Universal Scaling of Phase Diagrams of Polymer Solutions

Article Cite This: Macromolecules XXXX, XXX, XXX−XXX

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Universal Scaling of Phase Diagrams of Polymer Solutions Chi Wu*,†,‡ and Yuan Li† †

Department of Chemistry, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong Hefei National laboratory of Physical Science at Microscale, Department of Chemical Physics, The University of Science and Technology of China, Hefei, Anhui, China

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‡

ABSTRACT: A combination of microfluidic and small-angle laser light scattering enables us to map phase diagram of two polymers with different chain lengths in three solvents in an unprecedented speed. Each precisely measured phase diagram leads to a pair of critical temperature (Tc) and volume fraction of polymer (ϕc), in which ϕc is scaled to the chain length (N) as ϕc ∼ N−0.37±0.01. Accounting for a huge size difference of polymer chain and solvent molecules, an adjustable volume ratio (Rc) of solvent to polymer is introduced to generate a dimensionless reduced volume fraction ψ {= ϕ/[ϕ + Rc(1 − ϕ)]}, so that each skewed phase diagram is shifted and symmetrized with a symmetrical axis ψc = 0.325 ± 0.002. After normalized by a solvent and polymer (not chain length) dependent constant ψ0, all of the measured 17 phase diagrams collapsed into one master curve, |ψ − ψc|/ψ0 = ε0.325±0.007N0.152±0.004, where ε = |T − Tc|/Tc, a reduced temperature. As N → ∞, |ψ − ψc| ∼ |ϕ − ϕc|/ϕc nearby the critical point so that our results lead to |ϕ − ϕc| ∼ ε0.325±0.007N−0.22±0.01, where the scaling over the chain length is close to −2/9 predicted by Muthukumar. We have successfully placed the last jigsaw piece, i.e., the chain length dependence, in the phase diagrams of polymer solutions.

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INTRODUCTION The phase diagrams of polymer solutions are not only important to their applications but also significant in fundamental physics,1−3 namely, the critical phenomena, or more precisely whether there exists a universal scaling of ϕH − ϕL ∼ Nξεβ, where ϕH and ϕL are respectively higher and lower concentrations located on the coexistence curve for a given temperature (T) nearby the critical temperature (Tc), N is the Kuhn segments number, ε (= |T − Tc|/Tc) is the reduced temperature, and ξ and β are two scaling exponents.4−6 For binary mixtures of two kinds of small molecules, there is no chain length dependence; i.e., there is only one scaling exponent on the reduced temperature. Both theoretical and experimental studies had settled that β ≃ 1/3 a long time ago.6−16 Since the 1970s, more studies were done on polymer solutions wherein polymer chains and solvent molecules are greatly asymmetrical in their sizes. Nevertheless, both theoretical and experimental studies seem to agree that β remains a constant, close to 0.3265 predicted for small molecular binary mixtures because their space dimension (d = 3) and order parameter dimension (n = 1) are identical in the Ising model.7−11,17−19 On the other hand, ξ has a different story. In the earlier days, the mean-field theory20 predicted that ξ = −1/4 and β = 1/2. Later, de Gennes4 found that ξ = (β − 1)/2; namely, for β = 1/3, ξ becomes −1/3. More recently, Muthukumar21 showed that ξ = −2/9 after reasoning the significance of three-body interaction nearby the critical point when N is sufficiently large. Experimentally, there were only few reported values of ξ, ranging from −0.22 to −0.34, mostly © XXXX American Chemical Society

from narrowly distributed polystyrene standards in cyclohexane except for a few.22−29 Therefore, it was not possible to confirm whether there exists a scaling or, even if yes, what the correct value of ξ should be. The lack of reliable value of ξ is attributed to some experimental difficulties. Traditionally, a good phase diagram of a binary solution was obtained after it separated into two layers of clear and stable solutions with their respective concentrations (ϕH and ϕL) at different phase separation temperatures. At each temperature, ϕH and ϕL can be measured by different ways, but the light refraction is the most reliable method.28,29 For a mixture of two kinds of small molecules, the equilibrium between two solution layers can be reached within hours or days. However, such a time scale is prolonged to weeks or months or even years for polymer solutions, depending on the chain length and the starting concentration, because the diffusion of those entangled long chains is extremely slow. Moreover, one would never be sure whether a given phase separation has reached its equilibrium state, making the determination of reliable phase diagrams of polymer solutions rather difficult or painful if not practically impossible. This is why there are only a handful of true phase diagrams of polymer solutions reported in the literature.22−37 Received: June 13, 2018 Revised: July 11, 2018

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DOI: 10.1021/acs.macromol.8b01234 Macromolecules XXXX, XXX, XXX−XXX

Article

Macromolecules

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EXPERIMENTAL SECTION

Materials. A commercial poly(vinyl acetate) (PVAc, Mw = 2.56 × 105 and PDI = 2.39) sample and a series of narrowly distributed polystyrene samples were purchased from Sigma-Aldrich. Isobutanol (IBA) was obtained from Mallinckrodt, and both benzene and cyclohexane, from Sigma-Aldrich, were used without further purification. The commercial PVAc sample was fractionated into five narrowly distributed PVAc fractions by using a conventional precipitation method, where water as the precipitant was slowly added into a stirring PVAc solution in acetone at 35 °C until it became slightly turbid, and then the solution mixture was slowly heated until the solution became clear again. To obtain each narrowly distributed fraction, we slowly cooled the solution and stopped the cooling as soon as the solution became slightly tarnishing and then waited for a sufficiently long time before collecting the sample. The weight-average molar mass (Mw), polydisperse index (Mw/Mn), and degree of polymerization (N = Mw/M0) of five PVAc fractions and polystyrene (PS) standards used are summarized in Table 1.

Figure 1. Typical polymer volume fraction (ϕ) dependence of concentration-normalized relative scattered light intensity (⟨I⟩R/Cp) at 314.80 K, where increment of ϕ between two droplet is 2.5 × 10−3 and Cp is polymer mass concentration (g/mL); the dashed line schematically represents a coexistence curve. Schematic glass capillary with droplets in one phase (hollow) and two phases (with a center point or fully filled) shows where polymer chains start to undergo nucleation (metastable) and phase separation (unstable).

Table 1. Weight-Average Molar Mass (Mw), Polydisperse Index (Mw/Mn), and Degree of Polymerization (N = Mw/ M0) of Five PVAc Fractions and Polystyrene (PS) Standards Used, Where M0 Is Molar Mass of Monomer PVAc sample

Mw/(g/mol)

Mw/Mn

N = Mn/M0

PVAc-1 PVAc-2 PVAc-3 PVAc-4 PVAc-5 PS-1 PS-2 PS-3 PS-4 PS-5

× × × × × × × × × ×

1.09 1.08 1.08 1.10 1.11 1.03 1.03 1.05 1.01 1.01

600 1700 3800 6900 9650 450 930 2020 5000 15000

5.62 1.57 3.54 6.52 9.22 4.82 9.98 2.21 5.36 1.59

4

10 105 105 105 105 104 104 105 105 106

polymer concentration. For a given temperature, ⟨I⟩R/C should remain a constant in the one-phase region, and the two points at which ⟨I⟩R/C starts to increase mark the twophase region, namely, the two points on the coexistence curve. Therefore, by changing the temperature, we are able to map the coexistence curve. Also note that the top nearly flat region of the scattered light intensity indicates that those droplets are in the unstable spinodal region. Using this novel method, we mapped three chain-length-dependent phase diagrams of two polymers in different solvents: polystyrene (PS) in cyclohexane and poly(vinyl acetate) (PVAc) in isobutanol and in benzene. Figure 2 shows such obtained typical phase diagrams of PVAc with different chain lengths (N) in benzene. Fitting each

Instruments. Recently, combining microfluidic and small-angle lase light scattering, we developed a novel and quick experimental method to map the phase diagram of a polymer solution.38−40 For the benefit of readers, we outline its principle as follows. Using a microfluidic device, we are able to prepare and store a number of small polymer solution droplets (∼10 nL) with different concentrations (Cp) inside a glass capillary that is settled and maintained at a given phase separation temperature.39,40 The polymer concentration increment between two neighboring droplets is as small as 1.25 × 10−3 g/mL. Using a stepping motor, we can scan the time-average scattered light intensity (⟨I⟩) of each droplet by using a small-angle (∼4°) laser light scattering device. ⟨I⟩ is extremely sensitive to the chain association because it is proportional to the square of mass and number concentration of the scattering subject; namely, when on average two chains come together, the scattered light intensity doubles.41 Therefore, we can quickly know those droplets that are in the two-phase region from the sharp increase of their scattered light intensities instead of waiting for each droplet to reach its phase separation equilibrium because the concentration normalized scattered light intensity of those droplets in the one-phase region remains a constant. To make the microfluidic device, we used a semicrystallized Teflon plate, perfluoroalkoxy (PFA) with a thickness of ∼1.0 mm purchased from Yuyisong, Inc. Photoresist (SU-8) and PDMS prepolymer were respectively purchased from Microchem and Dow Corning (Sylgard 184). Fluorocarbon oil (FC3283) and a fluorosurfactant (1H,1H,2H,2H-perfluoro-1-octanol, PFO) were obtained from 3M and Sigma-Aldrich, respectively.

Figure 2. Phase diagrams of PVAc with different chain lengths in benzene, where N is degree of polymerization, proportional to chain length.

phase diagram, we can find its corresponding up critical solution temperature (Tc) and critical polymer concentration (ϕc), where ϕc is scaled to N as ϕc ∼ N−γ. The values of γ are listed in Table 2. Each phase diagram is skewed toward the lower concentration because polymer chains and solvent molecules are hugely different in size. Sanchez42 proposed a method to symmetrize it by introducing a dimensionless order parameter ψ to replace ϕ, i.e., ψ = ϕ/[ϕ + Rc(1 − ϕ)], where Rc

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RESULTS AND DISCUSSION Figure 1 shows a typical droplet (concentration) dependence of the relative scattered light intensity normalized by the B

DOI: 10.1021/acs.macromol.8b01234 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules Table 2. Summary of Critical Volume Fraction Scaling Exponent γ Between Critical Volume Fraction and Chain Length, Symmetrical Axis ψc, Normalization Factor ψ0, and Two Scaling Exponents b and β of Different Polymer Solutionsa polymer

solvent

γ

ψc

ψ0

b

β

PVAc PVAc PS PMMA16

benzene IBA cyclohexane 3-OCT

0.38 0.37 0.36 0.37

0.325 0.325 0.325 0.323

0.224 0.212 0.315 0.380

0.489 0.455 0.465 0.460

0.325 0.325 0.318 0.333

Uncertainties: γ: ±0.01; ψc: ±0.002; ψ0: ±0.005; b: ±0.009; β: ±0.006. a

physically represents an “interaction volume” ratio of a solvent molecule to a polymer chain and is obviously dependent on the chain length; namely, it decreases as the chain length increases. To minimalize the uncertainties and artificial errors, we fitted all the experimental data in Figure 2 to obtain Rc and ψc in one two-parameter nonlinear least-squares fitting, where ψc is chain length independent. Figure 3 shows such symmetrized phase diagrams. In such a symmetrized phase diagram and nearby the critical points

Figure 4. Symmetrized master phase diagrams of polymer solutions, where each symbol was used to represent different chain lengths in each type of polymer solution.

slightly deviating from the mean-field value of 0.5. As N → ∞, the polymer volume fraction in the symmetrized space is related to that in the original space of polymer volume fraction as (ψH − ψL) ∼ (ϕH − ϕL)/ϕc.42,43 Considering that the phase diagrams are symmetrical nearby the critical point (ϕc), ψH − ψL = 2|ψ − ψc| and ϕH − ϕL = 2|ϕ − ϕc| in the Landau theory,4 we have ξ = −γ + bβ. For three kinds of polymer solutions measured in the current study, ξ ranges from −0.21 to −0.23, close to −0.21 from the time-consuming refraction measurements of PMMA in 3-OCT27 and −0.23 from the simple scaling analysis of PS in MCH.22,23 In the mean-field theory,1,20 the high-order interactions are neglected, except at the theta point. de Gennes4 argued that when the two-body interaction vanishes or becomes insignificant near the theta condition, one should consider the three-body interaction. To account for such a dominant three-body interaction, Muthukumar21 simply refined the free energy expression under the framework of the mean-field theory and revealed the scaling law of the volume fraction dependence as |ϕ − ϕc| ∼ N−2/9 when N is sufficiently large. Our current results convincingly confirm such a universal scaling nearby the critical point. Physically, near the critical point, the composition or density fluctuates tremendously, and the correlation length significantly increases so that it is reasonable to argue that the three-body interaction becomes dominant. The recovery of such a density fluctuation in the explicit free energy expression certainly leads to a deviation from the mean-field predicted value of −0.25.

Figure 3. Symmetrized coexistence curves of PVAc with 4 higher degree of polymerization N in benzene (i.e., phase diagrams in Figure 2).

(εmaxN0.3 ≤ 0.075),42,43 |ψ − ψc| = ψ0(εΝb)β, where ψc is the symmetrical axis, ψ0 is independent of the chain length but varies with polymer and solvent, and β and b are two universal scaling constants.31−35 As shown in Table 2, we experimentally found that ψc = 0.325 ± 0.002, independent of the chain length, polymer, and solvent. The three-parameter Levenberg− Marquardt fitting of the symmetrized phase diagrams of each type of polymer solutions generates a set of ψ0, b, and β, also listed in Table 2. On the other hand, we also fit all of the four sets of data together to generate a set of averaged values of ψ0, b, and β, transforming 12 phase diagrams (with higher values of N) measured in the current study as well as 5 phase diagrams (with higher values of N) in ref 16, i.e., a total of 17 phase diagrams of different polymer solutions, into one master curve with β = 0.328 ± 0.009 and b = 0.47 ± 0.02, as shown in Figure 4. Note that each symbol represents 4 or 5 different chain lengths. The overall fitting of 17 phase diagrams as well as individual fittings of each kind of polymer solutions shows that β is very close to that predicted in the Ising model,7−11,18 and b is

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CONCLUSION A combination of microfluidic preparation of a set of small nanoliter droplets with different polymer concentrations and small-angle laser light scattering enables us to significantly shorten the time of mapping a phase diagram of a polymer solution from months or years to days with a much higher precision. Our current experimental results (two polymers in three solvents) together with previous literature results (one polymer in one solvent) clearly reveal that there exists a universal scaling exponent (ξ) over the chain length in the critical region of phase diagrams of polymer solutions. Our experimental results have confirmed the Muthukumar’s prediction21 and placed the last jigsaw piece in the whole picture of phase diagrams of polymer solutions. Using two universal scaling exponents (ξ = −0.22 ± 0.01 and β = 0.328 ± C

DOI: 10.1021/acs.macromol.8b01234 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules

(20) Flory, P. J. Principles of Polymer Chemistry; Cornell University Press: Ithaca, NY, 1953. (21) Muthukumar, M. Thermodynamics of polymer solutions. J. Chem. Phys. 1986, 85, 4722. (22) Dobashi, T.; Nakata, M.; Kaneko, M. Coexistence curve of polystyrene in methylcyclohexane. I. Range of simple scaling and critical exponents. J. Chem. Phys. 1980, 72, 6685. (23) Dobashi, T.; Nakata, M.; Kaneko, M. Coexistence curve of polystyrene in methylcyclohexane. III. Asymptotic behavior of ternary system near the plait point. J. Chem. Phys. 1984, 80, 948. (24) Shinozaki, K.; van Tan, T.; Saito, Y.; Nose, T. Interfacial tension of demixed polymer solutions near the critical temperature: polystyrene + methylcyclohexane. Polymer 1982, 23, 728. (25) Xia, K. Q.; Franck, C.; Widom, B. Interfacial tensions of phaseseparated polymer solutions. J. Chem. Phys. 1992, 97, 1446. (26) Chu, B.; Wang, Z. An extended universal coexistence curve for polymer solutions. Macromolecules 1988, 21, 2283. (27) Xia, K. Q.; An, X. Q.; Shen, W. G. Measured coexistence curves of phase-separated polymer solutions. J. Chem. Phys. 1996, 105, 6018. (28) Shultz, A. R.; Flory, P. J. Phase equilibria in polymer-solvent systems. J. Am. Chem. Soc. 1952, 74, 4760. (29) Wang, J.; Dan, Y.; Yang, Y.; Wang, Y.; Hu, Y.; Xie, Y. The measurements of coexistence curves and critical behavior of a binary mixture with a high molecular weight polymer. J. Mol. Liq. 2011, 161, 115. (30) Kita, R.; Dobashi, T.; Yamamoto, T.; Nakata, M.; Kamide, K. Coexistence curve of a polydisperse polymer solution near the critical point. Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 1997, 55, 3159. (31) Scholte, Th. G. Thermodynamic parameters of polymer-solvent systems from light-scattering measurements below the theta temperature. J. Polym. Sci. Part A-2 1971, 9, 1553. (32) Hashizume, J.; Teramoto, A.; Fujita, H. Phase equilibrium study of the ternary system composed of two monodisperse polystyrenes and cyclohexane. J. Polym. Sci., Polym. Phys. Ed. 1981, 19, 1405. (33) Rätzsch, M. T.; Krüer, B.; Kehlen, H. Cloud-point curves and coexistence curves of several polydisperse polystyrenes in cyclohexane. J. Macromol. Sci., Chem. 1990, 27, 683. (34) Melnichenko, Y. B.; Wignall, G. D.; Van Hook, W. A. Molecular weight scaling in critical polymer solutions. Europhys. Lett. 1999, 48, 372. (35) Yelash, L. V.; Kraska, T.; Imre, A. R.; Rzoska, S. J. Apparent exponents for the chain length dependence of the volume fraction in critical polymer solutions. J. Chem. Phys. 2003, 118, 6110. (36) Shangguan, Y.; Guo, D.; Feng, H.; Li, Y.; Gong, X.; Chen, Q.; Zheng, B.; Wu, C. Mapping phase diagrams of polymer solutions by a combination of microfluidic solution droplets and laser lightscattering detection. Macromolecules 2014, 47, 2496. (37) Mao, H.; Li, C.; Zhang, Y.; Bergbreiter, D.; Cremer, P. Measuring LCSTs by novel temperature gradient methods: evidence for intermolecular interactions in mixed polymer solutions. J. Am. Chem. Soc. 2003, 125, 2850. (38) Laval, P.; Lisai, N.; Salmon, J.; Joanicot, M. A microfluidic device based on droplet storage for screening solubility diagrams. Lab Chip 2007, 7, 829. (39) Shi, F.; Han, Z.; Li, J.; Zheng, B.; Wu, C. Mapping polymer phase diagram in nanoliter droplets. Macromolecules 2011, 44, 686. (40) Zhou, X.; Li, J.; Wu, C.; Zheng, B. Constructing the phase diagram of an aqueous solution of poly(n-isopropyl acrylamide) by controlled microevaporation in a nanoliter microchamber. Macromol. Rapid Commun. 2008, 29, 1363. (41) Chu, B. Laser Light Scattering: Basic Principles and Practice, 2nd ed.; Academic Press: San Diego, 1991. (42) Sanchez, I. C. A universal coexistence curve for polymer solutions. J. Appl. Phys. 1985, 58, 2871. (43) Sanchez, I. C. Corresponding states in polymer mixtures. Macromolecules 1984, 17, 967.

0.009), we are able to unify phase diagrams of polymer solutions into one master curve.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] (C.W.). ORCID

Chi Wu: 0000-0002-5606-4789 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We gratefully acknowledge Professor Bo Zheng (CUHK, Chemistry) for his instrumentation support and Professor Keqing Xia (CUHK, Physics) for helpful discussions. Dr. Yuan Li collected all the data and is fully responsible for those experimental data as well as their analysis presented in this manuscript. This work is supported by the National Natural Scientific Foundation of China Projects (51773192).

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REFERENCES

(1) Iwao, T. Polymer Solutions: An Introduction to Physical Properties; John Wiley & Sons: New York, 2002. (2) Tanaka, T. Experimental Methods in Polymer Science: Modern Methods in Polymer Research and Technology; Academic Press: San Diego, 2000. (3) Koningsveld, R.; Stockmayer, W.; Nies, E. Polymer Phase Diagrams: A Texbook; Oxford University Press: New York, 2001. (4) De Gennes, P. G. Scaling Concepts in Polymer Physics; Cornell University Press: Ithaca, NY, 1979. (5) Sanchez, I. C. Critical amplitude scaling laws for polymer solutions. J. Phys. Chem. 1989, 93, 6983. (6) Stanley, H. E. Introduction to Phase Transitions and Critical Phenomena; Oxford University Press: London, 1971. (7) Kadanoff, L. P. In Critical Phenomena, proceedings of International School of “Enrico Fermi”; Varenna, Green, M. S., Eds.; Academic Press: New York, 1971. (8) Fisher, M. E. The renormalization group in the theory of critical behavior. Rev. Mod. Phys. 1974, 46, 597. (9) Hocken, R.; Moldover, M. R. Ising Critical Exponents in Real Fluids: An Experiment. Phys. Rev. Lett. 1976, 37, 29. (10) Kumar, A.; Krishnamurthy, H. R.; Gopal, E. S. R. Equilibrium critical phenomena in binary liquid mixtures. Phys. Rep. 1983, 98, 57. (11) Fisher, M. E. Renormalization group theory: Its basis and formulation in statistical physics. Rev. Mod. Phys. 1998, 70, 653. (12) Guggenheim, E. A. The Principle of Corresponding States. J. Chem. Phys. 1945, 13, 253. (13) Levelt Sengers, J. M. H. From Van der Waals’ equation to the scaling laws. Physica 1974, 73, 73. (14) Stein, A.; Allen, G. F. An analysis of coexistence curve data for deveral binary liquid mixtures near their critical points. J. Phys. Chem. Ref. Data 1973, 2, 443. (15) Lai, H. H.; Chen, L. J. Liquid-liquid equilibrium phase diagram and density of three water + nonionic surfactant ciej binary systems. J. Chem. Eng. Data 1999, 44, 251. (16) Sengers, J. V.; Shanks, J. G. Experimental critical-exponent values for fluids. J. Stat. Phys. 2009, 137, 857. (17) Pelissetto, A.; Vicari, E. Critical phenomena and renormalization-group theory. Phys. Rep. 2002, 368, 549. (18) Albert, D. Z. Behavior of the Borel resummations for the critical exponents of the n-vector model. Phys. Rev. B: Condens. Matter Mater. Phys. 1982, 25, 4810. (19) Friedli, S.; Velenik, Y. Statistical Mechanics of Lattice Systems: A Concrete Mathematical Introduction; Cambridge University Press: Cambridge, 2017. D

DOI: 10.1021/acs.macromol.8b01234 Macromolecules XXXX, XXX, XXX−XXX