Ind. Eng. Chem. Res. 2007, 46, 2413-2422
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Analysis of Polymerization in Chaotic Mixers Using Time Scales of Mixing and Chemical Reactions Changdo Jung, Sadhan C. Jana,* and I. Sedat Gunes Department of Polymer Engineering, The UniVersity of Akron, Akron, Ohio 44325-0301
This study investigated the suitability of chaotic mixers as polymerization reactors by taking chain extension of thermoplastic polyurethanes as an example. The performance of chaotic mixers was analyzed in terms of the effects of mixer designs and mixer operating conditions on polymer molecular weight and properties. Specifically, the analysis involved computation of time scales of mixing and chemical reactions and finding their relationship to mixing torque, polymer molecular weight, and mechanical and thermal properties. It was found that the time scale of mixing has a strong dependence on the Liapunov exponent, a parameter used to characterize the degree of chaotic mixing. The results showed that the highest polymer molecular weight is obtained when the mixer operates under globally chaotic conditions and when the magnitudes of the time scales of mixing and chemical reactions are made comparable to each other. The study also showed that hard segment phase separation can reduce the value of the reaction rate constant and hence hinder the progress of polymerization. 1. Introduction Thermoplastic polyurethanes (TPU) are synthesized from diisocyanates, prepolymers, and short or long chain diols by step-growth polymerization. The prepolymer and short chain diols are not readily miscible in some polyurethane formulations. An example is presented in Figure 1, where the butanediol chain extender is seen dispersed as droplets in the prepolymer. In such cases, extended chain polymers are formed due to reactions between isocyanate (-NCO) and hydroxyl (-OH) functional groups across the material interfaces. Efficient mechanical mixing is necessary to generate material interfaces.1-4 The benefits of material interface generation in reactive polymer systems by mechanical mixing have been established in a number of experimental reports. A back-and-forth application of shear was found more efficient than one-way shear in achieving high conversion during esterification of the maleic anhydride (MA) residue in hydrogenated styrene-butadienestyrene block copolymer (SEBS-g-MA) with poly(caprolactone diol) (PCL) in a parallel plate rheometer.5 Shear-induced interface generation between immiscible polymers polystyrene (PS) and poly(methyl methacrylate) (PMMA) also significantly increased the extent of reactions between terminal -OH groups on PS and grafted -NCO groups on modified PMMA.6,7 These studies indicate that the exponentially fast intermaterial area generation feature of chaotic mixing can greatly augment the rate of polymerization and the rates of reaction between functional polymers. The exponentially fast intermaterial area generation feature of chaotic mixing was first established from initial work involving nonreactive passive tracers. It was also established that stable and unstable manifolds of hyperbolic periodic points govern the dynamics of mixing and that horseshoe type mixing microstructures indicate the presence of chaos.8-15 A major emphasis in the early work on chaotic mixing was given on the construction of chaotic flows in simple prototype mixing devices, for example, cavity flow,11 a journal bearing,16 vortex mixing flow,17 and continuous chaotic mixers.18 Subsequent * To whom correspondence should be addressed. E-mail:
[email protected]. Telephone: (330) 972-8293. Fax: (330) 258-2339.
Figure 1. Optical micrographs of droplets of BD chain extender dispersed in prepolymer of PPG-2000 polyol and methylene diisocyanate after the components were hand mixed at room temperature (a) without tin catalyst and (b) with 2.26 × 10-4 mol/L tin catalyst. The mixtures were later subjected to mixing and polymerization in the chaotic mixer.
studies considered blending of immiscible polymers19-23 in chaotic mixers and reported lamellae, fibrils, and droplet type morphological forms. A significant reduction in coalescence was also observed under chaotic mixing conditions.24 Reactive systems have been considered only in a handful of cases,25-34 a majority of which dealt with numerical experiments. Muzzio and co-workers25,29,32 and Cox33 solved convective diffusion-reaction equations to study evolution of reactant concentrations in two-dimensional model chaotic flows for bimolecular single,25,32 competitive-consecutive,25,33 and parallelcompetitive29 reactions. These investigators observed more efficient formation of desired products in chaotic regions. Chaotic mixing also enhanced reaction rates of elementary reactions as reported for two-dimensional27 and three-dimensional26 model chaotic flows. A recent experimental study also reported significant reaction rate increase in triiodide-thiosulphate oxidation-reduction reactions in a helical coil chaotic mixer reactor.34 Although it is widely accepted that higher polymer viscosity limits the scope of turbulent flows in polymerization reactors and that chaotic mixing presents a much better alternative, no single study considered polymerization reactions in chaotic flows. The following attributes make numerical studies of polymerization in chaotic mixers unwieldy. First, the viscosity and power-law index of polymeric fluids change with conversion which may lead to changes in hydrodynamic conditions, such as increase of Reynolds number and development of non-
10.1021/ie0613319 CCC: $37.00 © 2007 American Chemical Society Published on Web 03/20/2007
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Figure 2. Stretching of a typical droplet of chain extender in the chaotic mixer. (a) Initial droplet location. (b) Numerically computed stretched droplet after one period of chaotic flow. (c) The area in the rectangular block in part b is highlighted. Chemical reactions between prepolymer and chain extender occur at the interface. Local direction of stretching (x1) and the direction of concentration variation across the thickness of the droplet (x2) are shown.
Newtonian rheology. Second, increase of viscosity during polymerization deters molecular diffusion while most numerical studies on chaotic mixing-chemical reactions considered constant values of the molecular diffusion coefficient. Third, polymerization processes are non-isothermal because of the large exothermic heat of reaction; consequently, appropriate energy balance equations must be taken into account. Fourth, an experimental verification of species concentration distribution is difficult in most cases. To the best of our knowledge, only one study32 presented visual comparison of numerically computed mixing structures with experiments based on the acidbase neutralization reaction. In view of these difficulties associated with full scale numerical simulation of polymerization in chaotic mixers, we placed our emphasis more on experiments in this study. TPUs were synthesized in chaotic mixers, and the values of torque developed during polymerization, polymer molecular weight, and thermal and mechanical properties of the materials were analyzed using the numerically computed time scale of mixing and experimentally determined time scale of chemical reactions. As will be seen in the following section, the exponential intermaterial area generation feature of chaotic mixers is included in the time scale of mixing in the form of a mean value of the Liapunov exponent. 2. Time Scales of Mixing and Chemical Reactions The time scale of chemical reaction (trxn) between -NCO and -OH groups can be defined as
trxn )
1 kcat[NCO]0r
(1)
where [NCO]0 is the initial concentration of the isocyanate group coming from the prepolymer, r ≡ [NCO]0/[OH]0 is the sto-
ichiometric ratio, and [OH]0 is the initial concentration of -OH groups coming from the chain extender. The rate of urethane forming reactions is considered to be second-order, especially for aromatic isocyanates35,36 considered in the present work:
-
d[NCO] ) k[Cat]a[NCO][OH] dt
(2)
In eq 2, [OH] and [NCO] are respectively the molar concentrations of the -OH and -NCO functional groups and [Cat] is the molar concentration of catalyst. The apparent rate constant kcat can be expressed as kcat ≡ k[Cat]a, where a is a constant and k is the reaction rate constant with Arrhenius type dependence on temperature. These catalyzed chemical reactions can be quantitatively characterized by Fourier transform infrared (FT-IR) spectroscopy.37 Reflecting on the immiscibility of the prepolymer and chain extender seen in Figure 1, the rate of polymerization should depend strongly on the degree of mixing of the reactants, especially if molecular diffusion is slow. Figure 2 presents a typical scenario involving three competing processes, such as molecular diffusion, reaction, and mixing by stretching. The chain extender phase initially present as droplets (Figures 1 and 2a) is stretched by the application of chaotic flow into elongated filaments (Figure 2b). Note that, as a consequence, the concentration gradient inside the filament increases which induces higher diffusive flux of chain extender molecules to the interface (along the x2 direction in Figure 2c). In addition, fresh molecules of the chain extender are exposed to the interface via increased intermaterial area. Therefore, it is noted that the time scales of stretching and molecular diffusion both determine the concentration of chain extender molecules at the interface.
Raynal and Gence38 derived expressions for time scales associated with stretching and molecular diffusion for nonreacting systems. They considered a fluid element undergoing shear in two-dimensional chaotic flow with the principal stretching direction, x1, and compression direction, x2, and noted that molecular diffusion becomes important after a time when the value of dGi/dt in the fluid element vanishes. In this case, Gi ≡ ∂CA/∂xi, i ) 1, 2, is the concentration gradient of species A (e.g., -OH groups of chain extender in the present work). The mass balance of species A in the fluid element can be recast in the following form:
dGi ∂Vj ) -Gj + DAB∆Gi dt ∂xi
(3)
where Vj is the jth component of the velocity vector v, ∆Gi is the Laplacian of Gi, and DAB is molecular diffusion coefficient. A reaction term similar to eq 2 can be incorporated in eq 3 to obtain a form suitable for the present work:
( ) ( )
( )
2 ∂CA ∂ ∂ ∂ CA ∂ ∂CA + V2 ) DAB ∂t ∂x2 ∂x2 ∂x2 ∂x2 ∂x 2 2 ∂CA ∂CB - kcatCA (4) kcatCB ∂x2 ∂x2
where CA and CB are respectively concentrations of reactants A (e.g., -OH groups of chain extender) and B (e.g., -NCO groups of prepolymer) at time t and DAB is the molecular diffusion coefficient. It is assumed in eq 4 that the change in the concentration of A in the fluid element occurs only in the x2 direction, that is, thickness direction (see also Figure 2c). For initial concentrations of A and B, CA0 and CB0, respectively, and CA0 ) rCB0, where r is the stoichiometric ratio, eq 4 can be rewritten as
( )
( ) ( )
( )
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( )
2 ∂V2 ∂pA ∂ ∂ pA ) DAB - kcatCA0[(r - 1) + ∂x2 ∂x2 ∂x2 ∂x 2 2
The actual expression for tmix depends on the relative values of the terms on the right-hand side of eq 5. Each term in eq 6 has the units of 1/(m‚s), and an appropriate scaling can be found such as
( )
DAB kcatrCA0L )f V VL
∂CA (5) ∂x2
kcat[CA0(r - 1) + 2CA]
The left-hand side of eq 5 represents the rate of increase of concentration gradient in the fluid element (e.g., of chain extender in Figure 2) as the fluid element is stretched in the chaotic mixer. The three terms on the right-hand side of eq 5 represent respectively the rate of decrease of concentration gradient in the stretched fluid element due to molecular diffusion, increase due to stretching, and decrease due to consumption of A in the bimolecular reaction with B. If mixing is continued, a time is reached at which (d/dt)(∂CA/∂x2) ∼ 0 and the cumulative changes in concentration of A in the fluid element due to molecular diffusion, chemical reaction, and stretching vanish. This time can be used to define a time scale of mixing tmix, which can be interpreted as follows. The mechanical stirring of ingredients should be continued for time tmix by which time the domains containing reactant A become thin enough to trigger appreciable molecular diffusion of A and an appreciable rate of chemical reaction between A and B. At t ) tmix, eq 5 can be rewritten in terms of conversion of species A, pA ≡ (CA0 - CA)/CA0, as
(7)
where two dimensionless groups Π1 ) kcatrCA0L/V and Π2 ) VL/DAB are related by function f, with L and V as characteristic length and velocity. Raynal and Gence38 considered a case without chemical reactions and found the expression of tmix given in eq 8.
tmix ∼
( )
λL2 1 ln 2λ DAB
(8)
In the present work, both molecular diffusion and the rate of reaction were considered important. To derive an appropriate expression of tmix we consider a domain of reactant A (e.g., the chain extender), with an initial thickness d0, undergoing stretching at an exponential rate to the current thickness d at time, t:
d ) d0e-λt; λ > 0
(9)
In eq 9, λ is the Liapunov exponent which can be computed from the knowledge of mixing conditions and the mixer geometry. The time scale of mixing tmix for this case is given as (see the appendix)
tmix ∼
2 ∂V2 ∂CA d ∂CA ∂ ∂ CA ) DAB 2 dt ∂x2 ∂x2 ∂x ∂x2 ∂x2 2
∂pA (6) ∂x2
2(1 - pA)]
1 ln 2λ
[ ( )] L2/DAB 1 λ kcatrCA0
2
(10)
where the length scale L is the same as the initial fluid element thickness d0 or the initial diameter of chain extender droplets as seen in Figure 1. 3. Experimental Section 3.1. Materials. The prepolymer was synthesized from 4,4′diphenylmethane diisocyanate (MDI, Mondur M, Bayer, molecular weight (Mw) ) 250, Tm ) 40 °C) and poly(propylene glycol) (PPG) polyol. Two grades of PPG with number-averaged molecular weights of 1025 g/mol (Arcol PPG-1025) and 2080 g/mol (Arcol PPG-2000) were obtained from Bayer Materials Science (Pittsburgh). The chemical structure and some properties of polyols are presented in Table 1. Table 2 presents prepolymer molecular weight and shear viscosity measured at 80 °C using an ARES (TA Instruments) plate-plate rotational rheometer in oscillatory shear mode. The shear viscosity of prepolymer was found to remain constant up to a frequency of 100 rad/s. The chain extension reaction between prepolymer and chain extender 1,4-butanediol (BD, Fischer Scientific) was catalyzed by dibutyl tindilaurate (DABCO T-120, Air Products). The following concentrations of catalyst were used in an effort to vary the values of kcat and hence trxn: 0.56 × 10-4, 1.13 × 10-4, and 2.26 × 10-4 mol/L. Polyol and BD were dried at 80
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Table 1. Structure and Properties of Polyether Polyola
polyol grade
structure
hydroxyl viscosity Mw number at 25 °C (g/mol) (mg KOH/g) (cP)
Arcol PPG-1025
1025
110
155
Arcol PPG-2000
2000
55
335
a The data on molecular weight (M ), hydroxyl number, and viscosity w were obtained from the supplier.
Table 2. Prepolymer Molecular Weight and Complex Viscosity (|η*|)a source of polyol prepolymer Mn (g/mol) prepolymer Mw (g/mol) prepolymer |η*|; Pa‚s at 80 °C
PPG-1025
PPG-2000
2800 3800 0.92
4800 6500 0.82
a
Number-averaged (Mn) and weight-averaged (Mw) molecular weight were determined by GPC.
Figure 3. Sketch of chaotic mixer. (a) Side and (b) cross-sectional views are presented with circular rotors of radius Ri. Cross sectional view with Werner-Pfleiderer neutral kneading block elements is presented in part c. Mixing gap (d), speed (Ω1 and Ω2), and the direction of rotation of each rotor are shown.
°C under vacuum for 24 h and treated with 4 Å molecular sieves to remove moisture before use. The prepolymer was synthesized by reacting MDI with polyol in molar ratios of NCO/OH ) 2.1 at 80 °C under nitrogen purge in a three-neck round-bottom flask equipped with an oil bath, a mechanical stirrer, and a thermometer. The isocyanate content was monitored by titration with di-n-butylamine (ASTM D1638-74), and the reaction was found to be complete in 2 h in the case of PPG-1025 and 4 h in the case of PPG-2000. 3.2. Chaotic Mixing. A schematic of the chaotic mixer is shown in Figure 3a. The mixer produced two-dimensional chaotic mixing due to time-periodic motion of rotors with a circular cross section (Figure 3b) or due to rotation at a constant speed of the rotors with a non-circular cross section, as shown in Figure 3c. In the latter case, the ZSK30 neutral kneading block elements of Werner-Pfleiderer twin-screw extruders were used. The dimensions of the kneading block elements were 90° × 5 mm × 20 mm, which represented respectively the angle between two successive kneading disks, the thickness of each disk, and the width of the kneading block. The radii of the
mixing chamber (R0) and rotors of circular cross section (Ri) were respectively 40.6 mm and 12.7 mm, which created a mixing gap of 7.62 mm. In the rest of the paper, mixer 1 and mixer 2 will be used to denote respectively the mixers with circular rotors and kneading block rotors. Both mixers produced the same effective peak linear speed at the rotor surface; the “effective” diameter of the kneading blocks was the same as the diameter of circular rotors. The equivalent effective diameter of the kneading block was determined from the volume of the mixing gap which in turn was determined by the water displacement method. The non-intermeshing nature of noncircular rotors (Figure 3c) helped produce a two-dimensional shear flow in mixer 2. Chaotic mixing was produced by varying the speed of circular rotors in a sinusoidal waveform in that the angular speeds of rotor 1 (Ω1) and rotor 2 (Ω2) were varied with time t as
(
Ω1 ) Ω 1 + cos
2πt 2πt ; Ω2 ) Ω 1 - cos ; 0eteT T T (11)
)
(
)
In eq 11, Ω is the amplitude in degrees per second and T is the time period in second. In this paper, the results are reported for values of angular displacement of each rotor in a period θ (≡ΩT) of 720°, 1440°, and 2880°. The peak linear speed at the rotor surface was 0.106 m/s, and the time periods T were 3 s for θ ) 720°, 6 s for 1440°, and 12 s for 2880°. The peak rotor speed produced peak shear rates of 9.5 s-1 and 5.4 s-1 respectively at the rotor surface and at the mixing chamber wall. Accordingly, the time-averaged shear rates at the rotor surface and at the chamber walls were respectively 4.75 s-1 and 2.7 s-1 for circular rotors. The Werner-Pfleiderer neutral kneading rotors were rotated at constant angular speed such that the peak linear speed at the “effective” rotor surface was the same as for the circular rotors. Mixing was continued in each case for total times of 10 and 15 min. A torque transducer recorded torque on one of the rotors during mixing, and a thermocouple recorded the polymer temperature. A detailed description of the chaotic mixer used in this study can be found elsewhere.21 The prepolymer, chain extender BD, and catalyst were premixed by hand for 30 s at room temperature and poured into the mixer preheated at 80 °C. The molar ratio of NCO/OH was maintained at 1.05, which gave a value of r ) 0.95 and hard segment contents in extended polymer of respectively 38% and 23% in the system with PPG-1025 and PPG-2000. Figure 1 shows that BD droplets of 5-50 µm diameter were dispersed in prepolymer before chaotic mixing. Approximately 60 g of polymer was produced in one batch. 3.3. Kinetics of Chain Extension. The reaction rate constant of chain extension reactions between prepolymer and BD was determined at 60-90 °C from FT-IR spectroscopy of the hand mixed initial mixture of prepolymer, BD, and catalyst kept between two preheated KBr discs. The reaction temperature was controlled to within (1.0 °C using a proportional-integralderivative (PID) controller attached to a temperature-controlled cell (HT-32, Spectra-Tech). Excalibur, Series FTS 3000, Biorad/ Digilab FT-IR with a resolution 4 cm-1 was used in transmission mode to obtain IR spectra. 3.4. TPU Characterization. The molecular weight of polymer was determined by gel permeation chromatography (GPC). Thermal properties were determined using differential scanning calorimetry (DSC) with a Dupont TA calorimeter. The DSC scans were taken from -100 °C to 250 °C under nitrogen atmosphere at a heating rate of 10 °C/min. The soft segment glass transition temperatures (Tg) were recorded during the
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Figure 4. Evolution of absorbance of -NCO groups with time during chain extension reaction at 80 °C between prepolymer based on PPG-1025 and BD. Catalyst concentration was 1.13 × 10-4 mol/L.
second heating scan to ensure identical thermal histories in each case. Tensile properties were evaluated at 23 °C according to ASTM D 638 type V method with a cross-head speed of 50 mm/min. 4. Results and Discussion 4.1. Kinetic Constants. A representative FT-IR spectrum presented in Figure 4 shows that the absorbance of -NCO groups at 2270 cm-1 reduced during chain extension between prepolymer and BD. The absorbance of the urethane carbonyl band at 1705 cm-1 and -NH stretching band at 3350 cm-1 were also monitored and were seen to increase with time. The conversion of -NCO groups was computed by monitoring the changes in the area under the peak of the -NCO stretching band at 2270 cm-1 with time:
R)1-
ANCO/ACH ANCO,0/ACH
(12)
In eq 12, ANCO,0 and ANCO are respectively the areas under the -NCO peak at time t ) 0 and time t ) t, while ACH is the area under the peak of -CH2 stretching, which remained constant during reaction and was used as an internal standard. The reaction rate was determined from eq 2, and the value of R was calculated from eq 12 at temperatures between 60 and 80 °C for various catalyst concentrations. It was found that the value of a in kcat ) k[Cat]a is unity, and the values of kcat followed an Arrhenius relationship kcat ) A exp(-Ea/RT). The corresponding values of the pre-exponential factor (A) and the Arrhenius activation energies (Ea) are listed in Table 3. It is seen that at a given reaction temperature, the values of kcat are much lower for the polyol of molecular weight 1025 than for the polyol of molecular weight 2000, although the natures of alcoholic -OH functional groups are the same in both cases. Such differences in the values of kcat can be attributed to a higher degree of phase separation of urethane hard segments during chain extension reactions in the former, as is discussed below. It is known that hard segment phase separation due to hydrogen bond formation between urethane NsH and CdO groups is influenced by both hard segment content and hard segment structures.39 4.2. Hard Segment Phase Separation. The extent of hard segment phase separation was determined from the areas under the peaks of hydrogen bonded (Ab) and non-hydrogen-bonded
Figure 5. Evolution of free (1730 cm-1) and hydrogen bonded (1705 cm-1) carbonyl peaks with conversion of -NCO groups during chain extension of prepolymer of (a) PPG-2000 and (b) PPG-1000. Reaction temperature and catalyst concentrations were respectively 80 °C and 1.13 × 10-4 mol/ L. Table 3. Pre-Exponential Factor (A) and Arrhenius Activation Energy (Ea) of Apparent Rate Constant kcata type of polyol in prepolymer
catalyst concentration (mol/L)
T (°C)
kcat (L/(mol‚min))
Ea/R (K)
PPG-2000
0.56 × 10-4 0.56 × 10-4 0.56 × 10-4
70 80 90
1.2 2.5 4.1
7661 7661 7661
6.2 × 108 6.2 × 108 6.2 × 108
1.13 × 10-4 1.13 × 10-4 1.13 × 10-4
60 70 80
1.0 2.0 4.5
8831 8831 8831
3.2 × 1011 3.2 × 1011 3.2 × 1011
2.26 × 10-4 2.26 × 10-4 2.26 × 10-4
60 70 80
2.6 4.7 9.8
7788 7788 7788
3.6 × 1010 3.6 × 1010 3.6 × 1010
1.13 × 10-4 1.13 × 10-4 1.13 × 10-4
60 70 80
0.9 1.5 2.3
5518 5518 5518
1.42 × 107 1.42 × 107 1.42 × 107
2.26 × 10-4 2.26 × 10-4 2.26 × 10-4
60 70 80
1.3 2.2 3.3
5480 5480 5480
1.855 × 107 1.855 × 107 1.855 × 107
PPG-1025
a
pre-exponential factor A (L/(mol‚min))
The universal gas constant is R.
(Af) absorbance bands of carbonyl groups in FT-IR spectra as proposed by other investigators.40-42 The peaks for hydrogenbonded CdO groups at 1705 cm-1 and free CdO groups at 1730 cm-1 43 were monitored during chain extension reactions. Figure 5 shows how the absorbance of free and hydrogen-
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Figure 6. Variation of Ab/Af ratio for CdO stretching absorption bands with conversion during polyurethane formation between prepolymer of PPG polyols and BD at 80 °C. Catalyst concentrations was 1.13 × 10-4 mol/L.
bonded carbonyl groups changed with conversion of -NCO groups in chain extension of prepolymer based on PPG-1025 and PPG-2000 polyols. The values of conversion of -NCO groups were computed using eq 12. The hard segment contents in extended polymer were 38% and 23% respectively for the systems with PPG-1025 and PPG-2000. It is evident that the peak height of hydrogen-bonded CdO at 1705 cm-1 increased with reaction time in both cases, indicating that more urethane linkages participated in phase separation with the progress of chain extension reactions. Figure 6 shows almost a linear increase of the Ab/Af ratio with conversion. Figure 5 also shows that the tendency for phase separation is higher at higher hard segment content, that is, in the case of polymer containing PPG1025, which can be attributed to the incompatibility between the hard and soft segment phases. This is also evident from Figure 6: the Ab/Af ratio for the polymer of PPG-1025 is approximately 50% higher than that for the polymer of PPG2000. Now recall that the reaction rate constant for chain extension is much lower for prepolymer of PPG-1025 (Table 3), although the nature of alcoholic -OH functional groups are the same in both cases (see structure in Table 1). These indicate that the lower rate of chain extension reactions observed for prepolymer of PPG-1025 originated from a higher degree of phase separation. It is conceivable that phase separated domains trap some -NCO and -OH functional groups, which do not participate in chain extension reactions. This was earlier observed in the case of reaction of injection-molded polyurethanes.40 Note that the final conversion in Figure 6 was approximately 95% in both cases although reaction time was much longer for prepolymer of PPG-1025. It is evident from Figure 5b that phase separation of hard segments began very early, for example, after an -NCO conversion of 2%. Such phase separation also increased the viscosity as was noted by Castro et al.44 and hence caused additional reduction of diffusivity of functional groups to reaction sites. The trend observed in Figures 5 and 6 at 80 °C was also observed at 110 °C, a temperature at which most polymerization occurred in the chaotic mixers, although the set temperature during chaotic mixing was 80 °C. In view of these, it can be inferred that polymerization in the chaotic mixer was affected by phase separation. 4.3. Effect of Mixer Design and Catalyst Concentration. Representative plots showing variation of material temperature and mixing torque as a function of reaction time in mixer 1 and mixer 2 for two catalyst concentrations are presented in Figure 7. A value of θ ) 1440° was used in experiments
Figure 7. Representative plots of torque and temperature during chain extension reactions in mixer 1 and mixer 2. Prepolymer was based on PPG2000 polyol. (a) Catalyst concentration ) 1.13 × 10-4 mol/L, (b) catalyst concentration ) 2.26 × 10-4 mol/L. θ ) 1440° in the case of mixer 1. Table 4. Values of trxn and tmix Computed from kcat Values at 80 °C and Values of tmax Obtained from Torque vs Time Curvesa
polyol PPG-2000
catalyst concentration (×10-4 mol/L) 0.56 1.13 2.26
PPG-1025
1.13
mixer mixer 1 mixer 2 mixer 1 mixer 2 mixer 1 mixer 2 mixer 1 mixer 2
trxn (min) 0.51 0.28 0.129 0.55
tmix (min)
tmax (min)
0.03 N/A 0.079 N/A 0.139 N/A 0.027 N/A
11 11 6.3 7 3.5 5 1.7 2.3
a The initial concentration of isocyanate group: C A0 ) 0.75 mol/L. Ratio of isocyanate and hydroxyl group initial concentration: r )1.05. Molecular diffusivity DAB: 3.3 × 10-8 cm2/s. The value of λT ∼ 1.3 with T ) 6 s for mixer 1 with θ ) 1440°. The value of tmix was not computed for mixer 2 and is specified with an entry N/A.
involving mixer 1, which, as will be shown later, presented conditions for best chaotic mixing in mixer 1. The time at which maximum torque was reached (tmax) is a reflection of the rate of polymerization. The values of tmax are presented in Table 4. The torque in mixer 1 with circular rotors underwent timeperiodic fluctuations similar to the speed of the rotor as in eq 11. However, the peak torque value, corresponding to the peak shear rate, gradually increased with time as a result of an increase of viscosity, which in turn increased as a result of conversion of isocyanate groups into extended chain polymers and possibly as a result of hard segment phase separation44 as discussed in the previous section. A decrease in torque at later times can be attributed to such factors as higher density of
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polymer or partial slip between the rotor surface and the polymer. A partial slip, characterized by a large sudden drop in torque, did not occur in this study. The values of torque can be correlated to viscosity of the polymer by the following relationship:45
torque ) Cη0Nn
(13)
η0 ) B(T)Mw3.4
(14)
In eqs 13 and 14, Mw is the weight-averaged molecular weight, η0 is zero-shear viscosity, which is a function of temperature (T) and molecular weight (Mw), C is a characteristic constant which can be experimentally determined for a given mixer geometry, N is the rotor speed, n is a parameter in the powerlaw equation between shear stress (τ) and rate of strain (γ˘ ), τ ) Kγ˘ n, with K as consistency index, and B(T) is a temperaturedependent factor. It is assumed in eq 14 that polymer chains were entangled and that the viscosity varied with molecular weight as Mw3.4, although Castro et al.44 observed stronger dependence of viscosity on molecular weight. A stronger dependence may originate from additional entanglement in the form of physical cross-links presented by the phase separated hard segment domains. However, the contribution of phase separation to increased viscosity was not investigated in this study. Instead, we assumed that torque directly depended on molecular weight of the polymer and that such dependence may originate from chain entanglement both with and without hard segment phase separation. In view of this assumption and the results discussed in section 4.2 that the extent of hard segment phase separation increased with conversion, the values of tmax, that is, time to reach a maximum of torque, can be related to the extent of conversion of functional groups and, therefore, to the effectiveness of mixing. In view of this and the values of tmax reported in Table 4, several statements can be made about the polymer system considered. First, mixing became more effective at higher catalyst concentrations as a result of shorter time scales of chemical reactions. Note that the value of tmax was the same (11 min) for mixer 1 and mixer 2 for a catalyst concentration of 0.56 × 10-4 mol/L, indicating that chain extension reactions were kinetically controlled at this catalyst concentration and mixing had almost no effect. Second, it is apparent that mixer 1 was much more effective in the chain extension than mixer 2, especially at higher catalyst concentrations. For example, the time to reach maximum torque, tmax, in mixer 1 was 30% shorter than in mixer 2 at a catalyst concentration of 2.26 × 10-4 mol/L which corresponded to trxn ∼ 0.129 min for chain extension of prepolymer based on PPG2000. Although tmax and trxn were expected to decrease monotonically with catalyst concentration, the catalyst concentration was not increased beyond 2.26 × 10-4 mol/L to avoid side reactions, such as the formation of allophanates and biurets, especially in light of the polymerization temperature reaching 95-120 °C. We can now turn our attention to the relationship between tmix and trxn. The time scale of mixing (tmix) was obtained using a typical droplet diameter of 40 µm (Figure 1) as a representative length scale L in eq 10. The value of molecular diffusivity DAB of prepolymer based on PPG-2000 was accepted to be 3.3 × 10-8 cm2/s after modifying a typical value of DAB ) 1 × 10-6 cm2/s used in modeling of reaction injection molding46 with the ratio (∼30) of viscosity of prepolymer (0.8 Pa‚s at 80 °C, Table 2) and BD (0.03 Pa‚s at 25 °C). Note also that phase separation occurring in the early stages of reactions, for example,
with ∼10-20% conversion (Figure 5), may cause further reduction of molecular diffusivity of -NCO groups. The values of CA0 and r are given in Table 4. In light of the competing processes depicted in Figure 2, the expression of tmix in eq 10 also includes trxn as defined in eq 1 and the diffusion time scale, tdiff ∼ L2/DAB. However, such dependence is logarithmic and, therefore, weak; the values of tmix are primarily dictated by the value of the Liapunov exponent38 and hence by the degree of chaotic mixing. A mean value of Liapunov exponent (λ) was computed from the stretching distribution of passive tracers in the chaotic mixer following a procedure described in refs 21 and 47. A fluid element of initial length l0 is stretched to a length ln, after n periods of the chaotic flow. These two lengths can be related as
ln ) eλnT l0
(15)
where λ is the value of Liapunov exponent and T is the time period. Note that fluid elements experience different stretching depending upon the initial location in the mixer and due to the inhomogeneous nature of shear flows used in most chaotic mixers. Thus chaotic mixers are often characterized by distribution of Liapunov exponents.16 However, in the spirit of a simple approach adopted in this study, we resorted to using a mean value of Liapunov exponents computed from the data of ref 21: the mean values of the product λT were found to be 0.5 for θ ) 720° and 1.3 for θ ) 1440°. The veracity of such computation as applied to the present study was derived from the fact that the extended chain polymers formed at the interface between the droplets of BD and prepolymer act as “compatibilizers” (Figure 2c), and thus the droplets of BD can be safely assumed as passive tracers. In addition, the small volume fraction (
