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Depletion of dextran and PEO for latex particles with “hairy layers”: an electrophoretic study
Colloidsand Surfaces ELSEVIER
COLLOIDS AND SURFACES
A
A: Physicochemicaland EngineeringAspects 122 (1997) 33-42
Depletion of dextran and PEO for latex particles with "hairy layers": an electrophoretic study A. Krabi a, G. Allan b, E. Donath c, B. Vincent b a M P I of Colloids and Interfaces, Rudower Chaussee 5, 12489 Berlin, Germany b School of Chemistry, University of Bristol, Bristol, UK c Department of Biology, Humboldt University of Berlin, Berlin, Germany
Received 7 February 1996; accepted 5 August 1996
Abstract
The electrophoretic mobility of cross-linked polystyrene latex, stabilized with a grafted layer of poly(oxyethylene), and added dextran and PEO was measured as a function of ionic strength. The ratio of electrophoretic mobility in the absence of added polymer to that in the presence was less than expected from the Smoluchowski equation for dextran and higher for PEO. This seeming discrepancy could be explained by assuming polymer depletion with a consequent reduction in the viscosity near the interface and the additional effect of different flow penetrabilities into the grafted layer for the two polymers. The depletion effect was studied theoretically for the case of layers nonpenetrable and penetrable for hydrodynamic flow. The effect of adsorption on depletion was also analysed. Keywords: Depletion; Dextran; Electrophoretic mobility; Flow penetrability; Grafted latex; PEO; Viscosity
1. Introduction
The stability of particles with a "hairy" surface in the presence of free polymers is of both theoretical and practical interest [1-5]. A model system containing cross-linked polystyrene latex particles carrying terminally grafted, solvent-miscible polymer chains (PEO) has been designed previously (PS-g-PEO) [6]. The effects of the surface coverage of the grafted layer [7, 8] and of added polyelectrolytes [9] on the flocculation behaviour have been studied by Vincent et al. [10]. The results revealed depletion flocculation beyond a critical free polymer concentration [11 ]. Depletion of polymers takes place if the loss of configurational entropy near the interface is not balanced by positive interaction energy. Flocculation can occur when the osmotic attraction energy due to depletion outweighs the loss in configurational entropy [12].
In the so-called depletion zone the polymer segment density is lower than that in the bulk, resulting in a reduced viscosity in comparison with the bulk polymer solution. Electrophoretic studies on charged smooth particles in polymer solutions have been performed and generally revealed an increase in mobility compared with that predicted from the Smoluchowski equation [13,14]. These findings were attributed to the presence of a depletion layer. Donath and coworkers calculated the apparent thickness of depletion layers from electrophoretic data for human red blood cells and liposomes in dextran and poly(ethylene)glycol, assuming an exponential viscosity profile in a linear approximation [15-17]. This was possible for smooth and hairy surfaces at high ionic strength. The calculated thicknesses were in reasonable agreement with the radii of gyration of
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34
A. Krabi et al. / Colloids Surfaces A: Physicochem. Eng. Aspects 122 (1997) 33-42
the polymers [18,19]. The linearization failed when applied to the recent experimental results measured for liposomes in high molecular weight dextran in a wide range of ionic strengths. Therefore, in [20] a numerical integration of the position-dependent viscosity profile of the Navier-Stokes equation was performed. Good coincidence with the experimental results was demonstrated. It was shown that the hydrodynamic thickness of the depletion layer can be obtained. It was challenging to study the electrophoretic mobility of a hairy surface together with a depletion layer, since little is known about the structure of the depletion layer in front of a hairy surface. Also, it cannot be ruled out that the free polymer can influence the density, as well as the equilibrium distribution, of the terminally anchored polymers of the hairy layer. Previously, we investigated theoretically the depletion effect on the electrophoretic mobility in the presence of a hairy layer [21]. The electrophoretic behaviour of a hairy layer itself is by no means trivial. Among other factors it is quite important whether the electro-osmotic flow can be generated within the layer or whether the layer is too dense to allow for hydrodynamic flow inside the layer. Given, in addition, the different possibilities for the surface charge distribution, either within the layer or beneath the layer, the electrophoretic mobilities can be quite different even if the net surface charge densities of different particles are similar. In order to estimate quantitatively the hydrodynamic depletion effect for hairy particles we performed measurements of the electrophoretic mobility of latex particles with grafted PEO chains (M w~2000) in dextran as well as in PEO.
2. Theoretical background In the case of a depletion layer the viscosity q(x) of the solution is a function of the distance from the particle surface. We showed previously that, in the case of a smooth surface, integration of the Navier-Stokes equation together with the Poisson equation provides an expression for the electro-
phoretic mobility b [20]: b-
v(~) E
=
--E~Eo
id~ Jv,~xo) q[x(~U)]
(1)
v(oo ) is the electrophoretic velocity, E is the electric field strength, er and eo are the relative and absolute permittivities, ~(x) is the electric potential, 7t(xo) is the potential at the shear plane Xo, and x is the coordinate perpendicular to the surface. The inverse electric potential distribution x(~U) taking into account non-linearity is given by tanh zeo gs 1
4kT
k
zeo 7So
x ( ~ ) = - - In tanh
(2)
4kT
which is valid for symmetric electrolytes. ~Uois the surface potential at x = 0, z is the valence of the electrolyte ions, eo is the elementary charge, k is the Boltzmann constant, T is the absolute temperature and x the inverse Debye-Htickel length. In Ref. [20] we showed that the influence of the assumed specific viscosity profile on the fitted depletion layer characteristics is not too large since b is an integral over the inverse of that profile (Eq. (1). The thickness parameter A of the depletion layer, however, influences the measured mobility in a more sensitive manner. Therefore, it is reasonable to use, as an approximation, step viscosity profiles. In a sense the double-step viscosity profile with the solvent viscosity in a small layer of thickness A1 and with a larger viscosity in an adjacent layer of thickness A 2 is the simplest possible description of the depletion layer of polymers near the interface together with a slip layer.
q(x)=
r/s
x_<21
r/*
21
/~p
X > 2 1 -{-2 2
f
(3)
where qs is the solvent viscosity, qp is the viscosity of the bulk polymer solution. A1 and A 2 are the length parameters of the depletion layers. ~/* is the intermediate viscosity in the double-step profile. In contrast, in the case of hairy layers, an
A. Krabiet al. / ColloidsSurfaces A: Physicochem. Eng. Aspects 122 (1997) 33-42 analytical non-linear treatment is not possible. Recently, we presented an analytical formula assuming the linearized approximation for the electric potential distribution. This approximation is generally not too bad because, in the case of charged hairy layers, the electric potentials are smaller owing to the spatial distribution of the surface charge. For details the reader is referred to the original publication [21]. In that study only an exponential viscosity profile was analysed. However, it is not difficult to allow for arbitrary viscosity profiles, as is done in this paper. The solution presented in Ref. [21] can be divided into two parts, where the first part is responsible for the electrophoretic flow generated inside the layer as a result of the penetrating hydrodynamic flow, with the characteristic penetration length 1/a. The second part represents that part of the electrophoretic flow which stems from the contribution of the double-layer part outside the hairy layer. It is clear that, either for very thin hairy layers (or very low ionic strength) or for non-penetrable layers, the theoretical limit of a smooth surface is approached. For the specific case of the surface charge distributed in a plane underneath the hairy layer the mobility ratio is given by the following, simplified equation: n oo
bs
--
bp
=
t
us(x) e - ~ dx
o
(4)
{'~ Up(X)e -'x dx d0
where bs and by are the electrophoretic mobilities for the particles in polymer-free and polymercontaining solution respectively. The velocities of the liquid flow, inside and outside the hairy layer, for polymer-free and polymer-containing solutions, are given by sinh(ax) at/s cosh(a6) us(x) = ] x - 6 / qs
+ tanh(a6) ar/s
x<6 x>6
(5)
35
and f] uv(x) =
sinh(ax) aqs cosh(~6)
x<6
1 d~+ tanh(a6)
x>6
~/(~)
(6)
ar/s
where 1/a is the characteristic length of flow penetrability and 6 the thickness of the hairy layer.
3. E x p e r i m e n t a l
details
Two different types of PS-g-PEO particles, prepared with different initiators, were studied. They were prepared using the method described by Cowell and Vincent [6]. Styrene, PEO-methacrylate, and a small quantity of divinyl benzene, as a cross-linker, were co-polymerized in a watermethanol solvent mixture. The initiators used were 2,2'-azobisisobutyronitrile (latex A) and 4,4'azobis (4-cyanopentanoic acid) (latex B). Latex A, in theory, has a low concentration and latex B a higher concentration of carboxy-groups on the particle surfaces. Both sets of latex particles had a mean diameter of about 500 nm, as determined by transmission electron microscopy. The electrophoretic mobility of the PS-g-PEO latex particles was determined using both a Zetasizer 4 from Malvern and the phase analysis light scattering (PALS) technique [22] which is particularly suitable for measuring low mobilities. The electrophoretic mobility (bs) of the latices in the absence of polymer was measured as a function of the electrolyte concentration (NaC1) at pH 9.5 and used as a control. The chargecarrying groups of both latices are carboxy-groups which will be fully ionized at this high pH. This reduces any error in the measurement of bs. Dextran with molecular weights 110 kDa and 500 kDa was used. The dextrans were purchased from Fluka and used without further purification. Solutions with the appropriate dextran concentrations were prepared and their viscosities were determined with a Viscoboy capillary viscosimeter from Lauda. The weakly charged latex particles had velocities
36
A. Krabi et al. / Colloids Surfaces A." Physicochem. Eng. Aspects 122 (1997) 33 42
less than 1 I~m s-1. Given the small thickness of the electric double layer of the order of 1 nm, nevertheless high shear rates of the order of 103 s-1 are predicted within the double layer. Thus for the particular polymer concentration range of 2-8 wt.% we checked the newtonian behaviour by means of shear rate variation up to 2000 s-~. N o deviations from linearity were recorded. The mobilities be of the latex particles in the polymer solutions were measured and the results are presented as the ratio bs/bp.
10
o. .t3
9
~8 o
_=
IZ ~_
6
s o
4 r/piTs = 4.07
o e~
o
3
"~
2
~]
Lt,.I 1
'0
's
,'0
,'s
2'0
;5 ' ~'0
3,
Debye-H0ckel Length[nm] 4. Results
I f the effect o f the free polymer on the electro-
phoretic mobility consisted only of increasing the viscosity, Smoluchowski's formula ErEO~
b= - -
(7)
q
for the electrophoretic mobility predicts an electrophoretic mobility ratio bs/br, = qr,/qs. As Figs. 1 and 2 demonstrate, the measured electrophoretic mobility ratios for the latex particles in dextran were remarkably lower than the corresponding viscosity ratios r/p/~/s over the whole range o f ionic strength for the two dextran concentrations and molecular weights used. As discussed i
i
r
~
i
i
Fig. 2. Ratio of the electrophoretic mobility of latex B in 500 kDa dextran: O, 6 wt.%; [2, 4 wt.%. The corresponding viscosity ratios are shown as broken horizontal lines. previously [17], this is strong evidence for the existence o f a depletion layer, with a reduced viscosity near the particle particle surface, in dextran solutions. In the range o f high electrolyte concentrations the mobility ratio in 110 k D a dextran exhibited a steep increase with decreasing ionic strength, i.e. increasing D e b y e - H t i c k e l length. Towards lower salt concentrations this increase flattened. In Fig. 3 the mobility ratios for both latex A and B are plotted as a function o f the Debye length in 4% 20 k D a PEO. The mobility behaviour
i
5.0
,
~p/~/s = 4.45
o
4.0
n,
i
i
i
i
i
i
14
.................................................
"Q~ 4.5
..Q 0
12
3.5 10
"qp/r/s = 2.72 -~ o
3.0
o
2.5
.O O
.01
8
o ]1[
~6 1.51 I11 []
~t
i
1.0[
4
~D.I
2
I
I
I
0
0r~ 0 51
110
15
20
25
30
35
Debye-HOckelLength[nrn] Fig. 1. Electrophoretic mobility ratio of latex A particles in 110 kDa dextran as a function of ionic strength: II, 6 wt.% dextran; [], 4 wt.% dextran. The corresponding viscosity ratios are shown as dotted horizontal lines and were 2.72 (4%) and 4.45 (6%).
....
_t_ ........................................
'0
;
,o
'
~_(_~_
2'0'
' 3'0
Debye-H0ckel Length [nrn] Fig. 3. Electrophoretic mobility ratio o f latex A and B particles in 4 wt.% 20 k D a PEO as a function o f ionic strength: A , latex A; II, latex B; • . ., viscosity ratio r/v/qs = 1.88.
A. Krabi et aL / Colloids Surfaces A." Physicochem. Eng. Aspects 122 (1997) 33-42
is very different from the dextran solution data. It is evident that the mobility ratios here are higher than the viscosity ratios over the whole range of ionic strengths considered. It is remarkable that, at the highest ionic strength studied, the mobility ratio barely exceeds the upper theoretical limit for depletion, while at intermediate ionic concentrations the mobility ratio is an order of magnitude higher than the corresponding viscosity ratio. At relatively low ionic strengths (approximately 10-4 M ) the mobility ratio again becomes smaller.
5. Theoretical analysis and discussion
It is clear that there is no way that the PEO data can be fitted to the existing theories of the effect of depletion layers on the mobility ratio. This was a major contradiction to our expectations, since previous flocculation experiments with similar systems indicated that polymer depletion should be present [23]. Consequently, a different explanation for the present PEO data is clearly required. We shall firstly attempt to account for the dextran data within the framework of the existing theories for smooth as well as hairy surfaces, before we investigate alternative explanations for the PEO data. Although the model particles definitely have a hairy layer, it is still a reasonable first approximation to use the theories for smooth surfaces, since very little is known about the hydrodynamic properties of such hairy layers. The smooth surface concept will be completely valid if the layer is nonpenetrable for flow. In the case of a non-penetrable hairy surface the difference between the smooth and hairy layer concepts will be reflected in the values of the absolute mobilities, which in the case of a hairy layer depend in addition on the surface charge distribution. Since in this work only the mobility ratios are of interest, it is evident that the existing depletion layer theories of the electrophoretic mobility of smooth surfaces are completely applicable to hairy layers, provided that the hydrodynamic flow inside the layer can be safely neglected. Let us first compare the theoretical result, for an assumed single-step viscosity profile, with
37
different depletion layer thicknesses A, as shown in the insert of Fig. 4. Curves 1-3 in Fig. 4 demonstrate the generally observed increase in the mobility ratio with decreasing ionic strength. However, it is also obvious that, independently of the depletion layer thickness used, the theoretical slopes are quite different from the experimental slopes. At high ionic strengths the theoretical results do not change as strongly as the experimental data. If smaller values for the depletion layer thickness were used then the intermediate range of ionic strengths could not be fitted. Electrophoretic measurements provide information concerning the viscosity variation only within the electric double layer. The bulk viscosity as such is not important with regard to the electrophoretic mobility. Although this conclusion may seem to be contraintuitive, it is a consequence of Eq. (1); it is justified by the fact that the convective flow at larger distances from the particle surface obeys potentiality. From these considerations it follows that electrophoretic mobility values at high ionic strengths are sensitive to the viscosity at a distance of the order of 1 nm from the particle surface. To assess the viscosity at larger distances it is necessary to reduce the ionic strength. To prove whether the decreasing slope of the experi-
4.0 3.5 O := n,,
30 25
~5 0
2.0
o :,.=
1.5 1.0
o
05
m 0.0
I
[
0
5
i
10
I
115
20
Oistl~lnce
I
25
30
35
Debye-H0ckelLength[nm] Fig. 4. Effect of a single-step profile with different length parameters of the depletion layer (cf. insert) on the theoretical calculated electrophoretic mobility ratio: . • ', 4 wt.% dextran; qs, viscosity of the solvent solution; ~/p, viscosity of the polymer solution; curve 1, A I = 1.5 nm; curve 2, A1=0.7 nm; curve 3, A1 =0.1 nm. The experimental electrophoretic mobility ratios are given for 6 wt.% dextran ( R ) and 4 wt.% dextran (D).
A. Krabi et al. / Colloids Surfaces A: Physieochem. Eng. Aspects 122 (1997) 33-42
38
mental ratio towards smaller ionic strengths might be a consequence of an extended, but smaller in depth, viscosity profile, a double-step profile was compared with the experimental data (Fig. 5). The extension of the second step was varied. As expected, all three theoretical curves coincided at the highest ionic concentrations measured because in this region only the first step of the profile is of importance. At low ionic strengths the use of the second step resulted in a further increase in the depletion effect (further reduction in the mobility ratio). It may be concluded, therefore, that a double-step profile, and presumably any other form of a more extended monotonic profile, would also not describe the experimental data sufficiently well. We now consider the hairy surface layer concept in the context of depletion. In addition to variations in the depletion layer thickness and the profile it is now necessary to investigate the possible influence of flow penetrability in the layer on the electrophoretic depletion effect. In Fig. 6 we illustrate the influence of the depletion layer thickness, at a chosen value of the flow penetrability of 6/3, where 6 denotes the thickness of the hairy layer. We used for 6 a value of 10 nm which seems a reasonable approximation for the
i
,
i
,
i
,
i
,
[
,
i ......
2.5
Ill
!'" .,
.9
."
2,0
P(
=
.....
..
...'"
"
2
.......................11111111.
1
. ....
........
..ii........3 0 r 2.5
: j:: "
J5 1.5
......
..-
.:
, 3
.....................................
o :,=
m
1.0
8
2e~
s
> 05~
P o5 ~6
03
"1 )'1
W 0.0
0'
'
5
'o''
1
XI+X2 15
Distance
'0
2
25 '
'o
35
Debye-H0ckel Length [nm] Fig. 5. Effect of a double-step profile (cf. insert) on the theoretically calculated electrophoretic mobility ratio. The thickness parameter A1 =0.7 nm (for the first step) and Gg* = 1.86 mPa s. Curve 1, A 2 = 1 0 n m ; curve 2, A2=5 nml; curve 3, A z = 0 n m . As a comparison the experimental curve for latex A particles in 4 wt.% 110 kDa dextra~n as a function of ionic strength is shown (B).
i
i
,
i
i
l
i
4.0
0 nm /
~'0~3.5 o
=
/
/
~
~~
~-
,5 n m
X= 3 nm
if- 3.0 ._z-
-~ 2.5 ~
2.0
O r-
° t e~ 1.5
2
~
W
1.0
I
1
,
I
i
,
,
o
Debye-Heckel Length [nm] Fig. 6. Effect of different depletion layer thicknesses Gl for a given ftwo penetrability on the calculated electrophoretic mobility ratio. 6/3, flow penetrability of the hairy layer (6, thickness of the hairy layer). The experimental curve for latex A particles in 6 wt.% 110 kDa dextran as a function of ionic strength ( I ) is given for comparison.
extended length of a PEO chain consisting on average of 10 statistical segments (nominal PEO molecule molecular weight, 2000), having a length of 2.1 nm [24]. As expected, the theoretical depletion effect increases as the thickness of the depletion layers becomes larger. There is a specific feature of the penetrable hairy layer treatment which arises in the absence of a depletion layer; the predicted bs/bp values are still lower than the t/p/t/s limit. This is a consequence of the partial electrophoretic flow generation inside the hairy layer. This is also the explanation for the increase in the depletion effect with increasing flow penetrability as shown in Fig. 7. On inspecting the theoretical curves shown in Figs. 6 and 7 it is clear that, as in the case of smooth surfaces, the increase in the experimental bs/bp values with decreasing salt concentration is too rapid. It is intriguing that the mobility ratio data, obtained with PEO, have the same, although much more pronounced, unexpected dependence on ionic strengths. There must be an additional reason for the observed deviations of the fitted curves from the experimental data. One could make a number of suggestions:
A. Krabi et al. / Colloids Surfaces A." Physicochem. Eng. Aspects 122 (1997) 33-42 4.5
i
i
i
i
i
i
i
,
5/20
. ~ 4.0 .a o 3.5
b/3 2~
ne
.~, 3.0 0
2.5
O :~ 2.0 0 e-
~ 1.5
--~ 1.0 W I 0
,
I 5
'
110 '
115 '
2/
0
I 25
I 30
i
35
Debye-H0ckel Length [nm] Fig. 7. Effect of the flow penetrability on the calcuated electrophoretic mobility ratio. The thickness A o f the depletion layer was 2 nm. The experimental curve for latex A particles in 6 wt.% 110 kDa dextran as a function of ionic strength (11) is given for comparison.
(1) the depletion layer thickness could be dependent on the ionic strength; (2) a change in the surface charge density may occur as a function of the polymer concentration; (3) polymer adsorption might be present; (4) the structure, and hence the flow penetrability, of the hairy layer may change as a function of ionic strerlgth and polymer concentration.
the interaction with the bulk polymer the ion exclusion and thus the surface charge density changes as well. Current concepts of hairy layer electrokinetics do not quantitatively take into account ion exclusion. Further research is clearly needed to contribute to a better understanding of the mechanism of surface charge generation at porous or hairy surfaces. Adsorption, however, cannot be a priori ruled out either. Let us, therefore, explore the effect of a possible adsorption layer onto a smooth surface. This would lead to hydrodynamical screening of the electro-osmotic flow of the diffuse layer mobile charges, and the "plane of shear" would be shifted by the thickness A of the adsorption layer towards the bulk. It may be further assumed that the depletion layer starts just in front of the adsorption layer, as indicated in the insert of Fig. 8. It is obvious that the adsorption layer should lead to an apparent increase in the mobility ratio since the accessible charge density in the presence of the polymer is smaller. It is also clear that the influence of an adsorption layer will be more important at high ionic strengths than at low ionic strengths, since in the latter case the adsorption layer thickness is small compared with the Debye-HOckel length. These theoretical predictions are illustrated
i
Since the depletion layer thickness is primarily related to entropy effects associated with the soluble polymer chains, it is highly unlikely that the depletion layer thickness is a function of ionic strength. Although it also seems unlikely that the surface charge density is dependent on the bulk polymer concentration, since the relatively low concentrations of the added neutral polymer would not significantly modify the polarity of the solution, there is still the possibility that the small latex particle surface charge is at least partially the result of a different extent of ion exclusion from the surface layer. The hydrated cations, being larger than the less hydrated chloride, are less able to penetrate into the grafted layer as well as into the porous latex surface. This would result in a seemingly negatively charged surface. It is conceivable that on a layer structure change induced by
39
=
25
• 0 ~
f
2.0 mr'
" .':~'" ~c~ 0 o ~ 0
1.5
1.0
2e ~ o
If
,
i
,
,
,
i
.......................iiiiiiiiiiiii ...- ....
...'""
.
i
3
'
2
. ..............
...... .....
.-"i-.'"'"3.0
2.5~ ~ 2.0
~,0fl-
0.5
k~+~ W
Distance
00
Debye-H0ckel Length [nm] Fig. 8. Effect of an adsorption layer with a thickness 6 (cf. insert) on the theoretically calculated electrophoretic mobility ratio: • • . , plots with a single-step viscosity profile. The thickness parameter A x = l . 5 n m . Curve 1, 6 = 0 n m ; curve 2, 6 = 0.5 nm; curve 3, 6 = 1.10 nm. III, experimental data for latex A particles in 4 wt.% 110 kDa dextran as a function of ionic strength.
A. Krabi et al. /Colloids Surfaces A." Physicochem. Eng. Aspects 122 (1997) 33 42
40
in Fig. 8, where the adsorption layer thickness has been varied. The conclusion of adsorption leads to a non-monotonic behaviour of the theoretical mobility ratio, as occurs with the experimental data. The high sensitivity of the mobility ratio to adsorption at high ionic strengths is quite marked. Moreover, small changes in the adsorption layer thickness at high ionic strengths, in principle, could account for the observed behaviour of the PEOgrafted latex particles in dextran solution. To explore this hypothesis we solved the inverse problem of calculating the adsorption layer thickness which would be necessary to account for the experimental data. The results of these calculations are shown in Fig. 9. It could appear that a reasonable fit of the experimental data, for both 4 wt.% and 6 wt.% dextran, is produced only if unreasonably small values for the adsorption layer thickness are used, namely a gradual increase from approximately 0.1 nm at 90 mM NaCI to 1.3 nm at 0.1 mM NaC1, as shown in the insert. Moreover, to fit the PEO data as shown in Fig. 3, adsorption layer thicknesses of as much as 30 nm had to be taken. Thus, in both cases, it seems unlikely that adsorption, after all, is occurring. Finally, we consider the fourth suggestion, namely possible changes in the flow penetrability of the layer. One can imagine two causes for a change in flow penetrability. If the layer decreases t
t
i
i
i
i
in thickness then the friction coefficient for hydrodynamic flow should increase. However, penetration of bulk polymer molecules into the layer would also increase the hydrodynamic friction coefficient in the layer. It is clear that any resulting variations in the distribution of the surface charge would also play a very important role here. The effects of variations in the surface charge distribution are easily understood in qualitative terms, without referring to complex theoretical analysis [21]. If, for example, the surface charge groups were located only at the outer edge of the hairy layer, then any changes in the layer architecture inside the layer would have only a limited influence on the electrophoretic mobility. If, on the contrary, the charge is beneath the hairy layer, then the mobility of the particle would depend strongly on the hydrodynamic properties and extension of the layer, as in the case of an adsorption layer. Although we do not know the detailed surface charge distribution of the latex particles used, it is reasonable to assume that the surface charge is at the core surface, beneath the neutral grafted PEO. On that basis we calculated the effect of changes in the flow penetrability. The results are shown in Fig. 10. With decreasing flow penetrability, as compared with the control, large decreases in the i
o, ,.Q
i
i
i
i
i
i
i
6
..Q o. e~
4
o
5
~/10 6/5 (5/3
112
.a o
~O 55 ~o 2
3
o
~O
o
2
lie
r-
Q. O
<
o
@
o o0 o 0
W 0
I 0
I 5
110
o0s
o~0
0is
o2
0
I ElectrOlyte IC°ne~ntratifn Im°lllllel 15 20 25 30
rn 0
Debye-HOckel Length [nm]
Debye-H0ckel Length [nm] Fig. 9. Fits of the electrophorefic mobility ratios of latex A particles assuming electrolyte concentration dependent adsorption of dextran (cf. insert ). The length parameter A 1 of depletion was 1.5 nm. As a comparison the experimental data are shown: [], 6wt.% l l 0 k D a dextran; II, 4wt.% l l 0 k D a dextran.
Fig. 10. Effect of a change in the flow penetrability of the layer. Flow penetrability of the control, 1/6. The corresponding flow penetrabilities of the polymer-containing solutions were 6, 6/3, 6/10. The depletion layer thickness A was 2 nm. As a comparison the experimental data for 6wt.% l l 0 k D a dextran are given (11).
A. Krabi et al. / Colloids Surfaces A: Physicochern. Eng. Aspects 122 (1997) 33 42
mobility ratio can be produced. With low penetrability, the theoretical mobility ratio can be significantly larger than the theoretical viscosity limit. In addition, variations in the thickness of the layer or modification of the accompanying surface charge distribution would have a strong effect on theoretical curves. Having performed this comprehensive analysis of the influence of various parameters on the mobility ratio of a hairy particle in the presence of a depletion layer, it is obvious that there is much freedom to fit experimental data. It is not possible, at present, to draw any unambiguous conclusions. Nevertheless, the following suggestions seem reasonable. The electrophoretic mobility of latex particles with hairy layers is determined to a large extent by the electroosmotic flow inside the hairy layer. There may be a small decrease in the flow penetrability with added dextran which may be caused by a small compression of the hairy layer. Evidence for such an effect has been published [25]. Supported by the known immiscibility of PEO and dextran the latter forms a depletion layer in front of the grafted PEO. In the case of added PEO, any penetration of free PEO chains into the PEO grafted layer considerably reduces the flow penetrability and consequently would produce much lower values for bs/br, than predicted by the bulk viscosity ratio. It is not contradictory to the idea of the formation of a depletion layer of PEO in front of the grafted layer if the penetration of some low molecular weight PEO into the grafted layer is assumed. Together with osmotically induced layer compression this may cause a much more pronounced decrease in flow penetrability than with added dextran. The majority of larger molecular weight polymers still form the depletion layer in front of the grafted PEO. In future research special attention should be given to the problem of ion exclusion which may also modify the electrophoretic behaviour of weakly charged porous and hairy surfaces. Further experiments to investigate these effects are clearly necessary. In particular, it is desirable to use particles with a known, controllable surface charge distribution, as well as with hairy layers of variable thickness. This would be useful for a
41
better separation of surface charge effects and surface hydrodynamic effects on the electrophoretic mobility of hairy particles. We hope that this preliminary electrophoretic investigation of polymer depletion near hairy particles will lead to further developments in this area.
Acknowledgment This study became possible as a result of a DAAD-ARC exchange project. The theoretical foundations were supported by a DFG grant given to E. Donath (Do 410 1-1). G.C. Allan and B. Vincent acknowledge the support from the DTI Colloid Technology Programme.
References [1]D.H. Napper, Polymeric Stabilisation of Colloidal Dispersions, Academic Press, New York, 1983. [2] G.J. Fleer and J.M.H.M. Scheutjens, Colloids Surfaces, 51 (1990) 281. [3] F.K.R. Li-In-On, B. Vincent and F.A. Waite, Am. Chem. Soc. Symp. Ser., 9 (1975) 165. [4] C. Cowell and B. Vincent, J. Chem. Soc. Faraday Trans. I, 74 (1978) 337. [5] B. Vincent, P.F. Luckham and F.A. Waite, J. Colloid Interface Sci., 73 (1980) 508. [6] C. Cowell and B. Vincent, J. Colloid Interface Sci., 87 (1982) 18. [7] A. Jones and B. Vincent, Colloids Surfaces, 42 (1989) 13. [8] A. Milling, B. Vincent, S. Emmett and A. Jones, Colloids Surfaces, 57 (1991) 185. [9] M.R. B6hmer, O.A. Evers and J.M.H.M. Scheutjens, Macromolecules, 93 (1990) 2288. [10] N. Cawdery and B. Vincent, in R.D. Buscall and J.W. Goodwin (eds.), Latex Dispersions, Academic Press, New York, 1993. [ 11 ] B. Vincent, J. Edwards, S. Emmett and R. Croot, Colloids Surfaces~ 31 (1988) 267. [12] G.J. Fleer, J.M.H.M. Scheutjens and M.A. Cohen Stuart, Colloids Surfaces, 31 (1988) 1. [13] D.E. Brooks and G.V.F. Seaman, J. Colloid Interface Sci., 43 (1973) 670. [14] P. Snabre and P. Mills, Colloid Polym. Sci., 264 (1985) 494. [15] H. B~iumler and E. Donath, Stud. Biophys., 120 (1987) 113. [16] L. Pratsch and E. Donath, Stud. Biophys., 123 (1988) 101. [17] m. Krabi and E. Donath, Colloids Surfaces A: Physicochem Eng. Asp., 92 (1994) 175.
42
A. Krabi et al. / Colloids Surfaces A: Physicochem. Eng. Aspects 122 (1997) 33-42
[18] E. Donath, L. Pratsch, H. B~iumler, A. Voigt and M. Taeger, Stud. Biophys., 130 (1989) 117. [19] H. B~iumler, E. Donath, L. Pratsch, D. Lerche, J.F. Stolz, M. Donner and A.L. Copley (eds.), Hemorheologie et Agregation Erythrocytaire, Editions Medicales Internationales, Cachan, 1991, p. 24. [20] E. Donath, A. Krabi, G. Allan and B. Vincent, Langrnuir, 12 (1996) 3425. [21] E. Donath, P. Kuzmin, A. Krabi and A. Voigt, Colloid Polym. Sci., 271 (1993) 930.
[22] J.F. Miller, K. Sch/itzel and B. Vincent, J. Colloid Interface Sci., 143 (1991) 532. [23] C. Prestidge and Th.F..Tadros, Colloids Surfaces, 31 (1988) 325. [24] A.R. Rennie, R.J. Crawford, E.M. Lee, R.K. Thomas, T.L. Crowley, S. Roberts, M.S. Qureshi and R.W. Richards, Macromolecules, 22 (1989) 3466. [25] S. Bamberger, G.V.F. Seaman, J.A. Brown and D.E. Brooks, J. Colloid Interface Sci., 99 (1984) 187.
COLLOIDS AND SURFACES
A
A: Physicochemicaland EngineeringAspects 122 (1997) 33-42
Depletion of dextran and PEO for latex particles with "hairy layers": an electrophoretic study A. Krabi a, G. Allan b, E. Donath c, B. Vincent b a M P I of Colloids and Interfaces, Rudower Chaussee 5, 12489 Berlin, Germany b School of Chemistry, University of Bristol, Bristol, UK c Department of Biology, Humboldt University of Berlin, Berlin, Germany
Received 7 February 1996; accepted 5 August 1996
Abstract
The electrophoretic mobility of cross-linked polystyrene latex, stabilized with a grafted layer of poly(oxyethylene), and added dextran and PEO was measured as a function of ionic strength. The ratio of electrophoretic mobility in the absence of added polymer to that in the presence was less than expected from the Smoluchowski equation for dextran and higher for PEO. This seeming discrepancy could be explained by assuming polymer depletion with a consequent reduction in the viscosity near the interface and the additional effect of different flow penetrabilities into the grafted layer for the two polymers. The depletion effect was studied theoretically for the case of layers nonpenetrable and penetrable for hydrodynamic flow. The effect of adsorption on depletion was also analysed. Keywords: Depletion; Dextran; Electrophoretic mobility; Flow penetrability; Grafted latex; PEO; Viscosity
1. Introduction
The stability of particles with a "hairy" surface in the presence of free polymers is of both theoretical and practical interest [1-5]. A model system containing cross-linked polystyrene latex particles carrying terminally grafted, solvent-miscible polymer chains (PEO) has been designed previously (PS-g-PEO) [6]. The effects of the surface coverage of the grafted layer [7, 8] and of added polyelectrolytes [9] on the flocculation behaviour have been studied by Vincent et al. [10]. The results revealed depletion flocculation beyond a critical free polymer concentration [11 ]. Depletion of polymers takes place if the loss of configurational entropy near the interface is not balanced by positive interaction energy. Flocculation can occur when the osmotic attraction energy due to depletion outweighs the loss in configurational entropy [12].
In the so-called depletion zone the polymer segment density is lower than that in the bulk, resulting in a reduced viscosity in comparison with the bulk polymer solution. Electrophoretic studies on charged smooth particles in polymer solutions have been performed and generally revealed an increase in mobility compared with that predicted from the Smoluchowski equation [13,14]. These findings were attributed to the presence of a depletion layer. Donath and coworkers calculated the apparent thickness of depletion layers from electrophoretic data for human red blood cells and liposomes in dextran and poly(ethylene)glycol, assuming an exponential viscosity profile in a linear approximation [15-17]. This was possible for smooth and hairy surfaces at high ionic strength. The calculated thicknesses were in reasonable agreement with the radii of gyration of
0927-7757/97/$17.00 Copyright© 1997ElsevierScienceB.V. All rights reserved PH S0927-7757 (96)03824-1
34
A. Krabi et al. / Colloids Surfaces A: Physicochem. Eng. Aspects 122 (1997) 33-42
the polymers [18,19]. The linearization failed when applied to the recent experimental results measured for liposomes in high molecular weight dextran in a wide range of ionic strengths. Therefore, in [20] a numerical integration of the position-dependent viscosity profile of the Navier-Stokes equation was performed. Good coincidence with the experimental results was demonstrated. It was shown that the hydrodynamic thickness of the depletion layer can be obtained. It was challenging to study the electrophoretic mobility of a hairy surface together with a depletion layer, since little is known about the structure of the depletion layer in front of a hairy surface. Also, it cannot be ruled out that the free polymer can influence the density, as well as the equilibrium distribution, of the terminally anchored polymers of the hairy layer. Previously, we investigated theoretically the depletion effect on the electrophoretic mobility in the presence of a hairy layer [21]. The electrophoretic behaviour of a hairy layer itself is by no means trivial. Among other factors it is quite important whether the electro-osmotic flow can be generated within the layer or whether the layer is too dense to allow for hydrodynamic flow inside the layer. Given, in addition, the different possibilities for the surface charge distribution, either within the layer or beneath the layer, the electrophoretic mobilities can be quite different even if the net surface charge densities of different particles are similar. In order to estimate quantitatively the hydrodynamic depletion effect for hairy particles we performed measurements of the electrophoretic mobility of latex particles with grafted PEO chains (M w~2000) in dextran as well as in PEO.
2. Theoretical background In the case of a depletion layer the viscosity q(x) of the solution is a function of the distance from the particle surface. We showed previously that, in the case of a smooth surface, integration of the Navier-Stokes equation together with the Poisson equation provides an expression for the electro-
phoretic mobility b [20]: b-
v(~) E
=
--E~Eo
id~ Jv,~xo) q[x(~U)]
(1)
v(oo ) is the electrophoretic velocity, E is the electric field strength, er and eo are the relative and absolute permittivities, ~(x) is the electric potential, 7t(xo) is the potential at the shear plane Xo, and x is the coordinate perpendicular to the surface. The inverse electric potential distribution x(~U) taking into account non-linearity is given by tanh zeo gs 1
4kT
k
zeo 7So
x ( ~ ) = - - In tanh
(2)
4kT
which is valid for symmetric electrolytes. ~Uois the surface potential at x = 0, z is the valence of the electrolyte ions, eo is the elementary charge, k is the Boltzmann constant, T is the absolute temperature and x the inverse Debye-Htickel length. In Ref. [20] we showed that the influence of the assumed specific viscosity profile on the fitted depletion layer characteristics is not too large since b is an integral over the inverse of that profile (Eq. (1). The thickness parameter A of the depletion layer, however, influences the measured mobility in a more sensitive manner. Therefore, it is reasonable to use, as an approximation, step viscosity profiles. In a sense the double-step viscosity profile with the solvent viscosity in a small layer of thickness A1 and with a larger viscosity in an adjacent layer of thickness A 2 is the simplest possible description of the depletion layer of polymers near the interface together with a slip layer.
q(x)=
r/s
x_<21
r/*
21
/~p
X > 2 1 -{-2 2
f
(3)
where qs is the solvent viscosity, qp is the viscosity of the bulk polymer solution. A1 and A 2 are the length parameters of the depletion layers. ~/* is the intermediate viscosity in the double-step profile. In contrast, in the case of hairy layers, an
A. Krabiet al. / ColloidsSurfaces A: Physicochem. Eng. Aspects 122 (1997) 33-42 analytical non-linear treatment is not possible. Recently, we presented an analytical formula assuming the linearized approximation for the electric potential distribution. This approximation is generally not too bad because, in the case of charged hairy layers, the electric potentials are smaller owing to the spatial distribution of the surface charge. For details the reader is referred to the original publication [21]. In that study only an exponential viscosity profile was analysed. However, it is not difficult to allow for arbitrary viscosity profiles, as is done in this paper. The solution presented in Ref. [21] can be divided into two parts, where the first part is responsible for the electrophoretic flow generated inside the layer as a result of the penetrating hydrodynamic flow, with the characteristic penetration length 1/a. The second part represents that part of the electrophoretic flow which stems from the contribution of the double-layer part outside the hairy layer. It is clear that, either for very thin hairy layers (or very low ionic strength) or for non-penetrable layers, the theoretical limit of a smooth surface is approached. For the specific case of the surface charge distributed in a plane underneath the hairy layer the mobility ratio is given by the following, simplified equation: n oo
bs
--
bp
=
t
us(x) e - ~ dx
o
(4)
{'~ Up(X)e -'x dx d0
where bs and by are the electrophoretic mobilities for the particles in polymer-free and polymercontaining solution respectively. The velocities of the liquid flow, inside and outside the hairy layer, for polymer-free and polymer-containing solutions, are given by sinh(ax) at/s cosh(a6) us(x) = ] x - 6 / qs
+ tanh(a6) ar/s
x<6 x>6
(5)
35
and f] uv(x) =
sinh(ax) aqs cosh(~6)
x<6
1 d~+ tanh(a6)
x>6
~/(~)
(6)
ar/s
where 1/a is the characteristic length of flow penetrability and 6 the thickness of the hairy layer.
3. E x p e r i m e n t a l
details
Two different types of PS-g-PEO particles, prepared with different initiators, were studied. They were prepared using the method described by Cowell and Vincent [6]. Styrene, PEO-methacrylate, and a small quantity of divinyl benzene, as a cross-linker, were co-polymerized in a watermethanol solvent mixture. The initiators used were 2,2'-azobisisobutyronitrile (latex A) and 4,4'azobis (4-cyanopentanoic acid) (latex B). Latex A, in theory, has a low concentration and latex B a higher concentration of carboxy-groups on the particle surfaces. Both sets of latex particles had a mean diameter of about 500 nm, as determined by transmission electron microscopy. The electrophoretic mobility of the PS-g-PEO latex particles was determined using both a Zetasizer 4 from Malvern and the phase analysis light scattering (PALS) technique [22] which is particularly suitable for measuring low mobilities. The electrophoretic mobility (bs) of the latices in the absence of polymer was measured as a function of the electrolyte concentration (NaC1) at pH 9.5 and used as a control. The chargecarrying groups of both latices are carboxy-groups which will be fully ionized at this high pH. This reduces any error in the measurement of bs. Dextran with molecular weights 110 kDa and 500 kDa was used. The dextrans were purchased from Fluka and used without further purification. Solutions with the appropriate dextran concentrations were prepared and their viscosities were determined with a Viscoboy capillary viscosimeter from Lauda. The weakly charged latex particles had velocities
36
A. Krabi et al. / Colloids Surfaces A." Physicochem. Eng. Aspects 122 (1997) 33 42
less than 1 I~m s-1. Given the small thickness of the electric double layer of the order of 1 nm, nevertheless high shear rates of the order of 103 s-1 are predicted within the double layer. Thus for the particular polymer concentration range of 2-8 wt.% we checked the newtonian behaviour by means of shear rate variation up to 2000 s-~. N o deviations from linearity were recorded. The mobilities be of the latex particles in the polymer solutions were measured and the results are presented as the ratio bs/bp.
10
o. .t3
9
~8 o
_=
IZ ~_
6
s o
4 r/piTs = 4.07
o e~
o
3
"~
2
~]
Lt,.I 1
'0
's
,'0
,'s
2'0
;5 ' ~'0
3,
Debye-H0ckel Length[nm] 4. Results
I f the effect o f the free polymer on the electro-
phoretic mobility consisted only of increasing the viscosity, Smoluchowski's formula ErEO~
b= - -
(7)
q
for the electrophoretic mobility predicts an electrophoretic mobility ratio bs/br, = qr,/qs. As Figs. 1 and 2 demonstrate, the measured electrophoretic mobility ratios for the latex particles in dextran were remarkably lower than the corresponding viscosity ratios r/p/~/s over the whole range o f ionic strength for the two dextran concentrations and molecular weights used. As discussed i
i
r
~
i
i
Fig. 2. Ratio of the electrophoretic mobility of latex B in 500 kDa dextran: O, 6 wt.%; [2, 4 wt.%. The corresponding viscosity ratios are shown as broken horizontal lines. previously [17], this is strong evidence for the existence o f a depletion layer, with a reduced viscosity near the particle particle surface, in dextran solutions. In the range o f high electrolyte concentrations the mobility ratio in 110 k D a dextran exhibited a steep increase with decreasing ionic strength, i.e. increasing D e b y e - H t i c k e l length. Towards lower salt concentrations this increase flattened. In Fig. 3 the mobility ratios for both latex A and B are plotted as a function o f the Debye length in 4% 20 k D a PEO. The mobility behaviour
i
5.0
,
~p/~/s = 4.45
o
4.0
n,
i
i
i
i
i
i
14
.................................................
"Q~ 4.5
..Q 0
12
3.5 10
"qp/r/s = 2.72 -~ o
3.0
o
2.5
.O O
.01
8
o ]1[
~6 1.51 I11 []
~t
i
1.0[
4
~D.I
2
I
I
I
0
0r~ 0 51
110
15
20
25
30
35
Debye-HOckelLength[nrn] Fig. 1. Electrophoretic mobility ratio of latex A particles in 110 kDa dextran as a function of ionic strength: II, 6 wt.% dextran; [], 4 wt.% dextran. The corresponding viscosity ratios are shown as dotted horizontal lines and were 2.72 (4%) and 4.45 (6%).
....
_t_ ........................................
'0
;
,o
'
~_(_~_
2'0'
' 3'0
Debye-H0ckel Length [nrn] Fig. 3. Electrophoretic mobility ratio o f latex A and B particles in 4 wt.% 20 k D a PEO as a function o f ionic strength: A , latex A; II, latex B; • . ., viscosity ratio r/v/qs = 1.88.
A. Krabi et aL / Colloids Surfaces A." Physicochem. Eng. Aspects 122 (1997) 33-42
is very different from the dextran solution data. It is evident that the mobility ratios here are higher than the viscosity ratios over the whole range of ionic strengths considered. It is remarkable that, at the highest ionic strength studied, the mobility ratio barely exceeds the upper theoretical limit for depletion, while at intermediate ionic concentrations the mobility ratio is an order of magnitude higher than the corresponding viscosity ratio. At relatively low ionic strengths (approximately 10-4 M ) the mobility ratio again becomes smaller.
5. Theoretical analysis and discussion
It is clear that there is no way that the PEO data can be fitted to the existing theories of the effect of depletion layers on the mobility ratio. This was a major contradiction to our expectations, since previous flocculation experiments with similar systems indicated that polymer depletion should be present [23]. Consequently, a different explanation for the present PEO data is clearly required. We shall firstly attempt to account for the dextran data within the framework of the existing theories for smooth as well as hairy surfaces, before we investigate alternative explanations for the PEO data. Although the model particles definitely have a hairy layer, it is still a reasonable first approximation to use the theories for smooth surfaces, since very little is known about the hydrodynamic properties of such hairy layers. The smooth surface concept will be completely valid if the layer is nonpenetrable for flow. In the case of a non-penetrable hairy surface the difference between the smooth and hairy layer concepts will be reflected in the values of the absolute mobilities, which in the case of a hairy layer depend in addition on the surface charge distribution. Since in this work only the mobility ratios are of interest, it is evident that the existing depletion layer theories of the electrophoretic mobility of smooth surfaces are completely applicable to hairy layers, provided that the hydrodynamic flow inside the layer can be safely neglected. Let us first compare the theoretical result, for an assumed single-step viscosity profile, with
37
different depletion layer thicknesses A, as shown in the insert of Fig. 4. Curves 1-3 in Fig. 4 demonstrate the generally observed increase in the mobility ratio with decreasing ionic strength. However, it is also obvious that, independently of the depletion layer thickness used, the theoretical slopes are quite different from the experimental slopes. At high ionic strengths the theoretical results do not change as strongly as the experimental data. If smaller values for the depletion layer thickness were used then the intermediate range of ionic strengths could not be fitted. Electrophoretic measurements provide information concerning the viscosity variation only within the electric double layer. The bulk viscosity as such is not important with regard to the electrophoretic mobility. Although this conclusion may seem to be contraintuitive, it is a consequence of Eq. (1); it is justified by the fact that the convective flow at larger distances from the particle surface obeys potentiality. From these considerations it follows that electrophoretic mobility values at high ionic strengths are sensitive to the viscosity at a distance of the order of 1 nm from the particle surface. To assess the viscosity at larger distances it is necessary to reduce the ionic strength. To prove whether the decreasing slope of the experi-
4.0 3.5 O := n,,
30 25
~5 0
2.0
o :,.=
1.5 1.0
o
05
m 0.0
I
[
0
5
i
10
I
115
20
Oistl~lnce
I
25
30
35
Debye-H0ckelLength[nm] Fig. 4. Effect of a single-step profile with different length parameters of the depletion layer (cf. insert) on the theoretical calculated electrophoretic mobility ratio: . • ', 4 wt.% dextran; qs, viscosity of the solvent solution; ~/p, viscosity of the polymer solution; curve 1, A I = 1.5 nm; curve 2, A1=0.7 nm; curve 3, A1 =0.1 nm. The experimental electrophoretic mobility ratios are given for 6 wt.% dextran ( R ) and 4 wt.% dextran (D).
A. Krabi et al. / Colloids Surfaces A: Physieochem. Eng. Aspects 122 (1997) 33-42
38
mental ratio towards smaller ionic strengths might be a consequence of an extended, but smaller in depth, viscosity profile, a double-step profile was compared with the experimental data (Fig. 5). The extension of the second step was varied. As expected, all three theoretical curves coincided at the highest ionic concentrations measured because in this region only the first step of the profile is of importance. At low ionic strengths the use of the second step resulted in a further increase in the depletion effect (further reduction in the mobility ratio). It may be concluded, therefore, that a double-step profile, and presumably any other form of a more extended monotonic profile, would also not describe the experimental data sufficiently well. We now consider the hairy surface layer concept in the context of depletion. In addition to variations in the depletion layer thickness and the profile it is now necessary to investigate the possible influence of flow penetrability in the layer on the electrophoretic depletion effect. In Fig. 6 we illustrate the influence of the depletion layer thickness, at a chosen value of the flow penetrability of 6/3, where 6 denotes the thickness of the hairy layer. We used for 6 a value of 10 nm which seems a reasonable approximation for the
i
,
i
,
i
,
i
,
[
,
i ......
2.5
Ill
!'" .,
.9
."
2,0
P(
=
.....
..
...'"
"
2
.......................11111111.
1
. ....
........
..ii........3 0 r 2.5
: j:: "
J5 1.5
......
..-
.:
, 3
.....................................
o :,=
m
1.0
8
2e~
s
> 05~
P o5 ~6
03
"1 )'1
W 0.0
0'
'
5
'o''
1
XI+X2 15
Distance
'0
2
25 '
'o
35
Debye-H0ckel Length [nm] Fig. 5. Effect of a double-step profile (cf. insert) on the theoretically calculated electrophoretic mobility ratio. The thickness parameter A1 =0.7 nm (for the first step) and Gg* = 1.86 mPa s. Curve 1, A 2 = 1 0 n m ; curve 2, A2=5 nml; curve 3, A z = 0 n m . As a comparison the experimental curve for latex A particles in 4 wt.% 110 kDa dextra~n as a function of ionic strength is shown (B).
i
i
,
i
i
l
i
4.0
0 nm /
~'0~3.5 o
=
/
/
~
~~
~-
,5 n m
X= 3 nm
if- 3.0 ._z-
-~ 2.5 ~
2.0
O r-
° t e~ 1.5
2
~
W
1.0
I
1
,
I
i
,
,
o
Debye-Heckel Length [nm] Fig. 6. Effect of different depletion layer thicknesses Gl for a given ftwo penetrability on the calculated electrophoretic mobility ratio. 6/3, flow penetrability of the hairy layer (6, thickness of the hairy layer). The experimental curve for latex A particles in 6 wt.% 110 kDa dextran as a function of ionic strength ( I ) is given for comparison.
extended length of a PEO chain consisting on average of 10 statistical segments (nominal PEO molecule molecular weight, 2000), having a length of 2.1 nm [24]. As expected, the theoretical depletion effect increases as the thickness of the depletion layers becomes larger. There is a specific feature of the penetrable hairy layer treatment which arises in the absence of a depletion layer; the predicted bs/bp values are still lower than the t/p/t/s limit. This is a consequence of the partial electrophoretic flow generation inside the hairy layer. This is also the explanation for the increase in the depletion effect with increasing flow penetrability as shown in Fig. 7. On inspecting the theoretical curves shown in Figs. 6 and 7 it is clear that, as in the case of smooth surfaces, the increase in the experimental bs/bp values with decreasing salt concentration is too rapid. It is intriguing that the mobility ratio data, obtained with PEO, have the same, although much more pronounced, unexpected dependence on ionic strengths. There must be an additional reason for the observed deviations of the fitted curves from the experimental data. One could make a number of suggestions:
A. Krabi et al. / Colloids Surfaces A." Physicochem. Eng. Aspects 122 (1997) 33-42 4.5
i
i
i
i
i
i
i
,
5/20
. ~ 4.0 .a o 3.5
b/3 2~
ne
.~, 3.0 0
2.5
O :~ 2.0 0 e-
~ 1.5
--~ 1.0 W I 0
,
I 5
'
110 '
115 '
2/
0
I 25
I 30
i
35
Debye-H0ckel Length [nm] Fig. 7. Effect of the flow penetrability on the calcuated electrophoretic mobility ratio. The thickness A o f the depletion layer was 2 nm. The experimental curve for latex A particles in 6 wt.% 110 kDa dextran as a function of ionic strength (11) is given for comparison.
(1) the depletion layer thickness could be dependent on the ionic strength; (2) a change in the surface charge density may occur as a function of the polymer concentration; (3) polymer adsorption might be present; (4) the structure, and hence the flow penetrability, of the hairy layer may change as a function of ionic strerlgth and polymer concentration.
the interaction with the bulk polymer the ion exclusion and thus the surface charge density changes as well. Current concepts of hairy layer electrokinetics do not quantitatively take into account ion exclusion. Further research is clearly needed to contribute to a better understanding of the mechanism of surface charge generation at porous or hairy surfaces. Adsorption, however, cannot be a priori ruled out either. Let us, therefore, explore the effect of a possible adsorption layer onto a smooth surface. This would lead to hydrodynamical screening of the electro-osmotic flow of the diffuse layer mobile charges, and the "plane of shear" would be shifted by the thickness A of the adsorption layer towards the bulk. It may be further assumed that the depletion layer starts just in front of the adsorption layer, as indicated in the insert of Fig. 8. It is obvious that the adsorption layer should lead to an apparent increase in the mobility ratio since the accessible charge density in the presence of the polymer is smaller. It is also clear that the influence of an adsorption layer will be more important at high ionic strengths than at low ionic strengths, since in the latter case the adsorption layer thickness is small compared with the Debye-HOckel length. These theoretical predictions are illustrated
i
Since the depletion layer thickness is primarily related to entropy effects associated with the soluble polymer chains, it is highly unlikely that the depletion layer thickness is a function of ionic strength. Although it also seems unlikely that the surface charge density is dependent on the bulk polymer concentration, since the relatively low concentrations of the added neutral polymer would not significantly modify the polarity of the solution, there is still the possibility that the small latex particle surface charge is at least partially the result of a different extent of ion exclusion from the surface layer. The hydrated cations, being larger than the less hydrated chloride, are less able to penetrate into the grafted layer as well as into the porous latex surface. This would result in a seemingly negatively charged surface. It is conceivable that on a layer structure change induced by
39
=
25
• 0 ~
f
2.0 mr'
" .':~'" ~c~ 0 o ~ 0
1.5
1.0
2e ~ o
If
,
i
,
,
,
i
.......................iiiiiiiiiiiii ...- ....
...'""
.
i
3
'
2
. ..............
...... .....
.-"i-.'"'"3.0
2.5~ ~ 2.0
~,0fl-
0.5
k~+~ W
Distance
00
Debye-H0ckel Length [nm] Fig. 8. Effect of an adsorption layer with a thickness 6 (cf. insert) on the theoretically calculated electrophoretic mobility ratio: • • . , plots with a single-step viscosity profile. The thickness parameter A x = l . 5 n m . Curve 1, 6 = 0 n m ; curve 2, 6 = 0.5 nm; curve 3, 6 = 1.10 nm. III, experimental data for latex A particles in 4 wt.% 110 kDa dextran as a function of ionic strength.
A. Krabi et al. /Colloids Surfaces A." Physicochem. Eng. Aspects 122 (1997) 33 42
40
in Fig. 8, where the adsorption layer thickness has been varied. The conclusion of adsorption leads to a non-monotonic behaviour of the theoretical mobility ratio, as occurs with the experimental data. The high sensitivity of the mobility ratio to adsorption at high ionic strengths is quite marked. Moreover, small changes in the adsorption layer thickness at high ionic strengths, in principle, could account for the observed behaviour of the PEOgrafted latex particles in dextran solution. To explore this hypothesis we solved the inverse problem of calculating the adsorption layer thickness which would be necessary to account for the experimental data. The results of these calculations are shown in Fig. 9. It could appear that a reasonable fit of the experimental data, for both 4 wt.% and 6 wt.% dextran, is produced only if unreasonably small values for the adsorption layer thickness are used, namely a gradual increase from approximately 0.1 nm at 90 mM NaCI to 1.3 nm at 0.1 mM NaC1, as shown in the insert. Moreover, to fit the PEO data as shown in Fig. 3, adsorption layer thicknesses of as much as 30 nm had to be taken. Thus, in both cases, it seems unlikely that adsorption, after all, is occurring. Finally, we consider the fourth suggestion, namely possible changes in the flow penetrability of the layer. One can imagine two causes for a change in flow penetrability. If the layer decreases t
t
i
i
i
i
in thickness then the friction coefficient for hydrodynamic flow should increase. However, penetration of bulk polymer molecules into the layer would also increase the hydrodynamic friction coefficient in the layer. It is clear that any resulting variations in the distribution of the surface charge would also play a very important role here. The effects of variations in the surface charge distribution are easily understood in qualitative terms, without referring to complex theoretical analysis [21]. If, for example, the surface charge groups were located only at the outer edge of the hairy layer, then any changes in the layer architecture inside the layer would have only a limited influence on the electrophoretic mobility. If, on the contrary, the charge is beneath the hairy layer, then the mobility of the particle would depend strongly on the hydrodynamic properties and extension of the layer, as in the case of an adsorption layer. Although we do not know the detailed surface charge distribution of the latex particles used, it is reasonable to assume that the surface charge is at the core surface, beneath the neutral grafted PEO. On that basis we calculated the effect of changes in the flow penetrability. The results are shown in Fig. 10. With decreasing flow penetrability, as compared with the control, large decreases in the i
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I ElectrOlyte IC°ne~ntratifn Im°lllllel 15 20 25 30
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Debye-HOckel Length [nm]
Debye-H0ckel Length [nm] Fig. 9. Fits of the electrophorefic mobility ratios of latex A particles assuming electrolyte concentration dependent adsorption of dextran (cf. insert ). The length parameter A 1 of depletion was 1.5 nm. As a comparison the experimental data are shown: [], 6wt.% l l 0 k D a dextran; II, 4wt.% l l 0 k D a dextran.
Fig. 10. Effect of a change in the flow penetrability of the layer. Flow penetrability of the control, 1/6. The corresponding flow penetrabilities of the polymer-containing solutions were 6, 6/3, 6/10. The depletion layer thickness A was 2 nm. As a comparison the experimental data for 6wt.% l l 0 k D a dextran are given (11).
A. Krabi et al. / Colloids Surfaces A: Physicochern. Eng. Aspects 122 (1997) 33 42
mobility ratio can be produced. With low penetrability, the theoretical mobility ratio can be significantly larger than the theoretical viscosity limit. In addition, variations in the thickness of the layer or modification of the accompanying surface charge distribution would have a strong effect on theoretical curves. Having performed this comprehensive analysis of the influence of various parameters on the mobility ratio of a hairy particle in the presence of a depletion layer, it is obvious that there is much freedom to fit experimental data. It is not possible, at present, to draw any unambiguous conclusions. Nevertheless, the following suggestions seem reasonable. The electrophoretic mobility of latex particles with hairy layers is determined to a large extent by the electroosmotic flow inside the hairy layer. There may be a small decrease in the flow penetrability with added dextran which may be caused by a small compression of the hairy layer. Evidence for such an effect has been published [25]. Supported by the known immiscibility of PEO and dextran the latter forms a depletion layer in front of the grafted PEO. In the case of added PEO, any penetration of free PEO chains into the PEO grafted layer considerably reduces the flow penetrability and consequently would produce much lower values for bs/br, than predicted by the bulk viscosity ratio. It is not contradictory to the idea of the formation of a depletion layer of PEO in front of the grafted layer if the penetration of some low molecular weight PEO into the grafted layer is assumed. Together with osmotically induced layer compression this may cause a much more pronounced decrease in flow penetrability than with added dextran. The majority of larger molecular weight polymers still form the depletion layer in front of the grafted PEO. In future research special attention should be given to the problem of ion exclusion which may also modify the electrophoretic behaviour of weakly charged porous and hairy surfaces. Further experiments to investigate these effects are clearly necessary. In particular, it is desirable to use particles with a known, controllable surface charge distribution, as well as with hairy layers of variable thickness. This would be useful for a
41
better separation of surface charge effects and surface hydrodynamic effects on the electrophoretic mobility of hairy particles. We hope that this preliminary electrophoretic investigation of polymer depletion near hairy particles will lead to further developments in this area.
Acknowledgment This study became possible as a result of a DAAD-ARC exchange project. The theoretical foundations were supported by a DFG grant given to E. Donath (Do 410 1-1). G.C. Allan and B. Vincent acknowledge the support from the DTI Colloid Technology Programme.
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A. Krabi et al. / Colloids Surfaces A: Physicochem. Eng. Aspects 122 (1997) 33-42
[18] E. Donath, L. Pratsch, H. B~iumler, A. Voigt and M. Taeger, Stud. Biophys., 130 (1989) 117. [19] H. B~iumler, E. Donath, L. Pratsch, D. Lerche, J.F. Stolz, M. Donner and A.L. Copley (eds.), Hemorheologie et Agregation Erythrocytaire, Editions Medicales Internationales, Cachan, 1991, p. 24. [20] E. Donath, A. Krabi, G. Allan and B. Vincent, Langrnuir, 12 (1996) 3425. [21] E. Donath, P. Kuzmin, A. Krabi and A. Voigt, Colloid Polym. Sci., 271 (1993) 930.
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