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Model analysis on dispersion characteristics of fine particles in Newtonian molten polymer
Advanced Powder Technology 20 (2009) 139–144
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Advanced Powder Technology journal homepage: www.elsevier.com/locate/apt
Original Paper
Model analysis on dispersion characteristics of fine particles in Newtonian molten polymer Emi Hasegawa a, Hiroshi Suzuki b,*, Kananko Kameyama a, Yoshiyuki Komoda b, Hiromoto Usui b a b
Graduate School of Science and Technology, Kobe University, Kobe, Japan Department of Chemical Science and Engineering, Kobe University, Kobe, Japan
a r t i c l e
i n f o
Article history: Received 22 December 2007 Received in revised form 10 May 2008 Accepted 23 May 2008
Keywords: Particle agglomeration Cluster size distribution Thixotropy model Dispersion Shear breakup
a b s t r a c t In this paper, a model estimating the time-dependent distribution of cluster size consisting of fine particles in Newtonian molten polymer was proposed. The present model for agglomerative suspension was developed based on Usui’s thixotropy model, derived by taking the balance between shear breakup, shear coagulation and Brownian coagulation processes. The analysis was applied to the experimental results for silica/(ethylene methyl-meta acrylate copolymer) suspensions. The time variations of cluster size distributions, of viscosities and of the mean numbers of particles in a cluster were calculated and the results were compared with experimental results for the cases of shear rates from 0.2 to 10 s1 when the solid volume fraction was set at 0.15. From the results, it was found that the present model can estimated the steady state values of the slurry viscosity and the mean particle numbers in a cluster as well as Usui’s original model. However, the model could not sufficiently express the time variation of rheological characteristics due to the over-estimation of the contribution of shear rate when the cluster size is small. Ó 2008 The Society of Powder Technology Japan. Published by Elsevier BV and The Society of Powder Technology Japan. All rights reserved.
1. Introduction Dispersion characteristics of fine particles in solid–liquid slurries or suspensions are of importance in many industrial processes. The particles in solid-liquid slurries or suspensions coagulate due to the particle’s contact by the Brownian motions, the particle velocity difference induced by the fluid flow and by the other external forces, and so on. Smoluchowski [1] worked out the basic theory of the coagulation of the particles in such solid–liquid slurries or suspensions. Swift and Friedlander [2], and Friedlander and Wang [3] investigated theoretically and experimentally the simple shear coagulation in which Brownian coagulation occurs simultaneously. Higashitani et al. [4–6] also investigated the coagulation model and examined the validity of their model experimentally. On the other hand, Parker, Kaufman, and Jenkins [7] investigated the breakup mechanism in shear flows and showed the breakup mechanisms of surface erosion and splitting of a parent aggregate. Sonntag and Russel [8,9] suggested the relations between the floc size and the deformation rate experimentally. Among them, Usui [10] established a simple model considering Brownian and shear coagulations and shear breakup of clusters, in order to estimate apparent viscosity and bulk agglomeration characteristics of fine particle suspensions when the parent cluster of particles is splitting into smaller fragments by shear addition. It was developed for multi-modal systems (Usui et al. [11]) and applied to many * Corresponding author. E-mail address: [email protected] (H. Suzuki).
kinds of suspensions (Usui et al. [12], Mustafa et al. [13]). Komoda et al. [14] also revealed it can estimate solid-air systems. Kobayashi et al. [15] also developed shear breakup theory separately. However, these models assumed cluster size tends to be uniform with while the particles are coagulating and breaking up. In actual cases, the size of clusters including fine particles is widely distributed. Such distribution characteristics severely affect the functions and physical properties of some materials. Komoda et al. [16] experimentally investigated cluster distribution characteristics of fine particles in Newtonian molten polymer in order to clarify a shear breakup process of clusters. In this paper, a model estimating the time-dependent distribution of cluster size consisting of fine spherical particles in Newtonian fluids will be developed based on Usui’s thixotropy model [10] for data measured by Komoda et al. [16]. The results will be compared with their experimental data. From this, the present model will be evaluated.
2. Model analysis 2.1. Thixotropy model Usui [10] suggested a thixotropy model for analyzing suspension rheological characteristics as follows:
3 dk 4ab kb TN 0 4as /kc_ 3pd0 k k þ 1 gc_ 2 ¼ dt 4F 0 Nb 1 e 3g0 p
ð1Þ
0921-8831/$ - see front matter Ó 2008 The Society of Powder Technology Japan. Published by Elsevier BV and The Society of Powder Technology Japan. All rights reserved. doi:10.1016/j.apt.2008.05.001
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Nomenclature coagulation rate (m3 s1) diameter of particles (m) bonding energy (J) number of particles in a cluster (–) Boltzmann constant (J K1) total number of the primary particles per unit volume (–) number of bonds to be broken when the cluster is divided (–) concentration of the clusters composed of k particles (m3)
bij d0 F0 k kb N0 Nb nk
Here, t[s], k[–], kb[JK1], T[K], N0[m3], /[–], e[–], d0[m], g[Pas], g0[Pas] and c_ [s1] are time, the number of particles in a cluster, Boltzmann constant, absolute temperature, total number of particles in a unit volume, apparent solid volume fraction, void fraction in a cluster, diameter of particles, slurry viscosity, solvent viscosity and shear rate. ab[–] and as[–] are rate constants of Brownian coagulation and of shear coagulation obtained by Higashitani et al [4–6]. Fo[J] and Nb[–] are the bonding energy and the number of bonds to be broken when the cluster is divided into two clusters. This viscosity is estimated by Simha’s cell model. Simha [17] gave the following viscosity equation for a mono-modal dense suspension.
g 2 ¼ 1 þ kðcÞ/ g0 5
ð2Þ
where c is the ratio of the radius of spherical particle, a, to the cell radius, b. The function k(c) is given as follows:
kðcÞ ¼
4ð1 c7 Þ 10 4ð1 þ c Þ 25c3 ð1 þ c4 Þ þ 42c5
ð3Þ
Simha proposed a model that the cell radius was proportional to the difference between the average distance of particles, R12 , and particle radius, a. Defining f3 as the supposed volume fraction of total cell P expressed by f 3 ¼ 4=3p R12;i =ðtotal slurry v olumeÞ, the following relationship is obtained for the case of dense suspension system.
c3 ¼ h f3
/ i3 1 /1=3 =f
ð4Þ
When the solid volume fraction is the same as the maximum packing volume fraction, c becomes unity, and /p;max ¼ f 3 =8. When the apparent viscosity of a dense suspension is experimentally determined, the value of k is assumed as the first step. Then, the void fraction in a cluster is calculated. This model was established under an assumption that the number of particles in a cluster tends to be uniform. This assumption is also adopted by Swift and Friedlander [2] and Higashitani et al. [4– 6] in their analyses. However, the cluster size is widely distributed in an actual suspension as reported by Komoda et al [16]. In such a case, Smoluchowski’s traditional coagulation model [1] is required as follows: i¼j1 1 X dnk 1 X bij ni nj bij ni nk ¼ 2 i¼1 dt i¼1
ð5Þ
iþj¼k
Here, nk[m3] is the concentration of the clusters composed of k particles. bij [m3 s1] is rate of Brownian or shear coagulation. According to Smoluchowski’s theory [1], bij can be expressed with rate constant, ab or as as follows for each Brownian or shear coagulation:
ri T t
ab, as c e g, g0 /, /0
radii of the clusters including i particles (m) absolute temperature (K) time (s) Brownian and shear coagulation constants (–) shear rate (s1) void fraction in a cluster (–) slurry and solvent viscosities (Pas) apparent and actual solid volume fractions (–)
2kb T ab 1 1 : Brownian coagulation ðri þ r j Þ þ ri rj 3l _ s 4ca ðri þ rj Þ3 : shear coagulation bij ¼ 3 bij ¼
ð6Þ ð7Þ
Here, ri[m] and rj[m] are radii of the clusters including i and j particles, respectively. The shear breakup model of clusters can be developed from Usui’s model [10]. In the model, a cluster is assumed to be separated to two clusters, whose total have the same volume as that of the original cluster. Then, the concentration of k cluster is decreased by such a mechanism and is increased by the breakup of the cluster including twice particles, 2k. We also assumed that an even cluster break into two equal size clusters but that an odd cluster breaks into two clusters including (k + 1)/2 and (k1)/2 particles, respectively. Then, the breakup rate by shear addition can be obtained as follows:
3 dnk 3 p c_ 2 g d0 k 1 nk ¼ 4 F 0 Nb ðk 1Þ 1 e dt 3 3 p c_ 2 g d0 2k þ 1 n2k 2 F 0 N b ð2k 1Þ 1 e 3 3 p c_ 2 g d0 2k 1 þ 1 n2k1 4 F 0 N b ð2k 2Þ 1 e 3 3 p c_ 2 g d0 2k þ 1 þ 1 n2kþ1 4 F 0 N b 2k 1e
ð8Þ
The final form on the concentration change rate of k cluster can be derived with coagulation terms discussed above as follows:
i¼j1 dnk 1 X 2kb T 1 1 ðr i þ r j Þ þ ¼ ab ni nj 2 3g0 ri rj dt i¼1 iþj¼k
ab
1 X 2kb T 1 1 ni nk ðr i þ r j Þ þ 3g0 ri rj i¼1
i¼j1 1 X 4c X 1 4c þ as ðr i þ r j Þ3 ni nj as ðri þ rj Þ3 ni nk 2 i¼1iþj¼k 3 3 i¼1 3 3 p c_ 2 g d0 k 1 nk 4 F 0 N b ðk 1Þ 1 e 3 3 p c_ 2 g d0 2k þ 1 n2k 2 F 0 N b ð2k 1Þ 1 e 3 3 p c_ 2 g d0 2k 1 þ 1 n2k1 4 F 0 N b ð2k 2Þ 1 e 3 3 p c_ 2 g d0 2k þ 1 þ 1 n2kþ1 4 F 0 N b 2k 1e
ð9Þ
This equation is based on that the energy of Nb bonds were broken simultaneously, when the cluster was separated into two clusters.
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2.2. Silica/EMMA slurry In this paper, the model analysis has been applied to the experimental data obtained by Komoda et al. [16]. Fine particles of silica were dispersed in ethylene methyl-meta acrylate copolymer (EMMA) in their experiments. The diameter of the silica particle ranges from 2.25 to 2.75 lm by experimental measurements, but d0 is assumed to be uniform at 2.5 lm in the analysis. The viscosity of EMMA, g0, is 190.5 Pas and showed a constant shear viscosity independently of shear rate (Newtonian fluid). The volume fraction of silica particles, u0 [–], was set at 0.15. The constant shear was added to the mixture at 120 °C, which is higher than the melting point of EMMA at 67 °C, by using the stress controlled cone-and-plate type rheometer (SR-5, Rheometric Scientific). The cone fixture has the diameter of 4 cm and the cone angle of 0.04 rad. The initial condition of cluster distribution was obtained by applying shear at the shear rate at 0.1 s1 for 1000 s. After the establishment of the initial condition, the shear rate applied to the mixture was changed to 0.2–10 s1 immediately and maintained for a certain time by using the rheometer control software. After the application of shear, the rotation of the cone fixture was stopped and the mixture was cooled down to 25 °C in a several minutes. As a result, the thin film of the mixture, in which particles in various dispersed states depending on shearing conditions were included, was obtained. The thin film peeled from the cone fixture was cut off by using a microtome to observe the cross sectional area. From the scanning electron microscope (SEM) image of the sliced cross-section, the cluster distribution was obtained at 10, 100 and 1000 s after the initial condition.
the averaged particle number in a cluster at the shear rate of 10 s1 goes down to 1.00. It is very close to the cluster limitation of silica particles obtained by Usui et al. [11] in the case of water disperse media. Though this could affect shear breakup rate, the average value of 5.0 1013 J is adopted in this study in order to compare the results with those obtained by Usui’s original model. 3.2. Time variation of cluster distribution Fig. 2a,b and c shows the time variations of cumulative particle number distribution in a cluster in cases of shear rates of 0.2, 1.0 and 10 s1, respectively. Fig. 3a,b and c shows the time variations of the mean number of particles in a cluster in the corresponding cases of shear rate of 0.2, 1.0 and 10 s1, respectively. In these fig-
(a) Shear rate at 0.2s-1 100 Cumulative frequency [%]
When we know the bonding energy between particles in agglomerates with the experimental data of the steady state viscosity of the slurry, we can calculate the time variation of the mean number of particles in agglomerates for each case of shear rate.
80 60 40 Shearing time 10s
20
100s 1000s
Initial
Present model Experiment
0
0
10
20
30
Number of particles in agglomerate[-]
(b) Shear rate at 1.0s-1
3.1. Boding energy Before using the present model, the bonding energy between particles was obtained by Usui’s original model [10] from a steady-state experimental data. Fig. 1 shows the bonding energy for each shear rate. The bonding energy calculated from the data is not constant and tends to increase with shear rate as shown in the figure. When the shear rate increases, the particle number in a cluster becomes low. Usui et al. [11] pointed out that there exists the limitation of cluster breakup by shear addition and improved their model considering the cluster size limitation. This is considered to correspond to the cluster limitation. As described later,
Cumulative frequency [%]
10 100
3. Results and discussion
80 60 40 Shearing time 10s
100s 1000s
Initial
Present model
20
Experiment
0
0
10 20 30 Number of particles in agglomerate[-]
1.0E-11
100
=0.15 φ =0.15 average
Cumulative frequency [%]
Interparticle bonding energy, F0 [J]
(c) Shear rate at 10s-1
1.0E-12
1.0E-13 0.1
1
10 -1
Shear rate [s ] Fig. 1. Bonding energy.
80 60 40 Shearing time 10s
100s 1000s
Initial
Present model
20
Experiment
0
0
10 20 30 Number of particles in agglomerate[-]
Fig. 2. Time-variations of cumulative particle number distributions in a cluster.
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ures, both results obtained experimentally and analytically are plotted. When low shear rate of 0.2 s1 is added, the cluster beak-up is found to occur very rapidly in the first stage from the experimental results. At 10 s, the median value of the particle number in a cluster becomes almost the half of the initial value. The time variation of the cluster distribution becomes low after that. Corresponding to this, the mean number of particle in a cluster and the viscosity vary very quickly in the first stage and the time variation becomes low after that. On the other hand, the results calculated by the present model gradually vary with time compared with the experimental data as shown in Fig. 2a, though the particle number distribution catches up to the experimental data at 1000 s and agrees well. Corresponding to this, the mean number of particles
10 Present model
(a) Shear rate at 0.2s-1 10000
Viscosity [Pa s]
Experiment
5
0
0
200 2
400 600 Time [s]
800
1000
Experiment
100
Present model 10
1000
0
(b) Shear rate at 1.0s-1 10
200
Present model
5
0
0
200 2
400
600
800
1000
Present model 1000
100
10
1000
0
200
400
600
800
1000
Time[s]
(c) Shear rate at 10s-1
(c) Shear rate at 10s-1
10
10000 Present model
Viscosity [Pa s]
Experiment
Experiment 5
0
800
Experiment
Experiment
Time [s]
Mean number of particles in cluster [-]
400 600 Time[s]
(b) Shear rate at 1.0s-1
10000
Viscosity [Pa s]
Mean number of particles in cluster [-]
Mean number of particles in cluster [-]
(a) Shear rate at 0.2s-1
in a cluster calculated by the model shows higher values than those obtained in experiments as shown in Fig. 3a. In the case of the shear rate of 1.0 s1, the time variation of cluster distribution agrees rather well with the experimental data in the first stage at 10 s as shown in Fig. 2b. However, the clusters in the model break up slightly quicker than those experimental data after 10 s. Thus, cumulative distribution shifts upper in the figure. From Fig. 3b, it is found that the mean particle number in a cluster reaches a steady value around 300 s. This tendency can be observed in the corresponding experimental data though the value at the steady state obtained by the model becomes lower than that of experimental data. In the case of the highest shear rate of 10 s1, it is from Fig. 2c found that the time variation is much quicker in any stage and that the calculated distribution already becomes steady before 10 s. This quick time variation of the cluster breakup is also observed in Fig. 3c. Thus, it is found that the time variation of cluster breakup obtained by the present model is slower at lower shear
0
200 2
400 600 Time [s]
800
1000
Fig. 3. Time-variations of the mean number of particles in a cluster.
Present model
1000
100
10
0
200
400
600
800
Time[s] Fig. 4. Time-variations of viscosity of the slurries.
1000
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3.3. Steady state values Figs. 7 and 8 show the steady state viscosity and the mean number of particles in a cluster. As described above, the viscosity and the mean number of particles in a cluster in present cases does not reach steady values at 1000 s in the case at the shear rate of 0.2 s1, but the values at 1000 s are plotted in the figure for comparison with experimental data. In this figure, the results calculated by Usui’s original model [9] is also plotted. From these figures, it is found that the present model can estimate the steady state viscosity and mean size of cluster as well as
Ratio of chain-like clusters[%]
100
0.2s -1 80
1.0s -1 10s -1
60 40 20 0 0 2 4 6 8 10 Number of particles in agglomerate[-]
Fig. 6. Ratio of chain-like clusters among all clusters.
Usui’s original model. Especially, the viscosity calculated by the present model agrees well with the experimental data. From these results including the above discussions, it can be concluded that the present model can estimate the steady state values as well as Usui’s original model, but cannot sufficiently estimate the time variations of cluster breakup. This is due to the over-estimation of the shear rate contribution to the shear break-up when the cluster is small as pointed above. Usui et al [11] pointed out that the
10000
Present model Viscosity [Pa s]
rate and faster at high shear rate compared with experimental data. Fig. 4a,b and c shows the time variations of viscosity of the slurries in the corresponding cases of shear rate of 0.2, 1.0 and 10 s1, respectively. Symbols designate the experimental data and solid line does the calculated data in these figures. In the present model, the slurry viscosities were evaluated with Simha’s cell model [17] as well as Usui’s model. As described above, the contribution of shear rate to the cluster break-up is over-estimated in the present model. At the shear rate of 0.2 s1, the calculated viscosity slowly decreases with time, while the experimental data shows steady state around 500 s. Though the time-dependency of the calculated viscosity agrees rather well with that of experimental data only at the shear rate of 1.0 s1, the calculated viscosity much faster reaches a plateau level in the case at 10 s1 than that of experimental data. These behaviors of viscosity correspond to the time variation of cluster distributions. In the present model, the shear breakup occurs proportionally to the square value of the shear rate, but the present results indicate the present power index to the shear rate on the shear breakup is too large. This power index was analytically obtained by Usui under an assumption that the cluster shape is sphere. However, the cluster shape is not always sphere in the cases when the cluster is small. Some particles form a chain-like shape cluster as shown in Fig. 5 of a SEM photo. Fig. 6 shows the ratio of chain-like clusters among all clusters obtained experimentally. From the figure, it is found that the ratio of the chain-like clusters becomes higher as the cluster becomes smaller. These chain-like clusters easily rotate by shear flow and the most of the fluid motion energy added to the cluster is spend to this rotating motion of the cluster. It is considered that this effect should be taken in the breakup model when the cluster becomes small as the present cases.
Usui model Experiment Experiment 1000
100 0.1
1
10
Shear rate [s-1]
Mean number of particles in cluster [-]
Fig. 7. Apparent viscosity at a steady state.
20
Present model Usui model
15
Experiment Experiment 10
5
0
0.1
1
10 -1
Shear rate [s ] Fig. 5. Example of SEM photo of clusters in EMMA.
Fig. 8. Mean number of particles in a cluster at a steady state.
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particle number in a cluster in some cases approach not to one but to a certain value even if high shear rate is added to some suspensions. Then, they suggest a model for estimating rheological characteristics of such suspensions, which limits the breakup of the small clusters. In this case, such limitation of cluster breakup might occur, and then the mean cluster number of experimental data does not reach unity. The present model should be improved for limiting the effect of shear rate and for taking such break-up limitation when the cluster size is small. 4. Conclusions In this paper, a model estimating the time-dependent distribution of cluster size consisting of fine particles in a Newtonian fluid was proposed. The present model for agglomerative suspension was developed based on Usui’s thixotropy model, derived by taking the balance between shear breakup, shear coagulation and Brownian coagulation process. The analysis was applied to the experimental results taken by Komoda et al. for silica/EMMA suspensions. The time variations of cluster size distributions, of viscosities and of the mean number of particles in a cluster were calculated and the results were compared with experimental results for the cases of shear rates from 0.2 to 10 s1when the solid volume fraction was set at 0.15. From the results, it was found that the present model can estimated the steady state values of the slurry viscosity and the mean particle numbers in a cluster as well as Usui’s original model. However, the model could not sufficiently express the time variation of rheological characteristics due to the over-estimation of the contribution of shear rate when the cluster size is small. This should be improved in the near-future. Additionally, the effect of volume fraction will be discussed in the next report.
References [1] M. Smoluchowski, Versuch einer mathematischen theorie der koagulationskinetik kolloider loesungen, Z. Phys. Chem. 92 (1917) 129–168. [2] D.L. Swift, S.K. Friedlander, The coagulation of hydrosols by brownian motion and laminar shear flow, J. Colloid. Sci. 19 (1964) 621–647. [3] S.K. Friedlander, C.S. Wang, The self-preserving particle size distribution for coagulation by Brownian motion, J. Colloid. Interfaces. Sci. 22 (1966) 126–132. [4] K. Higashitani, T. Tanaka, Y. Matsuno, A. kinematic, Interpretation on coagulation mechanism of hydrophobic colloids, J. Colloid. Interfaces. Sci. 63 (3) (1978) 551–560. [5] K. Higashitani, S. Miyafusa, T. Matsuda, Y. Matsuno, Axial change of total particle concentration in poiseuille flow, J. Colloid. Interfaces. Sci. 77 (1) (1980) 21–28. [6] K. Higashitani, R. Ogawa, G. Hosokawa, Y. Matsuno, Kinetic theory of shear coagulation for particles in a viscous fluid, J. Chem. Eng. Jpn. 15 (4) (1982) 299– 304. [7] D.S. Parker, W.J. Kaufman, D. Jenkins, Floc breakup in turbulent flocculation processes, J. Sanitary Eng. Division 98 (SA1) (1972) 79–99. [8] R.C. Sonntag, W.B. Russel, Structure and breakup of flocs subjected to fluid stresses I. Shear experiments, J. Colloid. Interfaces. Sci. 113 (2) (1986) 399–413. [9] R.C. Sonntag, W.B. Russel, Structure and breakup of flocs subjected to fluid stresses III. Converging flow, J. Colloid. Interfaces. Sci. 115 (2) (1987) 390–395. [10] H. Usui, A rheological model for an agglomerative slurry of mono-modal silica particles, Kagaku Kougaku Ronbunshu 25 (1999) 459–495. [11] H. Usui, K. Kishimoto, H. Suzuki, Non-Newtonian viscosity of dense slurries prepared by spherical particles, Chem. Eng. Sci. 56 (2001) 2979–2989. [12] H. Usui, L. Li, S. Kinoshita, H. Suzuki, Viscosity prediction of dense slurries prepared by non-spherical particles, J. Chem. Eng. Jpn. 34 (3) (2001) 360–368. [13] Mustafa, H. Usui, S. Ibuki, H. Suzuki, T. Kobayashi, Y. Miyazaki, T. Ioroi, K. Yasuda, Dispersion control of nano-particles and the effect of the coating condition on the performance of proton-exchange membrane fuel cells (PEMFCs), J. Chem. Eng. Jpn. 37 (1) (2004) 31–39. [14] Y. Komoda, K. Nakashima, H. Suzuki, H. Usui, Viscosity measuring technique for gas-solid suspensions, Adv. Powder. Tech. 17 (3) (2006) 333–343. [15] M. Kobayashi, Y. Adachi, S. Ooi, Breakup of fractal flocks in a turbulent flow, Langmuir 15 (1999) 4351–4356. [16] Y. Komoda, K. Kameyana, E. Hasegawa, H. Suzuki, H. Usui, Behavior of fine particle agglomerates in Newtonian molten polymer under a shear flow, Adv. Powder Tech., in press. [17] R. Simha, A treatment of the viscosity of concentrated suspensions, J. App. Phys. 23 (1952) 1020–1024.
Contents lists available at ScienceDirect
Advanced Powder Technology journal homepage: www.elsevier.com/locate/apt
Original Paper
Model analysis on dispersion characteristics of fine particles in Newtonian molten polymer Emi Hasegawa a, Hiroshi Suzuki b,*, Kananko Kameyama a, Yoshiyuki Komoda b, Hiromoto Usui b a b
Graduate School of Science and Technology, Kobe University, Kobe, Japan Department of Chemical Science and Engineering, Kobe University, Kobe, Japan
a r t i c l e
i n f o
Article history: Received 22 December 2007 Received in revised form 10 May 2008 Accepted 23 May 2008
Keywords: Particle agglomeration Cluster size distribution Thixotropy model Dispersion Shear breakup
a b s t r a c t In this paper, a model estimating the time-dependent distribution of cluster size consisting of fine particles in Newtonian molten polymer was proposed. The present model for agglomerative suspension was developed based on Usui’s thixotropy model, derived by taking the balance between shear breakup, shear coagulation and Brownian coagulation processes. The analysis was applied to the experimental results for silica/(ethylene methyl-meta acrylate copolymer) suspensions. The time variations of cluster size distributions, of viscosities and of the mean numbers of particles in a cluster were calculated and the results were compared with experimental results for the cases of shear rates from 0.2 to 10 s1 when the solid volume fraction was set at 0.15. From the results, it was found that the present model can estimated the steady state values of the slurry viscosity and the mean particle numbers in a cluster as well as Usui’s original model. However, the model could not sufficiently express the time variation of rheological characteristics due to the over-estimation of the contribution of shear rate when the cluster size is small. Ó 2008 The Society of Powder Technology Japan. Published by Elsevier BV and The Society of Powder Technology Japan. All rights reserved.
1. Introduction Dispersion characteristics of fine particles in solid–liquid slurries or suspensions are of importance in many industrial processes. The particles in solid-liquid slurries or suspensions coagulate due to the particle’s contact by the Brownian motions, the particle velocity difference induced by the fluid flow and by the other external forces, and so on. Smoluchowski [1] worked out the basic theory of the coagulation of the particles in such solid–liquid slurries or suspensions. Swift and Friedlander [2], and Friedlander and Wang [3] investigated theoretically and experimentally the simple shear coagulation in which Brownian coagulation occurs simultaneously. Higashitani et al. [4–6] also investigated the coagulation model and examined the validity of their model experimentally. On the other hand, Parker, Kaufman, and Jenkins [7] investigated the breakup mechanism in shear flows and showed the breakup mechanisms of surface erosion and splitting of a parent aggregate. Sonntag and Russel [8,9] suggested the relations between the floc size and the deformation rate experimentally. Among them, Usui [10] established a simple model considering Brownian and shear coagulations and shear breakup of clusters, in order to estimate apparent viscosity and bulk agglomeration characteristics of fine particle suspensions when the parent cluster of particles is splitting into smaller fragments by shear addition. It was developed for multi-modal systems (Usui et al. [11]) and applied to many * Corresponding author. E-mail address: [email protected] (H. Suzuki).
kinds of suspensions (Usui et al. [12], Mustafa et al. [13]). Komoda et al. [14] also revealed it can estimate solid-air systems. Kobayashi et al. [15] also developed shear breakup theory separately. However, these models assumed cluster size tends to be uniform with while the particles are coagulating and breaking up. In actual cases, the size of clusters including fine particles is widely distributed. Such distribution characteristics severely affect the functions and physical properties of some materials. Komoda et al. [16] experimentally investigated cluster distribution characteristics of fine particles in Newtonian molten polymer in order to clarify a shear breakup process of clusters. In this paper, a model estimating the time-dependent distribution of cluster size consisting of fine spherical particles in Newtonian fluids will be developed based on Usui’s thixotropy model [10] for data measured by Komoda et al. [16]. The results will be compared with their experimental data. From this, the present model will be evaluated.
2. Model analysis 2.1. Thixotropy model Usui [10] suggested a thixotropy model for analyzing suspension rheological characteristics as follows:
3 dk 4ab kb TN 0 4as /kc_ 3pd0 k k þ 1 gc_ 2 ¼ dt 4F 0 Nb 1 e 3g0 p
ð1Þ
0921-8831/$ - see front matter Ó 2008 The Society of Powder Technology Japan. Published by Elsevier BV and The Society of Powder Technology Japan. All rights reserved. doi:10.1016/j.apt.2008.05.001
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E. Hasegawa et al. / Advanced Powder Technology 20 (2009) 139–144
Nomenclature coagulation rate (m3 s1) diameter of particles (m) bonding energy (J) number of particles in a cluster (–) Boltzmann constant (J K1) total number of the primary particles per unit volume (–) number of bonds to be broken when the cluster is divided (–) concentration of the clusters composed of k particles (m3)
bij d0 F0 k kb N0 Nb nk
Here, t[s], k[–], kb[JK1], T[K], N0[m3], /[–], e[–], d0[m], g[Pas], g0[Pas] and c_ [s1] are time, the number of particles in a cluster, Boltzmann constant, absolute temperature, total number of particles in a unit volume, apparent solid volume fraction, void fraction in a cluster, diameter of particles, slurry viscosity, solvent viscosity and shear rate. ab[–] and as[–] are rate constants of Brownian coagulation and of shear coagulation obtained by Higashitani et al [4–6]. Fo[J] and Nb[–] are the bonding energy and the number of bonds to be broken when the cluster is divided into two clusters. This viscosity is estimated by Simha’s cell model. Simha [17] gave the following viscosity equation for a mono-modal dense suspension.
g 2 ¼ 1 þ kðcÞ/ g0 5
ð2Þ
where c is the ratio of the radius of spherical particle, a, to the cell radius, b. The function k(c) is given as follows:
kðcÞ ¼
4ð1 c7 Þ 10 4ð1 þ c Þ 25c3 ð1 þ c4 Þ þ 42c5
ð3Þ
Simha proposed a model that the cell radius was proportional to the difference between the average distance of particles, R12 , and particle radius, a. Defining f3 as the supposed volume fraction of total cell P expressed by f 3 ¼ 4=3p R12;i =ðtotal slurry v olumeÞ, the following relationship is obtained for the case of dense suspension system.
c3 ¼ h f3
/ i3 1 /1=3 =f
ð4Þ
When the solid volume fraction is the same as the maximum packing volume fraction, c becomes unity, and /p;max ¼ f 3 =8. When the apparent viscosity of a dense suspension is experimentally determined, the value of k is assumed as the first step. Then, the void fraction in a cluster is calculated. This model was established under an assumption that the number of particles in a cluster tends to be uniform. This assumption is also adopted by Swift and Friedlander [2] and Higashitani et al. [4– 6] in their analyses. However, the cluster size is widely distributed in an actual suspension as reported by Komoda et al [16]. In such a case, Smoluchowski’s traditional coagulation model [1] is required as follows: i¼j1 1 X dnk 1 X bij ni nj bij ni nk ¼ 2 i¼1 dt i¼1
ð5Þ
iþj¼k
Here, nk[m3] is the concentration of the clusters composed of k particles. bij [m3 s1] is rate of Brownian or shear coagulation. According to Smoluchowski’s theory [1], bij can be expressed with rate constant, ab or as as follows for each Brownian or shear coagulation:
ri T t
ab, as c e g, g0 /, /0
radii of the clusters including i particles (m) absolute temperature (K) time (s) Brownian and shear coagulation constants (–) shear rate (s1) void fraction in a cluster (–) slurry and solvent viscosities (Pas) apparent and actual solid volume fractions (–)
2kb T ab 1 1 : Brownian coagulation ðri þ r j Þ þ ri rj 3l _ s 4ca ðri þ rj Þ3 : shear coagulation bij ¼ 3 bij ¼
ð6Þ ð7Þ
Here, ri[m] and rj[m] are radii of the clusters including i and j particles, respectively. The shear breakup model of clusters can be developed from Usui’s model [10]. In the model, a cluster is assumed to be separated to two clusters, whose total have the same volume as that of the original cluster. Then, the concentration of k cluster is decreased by such a mechanism and is increased by the breakup of the cluster including twice particles, 2k. We also assumed that an even cluster break into two equal size clusters but that an odd cluster breaks into two clusters including (k + 1)/2 and (k1)/2 particles, respectively. Then, the breakup rate by shear addition can be obtained as follows:
3 dnk 3 p c_ 2 g d0 k 1 nk ¼ 4 F 0 Nb ðk 1Þ 1 e dt 3 3 p c_ 2 g d0 2k þ 1 n2k 2 F 0 N b ð2k 1Þ 1 e 3 3 p c_ 2 g d0 2k 1 þ 1 n2k1 4 F 0 N b ð2k 2Þ 1 e 3 3 p c_ 2 g d0 2k þ 1 þ 1 n2kþ1 4 F 0 N b 2k 1e
ð8Þ
The final form on the concentration change rate of k cluster can be derived with coagulation terms discussed above as follows:
i¼j1 dnk 1 X 2kb T 1 1 ðr i þ r j Þ þ ¼ ab ni nj 2 3g0 ri rj dt i¼1 iþj¼k
ab
1 X 2kb T 1 1 ni nk ðr i þ r j Þ þ 3g0 ri rj i¼1
i¼j1 1 X 4c X 1 4c þ as ðr i þ r j Þ3 ni nj as ðri þ rj Þ3 ni nk 2 i¼1iþj¼k 3 3 i¼1 3 3 p c_ 2 g d0 k 1 nk 4 F 0 N b ðk 1Þ 1 e 3 3 p c_ 2 g d0 2k þ 1 n2k 2 F 0 N b ð2k 1Þ 1 e 3 3 p c_ 2 g d0 2k 1 þ 1 n2k1 4 F 0 N b ð2k 2Þ 1 e 3 3 p c_ 2 g d0 2k þ 1 þ 1 n2kþ1 4 F 0 N b 2k 1e
ð9Þ
This equation is based on that the energy of Nb bonds were broken simultaneously, when the cluster was separated into two clusters.
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2.2. Silica/EMMA slurry In this paper, the model analysis has been applied to the experimental data obtained by Komoda et al. [16]. Fine particles of silica were dispersed in ethylene methyl-meta acrylate copolymer (EMMA) in their experiments. The diameter of the silica particle ranges from 2.25 to 2.75 lm by experimental measurements, but d0 is assumed to be uniform at 2.5 lm in the analysis. The viscosity of EMMA, g0, is 190.5 Pas and showed a constant shear viscosity independently of shear rate (Newtonian fluid). The volume fraction of silica particles, u0 [–], was set at 0.15. The constant shear was added to the mixture at 120 °C, which is higher than the melting point of EMMA at 67 °C, by using the stress controlled cone-and-plate type rheometer (SR-5, Rheometric Scientific). The cone fixture has the diameter of 4 cm and the cone angle of 0.04 rad. The initial condition of cluster distribution was obtained by applying shear at the shear rate at 0.1 s1 for 1000 s. After the establishment of the initial condition, the shear rate applied to the mixture was changed to 0.2–10 s1 immediately and maintained for a certain time by using the rheometer control software. After the application of shear, the rotation of the cone fixture was stopped and the mixture was cooled down to 25 °C in a several minutes. As a result, the thin film of the mixture, in which particles in various dispersed states depending on shearing conditions were included, was obtained. The thin film peeled from the cone fixture was cut off by using a microtome to observe the cross sectional area. From the scanning electron microscope (SEM) image of the sliced cross-section, the cluster distribution was obtained at 10, 100 and 1000 s after the initial condition.
the averaged particle number in a cluster at the shear rate of 10 s1 goes down to 1.00. It is very close to the cluster limitation of silica particles obtained by Usui et al. [11] in the case of water disperse media. Though this could affect shear breakup rate, the average value of 5.0 1013 J is adopted in this study in order to compare the results with those obtained by Usui’s original model. 3.2. Time variation of cluster distribution Fig. 2a,b and c shows the time variations of cumulative particle number distribution in a cluster in cases of shear rates of 0.2, 1.0 and 10 s1, respectively. Fig. 3a,b and c shows the time variations of the mean number of particles in a cluster in the corresponding cases of shear rate of 0.2, 1.0 and 10 s1, respectively. In these fig-
(a) Shear rate at 0.2s-1 100 Cumulative frequency [%]
When we know the bonding energy between particles in agglomerates with the experimental data of the steady state viscosity of the slurry, we can calculate the time variation of the mean number of particles in agglomerates for each case of shear rate.
80 60 40 Shearing time 10s
20
100s 1000s
Initial
Present model Experiment
0
0
10
20
30
Number of particles in agglomerate[-]
(b) Shear rate at 1.0s-1
3.1. Boding energy Before using the present model, the bonding energy between particles was obtained by Usui’s original model [10] from a steady-state experimental data. Fig. 1 shows the bonding energy for each shear rate. The bonding energy calculated from the data is not constant and tends to increase with shear rate as shown in the figure. When the shear rate increases, the particle number in a cluster becomes low. Usui et al. [11] pointed out that there exists the limitation of cluster breakup by shear addition and improved their model considering the cluster size limitation. This is considered to correspond to the cluster limitation. As described later,
Cumulative frequency [%]
10 100
3. Results and discussion
80 60 40 Shearing time 10s
100s 1000s
Initial
Present model
20
Experiment
0
0
10 20 30 Number of particles in agglomerate[-]
1.0E-11
100
=0.15 φ =0.15 average
Cumulative frequency [%]
Interparticle bonding energy, F0 [J]
(c) Shear rate at 10s-1
1.0E-12
1.0E-13 0.1
1
10 -1
Shear rate [s ] Fig. 1. Bonding energy.
80 60 40 Shearing time 10s
100s 1000s
Initial
Present model
20
Experiment
0
0
10 20 30 Number of particles in agglomerate[-]
Fig. 2. Time-variations of cumulative particle number distributions in a cluster.
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ures, both results obtained experimentally and analytically are plotted. When low shear rate of 0.2 s1 is added, the cluster beak-up is found to occur very rapidly in the first stage from the experimental results. At 10 s, the median value of the particle number in a cluster becomes almost the half of the initial value. The time variation of the cluster distribution becomes low after that. Corresponding to this, the mean number of particle in a cluster and the viscosity vary very quickly in the first stage and the time variation becomes low after that. On the other hand, the results calculated by the present model gradually vary with time compared with the experimental data as shown in Fig. 2a, though the particle number distribution catches up to the experimental data at 1000 s and agrees well. Corresponding to this, the mean number of particles
10 Present model
(a) Shear rate at 0.2s-1 10000
Viscosity [Pa s]
Experiment
5
0
0
200 2
400 600 Time [s]
800
1000
Experiment
100
Present model 10
1000
0
(b) Shear rate at 1.0s-1 10
200
Present model
5
0
0
200 2
400
600
800
1000
Present model 1000
100
10
1000
0
200
400
600
800
1000
Time[s]
(c) Shear rate at 10s-1
(c) Shear rate at 10s-1
10
10000 Present model
Viscosity [Pa s]
Experiment
Experiment 5
0
800
Experiment
Experiment
Time [s]
Mean number of particles in cluster [-]
400 600 Time[s]
(b) Shear rate at 1.0s-1
10000
Viscosity [Pa s]
Mean number of particles in cluster [-]
Mean number of particles in cluster [-]
(a) Shear rate at 0.2s-1
in a cluster calculated by the model shows higher values than those obtained in experiments as shown in Fig. 3a. In the case of the shear rate of 1.0 s1, the time variation of cluster distribution agrees rather well with the experimental data in the first stage at 10 s as shown in Fig. 2b. However, the clusters in the model break up slightly quicker than those experimental data after 10 s. Thus, cumulative distribution shifts upper in the figure. From Fig. 3b, it is found that the mean particle number in a cluster reaches a steady value around 300 s. This tendency can be observed in the corresponding experimental data though the value at the steady state obtained by the model becomes lower than that of experimental data. In the case of the highest shear rate of 10 s1, it is from Fig. 2c found that the time variation is much quicker in any stage and that the calculated distribution already becomes steady before 10 s. This quick time variation of the cluster breakup is also observed in Fig. 3c. Thus, it is found that the time variation of cluster breakup obtained by the present model is slower at lower shear
0
200 2
400 600 Time [s]
800
1000
Fig. 3. Time-variations of the mean number of particles in a cluster.
Present model
1000
100
10
0
200
400
600
800
Time[s] Fig. 4. Time-variations of viscosity of the slurries.
1000
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3.3. Steady state values Figs. 7 and 8 show the steady state viscosity and the mean number of particles in a cluster. As described above, the viscosity and the mean number of particles in a cluster in present cases does not reach steady values at 1000 s in the case at the shear rate of 0.2 s1, but the values at 1000 s are plotted in the figure for comparison with experimental data. In this figure, the results calculated by Usui’s original model [9] is also plotted. From these figures, it is found that the present model can estimate the steady state viscosity and mean size of cluster as well as
Ratio of chain-like clusters[%]
100
0.2s -1 80
1.0s -1 10s -1
60 40 20 0 0 2 4 6 8 10 Number of particles in agglomerate[-]
Fig. 6. Ratio of chain-like clusters among all clusters.
Usui’s original model. Especially, the viscosity calculated by the present model agrees well with the experimental data. From these results including the above discussions, it can be concluded that the present model can estimate the steady state values as well as Usui’s original model, but cannot sufficiently estimate the time variations of cluster breakup. This is due to the over-estimation of the shear rate contribution to the shear break-up when the cluster is small as pointed above. Usui et al [11] pointed out that the
10000
Present model Viscosity [Pa s]
rate and faster at high shear rate compared with experimental data. Fig. 4a,b and c shows the time variations of viscosity of the slurries in the corresponding cases of shear rate of 0.2, 1.0 and 10 s1, respectively. Symbols designate the experimental data and solid line does the calculated data in these figures. In the present model, the slurry viscosities were evaluated with Simha’s cell model [17] as well as Usui’s model. As described above, the contribution of shear rate to the cluster break-up is over-estimated in the present model. At the shear rate of 0.2 s1, the calculated viscosity slowly decreases with time, while the experimental data shows steady state around 500 s. Though the time-dependency of the calculated viscosity agrees rather well with that of experimental data only at the shear rate of 1.0 s1, the calculated viscosity much faster reaches a plateau level in the case at 10 s1 than that of experimental data. These behaviors of viscosity correspond to the time variation of cluster distributions. In the present model, the shear breakup occurs proportionally to the square value of the shear rate, but the present results indicate the present power index to the shear rate on the shear breakup is too large. This power index was analytically obtained by Usui under an assumption that the cluster shape is sphere. However, the cluster shape is not always sphere in the cases when the cluster is small. Some particles form a chain-like shape cluster as shown in Fig. 5 of a SEM photo. Fig. 6 shows the ratio of chain-like clusters among all clusters obtained experimentally. From the figure, it is found that the ratio of the chain-like clusters becomes higher as the cluster becomes smaller. These chain-like clusters easily rotate by shear flow and the most of the fluid motion energy added to the cluster is spend to this rotating motion of the cluster. It is considered that this effect should be taken in the breakup model when the cluster becomes small as the present cases.
Usui model Experiment Experiment 1000
100 0.1
1
10
Shear rate [s-1]
Mean number of particles in cluster [-]
Fig. 7. Apparent viscosity at a steady state.
20
Present model Usui model
15
Experiment Experiment 10
5
0
0.1
1
10 -1
Shear rate [s ] Fig. 5. Example of SEM photo of clusters in EMMA.
Fig. 8. Mean number of particles in a cluster at a steady state.
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particle number in a cluster in some cases approach not to one but to a certain value even if high shear rate is added to some suspensions. Then, they suggest a model for estimating rheological characteristics of such suspensions, which limits the breakup of the small clusters. In this case, such limitation of cluster breakup might occur, and then the mean cluster number of experimental data does not reach unity. The present model should be improved for limiting the effect of shear rate and for taking such break-up limitation when the cluster size is small. 4. Conclusions In this paper, a model estimating the time-dependent distribution of cluster size consisting of fine particles in a Newtonian fluid was proposed. The present model for agglomerative suspension was developed based on Usui’s thixotropy model, derived by taking the balance between shear breakup, shear coagulation and Brownian coagulation process. The analysis was applied to the experimental results taken by Komoda et al. for silica/EMMA suspensions. The time variations of cluster size distributions, of viscosities and of the mean number of particles in a cluster were calculated and the results were compared with experimental results for the cases of shear rates from 0.2 to 10 s1when the solid volume fraction was set at 0.15. From the results, it was found that the present model can estimated the steady state values of the slurry viscosity and the mean particle numbers in a cluster as well as Usui’s original model. However, the model could not sufficiently express the time variation of rheological characteristics due to the over-estimation of the contribution of shear rate when the cluster size is small. This should be improved in the near-future. Additionally, the effect of volume fraction will be discussed in the next report.
References [1] M. Smoluchowski, Versuch einer mathematischen theorie der koagulationskinetik kolloider loesungen, Z. Phys. Chem. 92 (1917) 129–168. [2] D.L. Swift, S.K. Friedlander, The coagulation of hydrosols by brownian motion and laminar shear flow, J. Colloid. Sci. 19 (1964) 621–647. [3] S.K. Friedlander, C.S. Wang, The self-preserving particle size distribution for coagulation by Brownian motion, J. Colloid. Interfaces. Sci. 22 (1966) 126–132. [4] K. Higashitani, T. Tanaka, Y. Matsuno, A. kinematic, Interpretation on coagulation mechanism of hydrophobic colloids, J. Colloid. Interfaces. Sci. 63 (3) (1978) 551–560. [5] K. Higashitani, S. Miyafusa, T. Matsuda, Y. Matsuno, Axial change of total particle concentration in poiseuille flow, J. Colloid. Interfaces. Sci. 77 (1) (1980) 21–28. [6] K. Higashitani, R. Ogawa, G. Hosokawa, Y. Matsuno, Kinetic theory of shear coagulation for particles in a viscous fluid, J. Chem. Eng. Jpn. 15 (4) (1982) 299– 304. [7] D.S. Parker, W.J. Kaufman, D. Jenkins, Floc breakup in turbulent flocculation processes, J. Sanitary Eng. Division 98 (SA1) (1972) 79–99. [8] R.C. Sonntag, W.B. Russel, Structure and breakup of flocs subjected to fluid stresses I. Shear experiments, J. Colloid. Interfaces. Sci. 113 (2) (1986) 399–413. [9] R.C. Sonntag, W.B. Russel, Structure and breakup of flocs subjected to fluid stresses III. Converging flow, J. Colloid. Interfaces. Sci. 115 (2) (1987) 390–395. [10] H. Usui, A rheological model for an agglomerative slurry of mono-modal silica particles, Kagaku Kougaku Ronbunshu 25 (1999) 459–495. [11] H. Usui, K. Kishimoto, H. Suzuki, Non-Newtonian viscosity of dense slurries prepared by spherical particles, Chem. Eng. Sci. 56 (2001) 2979–2989. [12] H. Usui, L. Li, S. Kinoshita, H. Suzuki, Viscosity prediction of dense slurries prepared by non-spherical particles, J. Chem. Eng. Jpn. 34 (3) (2001) 360–368. [13] Mustafa, H. Usui, S. Ibuki, H. Suzuki, T. Kobayashi, Y. Miyazaki, T. Ioroi, K. Yasuda, Dispersion control of nano-particles and the effect of the coating condition on the performance of proton-exchange membrane fuel cells (PEMFCs), J. Chem. Eng. Jpn. 37 (1) (2004) 31–39. [14] Y. Komoda, K. Nakashima, H. Suzuki, H. Usui, Viscosity measuring technique for gas-solid suspensions, Adv. Powder. Tech. 17 (3) (2006) 333–343. [15] M. Kobayashi, Y. Adachi, S. Ooi, Breakup of fractal flocks in a turbulent flow, Langmuir 15 (1999) 4351–4356. [16] Y. Komoda, K. Kameyana, E. Hasegawa, H. Suzuki, H. Usui, Behavior of fine particle agglomerates in Newtonian molten polymer under a shear flow, Adv. Powder Tech., in press. [17] R. Simha, A treatment of the viscosity of concentrated suspensions, J. App. Phys. 23 (1952) 1020–1024.