The Glass Transition Temperature of Heterogeneous Biopolymer Systems

Biopolymers are abundant, renewable, and biodegradable resources. However, bio-based materials often require toughening additives, like (co)polymers or small plasticizing molecules. Plasticization is monitored via the glass transition temperature versus diluent content. To describe this, several thermodynamic models exist; nevertheless, most expressions are phenomenological and lead to over-parametrization. They also fail to describe the influence of sample history and the degree of miscibility via structure–property relationships. We propose a new model to deal with semi-compatible systems: the generalized mean model, which can classify diluent segregation or partitioning. When the constant kGM is below unity, the addition of plasticizers has hardly any effect, and in some cases, even anti-plasticization is observed. On the other hand, when the kGM is above unity, the system is highly plasticized even for a small addition of the plasticizer compound, which indicates that the plasticizer locally has a higher concentration. To showcase the model, we studied Na-alginate films with increasing sizes of sugar alcohols. Our kGM analysis showed that blends have properties that depend on specific polymer interactions and morphological size effects. Finally, we also modeled other plasticized (bio)polymer systems from the literature, concluding that they all tend to have a heterogeneous nature.


APPENDIX A
In this section, we give a short summary of the glass transition theory of mixtures and show how the Generalised Mean model can be derived from a quasi-second order thermodynamic transition. Couchman and Karasz (1978) demonstrated an interesting model for the effect of the composition of binary mixtures on Tg 1 . The model was successful for compatible polymer-polymers blends and also polymer-diluent systems 2,3 . The model derivation was based on writing that Tg is not a firstorder thermodynamic transition; more specifically, on the property of continuity of a system's specific entropy at Tg. Hence, the integration of Δ ! of a polymer blend results in the following simplified expression:

Glass Transition Modelling
where x, Tg, ! , l, g denote molar (or volume) fraction, glass transition, specific heat, supercooled liquid and glassy states of pure components 1 and 2, respectively. Originally, the authors were also careful to mention that the commonly neglected entropic terms just above and below the transition . From the perspective of Gibbs free energy, it is true that an enthalpic relationship of eq S1 also holds. Irrespective of the form (entropy or volume continuity conditions), note that for most systems the ! property of components undergoes a finite discontinuity at the transition.
All models derived from eq S1 have the activation or mobility of chain units as a fundamental principle, corrected through Δ ! values. Another ubiquitous Tg model is the expression earlier proposed by Gordon and Taylor (1953) 4 , which can also be rearranged into another simplification of eq S1. To derive this relationship, we must rewrite Δ !& /Δ !" into a constant, kGT, so that: In fact, the thermodynamic parameter kGT is related to a constant coefficient of expansion (volume) during the transition. However, this phenomenological solution was conceived for ideally mixed copolymers. Thus, a simplification (kGT = 1) results in a linear averaging of the Tg, which to our knowledge never occurs.
The most frequently used equation for the change in Tg is the phenomenological Fox (1956) From equation (S1), it assumes that Δ !& *Tg1/Δ !" is a constant. This way, the predicted Tg becomes a simple rule of mixing of the entropic contributions of the pure components. Hence, simply put, it assumes no effect from enthalpic interactions upon mixing.
However, when we analyse Tg changes with an x2 component, a large portion of experimental results show large deviations from those ideal mixture predictions. The most used forms of the discussed models either neglect excess thermodynamic property from mixing or make it elusive to work around such values since Δ ! corrections are not straightforward to interpret. Several other models have been proposed to tackle this, such as the models of Kwei (1984) 9 , additional terms expanding Couchman-Karasz (eq 3) 3,10-12 , and models based on virial coefficients 13 . Yet most of these approaches can easily result in over-parameterisation, and outcomes are hard to interpret. Inspired by the simple harmonic mean expression proposed by Fox, we have developed a new working model for the Tg property in blends. Similar to previous studies, we assume Tg is a quasisecond-order transition according to the Ehrenfest classification. Thus, it can be written as $ = , % -.

Generalised Mean Model
, % -/ , where ∂H and ∂S are, respectively, second or nth-term partial derivatives of enthalpy and entropy at transition. If we further expand on this equality, we can take the relation in thermodynamic properties to be weighted in the form of a generalised or power mean instead of the Fox-like harmonic mean. It follows that the Generalised Mean (GM) model becomes: where ϕi,, α, β are volume or mass fraction of components 1 or 2, entropy exponential, and enthalpy exponential, respectively.
The expression S4 can be further arranged if we couple some of the partial derivative terms into a constant, kGM. For convenience, kGM is obtained by substituting the ratio of partial derivative enthalpies ∂H of components ( 3 ∂Hi). However, a similar output could have been obtained by using ∂S. Considering that the Tg of each component is 3 ∂Hi/ 3 ∂Si and that kGM can be expressed as 3 ∂H2/ 3 ∂H1, the full GM equation can be manipulated into the applicable form: where ϕi, Tgi, kGM, α, β are volume or mass fraction of components 1 or 2, glass transition of components 1 or 2, model constant, entropy exponential, and enthalpy exponential, respectively.
This model has five degrees of freedom and can resolve into nontrivial parabola or S-shaped curves by tuning the α and β exponents. Previously, such S-shaped data have been previously modelled using virial Tg models, e.g., for tetramethyl bisphenol-A polycarbonate-(polystyrene) blend 13 .
Nevertheless, we noticed that most data take a simple form, and we can do a linearisation (α, β = 1). Hence, the linearised Generalised Mean model (GML) becomes: From this form, the model can also revert to the Fox equation (eq S3) if kGM is 1, which explains our choice for defining kGM from the enthalpy ratio. The constant kGM can also be interpreted as a partitioning factor correcting the volume fraction of diluent (ϕ2). Hence, it is a static measure of system partitioning or heterogeneity. The GML version resolves most nonlinear cases and is mathematically analogous to the Gordon-Taylor model (appears from assuming a ratio of partial derivative entropies ∂S). Moreover, GML is useful because excess property and possible structural changes can be easily monitored with one factor, kGM.
The values and morphological states implied from fitting the GM(L) model are heavily influenced by sample history. Hence, drying and cooling rates will likely influence the assessed diluent partitioning. In particular, the cooling rate effect on sample history should be accounted for, since quenching can suppress the difference in Tg(s). It is also worth underlining that sensible experimental kinetic rates should be used for determining the thermal transition. The experiment observation times should obviously be probing the relaxation times of system components.
Furthermore, the Tg property is known to broaden and increase logarithmically with the quench rate 14 . Although the time-dependent effects are not explored, the model is able to fit accordingly the Tg curves over composition, irrespective of the studied fabrication method or experimental rates employed.
Reliable Tg measurements are crucial for the findings of GML model fit to be valid. This should be ensured by selecting a sensitive enough technique, for instance DMTA, dielectric spectroscopy and modulated differential scanning calorimetry. These methods are less affected by broadening effects at transition 15 . Yet Tg variations among methodologies as high as 20 °C are normally expected. In addition, adequate machine calibration and experimental conditions (environment, rate, strain, oscillatory parameters) need to be explored. Lastly, the data analysis step should be well reported for there are multiple standard ways to obtain Tg.

Summary of Thermodynamic Relations
The Gibbs energy G(T, P, {n}) is a continuous function of its natural variables temperature T, pressure P, and composition, {n}, and has as derivatives over temperature entropy = − - At a second order transition, the enthalpy and entropy are continuously differentiable everywhere except at the phase transition, where the slope changes so that the heat capacity has a jump value , where the subscript s denotes that the values are to be taken at the secondary transition. If a material's glass transition is assumed to be a (pseudo) second order phase transition, the same relation should apply to the jump-values of ∂H and ∂S. Nevertheless, the relationships mentioned above do not work for systems with ! close to infinity at transition, i.e., lambda transitions, as is the case of systems with order-disorder evolution.
The blending of components will also involve thermodynamic mixing functions, i.e., the entropy of mixing ΔmixS and enthalpy of mixing ΔmixH. For ideal miscible mixtures, the enthalpy of mixing is zero, and the entropy of mixing is positive from increased disorder. We can imagine entropy as the main driving force arising from the dispersion of components. Real mixtures often contain excess interaction of any of the pairs of components. This specific interaction(s) might result in enthalpy changes or additional entropy upon mixing, for instance, from molecular clustering.
Depending on the magnitude of the enthalpy step from energetic interactions or adverse entropy, the total Gibbs energy becomes positive, and the system phase separates spontaneously. This would cause miscibility up to a certain composition (partial miscibility) or full immiscibly in a polymer blend. Excess functions can be calculated as the difference between ΔmixS (or ΔmixH) of real and ideal solutions to investigate nonideal cases.

Experimental design
Chart S1. Experimental design used in fabrication of Na-alginate-(sugar alcohol) films 0 to 50 wt.% plasticizer, from C 2 to C 6 , e.g. crystallinity and, in blends, also to the degree of miscibility. The samples studied actually showed different degree of haziness depending on the rate of drying and type of sugar-alcohol used ( Figure   S1A). Haziness appeared specially when high fractions of polyol where present and at low drying rate (in a laminar air flow hood). Therefore, we have used this observation to screen which films were best uniformly blended and should be used in our study. Additionally, by analysing our Na-Alginate polymer with x-ray diffraction (XRD), we confirmed that the semi-crystalline structure is formed in solvent-casted films (2θ peaks between 10° and 20°, Figure S1B) in contrast to flashdried samples, for instance, with freeze drying.

Photos of Na-alginate-(polyol) films
Free-standing films could be successfully produced by solvent-casting up to a plasticizer's compatibility limit. Under ambient drying conditions, it was possible to fabricate homogeneous transparent alginate films containing up to 50 wt.% plasticizer for most polyols (Figure S2). These films were pliable at ambient conditions from 15 wt.% polyol content. C4 polyol had a high tendency for recrystallization and fast drying of the film under vigorous N2 flow was required ( Figure S3). Films could only be produced up to a 30 wt.% C4 threshold. Hence, we can assume film drying conditions and storing played a role in the resulting properties. If such conditions are not handled properly, partial de-mixing, exudation and/or crystallization can occur. This compatibility limit is actually dependent both on specific interactions and sample processing (e.g., mixing and drying methods). For alginate-polyol, unwanted macroscopic effects such as crystallization and plasticizer migration had to be prevented, respectively, by fast evaporation and storing the films in a dry chamber. In fact, we have observed with optical microscopy that polyols showed a tendency to partially mix or crystallize at different concentrations ( Figure S4). The polyol mannitol crystallized at lower concentrations than sorbitol, even though these compounds are isomers. Therefore, there were plenty signs of different internal structures depending on type and amount of diluent.  The remaining content of C2 polyol was determined by subtracting weight loss values for pristine Na-Alginate from the ones obtained for the blends of Na-Alginate-(C2). The concentrations found are shown in legend of Figure S5 as wt.%. have the lower Tg and vice-versa. It is worth mentioning that the lower Tg component will have segmental mobility closer to its own bulk. In contrast, the higher Tg one experiences the dynamics of an averaged system 16 . Since the Tg(s) estimated from DMTA are a result of changes in segmental mobility on elastic and loss moduli, this technique will also reflect an averaged environment -as in the experienced by the slower or higher component Tg. This supports the Tg discrepancies normally found between methodologies, e.g., with differential scanning calorimetry.
In short, the DMTA analysis will likely show relatively higher Tg values and, thus, large positive deviation from the mixture's Tg predicted by rules of mixing.
After the Tg event, the rubbery modulus of alginate samples varied with type and size of added sugar-alcohol. Surprisingly, values as high as 1 GPa are obtained for short plasticizers (C2, C3) whilst decreased up to 0.01 GPa order of magnitude for C5 and C6. It might be that the plasticizer type (size, chirality) influences the degree of semi-crystallinity of alginate blocks under the same casting conditions.
We note that the sample containing C4 as plasticizer showed signs of separation into two relaxation moduli. The appearance of two Tg-like relaxations is often attributed to partial miscibility of a blend. Furthermore, most samples also show signs of residual water evaporation after the boiling point is reached (appearance of additional secondary relaxations), which is understandable considering hygroscopic samples should still have tightly bound water present.      DMTA and TGA of Na-alginate, C3, and C6 polyol films equilibrated to ambient moisture Water is a natural plasticizer and can be introduced into hygroscopic samples via air humidity.
The water content in humidified films, ~50RH, was measured via TGA (Table S1) Figure S12). Additionally, we also find relaxation spectra that are close to that composed of two alpha relaxations: one encompassing the plasticizer-rich phase with bound water and yet another for the polymer-rich phase. However, additional investigations would be needed to verify this partitioning, i.e., neutron and x-ray scattering.  Residuals plot over independent variable for Fox and GML models for alginate-polyols  systems.

GML B
Sugar alcohol glass transition temperature Full GM model applied to datasets from literature The full derivation of the Generalised Mean model, GM, can also be used to fit datasets of systems with higher degree of phase separation. This is accomplished through the exponents alpha and beta, in the expression: where ϕi, Tgi, kGM, α, β are volume or mass fraction of components 1 or 2, glass transition of components 1 or 2, model constant, entropy exponential, and enthalpy exponential, respectively.
In Figure S15, the selected systems from literature have regions of partial miscibility to immiscibility depending on diluent concentration. Therefore, they were fitted with the full GM model (Table S3). Other equations based on Couchman-Karasz theory have been previously adapted to fit such mixtures, with some degree of success. However, our general recommendation is to interpret such systems with the rationale of a phase inversion or plasticizer migration phenomena. Hence, we advise to first do a refined fitting and analysis over the (partially) miscible diluent range, using the linearized GM model. Secondly, analyse the remaining phase-separated system with another GML fitting.  * Values used to fit models were estimated by this study for illustrative purposes