5530
J . Phys. Chem. 1985,89, 5530-5536 a narrow temperature range (ACfim = 0) and the effect of the volume expansivity of the mobile phase is negligible. For this case, solute retention is dominated by the partial molar enthalpy of The volume expansivity solute transfer at infinite dilution and ACpmbecome dominant near the critical temperature of the solvent, contributing to the physical and chemical phenomenon controlling retention in the SFC regime. In deriving this relationship we have made the simplifying assumption that ARim( To) is the same in both the high-temperature (gas chromatography) and low-temperature (liquid chromatography) regions. While this assumption is almost certainly incorrect, and will be addressed in future studies, it provides a basis for treatment of the intermediate supercritical fluid regime. Further refinements in the model which are being undertaken include using the Peng-Robinson EOS instead of the RedlichKwong EOS for evaluation of the partial derivative in the volume expansivity, because of its greater accuracy near the critical point. Deiters and SchneideP have added an adjustable parameter into the Peng-Robinson EOS to account for large size disparities between solvent and solute molecules. Alternatively Kurnik et aLZ9have shown that a single adjustable binary interaction parameter can be added to the Peng-Robinson EOS to account for specific interactions between the solute and solvent, resulting in a relation that adequately models the solubility of a solid in a fluid. Both of these parameters would impact the volume expansivity in eq 15 and better describe the experimental data. A mathematical description of the heat capacity of the fluid near its critical point is being undertaken and will be discussed in a future work. These changes and further investigations into the variation of AHiover the relevant ranges of temperature and pressure are in progress and will serve to guide future experimental and theoretical developments.
(aim).
-2,6h jJ
-3.4
-4.21
0
'
I
20
w
'
, I 1 I 40 60 80 Temperature, OC
I
I
100
1
I
120
Figure 8. Experimental data for hexadecane on OV-17 with C02, 76.5 = -10.6 kcal/mol, a = 0.90. atm, and theoretical model (solid line) ai"
decane; modeling was undertaken using a ARim(To)value of -10.6 kcal/mol (see Table I) obtained from the Van? Hoff plot of heptadecane on OV-17. These data are plotted in Figure 8, with a fit parameter value for hexadecane of a = 0.90. Once again the fit of the experimental data by the model is quite good.
Conclusion The thermodynamic relationship developed in this work has been shown to describe the features of solute retention as a function of temperature at constant pressure quite well for conditions which include gas, liquid, and supercritical fluid chromatography. The dependence of retention upon temperature can apparently be ascribed to a combination of two effects. These effects include the rapid change in the number of intermolecular interactions of the solutesolvent molecules as one progresses through the critical p i n t for the solvent (ACpm)and on the volume expansivity of the solvent. The right-hand side of eq 15 reduces to the limiting case for is a constant over liquid and gas chromatography, where Mim
Acknowledgment. The authors acknowledge the financial support of the U S . Department of Energy, Office of Basic Energy Science, under Contract DE-AC06-76RLO-1830. (28) Deiters, U.; Schneider, G . M . Ber. Bunsenges. Phys. Chem. 1976, 12, 1316. (29) Kurnik, R. T.; Holla, S. J.; Reid, R. C. J . Chem. Eng. Dara 1981, 26, 47.
Electrochemical Behavior of Dispersions of Spherical Ultramicroelectrodes. 1. Theoretical Considerations Martin Fleischmann,+Jamal Ghoroghchian,*and Stanley Pons* Department of Chemistry, University of Southampton, Southampton SO9 5NH, England, and Department of Chemistry, University of Utah, Salt Lake City, Utah 841 1 2 (Received: April 8, 1985: In Final Form: July 30, 1985)
The bipolar electrolysis of suspensions of spherical ultramicroelectrodes is discussed. It is shown that the reactions at the surface will be rate controlling over a wide range of conditions in view of the high rates of mass transfer to the electrodes. The effects of diffusion can be taken into account in a straightforward manner in view of the absence of discontinuities in the spherical coordinate system. Bipolar electrolyses on ultramicroelectrodes can also be used for kinetic measurements and synthesis in solutions containing no deliberately added support electrolyte. Estimates are made of the effects of the coupling of diffusion and migration; exact predictions of the asymmetric polarization of the particles under these conditions will require numerical analysis.
Introduction The construction and behavior of microdisk (see, e.g., ref 1-19), m i c r ~ r i n g , ' and ~ - ~microsphere (see,e.g., ref 19,21-23) electrodes have been discussed extensively. The development of spherical diffusion fields in the bulk of the solution surrounding these University of Southampton. 'University of Utah.
0022-3654/85/2089-5530$01.50/0
electrodes leads to high steady-state rates of mass transfer to the electrode surfaces so that measurements can be made on fast (1) Z . G. Soos and P. J. Lingane, J . Phys. Chem., 68, 3821 (1964). (2) J.-L. Ponchon. K. Cesuuelio. F. Gunon. M . Jouvet. and J.-F. Puiol. Anal.' Chem., 51, 1483 (1979'). (3) M . A. Dayton, A. G. Ewing, and R. M . Wightman, A n d . Chem., 52, 2392 (1980).
0 1985 American Chemical Society
The Journal of Physical Chemistry, Vol. 89. No. 25, 1985 5531
Spherical Ultramicroelectrodes
I
-ve
s
Fender
Figure 2. Representation of the effect of the shunt resistances due to bipolar particles on the resistance between the anode and cathode feeder electrodes;R, and R, are the equivalent cathode and anode resistances of a bipolar particle, & the resistance of a volume element of the solution.
-c
.L
m
a 0 a
W
w
The modest scale-up of reactions at microelectrodes requires the use of special structures such as of embedded reticulated foams25 or fibers.26 Exploitation of the special advantages of microelectrodes for synthesis (e.g., the ease of work-up and the extension of the solvent range in the absence of support electrolyte) however requires specialized electrode and cell design such as the use of three-dimensional electrodes. Here we report on one such design, bipolar electrolysis on dispersions of spherical particles. These systems represent an extension of electrolysis in bipolar fluidized bed electrodes27to the area of ultramicroelectrodes. The mathematical description of the systems is developed for a number of limiting conditions; applications to synthesis will be described elsewhere.28
7-ve
Feeder
Figure 1. A representative particle between the anode and cathode feeder electrodes and the coordinate system for a < 0.5.
electrode reactions in steady state or quasi-steady-state conditions;5J9-21+22 the kinetics of fast reactions in solution coupled to the electrode reactions can be explored in novel ways.i4.15Equally, the spherical potential field leads to a decrease of charging times so that transient measurement techniques are simplified (the time constants are proportional to the radius of the electrode); cyclic voltammetric measurements have been reported a t sweep rates as high as lo5 V s-l.13 The low Ohmic losses in solution have also allowed measurements to be made in solutions containing only low concentrations of support electrolytei6 as well as in glasses at low temperature;” the range of potentials of solvents such as acetonitrile accessible to measurement is greatly increased in the absence of deliberately added support electrolyte so that for example inert substrates such as CHI, 02,N2,Ar, Kr, and Xe can be activated by anodic o x i d a t i ~ n . ~ ~ , ~ ~
(4) M. Kakihama, H. Ikeuchi, G. P. Sato, and K. Tokuda, J . Electroanul. Chem., 108, 381 (1980). ( 5 ) R. M. Wightman, Anal. Chem., 53, 1125A (1981). (6) K. B. Oldham, J. Electroanal. Chem., 122, 1 (1981). (7) K. A o k i and J. Osteryoung, J. Electroanal. Chem., 122, 19 (1981). (8) J. Heinze, J. Electroanal. Chem., 124, 73 (1981). (9) K. A o k i and J. Osteryoung, J . Electroanal. Chem., 125, 315 (1981). (10) B. Scharifker and G. J. Hills, J. Electround. Chem., 130,81 (1981). (11) D. Shoup and A. Szabo, J . Electroanal. Chem., 140, 237 (1982). (12) T. Hepel, W. Plot, and J. Osteryoung, J. Phys. Chem., 87, 1278 (1983). (13) J. 0. Howell and R. M. Wightman, Anal. Chem., 56, 524 (1984). (14) M. Fleischmann, F. Lasserre, J. Robinson, and D. Swan, J . Electroanal. Chem., 177, 97 (1984). (15) M. Fleischmann, F. Lasserre, and J. Robinson, J. Elecfroanal. Chem., 177, 115 (1984). (16) A. M. Bond, M. Fleischmann, and J. Robinson, J . Electround.
Chem., in press. (17) A. M. Bond, M. Fleischmann, and J. Robinson, J . Electround. Chem., in press. (18) J. Cassidy, S. B. Khoo, S. Pons, and M. Fleischmann, J. Phys. Chem., in press. ( 1 9) A. M. Bond, M. Fleischmann, and J. Robinson,“Extended Abstract”, 165th Meeting of the Electrochemical Society, May 1984, p 523, Electrochemical Society: Pennington, NJ. A. M. Bond, M. Fleischmann, S.B. Khoo, S.Pons, and J. Robinson, submitted for publication. (20) M. Fleischmann, S. Bandyopadhyay, and S. Pons, J . Phys. Chem., in press. (21) P. Bindra, A. P. Brown, M. Fleischmann, and D. Pletcher, J . Electroanal. Chem., 58, 31 (1975). (22) P. Bindra, A. P. Brown, M. Fleischmann, and D. Pletcher, J . Elecfroanal. Chem., 58, 39 (1975). (23) A. M. Bond, M. Fleischmann, and J. Robinson, J. Elecfroanal. Chem., in press.
Mathematical Description of Dispersion Electrolysis Dispersion electrolysis will normally be carried out using low or zero concentrations of support electrolyte. For example, the generation of a cation from an uncharged substrate (A)
O++e*R or the converse case of the generation of an anion
(B)
O+e*R-
The charge density generated in these reactions will be compensated by counterions either from deliberately added low concentrations of support electrolyte or from impurities or ions generated by autbionization of the solvent. It has been shown that the behavior of such systems can be discussed as examples of “two-ion cases”; the more common “three-ion cases” reduce to the two-ion case under the likely experimental conditions.23 In the discussion presented here we take into account a progressively wider range of phenomena. Control by the Interfacial Reactions. We consider the behavior of a representative test particle between plane parallel feeder electrodes, Figure 1. For the likely range of particle radii, a, cm < a < cm the steady-state mass-transfer coefficient, k ,
k, = D / a
(1)
is so high that the surface reaction is likely to be rate controlling at low and intermediate overpotentials. We assume that the reaction on the cathodic and anodic parts of the particle does not significantly perturb the equipotential planes so that the overpotential at the position on the surface (a$) is given by 9
= -A$s = a(-V,,,,/L)(cos
I9
+ cos A)
(2)
where V, is the applied voltage and L the distance between the feeder electrodes (see Figure 1 for coordinate system). Then the current density at (a$) in the absence of concentration changes in the solution is given for a single redox couple by i = io(exp(ay cos A) exp(ay cos 0) exp((a - 1)y cos A) exp((cu - l ) y cos 6 ) ) (3) (24) J. Ghoroghchian, S. Bandyopadhyay, J. Cassidy, S. Pons, and M. Fleischmann, to be published. (25) N. Seeszynski, J. Osteryoung, and M. Carter, Anal. Chem., 56, 130 (1984). (26) D. Shoup and A. Szabo, J . Electroanal. Chem., 160, 19 (1984). (27) (a) M. Fleischmann, F. Goodridge, and C. J. H. King, Brit. Pat. Appl. 16765, 1974. (b) F. Goodridge, C. J. H. King, and A. B. Wright, Electrochim. Acta, 22, 1087 (1977). (28) J. Ghoroghchian, S. Pons, and M. Flesichmann, to be published.
5532 The Journal of Physical Chemistry, Vol. 89, No. 25, 1985
Fleischmann et al.
where FVappla
Y=RTL
(4)
1
r2 sin 8
I = 2aa2L"-'i sin 0 d0
-
where the division between the cathodic and anodic areas is defined by the equality of the net cathodic and anodic currents: ( a - 1) exp(ay cos X) sinh (cry) = a exp((a - l)y cos X) sinh ( ( a - 1 ) y ) ( 6 )
= 0 (16)
With we obtain
At small polarizations eq 5 reduces to rioFVappla3 (1 RTL
a2ci ax2
If the gap between the feeder electrodes is small compared to the width and length of the electrodes then both the reaction at the surface of the particles and mass transfer will be independent of X and
2aioRTLa ( ( a - 1) exp(ay cos A) exp(ay) FVappla(a - 1) a exp((a - 1)y cos A) exp((a - 1 ) y ) + 1) ( 5 )
I==
-1
-
The current into the cathodic area of the particle is
+ cos X ) 2
(7)
For the special case
The solution has the form w
m
+ n=O
Ci = CAnr"Pn(p) xBnr-@+I)Pn(p)
we obtain
n=O
(9) and, at small polarizations, T I;=
~
~
F
V
~
~
~
~
(20)
where P,(p) denotes the Legendre polynomial and the coefficients An and Bn are derived from the appropriate boundary conditions. Here we restrict attention to simple examples, namely irreversible and reversible reactions close to equilibrium and reversible reactions far from equilibrium. For a redox process such as (A) ~ ~ or (B)
RTL
In the dispersion, each bipolar particle presents a shunt resistance to the overall solution resistance, Figure 2. If we consider a slice of thickness 2a through the dispersion, the total shunt current at low particle concentrations, N , is
CI=
2aN
4aioRTLa2N ( ( a - 1) exp(ay cos A) exp(ay) FVappla(a - 1) a exp((a - 1)y cos X) exp((a - 1)y) + 1) (11)
which is produced by the voltage 2aVappl/Lapplied to the slice. The shunt resistance of the dispersion in the slice therefore is R , = F p a p p , a ( a- l)/27rioRTL2Nal(a- 1 ) exp(ay cos X) X exp(ay) - a exp((a - 1 ) y cos A) exp((a - 1)y) I } ( 1 2 )
+
The total resistance of the slice, R, is given by
where the solution resistance R , is determined by the solution resistivity p s . The total observed current is I, =
For a reversible reaction
where Coo and CRaare the concentrations at the surface. Equations 23 and 24 show that
2a V*ppl LR
-
vappl 4aioRTLNa2
=-
PJ
+
( ( a - 1) exp(ay cos X) exp(ay) FVappla(a - 1 ) a exp((a - 1)y cos A) exp((a - 1)y) + 11 ( 1 4 )
For the special case (8) and at low polarizations
- - V+PSL3 p p ~
IT -
cow, (A,,o) = 0
(27)
Go = CR", (G"+o) = 0
(28)
A0
=
With the assumption
Do = DR = D eq 25 gives B, = -Hn
R TL
The Effect of Concentration Changes in the Solution. The distribution of potential over the surface of the particle, Figure 1 , leads to changes in the concentration with position at high reaction rates. The concentration in the steady state of a given species i is governed by
(29)
The Journal of Physical Chemistry, Vol. 89, No. 25, 1985 5533
Spherical Ultramicroelectrodes
For kinetic control at the surface we replace eq 26 by
Equations 31 and 32 with 26 and 4 give
The coefficients B, can therefore be determined by using the orthogonality of the Legendre polynomials: B, =
For sufficiently small perturbations, neglecting second-order small quantities and setting Co- = CRm For small cathodic polarizations
x:
B, = --(2n + ) a("+l)COm lypP,(p) dp 2 Bo = B2 = BS = ... = 0
giving
B, = -7a2COm/2
so that As in the case of reversible behavior, the only nonzero term is
or
c, = co-( 1 -
$)
(47)
The total rate is
giving a current
waDioyaCo"
I=
For the case of low polarizations eq 37 gives
I = rDFyaCo"
(41)
As the parameter y gives the maximum concentration change, AC, which we observe on the particle at the point (a,O) 7=
c o o - C R ~ 2AC =5-
COa
c0-
(42)
we obtain
I = 2rDFaAC
(43)
This current is the same as that which is observed for the uniform flux to a hemisphere with a concentration shift AC from the bulk concentration. It is evident therefore that mass transfer is considerably enhanced for bipolar operation of the spherical particles: if AC in eq 43 is replaced by the mean concentration over the hemisphere, AC/2, then the mass transfer can be seen to be double that for the unipolar polarization of the particle. At higher polarizations the integrals for eq 33 become very complicated and are best evaluated numerically. Figure 3 shows that with increasing polarization an increasing number of terms must be taken into account in the Legendre polynomial expansion. The concentrations reach their limiting values over most of the surface and show a rapid transition in the region of the stagnation plane (r,r/2). The apparent mass transfer is enhanced by a factor = 3.22 compared to that observed for unipolar operation of the hemispheres. A sharp transition between zones at which the anodic and cathodic reactions are at their limiting rates has been assumed previously in the modeling of bipolar packed bed electrode^.^^^^^ (29)M.Fleischmann, J. W. Oldfield, and C. L.K.Temakoon, Ins?.Chem. Eng. Symp. Ser., 37, 1.53 (1971).
For io/F >> DCom/a eq 48 reduces to eq 41. The opposite limit DCo-/a >> io/F gives
I = ra2yio =
rioFVappla3
RTL
The kinetically controlled current is half of that observed for uniform polarization of a hemisphere because of the distribution of potential over the surface. The Coupling of Diffusion and Migration. In the discussion so far we have assumed that the reactions on the bipolar particles are driven by the mean field between the feeder electrodes, the rate being controlled by the heterogeneous reactions, by diffusion, or by reaction and diffusion. However, in many cases, such as the generation of charged species from uncharged substrates, the reactions will be partly controlled by migration. We next examine two limiting cases of rate control by the Ohmic potential drop in the solution and then the coupling of the surface reactions with diffusion and migration. Ohmic Control of the Reactions. The voltage in the solution around a spherical electrode is determined from
provided the A-dependence can be neglected (cf. eq 16 and 17). The voltage will therefore be given by
In the simplest model we assume no polarization at the particle, i.e. V=O, r = a
(51)
5534
Fleischmann et al.
The Journal of Physical Chemistry, Vol. 89, No. 25, 1985 GAMMA = 0.1
-Er cos 8 = -Erp =
, "a
(55)
I .I_
1.04
-
1.01
-
1.05
1.02 1.01
since all the terms in the second series are zero. The only nonzero term in eq 5 5 is for n = 1 giving
-
A , = -E,
B , = Ea3
(56)
Ea3 + -p r2
(57)
Thus
1 -
0.90 0.88
V(r,8) = -Erp
0.98 0.97
0.85
The field at the surface of the particle is
,
0.94-.,
-
-
~,
0.4
0
-
PO-1
, , ,
,
1.2
1.6
0.0 p0-2
,
,
,
TL'W-YS
,
,
2
2.4
,
, 2.0
- PO-4
3.2 ~
- en -L
and the total current is 67ra2 --EJ
GAMMA = 1.0
0
pdp=
Ps
The total current is therefore
provided the currents into all the particles are additive, the field at infinity being reached in a distance = N-'I3: in effect the next layer of particles screens the potential due to the bipolar particle. The maximum screening effect would be observed if the particle were surrounded by an equipotential at the potential
Then the solution of d2V - + -2- d=Vo dr2 r dr
GAMMA = 10.0 (To 6th ordw appmxlmatlon)
2.1
gives for the total current
2.4
\\'\ '\
-._
-0.4
-0.1, 0
-Po
.
\ \'
~
, PO-I
,
, , ,
I
0.4
0.0
1.2
ra-.
,
,
,
1.0
, , 2
THETA
, , ,
, 2.4
2.0
3.2 PO--8
po - 6
Figure 3. Effect of y (= F V,,,a/RTL) on the normalized concentration distribution 1 f AC/C over the surface of the particles for a reversible system. A y = 0.1; B: y = 1.0; C : y = IO. The figures show the effect of the inclusion of successively higher terms in the Legendre polynomial
expansion. which is analogous to primary current distribution in electroplating. Furthermore we assume in this model that the resistivity of the solution is constant (presence of excess support electrolyte) ps
= constant
(52)
Equations 50 with 49 and the orthogonality of P n ( p ) gives B, = - A , ~ ( Z ~ + I )
(53)
-
u& av u+C* av -A -- ( 6 4 ) r
a8
rsin8 ah
The divergence of the fluxes (eq 6 4 ) in the steady state give
At distances far from the particle we have V = AEr cos 8 , r
The current given by eq 63 is within an order of magnitude of that given by eq 60 for the likely values of N, screening by a layer of particles at the position z = b would lie between the two limits in eq 60 and 63. Both equations show that the addition of particles leads to only a small increase in the current Vappl/psLobserved in the absence of the particles. The large increases experimentally observed30are therefore not controlled by resistive effects in the solution but by the combined effects of the surface reactions and diffusion (see above) or by the surface reactions and diffusion coupled to migration. Diffusion and Migration. We consider only the simple case of the generation of charged species such as 0' from neutral substrates (e.g., see eq A or B) the charged species regenerating the neutral substrate at the opposite side of the bipolar particle (no deliberately added support electrolyte). More complex reactions will be discussed e l ~ e w h e r e . ~ ~ The positive and negative ion fluxes at a given point on a single spherical particle are given by
(54)
the sign depending on which side of the particle we are considering; E is the uniform field at infinity. We take
(30) J. Ghoroghchian, S. Pons, and M . Fleischmann, to be published. (31) J. Ghoroghchian, S . Pons, and M. Fleischmann, to be published. (32) J. S. Newman and C. W. Tobias, J . Electrochem. SOC.,109, 1883 (1962).
Spherical Ultramicroelectrodes
D*
a2ct +--+-2 0 + ac, ar2 r ar
D* azc, -? sin2 8 a X2
f
The Journal of Physical Chemistry, Vol. 89, No. 25, 1985 5535 dC0 dr
i = F(N+ - N-) = -F(D+ - D-) - - FCo(u+ - u-)
r2 sin 0 a 0
a2v ac, a v f 2uci a v f u+c* - f U& - - -ar2
dr
rr
r
ar
dV dr (75)
dV/dr derived from eq 75 gives with N+ for spherical symmetry (see eq 64)
where The general case requires the solution of the two equations (65) together with that for the uncharged species (eq 17) and of Poisson's equation in the presence of current flow (compare eq 23); the boundary conditions include Gauss's equation at the particle-solution interface. Here we restrict attention to electroneutrality
c+= c- = c,
(66)
D+ = D- = Do
(67)
t+ =
u+ u+ + u-
D'=
2D+DD+ - D-
(77)
is the binary diffusion coefficient. A boundary condition a t r = a is derived from eq 26
Addition of the set (eq 65) gives where DR* is defined by eq 77. The second boundary condition is given by the reaction at the surface. For a reversible process we obtain the polarization curve on a hemisphere and, for the case of no X-dependence, subtraction of the set gives co
a2v ac0 av
2Co
av
-+ -- + -- + a- . 4
dr ar
r
L{
rz sin O Co &(sin
0
z) dr
+
where we also assume that the Nernst-Einstein relations apply: RT D* = TU' (70) Equations 68 and 69 have to be solved simultaneously. A general solution of the form (eq 20) used in eq 68 gives rise to a differential equation containing complex (r,e) dependences which must be solved n~merically.~'This applies even to simple situations such as polarization at high potentials where the concentration becomes effectively constant or zero over most of the surface, Figure 3. Here we can write
C = c" - c"a/r a2V ar2
dV
- + 2r - +ar
r3
a ( r - a) dr
,(cos 1
8
&(sin
0
5)]
= 0 (72)
The general case will be discussed elsewhere.31 Here we note that the shielding of a given particle at distances of the order r = b = N-'I3 will lead to an approximately spherical distribution of potential over substantial parts of the particle surface. The case of coupled diffusion and migration at spherical electrodes has been discussed previously23and can be used as an indication of maximum effects likely to be observed at the essentially hemispherical anode and cathode areas. For spherical symmetry eq 68 becomes (73) which must be solved together with the equation governing the diffusion of R
-d2cR + - - = o2 dCR dr2
= JbaF?(u+
ia2 d r + UJCO
for a
