ELECTROKINETIC PHENOMENA IN CHARGED MICROCAPILLARIES
August, 1963
characterizes the relationship between valence tautomers is the comparatively high rate of conversion from the less to the more stable form. This implies a lower activation energy for this unimolecular process than for other types of isomerization. It docs not involve excited electronic states. The rationale for this is that the adjustment of the skeleton geometry and the re-shuffling of multiple bonds occur simultaneously. A reasonable estimate for the activation energy is ‘/a
(Dc=c - Dc-c)
In contrast, for a cis-trans isomerization about a double bond the activation energy is considerably higher, being that required to convert a C=C double bond to a single bond as when compressed to 1.33 A. Other isomerizations require the rupture of a C-H bond concurrent with changes in the bonding between the carbon atoms. Finally, the appearance potentials of C7H7+ in toluene and cvcloheDtatriene (Table VI) were not found to be in considerable excess of the thermochemi-
1636
cally calculated values based on the heats of formation of the c7138 isomers. However, if the difference in appearance potentials of the parent C7H8+and product C7H7+ is assumed to be the activation energy for the process3 C7H8+ +C7137+
+H
an unexpectedly high activation energy is deduced for this reaction (2.8 e.v. in one case 2.0 e.v. in the other). For molecules with 39 oscillators, such as these are, calculations based on the statistical theory would give a small rate constant for the reaction a t the theoretical lower limit of excitation. 27 These processes appear to be sufficiently probable so that the threshold energies are not considerably in excess of the theoretical lower limits. Acknowledgments.-We wish to thank Professor F. P. Lossing and Professor M. J. Goldstein for valuable discussions and Drs. 0. L. Chapman and Y. Meinwzlld for the compounds used. (27) M. Wolfsberg, J . Chem. Phys., 96, 1072 (1962).
ELECTROKIKETIC PHENOMENA I N CHARGED MICROCAPILLARIESl B Y LAWRENCE DRESNER Oak Ridge National Laboratory, Oak Ridge, Tennessee Received January 19, 1965 Electrokinetic phenomena in small surface-charged capillaries have been studied. Fluid flow has been described by the Navier-Stokes equation, diffusion by coupled equations linear in the electrochemical potentials, and the electric field by Poisson’s equation. In the case of good co-ion exclusion from the capillary, the coefficients in the linear laws relating solution flow and counter-ion current to pressure, concentration, and electrical gradients have been explicitly calculated. Some numerical calculations of interest in water desalination are reported.
1. Introduction Electrokinetic phenomena have been known for more than a century and a half-electroosmosis, for example, was discovered in 1807. The first theory of these phenomena was the work of Helmholtz,2 who introduced the concept of the electrical double layer. Helmholtz’s theory was later refined by the introduction of a diffuse double layer in the researches of C h a ~ m a nand , ~ Stem6 When electrokinetic phenomena are produced in macroscopic capillaries, the double layer thickness is small compared to the capillary diameter, and charge separation in the fluid fiIling the capillary occurs only in a thin layer near the wall. Thus the bulk of the fluid is electrically neutral. I n microcapillaries, on the other hand, the double layer thickness is comparable to the capillary diameter and the bulk fluid is charged. I n general, the radial distribution of the fluid charge is strongly nonuniform. As Mackay and Meares6 first pointed out, the details of the spatial nonuniformity of the fluid charge affect electrokinetic phenomena in the (1) Work performed for the Office of Saline Water, U. S. Depart,ment of the Interior, a t the Oak Ridge National Laboratory, Oak Ridge, Tennessee, operated b y Union Carbide Corporation for the E. 6 . Atomic Energy Commission. (2) €1. Helmholtz. W i d . Ann., N.F., 7 , 339 (1879). (3) (a) G. Couy, J . Phys., 141 9, 357 (1910); (b) Ann. Phgsik, 191 7 , 129 (I9 17). (4) D. L. Chapman, Phil. Mug.. I61 25, 475 (1913). (5) 0. Stern, 2. Elelctrochem., SO, 508 (1924). (6) D. Maekay and P. Mearea. Trans. Faraday Soc., 66, 1221 (1959).
microcapillary. Mackay and Meares studied the effects of charge nonuniformity on electroosmosis in capillaries by assuming a power series with unspecified coefficients for the radial charge distribution in the capillary. I n the resent work, a consistent theory is used to calculate t e radial co- and counter-ion distributions and so to remove the arbitrariness present in Mackay and Meares’ work. This theory is based on the use of the h’avier-Stokes equation to describe fluid flow, coupled equations linear in the gradients of the electrochemical potentials to describe diffusion, Poisson’s equation to describe the electric field, and various appropriate equations of continuity and electroneutrality. I n sections 2, 3, and 4 the theory is developed in the case where there are no concentration gradients along the length of the microcapillary and only electrical and pressure gradients are considered. I n section 5, a variety of numerical results are displayed and discussed, so that the reader may get some idea of the size of the various effects. Following section 5, the theory is generalized to admit axial concentration gradients in the microcapillary. 2. The Equations for the Streaming Potential.Let us consider a long cylindrical microcapillary of radius R upon whose inner surface a charge of surface density resides. Let r measure the radial distance of a point from the capillary axis; let x measure the distance along the axis. Suppose now that the ends of
g
LAWRENCE DRESNER
1636
the microcapillary are immersed in two reservoirs of electrolyte of equal concentration c and fluid is allowed to enter it. After a time thermodynamic equilibrium will be reached. Suppose now that we create a pressure difference A p between the two reservoirs. Fluid then flows through the capillary in the opposite direction to the pressure gradient. This produces a separation of charge, the high pressure reservoir taking the charge of the eo-ions and the low prmsure reservoir taking the charge of the counter-ions. This separation of charge creates an electric field E that causes diffusion of the counter-ions oppositely directed to the flow of the fluid. When a steady state is reached, the velocity profile of the fluid flow and the magnitude of the electric field E have adjusted themselves so that the effluent fluid is electrically neutral, The potential difference across the capillary created by the electric field is called the streaming potential. Equations describing transport processes in multicomponent systems have been given by Bearman and K i r k r n o ~ dand ~ ~ by de G r ~ o t . ' ~These equations describe (i) diffusion of the various components Iyith respect to the local center of mass and (ii) the motion of the local center of mass. The isothermal diffusion of co- and counter-ions and water with respect to the local center of mass in the presence of an external electric field is specified by the equation
k=l
where i, is the vector mass-current of component i (g. cm.-2 sec.-l)
with respect t o the local center of mass and is defined by P , ( V , - v) pl is the local density (g. cm.-3) of component i vi is the local velocity (cm. sec. -1) of component i v is the velocity of the local center of mass, given by v =
T'ol. 67
and 3 i=l
J k=l Dllk is a phenomenological coefficient i v k is the number of particles (ions or molecules) per g. of component k z k is the valence of component k e is the charge of the proton 4 is the electrical potential pk is the chemical potential per particle of component k, and i = 1 denotes counter-ions, i = 2 denotes co-ions, and i = 3 denotes water. ]C=I
According to Onsager's theorem,7bthe coefficients in eq. 1' are symmetric, i.e., %!ik = 9llk;. Furthermore, 3
since
3
il
=
%=I
0, it can easily be shown that
%k ]!
=
2-1
If we replace ii by j J N l , where j i is the particle current (ions or molecules cm.-2 set.-'), eq. 1' can be rewritten as
0 for all k.
where
(7) (a) R. J. Bearman and J. G. Kirkaood, J . Chem. Phye., 28, 136 (1958); (b) €3. R. de Groot, "Thermodynamics of Irreveralble Procesm~,"NorthHolland Publishing Coo,Amsterdam, 1958,
0
(IC)
The motion of the local center of mass is given by a form of the Navier-Stokes equation, vie.
where 3
p
is the total fluid density, given by
p =
pk
k=l
p is the pressure 7 is the fluid viscosity Ck is the concn. of component k in particles (ions or molecules) per cubic centimeter
In steady-state flow in the microcapillary, the lefthand side of eq. 2' vanishes. The derivative dv/& vanishes by definition of the steady state; the term v . Vv vanishes because v is everywhere axially directed but oiily changes in the radial direction. Thus in all the applications made of it in this paper, eq. 2' takes the form dp -dx
+ - dr-d 7 T
T
dv dt$ - - e - (xlcl dT dx
+ z2c2)
=
0 (2)
In a steady state, the various flows j, and the velocity v are connected by conditioiis of continuity. Since oiily two of the flows ji are independent, there are three independent continuity conditions, which in the case of axial flow in a niicrocapillary, me take as V.v V.(jl V.(j,
-
=
Mik/Ni
=
(34
0
+ c,v) 0 + czv) = 0
(3b)
=
(3c) The electric potential is related to the ionic concentration through Poisson's equation
V 2 4 = - (zlecl
+ z2ec2)/c
(4)
where E is the dielectric constant. The electroneutrality of the fluid leaving the microcapillary can be written ([zdi,
+ c1v) +
zZe(j2
+ c~v>la) 0 =
(5)
where n is a unit vector parallel to the capillary axis, and the braces denote an average over the capillary cross section. Finally, the over-all electro-neutrality of the capillary interior can be written as (zlecl
+ z2ec2 + zoeco) = 0
(6)
Here the subscript zero denotes the fixed charges re siding on the capillary wall. They must be taken into account in eq. 6, but they do not enter eq. 4 directly since they are not present in the bulk of the fluid in the microcapillary. They do, however, affect the electrical potential + by influencing the boundary conditio11 the electric field must satisfy a t the capillary wall. If a suitable boundary condition is added to eq. 1-6, a unique solution will be specified. For this boundary condition the requirement has been taken that the fluid
ELECTROKINETIC PHENOMENA IN CHARGED MICROCAPILLARIES
August, 1963
in either end of the capillary is in thermodynamic equilibrium with the fluid of the reservoir into which it intrudes. 3. Solution of the Equations.-If ji = v p = v = 0, the concentrations and the electric potential are given by ci
= Cii0)
(4
(74
d, = # P ) ( r )
(7b)
where the superscipt zero denotes the equilibrium value of the quantity being considered. Now choose as solutions in the steady-state case under consideration the values ci = Ci(O)(?) q!~ = d,(’)(~)
- EX
(84 (8b)
where E is the constant value of the axial electric field. The choice (8b) of E constant and pointing along the axis is based on imagining the microcapillary to be one of many parallel pores through a thin membrane. It is plausible to suppose that the initialIy separated charges, both signs being mobile, will spread themselves uniformly over the membrane surfaces, producing a constant field within. I n thermodynamic: equilibrium only eq. 4 and 6 are nontrivial*; since the equilibrium solutions (7) simultaneously satisfy these equations, the assumed solutions (8) also do. The solutions (8a) moreover satisfy the boundary condition given a t the end of section 2. The continuity equations (3a,b,c) will be satisfied if we choose v and ji to be axially directed and functions of r only. Such an assumption for ji is consistent with eq. 1, since substitution of solution (8) into eq. 1 yields the result
1637
wi = &i/dp is the specific volume per particle of component i w = w1 - (Na/iV1)Ws fM11 = Cl(O’a).,/kT 91is the diffusion constant of the counter-ions in the fluid (em.$ sec. -1) k is Boltsmann’s constant T i s the temperature
If we now substitute solution (8) into eq. 2 we get p zleEcl(’J)(r) - ddx
d dv + -17r r-=0 dr dr
(10)
From (10) it follows that
By substitution of eq. 9 and 11 into eq. 5 and some rewriting, the following solution for the field E is obtained xleE = dP -X dx
Dresiier and Krausg have derived the formula
for c1(O)(r)in the case of good co-ion exclusion, where a2 = 1
+f ~ ~ / 8
f~ = KR f =
in the axial direction, while in the radial direction bot,h sides of eq. 1 vanish identically. I n this paper we shall only be interested in the case of good Donnan exclusion of the co-ions from the capillary.9 If we make the idealization that c2 = 0 in the microcapillary, then with the help of eq. lb,c eq. 9’ can then be written
K~
=
KT
e2(co)/lcTe
With this formula for c,(O)(r) the integrals in eq. 11 and 12 are simple ones. Carrying them out leads to the results
-_21eE
-
d PldX
(9”) where ( 8 ) I n equilibrium, eq. 3 and 5 are trivially satisfied. Equation 2 is likewise trivially satisfied in the axial direction, there being no axial flow, no axial pressure gradient, a n d no axial electric field. Equation 2 must also be satisfied in the radial direction: this means that a radial pressure gradient E~CZ). I n equilibrium, the electrochemical potenexists equal to -eV+(zicl tial of any component must be constant from point to point in the microcapillary. This means that the parentheses in eq. 1 vanish, so that it, too, is trivially satisfied. The constancy of the electrochemical potential requires a connection between the concentration and the electric potential which in the case of ideality is equivalent to the proportionality of the concentration to the Boltzmann factor expr - e i e + / k T ) . This proportionality, together with eq. 4 and 6 and the boundary conditions, serves t o determine the equilibrium concentrations and the equilibrium electric field (sea ref. 9). (9) L,Dresner a n d K . A. Ylraus, J. Phus. Chem., 67, 990 (1968).
+
4. Other Phenomena.-In addition to the streaming potential, various other electrokinetic quantities such as the streaming current and t h e electroosmosis are of interest.7b These quantities are calculable on the basis
LAWRESCF, DRESNER
1658
of the model described in sections 2 and 3, but such calculations are unneces.sary since all isot hernial elect rokinetic quantities can be ohtained from the results given in eq. I Z by usc of the laws of irrcvcrsible therniodynamics.7b Suppose that q represents t h t total w h t r n ~ t r i cflow through the microcapillary and J represents the total countcr-ion ciirrent. The entropy production d S dt in the capillary is given by
-T(dS/dl)
qAp
=
+ JzleA+
Vol. 67 1
+
-1
VB1WK' - -
2k7'
(21)
If A p = 0 and an extcrnal potcntial dil'fcrence is supplied, h' is diffrwnt from zei'o, and the appropriate rxprcssion for I' k)clonging i n (17) is givrn by thc hrst tcrni of (14t)). Thus
(13)
where henceforth + is the potential of either reservoir arid A denotes the difference between the two reservoirs The linear laws relating the flows and forces are q = LllAp 4-Id1?zleA4
J L1e
=
L?lAp
+
Ll?~laA+
(164 (1GI)) Wc)
= I121
Equation 1Gc is Onsager's relation.7b Equations 16 constitute a full thermodyiiamic description of the microcapillary system in the case of good co-ion exclusion from which all electrokinetic quantities can be obtained. The important I,-coefficients can be obtained from the results of eq. 14 as follows. When A+ = 0, B = 0 and the velocity 1) is given by the second term in cq. 14b. q is always given by
q
=
hR[(jl +
CIV)WI
+ ( j , + c~v)w~l2?rrdr
Since zlel?'d = --,PA+ in the casc at hand, comparison of eq. 16, 21, and 22 shows that 1.21 = as it must. E'inally, when A p = 0
J
[j,
= JOJi
where Thus
21
+
c ~ ( " ~ ( r ) i2ar l]
dr
=
is again given by the first term i n eq. 141).
(17)
Substituting eq. 9 and 9" for j, and j 3 and using the c,3w:i = 1 (Gibbs-Duhem equation) identity clwl this beconics
+
2s(~a)~R' zleE [I - ( 8 , ' t ~ I~n) (I
From (18) it follows that
K2ll
+ {n2,'8)]
(21b)
where d is the length of the capillary. The first term in eq. 19 is I'oisseuille's law; the second term arises from diflzision relative to the local center of mass caused by the pressure gradient. J is always given by
J =
+ c l v ) 2 sdr~
(jl
(20a)
which in the casc A+ = 0 becomes
(20b)
Thus
If in eq. 1Gb J is set equal to zcro, the result of eq. 14s for 13 is recovered. 5. Discussion and Numerical Examples.-h his discussion of elcctrokiiietic phenomena dc inentioris eight quantities all of which arc studicd experimrntally. The names and definitions of these quantities, as \vel1 as their expression in ternis of tlic Lcoefficients of eq. 16, are shown in the first eight lints in Table I. Two other quantitics not mentioned b?l dc Groot are shown in the last two lints. I3ecausc of Oiisager's relation (ltic), the first eight quantities in Table I are cqual by pairs, save for a possible sign change. Thus the streaming current equals (minus) the electroosmotic pressure, the streaniirig potential equals (minus) the electroosmosis, thc second streaming current equals the second rlectroosmosis, and the second electroosmotic prcssurr equals t h c second streaming potential.
ELECTROKISETIC PHENOMEKA
August, 1963
IN CHaRGED
MICROCAPILLARIES (~0)- 40
1639
moles/liter.
IO'"
10'7
W
40-'* 0
io
20
30 R
0
10
20
30 R
50
Fig. 1b.-Various electrokinetic quantities plotted against the capillary radius R in the case of constant fixed charge density
50
40
40
(8).
(ea).
cx,,
Fig. la.-Various electrokinetic quantities plotted against the capillary radius R in the case of constant fixed charge density
Bl = 2 X 10-5
T
(eo).
w
= 15' = 0.01 g. c1n-l
=o
see.-'
set.-'
TABLE I THEKAMESAND DEFINITIONS OF THE VARIOUS ELECTROKINETIC The radii have been limited to values 5 5 0 k . because QUANTITIES,AS WELLA S THEIR EXPRESSION IN TERMS OF THE this is the region in which one must work to achieve L-VALUESOF EQ.16 good exclusion ((cz)/c < 10%) with reservoir electroName
Definition
Expression in terms of the L'a
Streaming c u n e n t Electroosmotic pressure Streaming potential
(eJ/q)*+ = 0 (Ap/A+)* = (A+/Ap)j=o
eL2i/Lii -eL~z/Li. --L?l/eL??
Electroosmosis Second streaming current
(q/eJ)bp = o (eJ/Ap)A+ = o
Liz/eL?z eLz1
Second electroosmosis Second electroosmotic pressure
(P/&)Al, = 0
d l 2
(Ap/eJIq
Second streaming potential
(A+/q)JEo
(eLir) -1
Hydrodynamic resistance
( q / A p ) ~=
LI1( 1
Electrical resistance
(A+/eJ),,
=o
=0
(eh)-l( 1
(
1
-
L??Lll - -) LXlLl?
-1
z)
- __
-l
;2)
(e2Lzz)
Plotted against the capillary radius in Fig. la,b are' values of the various electrokinetic quantities calculated for the following typical choice of parameters (co) €/EO
eo
= 10 moles/l.
80 = dielectric constant of a vacuum (8.854 X farads/m.)
=
lyte concentrations 20.02 mole/l. (such as are of interest in the desalination of brackish water^).^ w has been taken to be zero for simplicity, but extensive numerical calculations have shown that when w varies in the range from to 0 to 30 A.f, the various electrokinetic quantities change only by amounts of the order of 2070. In Fig. 2 velocity profiles for R = 10, 25, and 50 k. are shown which correspond, respectively, to the situations J = 0 and A p = 0, Le., to the processes of reverse osmosis and electrodialysis. I n Table I1 the quantities hid, Ll& and Lz2dfor the same three radii are given. An average fixed charge concentration (co) of 10 moles/l. in a microcapillary with a radius Ro of 10 A. can be produced by a surface charge density u = (l/*) Ro (co) of about 3 X loL4monovalent ions per cm.2. If this cbarge density is kept fixed, in a capillary with a 25 A. radius the average fixed charge concentration will only be 4 moles/l. The data in Fig. 1 do not apply to such a microcapillary. However, when u: = 0, these data can be interpreted as follows.
1640 u = 3.012 X 10i4 cm2,
5 . ...
-I--.
0
0 2
-
..
_ . -
Ob
04
08
40
J
= 0)
rI
IOi6
0
5
2 40'~
\
l
I
I
I 5
. +..
t-.
2 10'~
5
5
2
2
2
2
0
too--
I
'
50
'50
400 R
(a).
200
I
250
Fig. 3n.-\'arious clcctrokinc~tic~ quaiititirs plotted against the c:j,pill:try radius Ii' in t h e (':tsr of constant surfavc c.ti:trgci density U.
Let 11s consider one series of capillarics ivith radii denoted by ZZ' for which u is held fixed and a second series with radii denoted by I2 for which (c") is held fixed. Let (co) = 2u/Rg. If R' = Rz/Rc,then
50
400
450
200
250
R ix1.
Fig. 3c..--T'nrious cilcctrokinc&c quztnt ities plottrd against the capillary radius It in tlic ( * i t 9 ~of constlint surlncr c.li:trgc. drnsity u.
ELECTROKINETIC PHENOMEKA IS CHARGED MICROCAPILLARIES
August, 1963
since in the capillaries with fixed surface charge (I,the average fixed ion concentratioii is 2g/R'. Since the constant 7,7~~331/2(c0) is actually independent of (co), it follows from eq. 25, for example, that the same value of Lzz which refers to a radius R and an average fixed ion concentration (co) also refers to a radius R' = Rz/Ro and a fixed surface ion density of B = Ro(co)/2. This equivalence is only one of several which may be summarized as follows. The quantities shown below in the first column of Table I11 refer to fixed (co). If they are multiplied by the quantities next to them in column 2, corresponding values are obtained which refer to fixed (I, B and (co)being related by (co) = 2a/Ro. Finally,
so that when plotted against the fractional radius the velocity profiles of Fig. 2 apply. Shown in Fig. 3a-c are the data of Fig. 1 replotted to TABLE I1 THE COEFFICIENTS b l d J LI2d,Lzzd CALCULATED FROM Eo. 19, 21, AND 25 FOR THREERADIIAND THE CONSTANTS SHOWN IX EQ.26 R
(L.1 10 25 50
3 927 X 1 534 X 10+5 2 454 X
-L22d
-Lid
-Liid (g.-l cm.5 sec.)
(g,-1
cm.2 ser.)
9 388 X 1 236 X 7.205 X
(g.-1 cm.-1 sec.)
1 244 X 8 327 X 3 391 X
(qleJ)ap=a ( e J / A p )A+ == 0
( RIRo)'
(a/&)Ap=O
(RIRo)'
(Ap/eJ),=o
(R/'Ro)-* (R/Ro)-'
(A+/'q)J=O
(QlAP)J=O
(A+/eJ)ap==o
(RIRo)'
(RIRO)' 1
apply to the case of B = 3.012 X 1014 cm.-Z. I n but, as before, only this case, the radii extend to 250 8., for radii up to 50 8.can 90% or better exclusion be achieved with electrolyte concentrations >, 0.02 mole/l. For the larger radii smaller values of c are required to produce good exclusion; when R = 250 8., c must not exceed 6 X mole/l. in order to have (cz)/c < 10%. 6 . Reservoirs with Unequal Electrolyte Concentrations.-In case the electrolyte concentrations iii the two reservoirs are not equal, axial concentration gradients appear along the length of the capillary. The set of eq. 16 is then no longer adequate to describe the flow of counter-ions and water. However, if the concentration gradients are not too large, linear phenomenological equations can be written down whose eoefficients are uniquely determined by the Lik already calculated. If we again ignore the presence of co-ions, the entropy production in the capillary can be written7b -T(dX/dt)
= JwAp3
+ JAp1
(29)
where as before J is the total counter-ion current, J, is the total current of water molecules, Ap1 is the difference in the electrochemical potential per counter-ion between the ends of the capillary, and Apa is the difference in the chemical potential per water molecule between the ends of the capillary. Appropriate h e a r laws relating the flows and forces are
lOI7
Jw
lo1*
TABLE I11 EQUIVALENCE OF THE CASE OF FIXED (eo) A N D THE CASE OF FIXED u WHEN w = 0. The quantities in the first column refer to fixed {eo). If they are multiplied by the i*atios next t o them in the second column, corresponding values are obtained which refer to fixed uJ u and (co) being related by (eo) = 2u/Ro. R RIRo L11 (R/Ro)4 LlZ (R/Ro1 Lzz 1 ( e J / p )A4 =0 (RIRo)-' (AplA+),=o ( RIRo) -' ( A + / A P ) J =0 (R i m
1641
=
J
Kllb3 K2iAp3
+ KlzAPl + KzzAPi
Ki, = Kzi I n the case of identical reservoirs Apl
=
wlAp
+ xleAq5
(304 (30b) (304 (314
w3Ap (31b) If these equations are substituted into eq. 30, the latter becomes identical in content with eq. 16. If we note that AM
Q =
WJ
+
w3Jw
(32)
we can determine the Kij in terms of the Lij. The result of this determination is
KII
=
K1z
&I
=
+
- 2 w J ~ W ~ ~ L ? Z )(33a) /W~~ Kzl
=
( L I ~- W I L Z ~ ) / W ~(33b)
Kzz
=
L22
(33c)
