J . Phys. Chem. 1990, 94, 2671-2679
2671
Application of Stokes’ Law to Ions in Aqueous Solutiont P. C. F. Pau,t J. 0. Berg,$ and W. G. McMillan* Department of Chemistry and Biochemistry, University of California, Los Angeles, California 90024 (Received: February 12, 1985; In Final Form: July 28, 1989)
A comparison of the so-called Stokes radii of ions in aqueous electrolyte solutions with the corresponding ion crystal radii gives order-of-magnitude agreement but with a tantalizing discrepancy: some Stokes radii are found to be smaller than the corresponding crystal radii, contrary to expectation for hydrated ions. Although the extrapolation of continuum hydrodynamics to the atomic domain would seem to be a prime source of this discrepancy, the close agreement with experiment given by the Stokes-Einstein diffusion constant for uncharged molecules (using the slip boundary condition) encourages the view that the discrepancy for ions may result less from the assumption of continuum hydrodynamicsthan from some neglected interaction with the solvent. Within this context, this paper reexamines the “stick” vs “slip” boundary conditions for Stokes’ law with specific reference to ions in aqueous solution. Further, a new concept is introduced that involves an enhanced electrical force acting on an ion as a consequence of local dielectric saturation of the solvent environment-which, so to speak, allows the applied external field to “shine through”. Inclusion of this force enhancement, together with the slip boundary condition, considerably improves the predictions of Stokes’ law.
1. Background and Preview It often happens that the application of a theoretical result gets extended beyond the limits for which it was originally designed. Sometimes such extension gives surprisingly good agreement with experiment, which then poses the question of how to justify the extension outside the apparent range of validity. Conversely, when such extended application is found to disagree with experiment, the question then is how to modify the theory to reestablish agreement. A case in point is the extrapolation of Stokes’ law’ for the drag force exerted by a viscous continuum hydrodynamic medium on a spherical particle, to the motion of particles of atomic dimensions in solvents composed of discrete molecules of comparable size. For the diffusion of neutral molecules driven by a concentration gradient, the Stokes-Einstein equation often gives remarkably close agreement with the experimental diffusion constant. However, when applied to the conductivity of ions in electrolytes, Stokes’ law encounters many inconsistencies. To summarize the existing situation and the resulting puzzle, we imagine the measurement of the conductivity of a single ionic species having charge Ze to be carried out in a conductance cell with parallel plates of area A and separation 1. An emf of E volts applied across the cell leads to current J amperes [=(current density j ) X (area A ) ] flowing against the cell resistance R ohms (= l/A): E = JR = j A / A = j V / A l (1.1) where V (=AI) is the volume between the plates of the cell and A is the conductance. When the plate separation 1 is 1 cm and tJe volume contained between the plates is the equivalentZvolume I(of the ion in question, A becomes the ion equivalent conductance A. To avoid having to correct for ionZion interactions, one employs the limiting equivalent conductance A. at zero concentration. The current density j is simply the product of the charge density and the ion drift speed u: j = $v/F (1 4 where $ is the faraday-Le., 1 mol of electronic charge. The steady-state ion drift speed v results when the electrical force on the ion is balanced by the viscous drag. Conventionally, the electric force is assumed to be the product of the ion charge Ze with the average potential gradient E/I across the cell:2 ZeE/I = ( Z e $ / & ) v (1.3) ‘Based on the doctoral thesis of Paul Chi Fu Pau, Department of Chemistr and Biochemistry, UCLA. ?Present address: Hyperion Treatment Plant, Playa del Rey, CA 90293 (Present address: TRW R1/1184, 1 Space Park, Redondo Beach, CA 90278.
0022-3654/90/2094-267 1$02.50/0
TABLE I: Comparison of Aqueous Ion Radii from Stokes’ Law with Crystal Radii
ion radius, A
ion Li+ Na+ K+ Rb+
cs+ F
CIBrI-
Ao?
ohm-’
crystalb
38.6 50.1 73.5 77.8 77.2 55.4 76.4 78.1 76.8
0.78 0.98 1.33 1.49 1.65 1.33 1.81 1.96 2.20
Stokesc stickdan slipd ad 2.39 1.84 1.25 1.18 1.19 1.66 1.21 1.18 1.20
3.58 2.76 1.88 1.78 1.79 2.49 1.81 1.77 1.80
OLimiting equivalent conductances in water at 25 OC, taken from Robinson and Stokes.’ bCrystal radii, from Pauling.4 cPhysical constants employed: 7 (water) = 0.8903 X lo-* P (25 “C); 9 = 96487 C/mol; e = 4.8043 X esu. dThe “stick” vs “slip” boundary conditions are treated in detail in section 8. The magnitude of the drag force is assumed to be given by Stokes’ law Fdrag
= ymav
(1.4)
wherein a is the radius of the migrating ion entity (assumed spherical) and 9 is the macroscopic shear viscosity of the pure solvent. On the assumption that the fluid “sticks” to the particle surface with frictional coefficient ~1 = m, so that the tangential component of relative velocity vanishes at the particle surface, Stokes obtained the coefficient y = 6, which thus defines the classical “Stokes’ law”. The other extreme of zero surface friction (k = 0) corresponds to the “slip” condition, with coefficient y = 4. Equating the forces (1.3) and (1.4) allows the ion speed to cancel and yields what is commonly referred to as the Stokes ion radius:
a7 = Ze$/300ym&
(1.5)
wherein the factor 300 converts the potential difference from (1) Stokes, G. G. Trans. Cambridge Philos. SOC.1845, VIII,287. (2) When equivalent quantities (designed by the tilde) are used, the plate separation I = 1 cm and thus drops out, leaving behind only its length dimension (like the smile of the Cheshire cat!) (3) Robinson, R. A.; Stokes, R. H. Elecfrolyte Solutions; Butterworths: London, 1959. (4) Pauling, L. The Nature of the Chemical Bond Cornell University Press: Ithaca, NY, 1942.
0 1990 American Chemical Society
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The Journal of Physical Chemistry, Vol. 94, No. 6 , I990
Pau et al.
TABLE 11: Motivation for Hydrodynamic Treatment of Viscous Fluid from Analogy with Elastic Solid physical quantity elastic solid viscous fluid notes equation P f = F prescription: replace S by (=a) except 2. I relative position (velocity) retain also @VS 4 relative strain (rate) = ;grad? do = dr grad.? grad.cc7 is the dyad 0.; 2.2 @ = 1/2[V.S (O.?),] 6 = 1/*[P.C (V.C)J strain (rate) tensor antisymmetrical counterpart discarded 2.3 as irrelevant rigid rotation dilation strain (rate) Tr Q, = FS Tr 6 = Pa contraction of strain (rate) tensor 2.4 $1 = AVTl + 2p@ stress linear i n strain (rate) rlr = ( B V f + p C ) 1 + first-order approximation for small 2.5 (A, p are Lame constants; 2 7 [ 4 - (PL’/3)1] strain (rates); 7 is the shear viscosity, 1 is the unit tensor) [ t h e compressional (or “second”) viscosity; the shear term is traceless static pressure p = -@OS; dynamic mean hydrostatic pressure 2.6 (A, p determined from static pressure = -pC moduli) refined stress tensor 2.1 VC = 0 for incompressible flow \k = -@ + 2/,pvF)1 2p@ P = [ - p ([ - */,7)PC]l 276
-
d
+
+
+
+
practjcal volts to statvolts. With the values e = 4.8043 X esu, 9 = 96 487 C, t) = 0.8903 CP for water at 25 OC, and A. in ohm-’. the ion radius in angstrom units becomes
a6 = 92.07Z/x0 a4 = 138.11Z/xo
(“stick”) (“slip”)
(1.6)
Table I shows a comparison of the Stokes “stick” and “slip” ion radii in aqueous solution (columns 4 and 5 ) calculated according to (1.6), for the alkali-metal and halide ions with the corresponding ion radii in crystals (column 3). Although the Stokes radii are the right order of magnitude, there are serious discrepancies: considering that ions in aqueous solution carry along a number of “solvated” water molecules, the mobile ionic unit can scarcely have radius less than that of the ion in the crystal-as occurs for the bromide and iodide ions. In view of the numerous assumptions mentioned above, perhaps these discrepancies should not be surprising. Nevertheless, the Stokes radii are tantalizingly close to the values that might be expected on physical-geometric grounds and thus constitute a theoretical challenge. Contributions to these discrepancies could come from several sources: (1) the lack of applicability of macrocopic continuum hydrodynamics (from which Stokes’ law is derived) to the atomic regime of ions in s ~ l u t i o n (2) ; ~ the choice of boundary condition on the relative fluid velocity at the ion surface, whether “stick” or “slip”, or some intermediate condition; and (3) the neglect of electrical interactions between the ion and solvent, inherent in the derivation of Stokes’ law. Although the first source-the extrapolation of continuum hydrodynamics to the atomic regime-would seem to be a prime suspect, the close agreement with experiment given by the Stokes-Einstein diffusion constant5 for uncharged molecules (using the slip boundary condition) encourages the view that the discrepancy for ions results less from the assumption of continuum hydrodynamics than from the third source, Le., the neglected electrical interactions with the solvent. The most important and obvious consequence of these electrical interactions is the solvation of ions, which results in an increase in effective radius of the migrating unit. A more subtle consequence is the so-called dielectric friction, first proposed by Born6 in 1920. Born’s idea was ignored, however, until it was rediscovered by Fuoss’ in an attempt to explain the failure of Walden’s rule. The heuristic nature of Fuoss’ argument left rmm for further theoretical investigation, which was undertaken next by Boyd* and somewhat later by Z ~ a n z i g .The ~ most thorough treatment of dielectric friction now available is that of Hubbard and Onsager,I0 but this is still based on continuum hydrodynamics. (5) See, for example: Alder, B. J.; Alley, W. E.; Pollack, E. L. Ber. Bunsen-Ges. Phys. Chem. 1981, 85, 944. (6) Born, M. Z . Phys. 1920, I , 45. (7) Fuoss, R . M. Proc. Natl. Acad. Sci. U.S.A. 1954, 45, 807. (8) Boyd, R. H. J . Chem. Phys, 1961, 35, 1281. (9) Zwanzig, R. J . Chem. Phys. 1962, 38, 1803. (IO) Hubbard, J. B.; Onsager, L. J . Chem. Phys. 1977.67.4850, Hubbard, J. B. J . Chem. Phys. 1978, 68, 1649.
+
An approach that attempts to extend to the atomic (discrete) regime the macroscopic domain in which ordinary continuum hydrodynamics applies has recently been described by Alder et al.,53” under the term “generalized hydrodynamics.” Although this approach is still in the early phases of development, it shows great promise for illuminating the applicability to transport problems of continuum hydrodynamics in comparison with the more realistic situation of molecules continually interacting through van der Waals forces under the dense packing that characterizes liquids. Another important issue in the derivation of Stokes’ law is the degree of adhesion at the particle/fluid interface. In application to ionic mobility in electrolytes, it is clear qualitatively that there must be some lack of adhesion-or “slip”-between the solution and the ionic unit (i.e., the ion plus its firmly attached bodyguard of solvent molecules): for supposing there were no such slip at the first molecular layer of solvent molecules, we move the boundary surface outward to encompass the next layer and continue the process until a surface is encountered at which slip does occur. That such a surface must exist is guaranteed by the observation of the relative translation of the ionic unit and the (distant) solvent, which would be impossible if all the solvent were rigidly attached to the ion. Unfortunately, this qualitative existence theorem is of little practical use in estimating the magnitude of the slip, since a microscopic slippage at each solvation layer would easily allow the observed ionic mobilities. In view of its central importance for the case of ions in electrolyte solutions, a reexamination of the Stokes’ derivation in the light of the question of particle-fluid adhesion will be a major concern of this paper. This derivation serves to establish both the concept and the Stokes results but also has the pedagogic function of making available in convenient form the hydrodynamic basis of the widely used Stokes’ law. In this sense, although the form of the derivation is new, the early sections are largely expository. Use of vector notation, of course, greatly condenses both the derivation and the equations involved and helps to make the underlying physics visible through the lesser complexity of the equations. In theblater sections, a new concept is introduced that involves an enhanced electrical force acting on an ion as a consequence of local dielectric saturation of the solvent environment-which, so to speak, allows the applied external field to “shine through”. Inclusion of this force enhancement, together with the slip boundary condition, considerably improves the predictions of Stokes’ law. 2. Hydrodynamics in a Viscous Fluid The treatment of hydrodynamic motion for a viscous fluid conventionally proceedsI2 in analogy with the corresponding development for the elastic solid. The deformation of an elastic solid is illustrated in Figure 1, which depicts the strain in terms of the (1 I ) Alder, B. J.; Alley, W. E. Phys. Today 1984, 37(1), 56. (12) Page, L. Introduction to Theoretical Physics; Van Nostrand: Princeton, NJ. 1952.
Application of Stokes' Law to Ions in Aqueous Solution
The Journal of Physical Chemistry, Vol. 94, No. 6,I990
2673
incompressible flow Vu' = 0, V(vV2.u') = vV2.Vu' = 0, so that the pressure p obeys Laplace's equation V2p = 0. Still another approach, used by Landau and Lifschitz,15 notes that from Vi? = 0, u' can be written as the curl of some vector.] To this point we have considered only the forces that act upon a parcel of the fluid. The equation of motion of a body embedded in the fluid also follows directly from Newton's second Jaw. Denoting the mass of the body by M and its velocity by V we obtain the analogue of (2.8) as M d p / d t = $ + $9 d S Figure 1. Deformation of positions in an elastic solid.
shifts in position of two neighboring points of the medium, initially at P and P', to their new positions Q and Q . With reference to Table 11, the usual results for an elastic solid are given in column 2. As shown in the parallelism of column 3 with column 2, the results for the viscous fluid are obtained directly from those of the elastic solid by replacing all strains by strain rates [in laboratory (Eulerian) coordinates], except that for generality also a volumetric compression term is retained in the form @ div S, where @ is the bulk modulus. For convenience we follow the tensor notation of J0os,l3 which uses a period to separate the factors of a dyad. Qenoting the sum of any external body forces (e.g., gravity) by F, Newton's second law applied to a fluid parcel (over which ranges the differential volume d r ) takes the form l(du'/dt)p dr =
F + $9
dS
(2.8)
where p is the fluid density and the stress tensor 9 operates on dS, the (outwardly directed) infinitesimal vector element of the surface that bounds the parcel, to give the differential force acting on that element. If the only body force acting on the fluid is due to the acceleration g of gravity (2.9) Then the application of Gauss' theorem to the surface integral yields S(du'/dt)p d r = l i p d r
+s
V 9dr
(2.10)
The convective (Lagrangian) time derivative of the velocity can be expressed in terms of the (partial) time derivative at a fixed (Eulerian) point plus its rate of change attending the displacement of the fluid parcel. Then, since the volume of integration is arbitrary, the integrands themselves can be equated:
+ uV.5) = p i + v 9
p(dv'/dt
(2.1 1)
For stationary flow, au'/at = 0. If the velocity is sufficiently small, as we shall assume, the second-order term uV-u'can be neglected. Further, ignoring the small effects of gravity, and employing (2.7)
0 = v\k = -vp
+ vv2.v' + ({ + ?7/3)V.VU'
(2.12)
Finally, if the flow is incompressible, Vv' = 0, and $72.5 = vp
(2.13)
We are therefore led to seek solutions of the flow field obeying the equation
(2.16)
wherein @isthe sum of the external forces and the integral extends over the surface of the body. The hydrodynamic drag may be treated by two equivalent pictures: either where the bulk of the Quid is at rest, with the body moving through it (with velocity V), or where the b2dy i_s at rest with the fluid moving around it [having velocity -V (=ku,) a t great distance]. For our purpose, {he second picture is more convenient. Thus the external force F needed to maintain the body in a fixed position in the moving fluid is simply
@ = -$9 & = -a2$9il
dR
(2.17)
wherein the last formjs specialized to a spherical body, for which the surface element dS = 3,aZdR, dR being the differential element of solid angle. From (2.7) with Vu' = 0, the stress tensor is 9 = -pl v[V.v' (V.5),] (2.18)
+
+
where the subscript c denotes the tensor conjugate. The integrand of (2.17) then becomes 9 3 , = -pi,
+ ?)[v.z,k + ilv.v']
(2.19)
Here the subscript k (for "konstant") on the unit vector means that FIkis not subject to the V operator preceding it. This same combination occurs in the triple vector product 7, x (V x 3) = v . 8 l k - 7,V.u' (2.20) which allows rewriting the integrand (2.19) as 9 3 , = -p?,
+ q [ i , X curl v'+
2 av'/ar]
(2.21)
Thus we need the fluid velocity u' and the pressure p .
3. Flow Field around a Spherical Body In applying the foregoing results to the flow around a stationary sphere, we imagine the sphere held fixed by an external force F (to be evaluated) while the fluid-extending to infinity in all directions-streams around it in the upward ( k ) direction. (The shapes of the stream lines of the flow field are discussed in section 7.) We shall thus use either cylindrical ( p , 4, z ) or spherical-polar (r, 0, 4) coordinates as dictated by symmetry and convenience. The cylindrical symmetry of the flow field requires that a given stream line lie on a plane through the z axis having fixed azimuth 4. Thus, curl v' is collinear with (but antiparallel to) the unit vector 4,.The magnitude of curl Cis independent of the azimuthal angle 4, a condition we denote in cylindrical coordinates by setting it equal to a function of p and z only. For later convenience this function is chosen as the partial derivative aflp,z)/ap of a function f(p,z) to be determined: curl u' = -& df(p,z)/ap (3.1)
curl V.5 = 0
(2.14)
V2.curI 0' = O
(2.15)
(Here p is the cylinder radius, not to be confused with the fluid density used above.) In the cylindrical unit vector set (&,&, i ) ,(3.1) can be transformed to give curl v' = -(i X z,)(df/dp) = -$ X Vf = curl if (3.2)
[An alternative approach, used by Lamb,14 notes that since for
wherein the conversion to Vf has utilized the independence off
or equivalently
(13) JOOS, G . Theoretical Physics; Hafner: New York, 1958. (14) Lamb, H . Hydrodynamics; Dover Publications: New York, 1932.
(15) Laudau, D.; Lifschitz, E. M. Fluid Mechanics; Pergamon: London, 1959.
2674
The Journal of Physical Chemistry, Vol. 94, No. 6, 1990
on 6,together with
i
X
i
= 0. Thus
p =p,
u'=if+Vx (3.3) in which the velocity potential x is a scalar function again independent of Returning to (2.15), the requirement
+.
o = V.cur1 U' = v x ( 0 2 . i ~ = [ V ( V ~ Ax] i
(3.4)
shows that V2fmust be a function of z alone, say V2f = -4*h(z)
(3.5)
However, Poisson's equation in this form applied to an infinite medium has no physically admissible (i.e., finite) solution for f unless the source density h is zero everywhere-except possibly at the origin (the sphere center). Thus f must obey Laplace's equation with a 6-function source: V2f = -4a[-2A6(7)]
(3.6)
in which the coefficient of the &function has been chosen as -2A for later convenience. The functionfcan now be taken to be spherically symmetric by allowing the as-yet-undetermined potential x to absorb any dependence on the polar angle 8. Thus the solution of (3.6) can be obtained immediately by analogy with Coulomb's law of electrostatics:
f = -2A/r
+ u,
(3.7)
The conStant of integration is chosen as u, to conform to the velocity ku,, (3.3). of the distant fluid. With this result forx (3.2) now yields curl u' = -4,(2A sin 8 / 1 2 )
(3.8)
Turning to the determination of x , the condition of incompressibility, Au' = 0, applied to ( 3 . 3 ) usingffrom ( 3 . 7 ) yields V 2 x = 2 A h ( 1 / r ) = -2A cos 8/12
(3.9)
This equation has as a particular solution A cos 8, to which must be added the solutions of the homogeneous equation V2x = 0, namely, the solid zonal harmonics:
x = A cos 8 + C(A,r' + B,/r'+')P,(cos
8)
(3.10)
It0
where PIis the Legendre polynomial of degree 1. Since the velocity of the distant fluid has the finite value ku,, which has already been incorporated into the function evidently we must set A, = 0 for all 1. Similarly, since the radial component u,(r,O) of the velocity must vanish everywhere on the particle surface at r = a, evidently Bl = 0 for all 1 # 1. Thus, noting that P,(cos 8 ) = cos 0 and renaming B, as B
x = ( A + B / r 2 ) cos 6
( 3 . 1 1)
Inserting this result, together withffrom (3.7), into ( 3 . 3 ) now gives for the velocity in spherical-polar coordinates:
u'= ?,[(om- 2 A / r - 2 B / r 3 )cos 81 - i l [ ( u , - A / r
+ B/?)
sin 61 (3.12)
Thus, the velocity flow field involves only the two parameters A and B, which are to be determined from the boundary conditions on the surface of the sphere. (The requirement that the radial component of velocity vanish at r = a can be used to eliminate B in terms of A, but the subsequent albegra is simplified by deferring this elimination.) Turning to the pressure p within the fluid, the requirement 09 = 0, (2.12), together with ( 3 . 3 ) and (3.6), leads to vp = vv2.u' = 7v.v2x Thus
v(p - 7V2X) or
Pau et al.
=0
+ 7V2x = p ,
- 2A7 cos 8 / r 2
(3.13)
in which p , is the pressure in the distant fluid, and (3.9) has been substituted for V2x. We thus have obtained all the pieces needed to calculate the surface drag force Wl of (2.21).
4. The Viscous Drag Force In what follows it will be useful to resolve the force exerted on the particle surface into radial and tangential components. To this end we invoke (2.21) W I= -p7, + ~ ( 7 X, curl u'+ 2 du'/dr) using the expressions (3.13) for the pressure p , (3.12) for the velocity u', and (3.8) for curl 6thus 9'7, = [-p,
+ (67 cos 8 / r 2 ) ( A+ 2B/12)]r'l + [(37 sin 8 / r 2 ) ( 2 B / r Z ) ] &(4.1)
Since_by Symmetry the drag force of (2.17) must lie along the z axis, F = kF,, where F, is given by the integral over the particle surface F, = -a2$Z97,
dfl
(4.2)
Then, since by definition i = 7, cos 8 - 3, sin 6, the integrand in (4.2) for general r is given by i\k7, = [-p, cos 8
+ (67 cos2 8 / r 2 ) ( A+ 2 B / r 2 ) ] [(37 sin2 8 / r 2 ) ( 2 B / r 2 )(4.3)
The two terms on the right again correspond to the radial and Noting that over a spherical surface the tangential parts of W,. angle averages in (4.2) are (cos 8) = 0,
(cos2 8) =
y3,
and
(sin2 6) =
y3 (4.4)
evidently in the angle integration of (4.3) the terms in B cancel, and the drag force becomes simply16 F, = - 8 7 4
(4.5)
The negative sign, of course, means that the applied force needed to keep the sphere stationary in the upward-moving fluid is in the negative z direction. In view of the detailed expression (3.12) for the radial and tangential components of the fluid velocity, both of which depend upon both parameters A and B, and the fact that as yet no boundary conditions have been invoked, (4.5) is a remarkable result. Although to determine the numerical value of A, boundary conditions are required on both u,(a) and ue(a), the value of A by itself is insufficient to specify these boundary conditions. Thus the drag force expression (4.5) is the same independently of the boundary condition on ue(a) and thus encompasses all allowable degrees of slip, of which the "stick" case is the no-slip extreme.
5. Viscous Drag with Slip In order to treat the general case involving slip explicitly, the radial component of the force Wl of (4.1) is retained, but the tangential component is replaced by pe(a), where p is the "friction coefficient" between the (stationary) spherical surface and the adjacent fluid moving with the (relative) tangential slip velocity 61DO( a ):
= [-p,
+ ( 6 7 ~ -COS O/a)(A/aU, + 2B/du,)]r', pu,[(l - A/au, + B/a3u,) sin 814
(5.1)
The integral, (4.2), of the z component is given by (16) It is interesting that this result, involving only the coefficient A , can be obtained directly from the Laplacian Vtf= 4r[-2Ab(r)] without reference to the velocity potential x. For this purpose we note that whereas V I = 0 in the body of the fluid, this does not apply within the (rigid) spherical particle. Rather, F_=-#IdS = - s V I dr, where V I = -Vp + vV2u = -Vp + qkVY = -Vp + k8*Agb(i). Then, since the gradient of the average internal pressure p within the particle vanishes, the volume integral of the &function yields (4.5).
The Journal of Physical Chemistry, Vol. 94, No. 6, 1990 2675
Application of Stokes’ Law to Ions in Aqueous Solution -Fz(p) = a 2 $ h N l
= 8rav,[q(A/ao,
dQ
+ 2B/a30,) + (pa/3)(1- A/au,
+ B / a 3 0 m ) ]( 5 . 2 )
this evidently depends upon the three parameters A , B, and p . If we now require in (3.21) the boundary condition u,(a) = 0, B can be eliminated from (5.2) to give - F z ( p ) = 8rao,[q(l - A / a u , ) + ( p a / 3 ) ( 3 / 2 - 2A/au,]
A/au,
+ 2(1 + p a / 3 q ) (5.3)
The virtue of writing the drag force in this seemingly complicated form is immediately evident when it is equated to the general expression of (4.5): for since the first term is identical with 8rqA, the remainder of the expression must vanish-from which
(5.4)
viscosity.
(5.5)
Thus, to zero order in r / a , p = q / r , which corresponds exactly to the friction coefficient between parallel planes of the fluid separated by the molecular diameter T . Note that this result is independent of the parameter CY (although in this model it appears reasonable to take CY = 712, p = 3112).
Thus the force becomes
-F, = 6 r s a u m [ ( 2 s+ p a ) / ( 3 7 + pall
This expression includes the usual Stokes’ law for no slip by setting the frictional coefficient p = 03:
-F, = 6rqau,
(5.6)
The case for complete slip corresponds to the other extreme, p = 0, for which
-Fz = 4rqav,
(5.7)
The boundary condition u,(a) = 0 determines B in terms of A :
0 = u,(a)/u, = 1 - 2 A / a u , - 2B/a30,
Figure 2. Stream lines about a sphere: left for frictional coefficient p = 0; right for p = -. Lines correspond to radii r l / a = 1.0, 1.05, 1.1, 1.2, 1.3, 1.5, 1.7, 2.0, 2.5, and 3.0. Both patterns are independent of
(5.8)
7. The Stream Lines To round out the hydrodynamic description of the flow field around a spherical body, it remains only to determine the configuration of the stream lines. The tangent to a given stream line at any point corresponds to the unit velocity vector v’,: v’, = G / u = Tlu,/u & u e / u a 7, d r &r d0 (7.1)
+
+
in which the infinitesimals are taken along the stream line. Thus
r dO/dr = uo/u,
whence
B/a3u, = ( 1 - 2A/au,)/2 = -pa/4(3q
+ pa)
(5.9)
In turn, this result, together with A from (5.4), can be used to determine the boundary condition on us(a), (3.12),as a function of the friction coefficient p:
ue(a)/um= sin 0(1 - A / a u ,
+ B/a3u,) =
6. Frictional Coefficient between Solvent Layers All of the foregoing discussion applies to a continuum hydrodynamic fluid that is structureless, in contrast with the molecularity of real liquid solvents. It is thus interesting to see what modifications are necessary for a molecular solvent. In order not to stress the continuum theory too severely, however, we consider the solvent molecular diameter T to be much less than the sphere radius a . Then since the integral drag force on every solvent shell must be the same, we can apply the foregoing results to the case in which the first solvent layer sticks to the particle surface [at radius ( a + a)]but the second layer [at radius (a + p ) ] slips over the first. As we shall see, this will prescribe the friction coefficient p between the first “stick” layer of solvent molecules and the second “slip” layer in terms of the solvent viscosity 7. To this end we simply equate the integrated force in the first layer, 6r7um(a+ a),to that in the second layer, ( 5 . 5 ) , using in place of a the radius ( a + P). Thus, apart from common factors that cancel and after some algebraic reduction, this yields P
=
[?/W
Now setting (3 - a =
cc =
- a)111 - 3 ( P - a ) l / ( a + PI1 T,
(6.1)
the diameter of a solvent molecule
(77/7)[1
Substituting the velocity components from (3.12), a simple rearrangement yields 2d[ln (sin O ) ] = -d(ln [rZ(l- 2 A / r v , - 2 B / r 3 0 , ) ] )
- 3(7/a)/(l + P/a)l
(6.2)
(7.3)
which integrates directly to give the equation for the stream line: $ ( 1 - 2 A / r v , - 2B/r3v,) sin2 0 = K
sin 0 [ 3 7 / 2 ( 3 q + p a ) ] (5.10)
Just as a physical check, evidently us(a) goes to zero as p goes to infinity and to ( u , / 2 ) sin 0 as p goes to zero.
(7.2)
(7.4)
The constant of integration K is most easily evaluated in terms of the radius r = b of the stream line in the equatorial plane, 0 = r / 2 . Using the values of A and B, (5.4)and (5.9), for the general case we thus obtain sin2
(7.5)
For the stick (1= respectively to
a) and
slip
(p
= 0) extremes, this reduces
sinZOrtick =
and
It is noteworthy that in both cases the stream lines are independent of both the viscosity q and the friction coefficient p. Figure 2 shows quadrant polar plots of these stream lines for any plane through the z axis. As might be expected, the main difference between
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Pau et al. 00
60
I Er 40
20
0
Figure 3. Electric potential
x-
I
vs distance X in a "sandwich" dielectric.
the plots is that the stream lines for no adhesion ( F = 0) acquire a somewhat greater radius as the polar angle decreases from a / 2 to 0. The stream lines shown in Figure 2 are as would be seen by an observer riding along with the body, Le., in a reference frame in which the body is at rest, with the viscous fluid moving around it. In contrast, Lambt4follows Stokes in focusing on the total flow through a planar circle (normal to the flow axis) as a function of the circle radius and the altitude above the equatorial plane. This total flow is, of course, an integral over a range of velocity stream lines and is thus much less meaningful to the intuition.
8. Stokes Ion Radii Revisited Having established the physical basis of the drag force on a spherical particle in a continuum hydrodynamic medium, we refer again to Table I, which shows a comparison of the Stokes radii for varous-ions calculated from the measured equivalent conductances A. at zero concentration using the stick (column 4) and slip (column 5 ) boundary conditions. Whereas the stick boundary condition leads to impossibly small radii, it is clear that the introduction of slip greatly relieves this disagreement, especially for the ions K+ through C1-. This is a clue that, for microscopic translational motion, the slip boundary condition is more appropriate. This view is supported by the recent work of Alder et al.,5J' Cukier et aI.,l7 and Masters and Madden.18 The change from the stick to slip boundary condition increases the Stokes radii for Br- and I- by 50%, although they are evidently still too small. In the next section, we propose a further modification. 9. Force on an Ion For an external applied potential difference we shall show that the reduction of the dielectric constant due to dielectric saturation about an ion in aqueous solution increases the effective field acting on the ion. To illustrate, consider the simple example of a dielectric medium in the form of a planar sandwich, Figure 3, with two slabs of the same dielectric constant t separated by a slab of lower dielectric constant t'(
