Dissecting Dynamics near the Glass Transition

Though the strong transformation in mechanical properties of glass-forming materials near the glass transition, T g , has been recognized and exploited for millenia, efforts to understand and predict this phenomenon at a molecular level continue to this day. Close to T g , where relaxation is considerably slower than predicted by the well-known Arrhenius equation, one of the most versatile and widely-used expressions to describe the dynamics or relaxation of glass formers is that of Vogel, Fulcher and Tammann (VFT). The VFT equation, introduced nearly 100 years ago, contains three adjustable fit parameters. In this work the dynamics of the polymer repeat units are related to macroscopic dynamics in polyelectrolyte complexes, which are hydrated amorphous blends of charged polymers. A simple expression, containing no freely adjustable fit parameters, is derived to quantitatively model relaxation from T g to temperatures well into the Arrhenius region. The new expression, which also fits a selection of three common neutral polymers, will advance the understanding and use of the glass-forming phenomenon.


Introduction
The mechanical relaxation rate of a glass-forming material, 1 such as a polymer in a melt or rubbery state, follows the well-known Arrhenius relationship at sufficiently high temperatures , [1] where , is the relaxation rate, ω0 is a prefactor and Eact an activation energy. 2 Such a dependence reflects well-localized dynamics with minimal cooperativity beyond a short length scale. On further cooling, the relaxation rate decreases faster than predicted by Equation 1 to, and past, the glass transition temperature, Tg, whereupon the material has reached a glassy state with significantly higher modulus. Figure 1 illustrates the temperature dependence of the relaxation rate (frequency) of a typical amorphous polymer using a classical set of lnω versus 1/T coordinates. A quantitative description of the glass transition using physically relevant and measurable parameters remains one of the greatest challenges in science. 3,4 Over the past several decades, many significant advances have been made in understanding the microscopic mechanisms of the glass transition. However, the frequency response of polymers as a function of temperature is still modeled or fit by a handful of equations which contain semiempirical fit parameters. The best-known fits that focus on the non-Arrhenius region near Tg are the Vogel-Fulcher-Tammann, VFT, with origins nearly a century old, [5][6][7] and Williams-Landel-Ferry, WLF, 8 equations, which are mathematically equivalent but highlight different aspects of polymer dynamics. The VFT fit includes a high frequency prefactor, ω0,VFT, commonly about 1 x 10 10 s -1 , and two additional fit parameters, D and To (Equation 2).
, , The temperature To, known as the Vogel temperature, implies divergent properties (ω = 0 at To) of the relaxation time 9 and has initiated vigorous debate over whether polymers continue to flow or relax below To, 10,11,12 invoking flow from materials > 10 7 years old. 13 The WLF fit, Equation 3, shifts data from frequencies measured at temperature T to a reference temperature Tr using a shift factor aT and focuses on constants C1 and C2 and whether they become "universal" if the reference temperature is selected to be Tg.
log [3] The dynamics of chains rely on the dynamics of the repeat, or monomer, units from which they are made. Thus, quantitative expressions for chain dynamics require understanding the dependence of repeat unit dynamics on temperature. The idea of a cooperatively rearranging region, CRR, discussed at length by Adam and Gibbs, 9 is often used to understand the molecular level picture of processes approaching Tg: the number of cooperatively rearranging units, CRUs, involved in the CRR is a minimum and remains constant in the Arrhenius region. As the polymer is cooled, this number grows 14 approaching Tg and the energy required to activate the growing CRR increases accordingly, pressing the slope in Figure 1 downwards.
In a recent study of ion transport in polyelectrolyte complexes -stoichiometric blends of oppositely-charged polymers -it was found that the rate of ion diffusion through the polymer was an excellent reporter of repeat unit dynamics. 15 Polyelectrolyte complexes, PECs, also termed coacervates when in a more liquidlike form, 16 The driving force for this association stems mainly from the release of counterions M + and Ainto solution, moderated by small enthalpy changes. The PEC may be "doped" with small "salt" ions, partially reversing Equation 4, 17 providing an ion-transporter with variable, high conductivity. 18 As with most ion-conducting polymers, the dynamics of ion hopping depend on the dynamics of the charged repeat units Pol + or Pol -. Prior work on a PEC at temperatures well above the Tg, i.e. in the Arrhenius region, showed that repeat unit relaxation times could be directly correlated with ion hopping, which was given by the ionic conductivity. 15,18 The entangled polyelectrolytes showed an unusual scaling of viscosity, η, as a function of number of repeat units N: η ~ N 5 . Using a theory of "sticky" reptation, 19 polymer chain dynamics were connected to those of the repeat unit 15 to obtain a quantitative relationship of η versus N.
In the present work, two PECs, both with a Tg near room temperature, were used to explore the relationship between repeat unit and polymer dynamics, and temperature: one was made from poly(diallyldimethylammonium), PDADMA, and poly(styrene sulfonate), PSS; the second from poly(vinylbenzyltrimethylammonium), PVBT, and poly(acrylamidomethyl propanesulfonate) PAMPS. PEC properties were always measured with an equilibrium content of water, maintained by contact with aqueous solutions, which provides reproducible properties and glass transitions near room temperature. Because the activation energy for rearrangement of Pol + Polmonomer pairs can be measured, the temperature dependence of the PEC near Tg can be rationalized in terms of a specific number of cooperatively rearranging pairs. This leads to a simple expression, containing no freely adjustable fit parameters, for the relaxation of PECs to and through the glass transition. Extension to a selection of common neutral polymers illustrates the breadth of validity for the new expression.

Results and Discussion
Compact PECs were prepared using pairs of oppositely-charged polyelectrolytes, one with a narrow molecular weight distribution, Đ, and one with a broad Đ. To obtain the relaxation behavior of polymer segments, ionic conductivity was first measured as a function of temperature. Using the Nernst Einstein equation, conductivities were converted to diffusion coefficients. 15 Diffusion was modeled using a simple nearest-neighbor hopping mechanism to obtain the relaxation rate, including the activation energy, of a Pol + Polcharge pair. PEC (Ω -1 cm -1 ), was converted to the ion diffusion coefficient, Di (cm 2 s -1 ), at temperature T using the Nernst-Einstein equation [5] where q is the charge of one ion (1.602 x 10 -19 C), C is the concentration of NaCl inside the PEC, Na is Avogadro's number, and k is Boltzmann's constant. Di shows Arrhenius behavior as a function of temperature (see Figure 2).
The prefactor ω0,i, here 8.12 x 10 12 s -1 , represents the hopping attempt frequency and the exponential term can be considered the probability, pT, that a hopping attempt will be successful at a particular temperature. It is also clear from Scheme 1 that the ion hopping rate is the same as the rate of Pol + Polpair breaking frequency, and Equation 7 directly provides the temperature dependence for this process which is fundamental to longer-range dynamics.

The Minimum Cooperative Rearranging Region (MCRR)
Cooperative rearrangement is the coordinated movement of a certain number of "units," such as chain segments, in a certain volume known as the cooperatively rearranging region. 9 Though the minimum number of units moving cooperatively is understood to be two, the identity of the units and how they move has historically been unclear. 9 Here, the most likely mode of rearrangement in PECs ensures ion pairing is preserved to minimize the electrostatic energy. To accomplish this, pair exchange, PE, was proposed. 20 In the PE mechanism, two neighboring Pol + Polpairs exchange places, as illustrated in Scheme 2

Scheme 2. The minimum cooperative rearranging unit in a polyelectrolyte complex. Two
Pol + Polpairs exchange places at a rate ωT,2 or ωT,arr . The activation energy is 2Ea,u.
Pair exchange allows polymer chains to move relative to each other. The displacement is minimal and the net energy is similar before and after exchange because all four charged units are paired before and after the event.
The MCRR is a thus two Pol + Polpairs, or a quad of charged units. In the Arrhenius region of viscoelastic response, PE is the only relaxation mechanism in operation, i.e. there is no cooperativity beyond exchanging quads. Recently, the mechanical properties of a more fluid-like PEC, PMAPTA/PMA, were studied at temperatures well above its Tg. 15 The viscoelastic properties followed an Arrhenius response with temperature. Importantly, the activation energy could be predicted knowing the (experimentally measured) activation energy for ion hopping, Ea,u, for breaking one pair (Scheme The slope of the line of a plot of lnωT,n at any 1/T is -nEa,u/R. i.e.
, , For the Arrhenius region, n = 2 , , , The deviation of lnωT,n from Arrhenius as T  Tg is modeled using some key concepts employed by Adam and Gibbs in their seminal paper on the CRR. 9 They expressed the relaxation rate in glass formers as a transition probability, W ∆ / [11] where A is a temperature independent prefactor, z is the number of monomer units (the "size") in the cooperative region, and Δμ is the energy barrier for rearrangement per monomer segment. Although the identity of the monomer unit is implied to be a segment, this was not specified and has, to date, been left open for interpretation. Equating z to n and Δμ to Ea,u the relative relaxation rates in the Arrhenius region (n = 2) and any other temperature is given by On a log plot, ωT,n shows clearly as a deviation from Arrhenius, illustrated in Figure 1. Similar conclusions can made starting from molecular entropy theories, 21,22 which are extensions of ideas presented by Adam and Gibbs.
Using Equations 9 and 12 along with the following boundary condition from n = 3 for Equation 12: an equation for , at any temperature may be derived (see Supporting Information for the   derivation): where Tc is the convergence temperature, defined at the point where , converges to a factor of 1/e of the Arrhenius value, which occurs at n = 3 (Equation 13). There are other ways to present Equation 14, but this form emphasizes the extent of deviation (second term on the right) from Arrhenius. Figure 3 compares the data to Equation 14.  Table 1. Panels C and D present the number of rearranging units in the CRR as a function of temperature for the PECs and neutral polymers respectively with n ~ 13 at Tg and n = 2 in the Arrhenius region at high temperatures.

Neutral Polymers
While the PEC data are predicted well by Equation 14,  Figure S4 for shift factors). The activation energies from the Arrhenius slope (= 2Ea,u), Tg and Tc for all the polymers studied are summarized in Table   1. As seen in Figure 3, Equation 14 was exceptional in describing the dynamics of the three neutral polymers as well.
Tc is between 60 and 100 degrees above Tg, comparable with the general statement that Arrhenius behavior is observed beyond Tg + 100. Partly in response to the controversy surrounding To, a characteristic temperature, Tx, for a transition from VFT to Arrhenius dependence has been sought in various analyses. 23 The crossover from VFT to Arrhenius depends on what is plotted. For example, a plot of logωT versus 1/T-To yields a Tx (also labeled an "onset" temperature 24,25 ) close to Tc seen here, 23 whereas a power law fit returns a lower Tx more aligned with mode coupling theory. 26,27 Comparison with VFT There are several significant aspects to Equation 14 in comparison to the VFT Equation, which also does a good job of fitting the data over the range Tg to Tg + 90 o C (see Supporting Information Figure S5 for a VFT fit). First, Equation 14 has no freely adjustable fit parameters: Tc is prescribed uniquely by Equation 13 (but the data must go to sufficiently high temperatures to obtain the Arrhenius slope; otherwise, Ea,u becomes a fit parameter). Second, instead of the highly-discussed Vogel temperature To in Equation 2, which implies divergence at this temperature, 9,11,12 the convergence temperature Tc avoids such a quandary.
Another feature of Equation 14 is the ability to fit both the Arrhenius and the non-Arrhenius region with the same expression and numerical constants. The VFT equation fails in this regard (see Supporting Information Figure S5 for an illustration of this) since it bends away from the Arrhenius slope at higher temperatures.

Why is Tg at Tg?
At any temperature, the CRR is composed of n CRUs, given by Equation 12 and plotted in Figure 3. The number of CRUs at Tg is consistently about 13 ± 1. The specificity of this n is in contrast to intensive efforts over the past few decades to identify both the cooperative unit and their number at Tg.
The fact that n = 13 at Tg is interpreted to reflect the fact that each pair is surrounded by approximately 12 nearest neighbors. Although static scattering methods show no order, packing simulations by Tanaka et al. suggest a "hexagonal-close-packed-like" arrangement at Tg. 28 If each pair experiences this condition, the entire material is percolated with n = 13 cooperatively rearranging regions and thus undergoes the glass transition. If the cooperatively rearranging unit were defined as the number of polyelectrolyte repeat units, n would be 26.
Significant insight would be provided by molecular dynamics simulations involving the correct number of pairs.
The "size" of the CRR has been a subject of much discussion and confusion. 29 Adam and Gibbs 9 cast the size as the number of cooperative "monomer segments," represented in Equation 11, here termed "units." When small structural units such as -CH3 are defined 30 as "beads" a universal value of 4-5 for the number of beads at Tg is inferred from heat capacity measurements. 29 Alternatively, a linear dimension ε, or volume VCRR (= ε 3 ) has been used to characterize the CRR. 14

Significant Parameters and "Universal" Constants
The Arrhenius parameter required for Equation 14 contains Ea,u and ωo,A. The simulated relaxation curves in Figure 3 show the influence of varying Ea,u with constant ωo,A and vice versa (deviations from Arrhenius are shown in Supporting Information Figure S6)..  Table 1). Alternatively, the "fragility" of a glass former describes the speed with which the relaxation rate changes at Tg. 32 Assuming n is 13 at Tg, the fragility parameter (steepness index), m, is given by , . [15] The m values from Equation 14 and from the VFT fits are compared in Table 1.
The analysis of the dynamics of numerous polymers using the WLF equation (Equation   3) has led to the (somewhat controversial) conclusion that the coefficients c1 and c2 have "approximately universal" respective values of 10-15 and 50-60 K when the reference temperature is Tg. 9 [18] Conclusions Equation 14 contains no prediction that dynamics should cease at any temperature above 0 K. Although the glass transition depends on the scan or deformation rate and is not itself a thermodynamic phase transition, underlying thermodynamic phenomena are often invoked. 28,34,35 Here, the small change in ion diffusion rate seen near Tg in Figure 2A is an example of a second order transition. According to the mechanism in Scheme 1 this relaxation mode remains fast and is not cooperative. The small decrease in Ea,u from 25.6 kJ mol -1 above Tg to 27.1 kJ mol -1 below Tg could indicate a transition from a more disordered to a less disordered system ("medium range crystalline order" 28 ) on falling through Tg, which would be consistent with a free volume transition.
The intercept ωo,A (Table 1)  evaporation. An axial force of 0.2 N was applied. Frequency sweeps were performed over a temperature range between 0 and 95 °C. The measurements were taken starting from the highest temperature to allow the complex to thermally equilibrate. The PEC was allowed to reach the target temperature by applying a ten-minute delay before each frequency sweep.
For neutral polymers, a 20 mm x 0.1 mm disk was hot pressed. The temperature was regulated between 45 and 350 °C using an TA Instruments ETC oven. The chamber was purged with N2.
The storage and loss modulus of the neutral polymers were recorded over a wide temperature range starting with T ~ Tg + 250 and going as low as Tg. Time-temperature superposition was performed using shift factors using Tg as the reference temperature according to Equation 3, shift factors given in Supporting Information Figure S4.
The Tg s were measured using dynamic mechanical analysis (rheometer) with the environment controlled as above. For PDADMA/PSS tanδ (= G''/G') was measured from 10 to 50 °C with a ramp rate of 2 °C min -1 under constant strain at eight frequencies between 0.01 and 2 rad s -1 . For PVBT/PAMPS and the three neutral polymers, a temperature ramp experiment was carried out at 0.62 rad s -1 with a ramp rate of 2 °C min -1 . The strain % was held constant throughout the experiment and forward and a backward temperature sweeps were recorded starting from the highest temperature. The forward and backward scans were averaged and the temperature where the maximum at tanδ (= G''/G') occurs was recorded as Tg.