Ind. Eng. Chem. Res. 2003, 42, 2415-2422
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Liquid Viscosity Model for Polymer Solutions and Mixtures Yuhua Song, Paul M. Mathias, David Tremblay, and Chau-Chyun Chen* Aspen Technology, Inc., Ten Canal Park, Cambridge, Massachusetts 02141
A simple liquid viscosity model for multicomponent mixtures containing polymers is presented. This model is essentially a new mixing rule for calculating the Newtonian viscosity of mixtures over the entire composition range using the pure-component viscosities. A modified MarkHouwink model is applied to calculate the Newtonian viscosity of pure polymer melts, and the Andrade/DIPPR correlation model is used to calculate the viscosity of pure conventional components. Two binary parameters, one symmetric and one antisymmetric, are introduced to capture nonideal mixing behavior of Newtonian viscosity in polymer solutions. Applications to polystyrene solutions and poly(ethylene glycol) solutions are presented. Introduction One of the most important transport properties in polymerization processes is the viscosity of polymer solutions. An accurate quantitative knowledge of the viscosity is essential for the detailed design and modeling of equipment, such as piping, pumps, and heat exchangers. The viscosity of mixtures that contain polymers is much more difficult to predict than the viscosity of mixtures with low molecular weight components. First of all, polymers, which are orders of magnitude larger than smaller molecules, have substantially more spatial conformations than small molecules. Second, polymers are essentially mixtures of polymer components with varying chain length, chain chemical composition, degree of branching, and so on. In other words, polymers are polydisperse. All of these factors result in the so-called non-Newtonian behavior in polymer solutions, which exhibit considerably stronger sensitivity to temperature, composition, and shear rate.1-3 Because of the complexity of describing the viscosity of polymer solutions, the shear-rate dependence is usually separated from other variables such as temperature, composition, and molecular weight. Therefore, for describing polymer solutions, the following form is generally accepted:
η ) η0(T,Mw,c) g(γ˘ )
(1)
where η is the viscosity of polymer solutions, η0 is the so-called zero-shear viscosity or Newtonian viscosity of polymer solutions, and g(γ˘ ) represents the shear-rate (γ˘ ) dependence on the viscosity. In eq 1, T is the temperature, Mw is the weight-average molecular weight of the polymer, and c is the polymer concentration in the mixture. Equation 1 is applicable to low and moderate shear-rate conditions (say, γ˘ e 50 s-1 for polystyrene (PS) solutions4 as shown in a later section) in which the shear-rate dependence on viscosity may not be significant compared to the composition effects.1,2 In this paper, we focus on the variation of the viscosity in response to polymer molecular weights, composition, * To whom correspondence should be addressed. Tel.: 617949-1202. Fax: 617-949-1030. E-mail: chauchyun.chen@ aspentech.com.
and system temperature, and we ignore g(γ˘ ) (set to 1). In other words, we focus on the Newtonian viscosity of polymer solutions. However, a later section is devoted to a brief discussion of the shear-rate dependence on viscosity. The description of the Newtonian viscosity of polymer solutions usually occurs at two different starting conditions: dilute polymer solutions (nearly pure solvents) and concentrated polymer solutions (polymer melts with small amounts of dissolved monomers). The two approaches are significantly different. For instance, the limiting viscosity number is usually used to describe the Newtonian viscosity of dilute polymer solutions in terms of the polymer concentration. In this category, the most popular model is the Huggins equation,5 in which the viscosity of the solvent is expressed as a polynomial of the polymer concentration and the limiting viscosity number:
ηs ) [η]c + kH [η]2c2
(2)
where ηs is the viscosity of the solvent, c is the polymer concentration, kH is the Huggins constant, and [η] is the limiting viscosity number:
[η] ) ηsp /c, when c f 0
(3)
with
ηsp )
η0 - ηs ηs
(4)
However, the Huggins model works well only at very dilute conditions and, therefore, many modifications have been proposed to improve its accuracy. For instance, Rudin et al.6 added higher terms to the Huggins equation for the limiting viscosity number and the polymer concentration. For concentrated polymer solutions, the well-known Mark-Houwink equation1 (or its many variations) for polymer melts is usually applied because the Newtonian viscosity of concentrated polymer solutions exhibits characteristics similar to those of polymer melts. The influences of parameters such as molecular weight and temperature on the Newtonian viscosity are largely similar. A power law of the polymer concentration is
10.1021/ie030023x CCC: $25.00 © 2003 American Chemical Society Published on Web 04/24/2003
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normally introduced to include the composition dependence on the Newtonian viscosity:1,3
η0 ∝ ηpcκ
(5)
where ηp is the Newtonian viscosity of polymer melts and κ is an index with a positive value; ηp can be calculated by a Mark-Houwink-type equation:
ηp ) ηcr
( ) ( ) Mw Mcr
R
exp
Eη RT
(6)
with
R ) R1
for Mw > Mcr
R ) R1
for Mw e Mcr
where ηcr is the critical viscosity, Mw is the weightaverage molecular weight of the polymer, Eη is the activation energy of viscous flow, and R is the universal gas constant. In eq 6, Mcr is the critical molecular weight of the polymer, at which the Newtonian viscositymolecular weight dependence changes as a result of entanglement. The parameter R is the exponential factor that accounts for the polymer molecular weight effect and may be R1 ()3.4) or R2 ()1) depending on whether the polymer is above or below the critical molecular weight.1 Within data regression, it can simply be treated as an adjustable parameter. Equation 6 has been successfully used to calculate the Newtonian viscosity of polymer melts as well as concentrated polymer solutions when the composition dependence is included (e.g., eq 5).1 Although a great deal of work has been devoted to describing the Newtonian viscosity of dilute polymer solutions and concentrated polymer solutions,1-3 relatively little attention has been paid to the broad range of polymer composition between these two extreme conditions. It is this region, however, that is important in polymerization processes, especially in solution polymerization processes where the polymer must be concentrated or dried to remove residual solvents. In this paper, we present a model for calculating the Newtonian viscosity of polymer solutions over the entire composition range. This model is essentially a new mixing rule for calculating the Newtonian viscosity of mixtures from the pure-component viscosities. The model assumes that the viscosities of all pure components in the mixture are already available. For polymer melts, we present a modified Mark-Houwink model to calculate the Newtonian viscosity. For solvents, we use the Andrade/DIPPR model7,8 to calculate the viscosity. Two binary parameters, one symmetric and one antisymmetric, are introduced to capture the nonideal mixing behavior of Newtonian viscosity in polymer solutions. The model has been successfully applied to a wide variety of mixtures, polymers, or nonpolymers, with any number of components. Applications to PS solutions and poly(ethylene glycol) (PEG) solutions are also presented. Modified Mark-Houwink Model We begin by introducing a modified Mark-Houwink model to describe the Newtonian viscosity of polymer melts. It is well understood that the Newtonian viscosity of polymer melts varies with the polymer structural
Figure 1. Newtonian viscosity of PS as functions of temperature at two weight-average molecular weights, Mw ) 79 000 and 299 000 g/mol. The data points are taken from the compilation by Kim and Nauman.9 The curves are calculated from eq 8 using the parameters and constants given in Table 1.
characteristics and system conditions. Among them, the polymer molecular weight and temperature dependencies on viscosity are most important.1,2As we already pointed out in the proceeding section, the well-known Mark-Houwink equation, or one of its variations (such as eq 6), is commonly used to calculate the Newtonian viscosity of polymer melts. However, in eq 6, the preexponential factor, i.e., the critical viscosity ηcr, is temperature-dependent and thus strongly coupled with the viscosity activation energy parameter Eη. This dependency makes eq 6 difficult to use in data regression. We introduce a reference state to separate the dependency between ηcr and Eη in eq 6. At this reference state, ηref, the polymer viscosity at Tref, reference temperature, and Mref, the reference polymer molecular weight, are known. Therefore, a more practical form of the Mark-Houwink model can be written as
ηp ) ηref
( ) [ ( Mw Mref
R
exp
)]
Eη 1 1 R T Tref
(7)
or
ln ηp ) ln ηref + R ln
( ) (
Eη 1 Mw 1 + Mref R T Tref
)
(8)
with
R ) R1
for Mw > Mcr
R ) R2
for Mw e Mcr
This modified Mark-Houwink model has two adjustable parameters, R and Eη, and four specified constants, Tref, Mref, ηref, and Mcr, for a given polymer. Literature data for the Newtonian viscosity of polymer melts are scarce. Here we apply eq 8 to correlate the Newtonian viscosity data for two polymers, PS and PEG. Figure 1 shows the results of the correlation of the Newtonian viscosity data for PS at two selected weight-average molecular weights: 79 000 and 299 000 g/mol. The experimental data of Newtonian viscosity of PS melts are taken from the compilation by Kim and
Ind. Eng. Chem. Res., Vol. 42, No. 11, 2003 2417 Table 1. Model Parameters and Constants (Equation 8) for PS parameter
value
parameter
value
Tref (K)a Mref (g/mol)a ηref (Pa‚s)a Mcr (g/mol)b
423.10 52 000 22 996.01 35 000
R1 (Mw > Mcr)c R2 (Mw e Mcr)d Eη (J/mol)c
3.1751 1 190 927.79
a
The reference state is arbitrarily chosen as a single data point;9 this specification does not affect the values of adjustable parameters, R and Eη. b The critical molecular weight is taken from van Krevelen’s book.1 c The values for R1 (Mw > Mcr) and Eη are determined by regression. d Because there are no data at Mw e Mcr, the value for R2 is set ) 1.
Table 3. Commonly Used Forms (Equation 9) for Calculating the Viscosity of Mixtures 1 2
f(η)
f(ηi)
η ln η
ηi ln ηi
Table 2. Model Parameters and Constants (Equation 8) for PEG parameter
value
parameter
value
Tref (K)a Mref (g/mol)a ηref (Pa‚s)a Mcr (g/mol)b
298.15 223.68 48157 3400
R1 (Mw > Mcr)c R2 (Mw e Mcr)c Eη (J/mol)d
3.4 1 40499.22
a The reference state is arbitrarily chosen as a single data point;10 this specification does not affect the values of adjustable parameters, R and Eη. b The critical molecular weight for PEG is not available from the literature and is set equal to the value for poly(ethylene oxide).1 c The values for R1 and R2 are preset. d The value for Eη is determined by regression.
Nauman9 for both PS melts and solutions. The model parameters, which are regressed from 63 data points, and constants are summarized in Table 1. The fit is cal fairly good, and the average error, |(ln ηexp p - ln ηp )|, is exp 9 0.538 (where ln ηp is calculated from the data and ln ηcal p is calculated from eq 8). Figure 2 shows the correlation of the Newtonian viscosity data for PEG at two weight-average molecular weights: 223.68 and 447.168 g/mol. The 14 experimental data points of Newtonian viscosity are taken from the recent publication of Ottani et al.10 The model parameters and constants in eq 8 are listed in Table 2. Because the viscosity is measured at low molecular weights, the value for R is preset and only the value for Eη is adjusted in regression. The agreement is good, and - ln ηcal the average error, |(ln ηexp p p )|, is only 0.024.
f(η)
f(ηi)
1/η 1/ln η
1/ηi 1/ln ηi
Composition Dependence on the Polymer Solution Viscosity The composition dependence on the Newtonian viscosity of polymer solutions is more difficult to analyze. For concentrated polymer solutions, the Newtonian viscosity exhibits characteristics similar to those of polymer melts. The influences of parameters such as molecular weight and temperature on the Newtonian viscosity are largely similar. As stated in the Introduction, however, we want to develop a model to calculate the Newtonian viscosity of polymer solutions over the entire range of composition using the viscosities of pure components in the mixture and a minimal number of adjustable parameters. This approach usually involves the use of mixing rules to obtain the actual value of the mixture. We begin by analyzing some mixing rules used for calculating the viscosity of liquid mixtures of nonpolymer systems. Linear Mixing Rules for the Viscosity of Liquid Mixtures. The most widely used mixing rule for calculating the viscosity of liquid mixtures from purecomponent viscosities is the so-called ideal linear mixing in terms of the mole fractions:
f (η0) ) Figure 2. Newtonian viscosity of PEG as functions of temperature at two weight-average molecular weights, Mw ) 223.68 and 447.168 g/mol. The data points are taken from the measurement by Ottani et al.10 The curves are calculated from eq 8 using the parameters and constants given in Table 2.
3 4
∑i xi f (ηi)
(9)
where f is a form to be determined for the viscosity and xi is the mole fraction of component i in the mixture. Table 3 lists some commonly used forms for f. Equation 9 usually gives a reasonable prediction for mixtures only when the nonideal mixing effect is not important. This applies to mixtures containing similar components, such as paraffins of similar size, and it is sometimes called the “ideal-solution viscosity”. In practice, form 2 from Table 3 is the most commonly used. However, it has been shown that the linear mixing rule given by eq 9 leads to large errors when applied to mixtures containing very different components, especially in terms of different molecular weights and sizes. This is not surprising, however, because eq 9 completely misses the contribution from the nonideal mixing effect to the viscosity of mixtures. Application of Linear Mixing Rules to Polymer Solutions. It is obvious that the linear mixing rule in eq 9 is not applicable to polymer solutions where there is a large difference in molecular size between polymers and solvents. However, we use eq 9 as a starting point to analyze the nonideal mixing effect to the Newtonian viscosity of polymer solutions. For both practical and accuracy purposes, it is more appropriate to use the weight fractions instead of mole fractions for polymer solutions. Therefore, we rewrite the linear mixing rule in eq 9 for polymer solutions in terms of the weight fractions:
f (η0) )
∑i wi f (ηi)
(10)
where wi is the weight fraction of component i in the mixture. Following the general conclusion of liquid
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Figure 3. Newtonian viscosity of PS-styrene solutions at T ) 313.15 K and Mw ) 366 000 g/mol. The data points are taken from the compilation by Kim and Nauman.9 The dashed line is the ideal linear mixing from eq 11. The curve is the correlation by the current model, eq 22, with a single set of two binary parameters: k12 ) -1.7649 and l12 ) 1.8556.
Figure 4. Newtonian viscosity of PS-styrene solutions at T ) 333.15 K and Mw ) 366 000 g/mol. The data points are taken from the compilation by Kim and Nauman.9 The dashed line is the ideal linear mixing from eq 11. The curve is the correlation by the current model, eq 22, with a single set of two binary parameters: k12 ) -1.7649 and l12 ) 1.8556.
mixtures, we use form 2 of f from Table 3 for polymer solutions:
ln η0 )
∑i wi ln ηi
(11)
We expect that eq 11 works reasonably well only in two extreme conditions: very dilute and highly concentrated polymer solutions. Under these conditions, the nonideal mixing effect between polymers and solvents becomes negligible. Figures 3 and 4 show the Newtonian viscosity of PSstyrene solutions at temperatures of 313.15 and 333.15 K. The dashed lines are the results from the ideal linear mixing, as calculated from eq 11. The Newtonian viscosity of PS is calculated using eq 8 with the parameters and constants given in Table 1. The viscos-
Figure 5. Newtonian viscosity of PEG-1,3-dioxolane solutions at T ) 303.15 K and Mw ) 223.68 g/mol. The data points are taken from the measurement by Ottani et al.10 The dashed line is the ideal linear mixing from eq 11. The curve is the correlation by the current model, eq 22, with a single set of two binary parameters: k12 ) -0.7689 and l12 ) 0.0378.
Figure 6. Newtonian viscosity of PEG-1,3-dioxolane solutions at T ) 303.15 K and Mw ) 447.168 g/mol. The data points are taken from the measurement by Ottani et al.10 The dashed line is the ideal linear mixing from eq 11. The curve is the correlation by the current model, eq 22, with a single set of two binary parameters: k12 ) -0.7689 and l12 ) 0.0378.
ity of styrene is calculated using the Andrade/DIPPR correlation.8 The data points of the Newtonian viscosity for PS-styrene solutions are taken from the compilation by Kim and Nauman.9 The large difference between the linear mixing assumption and data, especially at the middle range of composition, is obvious at both temperatures. Figures 5 and 6 show the Newtonian viscosity of PEG-1,3-dioxolane solutions at a single temperature of 303.15 K but at two different weight-average molecular weights: 223.68 and 447.168 g/mol. The notations are the same as those for PS-styrene solutions in Figures 3 and 4. Namely, the dashed lines are the results from the linear mixing, as calculated from eq 11. The Newtonian viscosity data of pure PEG and 1,3dioxolane, as well as their solutions, are all taken from the same source by Ottani et al.10 Once again, the large
Ind. Eng. Chem. Res., Vol. 42, No. 11, 2003 2419
Figure 7. Nonideal mixing behavior of the Newtonian viscosity for PS-styrene solutions at T ) 313.15 K and Mw ) 366 000 g/mol. The points are calculated from eq 14, ∆(ln η0). The Newtonian viscosity data of PS-styrene solutions are taken from the compilation by Kim and Nauman.9 The units of viscosities used in eq 14 are Pa‚s.
Figure 8. Nonideal mixing behavior of the Newtonian viscosity for PS-styrene solutions at T ) 333.15 K and Mw ) 366 000 g/mol. The points are calculated from eq 14, ∆(ln η0). The Newtonian viscosity data of PS-styrene solutions are taken from the compilation by Kim and Nauman.9 The units of viscosities used in eq 14 are Pa‚s.
difference between the linear mixing assumption and data, especially at the middle range of composition, is obvious at both molecular weights. Nonideal Mixing Behavior of the Polymer Solution Viscosity. To analyze the nonideal mixing behavior, we modify eq 11 by introducing an extra term as follows:
ln η0 )
∑i wi ln ηi + ∆(ln η0)
(12)
where ∆(ln η0) represents the nonideal mixing effect to the Newtonian viscosity of polymer mixtures. If we apply eq 12 to binary polymer-solvent solutions, we obtain
ln η0 ) (1 - wp) ln ηs + wp ln ηp + ∆(ln η0) (13) or
∆(ln η0) ) ln η0 - [(1 - wp) ln ηs + wp ln ηp] (14) where wp is the weight fraction of the polymer, ηs is the viscosity of the solvent, and ηp is the Newtonian viscosity of the polymer. The nonideal mixing behavior can be easily calculated by eq 14 by using the Newtonian viscosity data of polymer solutions and subtracting the contribution from the linear mixing. Figures 7 and 8 show the nonideal mixing term of the Newtonian viscosity for PS-styrene solutions at temperatures of 313.15 and 333.15 K, respectively. Both figures display similar behavior. That is, except at very dilute solutions, the nonideal mixing effect reduces the viscosity from the ideal linear mixing. The reduction reaches a maximum value (negative) in magnitude at a composition of wp > 0.5, and then the nonideal mixing effect decreases when the polymer concentration increases. At very high polymer concentrations, the nonideal mixing effect fades and the Newtonian viscosity of polymer solutions reaches the Newtonian viscosity of polymer melts. In both Figures 7 and 8, the positive deviations from the ideal linear mixing at the very dilute
Figure 9. Nonideal mixing behavior of the Newtonian viscosity for PEG-1,3-dioxolane solutions at T ) 303.15 K and Mw ) 223.68 g/mol. The points are calculated from eq 14, ∆(ln η0). The Newtonian viscosity data of PEG-1,3-dioxolane solutions are taken from the measurement by Ottani et al.10 The units of viscosities used in eq 14 are Pa‚s.
polymer solution region may be caused by the influence of changing from isolated polymers in solutions to entangled polymers in solutions. Figures 9 and 10 show the nonideal mixing behavior of the Newtonian viscosity for PEG solutions at a temperature of 303.15 K but with two Mw values: 223.68 and 447.168 g/mol, respectively. Both figures also display behavior similar to that shown for PS-styrene solutions in Figures 7 and 8. Description of Nonideal Mixing Behavior. As shown in Figures 7-10, the nonideal mixing behavior has a significant contribution to the Newtonian viscosity of polymer solutions. In both polymer solutions, the nonideal mixing terms all show a maximum value in magnitude in terms of the polymer weight fraction at the given temperature and molecular weight of the
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However, as Mathias et al.11 pointed out, a form like eq 19 cannot be directly generalized to multicomponent mixtures because it is not invariant when a component is divided into two or more identical subcomponents. They proposed a formulation to overcome the problem, and we use their formulation to account for the asymmetric shape of the nonideal mixing behavior of the Newtonian viscosity for multicomponent mixtures. Using the notation in this paper, we can write the formulation as follows:
D)
Figure 10. Nonideal mixing behavior of the Newtonian viscosity for PEG-1,3-dioxolane solutions at T ) 303.15 K and Mw ) 447.168 g/mol. The points are calculated from eq 14, ∆(ln η0). The Newtonian viscosity data of PEG-1,3-dioxolane solutions are taken from the measurement by Ottani et al.10 The units of viscosities used in eq 14 are Pa‚s.
polymer. Therefore, a quadratic (or higher order) mixing rule is needed to account for the influence of nonideality on the Newtonian viscosity of polymer solutions. We first consider the standard quadratic form in terms of weight fractions:
∆(ln η0) )
kij wi wj ln ηij ∑ j>i
(15)
where kij is a symmetric binary parameter and ln ηij is a cross binary term from viscosities of pure components i and j. A proper form for ln ηij is chosen as follows:
ln ηij ) |ln ηi - ln ηj |/2
(16)
so that ln ηij f 0 when ηi f ηj . Applying eqs 15 and 16 to binary polymer (1)-solvent (2) solutions, we obtain
∆(ln η0) ) k12(1 - wp)wp(ln ηp - ln ηs)/2
(17)
Equation 17 gives a maximum value in magnitude at wp ) 0.5. In other words, the nonideal mixing effect described by eq 17 shows a symmetric shape over the entire composition. As shown in Figures 7-10, however, the shape of the nonideal mixing behavior over the composition is asymmetric and its maximum position in magnitude is shifted to the polymer-rich side. Obviously, we need to modify eq 17 to account for the asymmetric shape of the nonideal mixing behavior. One way to accomplish this is to include the higher order composition dependence in eq 17. For binary polymersolvent solutions, we can include an additional term in eq 17 as follows:
∆(ln η0) ) k12(1 - wp)wp(ln ηp - ln ηs)/2 + D (18) with
D ) (1 - wp)wp[(1 - wp)l12 + wpl21](ln ηp - ln ηs)/2 (19) where l12 is an antisymmetric binary parameter, i.e., l21 ) -l12.
[
wj (lij ln ηij )1/3 ∑i wi ∑ j*i
]
3
(20)
where lij is the antisymmetric binary parameter of the unlike pair of components i and j, i.e., lji ) -lij. It can be easily shown that eq 20 reduces to eq 19 for binary mixtures, and it is invariant when a component is divided into two or more identical subcomponents. Combining eqs 11, 15, and 20, we obtain the general expression for calculating the Newtonian viscosity of multicomponent mixtures:
ln η0 )
kij wi wj ln ηij + ∑i wi ln ηi + ∑ j>i
[
wj (lij ln ηij)1/3 ∑i wi ∑ j*i
]
3
(21)
where the first term is the ideal linear mixing, the second term is the binary symmetric-quadratic mixing, and the last term is the binary antisymmetric mixing. For binary polymer (1)-solvent (2) solutions, eq 21 reduces to
ln η0 ) (1 - wp) ln ηs + wp ln ηp + [k12 + l12(1 - 2wp)](1 - wp)wp ln η12 (22) with
ln η12 ) (ln ηp - ln ηs)/2
(23)
That is, each polymer-solvent solution has two binary parameters, k12 and l12. Application to Polymer Solutions PS Solutions. The solid curves in Figures 3 and 4 are the correlations of the Newtonian viscosity of PSstyrene solutions given by eqs 22 and 23 at temperatures of 313.15 and 333.15 K. The weight-average molecular weight of PS at both temperatures is 366 000 g/mol. A single set of two binary parameters is derived from experimental data at two temperatures as follows: k12 ) -1.7649 and l12 ) 1.8556. The agreement between data and correlation is excellent. The dashed lines represent the prediction provided by the ideal linear mixing rule from eq 11. PEG Solutions. The solid curves in Figures 5 and 6 are the correlations of the Newtonian viscosity of PEG1,3-dioxolane solutions given by eqs 22 and 23 at the same temperature of 303.15 K but at two different weight-average molecular weights: 223.68 and 447.168 g/mol. A single set of two binary parameters is derived from experimental data at two molecular weights as follows: k12 ) -0.7689 and l12 ) 0.0378. The agreement between data and correlations is excellent. The dashed lines represent the prediction provided by the ideal linear mixing rule from eq 11.
Ind. Eng. Chem. Res., Vol. 42, No. 11, 2003 2421
Temperature Dependence on Binary Parameters. As shown in Figures 3 and 4, a single set of two binary parameters is sufficient for the correlation at two different temperatures for PS-styrene solutions. However, if the data cover a wide range of temperatures, it may be necessary to have a temperature dependence on these binary parameters. Commonly used forms shown below can be adopted:
kij ) aij +
bij T
(24)
lij ) cij +
dij T
(25)
where aij, bij, cij, and dij are temperature-independent constants for a given polymer solution and T is the system temperature. Polydispersity on Binary Parameters. Because of the polydisperse nature of polymers, it is always more appropriate to use the weight-average molecular weight (instead of the number-average molecular weight) of the polymer when calculating the Newtonian viscosity of polymer melts. This has been demonstrated in the literature by the Mark-Houwink equation and its many variations. As shown in Figures 5 and 6, a single set of two binary parameters is sufficient for the correlation at two different molecular weights for PEG-1,3-dioxolane solutions. This indicates that the current model is capable of representing polydisperse polymers through the modified Mark-Houwink model, and the two binary parameters can be considered as independent of polymer molecular weights or polydispersity. Shear-Rate Dependence on the Viscosity. Although the current model is specifically designed to calculate the Newtonian viscosity for mixtures containing polymers, it is interesting to investigate the shearrate range in which the model can be applied when a separate shear-rate contribution to the viscosity is included. In general, polymer solutions or polymer melts exhibit a shear-rate-dependent viscosity,4,12,13 especially at high shear rates. That is, increasing the shear rate results in a decrease in the viscosity. At low to moderate shear rate, the Bird-Carreau model4 is often used to model the shear-rate dependence on the viscosity. The Bird-Carreau model can be written as follows:
g(γ˘ ) )
η η0
[ ( )]
) 1+
η0γ˘ τ0
2 (n-1)/2
Figure 11. Shear-rate contribution to the non-Newtonian viscosity of PS-styrene solutions at T ) 313.15 K and the PS weight fraction of 0.9. The Newtonian viscosity, η0, is calculated from the current model, eq 22, and the non-Newtonian viscosity data are taken from the compilation by Kim and Nauman.9
Figure 12. Viscosity of methanol-water mixtures at T ) 313.15 K. The curve is the correlation by the current model, eq 22, with a single set of two binary parameters: k12 ) 11.9544 and l12 ) 6.3828. The experimental data are from Carton et al.14
(26)
where τ0 is a parameter in units of pressure, n is an index number, and both τ0 and n are constants for a given polymer. Notice that eq 26 may apply to polymer solutions as well as polymer melts. Figure 11 shows the results of η0/η as a function of the shear rate (γ˘ ) for PS-styrene solutions at a temperature of 423 K and the PS weight fraction of 0.9. The weight-average molecular weight of PS is 259 000 g/mol. The data of the non-Newtonian viscosity, η, are taken from the compilation by Kim and Nauman.9 The Newtonian viscosity, η0, is calculated from eqs 22 and 23 for PS-styrene solutions with the same parameters and constants as those reported in the earlier sections. The solid curve in Figure 11 is the correlation of eq 26, with τ0 ) 40069.58 Pa and n ) 0.1423. Some conclusions can be drawn from Figure 11. If we set g(γ˘ ) ) 1, the current model for the Newtonian
viscosity can be applied to the range of shear rates, γ˘ e 2 s-1, within 10% error. Including the shear-rate contribution from eq 26, the current model is applicable to the condition of shear rates up to 50 s-1 within 10% error. The discrepancy becomes larger at higher shear rates, and the error in |[η - η0g(γ˘ )]/η| at the shear rate of 340 s-1 reaches about 40%. Application to Nonpolymer Systems. Although this paper focuses on the Newtonian viscosity of polymer solutions, the current model requires no modification when applied to other systems. We have successfully applied this model to correlate the viscosity data for many nonpolymer systems, including highly polar mixtures in which strong nonideal mixing exists. As an illustration, Figure 12 shows an excellent correlation of liquid viscosity data14 for a methanol-water mixture at a temperature of 313.15 K. The large values for both binary parameters, k12 ) 11.9544 and l12 ) 6.3828, are
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indicative of the very large nonideal mixing effect on the viscosity. Conclusions A simple liquid viscosity model is presented for multicomponent mixtures containing polymers. This model is essentially a new mixing rule for calculating the Newtonian viscosity of mixtures over the entire composition range from the pure-component viscosities. It assumes that the viscosities of all pure components in the mixture are already available as input. For polymer melts, we present a modified Mark-Houwink model to calculate the Newtonian viscosity. For solvents, we use the Andrade/DIPPR model7,8 to calculate the viscosity. Two binary parameters, one symmetric and one antisymmetric, are introduced to capture nonideal mixing behavior of viscosity in polymer solutions. The model is applicable to mixtures containing any number of components. Acknowledgment This paper is the result of experience helping Polymers Plus users solve industrial problems related to transport properties in polymer process simulation. We are grateful to John Franjione and Suphat Watanasiri for helpful discussions and suggestions. Literature Cited (1) Van Krevelen, D. W. Properties of Polymers, 3rd ed.; Elsevier: Amsterdam, The Netherlands, 1990. (2) Biesenberger, J. A.; Sebastian, D. H. Principles of Polymerization Engineering; R. E. Krieger Publishing Co.: Malabar, FL, 1993; Chapter 5.
(3) Grigorescu, G.; Kulicke, W.-M. Prediction of Viscoelastic Properties and Shear Stability of Polymers in Solution. Adv. Polym. Sci. 2000, 152, 1. (4) Boildin, M.; Kulicke, W. M.; Kehler, H. Predictions of the Non-Newtonian Viscosity and Shear Stability of Polymer Solutions. Colloid Polym. Sci. 1988, 266, 793. (5) Huggins, M. L. The Viscosity of Dilute Solutions of LongChain Molecules. IV. Dependence on Concentration. J. Am. Chem. Soc. 1942, 64, 2716. (6) Rudin, A.; Strathdee, G. B.; Brain Edey, W. Practical Estimation of Dilute Solution Parameters. J. Appl. Polym. Sci. 1973, 17, 3085. (7) Andrade, E. N. da C. The Viscosity of Liquids. Nature 1930, 125, 309. (8) Evaluated Process Design Data; Design Institute for Physical Properties (DIPPR) Project 801; American Institute of Chemical Engineers: New York, 2001. (9) Kim, D.-M.; Nauman, E. B. Solution Viscosity of Polystyrene at Conditions Applicable to Commercial Manufacturing Processes. J. Chem. Eng. Data 1992, 37, 427. (10) Ottani, S.; Vitalini, D.; Comelli, F.; Castellari, C. Densities, Viscosities, and Refractive Indices of Poly(ethylene glycol) 200 and 400 + Cyclic Ethers at 303.15 K. J. Chem. Eng. Data 2002, 47, 1197. (11) Mathias, P. M.; Klotz, H. C.; Prausnitz, J. M. Equationof-State Mixing Rules for Multicomponent Mixtures: the Problem of Invariance. Fluid Phase Equilib. 1991, 67, 31. (12) Dunleavy, J. E.; Middleman, S. Correlation of Shear Behavior of Solutions of Polyisobutylene. Trans. Soc. Rheol. 1966, 10, 157. (13) Ashare, E. Rheological Properties of Narrow Distribution Polystyrene Solutions. Trans. Soc. Rheol. 1968, 12, 535. (14) Carton, A.; Ssobron, F.; Bolado, S.; Gerboles, J. I. Viscosity, Conductivity, and Refractive Index of Saturated Solutions of Lithium Sulfate + Water + Methanol. J. Chem. Eng. Data 1995, 40, 980.
Received for review January 9, 2003 Revised manuscript received March 20, 2003 Accepted March 21, 2003 IE030023X
