PL 01 Grafov

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RSE-SEE 2 GIBBS FLUCTUATION THEORY IN CONTEXT OF ELECTROCHEMICAL EQUILIBRIUM NOISE Boris M.Grafov A.N.Frumkin Institute of Physical Chemistry and Electrochemistry Russian Academy of Sciences, Moscow, Russia THE MAIN STATEMENTS The Gibbs fluctuation theory is the general foundation for the electrochemical elementary act theory. The Gibbs ergodic idea is splendid and paradoxical simultaneously. The Gibbs ergodic idea works perfectly with respect to the pair correlations of thermal fluctuations. The Gibbs ergodic idea does not work with respect to the triple correlations and the correlations of higher order. The noise version of the electrochemical charge transfer theory should be developed. The high order noise correlation technique may be useful in the nanoelectrochemical technologies. 1

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CONTENT SIMPLEST ELECTROCHEMICAL CIRCUIT GIBBS FLUCTUATION THEORY NOISY ELECTROCHEMICAL CIRCUIT LANGEVIN STOCHASTIC EQUATION LANGEVIN APPROACH TO THERMAL FLUCTUATION CONCLUSION REMARKS THE MAIN IDEAThe Langevin stochastic equation provides one the possibility to verify the Gibbs fluctuation theory. 2 RSE-SEE 2

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THE SIMPLEST ELECTROCHEMICAL CIRCUIT Fig.1. The macroscopic electric circuit for the electrode with the Faradaic process. The I=I(E) is nonlinear current-potential relation for the Faradaic process. The Q=Q(E) is the nonlinear charge-potential relation for the electric double layer. The q(t) is the fluctuating (changing with time t) component of the electrode charge. Point 1 corresponds to the bulk of electrode under study. Point 2 corresponds to the reference electrode.The bold vertical line marks the nonlinear electric element. 3 RSE-SEE 2

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GIBBS FLUCTUATION THEORY 1 The Gibbs basic idea is striking. It is assumed that any system of the Gibbs equilibrium ensemble transfer permanently from one state to another due to the random interaction with the environment. At the same time it is assumed that the distribution function in the Gibbs equilibrium ensemble does not depend of character and parameters of random interaction with the environment. In our case the role of the electric environment plays the Faradic process which supplies the electric double layer by the random quantity of electricity. In line with the Gibbs idea, any stochastic moment of random charge does not depend of stochastic properties of the Faradaic process. 4 RSE-SEE 2

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For variance of electrode charge the Gibbs fluctuation theory predicts : In (1) the angular brackets denote the average with respect to the Gibbs equilibrium ensemble, the k is Boltzmann constant, the T is temperature, and the C is the differential capacity of the electric double layer at the equilibrium potential. GIBBS FLUCTUATION THEORY 2 Eq. (1) shows that the charge variance is proportional to differential capacitance C of the double layer. It is of importance that the nonlinearity of double layer capacity does not influence the charge variance. 5 RSE-SEE 2

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The nonlinear properties of double layer capacity control the charge stochastic moments of high orders. The Gibbs fluctuation theory predicts: GIBBS FLUCTUATION THEORY 3 The (3) shows that the charge cumulant of 3-rd order is proportional to the derivative of the double layer capacity with respect to potential. The (4) shows that the charge cumulant of 4-th order is proportional to the second derivative of the double layer capacity with respect to potential. The (5) shows that the charge cumulant of 5-th order is determined by the third derivative of the double layer capacity with respect to potential. The Faradaic process is the only source of the charge fluctuations. However, any parameter of the Faradaic process is not involved in the Gibbs fluctuation relations. 6 RSE-SEE 2

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THE NOISY ELECTROCHEMICAL CIRCUIT In context of nanoelectrochemistry we have to take into account the current white noise i(t) of the Faradaic process. It results in the following noisy electrical circuit for the electrochemical cell under study: In line with the Nyquist fluctuation dissipation theorem, the linear electrical elements are only involved into the noisy electric circuit. 7 RSE-SEE 2

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THE LANGEVIN STOCHASTIC EQUATION 1 We carry out the noise analysis for the electric circuit of Fig.2 by use of the Langevin stochastic equation. To be in compliance with the Nyquist fluctuation dissipation theorem, the Langevin equation for thermal noise must have the form of the Kirhgoff linear equation: The linearity of the Langevin equation (6) for internal equilibrium noise reflects the essence of the Nyquist fluctuation dissipation theorem. 8 RSE-SEE 2

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THE LANGEVIN STOCHASTIC EQUATION 2 The Langevin equation (6) may be read as a mapping equation. The (6) maps the ensemble of the current noise realizations {i(t)} to the ensemble of voltage noise realizations {e(t)=q(t)/C} where e(t) is the random component of the electrode potential. The stochastic properties of the Faradaic current white noise i(t) define completely by the set of cumulant functions. These first four cumulant functions are as follows: In (7)-(8) the d(t) is the Dirac delta-function and the symbols i(2), i(3), i(4), i(5) stands for the intensities of cumulant functions. These intensities reflect the noise properties of Faradaic process. REPETITION 9 RSE-SEE 2

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LANGEVIN’S APPROCH TO THERMAL FLUCTUATIONS 1 The Langevin stochastic equation (6) has the same character as the Langevin equation in the theory for the Einstein diffusion. The solution of (6) is well known: where the response function H(t) is given by equation: REPETITION 10 RSE-SEE 2

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LANGEVIN’S APPROCH TO THERMAL FLUCTUATIONS 2 Combining (11) and (7) yields: In line with the Nyquist fluctuation dissipation theorem one has Therefore we receive instead (13) The relation (15) is remarkable. The (15) is identical with the corresponding equation of the Gibbs fluctuation theory (equation(1)). This means that the paradoxical Gibbs idea works perfectly in respect of the second order correlations of thermal fluctuations. However, the situation with the Gibbs description of high order correlations is completely different. The angular brackets denote averaging over the temporal realizations of random process. 11 RSE-SEE 2

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LANGEVIN’S APPROCH TO THERMAL FLUCTUATIONS 3 Combining (11) and (8), (9), and (10) gives: Let us assume that the statistics of Faradaic exchange current is Poissonian. In this case we have for the intensities of current cumulant functions: where the e is the elementary charge and the I0 is the exchange current. We have instead (14): Comparison between the Gibbs formula (3) and formula (20) of Langevin’s theory shows that the Gibbs fluctuation formula for triple charge correlations works if and only if the charge minimum or charge maximum takes place at the equilibrium potential. One may see that the Gibbs fluctuation relations (3)-(5) are not in compliance with the Langevin relations (14)-(16). 12 RSE-SEE 2

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LANGEVIN’S APPROCH TO THERMAL FLUCTUATIONS 4 The discrepancy between the Gibbs and Langevin approaches there also exists in respect of the charge correlations of fourth and fifth orders. In frame of the Langevin approach one has One may see that the Gibbs fluctuation relations (4) and (5) are in contradiction with the equation (21) and (22) of Langevin’s stochastic theory. At the same time the Gibbs fluctuation theory predicts The Gibbs formulae for the high correlations of thermal fluctuations are outside the ergodic hypothesis. 13 RSE-SEE 2

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CONCLUSION REMARKS The electrochemical noise verification of the Gibbs ergodic idea shows: This work is supported by Russian Foundation for Fundamental Investigations in frame of project 08-03-00051-a. It is my pleasure to express the deep gratitude to professor Vesna Miskovic-Stankovic and professor Branislav Nikolic for their kind invitation to come to Belgrade and deliver this lecture. The Gibbs ergodic idea works perfectly with respect to the pair correlations of thermal fluctuations. The Gibbs ergodic idea does not work with respect to the triple correlations and the correlations of higher order. The noise version of the electrochemical charge transfer theory should be developed. The high order noise correlation technique may be useful in the nanoelectrochemical technologies. 14 RSE-SEE 2