Langmuir 1993,9, 2077-2083
2077
Influence of Addition of Electrolyte and/or Increase of Temperature on the Viscoelastic Properties of Concentrated Sterically Stabilized Polystyrene Latex Dispersions W. Liang and Th. F. Tadros* Zeneca Agrochemicals, Jealott’s Hill Research Station, Bracknell, Berkshire RG12 6EY, United Kingdom
P. F. Luckham Department of Chemical Engineering and Chemical Technology, Imperial College, London S W7 2A2, United Kingdom Received January 11, 1993. In Final Form: May 26,1993 The viscoelasticpropertiesof aqueousstericallystabilizeddispersions have been investigated as a function of Na2S04concentrationto establish the critical flocculation concentration (CFC) at 25 OC. The critical flocculation temperature (CFT) was also determined from the temperature dependence of the rheological parameters at fixed NazSO4 concentration. The results showed that the dispersionbecame significantly more pseudoplastic at and above the CFC and CFT. Both the CFC and CFT were independent of the volume fraction of latex dispersions,&,over the range studied (0.35-0.6). Below the CFC and CFT, the yield values and modulishowed a slight decreasewith increase in electrolyteconcentrationand temperature. This was accounted for by the reduction in the adsorbed layer thickness as the solvency of the medium for the chains was reduced. However, above the CFC and CFT all rheological parameters showed a sharp increase with increase in both electrolyteconcentrationand temperature. A scaling relation between yield value or storage modulus and volume fraction of latex particle was established,which demonstrated that a more open structure of flocs may be formed when the concentrationof electrolyteat a given temperature or the temperature at a given concentration of electrolyte are well above the CFC or CFT. The elastic floc model was used to estimate the radius of the flocs above CFC as a function of particle volume fraction from the rheological data. The results showed an increase in the floc radius with increase in A, at a given Na2SOl concentration. At any 4*,the floc radius also increases with an increase in Na2804 concentration. Introduction Sterically stabilized particles can be flocculated if the solvency of the stabilizing moiety is sufficiently reduced. For aqueous sterically dispersions stabilized containing poly(ethy1ene oxide) (PEO)chains, flocculation may be produced above a critical electrolyte (e.g. Na2S04) concentration or above a critical temperature at constant electrolyte concentrations.lP2 Correlations between the critical flocculation concentration (CFC) or critical flocculation temperature (CFT) and the theta conditions have been demonstrated by Napper et al.3~4 for a large number of model sterically stabilized systems. However, most of Napper’s studies have been carried out with dispersions of terminally anchored stabilizing chains of relatively high molecular weight, M (generally >lo4). For dilute dispersions, turbidity methods can be used to determine the CFC or CFT. This method cannot be applied for large particles and/or high particle number concentrations (concentrated dispersions). As shown by Neville and Hunter: rheological measurements can then be used to follow flocculation. At low particle volume fractions, stable dispersions exhibit Newtonian behavior, whereas flocculated systems exhibit pseudoplastic flow. Flocculation is thus accompanied by a large increase in both yield value, 78, and storage modulus of the dispersion. ~
~
* Author to whom correspondence should be addressed.
(1) Napper, D. H. Polymeric Stabilisation of Colloidal Dispersions; Academic Presa: London, 1983. (2) Tadros, 7%. F.;Vincent, B. J. Phys. Chem. 1980,84,1575. (3) Napper, D. H. J. Colloid Interface Sci. 1970, 33, 384. (4) Napper, D. H. J. Colloid Interface Sci. 1977,58, 390. (5) Neville, P.; Hunter, P. C. J. Colloid Interface Sci. 1974, 49, 204.
Cowell and Vincent6*’have used low shear viscometry to study weak flocculation of polystyrenelatices, sterically stabilized by low molecular weight (1500) poly(0xyethylene). Recently Tadros and Hopkinson8 and Kim and Luckhams have studied the rheological properties of polystyrene latex stabilized by physically adsorbed PVA under conditions of incipient flocculation. In this paper, the influence of addition of NazSO4 and/ or increase of temperature on the viscoelastic properties of concentrated polystyrene latex dispersions containing grafted low molecular weight poly(ethy1eneoxide) chains (M 2000) have been investigated as a function of latex volume fraction. Both CFC and CFT were determined using rheologicalmeasurements. Above the CFC or CFT, scaling laws were established between the rheological parameters and the volume fraction of the suspension. The elastic floc model proposed by Firth and Hunter1b12 was applied to obtain the floc radius as a function of & at two NazSOc concentrations.
-
Experimental Section Materials. Water waa doubly distilled from all glass apparatus. Sodiumsulfate was analytical grade and used as received. Ethanol waa commercial grade and wed Without further puri(6) Cowell, C.; Li-Lin-On, F.K. R.; Vincent, B. J. Chem.Soc.,Faraday Trans. 1 1978, 74,337. (7) Cowell, C.; Vincent, B. In The Effect of Polymers on Dispersion Properties; Tadros, Th.F.,Ed.;Academic Pres: London, 1982. (8) Tadros, Th.F.;Hopkinson, A. Faraday Diecuss. Chem. SOC. 1990, 90,41. (9) Kim, I. T.; Luckham, P. F.J. Colloid Interface Sci. 1991,144,174. (10) Firth,B. A.; Hunter, R. J. J . Colloid Interface Sci. 1976,57,248. (11) Firth, B. A. J. Colloid Interface Sei. 1976,57, 267. (12) Firth, B. A. and Hunter, R. J. J. Colloid Interface Sci. 1976,57, 266.
0743-7463/93/2409-2077$04.00/00 1993 American Chemical Society
Liang et al.
2078 Langmuir, Vol. 9,No. 8,1993 60
40
I
dS=O.35 50
-
d
30
5
Br" m 2
.I
a
-
$iL1
dS =O. 52
m
I
T
40
-
I
7
20
h
a
I/ I
i
%z
-
dS =0.45
m
-u
pIs=O.40
s6s=0.50
--c
30 -
ds=0.54
0,
2
d,=O. 56 ----c
10
1/11Ji 0
0.1 0.2 0.3 0.4 0.5 0.6 Concentration of Na,S04 (aq)/mol dm-3 Figure 1. Plot of Bingham yield value as a function of NaESO. concentrations/mol dm-9 at various latex volume fractions at 25
OC.
fication. Styrene used was a B.D.H. material which was purified bystirringitwith5% (w/w)freshlypreparedFuller'sEarth (which had been heated at 500 O C in an oven for 4 h) for 20 min and filtered before used. This procedure removed the stabilizer from the styrene. Methoxy-poly(ethy1ene oxide) (2000) methacrylate was supplied by I.C.I. Paints Division (Slough, U.K.). The initiator azodiisobutyronitrile (ADIB) was also supplied by I.C.I. Paints Division. Preparation of Latex Dieperaione. Polystyrene latex dispersion with grafted poly(ethy1ene oxide) was prepared wing the *aquersemernprocess described before.Ia14 Basically, styrene is polymerized in an alcohol-water mixture in the presence of methoxy poly(ethy1eneoxide) methacrylate with PEO molecular weight of 2000. The z-average particle diameter was determined using a Malvern photon correlation spectroscopy apparatus (Malvem Instruments, U.K.). The results showed that the latex had a narrow particle size distribution14Je with a z-average diameter of 435 nm.
Rheological Measuremente. Steady-State Measurements.
Steady-state shear strm-shear rate curves were obtained using a Bohlin rheometer (Bohlii Reologi, Loud, Sweden) interfaced with an IBM microcomputer. The dispersions were initially sheared at 46 s-* for 10 min and left to stand for another 10 min and then a shear rate sweep was applied in the range of 13-600 s-I. From the shear rate sweep ry curves were obtained. The yield values were obtained by using the Bingham equation (1) 7 = 78 + f1-r from which the yield value, 78, was obtained by extrapolatjon of the linear portion of the shear stress-shear rate curve to 7 = 0, and the plastic viscosity, qpl, was obtained from the slope of the linear portion of the curve. The critical shear rate ydt above which the curve became linear was also estimated. Oscillatory Measurements. A Bohlin VOR rheometer was used for such measurements. For the present measurements a (13)Bromley, C. Colloid Surf. 1986,17, 1. (14)Linng, W.;Tadros, Th.F.;Luckham,P.F. J. Colloid Interface Sci. 1992, 153, 131. (16)Van der Boomgaard, Th.;King, T. A.;Tadroe, Th.F.;Tang, H.; Vincent,B. J. Colloid Interface Sci. 1978,61,68.
v -
0
0.1
0.2
03
0.4
0.5
0.6
Na,SO, (as) concentration/ mol dm-3 Figure 2. Plot of storage modulus as a function of NaaO,(aq) concentrations/mol dm-9 at various latex volume fractions at 25 OC.
frequency range of 0.01-5 Hz was chosen wing a strain of 0.002 or less to ensure that one is operating in the linear viscoelastic region. The complex modulus G I , the storage modulus G', and the loss modulus G" were calculated from the strew and strain amplitudes (TO and 70,respectively) and phase angle shift (6) by the following equations
P*l= rdro
(2)
G' = (G*Jcos 6 G" = JG*(sin S G* = G' + iG"
(3) (4) (5)
where i is equal to (-l)l/*. In oscillatory measurements, one initially fiies the frequency at 0.1 Hzand measurea the moduli as function of strain amplitude. Thisenables one to determine the linear viscoelasticregion where G*, G', and G" are independent of applied strain at the given frequency. Once the linear region is established, then measurements are made as a function of frequency at a fixed amplitude.
Results and Discussion Influence of Addition of Electrolyte (NatsO4). Figure 1shows the variation of the Bingham yield value, 75, with concentration of NazSO4 at 24 "Cfor various latex volume fractions. It is clear that when < 0.52, 75 is virtually equal to zero up to 0.3 mol dm4 NazSO4 above which it shows a rapid increase with further increase in NazSO4 concentration. When 6, > 0.52,a small yield value is obtained below 0.3 mol dm" NazSOd, which may be attributed to the possible elastic interaction between the absorbed layer when the particle-particle separation is less than 26 (where 6 is the absorbed layer thickness)." However, above 0.3 mol dm4 Na2S04, there is a rapid increase in 75. Thus, the CFC of all the concentrated latex dispersions is around0.3mol dma. It should be mentioned
Viscoelastic Properties of Aqueous Dispersions
Langmuir, Vol. 9,No. 8, 1993 2079
30
Figures 2 and 3 show the results of the storage modulus
G‘ and loss modulus G“ as function of Na2SO4 concen-
25
-
m 20
@s = 0.50
a
>:
s *
r(
1
2 15 E v)
I;
c;l
10
-
Na,SO,
(aq) concentration/ mol dm -3
Figure 3. Plot of loss modulus aa a function of NaBOl(aq) concentrations/moldmq at various latex volume fractions at 25 OC.
16
14
12
G’
i
tration. The results from the storage modulus show roughly the same trend as the yield values, i.e. an initial reduction in G’due to the reduction in effective volume fraction, followed by a sharp increase above the CFC (which is also -0.3 mol dm-9. The loss modulus results (Figure 3) also show the same abrupt increase above the CFC, although the slight reduction in modulus below the CFC was not observed. Figures 4-6 show the variation of the moduli (G*, G’, and G”) with temperature at 0.2, 0.3, and 0.4 mol cm-3 NazSO4. All the results show the same trend, namely a rapid increase in all moduli values above a certain critical temperature (CFT). The CFT values are 50,35, and 15 “C for 0.2,0.3, and 0.4mol dm3 NazSO4, respectively. The results show the expected trend, namely a reduction in CFT with increase in NazSO4 concentration. However, careful inspection of the results of Figures 4-6 show other important trends. For example, below the CFT, there is a systematic reduction in moduli values with increase in temperature. This, as discussed above, is the result of reduction in solvency of the chains.15 Such reduction occurs by increasing temperature which results in breakdown of the hydrogen bonds between the PEO chains and water molecules. Another observation is the relative values of G’ and G”. In 0.2 mol dm3 NazSO4, G’seems to be close to G” even above the CFT, whereas in 0.3 and 0.4 mol dm-3NazSO4, G’> G” above the CFT. A comparison between the three electrolyte concentrations is given in Figures 7 and 8, whereby G’and G’IG” are plotted versus temperature. In Figure 7, a log scale for G’was used to illustrate the reduction in G’with increase of temperature below the CFT. Figure 8 also shows a reduction in G’/G” with increase of temperature below the CFT. Indeed both Figures 7 and 8 give a more clear location of the CFT. Scaling Relationship between 78 or CT and 4,. loglog plots of 78 or G’vs & for latex suspension at various NazSO4 concentrations are given in Figures 9 and 10. All the data are described by the following scaling equations
G“
m 10
a
>
k 8
*i7. c7
6
4
2
n ”
30 40 50 60 70 Temperature/ C Figure 4. Plot of moduli aa a function of temperature for latex (D= 435 nm, @, = 0.550) in 0.20 mol dm4 Na*O4(aq). 0
10
20
that at Na2SO4 concentrations below the CFC, 78 showed a measurable decrease with increase in the Na2S04 concentration. We believe that this is due to the reduction of the effective radius of the latex particle as a result of reduction in solvency of the medium for the chains.lS This accounts for a reduction in the effective volume fraction of the dispersion which is accompanied by a reduction in 78.
= k$”
(6)
T@
G’= k’$” (0.35 < 4 < 0.53) (7) where k and k’ are constants and m and n are the exponents.
- -
The values of m and n are listed in Table I. These clearly show a sudden drop in m from a value of 31 to 9.4 and of n from -30 to -12 as the Na2S04 concentration is increased from 0.3 to 0.4 mol dm3. With further increase in Na2S04concentration from 0.4 to 0.5 mol dms, m drops from 9.4 to 2.8, whereas n drops from 12 to 2.2. This low exponent is an indication that an open network floc structures with low fractal dimensions is formed. The exponent of 2.8 or 2.2 at 0.5 mol dmq Na2SO4 is just in the range of the reported values in the literature. Many a u t h o r ~ * * ~have J & ~reported ~ that the exponent for flocculated suspensions is in the range 2.0-4.5. However, ~
~~
(16) Liang, W.; Tadros, Th.F.; Luckham, P. F. J.Colloid Interface Sci. 1993,156,166. (17) Zoeel, A. Rheol. Acta 1982,21,72. (18)Buscall, R.; McGowan, I. J. M.;Mille, P. D.; Sutton, R. F.; White, L. F.; Yates, L. F. J. Non-Newtonian Fluid Mech. 1987,24,183. (19)Russel, W. B. Powder Technol. 1987,61, 15. (20) Sonntag, R. C.; Ruseel, W. B. J. Colloid Interface Sci. 1987,116, &. (21) Shih,Wei-Heng; Shih,Wan, Y.; Kim, Smng-Q Liu, Jun;Akaay Ilhan, A. Am. Phys. SOC.1990,42,4772. (22) Buscall, R.; Will, P. D. A. J. Chem. SOC.,Faraday Trans. 1 1988, 84,4249. (23) Ball,R.; Brown, W. D. Personal communication.
Liang et al.
2080 Langmuir, Vol. 9, No. 8, 1993 1,000
30
0.2 mol dni' N%S04
25 100
20
m
m
a
%1
>
$
10
d
1
3t7 15
5
p Q,
M
d
*
10
1
I %
5
0.1
n -
1
0
10
20
30
40
50
60
Temperature / 'C
Figure 5. Plot of moduli as a function of temperature for latex (D= 435 nm, = 0.550) in 0.30 mol cm3 NazSOdaq). 100
I
0.01
I
I
I
I
I
Temperature / C '
Figure 7. Plot of storage modulus as a function of temperature for latex (D = 435 nm, 4n= 0.550) at various NaBO,(aq) concentrations. 3.5
G*
80
m
a ?
G'
3-
6o
c?
."u
40
"
,G"
5
10
15 20 25 Temperature/'C
30
35
Figure 6. Plot of moduli as a function of temperature for latex (D = 435 nm, 4, = 0.550) in 0.40 mol dm-S NazSOd(aq).
the value of the exponent depends to some extent on the treatment to which a coagulated suspension has been subjected before the measurements were made. Shih et al.21 have reported that the storage modulus exhibited a power law behavior with respect to particle concentration 4 of colloidal gels, i.e., G' 44.M.2. The exponent 2.2 for G' is comparable to that for colloidal silica suspension22
-
0
I
I
10
20
I
I
I
30 40 50 Temperature/ "C
I
I
60
70
Figure 8. Plot of G'/G" versus temperature for latex (D= 435 nm, 4n= 0.550) at various NazSO4(aq) concentrations.
which is 3.5 f 0.2 for diffusion-limited aggregation and agrees well with that predicted by Ball and Brown29 by
Viscoelastic Properties of Aqueous Dispersions
assuming that the clusters comprising the network are fractals, although in our case the flocculation is weaker. Influence of solvency on the Adsorbed Layer Thickness. As shown above, below the CFC or CFT both yield value and storage modulus decreased with increase in Na2S04 concentrations. This is due to the compression of the adsorbed layer 6 as a result of reduction of solvency of the medium for the chains.16 At any given volume fraction of the suspensions, reduction in 6 means a reduction in the effective volume fraction, &tt. This reduces the overall interaction between the particles in the dispersion and hence a reduction in 78 or G’ is the obvious result. The actual change of the adsorbed layer thickness may be evaluated from the reduction in viscosity or modulus. However,since the reduction in 6 is relatively small (1-2 nm), it is difficult to give a quantitative estimation of these reductions. The same results were obtained for increasing temperature (below CFT). In this case it is more difficult to obtain a quantitative estimate of the reduction in 6 since by increasing temperature the thermal expansion of the medium would be different from that of latex and this makes any estimate inaccurate. Attractive Force Contribution. As mentioned above, in many cases, the CFC and CFT may coincide with the theta point for the stabilizing chain. At the theta point the mixing term in the steric interaction is zero and any yield value measured should correspond to the residual van der Waals attraction. The energy arising from van der Waals attraction at this point where the absorbed polymer layers just begin to overlap may be calculated from the following equation
where a is the particle radius, Hois the particle separation that is taken to be equal to 26, A11 is the Hamaker constant of the particle, and A22 that of the absorption polymer layer, which is given by
A,, =
+ q5mA,’/212
where +,is the volume fraction of polymer in the adsorbed layer, 4m that of the medium, and A, and A m are the Hamaker constants of the polymer and medium (solvent), respectively. Using the following values, A11 = 19kT, A, = 16.6kT, A m = 9kT,%and 4, = 0.25, GAwas found to be 1.65kT when HO= 26 = 12 nm. The latter is twice the adsorbed layer thickness for the compressedchains. This value of GAcorresponds to a yield value of -0.05 N m-2 (see below). This is certainly a very small value. Thus, the electrolyte concentration at which a measurable value of 78 is obtained is very close to the concentration corresponding to the CFC. Beyond that point the measured yield value is mainly due to the attraction arising from the mixing term of the steric interaction which is negative above the 0 point. Any contribution from the van der Waals attraction is relativelysmall at the distances of interpenetration of chains considered. Interpretation of the Rheological Results above CFC. Twomodels may be used to interpret the rheological results of the present floccculatingsystem. Both models have been developed by H u n t e r a n d his coworkers.5J0-1212k2* In the first model Frith, Neville, and HunterslB introduced a doublet floc rupture model to deal (24) Visser, J. Adu. Colloid Interface Sci. 1972, 3, 331. (26) Frith, B. A.; Neville, P. C.; Hunter, R. J. J. Colloid Interface Sci. 1974, 49, 214.
Langmuir, Vol. 9,No. 8, 1993 2081 100
30
10 m
a” J
z 3
I
in water
a , m
.2 l-J h
i
in 0.1 M
1
3EI
in 0.2 M
G
0.3
in 0.3 M in 0.4 M
* ‘
0.1 in 0.5 M
0.03 Latex volume fraction
@s
Figure 9. log-log plot of G” versus 4, for latex (D= 435 nm) suspensions at various NazSO, concentrations. with stericallystabilizeddispersionswhich have undergone reversible flocculation. They assumed that the major contribution to the excess energy dissipation in such pseudoplastic systems comes from the shear field which provides energy to separate contacting particles in the flocs. Thus, the extrapolated yield value can be expressed as
where 41.iis the hydrodynamic volume fraction of the particles (equal to the effective volume fraction #&, i.e., 4~ = 4, [l + (6/a)I3),(a + 6) is the interaction radius of the particle, and E, is the energy needed to separate a doublet which is the sum of the van der Waals and steric intractions
EWp= (Aa/lNo)+ G,
(11) As discussed before, at a particle separation of 26 (12nm), the van der Waals energy is very small (1.65kT) and the contribution from G,to the attraction is significantlylarger than van der Waals attraction. Therefore, E, may be approximated to G,. Thus, from eq 10 one can estimate E,, from the yield value 78 for the flocculated suspensions. The results are summarized in Tables I1 and I11 which show an increase in Esepwith increaee in &. The values are unrealisticallyhigh since the flocculationin the present concentrated dispersions is not reversible within the conditions of the rheological measurements. It must be mentioned that these data should only be considered as (26) van der Ven, T. G. M.; Hunter, R. J. J.Colloidlnterfoce Sci. 1979, 68,136. (27) Hunter, R. J. A d a Colloid Interface Sci. 1982,17, 197. (28) Friend,J. P.; Hunter, R. J. J. ColloidInterface Sci. 1971,37,648.
Liang et al.
2082 Langmuir, Vol. 9,No. 8, 1993
Table 111. Results of & Calculated from rp for a Flocculated Sterically Stabilized Polystyrene Latex Suspension at Various Latex Volume Fractions in 0.5 mol dmJ NafiO4
50
20
0.515 0.487 0.465 0.440 0.409 0.371 0.330 0.288 0.246
10 m
a $
5
I
a
in water
-8
.--t
;&
in 0.1 M
P)
M
m
0"
2
d
in 0.2 M
$
b
1
in 0.3 M --c
in 0.4 M 0.5
. t
I
in 0.5 M d
0.2 Latex volume fraction
Q,
TO versus 4, for latex (D = 435 nm) suspensions at various Na&3O, concentrations.
Figure 10. log-log plot of
Table I. Scaling Relationship between and rp or G' at Various Nafi04 Concentrations NafiO, concentration (mol dm-3) m n 0.0 47 27 0.1 44 34 34 31 0.2 0.3 31 30 0.4 9.4 12 2.8 2.2 0.5 Table 11. Results of & Calculated from rp for a Flocculated Sterically Stabilized Polystyrene Latex Suspension at Various Latex Volume Fractions in 0.4 mol dmJ NaaSO4 6. 0.578 0.669 0.554 0.535 0.511 0.452 0.429
rg (N m-2)
17.4 9.1 7.3 5.3 3.3 2.4 1.3
E- ( k T ) 736 397 336 262 179 165 97
qualitative since the assumptions made in calculation of are not fully justified. In the second attempt the rheological results were analyzed using the elastic floc model of Hunter and co-workers.12*28-28In this model, the structural units are assumed to be small flocs of particles (called flocculi)which are characterized by the extent to which the particle structure is able to trap the dispersion medium (a floc is made from an aggregateof several flocculi). These flocculi may range from loose open structures (if the attractive forcesbetween the particles are strong)to very close packed structures with little entrained liquid (if the attractive forces are weak). In the system studied in the present investigation, the structure of flocculi depends on the volume fraction of the suspension and how far the system is from the CFC. Just above the CFC, the flocculi are
E,
28.3 23.1 21.1 17.0 14.1 11.4 7.4 5.4 3.5
1510 1380 1380 1240 1190 1170 961 910 804
probably close packed (with relatively small floc volume), whereas far above the CFC, a more open structure is found which entraps a considerable amount of liquid. Both types of flocculi persist at high shear rates, although the flocculi with weak interaction may become more compact by maximizing the number of interactions within the flocculus. A satisfactory description of the kind of flow of the present flocculated system (which is pseudoplastic) has been g i ~ e n ~ ~ @ by -analyzing ~ the energy dissipation process as suggested by Goodeve29and Gillespie.f"JThese authors suggested that the energy dissipation in flow can be separated into two componentsdue to the viscous flow of suspensionmedium around the flow unitsand the energy dissipated in overcoming the interaction between the particles. The totalenergy dissipated in flow, Ebt, is then given by
E,, = 7s; + 9,Ci2
(12)
Since the flocculated systems mostly show a linear 4 above a critical shear rate, ydt, it follows that qpl becomes constant above y d t and hence the degree of openness of the flow units must remain constant above ydt. In the elastic floc model, it is assumed that the flocculiestablished at high shear rate are able to associate to form flocs which have an essentially constant ratio of solid to trapped liquid. The degree to which liquid is trapped is measured by the floc volume ratio, Cm, given by the ratio of &/48 (where &is the volume fraction of flocs and that of the particles). At high volume fraction, & and hence Cm may be calculated from the Krieger equation:'
where 90is the viscosityof the medium, is the maximum packing volume fraction which may be taken as 0.74, and 171 is the intrinsic viscosity (taken as 2.5). Thus, qlpl becomes infinitely large as 4Bapproaches the close-packing value.32 The values of C F calculated ~ using eq 13 are listed in Tables IV and V. Several other dissipative processes may be identified in the high shear rate regime, i.e. orientation, stretching, and compression of the flocs and transfer of flocculi from one floc to another. The most important assumption of the elastic floc model is that all dissipative processes can be simultaneously evaluated by considering in detail what happens when two flocs collide and are separated by the shear field. Detailed examinationof the various processes showed that only one contribution is important in the (29) Goedeve, C. F. Trans Faraday SOC.1939, 36,342. (30)Gdeapie, T.J. Colloid Sci. 1960,15, 219. (31)Krieger, I. M.Ado. Colloid Interface Sci. 1972,3, 111. (32)Michaeh, A. J.; Bolger, J. C . Ind. Eng. Chem. htndam. 1962, I, 153.
Langmuir, Vol. 9, No. 8, 1993 2083
Viscoelastic Properties of Aqueous Dispersions Table IV. The Floc Radiur Calculated from the Elastic Floc Model for Flocculated Sterically Stabilized Polystyrene Latex Suspensionr at Variour Latex Volume Fnctionr in 0.4 mol dmJ N d 0 4 6. 0.578 0.569 0.554 0.535 0.486 0.462
0.409
10% 1
70
(~m-18)
qr
317.5 161.4 117.7 71.0 34.1 24.4 14.7
282.7 143.7 104.8 63.2 30.4 21.7 13.1
("-7
17.4 9.09 7.30 5.33 3.38 2.39 1.27
h
aaoc
CW
0.707 0.693 0.684
1.223 1.217 1.234 0.666 1.246 0.630 1.299 0.608 1.344 0.567 1.386
YO
bm)
(8-1)
66 56 51 45 39 34 27
126 95 95 95 95 96 96
Table V. Floc Radiur Calculated from the Elastic Floc Model for Flocculated Sterically Stabilized Polystyrene Latex Surpensions at Various Latex Volume Fractions in 0.5 mol dmJ Na18O4 10% 1
9.
(Nm-g 8 )
0.515 0.465 0.440 0.409 0.371 0.330 0.288 0.246
62.3 42.5 33.6 27.0 22.8 16.6 14.2 9.8
7a
vr
(Nm-2)
h
Cm
69.9 47.7 37.8 30.4 25.7 18.6 15.9 11.0
29.0 21.1 17.5 14.1 11.4 7.4 5.4 3.5
0.666 0.648 0.636 0.623 0.612 0.588 0.574 0.538
1.292 1.394 1.446 1.524 1.660 1.781 1.993 2.185
a& bm) 62
60 60 57 54 61 47
46
YO (8-1) -
280 249 218 218 218 187 187 156
energy dissipation p r o c e s ~ , ~namely ~ ~ ~ ~the ~@ energy involved in movement of liquid into and out of the space between the adjoiningparticles insidethe flocs. This leads to the following equation for the Bingham yield value 78
= cr~Xto;/,(anoc)2A(~,)2C~/a3
(14)
where a0 is the orthokinetic capture efficiency which depends weakly on shear rate (010 a 9 9,/3 is a constant (=27/5), X is a correction factor of order unity, t o is the viscosity of the medium, anocis the radius of the flocs, and A is the distance through which bonds are stretched inside the floc by the shearing force. Thus, using eq 14 one can calculate the radius of the flocs from the experimental data if one chooses (YO = 1, X = 1/3, and A = 0.5 nmeUThe results for anoccalculated from eq 14 at 0.4 and 0.5 mol dm-J NazSOr are also listed in Tables IV and V, respectively. Figure 11shows the plot of the radius of floc as a function of latex particle volume fraction at various Na2SOc concentrations. It is clear from Figure 11 that the floc size increaseswith increasein NafiO4 concentrationabove CFC,indicating that the structure of the flocs becomes (33) Hunter, R. J.; Malarate, R.; Napper, D. H. Colloids Surf. 1983, 7, 1.
(34)van de Ven, T. G. M.; Huntar, R. J. Rheol. Acta 1977,16,534.
90
80
E
a2.
2
70
60
@+4
!2
.r(
aa
50
/
40
30
20 0.2
0.3
0.4
0.5
Latex volume fraction
Q,
0.6
Figure 11. Plot of the radiue of the floc as a function of latex particle volume fraction at above CFC.
more open when NaSO4 concentration is far above CFC. This is consistent with the scaling results as discussed before. The results also show that afloc increases with increase in 4,. This can be understood by assuming that the larger floca are formed by fusion of two floca and smaller flocs by "splitting" of larger ones. From simple statistical arguments, one can predict that a h will increase with increase in 48becauee in such a process larger flw are favored over small 0nes.u
Conclusions Rheological measurements can be used to determine the critical flocculation concentration (CFC)and critical flocculationtemperature (CFT)of concentratedsterically stabilizeddispersionswith reasonable accuracy. Both CFC and CFT were independent of latex volume fractionwithin the range studied. As expected the CFT decreased with increase in electrolyte concentration at the same latex volume fraction. A scaling relation may be applied to obtain the power exponent for the dependence of 78 or G' on 4 when the electrolyteconcentrationis above the CFC. The power exponentwas in the range 2.2-2.8 as previously obtained with many flocculated systems. It was found that increasing electrolyte concentration or temperature results in a decrease in ~g or G' below the CFC or CFT as a result of the reduction in solvencyof the medium for the chains and hence reduction of adsorbed layer thickness 6. The elastic floc model was applied to obtain the floc radius a h as a function of Nazi304 concentration and volume fraction of the suspension. It was found that &increased with both increase in NaSO4 concentration and volume fraction of the suspension above the CFC.
