Unveiling the dependence of glass transitions on mixing thermodynamics in miscible systems.

The dependence of the glass transition in mixtures on mixing thermodynamics is examined by focusing on enthalpy of mixing, ΔHmix with the change in sign (positive vs. negative) and magnitude (small vs. large). The effects of positive and negative ΔHmix are demonstrated based on two isomeric systems of o- vs. m-methoxymethylbenzene (MMB) and o- vs. m-dibromobenzene (DBB) with comparably small absolute ΔHmix. Two opposite composition dependences of the glass transition temperature, Tg, are observed with the MMB mixtures showing a distinct negative deviation from the ideal mixing rule and the DBB mixtures having a marginally positive deviation. The system of 1, 2- propanediamine (12PDA) vs. propylene glycol (PG) with large and negative ΔHmix is compared with the systems of small ΔHmix, and a considerably positive Tg shift is seen. Models involving the properties of pure components such as Tg, glass transition heat capacity increment, ΔCp, and density, ρ, do not interpret the observed Tg shifts in the systems. In contrast, a linear correlation is revealed between ΔHmix and maximum Tg shifts.

Glass transition is a typical liquid-solid non-equilibrium transition occurred in all kinds of materials, however, the physics of the phenomenon remains unsolved and debated123. The glass transition behaviors are found to be quite sensitive to even subtle modification in molecular structure and intermolecular interactions4. Binary glass forming mixtures are simple and appealing systems to investigate the glass transition due to the ease in modifying the intermolecular interactions. Miscible mixtures usually show a single glass transition temperature, T5, which might not necessarily be the mean value of the pure components' Ts produced by the ideal mixing rule, but is related to the mixture composition (e.g. mole fraction, x)67.

The composition dependence of T in mixtures has been extensively studied in a number of systems, and diverse observations in the T-x relationships have been reported, showing positive, negative and zero deviations from the ideal mixing rule568910111213. Several models are proposed in search of the explanation for such deviations14, such as the most widely used Couchman - Karasz (CK) and Gordon - Taylor (GT) models1516. The CK model takes the glass transition as an Ehrenfest second order transition, and assumes the changes upon mixing in enthalpy (ΔH), entropy (ΔS), and volume (ΔV) for both liquid and glass to be negligible at T. Moreover, the glass transition heat capacity increment, ΔC, is assumed to be temperature-independent. The GT model is proposed based on ideal volume of mixing and the linear change in volume with temperature. The two models have the similar expression as, T = (x1T1 + k2T2)/(x1 + k2), where k = ΔC1/ΔC2 is determined for the CK model while for the GT model k = ρ2T2/ρ1T1, and x and ρ (i = 1, 2) denote the mole fraction and density of pure components17. Obviously, the two models emphasize the significance of two components' properties including T, ΔC, and ρ. Although the two models have successfully explained some experimental measurements1819, there are a number of systems that show pronounced discrepancy between experimental measurements and the theoretical calculation made by the models61320. Alternatively, mixing thermodynamics is argued to determine the glass transition behaviors in mixtures, and studies found that positive enthalpy of mixing generally corresponds to negative T deviation5, and even a value of ΔH as low as 200 J/mol can result in a remarkably negative T shift21. However, the evaluation of the weighted contributions to the T shifts from both mixing thermodynamics and the properties of pure components have not been available. Moreover, systematic studies of the dependence of T on mixing thermodynamics have not been done. Therefore, the studies of the impact of ΔH on the T shifts in mixtures by changing both magnitude (small vs. large) and sign (positive vs. negative) would help gain insight into the glass transition behaviors in mixtures. Unfortunately, such studies are not accessible, in particular, the effect of small ΔH.

In this work, we studied the glass transitions in binary glass forming molecular systems, and systematic comparison is shown with typical scenarios of small vs. large and positive vs. negative ΔH. A global picture of the ΔH effect on the T shifts in mixtures is presented. The results are expected to benefit the understanding of the glass transition as well as the precise evaluation of T in poor glass formers.

Results

Fig. 1 presents the C curves around T for the glass forming o- vs. m- MMB and o- vs. m- DBB systems. The composition dependence of ΔC obtained from all the C curves are plotted in Fig. 2(a) and (b) for the two systems. The ΔC values of the pure components are estimated by extrapolating the mixtures' ΔC, and 0.619, 0.673, 0.35 and 0.38 J/(g–K) are obtained for o-MMB, m-MMB, o-DBB and m-DBB. The composition dependences of T are plotted in Fig. 3(a) and (b). A markedly negative T deviation is immediately visible for the mixtures of o- vs. m- MMB, although the deviation is small. In contrast, the evaluation of the T shift for the system of o- vs. m-DBB in Fig. 3(b) is of a bit difficulty. The fitting with marginally positive deviation or a linear fitting seems to be able to explain the experimental points, as shown by the black solid line and the light cyan solid line. The bottom line for o- vs. m-DBB is that the deviation is, by no means, negative. The pure components' Ts can be evaluated by the numeric extrapolation in Fig. 3, yielding 156.1, 147.5, 160 and 157.2 K for o-MMB, m-MMB, o-DBB and m-DBB. The ΔH values are also presented in Fig. 3(c) and (d), reported for the mixtures of o- vs. m- MMB and o- vs. m- DBB, with ΔH maxima to be 3.42 and −4.98 J/mol, respectively22. Fig. 4(a) and (b) show the composition dependences of ΔC and T for the system of 12PDA vs. PG over the whole composition, where ΔH maximum of 12PDA vs. PG is recorded to be −3876 J/mol23. A remarkably positive T deviation is seen in the system. The ΔC values for 12PDA and PG are determined to be 1.36 and 0.86 J/(g–K) while the Ts to be 146.8 and 169.7 K.

Figure 1

Glass transition heat capacity C curves of two isomeric binary systems for the glass forming compositions.

(a) o- vs. m- methoxymethylbenzene (MMB); (b) o- vs. m- dibromobenzene (DBB). The curves are determined at a heating rate of 20 K·min−1.

Figure 2

Composition dependence of the glass transition heat capacity increases, ΔC, in two isomeric mixtures.

(a) o- vs. m- methoxymethylbenzene (MMB); (b) o- vs. m- dibromobenzene (DBB). The extrapolations for the mixtures' heat capacity increment, ΔC, are given by the blue dash-dot lines.

Figure 3

Composition dependence of T and ΔH in the isomeric mixtures of o- vs. m- methoxymethylbenzene (MMB) and o- vs. m- dibromobenzene (DBB).

In the upper frames (a) and (b), the red solid squares and circles are experimental values. The fitting line for the o- vs. m- methoxymethylbenzene (MMB) is given by the black solid line, while the fitting line for the o- vs. m- dibromobenzene (DBB) is expressed by the black or the light cyan solid line. The predicted values by the CK and GT models are shown by the blue dotted (CK) and magenta dash-dot-dot (GT) lines. The lower frames (c) and (d) show ΔH values for the two isomeric systems recorded in Ref. 22. The horizontal axes show the mole fraction of m- MMB (x) and DBB (x).

Figure 4

Composition dependence of the heat capacity increment, ΔC (a), and T (b) of 1, 2- propanediamine (12PDA) vs. propylene glycol (PG).

The predicted values by the CK and GT models are shown by the blue dotted (CK) and magenta dash-dot-dot (GT) lines in frame (b).

The explanations of the T values in the mixtures using the CK and GT equations with the extrapolated T and ΔC values of the pure components are presented by the blue dotted (CK) and magenta dash-dot-dot (GT) lines in Fig. 3(a), (b) and Fig. 4(b). It appears that the calculated T curves in terms of the CK and GT models do not satisfactorily explain the experimental measurements. Similar observations have been reported in early studies of a number of glass forming mixtures1320. The large discrepancy suggests that, notwithstanding the importance of the properties (T, ΔC and ρ) of the pure components, the exact explanation of the T - x relationship for most cases asks for more considerations.

The relative T shift, ΔT/T, for the systems of o- vs. m- MMB, o- vs. m- DBB, and 12PDA vs. PG are shown in Fig. 5, where ΔT is the T difference between the measured values and the ones calculated in terms of the linear average of the pure components' Ts. Two more typical systems recorded in our recent work, benzil (BZL) vs. m-nitroaniline (MNA)21 and methyl m-toluate (MMT) vs. methyl o-toluate (MOT)24, are also included in Fig. 5 to represent the mixing scenarios of larger positive ΔH and ideal mixing, respectively. It is seen in Fig. 5 that the five systems differ greatly in the T deviations from the ideal mixing rule. The system of 12PDA vs. PG has a remarkably large and positive T deviation with the maximum ΔT/T being as high as 14.5%. In contrast, the system of BZL vs. MNA has a large and negative T deviation with the maximum of ΔT/T of −2.3%21. For systems with smaller positive or negative ΔH, ΔT/T becomes smaller.

Figure 5

Relative T differences, ΔT/T, for five typical glass forming systems, representing various mixing modes.

(1) ΔH ≫ 0 (benzil (BZL) vs. m- nitroaniline (MNA)), (2) ΔH ≥ 0 (o- vs. m- methoxymethylbenzene (MMB)), (3) ΔH = 0 (methyl m-toluate (MMT) vs. methyl o-toluate (MOT)), (4) ΔH ≤ 0 (o- vs. m- dibromobenzene (DBB)) and (5) ΔH ≪ 0 (1, 2- propanediamine (12PDA) vs. propylene glycol (PG)).

Discussion

The mixing thermodynamics for the BZL vs. MNA system (Fig. 5) is recently measured showing the ΔH maximum of ~200 J/mol21, while the MMT vs. MOT system is regarded as an ideal mixing case with ΔH = 02425. Accordingly, the five cases in Fig. 5 represent the typical model systems of ΔH ≫ 0 (BZL vs. MNA), ΔH ≥ 0 (o- vs. m- MMB) ΔH = 0 (MMT vs. MOT), ΔH ≤ 0 (o- vs. m- DBB) and ΔH ≪ 0 (12PDA vs. PG). It is therefore evident that systems with large absolute values of ΔH, either positive or negative, show remarkable negative or positive T shifts. Similar observations have been recorded in earlier experimental measurements. For example, the binary mixtures constituted by monoalcohols and amines with large and negative ΔH262728 generally show large and positive T shifts29. Likewise, large and negative ΔH is also detected in the mixtures composed of water and alkoxy alcohols, such as the aqueous systems of 2-methoxyethanol30 and 2-ethoxyethanol31, and large and positive T deviations are observed in such mixtures10.

The results in this work allow us to check the presumably quantitative correlation between T and ΔH in binary mixtures. In addition to the experimental measurements reported here, more T821252932 and ΔH values2122233334353637 are collected. Fig. 6 shows a plot between ΔT/T and ΔH measured at 298 K for 13 systems with ΔH ranging from 1044 J/mol to −3900 J/mol. Strikingly, a nearly linear correlation is revealed, and positive ΔT/T values of mixtures are found to be enlarged with increasingly negative ΔH. Note that although systems with positive ΔH are relatively fewer, the available data in the three systems basically follow the correlation showing that increasingly positive ΔH corresponds to enhanced negative T deviation (ΔT/T). Note that in Fig. 6 there are a few systems that somehow deviate from the master line such as methanol vs. triethylamine2933. This may arise partly from the error in ΔH values, since several different ΔH values can be found for a mixing system38. However, the minor errors do not affect the demonstration of the explicit relationship between the thermodynamics and the glass transition behaviors in binary mixtures. Compared with the results in Figs. 3 and 4 expressed by the CK and GT models, Fig. 6 consequently implies that ΔH could be a more decisive factor in affecting the T behaviors in mixtures.

Figure 6

Correlation between ΔH and ΔT/T based on 13 binary molecular glass forming systems.

(1) 1- butanol vs. 1- bromobutane414344; (2) benzil (BZL) vs. m- nitroaniline (MNA); (3) o- vs. m- methoxymethylbenzene (MMB); (4) methyl m-toluate (MMT) vs. methyl o-toluate (MOT); (5) o- vs. m- dibromobenzene (DBB); (6) chloroform (CFM) vs. toluene834; (7) diethylether (DEE) vs. dichloromethane (DCM)837; (8) methanol vs. triethylamine2933 (9) 2-ethyl-1-hexanol (2E1H) vs. 2-ethyl-1-hexylamine (2EHA)25 (10) chloroform (CFM) vs. diethylether (DEE)836 (11) chloroform (CFM) vs. triethylamine (TEA)835 (12) 1,2- propanediamine (12PDA) vs. propylene glycol (PG); (13) 1,2- propanediamine (12PDA) vs. 1,3- propanediol (13PDO)2332. The correlation for the systems of (3), (4) and (5) is given in the insert.

Fig. 6 includes a typical glass forming binary system of 2-ethyl-1-hexanol (2E1H) and 2-ethyl-1-hexylamine (2EHA), and the accurate T data show a large and positive deviation with ΔT/T reaching 5.6%39. The enthalpy of mixing in the system is not available. Yet, the value could be roughly evaluated from the mixtures composed of the homologous liquids with similar molecular structures and chemistry. For example, the ΔH maximum at 298 K reaches −2905 J/mol for the system of butyl amine vs. 1-butanol26, which is comparable with the value of −2755 J/mol for the system of amylamine vs. 1-butanol27. It is seen that the absolute values of the negative ΔH for mixtures of monoalcohols and monoamines decrease with the carbon number in the backbones of either alcohols or amines, and also decrease with the enhanced steric hindrance due to the introduction of branched chains26272840. Approximately, the ΔH maximum for the mixtures of 2E1H vs. 2EHA is evaluated to be ~−2200 J/mol. The values of ΔH and ΔT/T (5.6%39) for the 2E1H - 2EHA mixtures appear to coincide well with the correlation shown in Fig. 6.

The mixing systems of positive ΔH are much less available. The system of 1-butanol vs. 1-bromobutane is reported to have a quite large and positive ΔH with the maximum reaching 1044 J/mol at 298 K41. The composition dependence of the kinetic glass transition temperature, T, defined by the temperature where the α-relaxation time approaches 100 s3942, can be obtained based on the studies of the dielectric relaxation dynamics for the pure components and the mixtures investigated by Goresy et al43 and by Murthy et al44. When combining the data from the two groups, a negative T deviation with the maximum of ΔT/T to be −3.2% is yielded. The result is comparable with the predicted one by the correlation in Fig. 6.

For ideally mixing systems with ΔH = 0, a common observation is the negative T deviation from the ideal mixing rule, as shown in Figs. 5 and 6 for the MMT vs. MOT mixtures. This is the typical feature driven by mixing entropy2945. According to the Adam-Gibbs' configurational entropy (S) theory46, τ(T) = τ0 exp(ΔμCAG/TS), where τ0 is the pre-exponent having the phonon time, Δμ is potential barrier hindering molecular rearrangement, and C is a constant474849, if the intermolecular interaction keeps unchanged (ΔH = 0), the variation in Δμ is negligible, and the increase in S during mixing at a fixed temperature would correspond to a decrease in τ(T), which makes the relaxation time curves move toward low temperature regions in activation plots (log τ(T) vs. 1/T), and finally leads to a decrease in T (or T). Furthermore, for the small and comparably-valued ΔH of opposite sign, the case with positive ΔH (for example, the o- vs. m- MMB system in the inset of Fig. 6) would correspond to relatively larger T shift (negative) than the shift (positive) in the case with negative ΔH (for example, the o- vs. m- DBB system).

In addition to the correlation between ΔH and the T shifts in glass forming mixtures, recent studies explored the liquid fragility in various mixtures with varying ΔH, which defines how rapidly the liquid dynamic properties (viscosity or relaxation time) change with temperature at T, and found a general trend of negative deviations from the ideal mixing rule25, somehow regardless of the enthalpy of mixing. Combining the impacts of ΔH on T and fragility in mixtures, it is evident that large and negative enthalpy of mixing would correspond to the high viscosity within supercooled liquid regions50, which further favors kinetically glass formation. This rationalizes the empirical criterion that the large and negative enthalpy of mixing is a crucial consideration in order to prepare multicomponent bulk metallic glasses5152.

Finally, the present results of the quantitative correlation between ΔH and the composition dependence of T in mixtures can provide a guidance to estimating the T for materials which are extremely difficult to be vitrified, such as benzene. For the mixtures containing benzene, although a complete dataset of both mixing thermodynamics and glass transition are not available, considering the fact that for most mixtures the composition dependences of T generally show a similar trend as the composition dependence of viscosity2953, the latter can consequently be used to roughly estimate the former. ΔH and viscosity for benzene vs. decane mixtures have been reported, showing a positive ΔH54 but negative deviation in the composition dependence of viscosity55, which could imply the negative deviation in the composition dependence of T. This agrees with the observation in the present work for the correlation between the mixing thermodynamics and glass transitions in various mixtures. It is therefore speculated that, provided that the enthalpy of mixing is given for the glass forming miscible mixtures containing benzene, the reliable extrapolation towards the T of pure benzene can be ensured based on the accessible T values of the mixtures. Such endeavor will be conducted in further study.

Methods

Chemicals

o- and m- isomeric systems of methoxymethylbenzene (MMB) and dibromobenzene (DBB) were selected in this study with the former system having small and positive ΔH and the latter system small and negative ΔH22. The system of 1, 2- propanediamine (12PDA) vs. propylene glycol (PG) was selected for the considerably large and negative ΔH generated during mixing23. For comparison, the results for the system of benzil (BZL) vs. m-nitroaniline (MNA) featured by the remarkable large and positive ΔH in our recent studies were used21. o- MMB (Alfa Aesar 99%), m- MMB (Alfa Aesar 99%), o- DBB (Alfa Aesar 98%), m- DBB (Alfa Aesar 97+%), 12PDA (Sigma Aldrich 99%), PG (Sigma Aldrich 99%) were used as received.

Thermodynamic measurements

A Perkin-Elmer differential scanning calorimeter (Diamond DSC) was used for the calorimetric determination of the heat capacity curves, and the glass transition temperatures, T. A sapphire sample (31.4 mg) was used for the C standard. The sample was initially quenched to low temperatures at the accessible highest cooling rate in the calorimeter to guarantee the complete vitrification, and a subsequent heating-cooling-reheating cycle was performed around the glass transition at a fixed rate of 20 K/min. The calorimetric quantities for the systems are determined from the reheating C curves. The sapphire and baseline heat curves were obtained following the same procedure. The DSC was calibrated using indium and cyclohexane prior to the measurements56.

Author Contributions

W.K.T., Y.X.W., P.Z. and L.-M.W. finished the experiments and wrote the main manuscript. X.L., S.H.J. and Y.J.T. provide experimental and writing guidance. All authors reviewed the manuscript.