Rheol Acta 35:446-457 (1996) © Steinkopff Verlag 1996
Stefan Kurzbeck Joachim Kaschta Helmut M~instedt
Received: 29 August 1995 Accepted: 15 July 1996
Dedicated to Prof. Dr. Joachim Meissner on the occasion of his retirement from the chair of polymer physics at the EidgenOssische Technische Hochschule (ETH), Ztirich. S. Kurzbeck (~) • J. Kaschta H. M~nstedt Department of Polymer Materials Institute of Materials Science Friedrich-Alexander-Universit/~t Erlangen-Ntirnberg 91058 Erlangen-Ntirnberg, Germany
Rheological behaviour of a filled wax system
The temperature dependent rheological behaviour of a pigment filled wax system is investigated in a cone-and-plate viscometer over a range of shear rates from 60 to 10000 s -1. A strong influence of water adsorbed by the pigment on rheological properties of the filled system can be found. The increase of the yield stress and the viscosity at low shear rates can be related to a build-up of pigment structures due to growing water content. The flow behaviour can be described by the Casson equation as well as by the Herschel-Bulkley equation. Abstract
Introduction Filled systems with a suspending medium showing Newtonian flow behaviour are of widespread use. Dyes, laquers and printing inks are some examples for industrial applications of filler-modified fluids. The knowledge of the rheological behaviour plays an important role in processing such materials. There have been many attempts to understand and describe the rheological behaviour of suspensions. Several models for the description of the increase of viscosity above that of the suspending medium and the dependence on the volume concentration of fillers are reported (Wildemuth and Williams, 1984; Quemada, 1977). Also, several models exist to describe the non-Newtonian flow behaviour dependent on filler concentration and shear stress or shear rate, respectively (e.g., Krieger and Dougherty, 1959; Casson, 1959; Cross, 1965; Quemada, 1978 a, b; Wildemuth and Williams, 1984, 1985). In these models the destruction of particle agglomerates with in-
Both formulations are compared and discussed. The Casson model is evaluated more closely by the calculation of characteristic structural parameters of the suspension which are critically discussed. Key w o r d s Pigmented wax v i s c o s i t y functions - yield stress -
Casson model - Herschel-Bulkley model
creasing stress is regarded to be responsible for viscosity changes. They differ, however, with respect to their basic assumptions. In some models a shear rate dependent packing fraction of the fillers, which determines the amount of suspending liquid trapped by the filler particle structure, is assumed. This causes an increase Of the effective filler concentration which influences viscosity and its dependence on shear rate (Wildemuth and Williams, 1984). In other models the non-Newtonian viscosity is considered to be dependent on the energy dissipation by destruction and rotation of filler particle agglomerates in the flowing medium (Krieger and Dougherty, 1959; Casson, 1959). Another reason for dispute is the prediction of a yield stress by some of the models (Casson, 1959; Wildemuth and Williams, 1985), while others predict a finite Newtonian viscosity at very low shear rates (Cross, 1965; Quemada, 1978). Some authors even claim that yield stress may only be a result of extrapolation and not a real feature of a filled system (Barnes and Wakers, 1985). The assumption of a build-up and destruction of > particle structures and agglomerates implies strong in-
S. Kurzbeck et al. Rheological behaviour of a filled wax system teractions between the components of a filled system. These can be interactions between the suspending fluid and the filler particles as well as manifold interparticle forces. They are influenced by properties of the filler particles such as shape (Ziegel, 1970), size, size distribution (Wildemuth and Williams, 1985), specific surface and chemical properties as well (Ullmann, 1989). For example a change of the rheological behaviour due to a manipulation of the surface chemistry can be induced by addition of surface active agents to the filled system as reported for pigment/oil suspensions and water as additive (Weltmann, 1960). The intention o f this work is to characterize a technically relevant suspension rheologically in a range relevant for processing. Changes of the rheological behaviour will be related to the condition of the filler and to changes of the particle structure. The gained insights offer the possibility to optimize the system with respect to its processability.
Experimental Description of the filled system The wax base consists of a mixture of a triglyceride with fatty acids and a neutral oil of a mass ratio of 9: 1. Melting of the wax base occurs between 35 ° and 37 °C. As filling material an inorganic pigment based on a hydrated chromium oxide (Cr203"2H20) was used. The volume concentration of the i3igment was 17%. The powdery filler was dispersed in the wax base by a three roll laboratory mixing mill. The filled system passed the machine four times in order to destroy large particle agglomerates and to disperse the filler homogeneously. All investigated suspensions were produced under equal processing conditions concerning roll rotary frequency, roll nip and processing temperature.
447
100 80.~ 6 0 4O 20
o
0
m
~
2
~
t
~
~
~
~
t
[
3 4 5 6 7 8 910 partiele size, /xm
Fig. 1 Cumulative mass distribution of the filler particles as function of particle size
by the extremely structured surface and porosity of the pigment particles which can be seen from the scanning electron micrograph of the particle surface (Fig. 2). It was found that take-up of water from the air by the pigment particles cannot be neglected. Therefore the adsorption of moisture was determined by weighing filler specimens which were stored in desiccators at constant temperature and constant relative humidity. The different constant humidities were achieved by using saturated aqueous salt solutions. The desorption of surface water was accomplished by a temperature treatment at 130°C and 100 mbar pressure.
Filler characterization and preparation To characterize the particle size distribution of the filler a Brookhaven (BIC) XCD Particle Analyser was used. Particle agglomerates in the aqueous suspension were destroyed by ultrasonic treatment. The analysis of the particle size yields particle diameters ranging from 1 to 10 gm with an average particle size of 5.1 gm (Fig. 1). Assuming spherical particles it follows a specific surface area of 0.4 mZ/g. In addition, the specific surface area was analysed by nitrogen gas adsorption (BET-analysis) using a Micromeritics Asap 2000 Analyser. The determined specific surface area from BET amounted to 147 m2/g. The enormous difference to the geometric value can be explained
Fig. 2 Scanning electron micrograph of the particle surface structure
448
Rheologica Acta, Vol. 35, No. 5 (1996) © Steinkopff Verlag 1996
Rheological measurements For the rheological characterization a cone-and-plate rheometer (Contraves Rheomat 115) was used with coneand-plate geometries (70mm cone diameter/0.5 ° cone angle and 50 mm cone diameter/2.0 ° cone angle). The experiments reported in the third section were measured with the 70mm/0.5°-geometry. The shear rate was increased from 60 to 104 s - i in 15 steps; after reaching a steady state the resulting torque was measured. Viscosities between 0.6 and 104 mPas and shear stresses between 5.5 and 5.5x102Pa can be measured. The shear rate range up to l04 s -1 was chosen, because shear rates up to 7×103s -1 occur during processing of such systems, especially when passing narrow capillaries. The uncertainty of the measured viscosities may reach 50°7o at low viscosity levels up to 50 mPas and low shear rates because of a torque lower than 1% of the maximum torque. The uncertainty decreases to 1% at high shear rates. At higher viscosity levels the variation of the values amounts to 5 070 at low shear rates (cf. Fig. 5). The uncertainty at high shear rates amounts to 1%. Using a cone-and-plate geometry at high shear rates poses some questions regarding the accuracy of the measurements. Effects like slippage, secondary flows and temperature changes by energy dissipation can principally influence the results. In order to get an insight into their magnitude and their importance for our experiments they are discussed in the following. Measurements with the two different cone geometries were carried out on a pigment filled system at a volume concentration o f 14%. There is good agreement between the measured viscosities of both geometries in the common range of shear rate (Fig. 3) that means slip effects can be excluded.
10 4
l
dried particles
lO3
101
~
....
,
101
.
•
90QC, c a n e 1
o
90°C, e o n e 2
........
i 10 2
........
i 10 3
........
i 10 4
shear rate ~, s-1 Fig. 3 Viscosity functions of a filled wax system at two different temperatures and two different cone geometries (Rl = 25 mm, /71 = 2°; R2 = 35 mm,/?2 = 0.5°)
Because of the lack of possibilities to measure normal forces it could not be observed directly whether inertiadriven secondary flows disturb the assumed laminar shear flow pattern significantly. The influence of secondary flows on the value of the measured torque, however, can be estimated (Macosko, 1994). Turian (1972) theoretically developed Eq. (1) which allows an estimation of the relative torque M/Mo as a function of the Reynold's number, where M is the measured torque influenced by secondary flows and M 0 is the theoretical torque due to ideal laminar flow. M -
1+
Mo
3
Re 2
(1)
4900
Re is given by Re
-- ~°(~°R2/~2
(2)
where Q is the density of the fluid, co the angular frequency, R the cone radius, fl the cone angle and t/the viscosity of the fluid. Whitcomb and Macosko (1978) found that Eq. (1) describes the relative torque up to values of 2 for Newtonian fluids and found in own experiments that Eq. (1) even holds for non-Newtonian polymer solutions. Application o f Eq. (1) to our data leads to a maximum increase of the measured torque due to possible secondary flows by 8 × 1 0 - 3 % . Another equation has been derived by Ellenberger and Fortuin (1985) regarding the influence o f secondary flows (Eq. (3)) M --
M0
0.309 Re 3/2 =
I ~
(3)
50+Re
These authors report a better description of their own experimental data and those of other authors for Newtonian fluids by Eq. (3) and correlate an increase of the relative torque over unity to the onset of significant distortion of pure laminar flow by secondary flows. The application of this calculation leads to a maximum increase of torque of 0.2°70 in our case. The influence of free surface and effects due to the geometry of the cone-and-plate arrangement on the measured torque is estimated by Waiters (1975) to be lower than 6070 in general and lower than 1070 for cone angles smaller than 2 °, which ties within the experimental inaccuracies of the apparatus used in our experiments. An effect which is also related to strong secondary flows is the expulsion of fluid from gap (Wildemuth and Williams, 1985). This, however, could not be observed in our experiments. Due to the high shear rates in our experiments the effect of shear heating on the measured viscosity has to be
S. Kurzbeck et al. Rheological b e h a v i o u r o f a filled wax system
taken into account. The dissipated energy per unit volume A W~V can be estimated by 6W
--
T'~)
=
/,/.~)2
.
10 2
(4)
000
V The increase of the average fluid temperature in the adiabatic case follows from AT_ AW At
VQ cp
_
/~)2
Q cp
Structural investigations The filler particle structure was investigated by means of a Leitz Orthoplan microscope equipped with a hot stage. The samples were prepared by putting a drop of suspension on the microscope slide, covering the suspension with a cover slip and letting the sample rest at least 1 h at a temperature of 50°C. For this experiment a reduced volume concentration of the filler of 6% was used to achieve transparence. The temperature of the hot stage during the experiment was 50 °C thus the samples could be investigated in the molten state.
Results
o
oo
~
n
~
V v V v Y ~ W W U D O Q ~ n
101
(5)
Here, Cp is the specific heat of the fluid at constant pressure and Q the fluid density~). For our experiments an increase of the temperature between 0.5 and 2.7 K/s at its maximum can be calculated for the highest shear rates at 50 ° and 90 °C. At high shear rates the torque was measured after 15 s. Although the temperature should increase by 40 K according to Eq. (5) a temperature rise measured at the under-side of the plate was only observed for shear rates higher than 10 3 S - 1 and never exceeded i K. This indicates that good heat conduction of the steel and the high heat capacity of the cone (130 g) and plate (310 g) keep the temperature increase low. It is assumed to be about 5 K at the maximum.
449
10 o
0
0
50°C
a
a
61°C
v
v
70°C
D
D
......
81°C i
10 z
10 3
10 4
s h e a r r a t e y, s -1
Fig. 4 Viscosity f u n c t i o n s o f t h e w a x at different t e m p e r a t u r e s
The dispersion of the pigment particles in the wax base increases the viscosity of the whole system by two orders of magnitude, therefore the uncertainty decreases from 50°70 for the wax base to less than 5% for the filled system at low shear rates (Fig. 5). The addition of filler particles changes the Newtonian flow behaviour of the wax system into non-Newtonian. The viscosity function shows a pronounced shear thinning behaviour. A decrease of viscosity b y a factor of 10 over the experimental range of shear rate is found. At high shear rates the viscosity is still nearly ten times as high as the Newtonian viscosity of the pure wax base (Table 1). It is well known from literature and practice that fillers may be able to adsorb water. This can result in a change of interactions between the particles. To investigate this
10 4
T = 50°C
10 3
ystem
Rheological behaviour The wax base shows Newtonian behaviour at all measuring temperatures and over the whole range of shear rates (Fig. 4). Deviations from a constant Newtonian viscosity at low shear rates have to be attributed to uncertainties in the torque readings at low torque values (see preceeding section).
102 wax base
101
......
10 ~ 1) For t h e m a t e r i a l investigated t h e densities are 1.026 g / c m 3 at 50 °C a n d 0.996 g / c m 3 at 90 °C, T h e specific h e a t c a n be a s s u m e d to be 2 J/gK.
,.
i
10 e
.....
i
. . . . .
10 a
I
104
s h e a r r a t e y, s -l
Fig. 5 Viscosity f u n c t i o n s o f t h e filled a n d unfilled wax
450
Rheologica Acta, Vol. 35, No. 5 (1996) © Steinkopff Verlag 1996
Table 1 Viscosities of all investigated fluids at different temperatures and shear rates t/, mPas Temperature
50 °C
70 ° C
)), s - 1 Wax base Uncond. filler Dried filler Filler with 6 wt.% water Filler with 13 wt. % water
60 24 2153 1688 2432
3200 24 176 188 179
2962
206
90 °C
60 13
3200 13 . . . 1233 121 1317 111
60 . 1007 1242
3200 -
2336
1804
108
145
81 82
found. Between the two samples containing water there is a difference of about 20°70 at the low shear rate of 60 s -I (Table 1). At high shear rates the viscosity levels of the three samples differ only slightly. From these findings it can be concluded that water adsorbed on the filler surface not only increases the viscosity level but the shear rate dependence in addition. The same behaviour is found at a higher measuring temperature of 90°C (Fig. 6). In all cases a constant viscosity has not been measured at even the highest achievable shear rates.
Thermo-rheological behaviour
influence the filler was exposed to climates of constant temperature and constant relative humidity (r. h.) to ensure definite conditions. Three conditions were chosen. One batch of filler particles was dried at 130°C and 100 mbar until mass equilibrium was reached. After drying two batches were exposed to 32°70 and 75°70 relative humidity (r.h.) at a temperature of 21 °C. Exposing the pigment to 32% relative humidity leads to a water content of 6wt.% and 75% relative humidity to a content of 13 wt.%. Comparing the viscosity functions significant differences occur as a result of differently conditioning the fillers (Fig. 6). All samples show non-Newtonian shear thinning behaviour but the viscosity level increases with increasing loading of the pigment with adsorbed water. Regarding the viscosities at low shear rates in comparison to the dried fillers a viscosity increase of 4007o for the filler with a low water content of 6 wt.% and of 8007o for the filler with a high water content of 13 wt.% has been
The temperature dependence of the viscosity function of the filled wax system containing dried filler particles is presented in Fig. 7. All viscosity functions approach a slope tan a = - 1 at low shear rates in the double-logarithmic plot versus shear rate. This result is compatible with the existence of a yield stress (e.g. Mfinstedt, 1981). The shift of the viscosity with temperature is dependent on the shear rate or on the shear stress respectively. This is indicated by the values of the shift factor at(T, To) at two shear stresses, one at the lower (183 Pa) and one at the upper end (540 Pa) of the shear stress range of the experiments at five different measuring temperatures. Figure 8 shows the temperature dependence of the shift factor in an Arrhenius plot for the two chosen stresses of 183 and 540 Pa in comparison to the shift factors of the suspending wax base. The shift factors of the suspension at the high shear stress differ only slightly from those of the wax base, while the shift factors at the low shear stress show higher values. Therefore the temperature dependence of the suspension is in some degree
10 4
10 4 condition of t h e filler
t
0
dried
v wet (6 ~ H~0) n wet (la ~ H~0) 103
103
**V,.
b 03
102
.~
102
T=50°C ....... T=90°C I0 i
I0 i i0 i
10 2
10 3 shear
10 4
rate y, s -I
Fig. 6 Viscosity functions of the filled wax system for different filler treatments measured at 50 ° and 90 °C
v
70°C
[]
80°C
o
92°C
. . . . . . . .
i0 i
j
10 2
. . . . . . . .
~
. . . . . . . .
s h e a r rate ~,
j
. . . .
10 4
i0 3 s -I
Fig. 7 Viscosity functions of the wax system filled with dried particles at different temperatures
S. Kurzbeck et al. Rheological b e h a v i o u r o f a filled w a x system
0.6 I"
0.4 d"
1
""
0.2 0-
451
significantly higher shift factors. A flow activation energy of 62.5 kJ/mol of the wet particle system at the low stress exhibits a remarkable deviation from the thermo-rheologically simple behaviour and a temperature dependence which is nearly double as high as that for the dried particle system.
O
-0.2 I .',t
~."
-0.4 -
~~
1"f f
-0.6
I 2.7
2.6
I 2.8
base waxwaxbase
Description of the rheological behaviour
~
-<> dried particles "r=183 P a
~.,
.u
dried particles T=540 P a
I P I 2.9 3 3.1 1 0 0 0 / T , K -~ - - ~
I 3.2
Fig. 8 A r r h e n i u s plot o f t h e shift factor a (T, To) o f t h e wax a n d t h e wax s y s t e m filled with dried particles at two different s h e a r stresses
more pronounced at lower shear stresses and the thermorheological behaviour of the dried particle system is not a simple one. A quantitative analysis marks a weak increase of the calculated Arrhenius flow activation energy from 27.3 kJ/mol for the wax base up to 31.0 kJ/mol at the higher stress due to the filler particles. The activation.. energy of 36.7 k J/tool at the low stress is significantly higher. The system with the high water content shows a more pronounced deviation from a thermo-rheologically simple fluid than the system with dried fillers. The comparison of the shift factors at similar shear stresses proves this and is underlined by the values of the flow activation energies. The temperature dependence at the high shear stress is very similar to that of the wax base and the activation energy of 32.0 kJ/mol is nearly equal to the activation energy of the dried particle system (Fig. 9). At the low shear stress, however, the adsorbed water results in
Due to the fact that all investigated suspensions indicated the existence of a yield stress, two theoretical models with a yield stress criterion have been chosen to describe the rheological behaviour. One is the so-called Casson equation (Casson, 1959), T1/2 = k0 + kl. ~ 1/2
(6)
which goes back to the assumption of interparticle attractive forces and that energy dissipation is based on desintegration and rotative motion of chain-like particle agglomerates in laminar shear flow in Newtonian media. /Co is a measure for the yield stress of the suspension and k I is the square root of its high shear viscosity. Both constants are related to characteristical parameters of the assumed filler structure in the pigmented wax (see in the following). The other equation is the empirical HerschelBulkley equation. (7)
r = ro+k.~)n
r0 describes the yield stress, k and n are constants of the rheological system corresponding to the consistency and the flow exponent of a power law flow equation. If the experimental data fulfill the so-called Casson plot, i.e. r 1/2 versus 2)1/2 give a linear relation,/Co and kl can easily be determined by linear regression. Figure 10
1200 0.6
/
0.4
t ~5 t~O 0
/
t
..1"
1000 800
0.2 0
~
-0.2
~
~ j . , . Y
-0.4
c-----a
w a x base
<> /"
o--
w e t p a r t i c l e s T=183 P a
]
~..-..4
/
d"'" /
-0.6 2.6
I 2.7
r
I 2.8
~>
.~
600
~b
400
To=70Oc
-o
dried
filler
T = 50°C
200
w e t p a r t i c l e s T=540 P a
I I I 2.9 3 3.1 I O 0 0 / T , K-1 -.-...-
I 3.2
Fig. 9 A r r h e n i u s plot o f t h e shift factor a (T, To) o f t h e wax a n d the wax s y s t e m filled with wet particles (water c o n t e n t 13 wt.°7o) at two different s h e a r stresses
i
J
i
i
i
i
i
i
0
10
20
30
40
50
60
70
;
/30
0
100
Fig. 10 C a s s o n plot o f t h e w a x system filled with dried particles. E x p e r i m e n t a l error as indicated by error bars
452
R h e o l o g i c a Acta, Vol. 35, No. 5 (1996) © S t e i n k o p f f Verlag 1996
Table 2
Parameters of the Casson and the Herschel-Bulkley equation
Filler c o n d i t i o n
Dried
Temperature
6wt.°70 water c o n t e n t
50°C
70°C
90°C
50°C
1 3 w t . % water c o n t e n t
70°C
90°C
50°C
70°C
90°C
Casson equation 253 9.2 64
ko, V m P a ~:o, P a
224 7.1 50
210 5.2 44
329 7.5 108
-
H e r s c h e l - B u l k l e y e q u a t i o n (fit with ~, Pa k
62 2445
49 1548
45 998
108 2262
shows that the measured data of the dried system can be described by Casson's equation. The uncertainty of the determination of/Co and k 1 can be estimated from Fig. 10 as I0% at the maximum. The description holds for all temperatures in the same way (Fig. 11). The system with the highest water content gives straight lines, too, over the whole range of strain rates and at all temperatures measured, if plotted according to Casson's equation (Fig. 11). The extrapolated values for k0 are significantly higher, however. Both systems show only slightly temperature dependent yield stresses. The adsorption of water seems to increase the temperature dependence of the extrapolated yield stress. The parameters k0 and k~ for all three suspensions are listed in Table 2. kl decreases with rising temperature. Under all conditions the dried system shows lower k~ values than the wet particle systems. The water content does not show a significant influence on k~, however. The parameter k0 and therefore the yield stress
-
253 4.5 64
n=
391 7.5 153
344 6.0 118
311 4.9 97
2/3) 70 884
158 2352
125 1559
105 1063
r 0 increase with growing water content. In contrast to the parameter k~ the influence of the water content on/Co is very pronounced. To fit the experimental data with the Herschel-Bulkley equation, Eq. (7), the simplex method was used with free parameters k and n. The resulting values for n varied between 0.6 and 0.7 but without any systematic correlation to the temperature. The values for parameter k also did not show a systematic relation to temperature. By using a fixed value of n = 2/3 for the fit a good description of the data within the measuring inaccuracies and a systematic relation of the parameter k to the temperature can be achieved in the case of the dried particles as well as for the wet particles containing 13% water (Fig. 12). A slight variation of the value of the exponent n between 0.64 and 0.68 does not improve the quality of the fit. The parameters k then decrease with temperature but are similar for dried and wet particles. Thus it seems, com-
I000 I 800
,O'"
Jt**~
.......
,~"
V
600 gWoO*
"
,., o
400 o
90°C
200 dried ....... 0
0
50°C
v
70°C
o
90°C
©
wet particles
I
i
t
I
I
n
I
i
i
10
20
30
40
50
60
70
80
90
~11~, s-llz
'
particles
10 4 100
,.
Fig. 11 C a s s o n plot o f t h e w a x s y s t e m filled with dried particles a n d wet particles (water c o n t e n t i3 wt.%) at three different t e m p e r a tures
......
~ 10 e
.........
'
dried
parHeles
....... wet particles ~ . . . . . . lO s 10 4
s h e a r raLe y, s -1
m
Fig. 12 Shear stress as a f u n c t i o n o f s h e a r rate for t h e wax s y s t e m filled with dried particles a n d wet particles (water c o n t e n t 13 wt.%) at three different temperatures. T h e lines represent t h e theoretical curves according to Eq. (7) with p a r a m e t e r s given in Table 2
S. Kurzbeck et al. Rheological behaviour of a filled wax system parable to k 0 of the Casson equation, that there is only the parameter r0 in the Herschel-Bulkley equation which depends directly on the temperature and the water content of the filler particles, as well. Another remarkable fact is the coincidence in the resulting values for the yield stresses for both flow equations. All calculated parameters are listed in Table 2.
453
10 9
I
I
T=50°C
i/,
I
10 8 . . . . . . . . . . . .
Hers hel-Bu kley
/-
C a s s l [1
10 7
o~
~6 10 6
zet par icles
O3
S
~c~ /
lO 5
Discussion of the results
dried
article
10 4
Non-Newtonian behaviour of suspensions of solid particles in Newtonian liquids is related to a disruption of particle agglomerates (Casson, 1959; Quemada, 1978; Ziegel, 1970; Wildemuth and Williams, 1984). The interpretation of the effect of these desagglomerations on the viscosity, however, is different. One interpretation uses the concept that a shear dependent destruction of the filler structure decreases the amount of suspending liquid which is trapped by particle structures. Therefore the effective volume fraction of the filler which determines the viscosity of the suspension decreases (Wildemuth and Williams, 1984). Other theories regard the energy dissipation due to rotation and disruption of particle agglomerates (Krieger and Dougherty, 1959; Ziegel, 1970) as the reason for a change in viscosity. Casson uses assumptions similar to the latter ones to calculate the rate of energy dissipation from which an equation for the viscosity function of the suspension is derived. This equation and the empirical Herschel-Bulkley equation of the shear rate dependence of the shear stress allow a remarkably good description of the measured rheological behaviour of the pigmented wax, as is shown in Figs. 11 and 12.
Comparison of the two flow equations At each measuring temperature the two yield stresses determined by extrapolation have very similar values (Table 2). The parameter kl of the Casson equation and the parameter k in the Herschel-Bulkley equation show a comparable tendency. They increase with temperature and are independent of the water content of the filler particles. Only at very high shear rates do the two formulations differ fundamentally and only then will there be evidence of which equation fulfills the experimental results. The Casson equation predicts a slope of 1 in a double logarithmic plot if kl.))1/2 is much higher than ko, that means for high shear rates. For high shear rates the Casson equation predicts a constant viscosity which is equal to k 2. It is not unreasonable to assume such a behaviour, because of the Newtonian flow behaviour of the suspending fluid. At very high shear rates at which all particle agglomerates can be assumed to be broken down to a size of the magnitude of the primary particles a shear
10 3
10 -1
10 °
101
l0 a
10 ~
10 4
shear
rate
y, s -1
10 5
10 8
10 v
,,
Fig. 13 Shear stress as a function of shear rate for the wax system filled with dried and wet particles (water content 13 wt.%) at 50 °C and theoretical curves according to Eqs. (6) and (7)
rate-independent viscosity should be observed. In a double logarithmic plot the Herschel-Bulkley equation predicts at high shear rates a slope of the value of the exponent n, in our case 2/3. Therefore no constant viscosity at very high shear rates is predicted by this model. As Fig. 13 shows for a temperature of 50 °C and for two different water contents the experiments do not give a hint which equation will be suitable in the high shear rate range. The experimental data points are described by both equations but the shear rates accessible to the experiments are not high enough to decide experimentally whether the slope 2/3 or 1 is valid. The Casson equation is derived from assumptions concerning energy dissipation mechanisms. It uses a model of a chain-like particle agglomerate structure that is built up in the flow field. Therefore, model parameters can be calculated and compared with the results of other investigations. The conclusions from the Casson model and their relevance will be discussed later in this work.
The effect of filler pretreatment on particle structures The experiments give similar viscosities at high shear rates under all conditions. This result indicates that neither water content nor temperature has a significant influence on the agglomerate structure at high stresses. It is not unreasonable to conclude that changes of interacting forces due to the given changes of temperature and water content are small compared to the stresses exerted at high shear rates. Another important result is the influence of the filler pretreatment on structural parameters at low shear rates. The viscosity level and the shear rate depen-
454
Fig. 14A
Rheologica Acta, Vol. 35, No. 5 (1996) © Steinkopff Verlag 1996
Micrograph of the wax system filled with dried particles
Fig. 14B Micrograph of the wax system filled with particles of a water content of 6 wt.% Fig. 14C Micrograph of the wax system filled with particles of a water content of 13 wt.°70
dence of the viscosity of the suspensions is the more pronounced the higher the water content. The shear rate dependence of the viscosity is ruled by the energy dissipation rate. Relating the energy dissipation to the destruction of particle agglomerates it can be assumed that an increasing water content implies larger interacting forces between the particles. A distinct difference in the build-up of particle structures becomes obvious in the micrographs in Fig. 14, which show the structures of the dried and moistened particles. The dried particles only show a weak tendency to agglomerate (Fig. 14a), while the moistened fillers tend to build up large agglomerates and clusters (Fig. 14b, c). Comparing the micrographs of the two fillers with different water contents it is evident that the higher water load causes a more tightened structure (Fig. 14 c). The tighter structures as a result of increasing water content can be related to higher attractive forces between the primary particles. Bearing in mind that the yield stress is defined as the cohesive strength that has to be overcome by the stress which is exerted on the filler particle structure during flow, the yield stress also should grow with an increasing amount of adsorbed water. The yield stresses that can be determined from an extrapolation of the
curves in Fig. 11 to the zero shear rate are listed in Table 2 and confirm the expected tendency. A qualitative correlation of the filler particle structure in the unsheared state with the rheological behaviour in the sheared state can be tried, because of the ability of the investigated system to rebuild an agglomerate structure at lower shear rates which has been destroyed by shear flow under high shear rates before. This feature is revealed by an experiment with at first increasing and afterwards decreasing shear rate (Fig. 15). The resulting loop of the viscosity function, based on steady-state viscosity values, does not show any remarkable hysteresis. From these findings it can be concluded that a rebuilding of the parti-
104 wet particles (13 w~ % water content)
1
ineresing ..... '7 decreasing 103
8 5~
T = 70°C 102 101
""~...x~ ,
. . . . .
,
102 103 s h e a r rate ~, s -1 ---~
Fig. 15 Viscosity functions of the wax system with wet particles (water content 13 wt.%) measured with increasing and decreasing shear rates
S. Kurzbeck et al. Rheological behaviour of a filled wax system cle structure from the partly desagglomerated state takes place. For this reason the low shear rates occurring during sample preparation for the microscope only may have a small influence on the particle structure because only a weak desagglomeration can have taken place and a rebuilding of structures at very low shear rates can be assumed.
455
1/2
(8)
1 - qs) aa-1
structural parameters of the suspension are expressed by /Co. Moreover,/Co is a measure of the yield stress -c0 of the suspension as follows from Eq. (6) setting ?) = 0.
The Casson model in focus In the light of the good description of the rheological behaviour of the filled wax system by the Casson formula, Eq. (6), the hypothetical assumption of chain-like particle structures in shear flow seems to be an approach to the real particle structure of the investigated suspension worthy to be discussed in more detail. The assumption of chain-like particle groups in shear flow is not unreasonable, because there are reports about chain-like aggregation effects of spherical particles in shear flow of polymer solutions, which describe the formation of chains of spheres (Michele et al., 1977). This also can be observed for dispersions of P M M A beads in silicone oil (Sauer, 1996). In our case the pigment particles are not ideally spherical but they can be considered to be similar to spherical particles. Originally developed for filled systems with Newtonian fluids as suspending medium like suspensions of pigments in oil Casson's equation describes the relation between shear stress and shear rate in a very simple form (Casson, 1959). A general assumption is the building and destruction of chain-like groups of filler particles under the effect of shear flow. The cohesive strength of these filler particle groups depends on the magnitude of the interparticle attractive forces. The groups will break down into smaller subunits if the disruptive stresses due to flow exceed the cohesive stresses. The disruptive stresses depend on the rate of shear and the size of the particle groups. In the Casson model the particle groups are treated like solid rods with an axial ratio J = L / R . L and R describe the half-length and radius of the rod. By adding filler particles to a suspending fluid the flow pattern is disturbed and therefore the dissipation of energy is increased. The change of energy dissipation depends on the size and the orientation of the chain-like filler particle groups. Due to the shear rate-dependence of the particle group size the rate of energy dissipation/~ according to the disturbance of the flow patterns by the particle groups is also shear rate dependent. Integration over all rates of energy dissipation of all existing rods at one rate of shear and a filler volume concentration q~ delivers the Casson flow equation (Eq. (6)). The constants k 1 and k 0 contain characteristic microstructural parameters of the suspension. While k I takes into account the properties of the suspending Newtonian fluid by its viscosity r/0 as
"((1-- tP) 1-aa/2- 1)
(9)
aa-I
The parameter a is a constant, depending on the orientation of the chain-like groups at the begin of flow varying between 0 and 12). The size of the particle groups is characterized by a and ft. They are related to the axial ratio J by J=
a + f l ' ( r l o f O - I/2
(10)
a is a measure for Jo~ = J(p--,oo) the limit of J at very high shear rates, while fl is a measure for the interparticle attractive forces. In addition, the shear rate dependent cohesive stress a A can be calculated which describes the cohesive force between two adjoining particles related to the cross-sectional area of the chain-like particle groups. aA
= 3r/0~)aJ 2
(11)
For very low shear rates and large values of J C a s s o n gives a limiting value for fl as 3 0"1/2 fl = - (48a) 1/2 Then
(12)
is given by
aA
f12.48a a A -- -
-
(13)
9
The experiments show similar viscosities in the range of the highest achievable shear rates independent of the filler condition. The Casson model leads to results for the axial ratios of the filler particle groups at this shear rate and at "infinite" shear, which are very close to each other (Table 3) at all water contents. Comparing the calculated axial ratio J(~)) at the highest applied shear rate of )J = 9130s -~ and the limiting value Joo for "infinite" shear rates 0 ) ~ co) in Table 3 there is a significant difference which can be explained by the assumption that the stresses applied to the agglomerates even at the highest
2) For calculations Casson proposed an average value of a
2 =
0.5.
456
Rheologica Acta, Vol. 35, No. 5 (1996) © Steinkopff Verlag 1996
Table 3 Characteristic structural parameters for different filler pretreatments and measuring temperatures Filler condition
Dried
6 wt.% water content
Temperature J 0 ) = 60 s -~) J(?) = 9130 s -1)
50°C 83 17
a = J~ = J(#~oo)
11
70°C 95 18 11
~o, Pa t7A, kPa
64 28
50 22
90°C 117 20 11 44 19
50°C 112 16 8
70°C -
108 61
-
shear rates are not strong enough to disintegrate the structures down to the size of the primary particles. The fact that the aspect ratios for "infinite" shear rates do not reach unity, as would be expected for spherical particles, could be explained by the existence of asymmetric filler particles as it is also discussed by Casson. The fillers used in the experiments are not spherical. An aspect ratio of ten, as it comes out from the calculations, is difficult to imagine, however. At low shear rates the filler conditioning has a strong influence on the viscosity and on the shear rate dependence of the viscosity. The micrographs of the suspensions at rest indicate significant differences in the agglomeration of the filler particles. The calculated axial ratios at 60 s - 1 (Table 3) show a corresponding tendency. Comparing the amounts of adsorbed water to the magnitude of calculated axial ratios at low shear rates there seems to exist an at least semiquantitative relation. The difference between axial ratios for particles with low water content and those of the dried ones is half of the difference between the values for the dried system and those with the high water content (Table 3). The absolute values, however, again appear to be overestimated by the Casson model as it became obvious for the limiting axial ratios at "infinite" shear rates. These values imply lengths of the chain-like particle groups which would not allow complete revolutions in the rheometer gap for the assumed rotational motions in the Casson model. This, however, is not necessary if one interprets the rotational motions as oscillations around an equilibrium orientation in the flow direction. As mentioned earlier, the larger and tighter structures due to higher water content correlate with higher yield stresses and therefore higher interparticle attractive forces. If the calculated cohesive stresses 3) are regarded as a function of the a m o u n t of adsorbed water a tendency similar to the dependence of the axial ratios on the water content is found (Table 3). The definition of the cohesive stress is given as the cohesive force per cross-sectional area of t h e assumed chain-like filler groups. Taking the average particle size of 5.1 g m as the diameter of the
3) The cohesive stress is calculated for ~)--*0according to Eq. (13).
13 wt.% water content 90°C 144 20 8 64 35
50°C 131 18 8
70°C 155 20 9
153 86
118 56
90°C 174 23 10 97 47
idealized cross-section of a filler particle chain the calculated cohesive forces range from 4 x 1 0 -7 to 2 x 10 .6 N. These values are in good agreement with data from Schubert (1977) calculated for adhesive forces between small particles in a suspending liquid to be in the range from 5 x 10 .7 to 5× 1 0 - 6 N for particle sizes between 1 and 10 gm. The temperature dependence of structural features is mirrored by the characteristic parameters of each suspension calculated from the evaluation of the Casson Eq. (6) in Table 3. Especially the axial ratios of the chain-like filler groups at low shear rates reveal a Significant increase with temperature. An increase of the axial ratio J = L / R indicates an anisotropic thermal expansion of the particle groups. The changes of the calculated values of J ( p ) with increasing temperature, however, are surprisingly high. The temperature increase effects larger particle to particle distances. Thus the interparticle attracting forces, which are reciprocal to a higher power of the interparticle distance, decrease. This consideration explains that the calculated cohesive stresses and the values of the extrapolated yield stress become less with rising temperature (Table 3).
Conclusions The rheological behaviour of a pigment-filled wax system was investigated in shear flow in a cone-and-plate viscometer. Addition of fillers to the wax causes a change from Newtonian to non-Newtonian shear thinning behaviour. The non-Newtonian viscosity can be related to a break-down of filler particle structures under shear stress. The temperature dependence of the particle structure causes systematic deviations of the filled system from a thermo-rheologically simple behaviour. Corresponding to the shear rate dependence of the dimensions of the agglomerate structure the temperature dependence decreases with increasing shear rate. Water adsorbed on the filler particle surface has a strong influence on the viscosity level of the filled system. From microscopic investigations of the filler agglomerate structure at rest and the results of the rheological experiments it is evident that
S. Kurzbeck et al. Rheological behaviour of a filled wax system adsorbed water must increase the interparticle attracting forces and therefore increase the dimensions o f the agglomerate structures. The influence, however, is restricted only to ranges o f low shear rates but is p r o n o u n c e d in the shear rate dependence o f the viscosity and in the values o f the yield stress o f the suspensions. The HerschelBulkley equation and the Casson equation are able to describe the rheological behaviour satisfactorily over the whole experimental range o f temperature and shear rate. B o t h equations give equal results if the prediction of the yield stress is concerned but differ in the prediction o f the high shear rate viscosity. The question o f which equation fulfills the experimental results in the high shear rate range c a n n o t be answered at this point because the distinctive high shear rate range is not accessible experimentally. The general assumption o f a chain-like filler structure in the Casson model has been used as a hypothesis in order to draw quantitative conclusions from the measurements on the pigment-filled wax system. Characteristic parameters o f the filled system like the shear rate dependent axial ratio o f the particle structures, the yield stress and the cohesive strength o f the agglomerates are calculated. Their dependence on shear rate, temperature and
457
water content can be understood in a qualitative way. Especially the change in the rheological behaviour due to water adsorbed on the particle surface and following from that an increase o f particle interactions is qualitatively well illustrated by the calculated parameters characterizing the filler structure. Concerning the size o f the assumed chain-like aggregates, the Casson model seems to overestimate the real dimensions, however. The assumption o f chain-like structures m a y be a step in the right direction. But it seems unlikely, however, that the particle aggregations will behave like rigid rods as hypothetically assumed by Casson. One reason for this assumption is the fact that the aggregates seen in the micrographs do not show any rod-like structure as would be expected f r o m rigid rods rotating under shear flow. Therefore, an energy dissipation mechanism other than that o f rotating rigid rods has to be required. It should be able to take into a c c o u n t flexible particle chains built up under flow and energy dissipation due to d e f o r m a t i o n and orientation o f these chains. Acknowledgement One of the authors (H.M.) would like to express his particular gratitude and appreciation to Joachim Meil3ner who introduced rheoiogy to him as a fascinating and versatile subject.
References Barnes HA, Waiters K (1985) The yield stress myth? Rheol Acta 24 (4):323- 326 Casson N (1959) A flow equation for pigment-oil suspensions of the printing ink type. In: Mill CC, Rheology of disperse systems. Pergamon Press, New York 84-104 Cross MM (1965) Rheology of non-Newtonian fluids; a new flow equation for pseudoplastic systems. J Coll Sci 20: 417 -437 Ellenberger J, Fortuin JMH (1985) A criterion for purely tangential laminar flow in the cone-and-plate rheometer and the parallel-plate rheometer. Chem Eng Sci 40 (1):111 - I16 Krieger IM, Dougherty TJ (1959) A mechanism for non-Newtonian flow in suspensions of rigid spheres. Trans Soc Rheol 3:137-152 Macosko CW (1994) Rheology - Principles, Measurements and Applications. VCH Publishers, New York Michele J, P~tzold R, Donis R (1977) Alignment and aggregation effects in suspensions of spheres in non-Newtonian media. Rheol Acta 16 (3):317-321
Mfinstedt H (1981) Rheology of rubbermodified polymer melts. Polym Eng Sci 21 (5):259-270 Quemada D (1977) Rheology of concentrated disperse systems. I. Minimum energy dissipation principle and viscosity-concentration relationship. Rheol Acta 16 (1):82-94 Quemada D (1978a) Rheology of concentrated disperse systems. If. A model for non-Newtonian shear viscosity in steady flows. Rheol Acta 17 (6):632-642 Quemada D (1978b) Rheology of concentrated disperse systems. III. General features of the proposed non-Newtonian model. Comparison with experimental data. Rheol Aeta 17 (6):643-653 Sauer S (1996) Diploma Thesis, University Erlangen-Niirnberg Schubert H (1977) Mechanische Verfahrenstechnik. Deutscher Verlag ft~r Grundstoffindustrie, Leipzig Turian RM (1972) Perturbation solution of the steady Newtonian flow in the cone and plate and parallel plate systems. Ind Eng Fundam 11(3):361- 368
Ullmann KC (1989) Stabilisierung von TiOz-Dispersionen durch nichtionische, " wasserl0sliche Makromolekiile. Ph.D. Thesis, University Stuttgart Weltmann R (1960) Rheology of pastes and paints. In: Eirich RF, Rheology Theory and Applications, Vol. 3. Academic Press, London Whitcomb PJ, Macosko CW (1978) Rheology of Xanthan Gum. J Rheol 22 (5): 493 -505 Wildemuth CR, Williams MC (1984) Viscosity of suspensions modeled with a shear-dependent maximum packing fraction. Rhe01 Acta 23 (6):627-635 Wildemuth CR, Williams MC (1985) A new interpretation of viscosity and yield stress in dense slurries: coal and other irregular particles. Rheol Acta 24 (1): 75-91 Ziegel K (1970) The viscosity of suspensions of large, nonspherical particles in polymer fluids. J Colloid Interface Sci 34 (2):185-196
