Viscosity of Polymer Solutions over the Full Range of Composition: A

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Viscosity of Polymer Solutions over the Full Range of Composition: A Thermodynamically Inspired Two-Parameter Approach Bernhard A. Wolf* Institut für Physikalische Chemie der Johannes, Gutenberg-Universität Mainz, Jakob Welder-Weg 11, D-55099 Mainz, Germany S Supporting Information *

ABSTRACT: The approach yields the following relation for the relative viscosity η rel as a function of polymer concentration c (mass/volume): ln ηrel = c̃/(1 + pc̃ + qc̃2). Reduced concentrations c̃ (defined as c̃ = c[η], where [η] is the intrinsic viscosity) are used instead of c to incorporate thermodynamic information. The parameters p and q account for changes in the free volume of the solvent caused by the polymer. The analysis of literature data for seven very dissimilar systems discloses the following common feature: p > 0 and q < 0. This means that the curves in the plots of ln ηrel as a function of c̃ are normally located below the tangent at low c̃ and above it at high c̃. The values of p and q correlate strongly with the temperature distance to the glasstransition temperature of the polymer (Tg). Beyond the mere modeling of viscosity data, the approach allows the determination of [η] from data at high polymer concentrations and provides information on the generalized intrinsic viscosity, {η}. Measurements for T < Tg give access to glass curves, i.e., to Tg(c). Moreover, the modeling helps to recognize systems with special behavior, such as solutions of poly(dimethyl siloxane) in its oligomers. where M is the molar mass of the polymer; η̅ and γ are adjustable parameters. The drawback of approach (i) lies in the fact that two types of relationships and four parameters are required to cover the entire range of composition. Approach (ii) is based on a viscosity vs c relation derived for concentrated polymer solutions, in terms of a simple freevolume theory.6 It fits the data well as long as a sufficient number of interchain entanglements occur. The drawback of this approach lies in its complexity and in the deviations from the experimental findings in the region of relative low polymer concentrations. Starting point of the approach presented here is the treatment of η as a variable of state and application of the laws of phenomenological thermodynamics. Under the normally well fulfilled preposition that the viscosity of the polymer solutions of interest is not dependent on the particular manner in which the variables of state are adjusted, one can define a generalized intrinsic viscosity {η} (hydrodynamic specific volume at arbitrary polymer concentrations c) as

1. INTRODUCTION Numerous relationships1−4 have been proposed to describe the viscosity of polymer solutions from the pure solvent up to the polymer melt or to the concentration at which the mixture solidifies as a glass or segregates crystals. The motivation for the establishment of still another approach originated from a study on the viscosity of dilute polyelectrolyte solutions. More specifically, we have developed a method5 for the determination of intrinsic viscosities [η] for this class of polymers in the absence of extra salt: [η] is obtained from the initial slope of the natural logarithm of the reduced viscosity as a function of polymer concentration (mass/volume). Here, we check whether this relationship can be successfully extended or modified to describe the composition dependence of viscosity for uncharged polymers over the full range of composition. The testing of the obtained relationship uses exclusively published data. 2. MODELING Two types of approaches can mainly be found in the literature for the modeling of the viscosity as a function of composition: (i) either a phenomenological description or (ii) a modeling by means of the concept of free volume. Approach (i) often uses4 stretched exponentials (eq 1) in combination with power laws (eq 2) to describe the viscosity η as a function of polymer composition c. The stretched exponential reads η = η0 exp(αc ν)

{η} =

(3)

with the temperature, pressure, and shear rate (γ̇) being constant. For Newtonian behavior and within the limit of infinite dilution, {η} becomes identical to the well-known intrinsic viscosity [η]. lim {η} = [η]

(1)

c→0 γ→ ̇ 0

where η0 is the viscosity of the solvent; ν is a scaling exponent and α is a scaling prefactor. The power law dependence can be formulated as η = η ̅ c νM γ

⎛ ∂ln η ⎞ ⎜ ⎟ ⎝ ∂c ⎠T , p , γ ̇

(4)

Received: March 4, 2015 Revised: April 3, 2015 Accepted: April 6, 2015

(2) © XXXX American Chemical Society

A

DOI: 10.1021/acs.iecr.5b00845 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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[η]•/[η]); B constitutes an viscometric interaction parameter. Comparing eqs 7 and 8 yields α = Br, β = B, and γ = 0. For the application of eq 7 to determine intrinsic viscosities of polyelectrolytes, this means that the parameter α is only required under special conditions under which the coil dimensions change pronouncedly with composition (i.e., in the absence of extra salt, shielding the electrostatic repulsion between the charged monomeric units). This finding corroborates the interpretation of α as measure for the composition dependence of the intersegmental friction between the components. For polyelectrolyte solutions, it is mainly caused by the variation of the spatial extension of the segments belonging to a given macromolecule. Moreover, the parameter γ can generally be set to zero for determination of the intrinsic viscosities. According to recent experiments (unpublished results) performed for polyelectrolytes with divalent counterions, all three parameters of eq 7 are necessary to model the behavior for insufficient extra salt. We are now coming back to solutions of uncharged macromolecules and to the description of their viscosity over the entire range of composition from the pure solvent to the pure melt. The application of the general relation (eq 7) to such systems shows that the parameter α can, in all cases studied so far, be set to zero. However, this does not imply composition-independent intersegmental friction; it is more likely that these contributions are small, compared to the changes in free volume, and that they can be incorporated into parameter β. For the sake of an unequivocal nomenclature, we rewrite eq 7 for the application to uncharged polymers as

For a given solvent and constant temperature, the dependence of [η] on the molar mass M of the polymer is given by the following well-known Kuhn−Mark−Houwink relation:

[η] = KM αKMH

(5)

where K and αKMH are system-specific constants. The present modeling starts from eq 3 and describes the composition dependence of the viscosity, in terms of the natural logarithm of the relative viscosity ηrel = ηsolution/ηsolvent. This matter of fact implies that the intrinsic viscosity of the polymer in the solvent of interest signifies the central parameter of the approach. Instead of the normally used composition variable (weight fractions, volume fractions, or surface fractions7), we therefore employ the reduced polymer concentration, which is defined as c ̃ = c[η]

(6)

c̃ is sometimes erroneously also called coil overlap parameter; however, this generally is not permissible, because c̃ does not account for changes in the coil dimensions with polymer concentration. Under the (unrealistic) assumptions that the friction between the solvent molecules and the monomeric units of the polymer is independent of concentration and that the free volume of the system similarly does not change with composition, ln ηrel should simply be proportional to c̃ over the entire concentration range. In this case, the high viscosity of the pure polymer would be a mere consequence of the large values of the maximum reduced polymer concentration, which can be calculated by multiplying the intrinsic viscosity with the density of the polymer in the pure liquid state. The simplest way to account for the composition dependence of the f riction between the components consists in the introduction of an additional term of the form αc̃2, where the parameter α quantifies these extra effects. In order to model the consequences of changes in the f ree volume of the mixture, we proceed in a similar manner. In this case, however, it is the denominator of the expression for ln ηrel as a function of c̃ that needs to be modified, because the viscosity is inversely proportional to the free volume. A series expansion with respect to c̃ must be performed up to the second power to describe all experimental findings. The denominator thus becomes 1 + βc̃ + γc̃2, where β and γ quantify the changes in the free volume of the solutions as compared with that of the pure solvent caused by the presence of polymer. The equation resulting from the above considerations for the composition dependence of ln ηrel thus becomes ln ηrel =

c ̃ + αc 2̃ 1 + βc ̃ + γc 2̃

ln ηrel =

(7)

c ̃ + Brc 2 1 + Bc ̃

(9)

by using p instead of β and q instead of γ. Equation 9, similar to eq 8, only contains two adjustable parameters. When studying systems for which the intrinsic viscosity of the polymer in the given solvent is not available, [η] can be treated as a third adjustable parameter, by plotting ln ηrel as a function of the unreduced polymer concentration (c), instead of the reduced concentration (c̃). Parameter p (larger than zero for thermodynamically good solvents) models the less than exponential growth of η with rising c caused by the shrinkage of the polymer coils under these thermodynamic conditions. Parameter q (normally less than unity) accounts for the fact that the chain molecules encounter increasing difficulties to change places as their concentration rises. This growth in viscosity is particularly pronounced for temperatures below the glass-transition temperature (Tg) of the pure polymer, where η tends toward infinity at a characteristic composition of the mixture. This “solidification” is mathematically reproduced by the condition that the denominator of eq 9 moves toward zero. In this context, it appears worthwhile to note that a series expansion of ln ηrel, with respect to c, up to the power of 10, is unable to describe this phenomenon. According to earlier work with polyelectrolytes,8 under certain conditions, it is possible to obtain information on the composition dependence of the dimensions of individual polymer coils from {η} as a function of concentration. Differentiation of eq 9, with respect to c, yields the following relationship:

A relation similar to eq 7 can be successfully employed for the determination of intrinsic viscosities of polyelectrolytes (univalent counterions) in the absence of extra salt and of uncharged polymers from the dependence of the relative viscosity on polymer concentration in the region of pair interactions between the solute. It reads5 ln ηrel =

c̃ 1 + pc ̃ + qc 2̃

(8) •

Here, r represents the ratio of a parameter [η] (becoming zero if the solvent contains sufficient amounts of extra salt and for uncharged macromolecules) and the intrinsic viscosity (r = B


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Industrial & Engineering Chemistry Research 1 − qc 2̃ {η} = [η] (1 + pc ̃ + qc 2̃ )2

In view of the fact that differences between solvent and polymer for the present system are confined to the dissimilarities of end groups and middle groups, in an earlier work, the data of Kataoka and Ueda were20 already used to check whether surface fractions would be more adequate than volume fractions for the modeling of the composition dependence of viscosities. Here, we make use of the data of Kataoka and Ueda to check the applicability of eq 9. Figure 1 shows the evaluation of the primary data according to this relation. The parameters modeling the different curves are collected in Table S1 in the Supporting Information.

(10)

which describes the relative change in the hydrodynamic specific volume as a function of the reduced polymer concentration c̃. For the application of the above relationship up to high polymer solutions, one must keep in mind that the calculated {η} represents only an effective specific volume and no longer yields information on the actual spatial extension of polymer coils.

3. RESULTS Because of the general interest in information concerning the viscosity of polymer solutions, as a function of polymer molar mass, polymer concentration, and temperature, ample data can be found in the literature. The characteristic parameters of the systems selected for the present study are specified in Table 1. Table 1. Collection of Systems under Investigationa systemb

ref

Mw (kg mol¬1)

DMS5/ PDMS MeNa/PB

9

1.15−490

10, 11 13

38−240

CCl4/PIB 0.1 M NaCl/ xanthan DEP/ PVAc DEP/ PMA DBE/PS

14

Mη (kg mol¬1)

5250

Tg,∞ (K)

∼2

30

81

22 000 can be successfully superimposed, according to a relation proposed by Ferry,1 and they attribute this observation to (i) the fact that the present system represents a mixture of homologues and (ii) the low Tg value of PDMS.

Figure 2. Joint evaluation of the results for the systems MeNa/PB and CCl4/PIB, according to eq 9.

0.1 M NaCl/Xanthan. The 0.1 M NaCl/xanthan system14 was chosen to demonstrate that the approach also works with aqueous solutions of polyelectrolytes if the extra salt concentration of the solvent is high enough. The results for this system are not shown graphically, because the dependencies appear to be very similar to those presented in Figures 1 and 2. All parameters required for the modeling of the three systems, for which viscosities were measured under isothermal conditions and the molar mass of the polymer was varied, are C


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Industrial & Engineering Chemistry Research collected in Table 2. For PIB, PB, and xanthan, the authors of the respective publications have determined the intrinsic Table 2. Parameters of the Systems Studied at Constant Temperature for Different Molar Masses of the Polymers system

c̃maxa

100p

100±

1000q

1000±

DMS5/PDMS CCl4/PIB methyl naphthalene/PB 0.1 M NaCl/xanthan

79 240 270 10

7.39 6.12 6.49 6.73

0.12 0.11 0.25 0.15

−0.33 −0.09 −0.11 −0.11

0.02 0.01 0.01 0.06

a

c̃max is the maximum value of the reduced concentration (c̃max = [η]ρpol).

Figure 3. Evaluation of the viscosity data reported for the diethyl phthalate/poly(vinyl acetate) system,16 according to eq 9.

viscosities themselves by independent methods. In the case of PDMS, the data stem from the present modeling of ln ηrel vs c̃, treating [η] as an adjustable parameter. For this system, Table 2 states the average parameters for the three samples of highest molar mass. 3.2. Constant Molar Mass. All these experimental data, referring to the influence of temperature on the composition dependence of η, are evaluated as a function of the polymer concentration c instead of c̃, because the authors did not report intrinsic viscosities; therefore, [η] is treated as an adjustable parameter. Diethyl Phthalate/Poly(vinyl acetate) and Diethyl Phthalate/Poly(methyl acrylate). The two systems diethyl phthalate/ poly(vinyl acetate)16 and diethyl phthalate/poly(methyl acrylate)17 are very similar. The solvents are the same, and the polymers have the same chemical formula and a C−C backbone; they differ only by the way the remaining C2H3O2 group is bound to the polymer backbone. With PVAc, the link is made by an O atom, whereas, in the case of PMA, the link is made by a C atom. Kishimoto16 studied the diffusion coefficient of methanol in poly(vinyl acetate) (PVAc) and the steady flow viscosity of the system DEP/PVAc for temperatures ranging from 10 °C to 100 °C. Kishimoto16 observed that the viscosity against the polymer weight fraction at a fixed temperature is convex upward at low concentration and then becomes concave upward at high concentrations, with an inflection point in the intermediate region. An analysis of the results on the basis of free volume theory6 showed that the data can be well-represented. However, considerable additional experimental and theoretical work is required for such modeling. For this reason, we check whether eq 9 can accomplish the same job with only two parameters. Figure 3 shows the result; the parameters are collected and presented in Table S2 in the Supporting Information. Fujita and Maekawa17 measured viscosities over the entire range of composition at temperatures between ∼20 °C and 110 °C, using a proper combination of capillary viscometers, a coaxial falling cylinder viscometer, and a tensile creep apparatus. The evaluation of their data, according to eq 9, is shown in Figure 4; the obtained parameters are collected and presented in Table S3 in the Supporting Information. Dibenzyl Ether/Polystyrene. The dibenzyl ether/polystyrene system18 takes a special position, because of the wide temperature range that was investigated and the fact that measurements were also performed far below the Tg value of the pure polymer. For the polymer sample of interest, a Tg value of 97.5 °C was obtained, according to eq 2 in ref 21. Table S4 in the Supporting Information lists the parameters of eq 9, describing the experimental findings. For this system, the

Figure 4. Evaluation of the viscosity data reported for the diethyl phthalate/poly(methyl acrylate) system,17 according to eq 9.

data were evaluated in two graphs: one for T ≥ Tg (Figure 5a) and one for T ≤ Tg (Figure 5b).

Figure 5. Evaluation of the viscosity data reported for the dibenzyl ether/polystyrene system,17 according to eq 9: (a) T ≥ Tg and (b) T ≤ Tg. The dotted line in panel b indicates the value of ln η at the glass transition.22 D


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4. DISCUSSION This section examines the following items: (i) The influence of molar mass on the parameters and the reliability of the intrinsic viscosities obtained from data in the region of high polymer concentrations. (ii) The effects of temperature on the system-specific parameters and possibilities to model the glass transition of solutions. (iii) The dependence of the generalized intrinsic viscosities {η} on polymer concentration. 4.1. Influence of Molar Mass. DMS5/Poly(dimethyl siloxane). Figure 6 shows the dependence of the intrinsic

Figure 7. System-specific parameters p and q (eq 9), as a function of the inverse degree of polymerization (DP) of PDMS for the DMS5/ PDMS system at 30 °C. Figure 6. Kuhn−Mark−Houwink plots (cf. eq 5) for the DMS5/ PDMS system (red solid line) (K = 39 × 10−3 mL/g, αKMH = 0.58), the butanone/PDMS (theta system, 20 °C)12 (red-brown dashed line) (K = 81 × 10−3 mL/g, αKMH = 0.50), and the toluene/PDMS system12 (25 °C) (black dashed line) (K = 2.4 × 10−3 mL/g, αKMH = 0.84).

become longer, with the exception of the lowest DP. This observation implies that the PDMS maintains its rheological polymer character down to values that are well below the molar mass of entanglement 23 (Mw,e = 35.2 kg/mol). These findings fit well into a debate on the question: “When does a molecule become a polymer?”.24 For the present system, this limit is strongly dependent on the experimental method of its determination. For the isolated molecule, it is obviously lower (DP ≈ 70) than that for the viscometric interaction between different chains in solution (DP ≈ 200), which, in turn, is again lower than the entanglement limit for the pure melt23 (entanglement DP ≈ 475). An evaluation of the data for PIB and PB, by analogy to that for PDMS, does not make sense. One reason lies in the limited variation of the molar masses; furthermore, the intrinsic viscosity data were measured by the authors and used for the evaluation but not cited explicitly. 4.2. Influence of Temperature. The measurement for the solutions of PVAc, PMA, and PS were performed in a wide range of temperatures and were (except for the experiments with PS at temperatures below the glass transition temperature of the pure polymer) precise enough to permit the determination of intrinsic viscosities from the reported data. Figure 8 shows the results of this evaluation for the different systems according to eq 9. For PVAc and PMA, having the same formula and being studied in the same solvent, the values of [η] and their dependence on temperature are expectedly very similar. Furthermore, the data lie well within the range observed with other solvents and at other temperatures. The experimental precision of the measurements appears high enough to document the existence of minima; at the corresponding temperature (∼70 °C), the solvent interacts most unfavorably with the polymer. Additional work would be necessary to discuss why the solvent quality is higher below and above this

viscosities in the DMS5/poly(dimethyl siloxane) system,9 as obtained from the evaluation of ln ηrel vs c on the molar mass of samples, in terms of a Kuhn−Mark−Houwink plot. Except for the sample with the lowest molar mass (only three times larger than the solvent), all data fall very well on a common line. This observation indicates that a DP = 70 suffices for the full establishment of polymer properties, with respect to intrinsic viscosities. Figure 6 shows that the pentamer DMS is a slightly better solvent for PDMS than butanone (theta solvent), but considerably worse than toluene. This means that (i) the mixture of homologues does not represent a theta system and (ii) the thermodynamic interactions between the components are less favorable than for the best solvents. The molar mass of the polymer not only governs its intrinsic viscosity but also modifies the viscosity η of the solution at a given reduced concentration (cf. eq 9). This effect, which is quantified by the parameters p and q, is dependent on the chain length of PDMS, as demonstrated in Figure 7, as a function of the inverse DP. For the extrapolation of the parameters to the infinitely long PDMS chains shown in Figure 7, the point for the sample of lowest molar mass is omitted, because the solute can no longer be considered as a polymer (DP = 15). In order to obtain an idea down to which chain lengths the molecules behave as a polymer, p and log(−q) were also plotted as a function of log DP. For p, one obtains a smooth curve, which starts at zero for the solvent and passes a distinct maximum at DP = 190. Roughly speaking, this means that PDMS loses its rheological polymer character as the number of monomeric units falls below ∼200. The log(−q) value decreases linearly as the chains E


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According to the interpretation of p and q presented in Modeling section, these parameters account for the changes in the free volumes of the solvent caused by the addition of polymer, i.e., by an increase in polymer concentration. Based on the reasonable assumption that the free volume of the polymer becomes larger as the distance between this temperature and the Tg value of the polymer increases, it seems natural to check how p and q are dependent on the reduced temperature Tred (Tred = T/Tg,∞ instead of T). Figure 10 shows this dependence

Figure 8. Intrinsic viscosities adjusted to the isothermal composition dependencies of η for the diethyl phthalate/poly(vinyl acetate),16 diethyl phthalate/poly(methyl acrylate),17 and dibenzyl ether/ polystyrene systems,18 as a function of temperature. The arrows indicate the [η] ranges reported12 for the present polymers (same molar masses) for different solvents and temperatures.

characteristic temperature, for which the mixing should be athermal. With the solutions of PS, the situation is more complicated than with PVAc and PMA. The evaluation of the data reported for temperatures below the Tg value of the pure polymer are very uncertain. The results for T > Tg, on the other hand, appear reliable. The assured [η] values indicate an uncommonly low solvent quality just above Tg. With rising T, the interaction becomes even more unfavorable. This observation is in accord with the experience that typical polymer/solvent systems approach lower critical solution temperatures if the free volume of the solvent surpasses a critical value. We are now going to discuss the temperature influences on the characteristic parameters p and q of eq 9 by means of Figure 9. The signs of the temperature dependencies of p and q turn out to be the same for all systems under investigation. The results for PVAc and PMA are again very similar, and, for PS, the statements of the last paragraph concerning the reliability of the date below Tg hold true again.

Figure 10. Dependence of the system specific parameters p and q on the reduced temperature Tred = T/Tg,∞ for 25 °C (for PDMS, T = 30 °C). Blue data represent the variation of M, red data represent the variation of T. The lines are only in place to guide the eye.

for T = 25 °C; Tg,∞ is the limiting glass temperature of the polymer for infinite molar mass. The results demonstrate that the correction terms p and q indeed become less important as Tred rises. This is particularly true for q, the second term in this series expansion, which can be neglected for sufficiently high Tred values. It would be tempting to use the dependencies of the parameters on Tred to forecast the viscometric behavior of polymer solutions. The known Tred value could give access to the corresponding characteristic values of p and q. The intrinsic viscosity of the polymer under the conditions of interest and the viscosity of the pure solvent would be the only additional data required for the application of eq 9. The reason why we refrain from such an analysis at the present time lies in the considerable scattering of the experimental points of Figure 10. One probable reason for this uncertainty, in addition to the effects of limited experimental accuracies, lies in the use of Tg,∞, instead of the Tg value of the actual polymer of molar mass (M). Even if we exclude values below 200 kDa, these influences remain too large to enable reliable predictions. Nevertheless, the above analysis discloses that the composition dependence of the viscosity of polymer solutions is (for a given solvent plus

Figure 9. Variation of the system specific parameters p and q (cf. eq 9) with temperature. F


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Industrial & Engineering Chemistry Research fixed values of T and [η]) primarily determined by the distance between this temperature and the Tg value of the polymer under investigation. 4.3. Influence of Concentration. Viscosity. The most versatile relation for the modeling of ln ηrel as a function of polymer concentration is eq 7, which requires the parameters α, β, and γ. For the uncharged macromolecules of present interest, α can be set to zero and the parameters p and q (eq 9) suffice for their modeling. It appears remarkable that p is always positive and q is always negative for the seven systems under investigation, despite the pronounced chemical differences between the individual solvents and polymers. In the light of considerations presented in the Modeling section, this would mean that the free volume of the mixture is in the region of low polymer concentrations larger than that of the pure solvent, whereas it becomes less than that value for large c values, i.e., in the vicinity of the (sometimes hypothetical) polymer melt. The latter observation can be easily rationalized in terms of the fact that the distance between the monomeric units of the polymer is fixed by the length of the chemical bonds and thus delimitates the fraction of unoccupied volume. The increase in the free volume of the system within the region of low polymer concentrations, on the other hand, is less easy to interpret. One could argue that a comparatively tight packing of the molecules in the pure solvent is disturbed by the presence of polymer coils; if that were indeed the case, it would mean an increase in the free volume. Further data and a more detailed theoretical analysis are needed to check to which extent the present findings are actually general. On the other hand, the combined effects of the parameters p and q undoubtedly govern the shape of ln ηrel(c̃). Despite the generally negative q values, the viscosity does not necessarily increase more than exponentially at high polymer concentrations. The reason lies in the reduced concentrations, which are physically only meaningful up to a certain maximum value. For instance, in the case of PDMS with DP = 3030, the highest reduced concentration that can be reached is 48.5, compared to the reduced concentration at the point of inflection of the dependence ln ηrel(c̃) calculated from eq 9, using the parameters of Table S1 in the Supporting Information, which would be 50.0. This means that the curves are always located below their tangent. In a more illustrative description, this observation means that the measurement temperature of 30 °C is still too far above the Tg value of the pure polymer. Glass Transitions. The amorphous solidification of liquids upon cooling sets in as their free volume falls below a certain critical value. The “dilution” of the pure polymer, via the addition of solvent, increases the free volume of the system and consequently reduces the Tg value of the pure polymer. The present modeling of the concentration dependence of ln rel also rests on free volume considerations. For this reason, it appears natural to check whether the modeling of viscosities according to eq 9 provides access to the concentration dependence of Tg. The data for the system DBE/PS are used for that purpose. The following analysis is based on the assumption that the viscosity of liquids under the conditions of solidification, i.e., at Tg, assumes a large characteristic value for a given system. According to eq 9, the viscosity approaches infinity upon augmentation of c as pc̃ and qc̃2 make a sum of −1. Figure 5b shows this situation for different temperatures below the Tg value of pure PS. If the viscosity under the solidification conditions is known, one can read the polymer concentration at

which this value is reached for the temperature of interest. The dotted line in Figure 5 is drawn using the general experience,22 according to which η should assume a value of 1012 Pa s at the glass transition. In accordance with the reported Tg of the pure polymer,21 the curve for 100 °C does not intersect the ln ηrel value, which is characteristic for solidification within the physically meaningful concentration range, i.e., for c < ρpolymer. Despite the uncertainties in the actual viscosity at the glass point, the concentrations of glassy solidification determined from the intersection of the dotted line in Figure 5 with the calculated concentration dependencies of ln ηrel vs c change very little, because of the steepness of this dependence. The obtained glass-transition curve shown in Figure 11 agrees very

Figure 11. Concentration dependence of the glass transition temperature (Tg) for the DBE/PS system obtained from Figure 5b, as described in the text. The Tg value of the pure polymer is taken from the literature.21

well with the results of conventional measurement reported for PS for various solvents.25 According to these published data, the efficiency of DBE, with respect to the reduction of Tg, is very similar to that of methyl acetate, ethyl acetate, and carbon disulfide. Generalized Intrinsic Viscosities {η}. From an analysis of the viscometric behavior of polyelectrolytes by means of eq 8, which is a simplified version of eq 7, we know that the generalized intrinsic viscosity {η} (defined in eq 3) can yield information on the composition dependence of coil dimensions.8 By means of Figure 12, we want to determine whether an analogous evaluation could also be useful for the solutions of

Figure 12. Plot of {η}/[η] (generalized intrinsic viscosity normalized to the intrinsic viscosity), as a function of the reduced polymer concentration, calculated according to eq 10 from the parameters tabulated for the different systems. The curves end at the c̃ values of the pure polymers. G


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where η rel is the viscosity of the solutions relative to that of the solvent. Only two adjustable parameters (p and q) are required to cover the entire concentration range in case the intrinsic viscosities are known from independent measurements or can be calculated by means of the Kuhn−Mark−Houwink equation from the molar mass of the polymer. In case such information is lacking, the intrinsic viscosity is accessible from the measured concentration dependence of η by treating [η] as a third adjustable parameter, in addition to p and q. Detailed literature data concerning the viscosities of polymer solution, as a function of composition reported in the literature for seven different systems in wide ranges of molar masses and temperatures, were evaluated according to the above relationship. In all of these cases (uncharged polymers and one polyelectrolyte in a saline solvent), the parameter p is positive and the parameter q is negative. For six of the seven systems, the values of p and q yield points of inflection in the composition dependence of the viscosities: At low polymer concentrations, η increases less than exponentially with rising c, whereas it increases more than exponentially for high concentrations. Solutions of poly(dimethyl siloxane) (PDMS) in its pentamer assume a special place in several ways; this exceptional position is tentatively explained in terms of the high flexibility of PDMS. A more detailed analysis of the data reveals that the values of the parameters p and q show a clear dependence on the distance of the measurement temperature to the glass-transition temperature (Tg) of the polymer, such that the viscosity of the solution becomes infinitely high at T ≈ Tg. For polystyrene solutions in dibenzyl ether, this feature was used to determine the concentrations at which the solutions solidify at temperatures below the Tg value of the pure polymer. The results are in good agreement with corresponding literature data. The analysis of the viscometric behavior according to the proposed relation also provides access to the generalized intrinsic viscosities {η}, i.e., to the effective hydrodynamic specific volume of the polymer at arbitrary polymer concentrations. Within the range of high dilution, it is possible to determine the coil dimensions as a function of c. However, once the reduced polymer concentration surpasses a critical value, the {η} data can no longer be considered to be characteristic for individual molecules. In that case, they must be reinterpreted as kinetically determined dimensions of flow units. In view of the general importance of quantitative information concerning the viscosities of polymer solutions, as a function of concentration, methods enabling forecasts would be of great help. The general features apprehended by the present analysis should be a first step in this direction. It goes without saying that many more systems must be studied, including solutions of co-polymers and of macromolecules with nonlinear architecture.

uncharged polymers of present interest. To this end, we plot {η}, normalized to the intrinsic viscosity, as a function of the reduced polymer concentration, as formulated in eq 10. For all systems, one observes an initial reduction of {η}/[η] with rising reduced polymer concentration. It is caused by the fact that the solvents are, in all cases, better than theta solvents. This situation leads to a reduction in the size of the individual polymer coils as c becomes larger, because of the less favorable interaction of the polymer segments with polymer segments belonging to other polymer molecules than with solvent molecules. According to thermodynamic considerations, this effect should subsist all the way to the pure melt, because it is general consensus that the coils should finally approach the theta dimension. Only the DMS5/PDMS system shows such a behavior, but the effect is by far too large to be explained along these lines. For that reason, we need to change our viewpoint on {η} from thermodynamic to rheological (i.e., kinetic) considerations. To this end, we make use of the concept of flow units,26 which was already introduced some 50 years ago as “a coherent mass of material that moves as a unit in a continuous flow field”. For the dissolved polymer molecules and at infinite dilution, the volume of a flow unit should be that of an individual coil, i.e., proportional to [η]M. As c becomes larger, the flow unit changes not only for thermodynamic reasons but also because of the increasing difficulty of the polymer molecules to move independently. This means that the measured {η} value is a conglomerate of both effects and must be interpreted as an effective hydrodynamic specific volume, which is, at large polymer concentrations, dominated by the geometrical obstacles impeding the viscous flow. According to these considerations, the glassy solidification should go along with extremely large flow units, i.e., high {η} values. The question remains why solutions of PDMS in its pentamer do not follow the behavior outlined above. According to Figure 12, the volume of the flow unit decreases continuously up to the pure polymers, in contrast to the usual behavior. This exception might be explained in terms of the observation that highly crystalline materials can be obtained27 by cooling PDMS oils or rubbers. If one tentatively assumes that such processes are promoted by the short-chain homologue solvent and are taking place on a molecular level (formation of molecular “crystals”), the volume of a flow unit could fall even below the value at infinite dilution. It would be interesting to validate this hypothesis by directed experiments.

5. CONCLUSIONS This contribution presents a simple thermodynamically inspired relation to model the viscosities of polymer solution, as a function of composition. The intrinsic viscosity of the polymer in the solvent of interest plays a central role for that purpose; it enables the introduction of a reduced polymer concentration (c̃ = c[η]), accounting for the spatial extension of the polymer coils in solution and for the thermodynamic quality of the solvent. Incorporating ideas concerning changes in the free volume of the mixture, compared to that of the pure solvent, yields the following relation: ln ηrel =

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AUTHOR INFORMATION

S Supporting Information *

The authors declare no competing financial interest. This material is available free of charge via the Internet at http:// pubs.acs.org. Corresponding Author

c̃ 1 + pc ̃ + qc 2̃

*Tel.: +49 6131-39-22491. E-mail: bernhard.wolf@uni-mainz. de. H


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Industrial & Engineering Chemistry Research Notes

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The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The author thanks Dr. J. Eckelt (WEE-Solve GmbH, Mainz, Germany) for fruitful discussions.

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REFERENCES

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Viscosity of Polymer Solutions over the Full Range of Composition: A

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