Polymer Coatings with a Destabilizing Action on Suspensions
Citation for published version (APA):
Laven, J., Huisman, F. J., Lalieu, L. J., & Stein, H. N. (1988). Polymer Coatings with a Destabilizing Action on Suspensions. Colloids and Surfaces, 31(1), 385-405. https://doi.org/10.1016/0166-6622(88)80040-4
DOI:
10.1016/0166-6622(88)80040-4
Document status and date: Published: 01/01/1988
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Colloids and Surfaces, 31 (1988) 385-405
Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands
385
Polymer Coatings with a Destabilizing Action on
Suspensions
J. LAVEN, F.J. HUISMAN*, L.J. LALIEU** and H.N. STEIN
Laboratory of Colloid Chemistry, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven (The Netherlands)
(Received 18 February 1987; accepted in final form 14 July 1987)
ABSTRACT
The rheologicsl behaviour of concentrated suspensions of spherical glass particles can be influ- enced drastically by surface pretreatment of the particles. This treatment consisted of silanization with mono- or difunctional organo-silicon compounds. Results from ESCA, IR and SEM analysis strongly suggest that silanixation with the reagent dimethyldichlorosilane (DMDCS) results, depending on the processing, in a more or less homogeneous layer of oligomer and/or polymer.
The particles were suspended in an apolar (DOP) and in a polar liquid (glycerol/water). All suspensions exhibited perfect or nearly Newtonian behaviour except DMDCS-coated glass in gly- cerol/water. With the latter, under oscillatory shear conditions, behaviour of a continuous coag- ulation network structure was found dynamic moduli are hardly dependent on frequency while network structure breakdown occurred already at the lowest accessible deformation ( 10m3). Under steady shear rate conditions pseudoplastic behaviour was found, with relative differential viscos- ities at high shear rates (10’ s-l) approaching the value of equivalent suspensions of uncoated spheres.
Results are interpreted with the giant floe model, according to which macroscopic shear is understood as microscopic slip along shear planes between coagulated domains. The origin of coagulation in these systems is not in van der Waals forces because bond strengths as judged from rheology are too high. Polymer bridging between coating polymers is suggested as flocculation mechanism.
1. INTRODUCTION
Polymer layers on a solid-liquid interface in a dispersion are usually formed by adsorption from the liquid phase, and either stabilize the dispersion con- cerned (see e.g. Ref. [ 1 ] ) , or induce bridging flocculation ( see Ref. [ 2 ] ) .
In the present communication, attention is focused on polymer layers formed by silanization. By this method, a hydrophilic surface (e.g. silicate glass) can be converted into a hydrophobic one; this leads to coagulation of a dispersion
*Present address: Philips Research Laboratories, Eindhoven, The Netherlands. **Present address: Philips B.V., C.F.T., Eindhoven, The Netherlands.
386 TABLE 1
Characteristics of glass samples
Sample No I II
Mass average particle diameter, $ (pm) 28.32 78.32
Number average particle diameter, &, (pm) 26.64 75.38
Quotient @i/4,, 1.063 1.039
Thickness of polymer layer (nm) 1.9 3.0
Specific mass 2.69 2.84
of particles covered by such a layer, in polar liquids [ 3-51. We compared the properties of dispersions of glass particles silanized by different methods in different media; on the basis of rheological, IR, SEM and ESCA data a mech- anism for the coagulation is proposed.
2. EXPERIMENTAL
2.1. Materials
2.1.1. Glass
The batches of glass beads were obtained from Tamson (Zoetermeer) . Ac- cording to SEM evidence they consisted of spherical particles. ESCA evidence
(see Section 3.1.3) showed that the glass contained, in addition to Si, Na and Ca, substantial amounts of Pb. The beads were divided into different particle size fractions either by dry sieving or by sedimentation in water. The specific mass of the glass varied with particle size (see Table 1) , apparently due to partial evaporization of PbO during the preparation of the spherical glass particles.
Silanization was performed by the following method:
A,: A suspension of 10 g of native glass in a mixture of 59.5 ml hexane (ex Merck, “reinst”) and 0.5 ml dimethyldichlorosilane (DMDCS, ex Janssen Chimica, reagent grade), was vigorously stirred for 2 h at 20” C. The super- natant was decanted, and the glass was washed 5 times with 20 ml portions of hexane. Then the glass was dried at 55°C at a reduced pressure for two days, and subsequently stored over P205.
A,: Treatment analogous to method Al, with trimethylchlorosilane (TMCS, ex Janssen Chimica, reagent grade) instead of DMDCS.
B,: Native glass was heated overnight at 145 ‘C, placed under vacuum (oil- pump) pressure, and afterwards exposed to DMDCS vapour at a temperature of 65’ C for 6 h. Then the DMDCS vapour was replaced by dry Nz. The treated glass was stored in a stoppered flask under normal atmospheric conditions.
387
temperature was 50’ C, in order to obtain a vapour pressure comparable to that used in method B,.
BB: Treatment analogous to method B1, with hexamethyldisilazane (HMDS, ex Janssen Chimica, reagent grade) instead of DMDCS, at a temperature of 100’ C (for realizing a vapour pressure comparable to those in methods B1 and B2).
C,: 150 g of native glass was dried for 15 h in a fluidized bed at 410°C by means of a PaOb-dried flow of Nz (1000 ml min-‘, tube diameter 45 mm). Afterwards, the glass was silanized for 4 h by passing the dry N, flow through liquid DMDCS (15°C) before entering the liquid bed.
Cz: Treatment analogous to method Ci, with TMCS instead of DMDCS. The temperatures were the same as those employed in method C,.
CB: Treatment analogous to method C1, with HMDS instead of DMDCS. Drying was performed at 410” C; during silanization the fluidized bed was kept at 65°C.
D,: Treatment and materials analogous to method Al, but the amounts of beads and chemicals were higher by a factor 10. Beforehand, the beads had been dried according to method C,.
2.1.2. Dispersion media
Glycerol: ex Merck (“Zur Analyse”), containing 13% m/m of water
( q293K = 152.05 mPa s).
Di-2-ethylhexylphthalate (DOP) : ex Fluka, > 99% (~293 ~~81.1 mPa s).
Phthalic acid bis-methylglycol ester (DMPG) : ex Fluka, “chemical purity”,
(t/293K= 29.12 mPa s).
2.2. Apparatus and procedures
2.2.1. IR spectra
Glass beads were investigated with a Bruker IFS 113V Fourier Transform Infrared Spectrophotometer with a globular light source. The number of scans ranged between 125 and 1000. The resolution was 4 cm-‘. Prior to analysis the glass samples were dried at 150°C (native glass) or 110°C ( silanized glass). A finely divided mixture of 10.0 mg glass (0.4-1.5 pm diameter) and 190.0 mg KBr was compressed at 0.74 MPa for 10 min; the pellets thus formed were stored over P205.
2.2.2. SEM
Scanning electron microscopy pictures were taken with the Cambridge In- struments electron microscopes Stereoscan MK 2A and Stereoscan S 200.
388
2.2.3. ESCA
For ESCA spectra use was made of a PHI 550 System (Perkin-Elmer) equipped with facilities for AES and XPS analysis, containing a double pass cylin~ical mirror analyzer.
2.3. Rheology
2.3.1. Steady shear flow
Most steady shear flow measurements were carried out with a Contraves Rheomat 15 (Epprecht AC., Zurich), measu~ng system C (coaxial cylinder type; radius of inner cylinder 6.77 mm, radius of outer cylinder 10.00 mm; ef- fective length 46.2 mm). Incidentally, a Contraves Rheomat 115 with DIN 145 coaxial cylinder system was used, with this instrument much lower shear rates were attainable than with the Contraves Rheomat 15. In all measurements centripetal acceleration was restricted to values of at most 0.93g (g is the ac- celeration due to gravity). If not mentioned explicitly, measurements were carried out at 20 ’ C.
Pastes were prepared on a mass basis. Two procedures were employed: (a) The suspensions were homogenized manually, thermostatted at 20.0 2 0.3 “C for an hour, stirred vigorously by mechanical means for 5 min. and thermostatted again for 15 min. Then the suspension was homogenized man- ually again and poured into the rheometer cell. With each suspension, four step-wise shear-rate sweeps were taken, i changing first from high to low val- ues, then from low to high values, then again from high to low values, and from low to high values. At each angular velocity, the torque was measured after 10 s. Between successive sweeps, the suspension was homogenized by hand.
(b ) After thorough homogenization by hand with a spatula, the suspension was stirred for 2 min at a stirring speed of 3000 rpm, in a 100 ml glass vessel of 43 mm internal diameter; the stirrer’s head (dimensions: 27 x 11 x 1 mm3)
made an angle of about 30” with the horizontal plane. The sample was then introduced into the rheometer as quick as possible and meas~emen~ were started immediately. A series of measurements performed on one sample, con- sisted of 3-4 angular-velocity scans. Each scan contained two step-wise an- gular-velocity sweeps: from the angular velocity at the highest measurable torque down to the lowest possible velocity (about 10 steps) and then back upto the highest velocity. Between successive scans, the suspension was ho- mogenized by hand. Each scan was performed in less than one minute.
2.3.2. Oscillatory shear flow
Oscillatory shear flow measurements we carried out with a Weissenberg rheogoniometer (R18, Sangamo Ltd) , equipped with a temperature-controlled cup and bob system, with a cup radius of 27.5 mm and a bob radius and height
389
0.15 - ABSORBANCE
0 I I I I
2800 3000 3200
WAVE NUMBER ,m_,
Fig. 1. Typical FT-IR spectrum of DMDCS-coated glass beads (method B,) .
of 26.0 and 48.5 mm respectively. The axial distance between bob and bottom of the cup was kept at 1.8 mm.
Measurements of the dynamic shear moduli of pastes at 20°C were per- formed in two ways:
- as function of the amplitude of the imposed shear deformation (“dynamic deformation”), at a fixed angular frequency (3.14 s-l ) .
- as function of the angular frequency at an approximately fixed level of dy- namic deformation ( N 4 x 10A3).
The pastes had been prepared according to procedure b of Section 2.3.1. Measurement time was limited, depending on the pastes, due to sedimentation in the measuring cell.
3. RESULTS
3.1. Data on the silane luyer
3.1.1. IR spectra
Spectra were recorded for native glass, and for glass treated by methods Al, B1, AZ, Bz, C,. A typical absorption spectrum for a DMDCS gas phase treated sample is shown in Fig. 1. The 2966 cm-’ peak is assigned to the symmetric C-H stretching mode of methyl groups [ 61. In addition, the antisymmetric C-H stretching mode of methyl groups at 2914 cm-l is seen. The peak height at 2966 cm-’ is shown in Table 2.
The 2966 cm-l peak of vapour phase coated particles was larger in the case of DMDCS (method B1 ) if compared with HMDS as vapour reactant (method B3). It is known that DMDCS is less reactive towards = Si-OH than HMDS
390 TABLE 2
IR-spectrometry: Peak heights at 2966 cm-’ for glass beads silanized by different methods Reactant Medium Code of method
(see text) Peak height (extinction units) DMDCS DMDCS TMCS TMCS hexane J% gas hexane Nz gas A, 0.016 f 0.005 B, 0.075 AZ 0.016 Bz 0.021
is at the same temperature. This difference is even more pronounced in our case, with reaction temperatures of 60°C and 100°C. This indicates that with DMDCS vapour phase treated glass the silane is not restricted to methyl or methylene groups but consists of a polymer (DMDCS contains two reactive groups which can, with H20 from the glass, form large condensation products). Vapour phase treatment appears to be more effective than liquid phase treat- ment, presumably because of the higher reaction temperature.
3.1.2. SEM
Figures 2 and 3 show typical data. The DMDCS vapour phase treated glass
is covered by an organic layer, which, however, is not always spread evenly
over the surface: in some cases the organic materials form drop-like accretions which are more pronounced than the deviations from a smooth surface in un- treated glass. Between these accretions, the glass surface is also covered by organic material as may be seen in some electron micrographs (Fig. 4) ; how- ever, this continuous layer cannot always be clearly distinguished. The role of glass-bound water in the formation of such coatings is evidenced by experi- ments with glass prepared according to method D, and with glass dried over Pz05 before DMDCS gas phase treatment. These glasses did not show the accretions seen in Fig. 3, but their coagulation properties were the same as those of the beads with accretions on their surface (see Section 3.3.).
3.1.3. ESCA
ESCA spectra for native glass and DMDCS liquid phase treated glass are shown in Figs 5 and 6, respectively. The native glass shows peaks of Na (1) ,
0
(2), Pb (3), Si (4), and K ( 6) ; in addition a C peak is found ( 5 ) , probably due to adhesive used for fixing the glass spheres while in the ESCA equipment. In comparison with this, the DMDCS treated glass (Fig. 6) shows less pro- nounced peaks (l-4,6) ; the C peak is stronger, and a Cl peak ( 7) is found. On the assumptions of a homogeneous coverage, and of equal Si concentrations in the glass and in the polymer coating, the average thickness of the silane layer
Fig. 2. SEM photograph of native glass beads of 12 p. One side equals 20 p.
Fig. 3. SEM photograph of DMDCS vapour phase treated glass beads of 12 p. One side equals 10 ,um.
Fig. 4. SEM photograph of vapour phase treated glass beads of 12 ,mn. One side equals 2Opm. Note the coating of (organic) material.
can, according to Ref. [ 71, be estimated from the decrease of the Pb peak heights (see Table 1).
3.2. Sedimentation behaviour of suspensions
Both initial sedimentation rates and solid volume fractions in the final sed- iment of dispersions (initial solid volume fraction 0.05-0.15) were observed. The glass was either untreated or coated by method A,. Coagulation was ap- parent both from a significantly larger initial sedimentation rate than expected for the primary particles, and from a lower solid volume fraction in the final sediment than expected for a random close packing of equal spheres (0.57). Coagulation was only found for suspensions of DMDCS coated glass particles in polar liquids (in glycerol, to a smaller extent in DMGP) [ 51.
3.3. Rheology
3.3.1. Steady shear flaw
From the results obtained with procedure a, strongly coagulating suspen- sions can be distinguished from suspensions with no distinct signs of coagu-
393 78 wi morrh 1 : : 3 6-. y :.
2
:
-5! ,-! : 0 .a, “A_: 5’ g zd
~~li:
x2- .,.. ! 0+ -1100 -550 binding energy/& 0Fig. 5. ESCA spectrum of native glass beads (78e). (1) Na; (2) 0; (3) Pb; (4) Si; (5) C; (6) K; (7) Cl.
i
-550 bind~g energy/eV 0
Fig. 6. ESCA spectrum of DMDCS liquid phase treated glass beads (78 w ) . See also legend to Fig. 5.
lation. Strong coagulation is evidenced primarily by deviations from Newtonian behaviour (shear thinning) at low solid volume fractions (
@c
0.30) [ 41. De- viations from Newtonian behaviour were, for reasons of surveyability, char- acterized by the properties at large 3 values, where the ~rque-alar velocity graph becomes linear. From the slope of the graph in this region a “plastic viscosity” (i.e., a limiting value of the differential viscosity for maximum 3 values) was calculated, and from its intercept with the axis for 3 =O, a r. value. Thus the latter is not a real yield value. By this criterion, DMDCS treated glass particles in glycerol (Fig. 7) and in DMGP showed the most pronounced394 glycerol / water 2- 7 coated 56 4 3 4 4 &&/ 2 O- Q p : I& O / native -2
1-01
,
0.15 0.2 ) oi3 4 0,L 0,5--4, I I I I I -1 -0.5 -L ‘o9Q)Fig. 7. Values of the quantity 7. for suspensions of glass beads in glycerol/water (87113% m/m).
(1) Native glass (43 pm); (2) native glass (12 e); (3) DMDCS liquid phase coated glass beads (method A,, 43 pm) ; (4,5) DMDCS vapour phase treated glass beads (method Br, 12 pm); (6) as curve 5 but before silanization the glass had been stored over P,05 for 1 month, (7) as curve 5 but before silanization the glass had been exposed to 93% relative humidity for 1 month.
Fig. 8. Values of the quantity 7,, for suspensions of glass beads (12 pm) in DOP. Curve 1: untreated,
native glass; curve 2: DMDCS vapour phase treated glass beads (method B, ) .
coagulation. Rheological effects caused by TMCS and HMDS treatment are small, although discernable.
Procedure b was applied as an additional check on the influence of sedimen- tation or centrifugal separation (if any). Generally, for coagulating suspen- sions during the later part of a scan (increasing 9 ) a slightly smaller torque was measured, at a given f value, than during the initial part of a scan (de- creasing j ). This excludes hysteresis in the building up and destruction of the coagulation structure, as a cause for this difference. Significant sedimentation did not occur, as shown by visual checks; neither is centrifugal separation prob- able, since the centripetal acceleration was for most angular velocities much smaller than the acceleration due to gravity. The difference between the initial and later parts of a scan can best be understood as progressive ordening of shear planes in the coagulation structure, with more prolonged shear. A pos- sible alternative may be the effect recently described by Leighton and Acrivos
395
I I
-1 0
1
lo&r)
2
Fig. 9. Flow curves of a 40% (v/v) dispersion of DMDCS-coated glass (method D,) in glycerol/ water (N/l3 m/m). measurement ~m~ratures are 2,12 and 2O”C, respectively.
[ 81, According to their data, suspended particles migrate under the influence of shear towards regions of low shear. In a coaxial cylinder geometry this would mean depletion of suspended particles from the gap between the cylinders, and their accumulation in the suspension outside this gap. However, according to Leighton and Acrives, this effect would be most pronounced with the largest particles (78 pm ) while we found the opposite trend.
Dispersions of native glass beads have, at Qi< 0.25, no measurable values of to. Above @ = 0.25 its dispersions in glycerol/water (Fig. 7) and in DMGP show low z. values. In the case of suspensions in DOP (Fig. 8) coagulation is slight but distinct in view of the larger r. values (compare Figs 7 and 8).
Glass beads, coated with DMDCS (methods A, B) exhibit coagulation if dispersed in appropriate media. In glycerol/water media the coagulation was very strong (Fig. 7). In DOP the z. values are so small that in that case coag- ulation is very slight or absent (see Fig. 8). In DMGP the z. values are slightly larger than in the case of native glass beads in DMGP, indicating slight but distinct coagulation in the case of coated beads. Coating via method C is inef- fective in generating coagulation, even in glycerol/water. We ascribe this to the fact that water was removed too effectively from the glass before silanization; no water molecules were left for generating polymerization of DMDCS.
With TMCS and HMDS treated glass dispersed in glycerol (methods Az, Bz, Cz, B3) the z. values are only slightly larger than those of uncoated beads indicating only minor coagulation. Also with method C3 coagulation was not significant.
The temperature dependence of the viscosity of strongly coagulating disper- sions can be judged from Fig. 9, where the shear stress of a 40 vol.% dispersion of DMDCS-coated glass beads in glycerol/wa~r is shown in a double log plot as function of the shear rate. If shifted horizontally according to the viscosities of the dispersion medium at the various temperatures, the curves cover each other perfectly.
3 loglG.G'/ Pa)
Fig. 10. Dynamic shear moduli G’ ( n , 0 ) and G” ( a,0 ) as function of the dynamic deformation y for 40% (v/v) dispersions of native 45 pm glass beads ( 0, 0 ) and for DMDCS liquid phase coated 45 pm glass beads (method D1; n , 0) in glycerol/water (87/13% m/m). The angular frequency was kept at 3.14 s-l.
log(G,G”/Pa) 3- 2- l- O- -1 I I I -1 0 1 2 log Ws-‘)
Fig. 11. Dynamic shear moduli G’ (
w,
0 ) and G” ( 0, 0 ) as function of angular frequency o for 40% (v/v) dispersions of native 45 pm glass beads ( 0, 0 ) and for DMDCS liquid ~b coated 45 pm glass beads (method D1; n , 0) in glycerol/water (87/13% m/m). The dynamic deformation was kept at * 4. 10W3.391 3.3.2. Oscillatory shear flow
Figure 10 shows plots of the dynamic shear moduli as function of the dy- namic deformation for a 40% (vol.) dispersion of glass beads in glycerol/water. The dispersion with DMDCS-coated glass (method D1) shows the typical be- haviour of a continuous network of “non-entropic” nature, i.e. the moduli in- crease with decreasing deformation, at very small deformations where the moduli of typical entropic polymer networks are constant. Such behaviour is found e.g. in dispersions of fat crystals in oil [ 91 and in acidified milk gels. Contrary to the latter systems, our dispersions have loss moduli G” larger than storage moduli G’ , possibly because our measurement conditions are still too far from the linear viscoelastic, low deformation limit for G’ and G” .
On the other hand, dispersions with native glass beads in glycerol/water are linear viscoelastic over the larger part of the deformation range studied indi- cating the absence of the continuous network of the type mentioned before.
The results on the dynamic deformation dependence are supported by the frequency dependencies of these systems at small deformations as shown in Fig. 11. The dispersion of native glass is almost entirely viscous and obeys, within experimental accuracy, the Kronig-Kramers relationship between the slopes of the G’ and G” curves, as valid for linearly viscoelastic systems [lo]. On the other hand, the dispersion of coated glass does not fulfill the Kronig-Kramer requirement for linear viscoelasticity that G” must be much smaller than G’ if the slopes of their curves are small.
4. DISCUSSION
The basic fact of the phenomena reported here is that pronounced rheolog- ical signs of coagulation are only found for suspensions of DMDCS treated glass in glycerol, with the additional requirement that the glass had not been dried extensively beforehand, or that the DMDCS treatment had occurred in hexane probably containing some water. Thus, only when a polymer layer is formed does coagulation become pronounced.
Figures 12 and 13 compare the relative differential viscosities ef= (dr/dj ) /Q for suspensions in glycerol/water of native glass and DMDCS liq- uid phase treated glass, for two particle sizes (mass-averaged diameters 28.9 pm and 78.3 pm, respectively). In the case of native 28.9 pm particles, only the limiting values of the differential viscosities for large i) values are shown.
For suspensions of native glass in glycerol/water, there is a slight tendency for the relative viscosity to decrease with increasing j, especially at high solid volume fractions. This is apparent from the small but non-zero values of r. in Fig. 7. This effect could be due to slight coagulation at low 9 values, where “slight coagulation” means that of all collisions expected on rectilinear ap- proach a small but non-zero fraction leads to pair formation. Against this pos- sibility it has been argued [ 4,5] that suspensions of native glass do not show
dif 9 rel
20 . .
10 __
Fig. 12. Relative differential viscosities of native (full symbols) and DMDCS liquid phase coated (open symbols; preparedwith methodll),) glass beads (28.9m) in glycerol/water (87/13% m/m). The volume fractions of the solid phase in the suspensions are 35.0% ( v, V ) ,37.5% ( n , a) and 40.0% (0,O).
T-Id”
rel10 ._
Fig. 13. Relative differential viscosities of native (full symbols) andDMDCS liquid phase treated (mathodD,; open symbols) glass beads (78.3~011) in glycerol/water (87/13% m/m). The volume fractions of the solid phase in the suspensions are 35.0% ( V V ) ,37.5% (m, 17 ) and 40.0% ( 0,
0).
any distinct difference in rheological behaviour in media of such different po- larity as glycerol/water, or DOP. An equality of colloid chemical interactions in glycerol/water, and in DOP would be quite fortuitous.
to be slightly higher than those obtained for suspensions of untreated glass in glycerol/water. The difference is hardly larger than the experimental uncer- tainty of the r. values; but the regular course of the z. versus $ graph (Fig. 8) gives us confidence that the difference is real. The most far-going statement possible at present is that coagulation indeed is absent as far as rheological evidence is concerned, in suspensions of untreated glass in glycerol/water. The former statement can be based not only on the very low r. values found, but also on the entirely viscous behaviour in oscillatory shear experiments (within experimental accuracy). The possibility that native glass particles in glycerol reside in a secondary energy minimum, can, however, not be excluded.
This means that the low z. values found during steady shear in suspensions of native glass in glycerol/water must be ascribed to alignment of particles in shear, being more pronounced at high 3. While a quantitative theory on this aspect cannot yet be given, it is noted that diffusion as disturbing the align- ment at low 9 [ 11,121 must be excluded in the case at hand: our particles need 2704700 h to diffuse over a distance equal to their own radius in the pure suspension medium, while y-’ is at most 0.1 s.
Suspensions of DMDCS treated glass in glycerol show a much more pro- nounced decrease in differential viscosity with increasing 9. Clearly, during the initial part of a scan (decreasing 9 ) a coagulation structure is formed, leading to 3dimensional regions in which the particles remain surrounded by the same neighbours, of increasing size with decreasing 9. These regions (“do- mains”) are partly destroyed again on increasing f during the later part of a scan.
In the “giant floe” model [ 61 this situation is schematized by assuming the formation of shear planes when the coagulated suspension is subjected to a shear. In calculations, these shear planes are considered to be flat and aligned in the direction of the motion, with an average distance between successive shear planes equal to A. A suspended particle occupies on the average, an area 4’ in this plane. When shear is applied, particles bordering a shear plane meet particles from the adjacent domain i and both particles are displaced over a distance 6, from their rectilinear course, in a time t,,.
If q is the average number of neighbours of a particle within a domain, a particle bordering a shear plane has q- 1 neighbours within its own domain. When a particle bordering a shear plane is forced out of its way over a distance ao, it entrains its q - 1 neighbours over a distance 6, x I (with 0 < I < 1) . These neighbours entrain their q- 1 other neighbours over a distance 6,x 1 2, etc. This leads to an energy dissipation per collision:
~,,=2~6n~~~b~f~{6~/t~+(q-l)Z16~/t~+(q-l)Z~4;/t~+ . . ...}
(1)
=127voX~Xf(w~o)
x1_(ql
1)z2 -400
where b is the particle radius, q. the viscosity of the suspension medium, f the frictional ratio, i.e. the quotient of the friction experienced over that expected for an isolated particle (comparable to the friction ratio introduced by Van der Ven and Hunter [ 131) . The friction on such a particle in the suspension arises from two effects: cooperative flow of assemblies of particles and relative move- ment of neighbouring particles. For both effects it has been shown (by Gluck- man et al. [ 141 and Brenner [ 151, respectively) that the friction can, with the aid of a friction coefficient, be expressed in a modified Stokes law. Thus, our approximation in the use of a friction coefficient is mainly in assuming addi- tivity of both effects and in taking a time-averaged value off.
The energy dissipated per unit of volume and time is obtained by multiplying e. by the number of particles in shear planes per unit volume ( 2/ (A xd2) )
divided by 2, and divided by the time between two successive collisions (~65% u/f xA) , where u is the angle between the direction of motion and the line connecting the centers of two particles in a shear plane which are met successively by one particle. In addition, r. = 6,/ (p x A x cos u) .
We obtain: bdoA
i=
127rq) x-f
A3 y- (q_&02 ~.=12lrx%xAxf
l- (q-1)12With increasing shear rate, primarily A will change; in addition, the factor
f/O- kW2)
may change but as long as there are S-dimensional domains the changes in this factor are thought to be negligible.This model is, of cause, an idealization. When shear starts the shear planes will initially not have the idealized alignment; neither will they be flat. How- ever, protruding parts of a domain will, due to stress concentration, be subject to an increased local stress; and in hollows of a domain particles from a neigh- bouring domain tend to become attached. Thus, with increasing duration of the shear a straightening and increased alignment of shear planes is expected. This is thought to be the cause of the lower torque, at a given 9 value, during the later part of a scan as compared with that measured during the initial part. Note that it is not to be expected that aligned shear planes extend infinitely. There always remain external disturbing factors (imperfect homogeneity of the macroscopically imposed shear; wall effects). An intrinsic disturbing fac- tor is due to particles which are not incorporated in, or are occasionally ex- pelled from shear planes. Such particles induce blocking and locally destroy the alignment structure, limiting the persistency length of this structure.
We first consider situations in which still a reasonably complete 3dimen- sional network exists within the domains (A >> 2b). On increasing 3 starting from low 9 values, the predominant aspect of structure breakdown is a lowering
401
of A in Eqn (3) rather than a change in f / (1 - (q - 1) 2 2). On the assumption that changes in f / (1 - (q - 1) 1 2, are negligible we can deduce the dependence of A of the shear rate. It appears that A -jen with n =0.25-0.30. Thus, the
number of shear planes per unit length in the direction of the velocity gradient, is proportional to 9°.25.
In the context of the present paper it is especially important that differences in relative differential viscosities between coagulating and non-coagulating dispersions tend to vanish at high 9. With dispersions of 78 pm particles this occurs at y N 100 s-‘; with 28 pm particles, the shear rate is increased by about a factor of 3.
Up to this limit (in the case of 78 pm particles), the differential viscosity decreases regularly if plotted versus log 3. Thus it is probable that up to this i, value only one structure breakdown effect is involved.
It is, however, very unlikely that the regularly decreasing character of the qdiff versus log 9 graph is preserved beyond this limit. For this would imply:
(a) that at one particular i) in this region, the differential viscosities in co- agulating suspensions would become lower than those of non-coagulating sus- pensions with the same size distribution of dispersed particles. This has never been observed.
(b ) that the differential viscosity of a concentrated suspension, at one 9 value in this region, would become lower than that of a dilute one with the same size distribution of dispersed particles. This again is very improbable.
The simplest interpretation of the data is that, at the 9 value at which &r becomes equal for coagulating and non-coagulating suspensions, breakdown to A = 2b occurs. Note that this does not imply complete structure breakdown in the coagulating suspension. At this shear rate, the relative viscosity itself of the coagulating suspension still surpasses that of the non-coagulating one: within the layers in the coagulating suspensions still remains of the coagula- tion structure persist.
On the basis of this interpretation, we can calculate the average number of contacts between adjacent layers which is just broken by the average shear stress acting at the shear rate concerned, and thus we can calculate the average force necessary to separate two adjacent particles [ 61. For this calculation a random distribution of vacancies over the available sites in the layers is as- sumed; within the layers, either a hexagonal or a cubical arrangement of the particles is introduced.
The use of an average value for the number of contacts between adjacent planes, may be questioned. Certainly there will be adjacent layers with more than the average number of contacts between them; this leads to resistant domains in the coagulation structure. However, during strain the stress is con- centrated in these regions. The use of the average shear stress as necessary for separating two adjacent layers with average occupation, for calculating the force necessary to separate two touching particles is equivalent to the assump-
402
TABLE 3
Bond strengths of DMDCS-coated glass beads, dispersed in glycerol/water. For calculation pro- cedure, see text
Averaged Volume fraction particle size (pm) in dispersion
Bond strength (1O-7 N) 28.9 35.0 3.0 37.5 2.9 40.0 3.0 78.3 35.0 18 37.5 16 40.0 11
tion that, when the average shear stress suffices to separate the neighbouring layers with an average degree of occupation, stress concentration takes care of regions connected by a larger number of bonds between the particles.
Our hypothesis leads to values for the average force (F) necessary to sepa- rate two touching particles, which are independent of the solid volume fraction $. Results are shown in Table 3. The range of volume fractions # covered by the data in this table is limited by difficulties in obtaining homogeneous sus- pensions at @ > 0.40, and by the consideration that for $ < 0.35 discrete floes are formed. In the latter case the Hunter approach [ 131 would be more real- istic than the giant floe model. In Table 3, the bond strengths for the 78.3 pm particles are calculated from the values of the average shear stress at the shear rate where the differential viscosities of coagulating and non-coagulating sus- pensions are equal. For the 28.9 ,um particles, extrapolated values are used.
These values provide insight into the mechanism of coagulation. The follow- ing mechanisms are envisaged:
(a) The Hamaker constant for the system polymer/( glycerol + water) /polymer is significantly larger than that for the system glass/ (glycerol + water) /glass. Reference [ 161 mentions, e.g., for C/H,O/C:
A Ham= 1 X 10-l’ J; for Si02/HzO/Si02: AHam= 1 X 10-20 J.
(b ) The Hamaker constant for glass/ (glycerol + water) /glass suffices for coagulation, but dissociation of surface Si-OH or Si-ONa groups on native glass in glycerol leads to electrostatic stabilization. Indeed, native glass in glyc- erol was found by electrophoresis to have a negative zeta potential.
(c ) Polymer chains protruding from the coatings in the case of silanized glass form bridges between the particles.
Among these alternatives, the last-mentioned one (c ) is the most probable one. Thus, alternative (a) can be excluded by the following consideration.
The theoretical value for the attractive force between two spherical particles which nearly touch, is given by:
403
FAEAHamXb
12xH2 (4)
where b is the radius of the particles; H the shortest distance between the surface of the particles. However, in case of suspensions with coated particles,
we are in reality dealing with the system glass/polymer/
( glycerol + water) /polymer/glass (g/p/l/p/g). If we neglect the contributions of the glass itself to the attractive energy (cf. the values for AH_ quoted), we can describe the system by an effective Hamaker constant:
A
gplpg
=
A@&,i
l+(l+:$y-(l+~)
(5)
where 6 is the thickness of the coating. Aeff is > 0.5 XAH- only if S/H> 0.7. This is not compatible with the 6 values found by ESCA (Table 1) and the roughness of surfaces (cf. Figs 2-4). If the contact between the particles is thought to be restricted to the globular accretions, then the attractive force becomes much smaller because the effective particle radius b in the relation for FA should then be equated to the radius of the globular accretion rather than that of the glass paticle. In this case unrealistic values are calculated for Hfrom Eqn (4).
Similarly, unrealistic values for H are found through alternative (b ) . This
amounts to the assumption that AHam for the system glass/
(glycerol + water) /glass suffices for coagulation if electrostatic repulsion is ab- sent. With AHam= lx 10-20 J, FA=1~10-6 N, and b=40 pm we calculate H- 2 x lo-” m. Again this distance is much smaller than the surface rough- ness. Thus alternative (c) is the most probable one.
From the conclusion that A = 2b at 9 - 100 s-l in 78pm particle suspensions, we can calculate the parameter combination as appears in Eqn ( 3
) :
f
l- (q-1)12
This parameter combination is the factor by which the Stokes friction of a particle in a shear plane must be multiplied in order to obtain the real energy dissipation. In this parameter combination are incorporated both the differ- ence in friction between a particle in a floe and an isolated particle (via f) , and the number of additional motions caused by entrainment of neighbouring par- ticles (via l/ (1 - (q - 1) 1 2, ) . This parameter combination can be calculated from Eqn ( 2 ) by assuming reasonable values for b/d (from a comparison with crystal structures) and for do/b (from Batchelor and Green trajectories [ 171,
404
and from deviations from rectilinear motion only occurring for avoiding direct steric overlays [ 61) . In this way values for f / (1 - (q - 1) 1”) are calculated of the order of 10, the largest uncertainty being connected with our ignorance of the degree of order effected by a high shear rate [ 61. In view of the fact that there is no distinct difference between lim (~&~o) i+oo in coagulating suspen- sions and in non-coagulating ones, the large value for f / (1 - (q - 1) I ” ) should be ascribed to a large value off rather than to a small value of the denominator. A large value off is indeed expected because the effects of all deviations from rectilinear motion in a very dilute suspension, including rotation, are combined in this parameter.
ACKNOWLEDGEMENTS
The authors wish to thank Ir. F.W.A.M. Schreuder for part of the rheological measurements and Dr A.D. Langeveld (Physics Department of Eindhoven University of Technology) for the ESCA spectra.
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DISCUSSION
G. FRENS (TU Delft, Delft, The Netherlands)
Could the adhesion between hydrophobic spheres not be brought about by the immersing liquid withdrawn from the point of contact when two particles have collided? This would conceivably enhance the strength of the adhesion contact considerably. This idea was suggested 20 years ago at a meeting (Not- tingham) of the Faraday Society.
J. LAVEN (Eindhoven University of Technology, Eindhoven, The Netherlands)
I once thought about this problem and then came to the conclusion that it just is an expression of the van der Waals forces and does not contribute an extra force. However, I do not remember how I came to my conclusion. Your remark is acknowledged and I will check whether the arguments used are really valid.
An experimental check of gas bridging was made in the following way. If gas bridges would occur, elimination of gas nuclei in the liquid (by compression) will probably reduce the effective number of bridges in a suspension and thereby its viscosity. However, such reduction was absent.
