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Properties of polarization curves for electrochemical cells described by Butler-Volmer kinetics and arbitrary values of the transfer coefficient
Yurij I.Kharkats
Artjom V.Sokirko#
and
Fritz.H.Bark#,
# Department of Mechanics, Royal Institute of Technology, 100 100 44 Stockholm, Sweden
The A.N.Frumkin Institute of Electrochemistry, Russian Academy of Sciences, 117071 Moscow, Russia
Abstract
Theoretical investigation of potentiostatic electrolysis of a metallic salt in three component electrolyte solution is carried out for a cell consisting of two identical parallel electrodes. Analytical and numerical results are given for polarization curves for electrochemical cells with arbitrary values of transfer coefficient a and exchange current density. Theoretical analysis of electrodiffusion problem, based on exact solution of the Nernst-Planck equations with boundary conditions of Butler - Volmer type, leads to a formula for the polarization curve that is similar to the Tafel equation but with an effective transfer coefficient aeff = a(1 - a).. It is shown that, under certain conditions, the polarization curve can have two inflection points.
Introduction
In recently published paper [1], a theoretical investigation of potentiostatic one-dimensional electrolysis of a metallic salt in a three component electrolyte solution was carried out for electrochemical cell having two identical parallel electrodes. The analysis led to an exact solution of the system of Nernst - Plank transport equations accounting ion transfer by diffusion and migration mechanisms [2-5]. The polarization curves were derived in parametric analytic form for the special case where the transfer coefficient a in the Butler - Volmer law is equal to 0.5 and for an arbitrary concentration of supporting electrolyte.
Qualitatively, the polarization curves found in [1] for a = 0.5 were similar to those found for wide class of different systems in which ionic diffusion and migration transport was analysed at the surface of one electrode [6 -10].In particular polarization curves of such type were studied in recently published papers [11 - 16]. On the other hand, for a 0.5, one may expect effects on the current - voltage curve due to the asymmetry of the reaction. In the present paper, this issue is examined in some detail for arbitrary values of the exchange current density.
Problem statement
Consider a binary electrolyte with ionic charge numbers z1 and z2 for cations and anions, respectively. The following reduction reaction occurs at the cathode
Mz1+ + z1e- -> M.
At the anode the same reaction proceeds in the opposite direction. We shall analyse the current - voltage characteristic of the cell assuming that there is no convective motion of the electrolyte in the space between electrodes and that ionic transport takes place due to diffusion and migration in the O(e;-;)X - direction, which is perpendicular to the electrodes surfaces at X = 0 (cathode) and X = L (anode).
Let the cathode electric potential be equal to 0 and the anode potential equal to V. We shall assume that there is no redistribution of the electric potential F in the diffuse double layers during current flow.
In the case of dilute electrolyte, the Nernst - Einstein relation for the diffusion and mobility coefficients can be used. The concentrations of cations and anions C1 and C2, respectively, and the electric potential F can then be computed from the following system of equations
D1f(dC1;dX) + z1C1 f(FD1;RT) f(dF;dX) = - f(i;Fz1), (1)
f(dC2;dX) - z2C2 f(F;RT) f(dF;dX) = 0, (2)
z1C1 - z2C2 = 0. (3)
Here D1 is the diffusion coefficient for cations, F is the Faraday constant, R is the gas constant, T is the absolute temperature, i is the electric current density.
From the Butler - Volmer law, the electric current density can be expressed in terms of overpotentials for electrode reactions, which are F(0) and V - F(1) for the cathode and anode, respectively :
i = i0BBC{(eSUP8(- F(az1FF(0);RT)) - F(C1(0);CS(0;1)) eSUP8(F((1-a)z1FF(0);RT))). (4)
i = i0BBC{(F(C1(0);CS(0;1)) eSUP8(F((1-a)z1F(F(L)-V);RT)) -eSUP8(- F(az1F(F(L)-V);RT)) ). (5)
In these formulae, i0 is the exchange current density, for a concentration of cations equal to CS(0;1). a is the transfer coefficient.
Taking into account that the reactions at anode and cathode have equal rates, conservation conditions for C1 and C2 may be written in the form
IIN(SDO5(0);SUP20(L);CmdX) = CS(0;m)L, m = 1,2, (6)
where CS(0;m) are the initial (homogeneous) concentrations of species in the electrolyte.
Equations (2)-(6) can be written in dimensionless form in terms of the coordinate x = X/L, the concentrations cm = Cm/CS(0;m), (m = 1,2) and the electric potential f = FF/RT:
f(dc1;dx) + z1c1 f(df;dx) = - f(j;z1), (7)
f(dc2;dx) + z2c2 f(df;dx) = 0, (8)
z1c1 - z2c2 = 0. (9)
The boundary conditions (4) - (5) can be written in the form
j = j0BBC{(c1(0) eSUP8((1-a)z1f(0)) - eSUP8(- az1f(0)) ), (10)
j =j0BBC{(eSUP8(- az1(f(1)-v)) -c1(1) eSUP8((1-a)z1(f(1)-v)) ). (11)
Where j = - iL/FD1CS(0;1), j0 = i0L/FD1CS(0;1) and v = V/RT.
For the normalization conditions (6), one obtains the following expressions
IIN(SDO5(0);SUP20(1);c1 dx) = 1, IIN(SDO5(0);SUP20(1);c2 dx) = f(z1;z2). (12)
Integration of Eqs.(7) - (9) after use have been made of conditions (12), gives the following expressions for the concentrations
c1(x) = 1 + f(jz2;z1(z1 + z2)) BBC((f(1;2) - x) = f(c2(x)z1;z2) . (13)
From e.g. Eq.(8) one then finds the expression for the total potential drop in electrolyte
Df = f(1) - f(0) = f(1;z2)ln f(c2(1);c2(0)) = f(1;z2)ln f(jl + j;jl - j) , (14)
where
c1(0) = 1 - f(j;jl ), (15)
c1(1) = 1 + f(j;jl ), (16)
and
jl = f(2z1(z1 + z2);z2) . (17)
jl is the limiting diffusion - migration current in the cell, corresponding to the condition c1(0) = 0.
Rewriting the boundary conditions (10) - (11) by using expressions (14) - (16) one obtains the following two relations between f(0), v and j:
j = j0BBC{(BBC((1 - f(j;jl ))eSUP8((1-a)z1f(0)) - eSUP8(- az1f(0)) ), (18)
and
j =j0SDO6(BBC{(eSUP15(- az1BBC((f(0)-v + f(1;z2)ln f(jl + j;jl - j)))-BBC((1 + f(j;jl ))eSUP15((1-a)z1BBC((f(0)-v + f(1;z2)ln f(jl + j;jl - j))))) (19)
Elimination of f(0) from these expressions leads to an implicit expression for the polarization curve j(v). A convenient form is obtained by solving for eSUP8(z1f(0)), which gives the following result
E = eSUP8(z1f(0)) = f(eSUP15( az1BBC(v - f(1;z2)ln f(jl + j;jl - j))) + 1; 1 - f(j;jl ) + BBC((1 + f(j;jl )) eSUP15((1-a)z1BBC((f(1;z2)ln f(jl + j;jl - j) -v))). (20)
Substitution into relation (18) then gives that
j =j0 E SUP8(- a )BBC[(BBC((1 - f(j;jl))E - 1). (21)
Expressions (21) and (20) define j as an implicit function of v.
Analysis of some limiting cases
1. For small values of j and v, expansion of expression (20) gives, to lowest order, that
E 1 + F(z1;2) BBC(v - f(2;z2) f(j;jl)). (22)
Substitution of this expression into expression (21) gives the explicit linearized form of the polarization curve
f(j;j0) F(z1;2) v - f((z1 + z2);z2) f(j;jl). (23)
Thus, the slope of j(v) curve at small values of v depends on the ratio between j0 and jl. In the case of j0 << jl expression (23) may be written in the form
j j0 F(z1;2) v, (24)
and in the opposite case, i.e. j0 >> jl , as
j jl F(z1z2;2(z1 + z2)) v. (25)
2. Consider now the case of small values of j0. Assuming that the values of v such that j0 << j(v) ~ jl are of the order 100 - 101 or larger, one obtains from expression (20) the approximate relation
E = eSUP15( az1v)BBC((1 - f(j;jl )) SUP15(f(az1;z2) - 1)BBC((1 + f(j;jl )) SUP15(- f(az1;z2)). (26)
For E >> 1, one then finds from expression (21) that the polarization curve is given by the following approximate expression
j j0 F(BBC((1 - f(j;jl )) SUP15(f(a(1-a)z1;z2) + a);BBC((1 + f(j;jl )) SUP15( f(a(1-a)z1;z2))) eSUP15( a(1-a)z1v). (27)
This result shows that for values of j that are not too close to jl, the polarization curve j(v) can be represented as a Tafel-like law with the effective transfer coefficient
aeff = a(1 - a) (28)
The maximum value of aeff is equal to 0.25 and the dependence of j(v) on a is symmetrical with respect to a = 0.25. Again, it should be stressed that this symmetry property is valid for small j0 values only.
3. For j0 << j(v) ~ jl formula (27) simplifies to
j j0eSUP15( a(1-a)z1v). (29)
A comparison between formulas (24) and (29) shows that in the transition region from the linear behaviour of j (v), which is independent of a, to the exponential one, there is an inflection point for values of a larger than 0.5. This inflection point is similar to that of the current - voltage curve for the reaction at one electrode, which is described by Butler - Volmer law. For j0 << 1 and j0 << j O(~;<)jl one then has that
j = j0BBC{(eSUP 8((1-a)z1v) - eSUP8(- az1v)). (30)
The position v = vc, say, of the inflection point in this case can be readily computed and one finds that
vc = F(2;z1) ln BBC((F(a;1-a)). (31)
It turns out that this expression can also be used as an estimate of the location of one of the inflection points of the polarization curve for the cell with two electrodes. The closer is a to 1, the higher is the accuracy of Eq.(31).
The second inflection point of polarization curve is found in the region of high values of v where the current limitation due to diffusion and migration becomes important.
4. For high values of the exchange current density, j in the left hand side of expression (21) can be neglected and the polarization curve is approximately determined by relation
j jl BBC[(1 - f(1;E(v ))), (32)
where E(v ) is given by expression (20).This relation is, of course, invalid when j is very close to the limiting current density.
The role of the supporting electrolyte in a system with three species.
In the presence of a supporting electrolyte, the system of dimensionless equations for the concentrations distributions of electroactive cations, indifferent cations, anions and the electric potential distribution reads as:
f(dc1;dx) + z1c1 f(df;dx) = - f(j;z1), (33)
f(dc2;dx) + z2c2 f(df;dx) = 0, (34)
f(dc3;dx) - z3c3 f(df;dx) = 0, (35)
z1c1 + z2c2 = z3c3. (36)
In these equations, the dimensionless concentrations are defined as
ci = f(Ci;CS(0;3)) (37)
and the electric current density is normalised according to the relation j = - iL/FD1CS(0;3.) It turns out to be convenient to introduce the parameter k = CS(0;1)/CS(0;3) . Then, CS(0;2)/CS(0;3) = z3/z2 - kz1/z2, 0 < k < z3/z1.
The case of the aqueous solution of CuSO4 with H2SO4 as a supporting electrolyte will be considered below. In this case at relatively low H2SO4 concentrations H2SO4 dissociates mainly into H+ and HSO4- and, correspondingly, z1 = 2, z2 = z3 = 1.
For the three component system under consideration, the dimensionless version of Butler-Volmer laws for the electrode reactions, i.e. formulas (4) and (5), read
j = j0 BBC{(F(c1(0);k ) eSUP8(2(1-a)f(0)) - eSUP8(- 2af(0)) ), at x = 0, (38)
j =j0 BBC{(eSUP8(- 2a(f(1)-v)) - F(c1(1);k ) eSUP8(2(1-a)(f(1)-v)) ), at x = 1. (39)
The dimensionless conservation conditions, c.f. formula (6), takes the form
IIN(SDO5(0);SUP20(1);c1 dx) = k , (40)
IIN(SDO5(0);SUP20(1);c2 dx) = 1-2k, (41)
IIN(SDO5(0);SUP20(1);c3 dx) = 1. (42)
In order to compute the polarization curve, some algebraic simplifications result if the solutions of Eqs.(34) - (35) are written in the following form:
c2 = c2(0)eSUP8(-BBC[(f(x)-f(0))), (43)
c3 = Nc3(0)eSUP8(BBC[(f(x)-f(0))), (44)
where c2(0) and N are two constants of integration. As in ref. [1], it is found expedient to express these constants in terms of the potential drop in solution Df = f(1) - f(0).
Combining Eqs.(33) - (36) we have
f(dx;df) = F(1;j) (3c3 - c2). (45)
Substituting of this equation into normalization condition (42) gives the following relation between c2(0), N and y = eSup8(2Df):
F(Nc2(0)2;2j) BBC[(3N (y-1) - lny ) = 1. (46)
A second relation between c2(0), N and y follows from the second conservation condition (41)
F(c2(0)2;2j) BBC[(3N lny + y- 1- 1) = 1 - 2k, (47)
Combination of reflations (46) and (47) gives N(y):
N = F(SUP8((2 - k)lny +R((2 - k)2ln2y + 3(y - 1)(y - 1- 1)(1 - 2k )));SDO15 (3(y - 1)(1 - 2k ))) (48)
Taking use the identity
IIN(SDO5(f(1));SUP50(f(0));f(dx;df) df) = 1, (49)
and Eqs.(43) - (44), one can express c2(0) and j in terms of y
c2(0) = F(SUP8(2(3N - y SUP8(- F(1;2)))(y SUP8(F(1;2)) - 1));SDO15 (N BBC[(3N (y - 1) - lny ) )), (50)
j = F(SUP8(2 BBC[(3N (y SUP8(F(1;2)) - 1) + y SUP8(- F(1;2)) - 1))SUP28( 2);SDO15(N BBC[(3N (y - 1) - lny ))), (51)
Combination of these expressions with the electroneutrality condition (36) and formulas (43) - (44), leads to the following expressions for c1(0) and c1(1) in terms of y:
c1(0) = F(1;2)(N - 1)c2(0), (52)
c1(1) = F(1;2) SDO4(BBC((N y SUP8(F(1;2)) - y SUP8(- F(1;2))))c2(0). (53)
Formulas (48) and (50) - (53) and expressions
j = j0BBC{(F(c1(0);k ) eSUP8(2(1-a)f(0)) - eSUP8(- 2af(0)) ), (54)
j =j0BBC{(eSUP8(- 2a(f(0) + R(y) - v)) - F(c1(1);k ) eSUP8(2(1-a)(f(0) + R(y) - v)) ), (55)
which follow directly from expressions (38) - (39) and the definition of y, determine the current - voltage characteristic j(v) of the cell.
Equating the right hand sides of expressions (54) and (55) one finds that
eSUP8(2f(0)) = F(SUP8(1 + eSUP8( 2a(v - R(y))));SDO25( F(c1(1);k ) eSUP8(2(1-a)(R(y) - v)) + F(c1(0);k ))). (56)
Substitution of this expression for eSUP8(2f(0)) into expression (54) leads to an implicit relation between v, j and y. For each value of j, this relation permits numerical computation of v as function of y. Together with formula (51), which determines j(y) this gives the polarization curve j(v) in parametric form for arbitrary values of the exchange current density j0 and the transfer coefficient a .
Numerical simulation and discussion
Some results of numerical calculations of polarization curves are presented in Fig.1 - 3. Fig.1 represents a set of curves for a binary electrolyte, calculated according to Eqs. (51), (54) and (56) and different values of a . One can see that the curves corresponding to values of a that are different from 0.5 are shifted towards higher v values with respect to the curve corresponding to a equal to 0.5.
The extremum property of the value a = 0.5 can be illustrated by the dependence of the value of the potential v1/2, which corresponds to the current density equal to one half of the limiting current density, on the values of parameter a. This relation is shown in Fig.2. The curve v1/2(a ) is practically symmetrical with respect to a = 0.5 in good agreement with theoretical analysis. It should be stressed that such symmetry appears only for small values of exchange current densities.
Fig.3 shows a set of polarization curves with the exchange current density equal to 0.1 and different values of transfer coefficient a . For a = 0.5 polarization curve has only one inflection point corresponding to the transition into the current region, where the diffusion limitation of electrode reaction becomes important. When the transfer coefficient a increases, a second inflection point appears on the polarization curves at relatively small current densities. This inflection point is related to the transition from region of linear "ohmic - like" behaviour of polarization curves at small reaction overvoltages, which is independent of a , to the exponential "Tafel - like" region of the curve, which depends strongly on a. The inflection point of this type exists on the polarization curve only for relatively small exchange current densities j0. At very small values of this parameter the position of this inflection point of the current - voltage curve corresponds to very low current density values.
For a binary electrolyte it should be noted, that in some special cases such as a = 2/3, a = 1/3 as well as in the case of a = 1/2, which was considered in [1], the polarisation curve can be found from expressions (18) - (19) in analytic form. Eq.(18) can be easily solved with respect to j:
j = f(SUP10(eSUP8((1-a)z1f(0)) - eSUP8(- az1f(0)));SDO25(f(1;j0) + f(1;jl ) eSUP8((1-a)z1f(0)))). (57)
Introducing notations
S = exp BBC[(z1 BBC((f(0) + f(1;z2) ln f(jl + j;jl - j) )), (58)
p = - f(j;j0) S S UP10(a), q = - BBC((1 + f(j;jl)) S, (59)
one finds after some algebra, that relation (19) for a = 2/3, can be reduced to the form
t3 + pt + q = 0, (60)
where t = eSUP8((z1v / 3)) > 0. The solution of Eq.(60) is,see e.g. [17],
t1 =R(3;- f(q ;2)+R(BBC(f(q ;2))2+ BBC(f(p ;3))3)) +R(3;- f(q ;2) - R(BBC(f(q ;2))2+ BBC(f(p ;3))3)), (61)
Taking f(0) to be an independent parameter one can find the parametric analytic representation of the function j(v) from formulas(57) and (61), the relation v = f(3;z1) ln t1 and the definitions of S, p and q.
In a similar way the parametric form of current - voltage curve j(v) can be found for case of a = 1/3.Some examples of analytical treatment of electrode reaction kinetics for fractional values of a were considered in [18].
Conclusion
An exact solution of the system of Nernst-Planck equations for a 1-dimensional electrodiffusion problem, with boundary conditions of Butler - Volmer type shows that the polarization curves for a cell having two identical electrodes have some interesting general properties. At small values of the exchange current density and current densities that are not close to the limiting current density jl,the polarization curve is of a form that is similar to the Tafel equation. One obtains that
j j0exp(aeffz1v)
with the effective transfer coefficient aeff related to the physical transfer coefficient a by the formula
aeff = a(1 - a).
Another general property of the polarization curves for the system under consideration is that, under certain conditions, two inflection points may appear. One of these is similar to that of the current - voltage curve for the reaction at one electrode, which is described by a Butler - Volmer law for the case of small exchange current density j0 and current densities j , such that j0 << j O(~;<)jl. The second inflection point can be observed on the j(v) curve in the region of relatively high values of v, where the current has values of the order of the limiting current density.
Acknowledgment
The financial support from the Royal Swedish Academy of Sciences for the participation of Yu.I.K. and A.V.S. in the present work is gratefully acknowledged. One of the authors (Yu.I.K.) would like to acknowledge the Department of Hydromechanics of the Royal Institute of Technology, Stockholm, where this work has been done, for financial support and hospitality as well as partial support from the Fund of Fundamental Researches of Russian Academy of Sciences.
References
1. A.V.Sokirko and F.H.Bark, (Submitted to Electrochem. Acta)
2. J.S. Newman, Electrochemical Systems, 2:nd ed. (1991),
Prentice-Hall Inc.
3. V.G. Levich, Physicochemical Hydrodynamics (1962),
Prentice-Hall Inc.
4. K. Vetter, Electrochemische Kinetik (1961), Springer-Verlag.
5. A.J. Bard and L.R. Faulkner, Electrochemical Methods, Fundamentals and Applications (1980), John Wiley & Sons, Inc.
6. B.B. Damaskin and O.A.Petrii. Introduction to Electrochemical Kinetics. (In Russian) Vysshaya Shkola, 1975.
7. M.D. Pritzker, J. electroanal. Chem.296, 1 (1990).
8. K.B.Oldham, J.electroanal.Chem., 337, 91 (1992).
9. J.C.Myland and K.B.Oldham, J.electroanal.Chem., 347, 49 (1993).
10.Yu.Ya.Gurevich, Yu.I.Kharkats, Elektrokhimiya 15, 94 (1979)
11.A.Milchev, J.electroanal.Chem.,312, 267 (1990)
12.A.Milchev, Electrochimica Acta,37, 2229 (1992)
13. D.R.Baker and M.W.Verbrugge , J.Electrochem. Soc.,137, 1832 (1990)
14. D.R.Baker and M.W.Verbrugge , J.Electrochem. Soc.,137, 3836 (1990)
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16.D.R.Baker,SIAM J.Appl.Math.,53, 419 (1993)
17. G. Korn.and T.Korn, Mathematical Handbook, McGrow-Hill,1961.
18.F.A.Posey,J.Electrochemical Soc., 111, 1173 (1964).
Captions to the figures
Fig. 1. Polarization curves j(v) determined by Eqs. (51), (54) and (56) corresponding to different a values. k = 0.4; jl = 6; jS(0;1) = 0.001; 1 - a = 0.5; 2 - a = 0.4; 3 - a = 0.6; 4 - a = 0.3; 5 - a = 0.7.
Fig. 2. Dependence of the potential v1/2 value corresponding to current density j = jl/2 on the transfer coefficient a for binary solution. jS(0;1) = 0.001; jl = 6.
Fig. 3. Polarization curves j(v) determined by Eqs. (20) and (21) corresponding to different a values. jl = 1; jS(0;1) = 0.01; 1 - a = 0.5; 2 - a = 0.6; 3 - a = 0.7; 4 - a = 0.8; 5 - a = 0.9; 6 - a = 0.95.
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