Microscopic versus Macroscopic Glass Transitions and Relevant Length Scales in Mixtures of Industrial Interest

We have combined X-ray diffraction, neutron diffraction with polarization analysis, small-angle neutron scattering (SANS), neutron elastic fixed window scans (EFWS), and differential scanning calorimetry (DSC) to investigate polymeric blends of industrial interest composed by isotopically labeled styrene–butadiene rubber (SBR) and polystyrene (PS) oligomers of size smaller than the Kuhn length. The EFWS are sensitive to the onset of liquid-like motions across the calorimetric glass transition, allowing the selective determination of the “microscopic” effective glass transitions of the components. These are compared with the “macroscopic” counterparts disentangled by the analysis of the DSC results in terms of a model based on the effects of thermally driven concentration fluctuations and self-concentration. At the microscopic level, the mixtures are dynamically heterogeneous for blends with intermediate concentrations or rich in PS, while the sample with highest content of the fast SBR component looks as dynamically homogeneous. Moreover, the combination of SANS and DSC has allowed determining the relevant length scale for the α-relaxation through its loss of equilibrium to be ≈30 Å. This is compared with the different characteristic length scales that can be identified in these complex mixtures from structural, thermodynamical, and dynamical points of view because of the combined approach followed. We also discuss the sources of the non-Gaussian effects observed for the atomic displacements and the applicability of a Lindemann-like criterion in these materials.


MAGNITUDES MEASURED BY THE SCATTERING EXPERIMENTS
As a representative example, Figure S1.1 shows the differential scattering cross section measured by SANS on the sample with 50/50 composition (50h). These data are combined with the information obtained from D7. Using both instruments, the Q-range from about 0.003 to 2.5Å -1 has been covered. This is equivalent to spatial scales (~1/Q) ranging from about the bond length to several nanometers. In addition to the diffraction experiments, we have also carried out EFWS on IN13 which cover the high-Q regime 0.52 ≤ Q ≤ 4.5 Å -1 . We address in the following the origin of the contributions to the scattered intensity, taking Figure S1.1 as illustration.
The interaction of a given nucleus with neutrons is characterized by the scattering length b. This magnitude depends on the relative orientation of the neutron-nuclear spin pairs and varies from one isotope to another. The huge difference of the scattering length values for hydrogen and deuterium (bH = -3.74 fm, bD = 6.67 fm; bD ~ bC = 6.65 fm) produces a high contrast between h/d isotopically labeled macromolecules. Therefore, SANS diffraction experiments accessing low values of Q -exploring thus large-scale properties-on mixtures where one of the components is protonated and the other is deuterated are highly sensitive to thermally driven concentration fluctuations (TCF). In such a Q-range also long-range density fluctuations contribute to the scattered intensity, though this contribution is usually expected to be much less important than that of TCF.
Both, concentration and density fluctuations, give rise to coherent scattering. Toward high-Q values -local length scales-the contribution of TCF tends to vanish, and coherently scattered neutrons reflect instead the (partial) structure factors revealing the short-range order in the sample. Its main manifestation in glass-forming systems like polymers is the broad peak usually appearing in the Q-range of about 1 Å -1 reflecting inter-molecular correlations. Figure S1.1. SANS (circles, scale on the left) and D7 (diamonds, scale on the right) results on the 50h-sample (50% hSBR / 50% dPS composition) around RT. Areas with different colors show the different contributions to the differential scattering crosssection: the Q-independent incoherent scattering (red) and the coherent contributions mainly dominated by concentration and long-range density fluctuations at low Q (yellow) and reflecting the short-range order at high Q (blue).
Superimposed to these coherent contributions incoherent scattering of very different nature is also present in the measured signal. The incoherent differential scattering cross section is Q-independent and appears as a flat background in diffraction experiments.
Incoherent scattering is particularly important by hydrogen nuclei. The incoherent scattering cross-section of H amounts to #$% & » 80 barn, while its coherent cross-section is #$% &' » 2 barn. In general, in hydrogenated samples, or samples containing hydrogens, the incoherent cross section (summed up over all nuclei of the system) is much higher than the coherent one (see Table 3 in the main text). However, this does not imply that in a given Q-region the incoherent scattering always dominates the spectrum, since -as we can see in the example of Fig. S1.1-coherent scattering strongly depends on Q. The neutron spin is flipped with 2/3 probability in incoherent scattering due to nuclear spin disorder, whereas no flip occurs in the case of coherent scattering 1 . Thanks to this property, polarization analysis of the scattered intensities allows distinguishing between these two kinds of phenomena. This is the principle applied in the diffraction experiments carried out by means of D7. As can be seen in Fig. S1.1, D7 results tell us that for the sample with 50% hSBR / 50% dPS composition the incoherent contribution dominates the scattered intensity in the high-Q range above » 0.5 Å -1 . This is the range explored by

IN13.
On IN13, no polarization analysis is performed. Therefore, the intensity recorded in the EFWS has both, incoherent and coherent contributions. From the D7 results also for other blends shown in Fig. 3 we can infer that in the IN13 Q-window the coherent contribution from TCF is much smaller than the incoherent one, and that the scattered signal is dominated by the incoherent contribution. The exception is the sample rich in the deuterated component, but only in the neighborhood of the structure factor peak (around  Moving to X-Rays, the weights of the contributing correlations to the structure factor measured by diffraction are the Q-dependent atomic scattering factors for X-Rays. Since this probe mainly interacts with the electrons, these weights are proportional to the atomic number and no sensitive to isotopic labeling. Therefore, due to lack of contrast, the low-  Comparison between D7 (filled symbols) and X-Ray (empty symbols) results for the coherent scattering of the 50d sample.

X-RAY DIFFRACTION RESULTS: LOCAL STRUCTURE
Since the X-Ray diffraction results are free from low-Q contributions from TCF, they were used to study the short-range order and possible nano-domain structuration of phenyl rings in the samples. Figure

SANS RESULTS: DETERMINATION OF Ts AND c-PARAMETER
As explained in the manuscript, the values of the spinodal temperatures Ts were obtained by extrapolating the law 56 78 (0) ∝ 78 to low temperatures and deducing the value at which 56 78 (0)=0. The construction is shown in Figure S3.1 for the samples with 20% and 50% SBR concentration considered in the manuscript and in addition for samples with 65% SBR content (not included in the manuscript because they were not investigated by IN13). The results on the 80% blends are out of the scale in this plot. From the SANS results the interaction parameter between the two components was also obtained from the measured amplitudes of the OZ contribution, applying the RPA expression: (∆>) ?
where Nx is the number of monomers of species x and the rest of the parameters are defined in the manuscript (see Figure S3.

DSC RESULTS AND THEIR MODELING
The approach proposed here to model the DSC behavior in the SBR/PS blends is based on a direct connection between DSC and broad band dielectric spectroscopy (BDS) experiments. We first analyze the glass transition and the dielectric relaxation of the neat components and afterward we model the calorimetric traces of the blends.  In order to analyze the contribution to the experimental DSC trace of the segmental dynamics responsible for the glass transition, first the glassy behavior has been accounted for with a linear function (for the sake of simplicity) and subtracted from the DSC cooling scan of the reversible heat flow ( Figure S4.1). We have used this procedure for the homopolymers as well for the blends. The resulting calorimetric traces for the hSBR/dPS and dSBR/hPS systems respectively, that will be used for the following analysis are shown in Figure S4.2 and will be referred to as segmental heat capacity, s-Cp. The behavior at temperatures well above ] for all samples nearly superimposes and can be approximately described by a power law ( 7$ ) with n=2. It is worth mentioning that the subtraction of a linear function does not alter the location of the inflection point. Calorimetric traces after the subtraction of the glassy part; same procedure has been applied on the neat components and the mixtures hSBR/dPS. The solid lines fitting the neat polymers data were obtained by using eq. S4.2 with parameters given in Table S1.  The dielectric α-relaxations can be described by means of the Havriliak-Negami (HN) equation; 9,10 and the necessary β-relaxation contribution has been taken in account by using a Gaussian function, following previous work 11 . Figure S4.3 shows that in this way a good description of the experimental data is obtained;

(ii) Dielectric Relaxation of Neat Components
the low-frequency increase of the data is due to conductivity effects, not considered in this analysis. The characteristic time at each temperature can be defined as the inverse of the angular frequency at the dielectric loss-permittivity maximum (τmax º ω -1 max) of the α-relaxation process as calculated from the HN fitting parameters. Figure S4 In the fits we kept constant the prefactor value τ∞=10 -13 s in the VFT equation. Table S1 includes the values obtained for the fragility parameter, D, and the Vogel temperature C .

(iii) Calorimetric Traces of the Neat Components
A simple full characterization of the homopolymers' DSC behavior has been done in order to establish the connection between the segmental dynamics time and the DSC data.
Following previous work, 11  where ∆Cpg is the heat capacity jump, d measures the width of the glass transition range and ] * is a characteristic temperature defined as the inflection point of the sigmoidal function. As can be appreciated in Figure S4.2, the description of the experimental data for the neat components, for example hSBR and dPS, is very good. The parameters determined by fitting the curves are given in Table S1.

(iv) Composition Dependence of the Glass Transition of the Blends
The glass transition processes of the blends manifest broad features in the range between the ] of the pure components, as can be observed in figure S4.2. Figure S4.5 shows the composition dependence of the glass transition temperatures defined from the inflection point. As expected ] decreases monotonously as we increase the content of SBR in the blends.
Following the scheme of a related work, 11 we have first described the whole set of data using the using the Gordon-Taylor (G-T) equation 15,16 ] where is the weight fraction of SBR and a fitting parameter ƒ7z is introduced. 17 The

(v) Modeling the Calorimetric Traces of the Blends
The model developed during recent years of the segmental dynamics of miscible polymer blends 11,21,22 is based on two major ingredients: the thermally driven concentration fluctuation (TCFs) and self-concentration concept; it is assumed that the TCFs evolve on a much longer time scale than that of the segmental relaxation. This entails that the polymer blend can be viewed as a set of sub-volumes 'i' each with a different SBR concentration, 0 ≤ # ≤ 1. This quasi-static distribution of concentration ( # ) in the blends can be described by a Gaussian function centered around the bulk concentration of the blend : where σ is the standard deviation of the distribution of concentration.
Within each region we consider the effective concentration •__,# describing the fact that the dynamics of a given polymer segment in a blend is controlled by the local composition in a small region around that segment. This makes the concentration felt by each specific component to be higher than the average in this region. This effect is reflected by the corresponding self-concentration parameter, 1•~_ . Thus, the effective concentration •__,# in each region for the SBR and PS components is given by: In this framework, the calorimetric behavior of SBR/PS blends is assumed to be the result of the superposition of contributions to the segmental heat capacity from different regions, and within each region the result of the individual contributions from the blend components. 11 The contribution of each component in a region i of the blend is taken having the shape and amplitude corresponding to the pure component and weighted by its concentration. Thus, the contribution to the segmental heat capacity as a function of temperature for each component, can be calculated as: where we have assumed that in the description of the segmental heat capacity the only parameter affected by blending is ],# * .
Therefore, the whole calorimetric signal can be obtained by summing up the respective contributions of SBR and PS: As a final step we will assume that the connections found for the homopolymer between concentrations. Particularly, a linear mixing rule is assumed for Di : For C,# we have used a Gordon-Taylor-like equation: After determining the ],# * values, the DSC curves can be described with the parameters above determined (Table S1). They were thus described in terms of those for the pure components and three fitting parameters: the self-concentrations of both components

(vi) Evaluation of the components' effective values
In Figure S4.9 the good agreement between the DSC trace and model is corroborated in the example for the n[\3 = 0.5 blend, when the temperature derivative is compared.
The good quality of the DSC data description is emphasized, both in peak position and in the breadth of the glass transition range. Figure S4. 10