J. Phys. Chem. 1993,97, 5019-5023
5019
Diffusion Coupling in Electrolyte Solutions. 2. Pseudo-Oscillations Induced by a pH Gradient Jean-Pierre Simonllq'qt Pierre Turq,' and Antonio Caladot Laboratoire d'Electrochimie. UniversitC Pierre et Marie Curie, Biftiment F, Tour 74, 7dme Stage, 8 rue Cuvier, 75005 Paris, France, and CECUL, C. Benito Rocha Cabral 14, 1200 Lisboa, Portugal Received: October 21, 1992; In Final Form: January 5, 1993
Using the closed capillary technique, we investigate diffusion coupling processes observed on ampholytic tracers under the action of a pH gradient in the region of the pK. Experimental results are compared to theoretical predictions based on numerical simulations on the one hand and on an approximate analytical treatment on the other hand. The latter is a new way of handling diffusion equations for a system with many species related by chemical reactions. The result is compared to previous results on the diffusion coupling of sulfate. The particular case of an amino acid (aspartic acid) is studied: it leads to an original pseudo-oscillatory behavior.
1. Introduction
In previous papers, we addressed the problem of the influence of ionic as~ociation~-~ in the Bjerrum or Manning sense on the transport of electrolytes and polyelectrolytes. One of the main results was that electrostatic associationdepends stronglyon ionic strength and is considerably reduced by adding a supporting electrolyte to the system. We also approached4 the problem of the influence of a gradient of pH on the transport of sulfate and phosphate ions. All experiments were performed with the convenient closed capillary technique.s*6 Sulfate and phosphate ions participate in several acid-base equilibria, but their ionic charge is always of the same (negative) sign. Therefore, when subjected to the internal electrical field generated insidethe solutionby the movement of some electrolyte,' each form moves in the same direction, and the transient concentration changes are of the same type as for a simple ion like chloride.* The results of our previous paper4 show that the behavior of these species is well described, near equilibrium, by the introduction of a hypothetical third species whose physical parameters (charge and mobility) simply are the mean values of the various forms. The same result was found for the case of ion pairing even when the ion pair, such as MgS040, is not charged. Moreover, satisfactory agreement was found with experiments with sulfate, starting from far-from-equilibrium conditions. These remarks led us to pose the following problem: what happens if we use a specieswhich can exist under forms of opposite ionic charge? Is it possible to define effective physical parameters as for sulfate and phosphate? And how does such a system react to the action of a pH gradient chosen in such a way that, at time zero, the species is under the positive form in one solution and under the negative one in the other solution? A peculiar behavior is expected because, at the beginning of the experiment, the internal electrical field (created mainly by the rapid motion of the hydronium ion) should create opposite forces, directed away from the interface, on the two forms of the tracer. However, after some short time, of the order of the diffusion time of the hydronium ion through the system (a few hours here, typically), the effect should be reversed under the influence of the diffusive forces on the tracer. For this purpose we chose to use an ampholyte species, I4Clabeled L-aspartic acid, in a pH gradient created by two buffers placed in a closed Capillary. This molecule, whose chemical formula is HOOCCHzCH(NHz)COOH, has three acid-base groups, for which the p 6 s are9 pKi = 2.10, pK2 = 3.86,and pK3 = 9.82.
* Author to whom correspondence should be addressed. +
Universitd Pierre et Marie Curie. CECUL.
The extreme pH values, considered here, lead us to take into account only the species AspH3+, AspHzO, and AspH-, where AspHzorepresents aspartic acid in neutral or zwitterionic form. 2. Experimental Section
The diffusion technique used here is the closed capillary technique, but it has been modified for the present purpose. Usually, it consists of two cylindrical sections of plastic glued together, through which a bore of 1-mm diameter is drilled in the center of the cylinder, and only the bottom section (of length L = 1.5 cm) is made from scintillating plastic (Altustipe), whereas the top section is made of ordinary plastic of the same length L. In the present case, two types of capillary were built, in which only one fourth of the cylinder is of scintillating plastic. The justification for that is given in the last section. We recall that, in coupling diffusion processes, the larger the difference between the mobilities of the anion and cation which create the electrical field, the bigger the effect observed. This is the case of acidic solutions in which the proton H+ has the highest mobility (in acid solutions, bromide ion is the second fastest ion, and its self-diffusion coefficient is 4.5 times lower than that of the proton). To create the gradient of pH, we used preferably two buffers instead of an acid such as HC1, because this way yields gradients which are more stable in time. Two buffer solutions, SA and SB, containing the same concentration of labeled aspartic acid as a tracer, were prepared. The compositions of these buffers were chosen in such a way that the equilibrium pH of the equimolar mixture (SA SB)is close to the isoelectric point of the aspartic acid. The reason for doing so appears in the last section of this paper. The concentration of aspartic acid in the buffers was about 4 X le5M. The buffers SA and SBwere prepared from trichloroacetic acid (pK = 0.7) and acetic acid (pK = 4.8), respectively, by an addition of concentrated NaOH. The total concentration of each buffer was 0.1 M. The solution SA had a pH value of 1.2,1.45,or 1.55 and SBa pH value of 4.8 or 5.6, depending on the equilibrium pH wanted. Solutions of inactive compounds were prepared gravimetrically using Merck analytical reagent grade products and deionized water from a Millipore system (Milli-Q). The labeled aspartic acid was purchased from the Commissariat a 1'Energie Atomique (CEA). We checked that no adsorption of the aspartic acid occurs by filling a given capillary successivelywith both active solutions SA and SB.The recorded activity was observed to be stable with time, and no remaining activity was detected after the capillary had been washed with water. At the same time we verified that both solutions had the same activity and that the latter is not pH-dependent. This last point was also checked by measuring the activity of the equimolar solution (SA + SB).
0022-365419312097-5019$04.00/0 0 1993 American Chemical Society
+
Simonin et al.
5020 The Journal of Physical Chemistry, Vol. 97, No. 19, 1993
In a typical experiment, the bottom half-section of the capillary is filled with the solution S A , using a Pasteur pipet, and the correspondingactivity is measured. The solution Se is then added on top as carefully as possible, to avoid mixing. The tube is then positioned vertically in a scintillating counter (Packard), and the activity is recorded as a function of time. The temperature in the counting chamber is 28 f 1 OC.
3. Theoretical Treatments The frame of the theoretical treatments is the Fickian diffusion process, without departures from ideality. This approximation was made also in our previous papers,2v4q8which yielded a fair agreement with experiments. The first treatment is the numerical simulation, or finitedifference treatment (FD), of the system. The ingredients of the algorithm were given previously.2J Basically, they are the following: symmetrization of the system at both ends to write properly the differential operators in the first and last cell; zero electric field at the ends to ensure the conservation of matter; equilibrationof the concentrationsof thespecies which participate in chemical reactions, after each time step; varying time step chosen so that the concentrations of determined species do not vary, in absolute value, by more than a given (small, 0.1 was taken here) proportion at each time step. Each simulation was performed with a number of cells n = 300. The second mathematical treatment is based on the nearequilibrium linearization of the equations of transport. This technique, which we call normal-mode analysis (NM), has been extensively used in our g r o ~ p . ~ - ~Though * * J ~ the approximation can seem rough (stating that the deviation of the concentration of each constituent is small compared to the equilibrium value), the agreement with finite differences is generally satisfactory, and this method provides an analytical solution. In the present case, the constituents are numerous: the four constituentsof the buffers, A- and AH, B- and BH; the hydronium ion, H+; the sodium ion Na+; and the three forms of the tracer, T+, TO, T-. This makes nine constituents, which are too many to handle. Then, for the present system, an adaptation of the method is required. It is presented in the following section.
4. Reduction of the Number of Variables for the NM We suppose first that the motion of the tracer is governed by the major species. This reduces the system to six constituents. There are also two chemical reactions for the buffers. Thus, the system can be considered as composed of four chemicalindependent constituents. The electric neutrality requirement is an additional condition on the concentrations of the ions. It is expressed by the so-called Debye relaxation mode which characterizes the local very fast relaxation of charge fluctuations toward equilibrium (the characteristic time is typically of the order of 1 ns in normal conditions). As the four new variables, we choosethe concentrationsof the radicals A and B of the buffers, of the sodium ion, and of the total H, and we will note, for the sake of convenience
+ [AH] B = [B-] + [BH] H = [H'] + [AH] + [BH] A = [A-]
N = [Na']
capillary. The diffusion equation of a species i reads
a,ci- D,a,,ci + D ~ z , ~ , ( c ~=E )
(1) where we write for convenience a, alar and ax2 = a2/ax2and Zi, Di, Ci, and ui are the charge, the diffusion coefficient,the local concentration at time t , and the chemical reaction rate of the species i. E stands for the reduced local electric field: E = j3eEL, with /3 = 1/k& the usual Lagrange multiplier, and e, the charge of the proton. The relation UA- + U A H = 0 allows us to make the sum of the two equations referring to the buffer, which yields bi
DA-d,(aAAE)
0 (2)
in which ai is the proportion of species i at position x and time t . For instance (YA-
= [A-] / A
(3)
Then we have U A - + AH = 1. In the following we will note the proportion of the acid AH which is dissociated
&A,
(YA = (YA-
We introduce now the deviations of the quantities from the equilibrium values and write, for example
SX=X-F (4) We suppose the system is close to equilibrium and wish to find in which way it relaxes, that is, its relaxation modes. Linearizing eq 2 leads to cancelling second-order terms such as 6aAax26A. This leads to
- DAd,,A
+ AqADAa,,6aA - DA-a?Aegt3J
= 0 (5)
with DA = azDA_
+ ( 1 - az)DAH
the mean diffusion coefficient of A at equilibrium ADA = DAH - DA-
normally a positive quantity, and because the electric field E is a second-order quantity, as shown by the Poisson equation
a$ =rCz,scj i
withy = 41rj3ezlt (ebeingthedielectricpermittivityofthesolvent), and since, at equilibrium, the electroneutrality makes the field vanish. Let K be the equilibrium constant of one of the buffers, h = [H+], and a the degree of dissociation. Then, since a local chemical equilibrium can be assumed because the proton exchange is very fast compared to the time scale of the experimental process, we write a = K/(h which gives, at the first order
+K)
6a = -aq( 1 - aq)6h/hq
(7) (8)
The term 6h can be expressed as a function of the new variables in the following way. From the definition of H,we have
H = h + (1 - a A ) A + ( 1 -a@
(9)
and so it is easy to prove, using eqs 8 and 9, that Let us take first the example of buffer A. The transport of all constituents occurs at one dimension x , which is the axis of the
6h = r [ 6 H - ( l - a ? ) 6 A - ( l - a 2 ) 6 B ]
(10)
Diffusion Coupling in Electrolyte Solutions
The Journal of Physical Chemistry, Vol. 97, No. 19, 1993 SO21
with
cationic, zwitterionic, and anionic forms. Then
l/r= 1
+ a?(l-
a?)Aq/hq
+ @ ( l - @)Bes/hq
a+ = h2/(h2+ K , h
+ K,K2)
(21)
The Poisson equation can be written as
ap = y(6h + 6~ - SC,_- 6cB_)
(11)
+
And it follows from the definitions of A, B, and H that
h - CA.- CB.= H - A - B
a- = K I K 2 / ( h 2 K , h
(12)
+ K1K2)
(23)
and thus
so that the Poisson equation reads
+
= y(6H 6 N - 6 A - 6 B ) (13) Thus, combining eqs 5,10, and 13 we get the diffusionequation of the buffer A d,aA - Dad,26A
+ PA( 1 - ~ ~ i ~ ) d , 2-6P~d,26H B X A [ 6 H + 6N-6A-6BI
6a- = -a-(aO
= O (14)
with
+ 2a+)6h/hq
(26)
In the following we will write
6ai = ai6h/hq (27) for i = +, 0, -. The sum of the three equations identical to eq 1 applied to each form and making use of eqs 13 and 10 leads to, after linearizing
d,T- D$,2T
and
D a = DA - PA( 1 - a?)
+ pT(1-
with
The same equation as eq 14 holds for buffer B, by replacing A by B and vice versa. The equation for the quantity H is obtained by first writing the transport equations for C A HC, B Hand , h. a!cAH
- DAHa~2CAH = 'AH
+
a2)aX*A pdl- a?)d,,B ~1$,2H + &[6H + 6N - 6A - 6B] = 0 (28)
D,=
apqD, i=+,O,-
(15) DJT=
- DBHax2CBH
= 'BH
(16)
aiDi i=+,O,-
AT = y(a?D+ The latter equation assumes that the autoprotolysisofthewater can be neglected, which can be done here because both buffers are acid. Summing these three equations, and using the same procedure as above, we obtain
- a_"LD-)TP
the Dfs being the diffusion coefficients of the three forms. The case of sulfate as a tracer is treated similarly, taking
DT = ( 1 - aq)D,
+ aqD2
DJT= aq(l - a q ) ( D , - D2) AT
with the definition expressed by eq 7 and the index 1 referring to HS04- and index 2 to S042-. The combination of eqs 14, 18,20, and 28 can then be solved by the normal-mode analysis.IO First, these equations are Fourier and Laplace transformed, on space and time, respectively, which yields a linear system that can be written
where ax = a?( 1 - a 7 ) P / h q and
= yDH+hq The equation for the sodium ion is readily obtained XH
a,bN-DNd,,dN+ AN[6H+ 6 N - 6 A - 6 B l = O
= -[( 1 - aCg)D,+ aqD2] P
(20)
with AN = Y D N P DN = DNa+ Each of the three forms of the aspartic acid obeys diffusion equations identical to eq 1. Let us denote by T the total local concentration of the tracer and a+,(YO, a- the proportions of the
Wq,46X(q,s) = 6X(q,t=O) (29) in which Mis a square matrix of rank five, q and s are the Fourier and Laplace variables, SX(q,s) is the column vector whose componentsare the Fourier-Laplace transforms of the deviations of the concentrations, and bX(q,t=o) is the Fourier transform at t = 0. The latter can be computed from the initial profiles of the species at t = 0. One can show that, for the bottom half of the capillary
GA(x,t=O)
-bB(x,t=O) = C / 2
GN(x,t=O) = b H ( x , t = O ) = N(xL,t=O) - P The Debye relaxation mode is obtained by solving det[M(q=O,s)] = 0 which leads to SD
xi
=-
0
(30)
i=A,B,H,N
with the approximation that AT Y 0 (tracer concentration). The diffusive modes Dj (j = 1-4) are given by
20
t "(&
60
80
Figure 1. Sulfate ion: p as a function of time t ; computed curves for sulfate ion. Curve 1: normal-mode result compared to the finite differences (continuous line below). Curve 2: normal-mode result for zero electric field compared to the finite differences (line below). Broken line exp: experimental points.
det[M(q,s=-q2Dj)] = 0 (32) in the limit q 0. As always, one of the modes is simply the diffusion coefficient of the tracer. Let +
D4 = D, (33) The other three roots are the solutions of a cubic. Since each form of the tracer has thesame intrinsicactivity on thescintillating plastic, the observed quantity in an experiment is the relative mean variation of the total tracer concentration. Respectively for the first fourth and the second fourth of the tube, we define (34) for the first fourth of the capillary and (35) for the second fourth. Equation 29 can be solved following the method explained in ref 10. In particular, the function 6T(q,t) can be found by the methods of standard linear analysis. This requires the calculation of the determinant of a matrix of rank five (identical to M,except that the first column is replaced with the vector 6X(q,t=O)) and a partial fraction decomposition. This calculation was made with the help of the symbolic calculus device Maple. The final result is m
with k = 1,2. The prefactor g i is a complicated function of the parameters of the system (diffusion coefficients, concentrations, ...) yielded by the above calculation. Also
jz(n) = sin(u,) sin(u,/2)/ui u, = n ~ / 2
and 4,= u,/L In the case of sulfate, we used capillaries in which the bottom half was made of scintillating plastic, and in this case the observable is P
= I/Z(PI + P J
(37)
5. Results and Discussion First we examine previous results on the diffusion coupling of sulfate ion in a pH gradient4 in the light of the present normal-
V
.
0
10
20
t (hi?
40
50
Figure 2. Aspartic acid: P I as a function of time t . First buffer, pH 1.45; second buffer, pH 4.8. Line f.d.: finite-difference result. Dashed line n.m.: normal modes.
mode analysis. Figure 1 shows the results of the application of eq 37 to the initial condition: lower compartment, buffer at pH 2.1; upper compartment, buffer at pH 3.8; concentration of the buffers, C = 0.1 M. Fair agreement between the normal-mode analysis (NM) and the finite-differencetreatment (FD) is found. For times longer than 1 day both curves are identical. Also, both curves are close to the experimental points. The two lower curves refer to the case of zero electric field, obtained by an addition of excess salt, such as NaCI. The experiment had yielded points located on the baseline. In this case too, NM is above FD and has the same order of magnitude. Then, concerning aspartic acid, we present three experiments. The function P I ( ? ) for the first quarter of the capillary has been measured experimentally in the case pHs, 1.45 and pHs, 4.8, C = 0.1 M (concentration of both buffers) (Figure 2). For the calculations we used diffusion coefficients which were measured by electrophoresis of aspartic acid in buffers of appropriate pH, for the charged forms, and by self-diffusion measurements with aclosedcapillaryS(ineachcasewetookn = 3),for thezwitterionic form. We obtained the values D+ = (0.58 f 0.02)le5cm2/s, D-= (0.55 f 0.02)10-5cm2/s, and DO= (0.65 f 0.04) l e 5cm2/s. For the other diffusion coefficients we took the self-diffusion values. The FD is in reasonable agreement with the experiment. Both the maximum of the effect and its timelocation are close to each other. The NM is much below these results. It is less than half the FD. It does not describe satisfactorily the observed effect, though its maximum is located at the same position as the experiment. For times larger than 30 h, the effect is seen to decrease more slowly than expected. In the same conditions, the function p2(f) was recorded. The result is shown in Figure 3. A slight initial oscillatory behavior is observed. This peculiarity is confirmed by the FD. The FD describes qualitatively the experiment, but it is noted that the effect is small and so the experiment is very sensitive to any initial disturbance. Nevertheless, the main features of the profile (amplitude and position of the extrema) are found by the FD, which indeed was our guiding line in this study. On the contrary the NM yields a profile which is completely different from the former. It begins with a fast increase in the first 2 h, as the FD
The Journal of Physical Chemistry, Vol. 97, No. 19, 1993 5023
Diffusion Coupling in Electrolyte Solutions
0
10
20
40
50
t (hl? Figure 3. Aspartic acid: p2 as a function of time 1. First buffer, pH 1.45; second buffer, pH 4.8. Line f.d.: finite-difference result. Dashed line nm.: normal modes.
I
67
I
-i-l , I
0
'
10
20
30
t (hr) Figure 4. Aspartic acid: p2 as a function of time t . First buffer, pH 1.2; second buffer, pH 5.6. Line f.d.: finite-difference result. Dashed line nm.: normal modes.
and the experiment, but decreases monotonously afterward. In terms of the NM it means that one of the relaxation modes is quite fast, of the order of some h-1. The NM leads to the positive real diffusive modes, DI = 5.27, D2 = 1.23, D3 = 1.13, expressed in 10-5 cm2/s units. By the way, this shows that the system does not relax in fact in a purely oscillatory manner, in which case we would find complex conjugate modes, but that the great enough number of modes allows the system to exhibit a time dependence that we may call pseudo-oscillatory. By virtue of eq 36 and with L = 1.5 cm, we find that these values correspond to relaxation times equal to 7i H 0.9/Dp, and N 4.7 h, 72 N 20 h, and r3 N 22 h. The first diffusion so coefficient D1 is close to DH (see eq 19, DH = 5.75 in the same unit), which represents the mean diffusion coefficient of the H species. In fact the maximum of the NM curve is reached at t H 2.5 h with an amplitude of 1.26%. We note also that the mean charge of the tracer, at equilibrium, is 0.021, which is quite low because the major form of the tracer is the neutral form (go%), the charged forms being in minor nearly equal amounts (10%) since the equilibrium pH is around 2.9, close to the isoelectric point of the amino acid. Increasing the pH of the first buffer from 1.45 to 1.55 and keeping the same pH for the other buffer yield similar results except that, concerning p2(r), the second extremum (the local minimum) has risen above the baseline. If the pH is still increased to 1.65, the latter nearly vanishes, thus canceling the oscillatory character of p2. We also carried out an experimentin which the tube was turned upside down at t = 0. This resulted in a sudden drop off of the
recorded activity in the first 5 h followed by a regular increase, with a characteristic time of about 1 day. This observation is similar to the one obtained with sulfate ion: which was attributed to an hydrodynamical instability first found by McDougallI and confirmed experimentally by Miller and Vitagliano.I2 In the last experiment we chose buffers at pH values of 1.2 and 5.6, leading to an equilibrium value of 3. Figure 4 displays the various results. Fair agreementis found in the first hours between the experiment and the FD, though both curves begin to increase for times of about 10 h, a larger discrepancyis observed, with the experimental curve increasing faster than the FD. The NM exhibits a still more pathological pattern than before, with a maximum value much below the FD and the experiment. Only the maximum is reached at 2 h, whereas it is at 1.5 h in the FD. Here the diffusive modes are DI = 4.75, D2 = 1.23, and D3 = 1.13, in the same units as above, and DH = 5.12. It is noted that the two slower modes are identical to the previous ones found with different buffers and that only the faster one is a bit lower, and so does DH. Lastly, the equilibrium mean charge of the tracer is here slightly negative, 4 . 0 1 2 5 . The reason, unlike the sulfate ion (which was also placed in a gradient of pH) and other previous systems studied in our group, the NM does not work here for the aspartic acid must be that, at equilibrium, the aspartic acid is neutral in the proportion of 8096,while much of the effect is due to the action of the internal electric field on the tracer. However, the NM is very useful as a means of physical insight into the relaxation of the system near equilibrium. If true oscillations occurred, complex modes would be found by the NM, since an oscillatory behavior would prevail, in particular close to equilibrium. In the present case, varying the pH of the buffers did not yield any. Thus, choosing pH values of 1.65 and 4.8, respectively, for the solutions SA and SBgives very close values of the modes DI and D2, but no complex (conjugate) modes are obtained. Acknowledgment. We are gratefully indebted to Y.Prulibre for technical assistance. References and Notes (1) Turq, P.; Orcil, L.; Chemla, M.; Barthel, J. Ber. Bunsen-Ges. Phys. Chem. 1981,85, 535. (2) Simonin. J. P.: Gaillard. J. F.: Tura. P.: Soualhia. E. J . Phvs. Chem.
1988, 92, 1696. (3) Simonin, J. P.; Tivant, P.; Turq, P.; Soualhia, E. J . Phys. Chem. 1990, 94, 2175. (4) Simonin, J. P.; Turq, P.; Soualhia, E.; Michard, G.; Gaillard, J. F. Chem. Geol. 1989, 78, 343. ( 5 ) Liukkonen, S.;Passiniemi, P.; Nosticzius, Z.; Rastas, J. J . Chem. SOC.,Faraday Trans. 1 1976, 72, 2836. (6) Simonin, J. P.; Mills, R.; Perera, A.; Turq, P.; Talet, F. J. Solution Chem. 1986, 15, 1015. (7) Robinson, R. A.; Stokes, R. H.Elecrrolyte solutions, 2nd ed.; Butterworth: London, 1959; p 286. ( 8 ) Mills, R.; Perera, A,; Simonin, J. P.; Orcil, L.; Turq, P. J . Phys. Chem. 1985,89, 2722.
(9) Weast,R.C.; Astle, M. J. CRCHandbookof ChemistryandPhysics; CRC Press: Boca Raton, FL, 1983. (IO) Turq, P.; Orcil, L.;Chemla, M.; Mills, R. J . Phys. Chem. 1982,86, 4062. (11) McDougall, T. J. J . Fluid Mech. 1983, 126, 379. (12) Miller, D. G.; Vitagliano, V. J . Phys. Chem. 1986, 90, 1706.
