2970
Langmuir 1991, 7, 2970-2980
Electrophoretic Fingerprinting of a Single Acid Site Polymer Colloid Latex B. J. Marlow? and R. L. Rowell” Department of Chemistry, University of Massachusetts, Amherst, Massachusetts 01003 Received March 6, 1991. In Final Form: October 10, 1991 The electrophoretic fingerprint of a system of colloidal particles is the contour diagram of the measured mobility as a function of the measured pH and the measured pX (logarithm of the specific conductance in S/m). In earlier work the electrophoretic fingerprint was shown to be a characteristic property of a system of charged colloidal particles. In the present work, the electrophoretic fingerprint of a carboxylic polystyrene latex is compared with theoretical calculations in order to explore the nature of the surface electrochemicalproperties and the structure of the double layer. Theoretical calculations include fingerprints (pH-pX contour plots) of the surface potential, the zeta potential, and the electrophoretic mobility. An explanationof the anomalous maximum in the zeta potential is offered and the sensitivity of electrokinetic properties to electrophoretic relaxation, shear plane expansion, and conductance within the shear plane is quantitatively investigated for the model carboxylic acid polystyrene latex.
Introduction The experimentally measurable electrokineticproperty of a system of colloidal particles is the electrophoretic mobility which has been shown to be a function of the characteristic measurable quantities pH and pX (the logarithm of the specific conductance, S/m) in our introductory work on electrophoretic fingerprinting.’i2 The electrophoretic fingerprint is a three-dimensional correlation of experimental data which may be used to characterize an unknown system of colloidal particles or, as in the present work, to explore the nature of the electrochemical surface by comparison with available theoretical models. The electrochemical fingerprint is obtained at low particle concentration so that the measurable pH and pX act as (semiindependent and independent, respectively) characteristic variables (analogous to state variables in thermodynamics). To probe higher particle concentrations, one may resort to electroacoustics.3-5 To explore the interpretation of the electrophoretic fingerprint in some depth, we analyze a single acid site polymer latex which has been measured in our laboratory.6 It is important to elaborate on the particular choice of a polymer colloid at this point. It is well-knownthat latex particles can be prepared uniform in size as is the one employed in this work. In fact, they are used as size standards for many different applications. Although they are a reproducible model system for size applications, the authors would like to point out that they are not a simple model for electrokinetics because of their complicated surface structure and other electrochemical effects discussed below. Thus, the athors thought this to be a challenging system for electrophoretic fingerprinting and it is shown how the technique can unravel some of the current controversies concerning the electrokinetics of polymer colloids. The electrokinetic properties of polymer colloids have recently been an area of great interest both theoretically + Present address: Pen Kem, Inc., Bedford Hills, NY 10507. (1)Marlow, B. J.; Fairhurst, D.; Schutt, W. Langmuir, 1988,4, 776. (2)Morfesis, A. A.; Rowell, R. L. Langmuir 1990,6, 1088. (3)Marlow, B. J.; Fairhurst, D.; Pendse, H. Langmuir 1988,4,611. (4)Babchin, A. J.;Chow, R. S.; Sawatzkv, R. P. Adu. Colloid Interface Sci. 1989,30,111. (5)O’Brien, R. W. J.Fluid Mech. 1988,190, 71. (6)Shiau, Shaw-Ji Ph.D. Thesis, University of Massachusetts, Amherst, MA, 1989.
and experimentally. Recent work has shown that electrokinetic data on polymer colloids can only be quantitatively explained by using models involving drastically different physical interpretations, namely, (i) anomalous surface conductance or ion mobility of a bound layer of ions within the shear (ii) expansion or contraction of surface layers (polymer chains) under changing electrochemical stresses, the so-called “hairy ball” model,+15 (iii)variation in surface charge density through preferential ion a d s o r p t i ~ n , ~ - ’ (iv) ~ J ~particle - ~ ~ size changes produced either homogeneouslyby osmotic swelling or through coreshell type of phenomena: and (v) crossing of the mobility/ { m a ~ i m u m .All ~ of these explanations come about from attempts to explain the observed phenomena that the electrokinetic potential { goes through a maximum when plotted versus ion concentration as shown by many researchers on polymer ~ o l l o i d s . ~ - ~ ~It~is~ shown ~ - ~ ’ J in (7)Zukoski, C. F.; Saville, D. A. J. ColloidInterface Sci. 1986,114,32. (8)Zukoski, C. F.;Saville, D. A. J.Colloid Interface Sci. 1986,114,45. (9)Midmore, B. R.;Hunter, R. J. J.Colloid Interface Sci. 1988,122, 521. (10)Goff, J. R.; Luner, P. J . Colloid Interface Sci. 1984,99,468. (11)Van der Put, A. G.; Bijsterbosch, B. H. J. Colloid Interface Sci. 1983,92,499. (12)Van den Hoven, Th. J. J.; Bijsterbosch, B. H. Colloids Surf. 1987, 22, 187. (13)Van der Linde, A. J.; Bijsterbosch, B. H. Colloids Surf. 1989,41, 345. ColloidInterface (14)VandeVen,T. G. M.;Dabros,T.;Czamecki,J.J. Sci. 1983,93,580. (15)Meijer, A. E. J.; van Megan, W. J.;Lyklema, J. J.ColloidInterface Sci. 1978,66,99. (16)Van den Hul, H. J. J. Colloid Interface Sci. 1983,92,217. (17)Van der Put, A. G. Ph.D. Thesis, University of Wageningen, The Netherlands, 1980. (18)Kamel, A. A.; El-Aasser, M. S.; Vanderhoff, J. W. J . Dispersion Sci. Technol. 1981,2, 183. (19)Kamel, A. A.; Ma, C. M.; El-Aasser, M. S.; Micale, F. J.; Vanderhoff, J. W. J . Dispersion Sci. Technol. 1981,2, 315. (20)Labib, M. E.; Robertson, A. A. J . Colloid Interface Sci. 1980,77, 151. (21)Elimelech, M., O’Melia, C. R. Colloids Surf. 1990,44,165. (22)Bijsterbosch, B. H.; van der Linde, A. J. Preprints, International Symposium on Surface Charge Characterization, 21st Annual Meeting of the Fine Particle Society, San Diego, CA, August, 1990;Fine Particle Society, Tulsa, OK; p 28. (23)Brouwer, W. M.; Zsom, R. L. J . Colloids Surf. 1987,24,195. (24)Takamura, K.;Chow, R. S. Colloids Surf. 1985,15,35. (25)Chow, R. S.; Takamura, K. J . Colloid Interface Sci. 1988,125, 226. (26)Bensley, C. N.;Hunter, R. J. J . Colloid Interface Sci. 1983,92, 448. (27)Zukoski, C. F.; Saville, D. A. J . Colloid Interface Sci. 1983,107, 332.
0 1991 American Chemical Society
Langmuir, Vol. 7, No. 12, 1991 2971
Electrophoretic Fingerprint of Colloidal Particles this work how electrophoretic fingerprinting can sort out which of these phenomena is occurringfor a model polymer colloid in a given electrochemical state. To further exemplify the need for electrophoretic fingerprinting when trying to unravel the compkexities of the electrokinetic behavior of polymer colloids, consider the following train of reasoning that has been used to arrive at the conclusion that basic electrokinetic theory has a flaw and cannot adequately explain the electrokinetic behavior of polymer colloids. Recent ~ork’98-2~ has shown that (calculated from electrophoretic mobility measurements did not compare to those calculated from relatively low frequency conductivity measurements under similar thermodynamic conditions. Since the conversion to (for both techniques involves the same underlying classical electrokinetics theories, then the discrepancy between the two techniques has led these authors to the conclusion that somehow the basic theory has a flaw. This flaw is shown to be minimized by introducing the “dynamic Stern layer” or movement of ions bound on the surface of the particle. Dukhin31was the first to quantify this motion and had shown that to some extent this effect may be compensated for by the diffuse ion atmosphere. Thus, there is some question as to whether this anomalous conductance completely eliminates the above-mentioned discrepancies. The authors contend that the discrepancies may result from this anomalous conductance but may further be complicated by other mechanisms as well which were not incorporated into the classical theory and that it may not be the theory that is flawed but rather the models employed. Also, there should be no reason why classical dc electrophoresis measurements, if performed correctly, cannot sort out which mechanisms are operative. We show that these discrepancies can be understood by making electrophoresis measurements as a function of both pH and pX and combining the data into one analysis using the electrophoretic fingerprint. Any differences that appear between the experimental data and calculations from the theory can now be used to explore the major electrochemical effects present. This work focuses on a polymer colloid (latex) having a chemically well-defined surface, Le., single acid sites composed of carboxylic acids. For this surface the potential is controlled by pH. The surface is electrophoretically fingerprinted using laser Doppler electrophoresis. Theoretical electrophoreticmobilities are calculated using a single acid site dissociation model that includes an accounting for conductance in the diffuse ion atmosphere (relaxation). The theoretical fingerprint thus obtained is compared to the experimental fingerprint using the technique of differential fingerprinting.
Experimental Section The polymer latex used in this work was obtained from Interfacial DynamicsCorp., IDC #lo-32-13,which had a reported mean particle radius of 0.49 pm with a 5.1% coefficientof variance and was supplied as a 3.5% solids stock solution. The “ultraclean” microspheres were prepared without the use of surfactants and were reported to have carboxylicacid groups covalently bound to the surface during the manufacturing process. The stock solution was diluted with doubly-distilled deionized water. (28) Bonekamp, B. C.; Alvarez, R. H.; De Las Nieves, R. J.; Bijsterbosch, B. H. Colloids Surf. 1986,21, 259. (29) Alvarez, R. H., De Las Nieves, R. J., Van der Linde, A. J., Bijsterboech, B. H. Colloids Surf. 1986, 21, 259. (30) James, R. 0.; Buscal, R., Stageman, J. F., Eds.; Polymer Colloids; Elsevier: Amsterdam, 1985. (31) Dukhin, S. S. Surf. Colloid Sci. 1973, 3, 83.
m 3 0
X
*
L
=!
$ E
PA Figure 1. Experimental U, = U,(ph,pH) template for IDC carboxyl microspheres, lot no. 10-32-13. Solutions were prepared by adding 2 drops of the latex dispersionto 250 mL of doubly-distilleddeionizedwater resulting in a particle concentration of approximately 50 ppm. The dispersion was stirred for approximately 10 min by magnetic stirrer. The pH of the dispersion was then adjusted with either HC1or KOH and electrophoresis measurements were performed on a very small aliquot (0.1 mL) of the dispersion. After pH adjustment, the same was allowedto equilibrate for a fewminutes before sampling. Electrophoretic mobility distributions weremeasured with the Pen Kem System 3000 automated electrokinetic analyzer which has been described in detail e l s e ~ h e r e . *Essentially, ~ ~ ~ ~ ~ ~the moving images of particles under an applied electric field are detected through a rotating optical grating of constant and known speed creating frequency shifts which are converted by a fast Fourier transform analyzer to give a reproducible measurement of the mobility based on hundreds of particles moving in the stationary layer. The mean mobility, used in this work, was reproducible to better than 3% . The mean electrophoretic mobility from the Pen Kem 3000 for a sample as a function of measured ph and pH is represented as a three-dimensional surfaceusing SURFER software described elsewhere.]
Results The experimental electrophoreticmobility template, U, = U,(pX, pH), for the carboxyl microspheres using the KCl/KOH/HCl electrolyte system is shown in Figure 1. The surface in Figure 1 is representative of the raw data and is not smoothed to any degree. The data sampling schemes that give a representative surface have been discussed elsewhere.34 Figure 2 shows the experimental electrophoretic mobility fingerprint for the carboxyl microspheres. A maximum is observed in the fingerprint with a peak mobility in excess of -5 X 10-8 S/m in the vicinity of the coordinates pH 5 , pX -2. Near zero mobility is asymptotically approached at the lowest pH and highest ionic strength. Discussion Conductance and Ionic Strength. In order to interpret the data, we first choose to relate the measured ph to the Debye length 1 / ~a ,central parameter of classical (32) Goetz, P. J. US.Patent 4,154,669. (33) Marlow, B. J.; Rowell, R. L. J. Energy Fuels 1988, 2, 125. (34) Rowell, R. L.; Shiau, S.-J.; Marlow, B. J. In Particle Size Assessment and Characterization; Provder, T., Ed.; ACS Symposium Series; American Chemical Society: Washington, DC, 1991.
Marlow and Rowel1
2972 Langmuir, Vol. 7, No. 12, 1991
-1
\ 4 6 8 10 12
- 42
PH Figure 2. Experimental U, = U,(pX,pH) fingerprint for IDC carboxyl microspheres, lot no. 10-32-13. electrokinetic theory. For a 1 : l electrolyte we have the well-known35relation 1 / K = (cotkT/2t2n0)1/2
(1) where to is the permitivity of free space, t is the dielectric constant of the bulk fluid, k is Boltzmann's constant, e is the electronic charge, and no is the bulk ion concentration. If the anion and cation mobilities are approximately equal, the conductivity X = 2n'ue (2) with ion mobility u , may be substituted for 2en0 to give
-51
,
I
I
4
2
,
1
6
,
a
,
,
I
10
1
12
PH Figure 3. Relationship between pX and pH defining experimentally accessible regions in pX-pH space for the KCl/KOH/ HCl electrolyte system: (0)DI water, (A)0.1 mM KCI, (A)1 mM KC1, (u) 10 mM KC1.
1-
0-
PA -111. = (totkTu/Xe)'l2 (3) For equal ion mobilities the diffusion coefficient D is given by35 D = ukT/e which may be combined with ( 3 ) to give
(4)
1 / K = (CoJ)/X)1/2 (5) An alternative form of eq 5 which is useful in the calibration procedure discussed below is
+
PA = -2 IOg ( 1 / K ) log €OED (6) This also shows that the log of the conductivity bears a negative linear relation to the log of the Debye length. Since K for aqueous 1:l electrolytes at 25 "C is related to the ionic strength I by35 K
= 3.288IIf2
(7)
we have X = 10.81totDZ
(8) If we define the index pX = log X and the index pZ = log I , then pX = p l + log (10.81t0J)) (9) so that a logarithmic conductance scale may be used in place of a logarithmic ionic strength scale. For a system of unknown ion composition, we use the conductance expressed by pX. For theoretical interpretation of a known system, we use the measured pX to obtain the ionic strength through ( 9 ) . Calibration in the pH-pX Domain. In Figure 3 we show a calibration plot in the pH-pX domain which defines the region that is experimentally accessible to the measurement of electrophoretic mobility. In distilled water (DOa roughly parabolic path defines the experimentally (35) Hunter, R. J. Zeta Potential in Colloid Science; Academic Press: New York, 1981.
-2-
-3-
-'t
- 5 - h -00
I
-5
14
I
-3
I
-2
,
-I
I
0
I
1
Log(KC1) Mole/liter Figure 4. Experimental dependence of pX on log KC1 concentration. accessible region. The curvature diminishes at higher electrolyte concentration but remains an important factor below pH 4 at 10 mM KC1. At the low particle concentrations used in the electrophoresis measurements, the curves are essentially the same since the contribution of the particles to the conductivity is negligible. The calibration curve for pX and pc is given in Figure 4. The molar concentration of KC1 is c and for a 1:l electrolyte, c = Z , the ionic strength. From Figure 4, we obtain log c = 1.059 log X 1.005 (10) which is consistent with the theoretical equation ( 9 ) .The calibration is required in using the theoretical model below. Single Acid Site Dissociation Model. We employ the single acid site dissociation model of Healy and White35136recast in terms of pX. The surface of a carboxylated polymer colloid develops charge from the dissociation of carboxyl groups according to
+
+
RCOOH + RCOO- HC and this equilibrium is described by
(11)
[RCOO-]a,(H+) (12) [RCOOH] where K , is the acid dissociation constant for the carboxyl
K, =
(36) Healy, T. W.;White, L. R. Ado. Colloidlnterface Sci. 1978,9,303.
Langmuir, Vol. 7, No. 12, 1991 2973
Electrophoretic Fingerprint of Colloidal Particles group bound to the surface which may differ considerably from the free acid used to make the polymer colloid, and a,(H+) is the activity of hydrogen ions in the surface region. The relationship between the activity of the hydrogen ions in the vicinity of the surface to that in the bulk can be found by equating the electrical and diffusional forces on these i o d 5 resulting in a Boltzmann type relationship expressed as a,(H+) = a#+) exp(-yo) (13) where ab(H+) is the bulk activity of hydrogen ions (measured by pH) and yo = e 9 o / k T . Since 90< 0, and yo < 0, then exp(-yo) > 1 showing that a,(H+) > ab(H+) or the hydrogen ion concentration is greater near the surface than in the bulk, i.e., they are accumulated in the surface region as counterions. The surface charge density uo is expressed as uo = -e[RCOO-]
h r
(14)
The total surface density of functional groups N, is
N, = [RCOO-] + [RCOOHJ By combining eqs 12, 13, 14, and 15, one obtains
(15)
-eN.
:!F
Figure 5. Single acid site dissociation model and a theoretical PO= Po(pX,pH)templateforpK, = 4.5andN, = 5 x 1013sites/cm2.
-2
The Gouy-Chapman model of the electric double layer35 assumes there is no bound layer of ions, i.e., specific ion adsorption is absent. In this model of the structure of the double layer, the charge density in the diffuse part of the double layer Ud is related to the surface potential by equating the surface charge to the net space charge in the double layer bd
= -(8~~cnOkT)"~ sinh
e)
Since the Gouy-Chapman model assumes that then equating eqs 16 and 17 gives
Ud
= -ad,
This expression can be rearranged to give pH = pKa - (0.0169\E0(mV))+ log [sinh (-0.0195\kO(mV))]log [5 - sinh (-0.01959,(mV))] (19) where 5 is a dimensionless parameter given as [=-
~ o ~ N , ~ 4Nac
where N, is in units of sites/cm2, K in cm-l, Na is Avogadro's number in mol-l, and c in mol/L. For aqueous systems at 25 "C,5 can be expressed as log 5 = log (1.36 X 10-'4N,) - 0.5 log C Thus, from eqs 21 and 10 one obtains
(21)
log 5 = log (1.36 X 10-14N,)- 0.5295pX + 0.5025 (22) From eq 22 one can see that the dimensionless term 5 combines site density N, and electrolyte concentration pX effects. This model predicts that at low pH and pX the surface potential asymptotically goes to zero where dQo/dpH
-3
P
4
8
6
10
12
PH Figure 6. Single acid site dissociation model and a theoretical PO= Po(pX,pH) fingerprint for pK. = 4.5 and N. = 5 X 1013 sites/cm2. shows a non-Nernstian behavior; Le., it does not equal -59.8 mV/pH units. As the pH increases the carboxylic groups dissociate and 90shows a rapid increase with pH centered around the pKa. 90then levels off to a plateau which depends on N, and pX. Theoretical Fingerprints. A theoretical fingerprint of 90= \ko(pX,pH) is produced by calculating 5 from pX and a given N, using eq 22 and then substituting this value into eq 19 using a given pKa to obtain 90as a function of pH for a fixed ph. Figure 5 shows the 90= *o(pX,pH) template for the KCl/KOH/HCl electrolyte system using a pK, = 4.5, N, = 5 X 10'3 sites/cm2 which corresponds to a parking area 1/N, = 200 A2/site. Since close packing of carboxylic acid groups on a surface would correspond to 20 A2/site,37the site density used here is relatively low. The fingerprint for this template is shown in Figure 6. The region of excluded data in Figure 6 corresponds to the regions that are experimentally inaccessible. Figure 7 shows \Eo-pH profiles taken at constant pX cuts. Notice that at lower conductance the span of 9 0 over pH decreases because of the inaccessible regions. The curves show that at pH values below the PKa, 90 asymptotically approaches zero but does not show an isoelectric point. As the pH becomes comparable to the pKa the surface groups begin to dissociate increasing 90until at higher pH values a plateau is reached, the value of which depends on both N, and pX. (37) Bangs, L. B. Uniform Latex Particles; Seragen Diagnostics, Inc.: Indianapolis, IN, 1985.
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2974 Langmuir, Vol. 7, No. 12, 1991
-50-
-100-
3. (mv)
-150-
-250
-300, 2
I
I
4
I
I
I
6
I
I
8
10
PH Figure 7. Single acid dissociation model and $o-pH profiles at constant pX for pKa = 4.5 and N , = 5 x l O I 3 sites/cm2for various PA: (0) -3, (A)-2, (0)-1, (0) 0, (X) +I.
,
Figure 9. Single acid site dissociation model and a theoretical { = {(pX,pH)template for pK, = 4.5, N , = 5 X 1013sites/cmZ, and 6 = 1 nm.
1
3.
\-
-2004
Figure 10. Single acid site dissociation model and a theoretical { = {(pX,pH) fingerprint for pKa = 4.5, N , = 5 X 1013sites/cmZ, and 6 = 1 nm. Figure 8 shows \ko-pX profiles taken at constant pH cuts. The figure shows that the surface potential is a linear function of the logarithm of the conductivity or ion concentration and a t lower pH it asymptotically approaches zero. The cuts taken at pH values of 8 and 10 are almost identical because they are both on the plateau of 00versus pH shown in Figure 7. The surface potential theoretical fingerprint can now be converted to a f fingerprint. To accomplish this we assume that the potential with distance from the surface decays exponentially as described by the Gouy-Chapman model from which one can calculate the f from 00by assuming a distance to the shear plane which for the first illustration is assumed to be 1 nm from the surface. Again, from the Gouy-Chapman theory, the potential a distance x from the surface q ( x ) is given as
where exp(y) -1
00(pX,pH) from eq 24 combined with eq 23 which incorporates the effect of electrolyte on double layer compression. The inverse Debye length K is calculated from eqs 5 and 9 where the constants are determined from the experimental data of (10). Figure 9 shows the theoretical template of f = f(pX,pH) calculated from 90in the template of Figure 5. The theoretical fingerprint is shown in Figure 10. The fingerprint o f f = f(ph,pH) looks almost exactly the same as ' k o = \ko(pX,pH)with the exception that the values of {for a given pX and pH are much lower than the \ko values. This is because {represents the potential a distance away from the surface where it has decayed from its surface value due to shielding by counterions (double layer compression). Figure 11 shows cuts taken from the f = f(pX,pH) fingerprint at constant pX or f-pH profiles. The trends are exactly the same as the 00-pH profiles with the exception that the shear plane potential f is much lower for a given pX and pH. Again, as pX decreases, the range of pH over which f is defined diminishes, and in fact, in distilled water it becomes a single point. Figure 12 shows c-pX profiles taken at constant pH cuts. Again, as with 00-pX profiles, the f decreases linearly as pX increases and at high pX asymptotically approaches zero. One should note that this model does not predict a maximum in the f-pX profile (see below).
'"=T exp -
+1
Thus, the potential 1nm from the surface, 0 ( 1 mm) = f, is calculated from the surface potential fingerprint 00=
Langmuir, Vol. 7,NO.12, 1991 2975
Electrophoretic Fingerprint of Colloidal Particles 0
-50
-=OI -300
\
1
I
8
4
2
10
12
PH
w
r
Figure 11. Singleacid site dissociationmodel and {-pH profiles at constant pX for pK. = 4.5, N , = 5 X 1013sites/cmZ, and 6 = 1 nm for various pi: (0) -3, (A)-2, (0) -1, (0) 0, (x) +1. 0
Figure 13. Single acid site dissociation model and a theoretical U,= U,(pX,pH) template for pK. = 4.5, NB= 5 X 10’3 sites/cmZ, 6 = 1 nm, and a = 0.49 pm.
-50
-100
(mv) -150
-2001 -250
-1
-300
0
fl
-2
-1
-3
-4
PA Figure 12. Single acid site dissociation model and {-PA profiles at constant pH for pK. = 4.5, N , = 5 X 1013sites/cm2,and 6 = 1 nm for various pH: (0) 4, (A)6, (0) 8,(0) 10. On the basis of the { = {(pX,pH) fingerprint one now can calculate a theoretical electrophoretic mobility fingerprint, Le., Ue = Ue(pX,pH),which one can compare directly to experimental data. In calculating the mobility from {, one must account for electrophoretic relaxation or conductance in the diffuse ion atmosphere using the most complete theory available, that due to O’Brien and White.% A simplified version and analytical form for this theory accurate to order l / K a , where a is the particle radius, i.e., valid for our purposes for Ka > 10, can be expressed as35
6
[
2
]
2
+ $exp[+)
(25)
(26) where 7 is the fluid viscosity, y = elflkT, and M = 1 + 3 m / z 2 ,where m is the dimensionless ion drag coefficient given as
2t0tNakT W
37z2e2 where the drag coefficient w is given by (38)O’Brien, R. W.; White, L. R. J. Chem. SOC.Faraday Trans. 2 1978, 74, 1607.
-4
-5 -6
-7
PA Figure 14. Data of Figure 13, rotated and tilted. N,e2z
(28)
A0
where I is a dimensionless reduced mobility defined as
m=
-3
w=-
- y(1exp(zy)) -
E=@-
-2
where bo is the limiting molar conductivity of the ions in solution at infinite dilution. For KCl,35the supporting electrolyte used in this work, m = 0.184 giving M = 1.55. Thus, one inputs 5; a, M ,and pX for a given pH to obtain U, = Ue(pX,pH). The theoretical UO= Uo(pX,pH)template for the KCl/KOH/HCl electrolyte system, where t was input from the data of Figure 9, and a particle radius a = 0.49 Km (see below),is shown in Figure 13. Interestingly, a maximum is produced in the template, i.e., as ph increases for pH vaues between 5 and 10, the mobility increases, goes through a maximum, and then decreases. Because of the data plotting routines this may be difficult to see in Figure 13, so the figure was tilted and rotated and replotted in Figure 14. Figure 14 clearly shows a “valley” that occurs between pX of 1 and 2. Thus, electrophoretic relaxation causes a maximum to be observed in the U,-pX profiles which shifts for different pH values and altogether
Marlow and Rowel1
2976 Langmuir, Vol. 7, No. 12, 1991
I
I
4
8
6
10
I
I 12
PH Figure 15. Single acid site dissociation model and a theoretical U, = U,(ph,pH) fingerprint for pKa = 4.5,N , = 5 X 1013 sites/ cm2, 6 = 1 nm, and a = 0.49 pm.
-B
1
t1
1
I
I
0
-1
-2
I
-3
-4
PA
0
Figure 17. Singleacid site dissociationmodel and U,-pXprofies at constant pH for pKa = 4.5,N. = 5 x 1013sites/cm2,6 = 1 nm, 4,(A)5, (0) 6, (0) 8,(X) and a = 0.49 pm at various pH: (0) 10.
-1 -2
-71 -8
-
-8j
a e 10 12 l PH Figure 16. Singleacid site dissociationmodel and U,-pH profiles at constant ph for pKa = 4.5,N , = 5 X 1013sites/cm2,6 = 1nm, and a = 0.49pm at various ph: (0)-3, (A)-2, (0) -1, (0) 0, (X) 2
4
6
+l.
disappears at some pH values as shown below. This maximum occurs at equivalent KC1 concentrations between 10-3 and M KC1 depending on the constant pH value chosen, similar to where the maximum has been observed e ~ p e r i m e n t a l l y . ~ - ~ ~ ~ ~ - ~ ~ Figure 15 shows the theoretical Ue = Ue(pX,pH) fingerprint where this maximum is clearly observed as “U-shaped” isomobility lines. Figure 16 shows Ue-pH profiles taken at constant pX cuts. A t high pX the curves have the same shapes as the f-pH profiles. As pX decreases, Uetends to increase with pH and then go through a slight maximum. As pX further decreases, the whole Ue-pH profile is shifted below those taken at somewhat lower PI and shows a more dramatic maximum and decrease in Ue with increasing pH. Figure 17 shows Ue-pX profiles taken at constant pH cuts. The curves show that at low pH, linear Ue-ph behavior results. As the pH increases the curves begin to show a maximum which gets more pronounced at higher pH values. The authors believe that these results are significant because it shows that the experimental observation of a maximum in a U,-pX plot or Ue-log ion concentration plot will depend on the pH value. At low pH no maximum is observed, at high pH the location of the maximum depends on the pH value, i.e., it shifts with pH. This will undoubtedly prove to be useful when analyzing literature data of Ue versus ion concentration. Some researchers report maxima, others maxima a t different ion concentrations. These discrepancies have led to a great deal of confusion as to the mechanism responsible for the maximum and it is difficult to analyze the literature data
because all the electrochemical state variables are not reported. It is obvious from this theoretical work that maxima in mobility-pX profiles can result from electrophoretic relaxation and their location are dependent on pH, N,, and pKa. Thus, the model predicts maxima in Ue-pX profiles due to electrophoretic relaxation but they are absent in l-pX profiles. Comparison with Experimental Data. The next step in the analysis was to generate a theoretical electrophoretic mobility fingerprint using the single acid site dissociation model discussed above to fit the experimental data of Figures 1 and 2. As adjustable parameters one has pKa and N, for a fixed shear plane distance 6 assumed here to be 1nm or 10A from the surface. These parameters were adjusted to give a minimum standard deviation between the experimental and theoretical fingerprint. Remember the PKa determines at what pH the surface begins to be charged and N, determines the value of the potential at the plateau of potential versus pH for a given PA. It is found that a pK, = 3.5 gives the best results which is typical for a carboxylic acid group but somewhat lower than that stated by the manufacturer of approximately 5. Interestingly, it is found that no one value of N, can be used to adequately explain the mobility-pH profiles over all ph values. To get the best minimum standard deviation between experiment and theory, a variable N, had to be utilized, i.e., N, = N,(pX). The form of the variable N, is shown in Figure 18. It was found that N, was independent of pH; i.e., for a given mobility-pH profile, N, had to be varied at a given pX to get a good fit. To cover all the pX-pH space N, had to be varied from 4 X 10l2to 3 X lo1* sites/cm2 or parking areas of 2500 to 33 A2/site. By use of the variable N , shown in Figure 18,a theoretical electrophoreticmobility template was generated using the above theory and is shown in Figure 19. The corresponding fingerprint is shown in Figure 20. If we now compare the experimental fingerprint in Figure 2 to the theoretical one in Figure 20, one notes remarkable similarities. This can further be shown by taking cuts at constant pX and PH. Figure 21 shows mobility-pH profiles taken from cuts at constant pX from both the experimental (Figure 2) and the theoretical fingerprint produced with variable N , (Figure 20). The dotted lines represent the cuts taken from the theoretical fingerprint for a given pX whereas the characters represent those from the experimental fingerprint for the same respective pX. Excellent fits are
Langmuir, Vol. 7, No. 12, 1991 2977
Electrophoretic Fingerprint of Colloidal Particles 0.
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Figure 21. Theoretical single acid site dissociation model UepH cuts (dashed lines) at constant pX using variable N,, pK, = 3.5,6 = 1nm, and a = 0.49 pm, compared to experimental cuts. -3,4 x 10l2;(A)-2, 5 x 1012; ph, N , values are as follows: (0) ( 0 )-i,i.24 x 1013; (0) 0, 2.6 x 1013; (XI +I, 3 x 1014.
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Figure 22. Theoretical single acid site dissociation model VepX cuts at constant pH using variable N,, pK, = 3.5,6 = l nm, and a = 0.49 pm, compared to experimental cuts. pH values are ( 0 )4, (A)5, (0) 6, and (0) 8. in the mobility-pX profiles for the single site acid surface. At higher pH cuts maxima are observed and the value of pX where it occurs is pH dependent. The overall quality of the fit can be shown by subtracting the experimental and theoretical electrophoretic mobility templates. This procedure is analogous to fitting two hats on top of one another. Figure 23 shows this “differential electrophoretic template”. It shows that in general the fit is better to 0.2 X m2/(V s) (10% or better over most pX-pH space) but that in the region of high pX and low pH a much greater deviation occurs (>30%)which is yet to be explained but has been observed with other polymer colloid surfaces.’ The question that now arises is whether the variable N , used in the theoretical fingerprint of Figure 20 to get a minimum standard deviation between experiment and theory has any real physical significance. The variable N , shown in Figure 18 reflects the fact that as pX increases or the supporting electrolyte concentration increases, the number of negative sites per unit are on the surface increases. One could postulate that this results from chloride ion (Cl-) adsorption as the concentration of KC1 increases. If this were the case, the experimental mobilitypH profiles would shift in terms of where the plateaus are reached for a given pX due to C1- adsorption. This would result in the requirement of a variable pK, which is not
Marlow and Rowel1
2978 Langmuir, Vol. 7, No. 12, 1991 10
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the case observed. Thus, chloride ion adsorption cannot explain the variable N,. This is also supported by the fact that others have demonstrated that co-ions are not specifically adsorbed on polystyrene l a t t i c e ~ . ~ J ~ * ~ ~ Another possibility that could cause the variable N , is particle expansion through coreahell type behavior.2This type of expansion is both pH and pX dependent, i.e., mobility-pH profiles at constant pX would go through a severe maximum. Since the mobility-pH profiles obtained here do not show this type of behavior, i.e., N , is dependent only on pX, then this type of behavior can be ruled out. Another type of expansion that could occur and cause a variable N , is homogeneous swelling of the particle caused by osmotic forces. In general, the particle would swell as pX increases or the ion concentration decreases causing the site density on the surface of the particle to decrease similar to the trend observed. Figure 24 shows a plot of expanded particle radius r/ro as a function of p i . To account for the observed changes in N , as a function pX, the particle size must change by a factor of 8. This is obviously an astronomical change in size that has never been observed experimentally in aqueous electrolytes for polymer colloids. In fact, changes of a factor of 2 are rarely (39) Midmore, B. R.; Diggins, D.; Hunter, R. J. J. Colloid Interface Sci. 1989, 129, 153.
seen. Thus, one can rule out osmotic swelling as a possible mechanism for the variation in N,. Considerationof a Variable Shear Plane. From the above discussion it is obvious that one cannot physically interpret or account for the changes in N , required to get theory and experiment to agree and, thus, the variable N , used here is physically unreasonable. This pitfall can be alleviated by fixing the parameters PKa and N , over all accessible pX-pH space and introducing a variable shear plane distance into the model. Normally, the shear plane distance 6 is fixed but as we will show there is real physical significance to its variation introduced here. Figure 25 shows the variable shear plane distance 6 as a function of pX required to get the experimental and theoretical fingerprints to agree. The shear plane distance from the surface varies from 0.5 nm (5 A) to 9 nm (90 A) for a PKa of 3.5 and N , of 3 X 1014sites/cm2 (33 A2/site), a value comparable to that obtained by surface titration by the manufacturer. Figure 25 shows a distinct break in the curve at a pX = -1 suggesting that two different mechanisms are responsible for causing the shift in shear plane; the same trends are seen in N , or r/ro versus pX. Using the variable shear plane distance shown in Figure 25, one can now produce a new theoretical electrophoretic mobility template shown in Figure 26 along with the corresponding fingerprint shown in Figure 27. Comparison of the theoretical fingerprint in Figure 27 with the experimental one in Figure 2 shows striking agreement. This can be further demonstrated by taking cuts at constant pX and pH. Figure 28 shows mobility-pH profiles taken at constant pX cuts for both the experimental and theoretical fingerprints using a variable 6. One notes that in general the fits are better using a fixed PKa and N , coupled with a variable shear plane distance than a variable N , with a fixed PKa and shear plane distance. Figure 29 shows mobility-pX profiles taken at constant pH cuts for both the experimental and theoretical fingerprints using a variable 6. There is a much better fit here than with the variable N,. Again, at low pH there is no maximum observed, but a maximum does appear at higher pH values and its pX location is somewhat pH dependent. Figure 30 shows the "differential electrophoretic template" between the experimental and theoretical template produced using a variable 6. Comparing this differential plot to that in Figure 19 again demonstrates that the variable 6 gives a better fit than the variable N , template. The fit is better than 10% over all pX-pH space. Also, as
Langmuir, Vol. 7, No. 12, 1991 2979
Electrophoretic Fingerprint of Colloidal Particles
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PH Figure 28. Theoretical single acid site dissociation model and U - H cuts at constant pX using variable 6, pK, = 3.5, N , = 5 X e l & sites/cm2,and a = 0.49 pm, compared to experimental -3,9 nm; (A) -2,5.2 nm; (0) -1,2.2 nm; cuts. PA, 6 values: (0) ( 0 )0, 1.2 nm; ( X ) +1, 0.5 nm. discussed below, a variable shear plane is physically acceptable and explainable whereas the variable N , required was physically unacceptable. On the basis of the above analysis it becomes clear that the maxima in the mobility-pX profiles at high pH values are the result of electrophoretic relaxation coupled with a variable shear plane. Also, whether or not a maximum
Figure 30. Differential U, = U,(pX,pH) template between experimental template and that produced with the single acid site dissociation model using variable 6, pK, = 3.5, N , = 5 X 1013 sites/cm2,and a = 0.49 pm. appears and the PI location will depend on the pH value that the mobility-pX profiles is determined. Conversion of the mobility-pX profiles to l-pX profiles removes the maximum if electrophoretic relaxation and shear plane expansion are taken into account. If the high pH mobilitypX profiles showed a linear dependence and a had a high mobility (>3 X lo-* m2/Vs) the conversionof such a profile to a [-PA profile would produce a maximum. Looking at the fingerprints it becomes obvious that alinear mobilitypX profile at high pH could be obtained if data were collected along a diagonal in pX-pH space artificially introducing a linear mobility-pX profile and hence introducing a maximum in the {-pX profile artificially. This could help explain some of the discrepancies in the literature concerning the mechanism of this maximU~.&10,21-30 As shown above, once electrophoretic relaxation is properly accounted for, the electrophoretic data of this simple polymer colloid surface can only be explained by introducing a shear plane distance 6 which is dependent on the particular pX value. At high pX or high ion concentrationthe shear plane is located close to the surface, approximately at 0.5 nm. At low pX or very low ion concentrations the shear plane moves away from the surface at a maximum distance of approximately 9 nm.
Marlow and Rowel1
2980 Langmuir, Vol. 7, No. 12, 1991 The particle radius for the microspheres is 490 nm, and a 9 nm change in the hydrodynamic radius would be less than a 2 ?6 change in the size of the particle which is difficult to detect by hydrodynamic particle sizing techniques such as photon correlation spectroscopy. Smaller particles could be used which would show a much larger change in hydrodynamic radius but may interfere with electrophoresis measurements, i.e., size limitations. The point is that this order of shift in shear plane distance is physically reasonable. The question now arises as to what is physically occurring on the surface that is causing the shear plane not only to move into the fluid at low ph but also to do so in a two-step process as suggested from the dependence of 6 on pX shown in Figure 25. The authors suggest the following mechanism which is consistent with the “hairy-ball” mode14-15 refined and slightly coupled with shear plane expansion and the dynamic Stern layer If one starts at very high pX, or high ion concentration, the polymer chains, loops, and tails on the surface are tightly coiled and the shear plane is close to the surface. As pX begins to decrease, or the ion concentration decreases, the carboxyl groups buried in the coiled chains on the surface become deshielded and tend to repel one another causing an uncoiling of the polymer chains shifting the shear plane from 0.5 nm out to 1-2 nm. These are reasonable shifts of the shear plane and have been observed by other^.^ Since N , must remain constant to get proper fits between experiment and theory as discussed above, this uncoiling must involve extension of mostly loops and tails, and segments containing the carboxylic groups stay close to the surface. As pX decreases further, Le., past a value of -1, a significant break occurs in the 6-pX curve suggesting the onset of a different mechanism. The authors interpret this result in the following manner. As pX further decreases, the shear plane may shift out somewhat further than 1-2 nm opening up a large volume between the surface and the shear plane. Under these conditions conduction can occur within this region, i.e., surface conductance becomes appreciable which increases as pX decreases or the ion concentration decreases. The effect of conduction
within the shear plane is to lower the measured electrophoretic To compensate for this lowered experimental mobility, the theory has to also be adjusted to produce a lower mobility which is accomplished by further shifting out the shear plane. In fact, the shear plane has to be shifted out significantly, to as much as 9 nm, since the above theory only accounts for conduction in the diffuse ion atmosphere (electrophoreticrelaxation) and not due to that within the shear plane.
Conclusions Thus, the physical picture that emerges from the above analysis is that there are several mechanisms operative concerning the surface layer structures and accompanying double layer of the polymer colloid investigated which have different degrees of significance under different electrochemical states. They include electrophoretic relaxation, shear plane expansion, and conductance within the shear plane. It has been shown how the technique of electrophoretic fingerprinting can be used as an “in situ surface analysis” tool and can sort out the electrochemical conditions under which different mechanisms controlling the chemistry and structure of the surface and accompanying double layer are operative. Note Added in ProoE While this manuscript was under review, our attention was drawn to an independent study by van der Linde and Bijsterbosch40who used a different approach and concluded that ”... a maximum in the zeta potential of polystyrene lattices as a function of ionic strength in all likelihood does not exist,” a finding that is consistent with the present work. Acknowledgment. This work was supported by grants from the CABOT Foundation and a contract with AMOCO Corporation. Technical assistance and instrument support were provided by Pen Kem, Inc. The experimental work presented here is based on a Ph.D. thesis by S.-J.Shiau, presently at DuPont Taiwan Ltd., 7th F1. International Bldg., 8 Tung Hua North Rd., Taipei, Taiwan, ROC. (40)Van der Linde, A. J.; Bijsterbosch, B. H.Croat. Chem. Acta 1990, 63, 455.
