreted as a two-step sequential charge-transfer reaction. However, multiple waves appear also by combinations of chemical reactions and adsorption. The peak current and the charge for this case are quite different from the predicted ones, as will be described in Section 3.2. Change in scan rates may be helpful for interpreting the multiple waves.
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3.2. Shape and values of peaks
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It is possible to predict theoretically a controlling step of voltammograms from their shape (a bell type corresponding to an adsorption wave or a draw-out type corresponding to a diffusion wave). However, the shape strongly depends on chemical complications, adsorption, and surface treatment of the electrodes. When redox species in solution is partially adsorbed on an electrode, the electrode process is far from a prediction because of very high concentration in the adsorbed state. A draw-out-shaped wave can be observed even for the adsorbed control. It is important to estimate which state the reacting species takes on the electrode. Potentials representing of voltammetric features do not express a controlling step in reality although the theory does. One should pay attention to the current. The peak current controlled by diffusion with one-electron transfer is given by Ip = 0.27 cAv1/2 μA (c, bulk concentration mM; A, electrode area mm2; v, potential sweep rate mV s−1). The microelectrode behavior sometimes comes in view at v < 10 mV s−1, A < 0.1 mm2, so the measured current is larger than the estimated value. On the other hand, the peak current controlled by adsorption is given by Ip = 1.6 Av nA when one redox molecule is adsorbed at 1 nm2 on the electrode. The voltammogram by adsorption often differs from the ideal bell shape due to adsorbed molecular interaction and DL capacity. Division of the area of the peak by the scan rate yields the amount of adsorbed electricity. Comparison of this with the anticipated amount of adsorption may be helpful for understanding the electrode process.
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3.3. Deviation of ΔEp from theoretical values
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The peak potential difference ΔEp between the oxidation wave and the reduction wave (Figure 1) has been used for a prediction of the reaction mechanism. For example, ΔEp = 60 mM suggests the diffusion-controlled current accompanied by one-electron exchange, whereas ΔEp = 30 mM infers a simultaneous reaction with two electrons. Then what would happen for 120 mV which is sometimes found? A half-electron reaction might not be accepted. Potential shift over 60 mV occurs by chemical complications. In contrast, the voltammogram by adsorbed species shows theoretically a bell shape with the width, E1/2 = 90 mV, at the half height of the peak (Figure 2). This value is based on the assumption of the absence of interaction among adsorbed species. However, adsorption necessarily yields such high concentrations as strong interaction.
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It is necessary to pay attention to the validity of analyzing ΔEp and E1/2. The peak potential is the first derivative of a voltammogram. Since ΔEp is a difference between the two peaks, it is actually the second-order derivative of the curves in the view of accuracy. In other words, the accuracy of ΔEp is lower than that of peak current. Furthermore, peak potentials as well as E1/2 readily vary with scan rates owing to chemical reactions and solution resistance. One should use the peak current for data analysis instead of the potentials.
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3.4. Criteria of diffusion-controlled currents
\n
Voltammograms of a number of redox species have been reported to be diffusion controlled from a relationship between Ip and v1/2. The redox species exhibiting diffusion-controlled current is, however, limited to ferrocenyl derivatives under conventional conditions. Voltammograms even for [Fe(CN)6]3−/4− and [Ru(NH3)6]3+ are deviated from the diffusion control for a long-time measurement. Why have many researchers assigned voltammograms to be the diffusion-controlled step? The proportionality of Ip to v1/2 in Eq. (6) has been confused with the linearity, Ip = av1/2 + b (b ≠ 0). The plot for the adsorption control (Ip = kv) also shows approximately a linear relation for Ip vs. v1/2 plot in a narrow domain of v, as shown in Figure 4B. The opposite is true (Figure 4A). Therefore, it is the intercept that determines a controlling step of either the diffusion or adsorption. Some may say that the intercept can be ascribed to a capacitive current. If so, the peak current should be represented by Ip = av1/2 + bv, which exhibits neither linear relation with v1/2 nor v.
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Figure 4.
Plots of Ip of (A) K3Fe(CN)6 and (B) polyaniline-coated electrode against v1/2 and v. Both plots show approximately linear relations.
\n
There is a simple method of determining a controlling step either by diffusion or adsorption. Current responding to diffusion-controlled potential at a disk electrode in diameter less than 0.1 mm would become under the steady state after a few seconds [8]. Adsorption-limited current should become zero soon after the potential application. Many redox species, however, show gradual decrease in the current because reaction products generate an adsorbed layer which blocks further electrode reactions.
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3.5. Plots of Ip against Ep
\n
It is well known that currents vary not only with applied voltage but also with the time. It is not popular, however, to discuss quantitatively time dependence of CV voltammograms. Enhancing v generally increases the current and causes the peak potential to shift in the direction of the scan. A reason for the former can be interpreted as generation of large current at a shorter time (see Eqs. (6) and (10)), whereas the latter is ascribed to a delay of reaction responses as well as a voltage loss of the reaction by solution resistance. Then the voltage effective to the reaction is lower than the intended voltage, and so the observed current may be smaller than the predicted one. Although Ip is related strongly with Ep, the relationship has rarely been examined quantitatively.
\n
A technique of analyzing the potential shift is to plot Ip against Ep, [9] as shown in Figure 5. If the plots on the oxidation side (Ip > 0) and the reduction side (Ip < 0) fall each on a straight line, the slope may represent conductivity. If values of both slopes are equal, the slope possibly stands for the conductivity of the solution or membrane regardless of the electrode reaction. The potential extrapolated to the zero current on each straight line should be close to the formal potential. Since this plot is simple technically, the analytical result is more reliable than at least discussion of time dependence of Ep.
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Figure 5.
Plots of Ip vs. Ep by CV of the first (circles) and the second (triangles) peak of tetracyanoquinodimethane (TCNQ), and ferrocene (squares) in 0.2 M (CH3)4NPF6 included acetonitrile solution when scan rates were varied, where triangles were displayed by 0.4 V shift.
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3.6. Meaning of n3/2 in the equation for peak current
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Most researchers have quoted the Randles-Sevcik equation, jp = 0.446 (nF)3/2c*(Dv/RT)1/2, for the diffusion-controlled peak current without hesitation, where n is the electron transfer number of the reaction. According to Faraday’s law, the electrolytic quantity is proportional to nc*. Why is the peak current proportional to n3/2 instead of n? Let us consider voltammetry of metal nanoparticles (about 25 nm in diameter) composed of 106 metal atoms dispersed in solution. Faraday’s law predicts that the current is 106 times as high as the current by the one metal atom. However, Randles-Sevcik equation predicts the current further (106)1/2 = 1000 times as large, just by the effect of the potential scan. The order 3/2 is specific to CV. The order of n for AC current and pulse voltammetry is 2 [10]. On the other hand, the diffusion-controlled steady-state currents at a microelectrode and a rotating disk electrode are proportional to n. Comparing the differences in the order by methods, we can predict that the time variation of the voltage increases the power of n.
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Let a potential width from a current-rising potential to Ep be denoted by ΔE. When an n-electron transfer reaction occurs through the Nernst equation at which F in Eq. (1) is replaced by nF, the concentration-potential curve takes the slope n times larger than that at n = 1 (see co/cr ≅ nF(E − Eo)/RT near E = Eo in Eq. (1)). Then we have (ΔE)n = (ΔE)n = 1/n. The period of elapsing for (ΔE)n becomes shorter by 1/n, as if v might be larger by n times. Then v in Eq. (6) should be replaced by (nv)1/2. Combining this result with the flux j/nF, the current becomes n3/2 times larger than that at n = 1. Therefore, the factor n3/2 results from the Nernst equation. This can be understood quantitatively by replacing F in Eq. (3) by nF. There are quite a few reactions for n ≥ 2 both for Nernst equation and in the bulk as stable species. The term n3/2 is valid only for a concomitant charge-transfer reaction, i.e., simultaneous occurrence n-electron transfer rather than a step-by-step transfer. Apparent two-electron transfer reactions in the bulk, for example, Cu, Fe, Zn, and Pb, cause other reactions immediately after the one-electron transfer.
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3.7. Area of counter electrode
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An electrochemical response is observed as a sum of the half reactions at the two electrodes. In order to extract the reaction at the working electrode, a conventional technique is to increase the area of the counter electrode so that the reaction at the counter electrode can be ignored. If the counter electrode area is increased by 20 times the area of the working electrode, the observed current represents the reaction of the working electrode with an error of 5%. Let us consider the experiment in which nanoparticles of metal are coated on a working electrode for obtaining capacitive currents or catalyst currents. Then, the actual area of the working electrode can be regarded as the area of the metal particles measured by the molecular level. Then, the area will be several thousand times the geometric area so that the observed current may represent the reaction at the counter electrode. This kind of research has frequently been found in work on supercapacitors. On the other hand, if the electrode reaction is diffusion controlled, the current is determined by the projected area of the diffusion layer. Then the current is not affected by the huge surface area of nanoparticles.
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It is important to examine whether or not a reaction is controlled by at a counter electrode. A simple method is to coat nanoparticles also on the counter electrode. Then the current in the solution may become so high that the potential of the working electrode cannot be controlled. It is better to use a two-electrode system. Products at the counter electrode are possible sources of contaminants through redox cycling.
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3.8. Functionality of reference electrode
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The Ag-AgCl electrode is most frequently used as a reference electrode in aqueous solution because of the stable voltage at interfaces of Ag-AgCl and AgCl-KCl through fast charge-transfer steps, regardless of the magnitude of current density. The “fast step” means the absence of delay of the reaction or being in a quasi-equilibrium. The stability without delay is supported with high concentration of KCl.
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When an Ag-AgCl electrode is inserted to a voltammetric solution, KCl necessarily diffuses into the solution, associated with oxygen from the reference electrode. Thus, the reference electrode is a source of contamination by salt, dichlorosilver and oxygen. It is interesting to examine how much amount a solution is contaminated by a reference electrode [9]. Time variation of ionic conductivity in the pure water was monitored immediately after a commercially available Ag-AgCl electrode was inserted into the solution. Figure 6 shows rapid increase in the conductivity as if a solid of KCl was added to the solution. Oxygen included in the concentrated KCl may contaminate a test solution. Even the Ag-AgxO electrode, which was formed by oxidizing silver wire, increased also the conductivity, probably because the surface is in the form of silver hydroxide. As a result, no reference electrode can be used for studying salt-free electrode reactions. If neutral redox species such as ferrocene is included in a solution, the potential reference can be taken from redox potential of ferrocene.
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Figure 6.
Time-variation of conductivity of water into which (circles) Ag|AgCl, (triangles) Ag|AgxO, and (squares) AgCl-coated Ag wire were inserted. Conductivity measurement was under N2 environment.
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3.9. Current flowing through the double layer
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When a constant voltage is applied to the ideal capacitance C, the responding current decays in the form of exp(−t/RC), where R is a resistance in series connected with C. It has been believed that a double-layer capacitance in electrochemical system behaves as an ideal capacitor, where R is regarded as solution resistance. However, any exponential variation cannot reproduce transient currents obtained at the platinum wire electrode in KCl aqueous solution, as shown in Figure 7. The current decays more slowly than by exp(−t/RC), because it is approximately proportional to 1/t. The property of non-ideal capacitance is the result of the constant phase element of the DL capacitance, as described in Section 2.3. The dependence of 1/t can be obtained approximately by the time derivative of q = V0C0t−λ for the voltage step V0.
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Figure 7.
Chronoamperometric curves when 0.2 V vs. Ag|AgCl was applied to a Pt wire in 0.5 M KCl aqueous solution. Solid curves are fitted ones by exp(-t/RC) for three values of RC.
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The slow decay is related with a loss of the performance of pulse voltammetry, in which diffusion-controlled currents can readily be excluded from capacitive currents. The advantage of pulse voltammetry is based on the assumption of the exponential decay of the capacitive current. Since the diffusion current with 1/t1/2 dependence is close to the 1/t dependence, it cannot readily be separated from the capacitive current in reality. A key of using pulse voltammetry is to take a pulse time to be so long as a textbook recommends.
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High-performance potentiostats are equipped with a circuit for compensation of resistance by a positive feedback. Unfortunately, the circuit is merely useful because voltammograms depend on intensity of compensation resistances of the DL capacitance. It should work well if the DL capacitance is ideal.
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3.10. Advantages of AC impedance
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AC techniques have an advantage of examining time dependence at a given potential, whereas CV has a feature of finding current-voltage curves at a given time. The former shows the dynamic range from 1 Hz to 10 kHz, while the latter does conventionally from 0.01 to 1 Hz. This wide dynamic range of the AC technique is powerful for examining dynamics of electrode reactions. Analytical results by the former are often inconsistent with those by the latter, because of the difference in the time domain. The other scientific advantage of the AC technique is to get two types of independent data set, frequency variations of real components and imaginary ones by the use of a lock-in amplification. The independence allows us to operate mathematically the two data, leading to the data analysis at a level one step higher than CV. An industrial advantage is the rapid measurement, which can be applied to quality control for a number of samples. The analysis of AC impedance necessarily needs equivalent circuits of which components do not have any direction relation with electrochemical variables.
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Data of the electrochemical AC impedance are represented by Nyquist (Cole-Cole) plots, that is, plots of the imaginary component (Z2) of the impedance against the real one (Z1), as shown in Figure 8. The simplest equivalent circuit for electrochemical systems is the DL capacitance Cd in series with the solution resistance RS. The Nyquist plot for this series circuit is theoretically parallel to the vertical axis (Figure 8A-a), but experiments show a slope of 5 or more (Figure 8A-b). This behavior, called constant phase element (CPE) and the power law, has been verified for combinations of various materials and solvents [6, 7, 11, 12]. The equivalent circuit for Eq. (12) is a parallel combination of capacitance and resistance (Figure 8B). Even without an electrode reaction, current always includes a real component.
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Figure 8.
(A) Nyquist plots for a RC-series circuit with ideal capacitor (a) and DL capacitor (b). (B) Equivalent circuit with the power-law of Cd. (C) Randles circuit.
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The equivalent circuit with the Randles type is a parallel combination of the ideal DL capacitor Cd with the ideal resistance Rct representing the Butler-Volmer-type charge-transfer resistance. Practically, the Warburg impedance (the inverse of Eq. (8)) due to diffusion of redox species is incorporated in a series into Rct (Figure 8C). Rct cannot be separated from the DL resistance because of the frequency dispersion. Since even the existence of Rct is in question (Section 3.12), it is difficult to determine and interpret Rct. The usage of a software that can analyze any Nyquist plots will provide values of R and C. Even if analyzed values are in high accuracy, researches should give them electrochemical significance.
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3.11. Residual currents depending on surface treatments
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Residual current varies with treatments of electrodes such as polishing of electrode surfaces and voltage applications to an extremely high domain. It can often be suppressed to yield reproducible data when the electrode is replaced by simple platinum wire or carbon rod having the same geometric area. Simple wire electrodes are quite useful especially for measurements of DL capacitance and adsorption. One of the reasons for setting off large residual current is that the insulator of confining the active area is not in close contact with the electrode, so that the solution penetrated into the gap will give rise to capacitive current and floating electrode reactions. Since the coefficient of thermal expansion of the electrode is different from that of the insulator, the residual current tends to get large with the elapse from the fabrication of the electrode. This prediction is based on experience, and there are few quantitative studies on residual currents.
\n
Unexpected gap has been a technical problem at dropping mercury electrodes. If solution penetrates the inner wall of the glass capillary containing mercury, observed currents become irreproducible. Water repellency of the capillary tip has been known to improve the irreproducibility in order to reduce the penetration. A similar technique has been used for voltammetry at oil-water interfaces and ionic liquid-water interfaces at present.
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3.12. Reversible or irreversible voltammograms
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Voltammograms are said to vary with electrode reaction rates, and the rate constants have been determined from time dependence of voltammograms. The fast reaction of which rate is not rate determining has historically been called “reversible.” In contrast, such a slow reaction that a peak potential varies linearly with log v is called “irreversible.” A reaction between them is called “quasi-reversible.” The distinction among the three has been well known since the theoretical report on the quasi-reversible reaction by Matsuda [1]. This theory is devoted to solving the diffusion equations with boundary conditions of the Butler-Volmer (BV) equation under the potential sweep. As the standard rate constant ks in the BV equation becomes small, the peak shifts in the direction of the potential sweep from the diffusion-controlled peak. Steady-state current-potential curves in a microelectrode [13] and a rotating disk electrode also shift the potential in a similar way. According to the calculated CV voltammograms in Figure 9, we can present some characteristics: (i) if the oxidation wave shifts to the positive potential, the negative potential shift should also be found in the reduction wave. (ii) Both the amounts of the shift should have a linear relationship to log v. (iii) The shift should be found in iterative measurements. (iv) The peak current should be proportional to v1/2.
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Figure 9.
CV voltammograms (solid curves) at a normally sized electrode and steady-state voltammograms (dashed curves) at a microelectrodes in 12 μm in diameter, calculated theoretically for v = 0.5 V s−1, D = 0.73 × 10−5 cm2 s−1, ks = (a) 0.1, (b) 0.01, (c) 0.001, (d) 0.0001 cm s−1. The potential shift of CV is equivalent to the wave-shift at a microelectrode through the relation, v = 0.4RTD/αFa2 (a: radius).
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The authors attempted to find a redox species with the above four behaviors. Some redox species can satisfy one of the four requirements, but do not meet the others. Most reaction rate constants have been determined from the potential shift in a narrow time domain. They are probably caused by follow-up chemical reactions, adsorption, or DL capacitance. For example, CV peak potentials of TCNQ and benzoquinone were shifted at high scan rates, whereas their steady-state voltammograms were independent of diameters of microdisk electrodes even on the nanometer scale [14]. The shift at high scan rates should be due to the frequency dispersion of the DL capacitance, especially the parallel resistance in the DL (Figure 8B). Values of the heterogeneous rate constants and transfer coefficients reported so far have depended not only on the electrochemical techniques but also research groups. Furthermore, they have not been applied or extended to next developing work. These facts inspire us to examine the assumptions and validity of the BV formula.
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Let us revisit the assumptions of the BV equation when an overvoltage, i.e., the difference of the applied potential from the standard electrode potential, causes the electrode reaction. The rate of the oxidation in the BV equation is assumed to have the activation energy of α times the overvoltage, while that of the reduction does that of (1 − α) times. This assumption seems reasonable for the balance of both the oxidation and the reduction. However, the following two points should be considered. (i) Once a charge or an electron is transferred within the redox species, the molecular structure changes more slowly than the charge transfer itself occurs. The structure change causes solvation as well as motion of external ions to keep electric neutrality. These processes should be slower than the structure change. If the overvoltage can control the reaction rate, it should act on to the slowest step, which is not the genuine charge-transfer process. (ii) Since a reaction rate belongs to the probability theory, the reaction rate (dc/dt) at t is determined with the state at t rather than a state in the future. In other words, the rate of the reduction should have no relation with the oxidation state which belongs to the future state. The BV theory assumes that the α times activation energy for the oxidation is related closely with 1-α times one for the reduction. This assumption is equivalent to predicting a state at t + Δt from state at t + 2Δt, like riding on a time machine. This question should be solved from a viewpoint of statistical physics.
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3.13. Contradiction of microscopic image with electrochemical data
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Development of scanning microscopes such as STM and AFM has allowed us to obtain the molecularly and atomically regulated surface images, which have been used for interpreting electrochemical data. Then the electrochemical data are expected to be discussed on a molecular scale. However, there is an essential problem of applying photographs of regularly arranged atoms on an electrode to electrochemical data, because the former and the latter include, respectively, microscopically local information and macroscopically averaged one. A STM image showing molecular patterns is information of only a part of electrode, at next parts of which no atomic images are often observed but noisy images are found. Electrochemical data should be composed of information both at a part of the electrode showing the molecular patters and at other parts showing noisy, vague images. Noisy photographs are always discarded for interpreting electrochemical data although the surfaces with noisy images also contribute electrochemical data.
\n
An ideal experiment would be made by taking STM images over all the electrodes that provide electrochemical data and by obtaining an averaged image. However, it is not only impossible to take huge amounts of images, but the averaged image might be also noisy. It may be helpful to describe only a possibility of reflecting the STM-imaged atomic structure on the electrochemical data.
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3.14. Surface wave by adsorption
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Voltammograms by adsorbed redox species, called surface waves, are frequently different from a bell shape (Figure 2). Really observed features are the following: (i) the voltammogram does not suddenly decay after the peak, exhibiting a tail-like diffusional wave; (ii) the peak current and the amount of the electricity are proportional to the power less than the unity of v; (iii) the oxidation peak potential is different from the reduction one; (iv) the background current cannot be determined unequivocally; and (v) voltammograms depend on the starting potential. Why are experimental surface waves different from a symmetric, bell shape in Figure 2?
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A loss of the symmetry with respect to the vertical line passing through a peak can be ascribed to the difference in interactions at the oxidized potential domain and at the reduced one. Since redox species takes extremely high concentration in the adsorbed layer, interaction is highly influenced on voltammetric form. When the left-right asymmetry is ascribed to thermodynamic interaction, it has been interpreted not only with Frumkin’s interaction [15] but also Bragg-Williams-like model for the nearest neighboring interactive redox species [16]. On the other hand, most surface waves are asymmetric with respect to the voltage axis even at extremely slow scan rates. This asymmetry cannot be explained in terms of thermodynamics of intermolecular interaction, but should resort to kinetics or a delay of electrode reactions. There seems to be no delay in the electrode reaction of the monomolecular adsorption layer, different from diffusion species. The delay resembles the phenomenon of constant phase element (CPE) or frequency power law of DL capacitance, in that the redox interaction may occur two-dimensionally so that the most stable state can be attained. This behavior belongs to a cooperative phenomenon [17]. A technique of overcoming these complications is to discuss the amount of charge by evaluating the area of the voltammogram. It also includes ambiguity of eliminating background current and assuming the independence of the redox charge from the DL charge.
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4. Conclusions
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The simplest theories for voltammetry are limited to the rate-determining steps of diffusion of redox species and reactions of adsorbed species without interaction. Variation of scan rates as well as a reverse potential is helpful for predicting redox species and reaction mechanisms. Furthermore, the following viewpoints are useful for interpreting mechanisms:
comparison of values of experimental peak currents with theoretical ones, instead of discussing ΔEp and E1/2;
examining the proportionality of Ip vs. v or vs. v1/2, i.e., zero or non-zero values of the intercept of the linearity;
a reference electrode and a counter electrode being a source of contamination in solution;
attention to very slow relaxation of DL capacitive currents;
inclusion of ambiguity in the equivalent circuit with the Randles type.
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\n\n',keywords:"current-voltage curves, cyclic voltammetry, AC impedance, double-layer capacitance, diffusion control, surface wave",chapterPDFUrl:"https://cdn.intechopen.com/pdfs/63917.pdf",chapterXML:"https://mts.intechopen.com/source/xml/63917.xml",downloadPdfUrl:"/chapter/pdf-download/63917",previewPdfUrl:"/chapter/pdf-preview/63917",totalDownloads:1585,totalViews:467,totalCrossrefCites:1,dateSubmitted:"June 15th 2018",dateReviewed:"September 5th 2018",datePrePublished:"November 5th 2018",datePublished:null,dateFinished:null,readingETA:"0",abstract:"Theories of cyclic voltammetry, AC-impedance techniques, and the double-layer capacitive currents are described concisely to touch their principles. Applications of the theory to experimental data do not always lead to reasonable interpretation consistent with other techniques. Several tips are presented not only in the experimental viewpoint but also in a perspective of the data analysis. Most of them are devoted to cyclic voltammetry. They include shape of voltammograms, information from peak currents and peak potentials, criteria of diffusion and adsorption controls, the static and the dynamic numbers of electrons, handling of reference and counter electrodes, usage of AC impedance, concepts of heterogeneous charge-transfer rates, and combination with data by scanning probe microscope. They belong partially to recommendation and prohibition.",reviewType:"peer-reviewed",bibtexUrl:"/chapter/bibtex/63917",risUrl:"/chapter/ris/63917",signatures:"Koichi Jeremiah Aoki and Jingyuan Chen",book:{id:"7348",title:"Voltammetry",subtitle:null,fullTitle:"Voltammetry",slug:"voltammetry",publishedDate:"June 12th 2019",bookSignature:"Nobanathi Wendy Maxakato, Sandile Surprise Gwebu and Gugu Hlengiwe Mhlongo",coverURL:"https://cdn.intechopen.com/books/images_new/7348.jpg",licenceType:"CC BY 3.0",editedByType:"Edited by",editors:[{id:"259515",title:"Dr.",name:"Nobanathi Wendy",middleName:null,surname:"Maxakato",slug:"nobanathi-wendy-maxakato",fullName:"Nobanathi Wendy Maxakato"}],productType:{id:"1",title:"Edited Volume",chapterContentType:"chapter",authoredCaption:"Edited by"}},authors:[{id:"263291",title:"Prof.",name:"Koichi",middleName:"Jeremiah",surname:"Aoki",fullName:"Koichi Aoki",slug:"koichi-aoki",email:"kaoki@u-fukui.ac.jp",position:null,institution:{name:"University of Fukui",institutionURL:null,country:{name:"Japan"}}},{id:"263292",title:"Prof.",name:"Jingyuan",middleName:null,surname:"Chen",fullName:"Jingyuan Chen",slug:"jingyuan-chen",email:"jchen@u-fukui.ac.jp",position:null,institution:{name:"University of Fukui",institutionURL:null,country:{name:"Japan"}}}],sections:[{id:"sec_1",title:"1. Introduction",level:"1"},{id:"sec_2",title:"2. Theory",level:"1"},{id:"sec_2_2",title:"2.1. Diffusion-controlled current",level:"2"},{id:"sec_2_3",title:"2.1.1. Linear sweep voltammetry by diffusion",level:"3"},{id:"sec_3_3",title:"2.1.2. AC voltammetry by diffusion",level:"3"},{id:"sec_5_2",title:"2.2. Adsorption-controlled current",level:"2"},{id:"sec_5_3",title:"2.2.1. Linear sweep voltammetry by adsorption",level:"3"},{id:"sec_7_2",title:"2.3. Capacitive current",level:"2"},{id:"sec_7_3",title:"2.3.1. Capacitance by AC impedance",level:"3"},{id:"sec_8_3",title:"2.3.2. Capacitive current by CV",level:"3"},{id:"sec_11",title:"3. Tips of voltammetric analysis",level:"1"},{id:"sec_11_2",title:"3.1. Understanding outline of voltammograms",level:"2"},{id:"sec_12_2",title:"3.2. Shape and values of peaks",level:"2"},{id:"sec_13_2",title:"3.3. Deviation of ΔEp from theoretical values",level:"2"},{id:"sec_14_2",title:"3.4. Criteria of diffusion-controlled currents",level:"2"},{id:"sec_15_2",title:"3.5. Plots of Ip against Ep",level:"2"},{id:"sec_16_2",title:"3.6. Meaning of n3/2 in the equation for peak current",level:"2"},{id:"sec_17_2",title:"3.7. Area of counter electrode",level:"2"},{id:"sec_18_2",title:"3.8. Functionality of reference electrode",level:"2"},{id:"sec_19_2",title:"3.9. Current flowing through the double layer",level:"2"},{id:"sec_20_2",title:"3.10. Advantages of AC impedance",level:"2"},{id:"sec_21_2",title:"3.11. Residual currents depending on surface treatments",level:"2"},{id:"sec_22_2",title:"3.12. Reversible or irreversible voltammograms",level:"2"},{id:"sec_23_2",title:"3.13. Contradiction of microscopic image with electrochemical data",level:"2"},{id:"sec_24_2",title:"3.14. Surface wave by adsorption",level:"2"},{id:"sec_26",title:"4. Conclusions",level:"1"}],chapterReferences:[{id:"B1",body:'Matsuda H, Ayabe Y. Zur theorie der Randles-Sevčikschen kathodenstrahl-polarographie. Zeitschrift für Elektrochemie. 1955;59:494-503\n'},{id:"B2",body:'Barsoukov E, Macdonald JR. Impedance Spectroscopy. New York: Wiley; 2005. ISBN: 0-471-64749-7\n'},{id:"B3",body:'Lasia A. In: White RE, Conway BE, Bockris JO’M, editors. Modern Aspects of Electrochemistry. Vol. 32. New York: Kluwer Academic/Plenum Publishers; 1999. pp. 143-248\n'},{id:"B4",body:'Brug GJ, Van Den Eeden ALG, Sluyters-Rehbach M, Sluyters JH. The analysis of electrode impedances complicated by the presence of a constant phase element. Journal of Electroanalytical Chemistry. 1984;176:275-295. DOI: 10.1016/S0022-0728(84)80324-1\n'},{id:"B5",body:'Zoltowski P. On the electrical capacitance of interfaces exhibiting constant phase element behaviour. Journal of Electroanalytical Chemistry. 1998;443:149-154. DOI: 10.1016/S0022-0728(97)00490-7\n'},{id:"B6",body:'Hou Y, Aoki KJ, Chen J, Nishiumi T. Invariance of double layer capacitance to polarized potential in halide solutions. Universal Journal of Chemistry. 2013;1:162-169. DOI: 10.13189/ujc.2013.010404\n'},{id:"B7",body:'Hou Y, Aoki KJ, Chen J, Nishiumi T. Solvent variables controlling electric double layer capacitance at metal|solution interface. Journal of Physical Chemistry C. 2014;118:10153-10158. DOI: 10.1021/jp5018289\n'},{id:"B8",body:'Zhang H, Aoki K, Chen J, Nishiumi T, Toda H, Torita E. Voltammetric determination of both concentration and diffusion coefficient by combinational use of regular and micro electrodes. Electroanalysis. 2011;23:947-952. DOI: 10.1002/elan.200900603\n'},{id:"B9",body:'Zhang C, Aoki KJ, Chen J, Nishiumi T. Blocking of two-electron reduction of non-charged species in the absence of supporting electrolyte at nanoelectrodes. Journal of Electroanalytical Chemistry. 2013;708:101-107. DOI: 10.1016/j.jelechem.2013.09.023\n'},{id:"B10",body:'Smith DE. In: Bard AJ, editor. Electroanalytical Chemistry. Vol. 1. New York, N.Y.: Dekker; 1966. pp. 1-155. DOI: 10.1002/ange.19680800321\n'},{id:"B11",body:'Aoki KJ, Wang H, Chen J, Nishiumi T. Formation of graphite oxide nano-disks by electrochemical oxidation of HOPG. Electrochimica Acta. 2014;130:381-386. DOI: 10.1016/j.electacta.2014.03.044\n'},{id:"B12",body:'Wang H, Aoki KJ, Chen J, Nishiumi T, Zeng Z, Ma X. Power law for frequency-dependence of double layer capacitance of graphene flakes. Journal of Electroanalytical Chemistry. 2015;741:114-119. DOI: 10.1016/j.jelechem.2015.01.008\n'},{id:"B13",body:'Aoki K. Evaluation technique of kinetic parameters for irreversible charge transfer reactions from steady-state voltammograms at microdisk electrodes. Electrochemistry Communications. 2005;17:523-527. DOI: 10.1016/j.elecom.2005.03.006\n'},{id:"B14",body:'Aoki KJ, Zhang C, Chen J, Nishiumi T. Heterogeneous reaction rate constants by steady-state microelectrode techniques and fast scan voltammetry. Journal of Electroanalytical Chemistry. 2013;706:40-47. 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The Open Access publishing model followed by IntechOpen eliminates subscription charges and pay-per-view fees, thus enabling readers to access research at no cost to themselves. In order to sustain these operations, and keep our publications freely accessible, we levy an Open Access Publishing Fee on all manuscripts accepted for publication to help cover the costs of editorial work and the production of books.
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