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Extension of the Concept of Intrinsic Viscosities to Arbitrary Polymer Concentration: From [η] via {η} to Intrinsic Bulkiness Bernhard A. Wolf*
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Institut für Physikalische Chemie der Johannes Gutenberg-Universität Mainz, D-55099 Mainz, Germany ABSTRACT: The capabilities of an alternative definition of intrinsic viscosities [η] published some years ago is being studied by means of comprehensive viscometric data reported in the early days of polymer science. It introduces the generalized intrinsic viscosity {η} as the specific hydrodynamic volume at arbitrary polymer concentration c. {η} quantifies the size of the flow unit and decreases monotonously for T ≫ Tg (glass transition temperature) as a function of c but passes a pronounced minimum as T approaches Tg. In the limit of the pure polymer melt, {η} becomes ; this newly introduced property is termed intrinsic bulkiness, by analogy to the intrinsic viscosity, and provides noncalorimetric experimental access to Tg; it also allows estimates of entanglement molecular weights based on the Newtonian flow behavior. Moreover, the molecular weight dependence of provides information on the contributions of endgroups to the flow behavior.
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old data, it turns out particularly helpful that η can be modeled over the full range of composition2 by means of a maximum three adjustable parameters, out of which one or even two can often be set to zero.
INTRODUCTION The intrinsic viscosity [η] is undoubtedly one of the most widespread quantities employed in polymer science. Among others, it provides quick access to molar masses (via the Kuhn−Mark−Houwink relation) and to the thermodynamic quality of solvents. Misleadingly, it has been named intrinsic “viscosity”, despite the fact that (although based on viscometric measurements), it is in reality the specific hydrodynamic volume of polymers in the limit of infinite dilution and not a viscosity. Traditionally, intrinsic viscosities are determined by measuring the viscosity of the solvent ηo and that of solutions at various polymer concentrations c (mass per volume) and plotting (η − ηo)/(ηoc) versus c, where [η] results from the extrapolation to c → 0. With uncharged macromolecules, this procedure works well; however, for polyelectrolytes dissolved in pure water, it fails. For that reason, a new method, based on a thermodynamically inspired approach, was developed.1 It generalizes the intrinsic viscosity to arbitrary polymer concentrations as {η} = ∂ln η/∂c and determines [η] by extrapolating {η} to vanishing polymer concentration. The present contribution deals with the question, whether as the limiting value of {η} attained at the introduction of the polymer end of the composition scale could help the understanding of the flow behavior of polymer solutions. For reasons explained in the course of the presentation, this quantity is termed intrinsic bulkiness of the polymer. To study the feasibility of the concept, use is made of abundant published experimental data with respect to two aspects. The first one concerns the influences of temperature on the concentration dependence of η for constant polymer molecular weights. The second one keeps the temperature constant but varies the molar mass of the solute. For this new evaluation of © XXXX American Chemical Society
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THEORETICAL BACKGROUND A new thermodynamically inspired procedure,1 developed to determine the intrinsic viscosities of polyelectrolytes in the absence of extra salt, understands the intrinsic viscosity [η] as i ∂ln ηrel yz i ∂ln η zy zz zz [η] ≡ lim jjj = lim jjjj z c→0 c→0 γ→ γ→ {T , p , γ ̇ ̇ 0 k ∂c {T , p , γ ̇ ̇ 0 k ∂c
(1)
where η is the viscosity of the solution for the polymer concentration c (mass/volume) at constant T and p in the limit of zero shear rate γ̇; ηrel is the viscosity of the solution normalized to that of the solvent. Applying the above relation to determine the intrinsic viscosities of charged and uncharged polymers, it turned out helpful to relinquish the limitation of vanishing polymer concentration and to introduce a generalized intrinsic viscosity1 {η} according to i ∂ln ηrel zy i ∂ln η yz zz zz {η} ≡ jjj = jjjj z k ∂c {c , T , p , γ ̇ k ∂c {c , T , p , γ ̇
(2)
3
In this manner, it is possible to compare the contributions of electrostatic shielding via extra salt with the effects of selfshielding caused by rising c. The introduction of {η} also enables the viscometric determination of coil dimensions as a Received: February 11, 2019 Revised: March 13, 2019
A
DOI: 10.1021/acs.macromol.9b00282 Macromolecules XXXX, XXX, XXX−XXX
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Macromolecules function of polymer concentration;4 the reported result agree well with the corresponding light-scattering data. In the context of the evaluation of measured viscosities with respect to the above equations, the following relation turned out to be in the position to model the concentration dependence quantitatively over the full range of composition2 ln ηrel =
c ̃ + αc 2̃ 1 + βc ̃ + γc 2̃
for infinitely large degrees M→∞ is the limiting value of of polymerization N; A and B represent system-specific parameters. The temperature dependence of for systems in the vicinity of their glassy solidification obeys the following relation
(3)
where T* is the temperature at which approaches infinitely large values upon cooling and T→∞ stands for the value the system would adopt at very high hypothetical temperatures far above Tg.
c̃ stands for the reduced polymer concentration defined as c ̃ = c[η]
(4)
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α, β, and γ constitute the system-specific parameters, where at least one of them can in most cases be set to zero. The fact that eq 3 contains system-specific parameters in the denominator impedes their direct molecular interpretation. This drawback can, however, be eliminated5 by expanding the right-hand side of eq 3 with respect to the reduced concentration up to the fourth power
RESULTS AND DISCUSSION In one of the preceding papers,2 we have modeled the viscosities of polymer solutions over the full range of composition using eq 3. In that context, we have (by means of published experimental data) investigated the viscometric behavior as a function of temperature for polymers, which solidify as glasses upon cooling. These polymers were poly(methyl acrylate)6 (PMA), poly(vinyl acetate)7 (PVAc), and polystyrene8 (PS). The system-specific parameters of eq 3 were reported in the Supporting Information of ref 2, together with the intrinsic viscosities of the polymers at the temperatures of interest. One of the central findings2 was the possibility to determine glass transition temperatures Tg as a function of polymer concentration by means of the composition and temperature dependence of η. The central question raised in the present work reads as follows: How do intrinsic viscosities [η] and intrinsic bulkiness change as T approaches the glass transition temperature? This issue will be studied by means of the systems specified in Table 1.
ln ηrel = c ̃ + (α − β)c 2̃ + (β 2 − αβ − γ )c 3̃ − (β 3 − αβ 2 − 2γβ + αγ )c ̃4 ...
(5)
and comparing the result with the direct series expansion of ln ηrel ln ηrel = c ̃ + M 2c 2̃ + M3c 3̃ − M4c ̃4 ...
(6)
That procedure enables the conversion of the different systemspecific parameters into the coefficients M2 to M4 values, quantifying the contributions of binary, ternary, and quaternary interactions to the viscosity of the solutions according to M 2 = α − β ; M3 = β 2 − αβ − γ ; M4 = −β 3 + αβ 2 + 2γβ − αγ
(7)
Table 1. Systems and Characteristic Parametersa,b,c
The leading term M1 is unity and refers to isolated solute molecules. Inserting eq 2 into eq 3 yields the following expression for the composition dependence of the generalized intrinsic viscosity {η} [η](1 + 2αc ̃ + (αβ − γ )c 2̃ ) {η} = (1 + βc ̃ + γc 2̃ )2
solvent
(8)
N
Tg/K
1510 1160 640
9
290 30210 37111
T/°C
T→∞/mL
20−100 10−100 0−270
7 eq 11 13 eq 11 10 eq 11
g−1
DEP DEP DEB
PMA PVAc PS
solvent
polymer
N
Tg/K
T/°C
M→∞/mL
DMS5 XL
PDMS PIB
15−6620 5700−71000
15012 19813
30 25
7.7 Figure 8 1.2 Figure 11
g−1
a
DEP: diethyl phthalate, DEB: diethyl benzene, XL: xylene, DMS5: penta dimethyl siloxane. bPMA: poly(methyl acrylate), PVAc: poly(vinyl acetate), PS: polystyrene. cPDMS: poly(dimethyl siloxane), PIB: polyisobutylene; N: degree of polymerization.
Equation 8 covers the entire concentration dependence of {η} from infinite dilution up to the pure melt. On the solvent end of composition scale, the expression simplifies to the conventional intrinsic viscosity of the polymer in this solvent. The limiting value for the other end of the concentration scale is reached as c approaches the density ρ of the pure polymer. This value, designated by , represents the hydrodynamic specific volume of the polymer in the limit of the pure melt. It is related to the basic system-specific parameters by eq 9
The dependence of following expression
polymer
In the following, we pursue the process of solidification in terms of the generalized intrinsic viscosity, as it varies with polymer concentration at different temperatures. Figure 1 displays, as an example, the results for the DEP/PMA. The system-specific parameters required for that purpose are documented in the Supporting Information of ref 2. In contrast to the behavior of systems at temperatures that are sufficiently larger than Tg (cf. the results for DMS5/PDMS in Figure 7), the generalized intrinsic viscosities as function of c do not decline continuously but exhibit minima if the temperature distance to Tg falls below a critical value. Variation of Temperature for Given Molar Mass. To answer the question concerning the changes in [η] (lim {η} for
on chain length can be modeled by the
B
DOI: 10.1021/acs.macromol.9b00282 Macromolecules XXXX, XXX, XXX−XXX
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Figure 1. Concentration dependence of the generalized intrinsic viscosity {η} (cf. eq 8) for the system diethyl phthalate/poly(methyl acrylate). can be red from the curves by equating the polymer concentration to the density of the pure melt6 at the respective temperature.
Figure 4. As Figure 2 but for the system diethyl benzene/polystyrene.
11, the intrinsic bulkiness should assume very low limiting values for temperatures far off Tg (cf. T→∞ values of Table 1). This finding is in accordance with the directly determined values resulting from the measurements far away from glassy solidification, as will be demonstrated in the next section. For a sufficiently large distance of T from Tg, [η] is still of the same order as , which means that the specific hydrodynamic volumes of the flow unit on the solvent side and on the polymer side of the composition range are comparable. The finding that < [η] for PMA in contrast to PVAc surprises if one keeps in mind that that the chemical composition of PMA and PVAc is identical. These polymers differ only by the fact the side groups are attached to the backbone by a C atom in the former case but by an O atom in the latter case. This difference could be due to dissimilar activation energies for the viscous flow as a function of polymer concentration. Recalling that constitutes the specific hydrodynamic volume of the fragment of a polymer molecule that moves independently as the solvent content of the solution approaches zero, the intrinsic bulkiness should necessarily interrelate with the glass transition. The last three graphs testify that this is indeed the case. However, the temperatures at which the values approach infinity are on the average approximately 30 °C lower than Tg. This behavior is most likely due to nonequilibria effects that show up in the vicinity of glass transition; they impede the extrapolation from the behavior in concentrated solutions to the pure melt. In other words, under these conditions, the viscosity is no longer a variable of state. This hypothesis is corroborated if one plots the inverse of as a function of Tg/T (Figure 5). extrapolates linearly According to Figure 5, the inverse of to Tg/T = 1 if the temperature is at least 30 °C higher than the corresponding glass transition temperature. From this finding, one can conclude that η remains a function of state within this temperature range and that the degree of nonequilibrium behavior outside this range varies from system to system. For the assessment of the results, one needs to keep in mind that the number of data points in the vicinity of Tg decreases from PS via PVAc to PMA. Furthermore, the glass transition temperatures reported for PMA vary considerably. To superimpose the dependence shown in figure for this polymer, its Tg value was assumed to be −3 °C instead of the reported +3 °C. Isothermal Variation of Degree of Polymerization. When studying the viscosity of polymer solutions over the full range of composition,2 one of the central questions concerns
c → 0) and in (lim {η} for c → ρ) on the approach of Tg, the corresponding endpoints of the dependencies of Figure 1 are in Figure 2 evaluated in terms of a reduced Arrhenius plot.
Figure 2. Normalized Arrhenius plot for the temperature dependence of the intrinsic viscosities and of the intrinsic bulkiness, , for the system diethyl phthalate/poly(methyl acrylate).
The next two graphs show the results for the other systems under investigation (Figure 3 and Figure 4).
Figure 3. As Figure 2 but for the system diethyl phthalate/poly(vinyl acetate).
The temperature dependencies of the intrinsic viscosities displayed in Figures 2−4 are in all cases approximately linear and testify exothermal heats of dilution. The intrinsic bulkiness on the other hand increases strongly as the temperature falls and approaches infinitely large values at the characteristic temperature T*; the curves shown for in the above modified Arrhenius plots are modeled according to eq 11 using Tg/T instead of 1/T as independent variable. As stated by eq C
DOI: 10.1021/acs.macromol.9b00282 Macromolecules XXXX, XXX, XXX−XXX
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Figure 5. Normalized Arrhenius plot for the inverse intrinsic bulkiness of the polymers indicated in the graph; the Tg values reported for PMA vary considerably (from 3 to 10 °C); to match the curve for PMA with that of PS and PVAc, a value of −3 °C was assumed for the plot.
Figure 7. Concentration dependence of the generalized intrinsic viscosity {η} (cf. eq 8) for solutions of PDMS samples differing in their degree of polymerization N in the pentamer DMS5. can be red from the curves by equating the polymer concentration to the density of the pure melt at the respective temperature.
the influence of the molar mass M of the polymer on the parameters of eq 3 at temperatures well above their Tg. This section deals with the question whether, and if so how, the intrinsic bulkiness varies with M. The two examples studied here are solutions of PDMS in its pentamer14 and of PIB in xylene.15 The evaluation of the primary viscosity data is for the system DMS5/PDSM shown in Figure 1 of ref 2. The following two parameters of eq 3 are required to model these curves, namely, β and γ; α can be set to zero. Figure 6 displays their dependence on the inverse degree of polymerization and testifies that limiting values are approached as the chains become infinitely large.
To check how depends on the chain length of the polymer, the generalized intrinsic bulkiness is in Figure 8 plotted as a function of the inverse degree of polymerization.
Figure 8. Intrinsic bulkiness of PDMS as a function of the inverse degree of polymerization.
, in The result proves the existence of a limiting value of accordance with the corresponding dependence of the parameters β and γ. For the present system, the intrinsic bulkiness increases as the chains get longer. The data published for the viscosity of PIB solutions as a function composition for different molecular weight15 refer predominantly to xylene as the solvent but also give some examples for decalin. The results of the present evaluation turn out to be so similar that only those for xylene are reported here. Figure 9 shows the composition dependencies of ln ηrel and their modeling according to eq 3 using the sample nomenclature of the authors.
Figure 6. Dependence of the system-specific parameter of eq 3 on the inverse degree of polymerization for the system DMS5/PDMS at 30 °C.
Figure 7 demonstrates that the generalized intrinsic viscosity decreases rapidly with rising polymer concentration for all representatives of this system. The endpoints (vanishing solvent content of the mixture) of the individual curves yield intrinsic bulkiness.
Figure 9. Evaluation of viscosity data published for the system xylene/ PIB15 according to eq 3. D
DOI: 10.1021/acs.macromol.9b00282 Macromolecules XXXX, XXX, XXX−XXX
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Macromolecules The limiting slope of the curves of Figure 9 for vanishing polymer concentration yields the intrinsic viscosities of the different samples according to eq 3. For both solvents, β suffices for the modeling; the dependence of β on the inverse of the degree of polymerization reads: β = 0.047 − 50/N; this means that β increases with rising molecular weight for both solvents, in contrast to the behavior of the system DMS5/ PDMS. The evaluation of the viscosity data published for the systems xylene/PIB and decalin/PIB yield the Kuhn−Mark− Houwink relation shown in Figure 10; it is the same for both solvents, and for decalin it is identical with that reported in the Polymer Handbook.13
reduce ). That of PIB, on the other hand, should have the ). opposite effect (increase The most striking feature of the present findings is the fact that the values resulting for systems far above Tg are in their order of magnitude identical with the T→∞ values extrapolated from measurements in the vicinity of Tg to high temperatures for other systems (cf. Table 1). This observation can be taken as an indication of the trustworthiness of the present approach. To check in more detail, whether the interpretation of as the intrinsic bulkiness of the polymer in the molten state is reasonable, we compare these values with the intrinsic viscosities [η] that are calculated by means of the Kuhn− Mark−Houwink relations for the particular systems13 from published Ment values. Doing so for PDMS, the reported Ment = 12 00016 yields [η] = 10.2 mL g−1 as compared with ranging from 4.4 to 7.7. For PIB, the corresponding Ment reads 10 50016 and yields [η] = 7.4 mL g−1 (cf. Figure 10) as compared with values ranging from 1.2 to 9.0. In view of the uncertainties (Kuhn−Mark−Houwink relation, Ment, and last but not least ) and the fact that this comparison ignores the chain length dependence of Ment, the agreement is unexpectedly good.
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CONCLUSIONS AND OUTLOOK Prior to a common discussion of all results, it appears expedient to recall the physical meaning of the different specific hydrodynamic volumes that can be obtained from the composition dependence of the viscosity of polymer solutions. [η], {η}, and are equally interpreted as measures for the size of the average flow unit or equivalently for the bulkiness of the solute molecules at the particular concentrations. [η] refers to c = 0, {η} to arbitrary concentration, and to c = ρpolymer. In the limit of infinite dilution, all segments are necessarily moving conjointly because of the bonds connecting them, where the value of [η] reflects the solvent quality; favorable interactions lead to higher and unfavorable interactions to lower values of the intrinsic viscosity. The generalized intrinsic viscosity {η} is normally less than [η] because the coils begin to overlap as c rises. The reason for this reduction lies in the formation of volume elements containing monomers of more than one macromolecule, which are separated by shear fields and thus reduce the bulkiness of the solute. As the polymer concentration increases still further and approaches the density of the pure polymer, the generalized intrinsic viscosity becomes the intrinsic bulkiness. refers to the pure polymer and should be modified by the thermodynamic quality of the solvent by analogy to [η]. Different features may result from the fact that the intrinsic viscosity refers to the establishment of a microphase equilibrium17 (polymer coils are in contact with a large surplus of solvent and change their size as a function of concentration); in contrast, the intrinsic bulkiness refers to isolated solvent molecules (with much less options to react on the variation of c), which are surrounded by a large surplus of polymer segments. The variation of {η} with polymer concentration depends primarily on the temperature distance to the glass transition temperature of the polymer: Far above Tg one observes a (cf. Figure 7). In monotonous decline of {η} from [η] to contrast to this situation, the generalized intrinsic viscosity passes a minimum as a function of c as the distance of T from Tg becomes smaller (cf. Figure 1). For low polymer
Figure 10. Kuhn−Mark−Houwink relation for solutions of PIB in xylene or in decalin, obtained from viscosity data15 up to approximately 40 wt %.
Figure 11 shows how the intrinsic bulkiness of PIB in xylene depends on chain length, together with the results for the
Figure 11. Intrinsic bulkiness of PIB and of PDMS as a function of the inverse degree of polymerization.
with falling N system DMS5/PDMS. The strong increase of is surprising; unfortunately, it is presently impossible to learn more about this behavior because of lacking data for lower molecular weights. Remarkably, the dependence of on the chain length differs qualitatively for the two systems under investigation. This dissimilar behavior ought to be attributed to the particular nature of the endgroups of the polymer chains; their influence can be rationalized in terms of their contribution to the free volume of the system or in terms of their steric structure. Unfortunately, information on the chemical nature of the endgroups of PDMS and PIB that were used for the reported measurement is lacking. In both cases, there exist too many options to guess. According to the present data, the endgroups of PDMS should increase the free volume of the system (i.e., E
DOI: 10.1021/acs.macromol.9b00282 Macromolecules XXXX, XXX, XXX−XXX
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(6) Fujita, H.; Maekawa, E. Viscosity behavior of system polymethyl acrylate and diethyl phthalate over complete range of composition. J. Phys. Chem. 1962, 66, 1053−1058. (7) Kishimoto, A. E. Diffusion + viscosity of polyvinyl acetatediluent systems. J. Polym. Sci., Part A: Gen. Pap. 1964, 2, 1421−1439. (8) Fox, T. G.; Flory, P. J. Viscosity-molecular weight and viscositytemperature relationships for polystyrene and polyisobutylene. J. Am. Chem. Soc. 1948, 70, 2384−2395. (9) Fernandez-Garcia, M.; Lopez-Gonzalez, M. M. C.; BarralesRienda, J. M.; Madruga, E. L.; Arias, C. Effect of copolymer composition and conversion on the glass-transition of methyl acrylatemethyl methacrylate copolymers. J. Polym. Sci., Part B: Polym. Phys. 1994, 32, 1191−1203. (10) Matsuoka, S. Thermodynamic theory of viscoelasticity. J. Therm. Anal. 1996, 46, 985−1010. (11) Rieger, J. The glass transition temperature of polystyrene Results of a round robin test. J. Therm. Anal. 1996, 46, 965−972. (12) Chou, C.; Yang, M. H. Structural effects on the thermalproperties of pdps pdms copolymers. J. Therm. Anal. 1993, 40, 657− 667. (13) Brandrup, J.; Immergut, E. H.; Grulke, E. A. Polymer Handbook, 4th ed.; Wiley: New York; Chichester, 2004. (14) Kataoka, T.; Ueda, S. Viscosity of PolydimethylsiloxanePentamer Systems. J. Polym. Sci., Part A-2: Polym. Phys. 1967, 5, 973− 986. (15) Johnson, M. F.; Evans, W. W.; Jordan, I.; Ferry, J. D. Viscosities of concentrated polymer solutions.2. polyisobutylene. J. Colloid Sci. 1952, 7, 498−510. (16) Fetters, L. J.; Lohse, D. J.; Milner, S. T.; Graessley, W. W. Packing length influence in linear polymer melts on the entanglement, critical, and reptation molecular weights. Macromolecules 1999, 32, 6847−6851. (17) Wolf, B. A. Making Flory-Huggins Practical: Thermodynamics of Polymer-Containing Mixtures. Adv. Polym. Sci. 2011, 238, 166 ISBN 978-3-642-17682-1.
concentration, the lubricating effect of solvent molecules dominates and the bulkiness decreases, but for sufficiently high polymer concentration, the reduction of segmental motion due to intersegmental interactions may lead to a dramatic increase in {η}, such that the intrinsic bulkiness can become substantially larger than [η] (cf. Figure 4). [η] and are modified by the endgroups of the polymer. The observed effects (cf. Figure 11) can be rationalized in terms of the free volume changes associated with the replacement of middle groups against endgroups. In the absence of significant chemical or structural dissimilarities between these unities, the free volume will become larger by this substitution because of the removal of a chemical bond. The intrinsic bulkiness should therefore increase as the chains become longer. On the other hand, spatial factors or favorable interactions of the endgroups with other components of the mixture may lead to a lowering of the free volume. In such cases, should decrease as M becomes larger; Figure 11 shows an example. Because of lacking information on the chemical nature of the endgroups, it is presently impossible to rationalize these observations. In summary, it can be stated that the newly introduced intrinsic bulkiness of polymers promises interesting additional insight concerning the chain mobility in the molten state. According to orienting calculations, it provides access to entanglement molecular weights (if an adequate Kuhn−Mark− Houwink relation is available) without the need to perform experiments within the non-Newtonian flow regime. Moreover, enables the noncalorimetric determination of glass transition temperatures (cf. Figure 5). It should, however, go without saying that further studies are required to validate the methods for routine applications. In particular, it appears mandatory to study additional polymer/solvent systems with diverse architecture of the macromolecules and well-defined endgroups. Furthermore, it is necessary to vary the thermodynamic quality of the solvents systematically and to analyze the predictive power of the present approach.
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. ORCID
Bernhard A. Wolf: 0000-0002-4051-7114 Notes
The author declares no competing financial interest.
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REFERENCES
(1) Wolf, B. A. Polyelectrolytes revisited: Reliable determination of intrinsic viscosities. Macromol. Rapid Commun. 2007, 28, 164−170. (2) Wolf, B. A. Viscosity of Polymer Solutions over the Full Range of Composition: A Thermodynamically Inspired Two-Parameter Approach. Ind. Eng. Chem. Res. 2015, 54, 4672−4680. (3) Wolf, B. A. Coil overlap in moderately concentrated polyelectrolyte solutions: effects of self-shielding as compared with salt-shielding as a function of chain length. RSC Adv. 2016, 6, 38004− 38011. (4) Suresha, P. R.; Badiger, M. V.; Wolf, B. A. Polyelectrolytes in dilute solution: viscometric access to coil dimensions and salt effects. RSC Adv. 2015, 5, 27674−27681. (5) Bercea, M.; Wolf, B. A. Intrinsic Viscosities of Polymer Blends: Sensitive Probes of Specific Interactions between the Counterions of Polyelectrolytes and Uncharged Macromolecules. Macromolecules 2018, 51, 7483−7490. F
DOI: 10.1021/acs.macromol.9b00282 Macromolecules XXXX, XXX, XXX−XXX
