JOUI~NziL OF COLLOID £ND INTERFACE SCIENCE 9.2, 78--99 (1966)

Calculation of the Electrophoretic Mobility of a Spherical Colloid Particle

P. H. WIERSEMA, A. L. LOEB, AXD J. TH. G. OVERBEEK

Van't Hoff Laboratory, University of Utrecht, The Netherlands. Ledgemont Laboratory, Kennecott Copper Corporation, Lexington, Massachusetts Received May 3, 1965

ABSTRACT A new calculation of the relation between the electrophoretic mobility and the/-- potential of a spherical colloid particle is presented. The model consists of a rigid, electrically insulating sphere surrounded by a Gouy-Chapman double layer. The ap- propriate differential equations (which account for both electrophoretic retardation and relaxation effect) have been solved without approximations on an IBM 704 com- puter. The theoretical assumptions and the basic equations ~re stated. A detailed account of the solutions of the equations is pub]ished elsewhere. The present paper contains the complete results. Comparison of these results with those of Overbeek ~nd of Booth shows that, %r high ~--potential ~nd 0.2 < ~a < 50, their approximations gen- erally overestimate the relaxation effect. The application to practical cases (espe- cially colloid particles in solutions of 1-1 electrolytes) is discussed extensively. Finally, the theoretical results are compared with experimental data published by others.

INTRODUCTION particle, it is valid only when ~a }> 1, The calculation of the ~-potential from where a is the radius of the particle and measurements of the electrophoretic mobil- is the reciprocal thickness of the surround- ity (E.M.) requires a theoretical relation ing ionic atmosphere. For moderate values between the two quantities. The oldest of ~a (say, 0.2 < ~a < 50), the so-called relation of this kind, which was derived by relaxation effect leads to important correc- yon Helmholtz (1) and improved by yon tions, which increase with increasing /'- Smoluchowski (2), reads potential. This was found by Overbeek (5) and, independently, by Booth (6). Both U e~ authors expressed the E.M. as a power series X - 4~" [1] in the ~-potential; however, because of mathematical complications they were able In this equation, U is the electrophoretie to calculate only a few terms of the series. velocity, X is the strength of the applied Quantitative validity of their results could d.-c. field (U/X is the electrophoretic therefore be claimed only for small ~'-poten- mobility); e and ~ represent the dielectric rials (~ < 25 my.). constant and the viscosity coefficient, The results of a more general calculation, respectively, of the solution surrounding which was carried out by means of an IBM the colloid particles. 704 digital computer, will be presented in More recent work by Hiiokel (3), Henry this paper. The physical assumptions, which (4), Overbeek (5), and Booth (6) has shown are the same as those used in the work of that the validity of Eq. [1] is rather re- Overbeek (5), are as follows: stricted. In the case of a spherical colloid a. The interaction between colloid par- 78 ELECTROPHORETIC MOBILITY OF A SPHERICAL COLLOID PARTICLE 79

X u

// // ii / k 3 / - II ~ kI=QX

\\\ \ \ /

FIa. 1. Forces in eleetrophoresis. The dotted line represents the ionic atmosphere. tides is neglected and only a single particle particle is neglected. (The order of magni- is considered. tude of this effect has been estimated by b. This particle (plus the adjacent layer means of a calculation that was published of liquid that remains stationary with re- elsewhere (Ta)). spect to it) is treated as a rigid sphere. This paper is written for readers who are c. The dielectric constant is supposed to primarily interested in results and applica- be the same everywhere in the sphere. tions. Therefore, the mathematical problem d. The electric conductivity of the sphere will just be stated here. The analytical and is assumed to be zero. This implies that the numerical methods that were used in solv- charge within the surface of shear does not ing the problem can be found elsewhere move with respect to the particle when the (7a-9). d.c. field is applied. e. The charge of the sphere is supposed STATEMENT OF THE PROBLEM to be uniformly distributed on the surface. In the remainder of this paper, the col- f. The mobile part of the electric double loid particle plus the adjacent layer of liquid layer is described by the classical Gouy- is called "the sphere." The electric charge Chapman theory. of the sphere is assumed positive; consistent g. The dielectric constant of the liquid sign reversal makes the results applicable surrounding the sphere is assumed to be in- to a negatively charged particle. dependent of position. Let us first consider the forces acting on h. Only one type of positive and one type the sphere when it is in uniform electro- of negative ions are considered to be present phoretic motion (Fig. 1). The force exerted in the ionic atmosphere. by the d.c. field, X, on the charge, Q, of the i. The viscosity coefficient of the liquid sphere is surrounding the sphere is supposed to be kl = QX. [2] independent of position. j. Because colloidal solutions follow The second force is the Stokes friction, Ohm's law at the moderate field strengths (a ks = --6v~aU, [3] few volts per centimeter) that are employed in electrophoresis experiments, only terms where ~ is the viscosity coefficient of the that are linear in the field are taken into liquid surrounding the sphere, U is the elec- account in the computation. trophoretic velocity, and a is the radius of /~. The Brownian motion of the colloid the sphere (i.e., the distance from the center 80 WIERSEMA, LOEB, AND OVEP~BEEK of the colloid particle to the surface of where dA is an element of the surface of shear). shear; ~ is the surface charge density de- The two remaining forces are caused by fined by 4¢a2~ = Q (where Q includes both the "ionic atmosphere." The d.c. field exerts the particle charge and other charges that on the ions in this atmosphere a force which may be bound within the surface of shear); is transferred to the solvent molecules. and h is the total electric potential caused The resulting flow of solvent causes a re- by the charges on the sphere and in the tarding force, ks, on the sphere (electro- liquid and by the external field. Hence, phoretic retardation). Furthermore, in the --VA is the total electric force on a unit stationary state which is present shortly charge. In Eq. [6], the subscript a denotes after the application of the d.c. field, the that we take the force at the surface of center of the ionic atmosphere lags behind shear, and x specifies the component of this the center of the particle. This causes an force in the direction of the d.c. field. electrical force, k4, on the particle, which is, The total hydrodynamic force can be ex- in most cases, also a retarding one (relaxa- pressed as tion effect). The two effects symbolized by k8 and k4 + = ff p~ dA, [7] are the same as those defined in the Debye- Hiickel theory of the conductivity of strong where p~ is the stress on an element of the electrolytes (10). In the latter theory, no surface of shear in the direction of the d.c. important error results when the two effects field. This stress is a rather complicated are superimposed linearly; for colloidal function of u, the velocity of the liquid particles the mutual interactions between with respect to the center of the sphere, and the effects are considerable. In this case, of p, the hydrostatic pressure in the liquid. ks must be calculated for an asymmetric It appears that more explicit expressions atmosphere and in the evaluation of k4 for the forces (and hence, for U) can be found the velocity distribution in the liquid must when A, u, and p are known as functions be taken into account explicitly. of position. The functions are governed by a In the stationary state the sum of all set of differential equations that will be forces acting on the particle is zero: given here without derivations, l~ore de- tails concerning these equations can be kl+k2+ k~+ k4 = 0. [4] found in the publications of Henry (4), Equations [2]-[4] can be combined to give Overbeek (5), and Booth (6), and in refer- ences 7a and 8. U - 1 (QX + k~ -~ k~). [5] In the formulation of the differential 67rya equations and boundary conditions, the center of the sphere is considered as the The forces k~ and k4 are functions of the origin of the coordinate system. ~-potential and of several other parameters, The total electric potential, A, which in- such as the radius of the sphere and the cludes the external d.c. field, is governed charges, concentrations, and mobilities of by Poisson's equation: the ions in the double layer. Because these other parameters are known in most cases, V2A = 4~__pp = 4~e (z+~+ -- z-v-) [8] the relation between U and f can be found by calculation of the forces ka and k4. As a starting point, we shall write down where p is the space-charge density in the expressions for the total electrical force, liquid, e is the dielectric constant of the kl + k~, and for the total hydrodynamic liquid, e is the elementary charge, z=~ are force, k2 + k3. The first expression reads: the valences of the ions, and ~ are their local concentrations (number/era)). It is essential that ~+ and v_ are not bulk concen- h~ ÷ k~ = ff ~(--VA)o~ dA, [6] trations and not equilibrium concentrations, ELECTROPHORETIC MOBILITY OF A SPHERICAL COLLOID PARTICLE 81 but ion concentrations in the distorted at- the left-hand side is the sum of the forces mosphere; by the introduction of these acting on a volume element of the liquid. quantities, the relaxation effect enters into The first term represents the friction be- the differential equations. tween this volume element and surrounding For a complete description of the electric portions of the liquid; the second term ex- potential, we Mso need an equation for presses that the volume element tends to A~, the potential within the sphere. This is move towards regions of low hydrostatic the Laplace equation, pressure; the third term is the electrical force on the ions in the volume element; v~a~ = 0, [9] this force is transferred to the liquid. The which expresses that there is no space- third term corresponds to the electrophore- charge within the sphere. tic retardation (ha); if only the first two The unknown concentrations ,+ and v_ terms were considered, the solution of Eq. are governed by two transport equations [11], combined with the integral of the that can be written in combined form as right-hand side of Eq. [7], would give just follows the Stokes friction (h2). Finally, we have [ v.z±e t~T ] Eq. [12], expressing that the fluid is incom- V. ~--~ VA--~-V~.-t-v±u =0[10] pressible: v.u = 0. [12] where f± are the friction coefficients of the ions, lc is Boltzmann's constant, and T is We now have eight differential equations the absolute temperature. The first term (Eqs. [8] and [9], the two equations [10], between the brackets represents the migra- the three components of Eq. [11], and Eq. tion of the ions in the electric field. The [12]) for the eight unknown functions A, second term is a diffusion term in which the Ai, v+, v_, the three components of u, diffusion coefficients of the ions are written and p. Hence, the problem is determinate as kf/f±, according to a theorem of Ein- except for the boundary conditions. Be- stein (11). The last term expresses that the cause all equations, except for Eq. [12], flow of the liquid gives an extra velocity to are of second order, we need fifteen condi- the ions. The sum of the three gives the tions. total flow of the ions (concentration times Far from the particle, the electric field velocity). In the stationary state, the ionic equals the external d.c. field. Hence, distribution around the sphere remains constant in time; hence, the divergence of lira A = - Xx, [13] r~oo the flow is zero. We observe that the so-called surface where r is the distance from the center of conduct±nee, insofar as it occurs outside the the particle and x is the value of the Car- surface of shear, is implicitly taken into tesian coordinate in the direction of the account by the transport equations. This external field. At the surface of shear, r - a, is so because the insertion of local, not bulk, the potential obeys the following relations: concentrations into the first term implies that the ionic conductivity in the double h = Ai at r = a; [14] layer differs from the bulk conductivity. 0A 0Ai Next we come to the hydrodynamic e0r ~i~-- 47r~ at r = a; [15] equations, which govern the functions u and p. First, we have where e~ is the dielectric constant of the ~V X V X u-l- Vp-t- oVA = 0, [11] sphere. Furthermore, everywhere within the sphere where v is the viscosity coefficient of the liquid; Eq. [11] is a Navier-Stokes equation Ai is finite or zero. [16] in which the acceleration terms were put equal to zero (cf. references 5 and 7a); In the solution far from the particle, the 82 WIERSEMA, LOEB, AND OVERBEEK ion concentrations are equal to n±, the ions in the solution. The parameter q0 bulk concentrations: is defined by

lim v_+ = nj:. [17] qo =- ~a/x, [2a] where Because the sphere is nonconducting, and, in the stationary state there is no accumu- 2 4~re2(n+z+2 + n--zff) lation of ions at the surface of shear, the ekT r-components of the flow vectors of the [24] 2 ions vanish at r = a: 4reN~c(z+ +z-) IO00&T Vq=v±z±e kT 1 and r. 1_ ~ vA - ~ w± + ~:~u [18] x ~ -- (z+ + z_)/2z_. [25] =0 at r =a. Here Na is Avogadro's number, c is the Far from the particle, the velocity of the average electrolyte concentration in equiv- liquid with respect to the particle equals alents per 1000 cm. 3. The factor X, which is minus the electrophoretic velocity: unusual, has been included for the following reason. From Eqs. [23]-[25] we find lira u = - U. [19] r~oo ~ 87re2cz_ N~ I/2 At the surface of shear, the fluid velocity q0 = \ 1000aT ] a. [26] is zero: This shows that at constant equivalent con- u = 0 at r = a. [20] centration, q0 is independent of z+, the Finally, because the effect of gravity is valence of the co-ions. The effect of z+ on the neglected, properties of the double layer (including the E.M.) is small. Using the parameter q0, we lira p = constant. [21] may compare numerical results for cases with the same c and z_, but different z+, Taking into account that Eqs. [17] and without having to account for trivial changes [18] each represent two equations, and that of q0 • Eqs. [19] and [20] each have three com- Furthermore, ponents, we have indeed fifteen boundary yo =- e~/kT, [27] conditions. and RESULTS AND DISCUSSION J~T f±. [28] A. DIMENSIONLESS VARIABLES m-4- ~ 6~_~7e2 The results will be reported in terms of The friction coefficients, f+ and f_, of the dimensionless variables in order to make the small ions can be expressed as tables and graphs applicable to various ex- perimental situations (e.g., different tem- f± = N~e ~ z~ [29] peratures or solvents). ~0 ' First, we define the function E by where X0 are the limiting equivalent con- E -- 6~r~e U [22] ductances of the ions. Hence, &T X' Na&T z~= [30] where U/X is the electrophoretic mobility ~t=t= ---- 6~-~- ~0" (E.M.). The parameters involved in the eompu- At this point we wish to mention that, in our tation are z+, z_, q0, Y0, m+, and m_ ; calculation, the eleetrophoretic mobility z+ and z_ are the valences of the small turns out to be independent of e~, the dielec- ELECTROPHOIRETIC MOBILITY OF A SPHERICAL COLLOID PARTICLE $3 tric constant of the particle (cf. references The function Xl*@a) is identical with 7a and 8). fl@a) in Eqs. [33]-[35]. Graphs of the func- tions X3*(~a), Y3*(za), Z3*@a), and Z4*(xa) B. ANALYTICALAPPROXI~ATIONS are given in Fig. 2 of Booth's paper (6b). In the discussion of the numerical results, In a recent paper (12), Pickard arrives at these results will be compared with analyt- a result which is essentially that of Htiekel ical approximations derived by previous (Eq. [32]). A more detailed discussion of authors. In order to facilitate this, we shall Piekard's calculation is given in reference 8. now write these approximations in terms of dimensionless variables. C. NUMERICAL RESULTS FOR By combining Eq. [1] (the approximation 1-1 ELECTr~OLYTES of yon Smoluchowski) with Eqs. [22] and The differential equations given in Part A [27], we obtain were solved by means of an IBM 704 com- E -- 3/~y0. [31] puter for a number of combinations of the parameters z+, z_, q0, Y0, m+, and m_. In In the same notation, the result of Hiickel this Part we shall give the computer results (3) reads for z+ = z_ -- 1. We shall first consider how the electrophoretic mobility depends on E = yo. [32] q0 and Y0 ; the effect of m+ and m_ (which is The approximation of Henry (4) for a non- relatively small) will be discussed later. conducting particle can be expressed as In Table I, E is given as a function of q0 and Y0. In all computations reported in E = yof~(~a). [33] this table, m+ and m_ were chosen equal to A graph of the function f(xa) =- (2/~)f~(~a), 0.184. In aqueous solutions at 25°C., this as well as analytical expressions for this corresponds to limiting mobilities of 70 function, can be found in Henry's paper (4). ohm-1 cm. 2 eq. -~. In Table I, the actual In the present notation, Overbeek's re- computer results (given in 4 digits) are sult (5) for symmetrical electrolytes is combined with interpolated values (given written as in 3 digits). No data were obtained for y0 > 6, because in this region the computer E = yorl(xa) - yo3[z2f3@a) program tailed to give convergent results. [34] The values for q0 ( 0.1 and for q0 > 50 were 4- 1/~(m+ + m_)f4(xa)], found by extrapolation, taking into account where z is the valence of both ions. For un- that Overbeek's equation [34] is valid in the symmetrical electrolytes, Overbeek found: limiting eases q0 ~ 0 and q0--~ ~. In the region q0 < 0.1, this extrapolatoin can be E = yofl(~a) - yo' (z- - z+)f~(~a) done easily, for, at q0 = 0.1, the computer 3 z+m+ + [35] results already coincide with those of Over- - yo z-m-f4(~a). z+ -l- z- beck (cf. Fig. 4). In the region q~ > 50, the extrapolations are less accurate, but it is un- Tables and graphs of the functions f~(~a) likely that the errors exceed a few per cent. were given by Overbeek (5). The data of Table I are graphically repre- The calculation of Booth (6b) is limited to sented in Fig. 2. symmetrical electrolytes. His result can be In Figs. 3 and 4, some values of E, taken written as: from Table I, are compared with the analyt- E = yoXl*(xa) -}- yo3 ical approximations [33], [34], and [36]. For all cases involved here, m+ = m_, with the • [z ~ { x~* (,~a) + Y?(,~a) } result that the last term of Booth's equation • [36] -t- 3(m+ + ~*v-)Za (na)] [36] vanishes. Therefore, the discrepancies between the curves labeled III and IV, repre- -t- Yo4{3z(m+ -}- ~w-)Z~*(na) }. senting Eqs. [34] and [36], are caused only by 84 WIERSEMA, LOEB, AND OVERBEEK

o~

a~

o

0

~d ELECTROPHORETIC MOBILITY OF A SPHERICAL COLLOID PAgTICLE 85

9 / yo=6 / /~- Yo--5

6 .----- Yo:4

5 -----.-- ~

,-,.,.,.,....,... ..~ I Yo: 2

yo--I

0 I Q02 0.05 0.1 0.2 0.5 I 2 5 I0 20 50 I00 200 500 I000

qo Fro. 2. E as a function of qo for different values of y0, z+ = z_ = 1, m+ = m_ = 0.184. For correspond- ing numerical data see Table I. the differences between the coefficients of electrophoretic retardation is remarkably y0~ in both equations. small. This is confirmed by the calculation of The curves labeled II in Figs. 3 and 4 have Booth (6b) ; the function X~*@a) in Eq. [36], been included for the following reason. The which (to his approximation) represents the numerical computations were carried out by nonlinear terms of the electrophoretic re- means of successive approximations; the tardation, is small compared to the other first approximation, in which the relaxation terms in the coefficient of y03. effect is neglected, is actually a numerical The conclusions of Overbeek (5) and of calculation of the electrophoretic retarda- Booth (6) concerning the relaxation effect tion (this point is discussed more extensively are at least qualitatively confirmed by the in references 7a and 8). It is therefore of numerical computations. The relaxation interest to compare this first approximation correction is appreciable and increases with the result of Henry (Eq. [33]), who sharply with increasing ~-potential. The also neglected the relaxation effect. The correction is largest for moderate values of differences between the curves I and II are q0 and is negligible when q0 is very snlall or caused by the fact that I-Ienry (4) had used very large. a Iinearized Poisson-BoRzmann equation, Figure 3 shows that, for moderately low whereas we included the nonlinear terms of values (up to y0 = 2.5) of the ~'-potential, this equation. Figures 3 and ~ show that the our numerical results are in quantitative contribution of these nonlinear terms to the agreement with the approximations of 86 WIERSEMA, LOEB, AND OVERBEEK

6 Z[ Z

5

4

V E 3

I

0 I L I 1 I I 2 3 4 5 '/o FIG. 3. E as a function of y0 for z+ = z_ = 1, q0 = 5, m+ = m_ = 0.184. I: Eq. [33] (Henry); II: first approximation of numerical computations; III: Eq. [34] (Overbeek) ; IV: Eq. [36] (Booth); V: numerical computations.

Overbeek and of Booth. The same is true the data of Table I in most cases. However, for other values of q0 • This provides a check the accuracy will be low in regions where the on the numerical computations. slope of the (E, y0) curve is small. For higher values of the f-potential, it ap- In order to find out how the E.5~. de- pears that the relaxation effect is overesti- pends on the mobilities of the small ions, mated by the analytical approximations [34] we carried out a few calculations with and [36]. For q0 = 5, the difference between m+ ~ 0.184 and/or m_ # 0.184 (cf. Eq. [28]). the approximations and the numerical re- Most of the results are given in Table II. sults becomes appreciable for Y0 > 3. How- The values m± = 0.0655 and m± = 0.0368 ever, this difference has a maximum at correspond to limiting equivalent conduct- qo = 5 (cf. Fig. 4); therefore, the approxi- antes of 196 and 350 ohm-1 era. 2 eq. -1, re- mations are more useful in cases when q0 is spectively, in aqueous solutions at 25°C. smaller or larger than 5. (under these circumstances, the values for It is also of interest to know whether the OH- and H + are 197 and 350, respectively). curve of E vs. Y0 passes through a maximum, Relevant data for m~= = 0.184, taken from for, if this is the case, two different values of the f-potential correspond to a given value Table I, are given in the third column of of the E.M. Such a maximum is implied by Table II. the equations of Overbeek and of Booth. From inspection of these results it follows According to the numerical results (Table that E depends linearly on m+ and m_, and I), E increases steadily with Yo, except for that, when both m+ and m_ differ from 0.184, q0 -- 20 and q0 = 50, where a maximum is the corrections may be linearly superim- observed. Therefore, when E is given, the posed. These conclusions are valid for the value of y0 is unambiguously determined by intervals 0.0368 -<_ m± =< 0.184. They are ELECTROPHORETIC MOBILITY OF A SPHERICAL COLLOID PARTICLE 87

8

E 4

2 i TTT

Oi I 0.01 0.1 I0 I00 I000 qo Fro. 4. E as a function of qo for z+ = z_ = 1, y0 = 5, m+ = m_ = 0.184. I: Eq. [33] (tIenry); II: first approximation of numerical computations; III: Eq. [34] (Overbeek); IV: Eq. [36] (Booth); V: nmnerical computations.

TABLE II for only one combination of q0 and y0, viz., VARIATION OF E WITH 7/2,+ AND ~_ FOR q0 = 10, y0 = 5. The results are shown in A F]~W COM]31NATIONS OF Y0 AND qo Fig. 5. The highest values of m+ and (1-1 ELECTROLYTES) m_ (m~ = 0.74) correspond to the limiting equivalent conductance of the lauryl sulfate E ion (17 ohm -1 cm? eq. -~) in aqueous solu- q~ tion at 25°C, In the larger region considered 0.0655 0.0368 0.184 0.184 0.0368 0.194 here, E depends linearly on m+, but not on 0.184 0.184 0.0655 0.0368 0.0368 0.184 m_. When both m+ and m_ differ from 0.184, the two corrections may be superim- 0.1 3 2.904 2. 921 2. 926 2.927 2.936 posed even for larger values of m+ and m_. 4.693 4.719 4. 807 0.1 5 4.58~ 4.6521 4.66~ Further detailed computations on the rela- 1 3 2.625 2.6571 2.662 2.693 2.710 2.746 tion between E and m_ (i.e., for other coin- 1 5 3.436 3. 506 3. 523 3.648 3.704 3.792 2 5 3.199 3.463 binations of q0 and y0) would have consumed 5 3 2.742 2.833 2.857, 2.890 too much computer time. Extrapolation of 5 5 3.169 i 3.225 3.402 3.463 i 3.533 the data of Table II will lead to significant 10 5 3.470 3.813 I errors only when the counter-ion is very 20 5 4.098 4.4761 slow. We shall return to this matter in Part 50 3 3.921 3.932 3.937 3.976 3.99¢ 4.004 E. 50 5 5.250 5.270 5.275 5.531 5.591 5.622 Values of OE/am+, computed from Table II with the assumption of linear dependence, supported by the analytical approximations are given in Table III. These results show [34] and [36]. that E always decreases with increasing A more detailed calculation, including m+ and m_, i.e., with increasing friction higher values of m+ and m_, was carried out factors of the small ions. This can also be 88 WIERSEMA, LOEB, AND OVERBEEK

expressed by stating that the slower ions In this equation, AE r is the Brownian mo- give the larger relaxation effect. This could tion correction to the E.M., expressed in be expected, because the time of relaxation dimensionless units (cf. Eq. [22]); f~ are the increases with decreasing ionic mobility. friction coefficients of the small ions (these The application of mobility corrections to coefficients were assumed to be equal); practical cases will be discussed in Part E. 6v,a is the friction coefficient of the colloid particle; a is a function of q0, of which the D. TIIE EI~FECT OF BROWNIAN following values are known: q0 = 0, a = 0; MOVEMENT q0 = 0.1, a = 0.004; q0 = 1, a = 0.01~; The more recent theories (13, 14) of elec- q0 = 5, a = 0.030. It follows that the cor- trolytic conductance take into account that rection is positive, as was to be expected. It the so-called central ion, by its own thermal should be observed that a large value of motion, takes part in the relaxation of its q0 (--xa/k) implies either a large value of a ionic atmosphere. When the central ion is a or a large ionic strength. In the first case, colloid particle, the Brown]an motion of this the correction is small because of the a in particle has a similar effect, which was not the denominator of Eq. [37]; the latter case considered in the computations reported in is unlikely to occur in stable colloid systems, Tables I and II. However, for 1-1 electro- or is, at any rate, inconsistent with a high lytes the upper limit of the Brownian mo- ~-potential (i.e., a high value of y0). This tion correction was calculated approximately argument explains why the region q0 > 5 (7a); in these special calculations, the values was not considered in these computations; of m+ and m_ were chosen equal to 0.184. it also leads to the conclusion that AE' is The numerical results, which are given in negligible in most practical cases. In addi- reference 7b, will not be repeated here in de- tion, there is another small effect which tail; they can be represented with good ac- leads to a negative correction to the E.M. curacy by Eq. [37]: At finite concentrations of the small ions, these ions themselves are retarded by their 0 < hE' < ayo~ f* [37] own atmospheres (electrophoretic retarda- f-~ -F 6~r~/a" tion + relaxation effect). In recent treat-

4.0

5.5

E

5.0

2.5 i i t i 0 0.2 0.4 0.6 0.8 m+ FIO. 5. O:Evs. m+ (m_ = 0.184); Q:Evs. m_ (m+ = 0.184);/k:m+ = m_ = 0.3; yl:m+ = m_ = 0.74. ELECTROPHOI%ETIC MOBILITY OF A SPHERICAL COLLOID PARTICLE 89

TABLE III OE/Offt+ AND OE/Ofrb_ ~kS ~q'~UNCTIONS OF q0 AND Yo FOR UNIVALENT ELECTROLYTES

-OE/Om+ --OE/Om- yo qo = 0.I 1 5 50 qo = 0.1 1 2 5 10 20 50

0.15 0.27 0.26 0.10 0.21 0.58 0.78 0.47 0.56 0.59 0.38 0.17 0.91 1.80 1.79 1.98 2.33 2.57 2.34

TABLE IV E AS A FUNCTION OF q0 AND yo FOR 2--2 ELECTROLYTES AND FOR FOUR TYPES OF UNSYMMETRICAL ELECTROLYTES

yo qo = 0.1 0.2 1 5 20 50

z+ = z-= 2 1 0.991 0.984 0.968 1.059 1.267 1.402 m+ = m_ = 0.368 1.5 1.417 2 1.924 1.865 1.637 1.619 2.056 2.456 2.5 2.246 1.826 1.693 2.176 2.737 2.75 2.164 3 2.798 2.601 1.943 2.124 2.800

z+ = 2; z_ = 1 1 1.191 m+ = 0.368 2 2.011 2.009 2.021 2.222 2.604 2.802 m_ = 0.184 2.75 2.79 3 2.971 2.933 2.78 3.58 4.009 4 3.835

z+ = 3; z_ -- 1 1 1.022 1.032 1.088 1.231 1.385 1.445 m+ = 0.552 1.75 2.064 2.352 m_ = 0.184 2 2.042 2.049 2.09 2.824

z+ = 1; z_ = 2 1 0.979 0.970 0.931 1.004 1.224 1.356 m+ = 0.184 2 1.887 1.821 1.547 1.495 1.948 2.377 m_ = 0.368 2.5 2.177 1.719 1.545 2.034 2.628 2.75 1.541 3 2.738 2.479 1.824 1.951 2.656

z+ = 1; z_ = 3 1 0.961 0.941 0.849 0.870 1.104 1.281 m+ = 0.184 12/~ 1.139 1.011 1.339 1.744 m_ = 0.552 2 1.827 1.661 1.208 0.992 1.278 1.772 ments of electrolytic conductance (14), this limit of this correction is known) and to use effect is accounted for; it is n@ected if one ~=~0, not X+(c), in Eq. [30]. computes the parameters m~ (cf. Eq. [30]) from the limiting equivalent conductances, E. OTHER TYPES OF ELECTROLYTE X~°, of the small ions. This effect can be in- Additional types of electrolyte that have eluded by using X±(c) (the equivalent con- been considered are: 2-2, 2-I, 3-i, I-2, and duetances at the given small ion concentra- 1-3 (2-1 indicates that the positive ion is tion) instead of X±°. A few calculations (7a) bivalent and the negative ion is monovalent, which were carried out using the data of etc.). The results are given in Table IV. In Tables II and III have shown that this ef- all these computations, the values of m+ and fect cancels the Brownian motion correction m_ were chosen equal to 0.184 z+ and 0.184 to a large extent. Therefore, in practical z_, respectively. Therefore, all data apply applications, the best one can do at present to ionic mobilities of 70 ohm-1 cm? eq. -1 at is to neglect zXE' (because only an upper 25°C. in aqueous solutions (cf. Eq. [30]). 90 WIERSEMA, LOEB, AND OVERBEEK

It should be noted that the data apply to E are plotted against qo' =- qo(z+/z_)U~ in- a positive colloid particle. In practical cases stead of against q0. This is done because where the colloid is negative, the numerical q0 is proportional to (cz_) ~j2 (cf. Eq. [26]), data for 1-2 electrolytes should be used when with the result that variation of z_ produces the actual electrolyte is Ba(NOa)2, etc. a trivial variation of the coordinate q0 • The Because of convergence problems, the variable q0' is proportional to (cz+) ~12. There- results reported in this section are limited fore, all data for a given q0' and different to rather low values of y0 • The variation of values of z_ (e.g., KC1 and K2SO4) corre- E with m+ and m_, and the Brownian mo- spond to the same equivalent concentration, tion correction, have not been considered, c, and can be conveniently compared. In because only a limited amount of computer Fig. 7, which represents the effect of z+, time was available. q0 is used because in this case q0 is constant For 2-2 electrolytes, the numerical results for constant c, even if z+ changes. and the differences between numerical and Comparison of Figs. 6 and 7 shows that approximated values follow the same pat- the valence of the counter-ions has much tern as those for univalent electrolytes. more effect on the E.M. than has the valence In Figs. 6 and 7, some data for different of the co-ions. The curves labeled IV in valence types of electrolytes are compared. Figs. 6 and 7 correspond to zero relaxation The effect of z_, the valence of the counter- effect. It follows that multivalent counter- ions, is represented in Fig. 6. The values of ions give a negative relaxation effect which

~.0

/ I" / 2.5 i / /1 I

7 / 7 f / 2.0

E

i

J o I

0.01 0.1 I I0 100 I000 qg FIG. 6. Effect of valence of counter-ions on the E.M.; Yo = 2. Drawn lines: numerical values (I: z+ = z_ = 1; II: z+ = 1, z_ = 2; III: z+ = 1, z_ = 3). Dotted lines: approximations for z+ = 1, z_ = 3 (IV: first approximation of numerical computations; V: Eq. [35] (Overbeek)). ELECTROPHORETIC MOBILITY OF A SPHERICAL COLLOID PARTICLE 91 5.0 f~

2.5 // J/'/ ///" / /~

j/ /

t f t j 2.0 ~

1.5 0.0[ QI 10 100 1000 qo FIG. 7. Effect of valence of co-ions on the E.M.; Y0 = 2. Drawn lines: numerical values (I: z+ = z_ = 1; II: z+ = 2, z_ = 1; III: z+ = 3, z_ = 1). Dotted lines: approximations for z+ = 3, z_ = 1 (IV: first approximation of numerical computations; V: Eq. [35] (Overbeek)). increases in absolute value with the valence of the electrophoretic mobility, U/X, has of the counter-ions. For multivalent co-ions, been obtained and that the radius, a, of the most of the data show a negative relaxation particles is known. Then the mobility values effect which decreases in absolute value with should be converted to the dimensionless increasing valence of these ions; the relaxa- quantity E (Eq. [22]). We shall give two tion effect is positive for 3-1 electrolytes examples (aqueous solutions at 20°C. and and 0.01 < q0 < 1. A positive relaxation 25°C.), using the following numerical values: effect was also predicted by Overbeek, but e = 4.803 X 10 -1° e.s.u. according to the computer results it is less v20 = 0.01005 poise. pronounced. Finally, we recall that, for v25 = 0.008937 poise. univalent electrolytes and y0 = 2, Over- E20 = 80.36. beek's equation [34] gives practically the e25 = 78.54. same results as do our computations. It k = 1.3805 ) 10 -z6 erg deg. -~. follows that Eq. [34] is a much better ap- 0 ° = 273.16°K. proximation than is Eq. [35]. This was to be When the mobility, U/X, is expressed in expected, because the coefficient of y0s in era. 2 volt -1 See. -1, Eq. [35] is incomplete, as was mentioned by E = 0.8387 X 104(U/X) at 20°0. [38] Overbeek (Ab). E = 0.7503 X 104(U/X) at 25°C. [39] F. APPLICATIONS TO PRACTICAL CASES Next, the value of q0 should be computed We shall now discuss the calculation of from the particle radius and the properties /'-potentials from E.M. measurements. The of the surrounding electrolyte. From Eq. discussion applies especially to univalent [26], if we use Na = 6.0226 X 1023 and the electrolytes; at the end of this Part we shall numerical values already listed: indicate to what extent it can be generalized q0 = (0.3279 X lOS)(cz-)l/2a at 20°C.; [40] to other valence types. Suppose that a set of experimental values q0 = (0.3286) lOS)(cz_)I/~a at 25°C,. [41] 92 WIERSEMA, LOEB, AND OVERBEEK

Here c = concentration in equivalents per must be calculated from the limiting mobili- 1000 cm2 aqueous solution. ties, ~,±0, of the small ions. Using Ec1. [30] At this stage it is advisable to decide and the listed numerical values, we find: whether some analytical approximation can be used. First, one should inspect Fig. 2 and m~ = 11.51 z_~ at 20°C.; [42] check whether the experimental combina- tion (E, qo) lies within a region where E does not depend on q0 • If this is the case, one may m~ = 12.86 z_~ at 25°C. [43] k± ° use either Eq. [31] (q0 >> ]) or [32] (q0 << 1). If not, the next step is to find out whether Here k± ° are given in ohm-1 cm. 2 eq. -i. When Henry's approximation (Eq. [33]) is valid. a mixture of univalent electrolytes is pres- This can be done by inspection of Fig. 8. ent, m+ and m_ may be calculated from the For combinations of E and q0 which lie number averages of k+ ° and k_°; this ap- above the drawn line (labeled I), Eq. [33] proximation is not based upon the theory, will lead to an error of more than 1 Inv. in but the error will be small because of the the ~-potential; for combinations (E, q0) small effect of m± on the E.M. lying above the dotted line (II), the error The next step is to decide whether Over- will exceed 2.5 InV. In this context, an error beek's approximation (Eq. [34]) can be means a difference between two values of used. This can be done by means of Fig. 8. ~" calculated by means of Eq. [33] and by The drawn line (III) corresponds to an error means of the computer results. The lines of of 1 Inv. in the ~-potential, the dotted line Fig. 8 apply to m+ = m_ = 0.184 (k± ° = (IV) to an error of 2.5 inv. These lines do 70 ohm-i cm. 2 eq.-0; for slower ions (espe- not shift appreciably when k+ ° and/or cially slower counter-ions), the lines would ),0 differ from 70 ohm -1 cm. 2 eq. -1, because shift to slightly lower values of E. the effect of k~ ° is included in Eq. [34] as If it is found that Eq. [33] is not suffi- well as in our computation. If it is found ciently accurate, the values of m+ and m_ that Eq. [34] can be used, it is preferable to

6

\ //

4

E 3 't

0 0.01 0.1 10 100 qo FIG. 8. Errors of analytical approximations. I and II: Eq. [33] (Henry); III and IV: Eq. [34] (Over- beck); V: limitations of computer results. For further explanation see text. I~LECTROPHORETIC MOBILITY OF A SPHERICAL COLLOID PARTICLE 93 do so, because the calculation of ~ by means curvature of the (E, m_)-plot is the same as of the present numerical data is somewhat in Fig. 5. more time-consuming. Finally, the corrected values of E are When our numerical data must be used, plotted against y0. From this graph one one may proceed as follows. First, several finds y0, using the value of E that follows curves of E vs. q0 are constructed from the from experiment. Because Yo = e~/kT, ~ is data of Table I (cf. Fig. 2). Next, from these found from the equations curves a set of values of E is derived for the value of q0 which follows from the experi- ~" = 25.26 Y0 my. at 20°C. [45] mental data. These values of E apply to /- = 25.69 y0 my. at 25°C. [46] m+ = m_ = 0.184 and must be corrected if necessary. At this point we wish to mention that the This correction can be found from the upper drawn line in Fig. 8 shows the limita- data of Part C; it should be observed that tions of the computer results for 1-1 elec- these data apply to a positive colloid par- trolytes. If an electrophoresis experiment ticle. It is best to assume that E varies implies a combination of E and q0 (e.g., linearly with m+ and m_ and to use the data q0 = 5, E = 4) which lies above this line, it of Table III, because no detailed curves of is not possible to calculate a ~-potential from E vs. m+ and m_ are available. As was men- the computer data, either because the com- tioned in Part C, this will lead to significant putations were not carried through far errors only when the counter-ion is very enough (i.e., for y0 =< 6 only) or because the slow. (An additional correction for that case E, y0 curve shows a maximum. will be suggested below.) The required The surface charge density, ~, can be ob- values of OE/Om. can be found by inter- rained using the data given in references 7 polation from Table III. First, OE/Om+ and and 9, where the relation between ¢ and ~" is OE/Om_ are plotted against qo for yo = 3 given in tabulated form. In the context of and Y0 = 5. From these plots one obtains electrophoresis, ¢ is defined by observing the differential quotients for the proper that 4~a2¢ is the charge contained within a value of q0 and for two values of y0. Next, sphere of radius a, where a is the distance OE/Om+ and OE/Om- are plotted against between the center of the particle and the Yo. In constructing these plots it should be slipping plane. observed that OE/Om:~ = 0 for y0 = 0, and We shall now indicate briefly how much that O/Oyo(OE/Om~:) = 0 for Yo = 0, as can of the preceding discussion can be applied to be concluded from Eqs. [34] and [361. Even nonunivMent electrolytes. Tim numerical then, these graphs will not be very accurate, data for these valence types are rather in- but this is not serious, because we are deal- complete and are therefore less suitable for ing with small corrections. From these plots accurate calculations of/~. one obtains, for every value of yo, the total The equations of this Part (except for Eq. correction, zXE, according to the equation [44]) are valid for any valence type. Hence, the calculations of E and q0 are straight- aE forward. However, in comparing computer AE = (m+ -- 0.184) -- data with analytical approximations, one Om+ [44] should remember that q0 equals ~a/h not ~a. OE -1- (~v- -- 0.184) am-~" For 2-2 electrolytes, the approximations of Henry and of Overbeek break down at much lower values of ya (and of E) than The algebraic values of AE must be added they do for 1-1 electrolytes. This also ap- to the values of E that were obtained for plies to unsymmetrical electrolytes when the m± = 0.184. For very slow counter-ions, it counter-ions are multivalent (cf. Fig. 6). is advisable to correct the value of OE/Om_ ; When the co-ions are multivalent, the situa- the correction can be estimated by assuming tion is different (Fig. 7). In this case, that, for all combinations of q0 and y0, the Henry's approximation is better for 2-1 94 WIEI~SEMA, LOEB, AND 0VERBEEK

and 3-1 electrolytes than it is for the 1-1 lying the present computations can be satis- type, and it also happens to be more ac- fied by the choice of the experimental sys- curate than Eq. [35] of Overbeek. tem. It is desirable that the colloid particles Because the variation of E with m+ and be rigid, impenetrable to the surrounding m_ was not computed for nonunivalent solution, spherical, and nonconducting. The types, it can only be estimated with the aid colloid concentration should be sufficiently of the analytical approximations. According low to avoid overlapping of ionic atmos- to Overbeek's equation [34] for symmetrical pheres. Mixtures of different valence types electrolytes, OE/Om+ and OE/Om_ do not should be avoided. Important parameters, depend on z+ and z_ (the valences enter into such as ~a, must be well defined; this ex- the relations between m~ and X~°, Eqs. [42] cludes experiments with particles of un- and [43]). Hence, for 2-2 electrolytes it known size or a wide size distribution. seems reasonable to calculate OE/Om=efrom Furthermore, it is desirable that certain Table III and to compute the mobility cor- parameters can be varied independently. rection from Eq. [44], replacing 0.184 by This requirement causes a few problems. 0.368. This correction can then be applied to When the surface potential of the particle is the values of E found from Table IV. changed by varying the concentration of the A similar procedure can be applied to potential-determining ions, the value of ~a unsymmetrical electrolytes, taking into ac- may also change. This problem is not serious count that, for this case, Overbeek's Eq. because it can either be avoided (by using a [35] predicts that OE/Om~=is proportional to relatively high concentration of indifferent z~=/(z+ + z_). For example, for a 1-3 elec- electrolyte) or be taken into account in the trolyte (positive colloid), the value of calculation of ~a. The variation of xa is more OE/Om_, computed from Table III, should difficult. When this is done by changing the be multiplied by a factor of (3~)/(1/~) = 3/~; ionic strength, the f-potential may also the result should be inserted into Eq. [44], change and a correction for this important replacing (m_ - 0.184) by (m_ -- 0.552). side effect requires additional theoretical as- Numerical data concerning the relation sumptions. Therefore, a better way of vary- between z and ~ for unsymmetrical elec- ing ~a is to work with different monodisperse trolytes can be found in reference 9. colloidal solutions of varying particle size. Of the many electrophoresis experiments G. COMPARISON OF T~EORY AND that were reported in the literature, very EXPERIMENT few (15-17) were carried out systematically The theoretical computations predict a with the purpose of testing the more recent relation between the E.M. and the f-po- theories. The other examples that will be tential. Of these two, the E.M. can be quoted were chosen according to the points measured directly, whereas the ~-potential of view given above, with a preference for (or changes of f) cannot. This makes it cases in which the relaxation effect is ap- rather difficult to find a really quantitative preciable. Only examples with 1-1 elec- experimental test. trolytes were considered because the present Following an indirect approach, one may theory for other valence types is rather in- try to find the ~-potential from other experi- complete. The E.M. values were converted ments, such as eleetroosmosis, streaming to the dimensionless variable E, in order to potential, determination of electrocapillary account for differences in temperature and curves, and titration. Although it is possible to facilitate the use of data (such as Fig. 8) to obtain additional information in this way, given previously in this paper. this discussion will be limited to measure- The more important parameters in the ments of the E.M. itself. theory are f and ~a. We shall first consider Before quoting examples from the litera- measurements of the E.M. as a function of ture, we shall discuss some requirements the concentration of potential-determining that should preferably be met by a suitable ions. This method of testing the theory is experiment. Some of the assumptions under- rather limited because it is not known quan- ELECTROPHORETIC MOBILITY OF A SPHERICAL COLLOID PARTICLE 95

TABLE V considered by Levine and Bell (24) as a COMPARISON OF EXPERIMENTAL AND THEORETICAL support of the theory of the discreteness-of- MAXIMUM VALUES OF THE ELECTROPHORETIC. charge effect.) MOBILITY OF SILVER HALIDE SOLS The silver halide sols are not entirely suitable for testing, because the particles Ka Sol Ref. I Added electrolyte (expt.) (theor.) are not spherical. This condition is more nearly satisfied by certain soap micelles. 3 -}-3.41 3.72 Stigter and Myse]s (25) measured the E.M. AgBr 20 2.5 mmoles ~.5 ~3.23[ 3.92 of sodium lauryl sulfate micelles as a func- { KNOa --3.60 tion of soap concentration and ionic strength. AgI 21 12 mmoles t +3.6 I 3.60 They extrapolated to the CMC as the EiM. KNOs was found to decrease with increasing AgI 22 -- i-5 micelle concentration. The extrapolated --3.14 values, converted to E, are given in Table Agl 23 -- 1.23 9+3.4 3.6 VI as a function of xa. The third colunm of this table gives the corresponding maxima titatively how f changes with the surface predicted by the theory of Overbeek. The potential, ¢0. However, one important next column shows the values of E (for Y0 = feature of the theory can be tested rather 6, i.e., f = 150 my.) that follow from the effectively in this way. The data of Table I present theory. The latter data are not predict that, for intermediate values of xa, maxima or limits, but the end points of the E.M. does not exceed certain maximum ascending theoretical curves (no computa- (or limiting) values, whatever the value of tions for Y0 > 6). The values in columns 3 may be. Hence, it is of interest to find out and 4 were corrected for the small ion mo- whether mobi]ities have been observed that bilities of the Na + and ]auryl sulfate ions do exceed these theoretical limits. that are present in the system. All experi- Most inorganic colloids do not show very high mobilities. Holliday (18) reports a TABLE VI value for a stable gold sol, which gives E = MOBILITY AND ~'-POTENTIAL OF SOAP 3.26 at va = 0.833. A number of experiments MICELLES on silver halide sols are summarized in t ~(mV) Table V. In all these experiments, the E.M. E (present theory, was measured as a function of pAg. 0nly ga [E (expt.) (OvEer~ek) Stigter yo = 6) and Present maximum values are quoted; the theoretical Mysels theory results apply to m+ = m_ = 0.184, but the mobility corrections are small, especially 0.61 3.41 2.53 3.37 101 162 when most of the ions are K + and NOa-. 0.86 3.20 2.80 3.46 92.3 116 The value E = -4.5 found by Troelstra 1.32 2.88 2.64 3.27 80.9 97.4 (21) exceeds the maximum (1El = 3.6) 1.69 2.72 2.56 3.15 75.0 90.5 that follows from the present theory. The 2.40 2.57 2.50 3.01 68.3 80.7 data for va = 4-5 are part of a complete curve of the E.M. vs. pAg, which was meas- TABLE VII ured by Parfitt and Smith (22). The E.M. MOBILITY OF POLYSTYRENE LATICES values, which were kindly sent to the authors I E [ (expt. limiting by Dr. Parfitt, show a maximum (E = values) J /~max. l Latex (theor.) -3.14) that exceeds the theoretical limit Dialyzed Undialyzed for ~a = 4 and just coincides with the limit for ~a = 5. (The values of ~, computed from A 2.22 2.13 2.40 3.09 this curve by means of the present, theory, B 3.80 2.22 3.01 definitely pass through a maximum as pi de- C 8.93 2.59 3.08 3.28 D 13.6 3.56 3.55 creases. This detail is mentioned here be- E 15.6 3.80 4.11 3.68 cause a maximum in the f-potentiM is 96 WIERSEMA, LOEB, AND OVERBEEK

mental values exceed the maxima predicted Next, we shall consider some experiments by Overbeek and by Booth. In the last two in which the E.M. was measured as a func- columns of Table VI, the values of f pub- tion of particle size, at constant composition lished by Stigter and Mysels (25) are com- of the medium surrounding the particles. As pared with those computed from the present indicated above, this is the best way of theory (the first value in the last column checking the theoretical relation between was found by a short extrapolation of the the E.M. and ~a. theoretical curve). The differences between Systematic measurements of this type the two sets of values are appreciable, but were carried out by Kemp (16) on sols of they do not seem to invalidate the conclu- gamboge and silica, with the purpose to test sions drawn by Stigter and 3/iysels from the theory of Henry (4). Because the mobili- their experiments. The interpretation of ties are rather low, the results are not suit- these experiments has been recently dis- able for testing the present theory. Next, we cussed again by Stigter (26). come to experiments of Mooney (15), Rather high mobilities are also found in Jordan and Taylor (31), and yon Stackelberg polymer latices. This can be concluded from and Heindze (17). All these authors have papers by March and co-workers (27), used emulsions; there is some doubt as to Voyutsky and Panich (28), and Sieglaff and whether emulsion droplets behave as rigid Mazur (29). These data are not suitable for particles, but recent publications are reas- a quantitative comparison with theory, be- suring on this point. Anderson (32) meas- cause 1-2 electrolytes (Na2HPO4-buffer) are ured the E.M. of n-octadecane dispersions present in the system (27), temperature and at a series of temperatures above and below ~a-value are not reported (28), or the values its melting point and did not find any dis- of ~a are too high (29) to obtain an appreci- continuity. From experiments on drops able relaxation effect. Recently, systematic rising or falling in water, Linton and Suther- E.M. measurements on monodisperse poly- land (33) conclude that inside circulation styrene latices of five different particle sizes occurs only in large droplets consisting of were carried out by Shaw and Ottewill (30), polar oils, and that small impurities (sur- who very kindly permitted us to quote from factants) at the interface are sufficient to their results. The polystyrene particles are stop this circulation. Mooney (15) has charged by built-in COOH groups and by reported some very high mobilities; e.g., in adsorbed laurie acid. The E.M. of the par- his Fig. 4, E -- 7.5 at ~a = 33, assuming a ticles increases with increasing pH until a temperature of 25°C. This value of E is much limiting value is reached at a pH of about 7. higher than the theoretical maximum for this The limiting values, measured at pK = 7.6 na (c/. Fig. 8), but this might be due to inner and 25°C. in 5 X 10-4 M NaC1, are given circulation, for the oils used by Mooney are in Table VII (dimensionless units). The polar and no stabilizers are mentioned. For values given in the third column of this this reason, and because no temperatures table were measured after the latices had are reported, no quantitative conclusions been dialyzed exhaustively; in these cases, will be drawn from this work; qualitatively, no laurie acid was present in the system. the dependence of the E.M. on particle size The last column of Table VII gives the is consistent with the theory of the relaxa- maximum values of E predicted by the tion effect. Jordan and Taylor (31) measured present theory (corrections for the mobilities the mobifities of deealine drops for two dif- of the small ions have been included). The ferent sizes (xa -- 2.7 and xa = 136). The values of the E.M. measured in latex E are mobility ratio was found to be in reasonable significantly higher than the predicted quantitative agreement with Overbeek's maximum. theory. Because ~ is not large in this case From the preceding discussion it follows (ca. 100 my.) the difference between Over- that experimental mobifities exceeding the beek's theory and the computer results does theoretical limits are rather exceptional, and not exceed 10 % at xa -= 2.7 and is negligible that the differences are relatively small. at ×a = 136. The detailed investigation by ELECTROPHORETIC MOBILITY OF A SPHERICAL COLLOID PARTICLE 97 yon Stackelberg and Heindze (17) was within the slipping plane was not accounted carried out with the purpose of testing the for, but this is not likely to occur, because theories of Overbeek and of Booth. The the ions in this region are probably at least authors conclude that theory and experi- as firmly bound to the surface as are the ment are in good qualitative agreement. molecules of the solvent. Because f < 100 my. and xa > 10 in these If the corrections mentioned so far were experiments, they also confirm the present taken into account, they would lead to theory, which gives about the same results lower theoretical mobilities. Since our model, for these parameter combinations. Similar as it is~ already leads to low theoretical conclusions can be drawn from the measure- values, we still need a positive correction in ments of Shaw and Ottewill (30) on poly- order to explain the discrepancies between styrene latices. These authors report that theory and experiment. The Brownian mo- the E.M. values for latices A, B, and C, tion correction is positive indeed, but it is measured in 5 X 10-s M NaC1, exhibit a far too small to account for the differences. minimum when plotted against xa; this Another source of error might be the as- minimum occurs in the region 1 < xa < 2, sumption of a continuous surface charge. in agreement with the theoretical prediction. The discreteness-of-charge effect certainly changes the relation between ~0 and/', and It. COS~CL~SmNS therefore the mobility at a given ¢J0. From From the preceding literature review it can a recent paper by Levine and Bell (24), be safely concluded that the theory pre- however, it seems correct to conclude that sented in this paper is qualitatively correct. this effect is mostly confined to the inner Furthermore, some critical experiments regions of the double layer, with the result (silver halide sols, soap micelles) show that that the relation between ~" and the E.M. the present theory explains more facts is not affected. than do the preceding theories. However, a Finally, there are several objections few experiments on AgI sols (21, 23) and, against the use of the classical Gouy-Chap- especially, certain high mobilities by Shaw man theory. La Mer (36) recently expressed and Ottewill (30) in polystyrene latices in- the view that the use of the complete dicate that our calculated mobilities are Poisson-Boltzmann equation is fundament- probably somewhat low. This implies that ally incorrect for laNe colloidal particles, one or more assumptions used in the theory and that a more advanced statistical treat- are not quite correct. A few possibilities will ment is to be preferred. Levine (37) con- now be considered briefly. eludes that the uncorrected Poisson-Boltz- In the present model it is assumed that mann equation can be applied at moderate e and v have bulk values in the region out- potentials (< 100 inv.) and small electrolyte side the slipping plane. These assumptions concentrations (<0.01 M for a univalent were examined by Lyklema and Overbeek electrolyte). At these limits, the various (34). The authors found that the correction corrections may already add up to 10 %- for decreased values of e in the double layer 20%. In our treatment of the relaxation is mostly negligible. It was estimated that effect, we have assumed the validity of the the correction for increased viscosity (the Poisson-Boltzmann equation in the mobile viscoelectric effect) might be appreciable part of the double layer, up to f-potentials (i.e., 20 % at high ~" and high ionic concen- of 150 my. It is quite possible that this as- tration). However, there are indications sumption has led to a slight overestimation that the viscoelectric effect is overestimated of the relaxation effect, but no definite con- by Lyklema and Overbeek, as has been clusion can be drawn as long as the various pointed out recently by Stigter (35). corrections have not been applied to specific It has already been mentioned that, as calculations of this effect. far as the region outside the slipping plane For a better understanding of electro- is concerned, surface conductance is in- phoresis, more systematic experiments, of eluded in our model. A surface conductance the type we have indicated, seem to be most 98 WIERSEMA, LOEB, AND OVERBEEK needed at this stage. If the theory is to be 7. (a) W~EaS~MA, P. It., Thesis, Utrecht, 1964. developed much further, it will probably be b. Ibid., p. 108. necessary to use an improved Poisson- 8. WIERSEMA, P. m., LOEB, A. L., AND OVER- Boltzmann equation or a new statistical ~3EEX, J. Tit. G., To be published. 9. LOEB, A. L., WIEaSEMA, P. H., AND OVEa- treatment of the double layer. In this con- EE~K, J. Tit. G., "The Electrical Double nection we wish to mention the work of Layer around a Spherical Particle." The Friedman on cluster expansions, which has M.I.T. Press, Cambridge, Massachusetts, been applied to equilibrium properties of 1961. strong electrolytes (38) and to electrolytic 10. DEBYE, P., AND I-I~CKEL, E., Physik. Z. 9.4, conductance (39), and a paper by Buff and 305 (1923). Stillinger (40), who have given a new treat- 11. EINSTEIN, A., "Investigations on the Theory merit of the ion and potential distribution of Browian Movement," Chapter 1. Dover in the double layer. It seems possible that Publications, New York, 1956. this type of approach will be adapted to the 12. P~cl~Altn, W. F., Kolloid-Z. 179, 117 (1961). 13. ONSAGER, L., Physik. Z. 28, 277 (1927). study of nonequilibrium phenomena in col- 14. Fuoss, R. M., AND ONSAG~IZ, L., Proc. Natl. loidal systems. Acad. Sci. U.S. 41, 274, 1010 (1955). 15. MO0NEY, M., J. Phys. Chem. 85, 331 (1931). ACKNOWLEDGMENTS 16. KEMI~, I., Trans. Faraday Soe. 31, 1347 (1935). The authors gratefully acknowledge the sup- 17. STACKELEEI.IG,M. x/ON, AND HEINDZE, H., Z. port received from the following sponsors: the Elektrochem. 61, 781 (1957). Stichting voor Fundamenteel Onderzoek der Ma- 18. HOLLIDAY, A. K., Trans. Faraday Soe. 46, terie (Foundation for Fundamental Research on 447 (1950). Matter), The Netherlands, the Nederlandse 19. DAvIEs, K. N., AND ItOLLIDAY, A. K., Trans. Organisatie voor Zuiver Wetenschappelijk On- Faraday Soc. 48, 1061, 1066 (1952). derzoek (Netherlands Organization for Pure 20. JONXER, G. H., Thesis, p. 33. Utrecht, 1943. Research); and the U.S. Army (Signal Corps), the 21. TROELSTI~A, S. A., Thesis, p. 105. Utrecht, U.S. Air Force (ARDC), and the U.S. Navy 1941. (Bureau of Ships and Office of Naval Research) 22. SMITg, A. L., Ph.D. Thesis, University of under contracts administered by the Research Nottingham, 1963. Laboratory of Electronics, Massachusetts Insti- PAtlFITT, G. D., AND SMITH, A. L., To be tute of Technology. Thanks are also due to the published. staff of the Computation Center at MIT, where 23. WATANABE, A., Bull. Inst. Chem. Res. Kyoto most of the computations were performed. The au- Univ. 38, 179 (1960); el. p. 194. thors are indebted to the Internationale Bedriifs- 24. LEWNE, S., AND BELL, G. M., J. Phys. Chem. machine Maatschappi] (International Business 67, 1408 (1963). Machines Corporation) at Amsterdam for provid- 25. STIOTER, D., AND MYSELS, K. J., J. Phys. ing the computer time required for a number of Chem. 59, 45 (1955). essential computational checks. 26. STIGTER, D., Paper presented at the 4th In- ternational Congress on Surface .Activity, REFERENCES Brussels, September, 1964. 1. H~LM~O~,TZ, If. yON, Ann. Physik 7, 337 (1879). 27. (a) MAlice, S. It., TUlZNEVL~, D., Ai~D ELD~I%, 2. SMOLUCI-IOWSKI,M. yON, Bull. acad. sci. Cra- M. E., J. Am. Chem. Soc. 70, 582 (1948). covie, (1903), 182. (b) MAtroN, S. H., AND BOWLEt~, W. W., J. 3. Hi~CKEL, E., Physik. Z. 25, 204 (1924). Am. Chem. Soc. 70, 3893 (1948). 4. H~N~?c, D. C., Proe. Roy. Soc. London A133, 28. VOYXZTSKY,S. S., Am) PAmcH, R. M., Colloid 106 (1931). J. (USSR) (English Transl.) 19, 273 (1957). 5. (a) OVEaBEEK, J. T~. G., Thesis, Utrecht, 29. SIEGLA~F, C. L., AND MAZVR, J., J. Colloid. 1941. Sci. 15, 437 (1960). (b) OV~I~EEEK, J. Ttt. G., Kolloidchem. Beih. 30. (a) SttAw, J. N., Doctoral thesis, University 54, 287 (1943). of Cambridge, 1965. (b) SitAw, J. N., AND (c) OVEltBEEII, J. Ttt. G., Advan. Colloid Sci. OTTEWILL, R. It., Nature 208, 681 (1965). 3, 97 (1950). 31. JORDAN, D. O., AND TAYLOR, A. J., Trans. 6. (a) BOOTH, F., Nature 161, 83 (1948). Faraday Soc. 48, 346 (1952). (b) BOOTtt, F., Proc. Roy. Soc. London A,203 32. ANDERSON, P. J., Trans. Faraday Soc. 55, 514 (1950). 1421 (1959). ELECTROPHORETIC MOBILITY OF A SPHERICAL COLLOID PARTICLE 99

33. LINTON, M., AND SUTHERLAND, K. L., Prec. Levine, S., and Bell, G. M., J. Phys. Chem. Intern. Congr. Surface Activity 2nd London 64, 1188 (1960). 1, 494 (1957). 38. FRIEDMAN, H. L., "Ionic Solution Theory." 34. LYKLEMA, J., AND OVERBEEK, J. TH. G., J. Interscience Publishers, New York, 1962. Colloid. Sci. 16, 501 (1963). 39. (a) FRIEDMAN, H. L., Physica 36, 509, 537 35. STIGTEa, D., J. Phys. Chem. 68, 3600 (1964). (1964). 36. LA M]~R, V. K., Discussion following a paper (b) FRIEDMAN, H. L., J. Chem. Phys. 42, 450, by Sawyer, W. M., and Rehfeld, S. J., J. 459, 462 (1965). Phys. Chem. 67, 1973 (1963), 40. BUFF, F. P., AND STILLINGER, F. H., J. Chem. 37. LEVIN~, S., Discussion following a paper by Phys. 39, 1911 (1963).