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Superposition Methods in Polymers and soft matter : Time-Temperature, Time-Concentration and Other. What do we learn ?
One of the most powerful tools for studying the dynamics of polymers as well as small molecule glass-formers has been time-temperature superposition (TTS)[1,2,3]. Furthermore, other works have suggested that the principle can be extended to other parameters such as aging time [4], pressure [1], or concentration[1,5]. As a result, the possibility that one can use superposition principles to describe the behavior of other soft matter systems such as colloids has become of considerable interest. It is also known that polymeric glass-formers that have a strong sub-glass relaxation (β-relaxation) will not follow time-temperature superposition, though it has been suggested that the individual relaxation mechanisms themselves superimpose[6]. Here we will examine some of the successes of superpositioning as well as some issues of failure of superpositioning. In many cases we will see a good level of success. However, as an example of the kind of care needed in using superposition principles we will show for both polymers and colloids that when there is a large β-relaxation either time-temperature (polymer case) or time concentration superposition (TCS) may break down in these materials. While the polymer results are reasonably well known, TCS in colloidal materials has only recently become an important area of study[7,8]. Here, we use both experiment (rheological measurements) and simulation (Brownian dynamics) in which the colloidal dispersions are forced into their final concentration by making rapid changes in the particle diameter at constant particle number density. This is done in the simulations of near-hard sphere particles while the experiments are performed using soft sphere, thermosensitive particles (polystyrene core/poly(N-isopropylacrylamide)). In the simulations, the mean square displacement of the system that has been "aged" into an intransient (metastable equilibrium) state after the particle size-jump induced concentration change is transformed into the dynamic moduli using the generalized Stokes-Einstein approach. The dynamic moduli G’(ω) and G’’(ω) are measured directly for systems that have achieved the intransient state through aging. We find that TCS does not hold for the full range of dynamics measured in either the simulations or the experiments because the glassy (long time or α-relaxation) and the high frequency β-relaxation show different concentration dependences. A further analysis of the results within the framework of the Baumgaertel-Schausberger-Winter (BSW) [3] relaxation function shows that the individual mechanisms themselves follow TCS with the α-relaxation showing typical glass-like strong dependence on the concentration while the β-relaxation shows a very weak dependence on the concentration. The significance of the results will be discussed and we will also consider time-aging time superposition in the colloids.
References
[1] J.D. Ferry, Viscoelastic Properties of Polymers, 3rd. ed., Wiley, New York, 1980.
[2] F. Schwarzl, and A. J. Staverman, “Time-Temperature Dependence of Linear Viscoelastic Behavior,” Journal of Applied Physics 23, 838 (1952).
[3] Y.H. Jeong, “Frequency-dependent shear modulus of glycerol near the glass transition,” Physical Review A, 36, 766 (1987).
[4] L.C.E. Struik, Physical Aging in Amorphous Polymers and Other Materials, Elsevier, Amsterdam, 1978.
[5] W.G. Knauss and V.H. Kenner, “On the hygrothermomechanical characterization of polyvinyl acetate,” J. Appl. Phys., 51, 5531 (1980).
[6] M. L. Cerrada and G. B. McKenna,"Physical aging of amorphous PEN : Isothermal, isochronal and isostructural results," Macromolecules 33, 3065 (2000).
[8] Y. H. Wen, J. L. Schaefer, and L. A. Archer, “Dynamics and rheology of soft colloidal glasses,” ACS Macro Lett. 4, 119
(2015).
[9] X. Peng, J. Galen Wang, Q. Li, D. Chen, R. N. Zia, and G. B. McKenna, “Exploring the validity of time-concentration superposition in glassy colloids : Experiments and simulations,” Phys. Rev. E., 98, 062602 (2018).
[10] M. Baumgaertel, A. Schausberger, and H. H. Winter, "The relaxation of polymers with linear flexible chains of uniform length," Rheol. Acta 29, 400 (1990).