Activation Thermodynamics and Concentration Scaling of the Viscosity

Article pubs.acs.org/Macromolecules

Activation Thermodynamics and Concentration Scaling of the Viscosity of Unimeric and Aggregated Poloxamer EO93PO54EO93 in Water François Quirion* Consultant R&D Physicochimie, 261 Boulevard Pereire, 75017 Paris, France S Supporting Information *

ABSTRACT: The viscosity of the aqueous solutions of Poloxamer EO93PO54EO93 (a triblock copolymer of ethylene oxide, EO, and propylene oxide, PO) was determined at concentrations ranging from 0.002 to 0.12 g·mL−1 at 0.9, 8.9, and 20.0 °C where only unimers are present, at 24.9 and 30.3 °C where the unimers are in equilibrium with aggregates and at 39.8 and 49.8 °C where only aggregates are present. The reciprocal of the intrinsic viscosity derived from Huggins, Kraemer, Martin and Phillies’s models is compared to the experimentally determined overlap concentration. A simple model that accounts for intermolecular interactions provides better estimates of the experimental overlap concentration. The concentration scaling exponents of the viscosity at low and high concentrations are compared to the predictions for entangled polymer chains. It is suggested that at low temperature, the Poloxamer molecules evolve from dilute unimers to semidilute unentangled unimers. At high temperature, Poloxamer molecules evolve from dilute aggregates to semidilute entangled aggregates. Finally, the apparent molar contribution of the Poloxamer to the activation free energy and enthalpy for viscous flow can be reproduced with a mass-action model corresponding to the cooperative association of unimers to form aggregates. The model confirms that the association of the Poloxamer molecules is endothermic with a low molecularity.

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molecules. Using 1H NMR relaxation, Cau and Lacelle4 conclude that P137 molecules are in the unimeric state below the transition and form aggregates with a PPO hydrophobic core above the transition. Kurumada and Robinson5 investigated the viscosity of Pluronic F127, EO100PO65EO100, which has a composition very similar to P317, and their results are consistent with unimeric molecules at low temperature and aggregates at high temperature. Using differential scanning calorimetry, PhamTrong et al.6 deduced that the fraction of Pluronic F127 molecules in the unimeric state decreases sharply as the temperature increases above the onset of the endothermic association of the copolymer. Quirion et al.7 successfully applied a mass-action model based on the cooperative association of Pluronic F127 molecules to simulate the concentration dependence of its apparent molar volume and heat capacity. Aqueous solutions of P317 are thus good candidates for the investigation of the impact of copolymer association on the viscosity. The concentration dependence was determined from 0.002 to 0.12 g·mL−1 at 0.9, 8.9, and 20.0 °C where P317 is in the unimeric state, at 39.8 and 49.8 °C where it is in the

INTRODUCTION Viscosity is sensitive to the molecular organization and interactions of polymer molecules in solutions.1a Its concentration dependence gives information on the hydrodynamic volume of the moving polymer chains and their interactions as they overlap and eventually entangle. The viscosity can also be expressed in terms of the activation free energy for viscous flow and its temperature dependence leads to the activation enthalpy and entropy for viscous flow. Jamieson and Simha2 have recently reviewed the concentration scaling of the viscosity of dilute, semidilute and concentrated polymer solutions. There seems to be few investigations dealing with the viscosity of associating copolymers and even less on the activation thermodynamics for the viscous flow of such copolymers. This article reports the concentration and temperature dependence of the viscosity of Poloxamer solutions. Poloxamers are triblock copolymers of ethylene oxide (EO = CH2CH2O) and propylene oxide (PO = CH(CH3)CH2O) and many of these copolymers are known to associate into aggregates as the temperature of their aqueous solutions is raised. It is the case for EO93PO54EO93, a Poloxamer that will be referred to as P317. For that copolymer, Williams et al.3 report a cooperative association around 24 °C that occurs with a positive change of the apparent molar volume of the copolymer. That transition would involve either many copolymer molecules or large segments of individual copolymer © 2015 American Chemical Society

Received: October 12, 2015 Revised: November 7, 2015 Published: November 19, 2015 8629

DOI: 10.1021/acs.macromol.5b02247 Macromolecules 2015, 48, 8629−8640

Macromolecules

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aggregated state and at 24.9 and 30.3 °C where P317 unimers are in equilibrium with aggregates. The activation free energy and enthalpy for viscous flow were determined below and above the overlap concentration from 10 °C where P317 is in the unimeric state to 50 °C where it is in the aggregated state.

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RESULTS AND DISCUSSION The temperature−concentration coordinates of the endothermic association of P317 unimers into aggregates have been reported by Dumont8 using expansibility and Lebeuf11 using differential scanning calorimetry, DSC, and compressibility experiments and their results are reproduced in Figure 2. The

EXPERIMENTAL SECTION

Poloxamer EO93PO54EO93 is a triblock copolymer of ethylene oxide, EO = (CH2CH2O), and propylene oxide, PO = (CH(CH3)CH2O) from Polysciences. Its number-average molecular weight, Mn = 11360 g·mol−1, and composition, xPO = 0.224 (27.6% w/w), were determined by Williams et al.3 using osmometry and 1H NMR, respectively. It could be described as Poloxamer 317 because the molecular weight of the central PPO segment is 3132 g·mol−1 and the weight percent of the PEO side chains is 72.4%. In this article, this Poloxamer will be referred to as P317. The aqueous solutions were prepared by weight and the final concentrations were expressed in g·mL−1 using the density of the solutions. The viscosity was determined using an Ubbelohde type viscometer (No. 1) thermoregulated in a water bath at ±0.02 °C. In this investigation the flow time, t, varied from 53.87 ± 0.05 s for water around 60 °C to 1645 ± 1 s for P317 at 0.12 g·mL−1 and 0.9 °C and the viscosity, η in mPa·s, was calculated with eq 1.

Figure 2. Temperature−concentration coordinates of the endothermic transition of the aqueous P317 solutions obtained from DSC11 with a Microcal MC2 at 1 °C/min (●), compressibility11 (▲), and expansibility8 (■). The peak (− − −) and the onset and endset (···) of the endothermic transition of Pluronic F127 were obtained by Pham Trong et al.6 with a Micro DSC III at 0.1 °C/min.

⎛ E⎞ η = k(t )ρt = ⎜D − 2 ⎟ρt ⎝ t ⎠ k(tw) =

⎛ E ⎞ = ⎜D − 2 ⎟ ρw , lit tw ⎝ tw ⎠

Article

ηw , lit

(1) −3

The density of P317 solutions, ρ in g·cm , was deduced from the data of Dumont8 (see the Supporting Information). The constant D is the viscometer’s constant at infinite flow time and the constant E reflects the kinetic energy correction that becomes necessary at short flow times.9 To obtain the viscometer’s constants D and E, k(tw) is calculated using the experimental flow time of water, tw, at a given temperature and the literature values10 of the viscosity, ηw,lit, and density, ρw,lit, of water at that temperature. Figure 1 shows k(tw) vs tw−2

transition of P317 is very close to the peak transition reported by Pham Trong et al.6 for Pluronic F127 in water. It is thus reasonable to assume that the onset and endset temperatures of the transition of Pluronic F127 will be similar to those of P317 and they are presented in Figure 2. The experimental data sets lie in three distinct regions of the phase diagram. At 0.9, 8.9, and 20.0 °C, P317 molecules are in the unimeric state. At 24.9 and 30.3 °C, the unimers are in equilibrium with the aggregates. Finally, at 39.8 and 49.8 °C P317 molecules are in the aggregated state. The analysis of the concentration dependence of the viscosity is presented in the first part of this section. The data are interpreted in terms of intrinsic viscosity, interaction parameter, concentration scaling and overlap concentration at given temperatures. The second part discusses the activation properties for viscous flow of P317 solutions extracted from the temperature dependence of the viscosity at 0.01 and 0.095 g·mL−1. The experimental values of (η vs c)T and (η vs T)c are reported in the Supporting Information. Concentration Dependence of the Viscosity at Constant Temperature. The concentration dependence of the viscosity gives information on the hydrodynamic volume, the intermolecular interactions and the overlap concentration between the dilute and semidilute regimes. In most investigations, the overlap concentration of a given polymer in solution is assumed to be the reciprocal of its intrinsic viscosity at infinite dilution, [η]0, determined in the same solvent and at the same temperature. While this is probably a reasonable assumption for systems that remain in the same state with small intermolecular interactions, it may not hold for systems experiencing strong attractions leading to molecular association. In this section, the intrinsic viscosity and the intermolecular interactions of P317 in water are determined using different models to represent the evolution of the viscosity with the concentration. The overlap concentration

Figure 1. Determination of the viscometer’s constants D and E using the literature values of the viscosity and density of water and the experimental flow time of water, tw, from 15 to 55 °C.

for water from 15 to 55 °C. The linear regression leads to D = 9.298 × 10−3 mm2·s−2 and E = 1.254 mm2. The water viscosities determined with eq 1 agree within ±0.2% with the literature values. Two types of experiments were performed. First, the viscosity was determined as a function of the concentration of P317 at 0.9, 8.9, 20.0, 24.9, 30.3, 39.8, and 49.8 °C. Second, water and two solutions at 0.01 and 0.095 g·mL−1 were analyzed as a function of temperature from 10 to 60 °C with steps of 5 °C for water and 2 °C for the P317 solutions. 8630


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Figure 3. (A) Huggins plots of ηred vs c. Symbols correspond to 0.9 (◆), 8.9 (■), 20.0 (▲), 24.9 (×), 30.3 (∗), 39.8(●), and 49.8 °C (+). Solid lines: Range used for the linear regressions. Dashed lines: extrapolations to c → 0 from the linear range. (B) Scaling over the entire concentration range using the normalized parameters ηred[η]H−1 and c[η]H.

Figure 4. (A) Kraemer plots of the inherent viscosity vs c. Symbols correspond to 0.9 (◆), 8.9 (■), 20.0 (▲), 24.9 (×), 30.3 (∗), 39.8 (●), and 49.8 °C (+). Solid lines: Range used for the linear regressions; Dashed lines: extrapolations to c → 0 from the linear range. (B) Scaling over the entire concentration range using the normalized parameters ηinh[η]K−1 and c[η]K.

Figure 5. (A) Martin plots of the natural logarithm of the reduced viscosity ln (ηred) vs c. Symbols correspond to 0.9 (◆), 8.9 (■), 20.0 (▲), 24.9 (×), 30.3 (∗), 39.8 (●), and 49.8 °C (+). Solid lines: Range used for the linear regressions; Dashed lines: extrapolations to c → 0 from the linear range. (B) Scaling over the entire concentration range using the normalized parameters ln(ηred[η]M−1) and c[η]M.

where ηr is the viscosity of the polymer solution relative to the viscosity of the solvent. When the polymer concentration, c, is expressed in g·mL−1, the intrinsic viscosity has units of mL·g−1 and it reflects the hydrodynamic volume occupied by 1 g of polymer at infinite dilution under zero shear. Equation 2 can be extended to higher concentrations by developing it into a polynomial of the normalized concentration, c[η]0:

deduced from these intrinsic viscosities is then compared to the experimentally determined overlap concentration. The overlapping of polymer chains is also associated with a change in the concentration scaling of the viscosity. The scaling exponents were thus determined before and after the overlap concentration and their values are compared with the scaling theories. Huggins, Kraemer, and Martin Models. The intrinsic viscosity of polymers at infinite dilution, [η]0, is usually defined as ⎛ η − 1⎞ lim⎜ r ⎟ = [η]0 c → 0⎝ c ⎠

ηr = 1 + [η]0 c + k 2([η]0 2 c 2) + k 3([η]0 3 c 3) + ...

(3)

where k1 = 1 and ki are constants that account for intermolecular interactions. The Huggins, Kraemer, and Martin models propose different approaches to linearize eq 3 at low concentration. Huggins defines the reduced viscosity, ηred:

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Table 1. Summary of the Intrinsic Viscosities at Infinite Dilution, [η], Interaction Constants, k, Scaling Exponents S, L, and H, and the Experimental Overlap Concentration, cexp*, Deduced from the Analysis of the Viscosity Data at Constant Temperaturea T, °C

kH

[η]H

0.9

23.3 0.56 (0.01−0.06) 22.1 0.55 (0.01−0.06) 20.5 0.54 (0.006−0.06) 18.4 0.62 (0.008−0.04) 15.5 0.89 (0.006−0.02) 11.1 2.24 (0.008−0.04) 10.2 2.24 (0.01−0.04)

8.9 20.0 24.9 30.3 39.8 49.8

[η]K

kK

24.2 −0.09 (0.02−0.10) 23.0 −0.10 (0.02−0.08) 21.0 −0.09 (0.01−0.08) 18.9 −0.03 (0.01−0.06) 15.6 0.21 (0.02−0.08) 12.8 0.46 (0.02−0.10) 11.5 0.54 (0.02−0.08)

kM

[η]S

S

Lb

Hc

cexp*

25.7 0.30 (0.02−0.12) 24.2 0.31 (0.02−0.12) 21.4 0.36 (0.01−0.10) 18.7 0.50 (0.008−0.06) 15.6 0.79 (0.006−0.04) 11.7 1.37 (0.008−0.08) 10.7 1.45 (0.008−0.08)

22.5

0.99

1.16

1.85

0.046

0.99

1.15

1.81

0.047

1.02

1.10

2.21

0.047

1.05

1.14

3.38

0.057

1.07

1.14

3.74

0.052

1.16

1.24

3.67d

0.054

1.18

1.25

3.82

0.055

[η]M

(≤0.04) 21.3 (≤0.04) 21.3 (≤0.02) 20.0 (≤0.02) 18.3 (≤0.06) 17.1 (≤0.06) 16.1 (≤0.08)

The ranges below the parameters correspond to the inclusive concentration ranges used to fit the data with the respective model. All [η] and aL are in mL·g−1 and all concentrations are in g·mL−1. bDetermined at c ≤ 0.02 g·mL−1. cExponent evaluated at 0.12 g·mL−1. dthe exponent H at 39.8 °C was evaluated using a specific viscosity extrapolated at 0.12 g·mL−1 from the data at lower concentrations.

a

Figure 6. (A) Temperature dependence of the intrinsic viscosity deduced from the different models: Huggins (◆), Kraemer (■), and Martin (●). The dotted line represents the average intrinsic viscosity for these three models, [η]HKM. (B) Temperature dependence of the interaction constants: kH (◆), kK (■), and kM (●).

ηred =

ηsp c

=

(ηr − 1) c

= [η]H + kH[η]H 2 c + ...

These three models all lead to [η]0 at infinite dilution so that [η]0 ≈ [η]H ≈ [η]K ≈ [η]M. However, the three models account differently for intermolecular interactions and kH ≠ kK ≠ kM. The Huggins, Kraemer and Martin plots are presented in part A of Figures 3, 4 and 5, respectively. The intrinsic viscosities, interaction parameters and the range of concentration used for the determination of these values are reported in Table 1. The evolution of the intrinsic viscosity with the temperature as deduced from the Huggins, Kraemer and Martin models is shown in Figure 6A. As expected, the three models lead essentially to the same values which can be represented by the average of the three models, [η]HKM. The slow decrease of [η] from 0.9 to 20.0 °C is coherent with a continuous loss of hydrophobic hydration of P317 resulting in a tighter packing of the unimers as the temperature increases. From 24.9 to 30.3 °C, the intrinsic viscosity decreases more steeply in agreement with the transition from the region of unimers to the region of aggregates. At 39.8 and 49.8 °C, the intrinsic viscosity continues a slow decrease, in accordance with further loss of hydrophobic hydration and tighter packing of the PPO core of P317 aggregates. The overall evolution of the intrinsic viscosity suggests that, at a given temperature, P317 molecules occupy a smaller volume in the aggregated state than in the unimeric state.

(4)

where ηsp is the specific viscosity and kH is the Huggins constant accounting for the intermolecular interactions between the polymer molecules. Knowing that at low concentration, ηsp ≈ ln(ηr), Kraemer defines the concentration dependence of the viscosity in terms of the inherent viscosity, ηinh: ηinh =

ln(ηr ) c

= [η]K + kK[η]K 2 c + ...

(5)

where kK is the Kraemer constant that accounts for the intermolecular interactions between the polymer molecules. Similarly, Martin uses the fact that at low concentration, (1 + k[η]c) ≈ e(k[η]c) and rewrites the Huggins equation as ηred = [η]M e(kM[η]M c)

(6)

where kM is an interaction constant that accounts for the intermolecular interactions. The Huggins and Kraemer model are often presented together for cross validation of the intrinsic viscosity. The Martin model is believed to fit the viscosity data over a wider concentration range than the Huggins model.2 8632


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Figure 7. (A) Double logarithmic plot of Phillies stretched exponential model, ln (ηr) vs c. Symbols correspond to 0.9 (◆), 8.9 (■), 20.0 (▲), 24.9 (×), 30.3 (∗), 39.8 (●), and 49.8 °C (+). Solid lines: Range used for the linear regressions. (B) Scaling over the entire concentration range as a function of the normalized concentration, c[η]S.

exponential model (Figure 7B) also presents two master curves, one for the unimers at low temperature and a second one for the aggregates at high temperature. This time, however, the two master curves are relatively close suggesting that the stretched exponential model is an improvement compared to the Huggins, Kraemer, and Martin models. At low concentration, the data at 24.9 and 30.3 °C are closer to the unimeric master curve while at high concentration they are closer to the aggregate master curve. This is in accordance with of shift of the equilibrium from unimers to aggregates as the temperature increases. Figure 8 shows how [η]S and S evolve with the temperature. At 0.9, 8.9, and 20.0 °C, [η]S is less than 10% lower than

The interaction parameters (see Figure 6B) show a steady state at T ≤ 20.0 °C and a sudden increase around 25 °C to higher values associated with strong attractions. The Huggins constant of P317 in the unimeric state corresponds well to the values obtained in an earlier study12 for Pluronic F38, P85, and P104 at 5 °C (0.56 < kH < 0.65). The increase of kH at high temperature was also observed for P85 and P104 at 45 °C which also form aggregates at high temperature. The evolution of the Huggins constant of P317 from 0.54 to 2.2 is thus coherent with a unimeric state at low temperature evolving toward the aggregated state at high temperature. This is better seen in part B of Figures 3, 4, and 5 where the three models are normalized with their respective value of [η]. The interest of normalization is to generate master curves that are universal for all conditions. Obviously, this is not the case for any of the three models presented. Nevertheless, the normalized Huggins, Kraemer and Martin models present a master curve for the data at 0.9, 8.9, and 20.0 °C, where P317 is in the unimeric state, and a second master curve at 39.8 and 49.8 °C, where it is in the aggregated state. In between, the data evolve from one state to the other. This suggests that the unimeric and aggregated states scale differently with P317 concentration. Phillies Stretched Exponential Model. On the basis of his work on the diffusion of polymer molecules in solution, Phillies13 suggested that the concentration dependence of the viscosity of polymer solutions is described by a stretched exponential model. S

Figure 8. Temperature dependence of the exponent S (◆) and the intrinsic viscosity deduced from the Phillies stretched exponential model (▲). The average intrinsic viscosity from Huggins, Kramer, and Martin models, [η]HKM, (---) is added for comparison.

S

ηr = e αPc = e([η]S c)

(7)

According to Phillies, αP is related to the strength of whole chain−whole chain hydrodynamic interactions so that it increases with the increase of the chain size. It has been suggested2 that αP is related to the intrinsic viscosity of the polymer molecules in solution. The exponent S is related to chain contraction and it is near unity for small rigid chains and decreases to about 1/2 for large flexible chains. In this investigation, it is assumed that αP = [η]SS so that eq 7 can be expressed in terms of the normalized concentration c[η]S. That model was applied to the viscosity of P317 solutions (see Figure 7A) and the parameters S and [η]S are reported in Table 1. The stretched exponential model is better at fitting the data at low concentration and it extends to fairly high concentration, in accordance with Phillies’s view13 that the model can fit the viscosity data from dilute to semidilute unentangled polymer solutions. The normalized stretched

[η]HKM. The assumption that αP = [η]SS thus seems reasonable, at least at low temperature where P317 molecules are in the unimeric state. At higher temperatures, [η]S is up to 50% higher than [η]HKM suggesting that the intermolecular interactions are reflected differently in the stretched exponential model compared to the HKM models for the aggregated state. According to Phillies et al.,13 small polymer chains, such as P317, are less subject to contraction and have higher values of S. The value of S ≈ 1 at low temperature is thus coherent with the behavior of small unimeric copolymer chains. The exponent S is also a measure of the compressibility of the polymer chain in the solution. The lower the value of S the more compressible is the polymer chain. Lebeuf11 reports a decrease of the apparent compressibility of P317 as the concentration increases at 52 °C. This suggests that the entanglement of the P317 aggregates at high temperature makes the solution stiffer, in 8633


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agreement with the observed increase of the exponent S at high temperature. Interestingly, the intermolecular interactions contribute to the value of [η]S while they do not for [η]HKM (see Supporting Information). For instance, at T ≤ 20.0 °C we find that S ≤ 1 and [η]S ≤ [η]HKM, an indication that the intermolecular interactions lead to the contraction of the hydrodynamic volume of P317 unimers. At T > 24.9 °C, S > 1 and [η]S > [η]HKM, an indication that the intermolecular interactions lead to larger and stiffer aggregates. Chain Dimensions. The evolution of the chain dimensions as the concentration increases can also be addressed through the effective intrinsic viscosity [η]c, i.e. a measure of the hydrodynamic volume at concentration c. [η]c 1 d⎣⎡ln(ηr )⎤⎦ = [η]0 [η]0 dc

c→0

lim(ηsp) = (aH c)H

c>c*

(9)

In eq 9, aL and aH are arbitrary constants and L and H are the scaling exponents at low and high concentrations, respectively. The exponent L and H were obtained by fitting the specific viscosity at low (c ≤ 0.02 g·mL−1) and high (c = 0.12 g·mL−1) concentrations with eq 9 and the results are reported in Table 1. Figure 10A presents double logarithmic plots of ηsp vs c at low concentration (c ≤ 0.02 g·mL−1) and Figure 10B presents the same data against the normalized concentration [c·aL]. Interestingly, there are still two master curves even at low concentration. One for the unimeric state at low temperature and one for the aggregated state at high temperature. As for Phillies’s model at low concentration, the data at 24.9 and 30.3 °C fall on the master curve for the unimers. Meilleur et al.12 determined the viscosity of Pluronics F38, P85, P104, P103, L122, and L101 in water and they found that these small molecular weight copolymers (3800 < Mw (g· mol−1) < 5900) are in the unimeric state at 5 °C (see Supporting Information for the original viscosity data). Figure 10C shows that the average scaling exponent of these unimeric copolymers is 1.146 ± 0.006 in excellent agreement with ⟨L⟩ = 1.14 ± 0.02 for P317 at 0.9 and 9.8 °C. These results are also in good agreement with the exponent 1.11 and 1.08 obtained by Hong et al.15 for PVA in water and in N-methyl pyrrolidone at 30 °C, respectively, and 1.2 obtained by Hwang and Shin16 for Chitosan in water below the overlap concentration at 25 °C. Figure 11 shows that the scaling exponent at low concentration increases slightly from L = 1.14 to 1.24 at T > 30 °C. Colby17 has reviewed the concentration scaling of the viscosity of semidilute unentangled and entangled regimes of neutral polymers in theta and good solvents. He reports a value of 1.3 for the concentration scaling exponent of the specific viscosity of semidilute unentangled polymer chains in good solvents. This suggests that in the range c ≤ 0.02 g·mL−1 used for the determination of L, P317 could be evolving from dilute unimers to semidilute unentangled aggregates as the temperature increases. At high concentration, P317 chains will eventually entangle and the concentration scaling will change from (aLc)L to (aHc)H. At low temperature, H ≈ 1.8, a value much lower than what is expected for the semidilute entangled regime. A possible explanation would be that P317 unimers do not entangle even at concentrations up to 0.12 g·mL−1. This is in agreement with the Graessley classification reported by Heo and Larson18 for the concentration regimes based on the polymer molecular weight. Based on Graessley’s work, at 11360 g·mol−1 and below 0.12 g·mL−1, P317 solutions can only evolve from the dilute to the semidilute unentangled regime. This is true at low temperature where P317 is known to remain in the unimeric state up to 0.12 g·mL−1. The absence of entanglement at high concentration and low temperature is also coherent with the absence of gel formation in these conditions. For instance, Lenaerts and Couvreur19 reported a much slower increase of the viscosity with the concentration of Pluronic F127 at 5 and 10 °C compared to the viscosity at T > 20 °C. Vadnere et al.20 observed that the gelation temperature of Pluronic F127 decreases as the

(8)

14

Weissberg et al. used eq 8 to evaluate the shrinking of polymer coils in good solvents. Values of the expansion parameter above unity indicate that the polymer molecules occupy a larger hydrodynamic volume than at infinite dilution. At the opposite, values below unity indicate that they occupy a smaller hydrodynamic volume. eq 8 was applied to the experimental viscosities and the corresponding expansion parameter is plotted against P317 concentration in Figure 9 (see Supporting Information for the calculation of the expansion parameter).

Figure 9. Expansion parameter of P317 solutions calculated using eq 8 at 0.9 (◆), 8.9 (■), 20.0 (▲), 24.9 (×), 30.3 (∗), 39.8 (●), and 49.8 °C (+).

At 0.9 and 8.9 °C, P317 remains in the unimeric state and its hydrodynamic volume shrinks as its concentration increases. At 39.8 and 49.8 °C, the increase of the concentration leads to the association of P317 molecules into larger aggregates in agreement with the increase of the expansion parameter. At 20.0, 24.9, and 30.3 °C, the initial decrease of the expansion parameter is followed by an increase at 0.08 and 0.04 and below 0.01 g·mL−1, respectively. These concentrations could be interpreted as the onset of aggregation of P317 in fair agreement with the diagram of Figure 2. Concentration Scaling Exponents. In the limit of very low concentration, eq 7 predicts a power law dependence of the specific viscosity with the concentration (see Supporting Information). A power law dependence is also expected at high concentration for the specific viscosity of entangled polymer solutions.2 This suggests that the dilute and concentrated regimes can both be represented by power laws having their respective scaling exponents. 8634


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Figure 10. A) Specific viscosity, ηsp, as a function of P317 concentration c ≤ 0.02 g·mL−1 at 0.9 (◆), 8.9 (■), 20.0 (▲), 24.9 °C (×), 30.3 °C (∗), 39.8 (●), 49.8 °C (+). (B) Specific viscosity, ηsp, as a function of the normalized concentration, caL, at c ≤ 0.02 g·mL−1. The broken lines represent the average power laws from 0.9 to 30.3 °C with ⟨L⟩ = 1.14 ± 0.02 (···) and from 39.8 to 49.8 °C with ⟨L⟩ = 1.243 ± 0.007 (− − −). (C) Specific viscosity, ηsp, as a function of the normalized concentration, c·aL, of six Pluronics12 in the unimeric state in water at 5 °C: F38 (○); P85 (−); P104 (−); P103 (△); L122 (□); L101 (◇). The dashed line corresponds to the average power law for the six Pluronics with ⟨L⟩ = 1.146 ± 0.006.

not entangle much at low temperature and concentrations up to 0.12 g·mL−1. At high temperature, the scaling exponent H ≈ 3.8 (see Figure 11) is in good agreement with the de Gennes’s prediction of 3.7521 using ν = 0.6 and 3.92 using ν = 0.58817,22 for entangled polymer chains in good solvents. It is also in fair agreement with the experimental exponents of H = 4.07 obtained by Adam and Delsanti22 for entangled polystyrenes in benzene, H = 4.07 obtained by Hong et al.15 for PVA in Nmethyl pyrrolidone at 30 °C and H = 3.94 obtained by Hwang and Shin16 for Chitosan in water at 25 °C. Note that these results are also in accordance with Graessley’s classification because the P317 aggregates have a higher molecular weight than the unimers, allowing for the entanglement at concentrations below 0.12 g·mL−1. Cau and Lacelle4 noted that at 0.09 g·mL−1, the PEO chains of P317 remain hydrated through the transition but experience a progressive entanglement as the temperature increases. This is also coherent with unentangled PEO chains for the unimers at low temperature and entangled PEO chains for the aggregates at high temperature. These results all suggest that at high concentration, P317 molecules evolve from semidilute unentangled unimers to semidilute entangled aggregates as the temperature increases. Overlap Concentration. There is no definite procedure to experimentally determine the overlap concentration of polymer solutions. Adam and Delsanti22 defined c* as the concentration at which the osmotic pressure no longer depends on molecular weight. Dupas et al.23 defined c* as the maximum concentration for which the Huggins equation is valid. Gu and Alexandridis24 calculated c* based on the experimental values of the hydrodynamic radius and molecular weight of copolymer molecules in solution. Hwang and Shin16 used the intercept of

Figure 11. Temperature dependence of the scaling exponents from the power law model at low, L = ▲, and high, H = ◆, concentrations.

concentration increases. It occurs around 25 °C at 0.17 g·mL−1 so that at T ≤ 10 °C, it should occur at c > 0.3 g·mL−1. Moreover, Kurumada and Robinson5 used the Quemada model to determine the gelation concentration of Pluronic F127 at given temperatures. The model generated gelation concentrations of 0.12, 0.13, and 0.14 g·mL−1 at 30, 40, and 50 °C, respectively. However, the solution at 0.15 g·mL−1 remained fluid at temperatures below 30 °C. When the Quemada model was applied to our data sets (see Supporting Information), it generated very similar results, i.e. gelation concentrations of 0.15, 0.15, and 0.16 g·mL−1 at 30.3, 39.8, and 49.8 °C, respectively. At lower temperatures, the Quemada model did not fit the data suggesting that below 30 °C and up to 0.12 g·mL−1 the P317 solutions are far from gelation. All these observations support the hypothesis that P317 chains do 8635


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the decrease and the increase of the expansion parameter (see Figure 9) at low and high temperatures, respectively. The stretched exponential model generates values of c* = [η]S−1 that are in better agreement with cexp*. This could be the consequence of the inclusion of the intermolecular interactions in the intrinsic viscosity determined using the Phillies model. Weissberg et al.14 and Yamakawa et al.26 both suggested that intermolecular interactions causes flexible polymer chains in a good solvent to occupy less and less volume as the concentration increases. Gmachowski27 came to the same conclusion from the analysis of polymer sedimentation data. This could explain why the intrinsic viscosity at infinite dilution does not properly predict the overlap concentration of P317 in water. One way to include the intermolecular interaction in the determination of the overlap concentration is to use the effective intrinsic viscosity at a concentration c instead of the intrinsic viscosity at infinite dilution. The definition of [η]c in eq 8 refers to the Kraemer model and when applied to eq 5, it leads to

the concentration scaling of the specific viscosity at low and high concentrations. The latter was used in this investigation and the extrapolation procedure is summarized in Figure 12 for

Figure 12. Extrapolation procedure used to determine the experimental overlap concentration, cexp*, at: 0.9 °C (◆) and 49.8 °C (+). Solid lines represent the power laws deduced from the data at c ≤ 0.02 g·mL−1, dashed lines represent the power law between 0.08 and 0.10 g·mL−1.The arrows indicate the intersection corresponding to cexp*.

cc*[η]K =

the data sets at 0.9 and 49.8 °C (see Supporting Information for the other temperatures). The change from dilute to semidilute is gradual so that the value of c* will depend on the range of concentration used for the power scaling at high concentration. This emphasizes the notion that the overlap concentration is more properly defined as a region of crossover between regimes1b and the experimental values reported in Table 1 and presented in Figure 13 should be regarded as cexp* ± 0.01 g·mL−1.

1 (1 + kK ψ )

(10)

−1

where cc* = [η]c is the overlap concentration corresponding to the reciprocal of the effective intrinsic viscosity, kK is the Kraemer constant that accounts for copolymer-copolymer interactions and ψ is an adjustable parameter that accounts for the higher order intermolecular interactions (see Supporting Information). Figure 13 shows that the values of cc* calculated with ψ = 0.96 and eq 10 agree within ±4% with the values of cexp*. This suggests that the reciprocal of the intrinsic viscosity at infinite dilution leads to fair estimates of the overlap concentration for polymer solutions experiencing weak intermolecular interactions. However, the overlap concentration of strongly interacting polymer solutions, such as associating copolymers, should be determined with a model that accounts for these interactions. Eyring’s Activation Thermodynamics for Viscous Flow. In this section, the temperature dependence of the viscosity is analyzed at two P317 concentrations. On the basis of the results of the previous section, in the solution at 0.01 g· mL−1, P317 evolves from unentangled dilute unimers to dilute or semidilute unentangled aggregates while at 0.095 g·mL−1, it evolves from semidilute unentangled unimers to semidilute entangled aggregates. The Eyring28 model is used to extract the activation free energy for viscous flow, ΔG‡, from the viscosity of P317 solutions.

Figure 13. Evolution of the overlap concentration, c*, with the temperature. Experimental values (◆), c* = [η]HKM−1 (▲), c* = [η]S−1 (∗), and cc* from eq 10 using ψ = 0.96 (□).

Colby17 mentions that the overlap concentration is often confused with the entanglement concentration. According to him, the relative viscosity at the overlap concentration is around 2 and it is greater than 10 at the entanglement concentration. In this investigation, the relative viscosity at cexp* is between 2 and 3, suggesting that cexp* corresponds to the overlap concentration. From 0.9 to 20.0 °C, cexp* = 0.047 ± 0.001 g·mL−1 and it slightly increases to 0.054 g·mL−1 from 30.3 to 49.8 °C. Holyst et al.25 report c* = 0.05 g·mL−1 for aqueous solutions of poly(ethylene glycol) of molecular weight 12000 g·mol−1, in fair agreement with our values. Figure 13 shows that cexp* ≥ [η]HKM−1 at low temperature and cexp* < [η]HKM−1 at high temperature, in accordance with

⎛ η⟨V ⟩ ⎞ ΔG‡ = RT ln⎜ ⎟ ⎝ hNA ⎠ ⟨V ⟩ =

(x PMP + x W MW ) ρ

MP = (x POMPO + x EOMEO)

(11)

In eq 11, h is the Planck’s constant, NA is the Avogadro’s number, and ⟨V⟩ is the average molar volume of the solution which is calculated using the density of the solution, ρ, and the mole fraction of P317, xP, and water, xW, in the solution. MW is the molecular weight of water, and MP is the molecular weight 8636


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Figure 14. Evolution of the apparent activation free energy of P317, ΔG‡ϕ,P. (A) As a function of temperature at copolymer concentrations of 0.01 g· mL−1 (■) and 0.095 g·mL−1 (◆). The dashed lines represent ΔG‡ϕ,A (− ·· − ) and ΔG‡ϕ,B (− − − ) at 0.01 g·mL−1. (B) As a function of P317 concentration at 0.9 (◆), 8.9 (■), 20.0 (▲), 24.9 (●), 30.3 (□), 39.8 (○), and 49.8 °C (△). The dotted lines (---) are only guides for the eye.

consistent with the passage from an unentangled to an entangled configuration at high concentration. Figure 14B plots the values of ΔG‡ϕ,P vs c calculated from the data sets of the previous section. At low temperature (T < 24.9 °C), ΔG‡ϕ,P decreases as P317 concentration increases. This was also observed for the apparent activation free energy of Pluronic F38 in water at 5 °C12 at concentrations where this copolymer remains in the unimeric state (see Supporting Information). This suggests that the work required for viscous flow is smaller for concentrated and unentangled unimers than for diluted unimers. At high temperature (T > 30.3 °C) ΔG‡ϕ,P increases with the concentration, suggesting that the flow is more and more difficult as the concentration rises, in accordance with the formation of an entangled network of large aggregates. In between, P317 evolves from one configuration to the other.

of the P317 units calculated using the mole fraction of PO and EO in the copolymer. The Gibbs−Helmholtz relation eq 12 is then used to get ΔH‡, the molar activation enthalpy for viscous flow, ΔH‡. ⎛ d(ΔG‡/T ) ⎞ ⎟ ΔH ‡ = ⎜ ⎝ d(1/T ) ⎠T

(12)

The activation free energy and enthalpy per mole of solution were calculated from the temperature dependence of the viscosity and the results are presented in the Supporting Information. These properties correspond to the difference between the activated solution and the solution at rest and they are relative to one mole of solution which is mainly composed of water. In order to emphasize the contribution of P317, these properties were used to determine the apparent molar contribution of P317 to the activation properties for the viscous flow of the solution. To do so, it was assumed that the molar activation free energy for the viscous flow of pure water, ΔG‡W, does not change with the addition of P317. The apparent contribution of P317 units to the activation free energy for viscous flow, ΔG‡ϕ,P, thus becomes ΔGϕ‡ , P = ΔGW‡ +

(ΔG‡ − ΔGW‡ ) xP

(13)

Once ΔG‡ϕ,P has been calculated with eq 13, ΔH‡ϕ,P is obtained with eq 12. Apparent properties have been used widely29 for the investigation of solute−solvent and solute− solute interactions. Monkos30 used a model very similar to eq 13 for the viscosity of biopolymers except that he assumed that the activation properties of the biopolymers did not depend on the concentration. For simplicity, the apparent contribution of P317 units to the activation properties for viscous flow will be referred to as apparent activation properties. Apparent Activation Free Energy. Figure 14A shows the evolution of ΔG‡ϕ,P with the temperature. At 0.01 g·mL−1, the transition is associated with a step down in ΔG‡ϕ,P starting around 25 °C and ending around 38 °C, in agreement with the onset and endset of the endothermic transition of Figure 2. At 0.095 g·mL−1, the transition is associated with a step up in ΔG‡ϕ,P starting around 21 °C and ending around 29 °C, once again in agreement with the onset and endset temperatures of Figure 2. This suggests that the aggregation at 0.01 g·mL−1 reduces the work required to induce the flow of P317 molecules while at 0.095 g·mL−1 it increases it. This is

Figure 15. Evolution of the apparent activation enthalpy of P317, ΔH‡ϕ,P, as a function of the temperature at copolymer concentrations of 0.01 g·mL−1(■) and 0.095 g·mL−1 (◆). The dashed lines (− − − ) are calculated using the mass action model described in the text (eq 16).

Apparent Activation Enthalpy and Mass-Action Model. The evolution of ΔH‡ϕ,P vs T is presented in Figure 15. The transitions observed in ΔG‡ϕ,P now appear as a maximum in ΔH‡ϕ,P at 0.01 g·mL−1 and as a minimum at 0.095 g·mL−1. These extrema are the result of the displacement of the equilibrium between the unimers and the aggregates characterized by an equilibrium constant K. As the temperature increases, the fraction of P317 in the unimeric state, α, decreases and the equilibrium shifts from the unimeric state A toward the aggregated state B. 8637


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Table 2. Summary of the Parameters Used To Fit ΔH‡ϕ,P vs T with eq 16 and the values of M and ΔHθP obtained at 0.01 and 0.095 g·mL−1 parameters and how they are obtaineda

value and range at 0.01 g·mL‑1

value and range at 0.095 g·mL‑1 units

ΔG‡0ϕ,A ΔS‡ϕ,A ΔH‡ϕ,A ΔG‡0ϕ,B ΔS‡ϕ,B ΔH‡ϕ,B M ΔHθP

ΔG‡ϕ,P vs T ΔG‡ϕ,P vs T (ΔG‡ϕ,P/T) vs (1/T) ΔG‡ϕ,P vs T ΔG‡ϕ,P vs T (ΔG‡ϕ,P/T) vs (1/T) best fit best fit

‡0

161.54 1043 446.52 129.02 813.2 350.94 5 0.66

12−22 12−22 12−22 46−52 46−52 46−52 12−52 12−52

ΔS‡ϕ,A or B

ΔG ϕ,A or B and are the extrapolation at T = 0 °C and the slope of the temperature range indicated. a

[BM ] M

[A]

=

(1 − α) Mα M

(14) 29

This mass-action model developed by Desnoyers et al. has been used in an earlier study for the analysis of the apparent molar volume and heat capacity of Pluronic F127 in water7 at 25 °C. As discussed in that paper, the molecularity M is a thermodynamic parameter that characterizes the cooperativity of the transition and it is different from the aggregation number which is a structural parameter. In this investigation, the mass action model is applied to the apparent molar properties of activation for the viscous flow of P317. First, ΔG‡ϕ,P is expressed as the sum of the temperature dependent apparent activation free energy of P317 in state A, ΔG‡ϕ,A, and (B) ΔG‡ϕ,B. ΔGϕ‡ , P = ΔGϕ‡ , B + α(ΔGϕ‡ , A − ΔGϕ‡ , B)

(15)

Then, ΔH‡ϕ,P is calculated by applying the Gibbs−Helmholtz relation to eq 15. ΔHϕ‡ , P = ΔHϕ‡ , B + α(ΔHϕ‡ , A − ΔHϕ‡ , B) + (ΔGϕ‡ , A − ΔGϕ‡ , B)

ΔH‡ϕ,A

α(1 − α)N ΔHPθ

(1 − α + Mα )RT

vs T and

ΔH‡ϕ,A or B

kJ·mol−1 J·mol−1·K−1 kJ·mol−1 kJ·mol−1 J·mol−1·K−1 kJ·mol−1

10−16 10−16 10−16 44−50 44−50 44−50 10−50 10−50

kJ·mol−1

is the slope of

(ΔG‡ϕ,P/T)

vs (1/T) over

used to determine the value of α as a function of the temperature at the concentrations investigated. To do so, one needs to know how ΔG‡ϕ,A and ΔG‡ϕ,B evolve with the temperature at these concentrations. Knowing that P317 exists only in state A at low temperature and only in state B at high temperature, then the trend of ΔG‡ϕ,P vs T corresponds to ΔG‡ϕ,A = ΔG‡0ϕ,A − TΔS‡ϕ,A at low temperature and to ΔG‡ϕ,B = ΔG‡0ϕ,B − TΔS‡ϕ,B at high temperature. As a first approximation, it is assumed that these trends apply from 10 to 50 °C (for an example, see the dashed lines of Figure 14A). The values of ΔG‡0ϕ,A, ΔS‡ϕ,A, ΔG‡0ϕ,B, and ΔS‡ϕ,B at 0.01 and 0.095 g·mL−1 are reported in Table 2. Using these values, the α vs T data sets were calculated with eq 15 and the results are shown in Figure 16. As expected, the fraction of unimers decreases steeply as the temperature increases over the endothermic transition. At a given temperature, α decreases with the concentration of P317, which is coherent with the decrease of the transition temperature as the concentration of P317 increases. The next step is to use the α vs T data sets to calculate the ΔH‡ϕ,P vs T. To do so, one also needs to know ΔH‡ϕ,A, ΔH‡ϕ,B, M and ΔHθP. The former two were assumed to be constant and equal to the slope of d (ΔG‡ϕ,P/T)/d(1/T) at low and high temperatures, respectively. The latter two were varied to fit the data at 0.01 and 0.095 g·mL−1. Table 2 summarizes the parameters used to calculate the best fit to ΔH‡ϕ,P vs T at 0.01 and 0.095 g·mL−1 and the results are shown as dashed lines in Figure 15.

MA ⇌ BM K=

ΔG‡ϕ,P

121.53 277.9 197.41 168.10 732.9 368.20 3 0.90

(16)

ΔH‡ϕ,B

In eq 16, and are the apparent activation enthalpy of P317 in state A and B, ΔHθP is the standard enthalpy of the transition from state A to state B per mole of copolymer unit and N is the overall degree of polymerization of P317. The complete derivation of eq 16 from eqs 14 and 15 is presented in the Supporting Information. The last term of eq 16 is the equilibrium displacement contribution to ΔH‡ϕ,P. As mentioned above, the transition at 0.01 g·mL−1 is associated with a step down of ΔG‡ϕ,P so that (ΔG‡ϕ,A − ΔG‡ϕ,B) is positive. At that concentration, ΔH‡ϕ,P goes through a maximum so that ΔHθP must be positive, in agreement with the endothermic nature of the aggregation of P317. At 0.095 g·mL−1, the transition is associated with a step up in ΔG‡ϕ,P so that (ΔG‡ϕ,A − ΔG‡ϕ,B) is negative. At that concentration, ΔH‡ϕ,P goes through a minimum so that ΔHθP must also be positive. This confirms that the transitions at low and high concentrations are both endothermic, in accordance with the hydrophobic association of P317 molecules. It is important to note that ΔG‡ϕ,P is independent of the molecularity. Hence, the experimental values of ΔG‡ϕ,P can be

Figure 16. Evolution of the fraction of copolymer molecules in the unimeric state, α, as a function of the temperature at concentrations 0.01 g·mL−1 (− × − ) and 0.095 g·mL−1 (−+−) in water as calculated with eq 15 and the parameters of Table 2. 8638


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It is suggested that the prediction of the overlap concentration from viscosity data should take into account both the intrinsic viscosity at infinite dilution and the intermolecular interactions. Such a model was derived from the Kraemer equation and the definition of the effective intrinsic viscosity. That simple model successfully predicts the experimental overlap concentration of weakly and strongly interacting P317 solutions. At low temperature and low concentration, the concentration scaling of the specific viscosity agrees with dilute unentangled copolymer molecules in the unimeric state (L = 1.14). At high temperature and low concentration, P317 forms aggregates that remain unentangled with a scaling exponent slightly higher than at low temperature (L = 1.24). At high concentration and low temperature, P317 molecules remain in the unimeric state and they do not seem to entangle much over the concentration range investigated with a scaling exponent remaining rather low (H = 1.8). At high temperature and high concentration, the concentration scaling corresponds to entangled copolymer molecules (H = 3.8). The activation free energy was extracted from the temperature dependence of the viscosity of water and of P317 solutions and used to compute the apparent contribution of P317 to the activation free energy and enthalpy for viscous flow of the solutions. These properties were then analyzed in terms of a mass action model for the temperature induced cooperative association of P317 molecules into aggregates. The model confirms that the thermally induced transition of P317 solutions is endothermic (ΔHθP = 0.8 ± 0.1 kJ·mol−1) with a rather small cooperativity (M = 4 ± 1). Interestingly, the massaction model generates essentially the same results when it is applied to the apparent activation properties for viscous flow and the apparent thermodynamic properties of P317 in water.

As observed in Figure 15, the mass-action model developed for thermodynamic properties applies very well to the activation enthalpy deduced from the application of Eyring’s model to the viscosity data at low and high concentrations. The values of M = 5 and 3 obtained at 0.01 and 0.095 g·mL−1 are in fair agreement with M = 2.5 obtained in an earlier investigation7 of the apparent molar volume and heat capacity of Pluronic F127 in water at 25 °C. That same investigation also led to ΔHθP = 1.2 kJ·mol−1 for Pluronic F127. Lebeuf11 investigated P317 solutions using DSC and he reports ΔHθP = 1.4 kJ·mol−1 at concentrations ranging from 0.005 to 0.01 g·mL−1. Cau and Lacelle4 estimated ΔHθP = 1.6 ± 0.3 kJ·mol−1 from the analysis of the concentration dependence of the transition temperature. These values are higher than ΔHθP = 0.7 and 0.9 kJ·mol−1 obtained at 0.01 and 0.095 g·mL−1. Nevertheless, the agreement is fair given the assumptions made for the estimation of ΔY‡ϕ,A and ΔY‡ϕ,B (with Y = G, H, or S).

■

CONCLUSIONS The findings of this investigation are summarized in Figure 17 where the state of P317 molecules is illustrated.

■

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.macromol.5b02247. Viscosity of the solutions, the method used to evaluate the density of the solutions, the analysis of the viscosity of some Pluronic solutions, and the derivation of some of the equations presented in the text (PDF)

Figure 17. Summary of the parameters determined for P317 solutions. The dotted line represents the endothermic transition from the unimer to the aggregated state. The dashed line represents the experimental overlap concentration. L and H are the scaling exponents at low and high concentrations, respectively. [η]H, kH, and cexp* are the intrinsic viscosity and the intermolecular interaction constant from the Huggins model and the experimental overlap concentration, respectively. ΔG‡ϕ,A or B is the contribution of P317 to the activation free energy for viscous flow in state A or B. ΔG‡ϕ,A → B is the change from the unimeric state A to the aggregated state B. ΔHθP is the standard enthalpy of association per copolymer units.

■

AUTHOR INFORMATION

Corresponding Author

*(F.Q.) E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

The concentration dependence of the viscosity of aqueous solutions of P317 tells us that the reciprocal of the intrinsic viscosity determined with the Huggins, Kraemer, and Martin models provides a fair estimate of the overlap concentration at low temperature where P317 experiences weak hydrodynamic interactions and does not associate. At high temperature, where P317 is associated into strongly interacting aggregates, the reciprocal of the intrinsic viscosity fails to predict the overlap concentration. The stretched exponential model proposed by Phillies can be rearranged to generate intrinsic viscosities that account for the hydrodynamic interactions. The reciprocal of the intrinsic viscosity generated by this model is in fair agreement with the experimental overlap concentration.

ACKNOWLEDGMENTS The author thanks Dr. Carmel Jolicoeur for suggesting this investigation and for providing the copolymer sample. The author also thanks Dr. Jacques E. Desnoyers for his comments and for stimulating discussions.

■

REFERENCES

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