Langmuir 1990, 6, 28-35
28
Use of Viscoelastic Measurements in Studying Interactions in Concentrated Dispersions? Th. F. Tadros ICI Agrochemicals, Jealotts Hill Research Station, Bracknell, Berks. RG12 6EY, U.K. Received September 12, 1988. I n Final Form: March 8, 1989 The use of viscoelastic measurements in studying interactions in concentrated dispersions has been demonstrated by using various model systems. These include charge-stabilized polystyrene latex dispersions, sterically stabilized dispersions containing grafted poly(ethy1ene oxide) chains, or physically adsorbed poly(viny1 alcohol). With electrostatically stabilized dispersions, the system became predominantly elastic when double-layer overlap occurred. This was the case at low electrolyte mol dm-3 NaC1) concentrations since the effective volume fraction +eff is significantly larger than the actual volume fraction 4. Increasing the electrolyte concentration to mol dmT3caused compression of the double layer, and the dispersion became relatively more viscous than elastic. With sterically stabilized dispersions, elastic interaction occurred as soon as the volume fraction reached a value whereby the grafted or adsorbed polymer chains start to overlap. Viscoelastic measurements could also be applied for studying flocculated systems. Three main types were investigated: weakly flocculated dispersions produced by addition of free (nonadsorbing) polymer (depletion flocculation), sterically stabilized dispersions flocculated by reduction of solvency to worse than @conditions (incipient flocculation), and strongly flocculated (coagulated) dispersions produced by addition of electrolyte to a charge-stabilized system. With weakly flocculated disperions, all rheological parameters show a rapid increase above a critical volume fraction of the added free polymer. In this case, it was possible to calculate the energy of separation of particles in the floc from the yield value. This was compared with theoretical calculation of the free energy of depletion flocculation. With sterically stabilized dispersions, reduction of solvency was produced by addition of electrolyte or increasing temperature at constant electrolyte concentrations. The results showed an initial reduction in modulus due to collapse of the chains, followed by a rapid increase above the flocculation point. Coagulated suspensions obtained by addition of electrolyte to an electrostatically stabilized system were investigated by using strain and oscillatory measurements. The strain sweep experiments indicated a sudden reduction in moduli value above a critical strain. This indicated breakdown of the structure above this value. The results also show that the modulus increases with the volume fraction 4 through a power law with an exponent in 4 of the order of 6. From the critical strain and the storage modulus in the linear region, the energy of cohesion in the floc, E,, was calculated. E, was also found to increase with through a power law with an exponent of 9 in 4.
Introduction The properties of concentrated dispersions have been the subject of numerous investigations in recent years.' One of the major objectives of these studies was to relate to the structure of the system on a microscale to that of its bulk properties such as osmotic pressure, rheology, etc. For this purpose, model dispersions that are relatively monodisperse and with well-characterized surfaces have been prepared. Information on the microstructure of the system was obtained by using smallangle neutron scattering (SANS),2 which allows one to obtain the internal structure of the particles and their arrangement in space. Clearly these studies form the basis of any quantitative relation between the microstructure and the bulk properties of the system. However, such measurements of SANS are only accessible in a small number of laboratories, and they require a great deal of experimental procedures both in the preparation of the system, e.g., for synthesis of deuterated latex systems, and in the measurement of scattering and analysis of the data. Presented a t the symposium on "Rheology of Concentrated Dispersions", Third Chemical Congress of North America, Toronto, June 5-10, 1988. (1) Ottewill, R. H. In Concentrated Dispersions; Goodwin, J . W., Ed.; Royal Society of Chemistry: London, 1982; Chapter 3, No. 43. (2) Ottewill, R. H. In Science and Technology of Polymer Colloids; Poehlein, G. W., Ottewill, R. H., Goodwin, J. W., Eds.; Martinus Nijhoff Publishers: The Hague, 1983;Vol. 11, p 503. +
0743-7463/90/2406-0028$02.50/0
A less rigorous (and less quantitative) procedure for studying interaction in concentrated dispersions is rheology. Apart from being accessible to many laboratories, the measurements do not require any special preparation of the dispersion. Moreover, qualitative information may be obtained on practical industrial systems. This makes rheology an attractive experimental technique, although great care must be taken in the measurements and interpretation of the results. Several rheological measurements may be performed depending on the nature of the system and the information required. It is perhaps convenient to classify rheological techniques into two main categories. The first is where measurements are made under small deformation such that the structure of the dispersion is not much disturbed from its equilibrium conditions. These measurements are informative for analyzing the structure of the system and the interaction between the particles. The second procedure involves measurements at relatively large deformation, e.g., in steady shear. T h e information obtained describes the system under flow conditions, and by carrying out measurements a t different shear rates one may be able to simulate the practical conditions involved in the application of the dispersion. In this paper, I will give some examples to illustrate the application of rheology for studying interaction in concentrated dispersions. Before mentioning the techniques that were applied, it is perhaps useful to say few words about the properties of concentrated dispersions. The latter are those whereby the translational motion of
0 1990 American Chemical Society
Interactions in Concentrated Dispersions the particles is restricted and where interparticle forces play a dominant role in determining the flow. The particles may undergo vibrational motion that is often large compared to the particle radius. Concentrated dispersions show time-dependent properties whereby spatial and temporal correlations may be considered. It is, therefore, essential to carry out rheological experiments within a well-defined time scale that is comparable to the relaxation time of the system. This is the basis of the dynamic (oscillatory)technique whereby measurements can be made over a well-controlled frequency range.3 This constitutes the first technique applied in the present investigation. This allows one to obtain information on the viscous and elastic components of the systems. By comparing such parameters, one is able to obtain information on interparticle interaction. A second low-deformation technique is that of shear wave propagation, whereby one measures the velocity of a low-amplitude shear wave through the dispersion. This allows one to obtain the shear modulus, i.e., the elastic component of the modulus, that may also be related to interparticle i n t e r a ~ t i o n . ~ Steady-state shear stressshear rate measurements are performed by using rotational concentric cylinder viscometers. To simplify the interpretation of the rheological data, a model latex dispersion with well-characterized surfaces was prepared. Both electrostatically and sterically stabilized dispersions were used under conditions of stability/instability requirements, applying the abovementioned rheological techniques. Experimental Techniques Materials. Two main types of latex particles were prepared. The first was an electrostatically stabilized polystyrene latex that was prepared by using the surfactant-free emulsion polymerization method described by Goodwin et aL5 The particle size of the latex was determined by using photon correlation spectroscopy (PCS) and the Coulter counter. Both methods (which agreed with each other) showed the latex to be fairly monodisperse with an average particle radius of 700 nm with a standard deviation of less than 5%. This charged latex was also used for studying steric interaction by coating the particles with a physically adsorbed poly(viny1 alcohol) layer. The weight-average molecular weight of the polymer, M,, is 45 000, and it gave an adsorbed layer thickness of 46 nm as determined by using PCS.‘ The second polystyrene latex was a sterically stabilized system containing grafted poly(ethy1ene oxide). It is prepared by using t h e dispersion polymerization technique in water/ alcohol mixtures.‘ The basic recipe consisted of styrene monomer, a macromonomer of poly(ethy1ene oxide) (M,= 2000) methacrylate, azobisisonitrile and peroxide initiators, and a water/ alcohol mixture (2:3 ratio). The z-average particle size of the latex was determined by using PCS. This showed that the latex was fairly monodisperse with an average radius of 175 nm.
Principle of the Measurements Dynamic (Oscillatory) Measurements. A Model VOR Bohlin rheometer (Bohlin Reologi, Lund, Sweden) interfaced with a Facit DTC2 or an IBM microcomputer was used for such measurements. The instrument can operate in the frequency range 10-3-20 Hz and has inter(3) Whorlow, R. W. Rheological Techniques; Ellis Horwood: Chichester, 1980. (4) Buscall, R.; Goodwin, J. W.; Hawkins, M. W.; Ottewill, R. H. J. Chem. SOC.,Faraday Trans. I 1982, 78, 2872, 2839. (5) Goodwin, J. W.; Hearn, J.; Ho, C. C.; Ottewill, R. H. Colloid Polym. Sci. 1974, 252, 464. (6) Garvey, M. J.; Tadros, Th. F.; Vincent, B. J . Colloid Interface Sei. 1976, 55, 440. (7) Bromley, C. Colloids Surf. 1985, 17,1.
Langmuir, Vol. 6, No. 1, 1990 29 changeable torsion bars covering a wide range of sensitivities. A coaxial cylinder (C25) with a moving cup of radius 27.5 mm and a fixed bob of radius 25.0 mm was used. The Bohlin rheometer performs oscillatory tests by turning the cup back and forth in a sinusoidal manner. The shear stress in the sample is measured by measuring the deflection in the bob which is connected to interchangeable torsion bars. The phase angle shift is automatically computed from the time displacement between the sine waves of stress and strain ( A t ) ,Le., 6 = uAt, where u is the frequency in radians ( u = 2 ~ wwhere , w is the frequency in hertz). The complex modulus G*, storage modulus G’, and loss modulus G” are calculated from the stress and strain amplitudes (r0 and yo, respectively) and the phase angle shift 6:
G* = T o / Y o (1) G’ = G* cos 6 (2) G” = G* sin 6 (3) G’= G’ + iG” (4) where i is a constant that is equal to (--l)’/’, In oscillatory measurements, one initially fixes the frequency and measures the rheological parameters as a function of strain amplitude. This enables one to obtain the linear viscoelastic region where G*, G’, and G” are independent of applied strain, at any given frequency. Once the linear region is indicated, then measurements are made as a function of frequency at a fixed amplitude. Many systems only show a linear region a t very low amplitudes, and therefore the measurements should be carried out at such low deformation. Shear Modulus Measurements. These are performed by using the Rank pulse shearometer (RankBrothers, Bottisham, Cambridge). Basically, the suspension is placed in cell that is fitted with two parallel metal or perspex disks, whose distance of separation d can be changes by using a micrometer attached to one of the plates. Each plate is connected to a piezoelectric crystal (LiC1, transducer). An electric generator is connected to the bottom crystal and is used to initiate a small amplitude (- lo4 rad) strain with a high frequency ( 200 Hz). This initiates a shear wave that propogates through the dispersion and is picked up a t the upper crystal. The latter is connected to an oscilloscope or microcomputer enabling one to display the damping of the shear wave. By measuring the time t , for a shear wave maximum to go to a minimum, a t a given distance d between the plate, one is able to measure the shear wave velocity u. One usually measures t, a t various d values and then plots d versus t , to obtain u from slope of the linear curve. The shear modulus G, (which is basically the elastic modulus a t the applied frequency of -200 Hz) is given by N
G, = v2p where p is the density of dispersion. S t e a d y - S t a t e M e a s u r e m e n t s . Shear stress (7)shear rate (i.)curves were obtained by using a Haake Rotovisco rotational viscometer fitted with concentric cylinder platens. For Newtonian systems, the viscosity was obtained from the slope of the linear curve between T and 4,which also passed through the origin. For nonNewtonian systems which showed pseudoplastic flow, the data were analyzed by using the Bingham3 equation or Casson’s’ equation: (8) Casson, N. Rheology of Disperse Systems; Hill, C. C., Ed.; Pergamon Press: Oxford, 1959; p 84.
30 Langmuir, Vol. 6, No. 1, 1990
Tadros
larger than those obtained in mol dmw3NaC1. This reflects the much longer range repulsion in mol dm-3 NaCl (with an extended double layer of 100 nm) when compared with that obtained in mol dm-3 NaCl (with a compressed double layer of 10 nm). Secondly, in mol dmT3NaCl, G’ > G” over the whole volume fraction range. At the lowest 4 value measured (0.465), G’ is still higher than G”, indicating strong double-layer interaction. This can be understood from a consideration of the effective volume fraction, which is 4 plus the contribution of the electrical double layer:
2OOOr
4erf= 4~ + ( 1 / w i 3 (8) Thus at such electrolyte concentration 4eff= 1.484, which
-
- 1
OL
03
0
05
06
Figure 1. Variation of G*, G‘, and G“ (at w = 1 Hz) with and mol dm-3 NaCI. 7
=
To
+ Vpl?
@J
at
(6)
,lP = r,112 + 4cl12y’12
(7) where r P is the extrapolated (to y = 0) yield stress and 7 the apparent or plastic viscosity that is the slope of t i e linear portion of the 7-9 curve. 7 , is the Casson yield value, while qC is the Casson viscosity.
Results and Discussion Three different systems were investigated: (1)electrostatically stabilized polystyrene latex dispersions, (2) sterically stabilized polystyrene latex dispersions containing grafted PEO or physically adsorbed PVA, and (3) unstable systems that are weakly flocculated by addition of “free” (nonadsorbing) polymer or strongly flocculated by reduction of solvency to worse than 0-conditions or by addition of sufficient electrolyte (above the critical flocculation concentration) for an electrostatically stabilized polystyrene latex dispersions. A summary of the results obtained and their interpretation are given below. Electrostatically Stabilized Polystyrene Latex Dispersions. Results were obtained as a function of the volume fraction of the dispersion at two different NaCl concentrations, namely, and mol dm-3. At the lower NaCl concentration mol dm-3), the doublelayer thickness ( 1 / ~ is ) 100 nm, and therefore the suspension should show “soft“ type interaction due to the presence of such extended double layers. In contrast at NaCl, 1 / is~ only 10 nm and is much smaller than the particle radius, and hence the dispersion should approach a hard-sphere type. Measurements of G*, G’, and G” (very low strain of 0.004 or 0.01 to remain as close as possible to the linear region) were made in the volume fraction range of 0.4650.525 at mol dm-3 NaCl and between 0.26 and 0.566 at mol dm-3 NaC1. Details of the results obtained at various 4 values and NaCl concentrations will be published elsewhere.’ Generally speaking, all dispersions were more elastic than viscous within the 4 range studied at mol dm-3. In contrast, dispersions in mol dm-3 NaCl were mostly more viscous than elastic within the 4 range studied. As an illustration, Figure 1 shows plots of G*, G’, and G” (at w = 1 Hz) versus 4 at the two NaCl concentrations studied. A number of striking features can be observed. Firstly, at any given 4 the moduli obtained in mol dm-3 NaCl are orders of magnitude (9) Hopkinson, A.; Tadros, Th. F., to be published.
means that, when 4 = 0.463, 4 0.7, which is near to the close-packed limit, and hence the latex dispersion behaves more elastic than viscous. A t the highest volume fraction studied, namely, 4 = 0.525, G’ becomes much higher than G” and is now close to G*. Under these conditions, $eff would be 0.77, which is above the maximum packing fraction. This means that double-layer overlap is significant, and the system behaves as an elastic solid. In contrast, the results in mol dm-3 NaCl show that G” > G’ over most of the volume fraction range studied. In this case, 1 / = ~ 10 nm and deff is only 1.044. This means that even at the highest 4 value studied, namely, 0.566, $eff is only 0.59, which is lower than the maximum random packing. This is typical of a hardsphere dispersion which will show elastic behavior at higher 4 values that are greater than 0.64, since in this case the contribution from the double layer is insignificant. A comparison of the double-layer interaction at the two electrolyte concentrations may be made from loglog plots of G* versus @. In mol dm-3 NaC1, such a plot was linear, showing the following power law dependence:
G* = k@20 (9) The high power in 4 denotes strong double-layer repulsion above 4 = 0.463, which as mentioned above is close to the maximum effective volume fraction packing limit (-0.7). mol dmT3NaCl were obtained over The results in a much wider 4 range, and log-log plots of G* versus 4 showed two distinct straight lines having two power laws:
> 0.2 G* = k430 0.57 > 4 > 0.5 G* = kd3
0.5 > 4
(10) The lower power in the first region indicates weak interaction between the particles which contain compressed double layers. A t 4 < 0.5, the double layers are fairly separated from each other. In contrast, at 4 > 0.5 the interaction becomes much stronger and the power in 4 becomes larger. Sterically Stabilized Polystyrene Latex Dispersions. 1. Dispersions with Short Grafted Poly(ethylene oxide) (PEO) Chains. Steady-state shear stressshear rate measurements showed the dispersions to be fairly Newtonian over a wide range of 4 values (0.330.57). In this case, the relative viscosity 77 was calculated at each 4 and the data compared with the theoretical predictions of the hard sphere model proposed by Krieger and Dougherty.lo.” Plots of vr versus 4 gave a curve that was significantly higher above the curve predicted for a hard-sphere dispersion. This was thought to be due to (10) Krieger, I. M. Adu. Colloid Interface Sci. 1972, 3, 111. (11) Krieger, I. M.; Dougherty, M. Trans. SOC.Rheol. 1959, 3, 137.
Langmuir, Vol. 6, No. 1, 1990 31
Interactions i n Concentrated Dispersions 10Lr
G’ fG’
2 3c i;l;
~
~
O’t IC‘
-lHz
Figure 2. Variation of G*, G’, and G” with
w
for polystyrene
latex dispersions containing grafted PEO chains at various 4 values. the contribution of the adsorbed layer. In other words, @eff is significantly larger than @, and it was decided to obtain @eff by fitting the qr results to be the DoughertyKrieger equation:’0v1’ where [a] is the intrinsic viscosity which was taken to be equal to 2.5 and @p is the maximum packing fraction which was calculated to be 0.71 from a plot of 1/q’l2 versus @ and extrapolation to l / q 1 / 2 = 0. From 4effand 4, the adsorbed layer thickness 6 was calculated by using the equation
+ (6/mi3 (12) Values of 6 at various @ values were calculated, and the results were published elsewhere.12 These showed that, at low @, 6 20 nm, which is a reasonable value for such molecular weight. Small-angle neutron scattering” showed that PEO chains on polystyrene are fairly extended forming long tails. Indeed for PEO with M w = 4800, 6 was found to be in the region of 30 nm. The results using rheology also showed that 6 becomes smaller as @ is increased, reaching a value of 9.2 when @ = 0.57. This is due to the compression of the chains on increasing the volume fraction. The results of G*, G’, and G” versus frequency w (Hz) at various polystyrene volume fractions are shown in Figure 2. At @ = 0.44, G“ > G’, and the dispersion behaves as a viscous fluid. This reflects the relatively weak interaction a t such a volume fraction since the surface-surface separation is larger than twice the adsorbed layer thickness. When @ is increased to 0.465, G“ is still higher than G’, and the modulii values increase by about a factor of 2, compared to the values at 4 = 0.44. However, when 6 is increased to 0.5, G’ now becomes larger than G” and the modulii values increase. At such @ value, the interparticle distance (assuming random packing) is in the region of 30 nm, which is now less than 26, and relatively strong interactions take place between the particles. On further increase of @ to 0.575, G’ becomes much =
-
(12) Prestidge, C.; Tadros, Th. F. J . Colloid Interface Sei. 1988, 124, 660. (13) Cosgrove, T.; Crowley, T. L.; Vincent, B.; Barnett, K. G.; Tadros,
Th. F. Faraday Symp. Chem. SOC.1981,16,101.
I 1
I
CLL O L 6
I
CLB
I
1
1
I
I
050 C 5 2 C5L 056 058
Q Figure 3. G*, G‘, and G” (at w = 1 Hz)versus 4 for aqueous polystyrene latex dispersions ( R = 175 nm) containing grafted PEO ( M , = 2000).
larger than G”, and it closely approaches G*. The dispersion behaves as a near elastic body as a result of interpenetration and/or compression of the chains. Note that at such a @ value the interparticle separation is about 12 nm, which is now significantly smaller than 26. Figure 3 shows plots of G*, G’, and G” a t 1 Hz versus 4. The crossover point a t which G’becomes equal to G” denotes the start of the elastic interaction between the PEO tails. Above this volume fraction, G’ increases rapidly with @ and approaches G* very closely. Moreover, both G* and G‘ reach very high values whereby the latex behaves as an elastic body. This results from interpenetration and compression of the PEO chains. 2. Dispersions with Physically Adsorbed PVA Chains. Physically adsorbed PVA chains form long dangling tails resulting in a large hydrodynamic thickness 6, of the adsorbed layer. For the PVA sample studied with M , = 45 000,6, was found to be 46 nm. This is the result of the presence of long dangling tails. Interaction between such tails would be expected to occur at particle-particle separation distances in the region of 92 nm. This is reflected in the rheology of the dispersion, which is more elastic than viscous at @ > 0.5, even though the particle radius in this case is large (700 nm). Details of the viscoelastic results will be published e1~ewhere.l~Plots of G*, G’, and Grf (at w = 1 Hz) versus @ are shown in Figure 4. It is clear that both G* and G’ increase rapidly above @ = 0.53 whereas Grf remains low over the whole volume fraction of the dispersion. This corresponds to an effective volume fraction of 0.64 that is the maximum random packing fraction. Thus, the rheology curves reflect the strong steric interaction that occurs when the interparticle distance becomes smaller than 26. Flocculated and Coagulated Dispersions. 1. Weakly Flocculated Dispersions. These are exemplified by the case whereby a “free” (nonadsorbing) polymer is added to, say, sterically stabilized dispersions. Recent experiments were carried out by using the polystyrene latex dispersion with grafted PEO, to which PEO with various molecular weights was added to the contin(14) Hopkinson, A,; Tadros, Th. F., to be published.
32 Langmuir, Vol. 6, No. I , 1990
Tadros
i‘
PEO 20030
Ib)
1
/
10 l2I
Figure 4. G*, G‘, and G“ (at w = 1 Hz)versus 4 for aqueous polystyrene latex dispersions ( R = 700 nm) containing physically adsorbed PVA (M,= 46 000). Table I. Results for Flocculation of Polystyrene Latex Dispersions (4 = 0.3) with PEO for Various Molecular Weights Mw 20 000 35 000 90 000
[?I
R,, nm
%*
&*
0.351 0.519
5.52 7.59
0.02 0.01
0.029
1.013
12.90
0.005
0.020 0.010
Figure 5. Variation of r8, T ~ and , G with 4p of PEO (M,= 20 OOO) for a polystyrene latex dispersion contaming grafted PEO (4 = 0.3). nz
=5.10
uous medium.15 In these experiments, the volume fraction of the latex 4 was kept constant at 0.3, while the volume fraction of the free polymer 4 was gradually increased. Three molecular weight PEO samples, namely, 20 000, 35 000, and 90 000 were used. The flocculation was followed by using three rheological techniques, namely, steady-state shear stress-shear rate, shear modulus, and oscillatory measurements. This enabled one to measure the Bingham yield value 78,the Casson yield value 7c, the shear modulus G , and complex modulus G*, and the storage modulus G’ and loss modulus G” as a function of g5p. The results obtained from all rheological techniques were self-consistent, showing a sudden change above a critical volume fraction of the free polymer @p+, which decreased with increase of M,. &+was also near to @?*, i.e., the semidilute region where the free polymer coils began to overlap. A summary of the results is given in Table I, which also gives the physical characteristics of the free PEO chains. Below &+,the dispersions showed Newtonian flow and gave either a very low modulus or an immeasurable value. Above 4p+,all rheological parameters showed a rapid increase with further increase in 4p. As an illustration, the results obtained with PEO, M, = 20 000, are shown in Figure 5, which gives the variation of r8, T,, and G , with 4p. This clearly shows the rapid increase above C$ = 0.02. Similar trends were obtained when G*, G‘, and G” were plotted versus 4p as shown in Figure 6. Results for the other molecular weights also showed the same trends, and such data were published e1~ewhere.l~ The rheological parameters may be related to the interaction energy between the particles. For example, the Bingham yield value, r p ,may be related to the energy of separation between the particles in a floc, Esep,by the equation16 70
= NEsep
(13)
where N is the total number of contacts between the par-
0 0
0
0 02
OGL
GO6
008
@P
Figure 6. Variation of G*, G‘, and G” with $,of PEO (M,= 20 OOO) for a polystyrene latex dispersion containing grafted PEO ((J = 0.3).
ticles that is given by the equation
N = -l ( 34n) 2 4irR3 where n is the number of contacts in a floc, i.e., the coordination number. Combining eq 13 and 14, one obtains
To calculate Esepfrom r8, one has to assume a value for n. This range is from a maximum of eight for random close packing of the particles in a floc to a value of about four for open flocs. E,, values calculated by using eq 13 and assuming two varues of n of 8 and 4 are given in Table 11. Esep may also be equated to G,, the energy minimum in the particle interaction-$stance curve produced by addition of free polymer, Le., the depletion free energy. The latter is simply related to the osmotic pressure of the free polymer solution and has a range comparable to the radius of gyration R of the free polymer. For example, Asakura and Oosawaf7 derived the following single expression for Gdep:
~~
(15) Prestidge, C.; Tadros, Th. F. Colloids Surf 1988, 31, 325. (16) Gillespie, T. J. Colloid Sci. 1960, 15, 219.
where g52 is the volume concentration of the polymer that
Langmuir, Vol. 6, No. 1, 1990 33
Interactions in Concentrated Dispersions
Table 11. Results of E,, Calculated from Experimental r8 Vafues and GdaD
PEO M, = 20 000 0.025 0.003 0.004 0.006 0.008
2.0 2.8 2.8 5.8 13.1
9.1 12.7 17.8 26.4 59.6
18.2 25.4 34.6 52.8 111.2
25.3 30.3 40.5 60.7 80.9
81.9 102.9 149.5 261.1 397.4
28.6 38.1 57.1 76.2
54.1 78.0 117.0 203
55.0 82.6 110.1 137.6
48.4 85.6 131.3 185.7
PEO M, = 35 000 0.015 0.02 0.03 0.04
2.3 4.4 7.0 11.7
0.01 0.015 0.02 0.025
1.2 2.8 4.4 5.9
10.5 20.0 31.9 33.2
21.0 40.0 63.8 66.4
I
PEO M, = 90 000 5.5 12.7 20.0 26.9
11.0 24.5 40.0 53.8
1
"AO, Asahura and Oosawa (ref 17). FSV, Fleer, Scheutjens, and Vincent (ref 18).
is equal to (4/3)aA3N/ V, with A being the depletion thickness (A3R ) N2 is the total number of polymer molecules, and 8; is the total volume of the solution. P is equal to (RIA), whereas x is equal to [A - ( h / 2 ) ] / Awhere , h is the surface to surface separation. Clearly, when h = 0, i.e., a t the point where the free polymer chains are squeezed at, from between the particles, x = 1. Values of GdeP based on Asakura and Oosawa's model" are given in Table 11. This model was extended by Fleer et a1.,18 who arrived at the following expression (when A
