Dynamic Heterogeneity and Cooperative Length Scale at Dynamic

Article pubs.acs.org/Macromolecules

Dynamic Heterogeneity and Cooperative Length Scale at Dynamic Glass Transition in Glass Forming Liquids Bidur Rijal, Laurent Delbreilh, and Allisson Saiter* AMME-LECAP EA 4528 International Laboratory, Normandie Université, Université et INSA de Rouen, Av. de l’Université BP 12, 76801 Saint Etienne du Rouvray Cedex, France ABSTRACT: Understanding the evolution of the cooperative molecular mobility as a function of time and temperature remains an unsolved question in condensed matter research. However, recently great advances have been made within the framework of the Adam−Gibbs theory on the connection between cooperatively rearranging regions, or dynamic heterogeneities, i.e., domains of the supercooled liquid whose relaxation is highly correlated. The growth of the size of these dynamic domains is now believed to be the driving mechanism for different experimental parameters like relaxation times and viscosity of supercooled liquid approaching the glass transition. Recent studies have shown the evolution of cooperative motions in supercooled liquids using different experimental tools and models. In this work, broadband dielectric spectroscopy and modulated temperature differential scanning calorimetry were carried out on six different amorphous glass-forming systems in order to scan a wide range of relaxation times and temperatures. Two different models based on four point dynamic susceptibilities and the thermodynamic fluctuation approach, have been used to compare the temperature evolution of the number of molecules dynamically correlated during the α-relaxation process. Divergences and convergences between these two models are discussed.

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dynamics near the glass transition. Adam and Gibbs13 introduced the notion of cooperatively rearranging region (CRR) defined as a subsystem, which can rearrange its configuration to another one independently of its environment. Each CRR presents subsystem having its own glass transition temperature and its own free volume, linked to its own relaxation time. According to Donth et al.,16,17 the cooperativity length at the glass transition ξα corresponds to the size of a CRR. Different methods have been proposed to quantify the number of correlating units. Other determinations are obtained from high-order correlation functions, such as the four-point dynamic susceptibility of the density fluctuations.18 The four-point dynamic susceptibility χ4(t) quantifies the correlation in space and time of the molecular motions, and although it can be expressed as the variance of the selfintermediate scattering function.19 Berthier et al.,20,21 derived an approximation for four-point dynamic susceptibility χ4(t) in terms of temperature derivative of a two-point dynamic susceptibility χT(t). χ4(t) shows a maximum at a time t = τα, the height of the peak in the dynamic susceptibility being proportional to the number of dynamically correlated molecules Nc.20,22,23 According to Berthier et al.,10,20,24 the dynamic susceptibility of real molecular systems is expressed in the following approximate way:

INTRODUCTION The glass transition is a phenomenon in which the dynamics of a liquid slow down as the temperature decreases. The associated dynamic phenomenon known as the α-relaxation has been widely studied in bulk glass formers, including polymers. However, this important aspect of condensed matter research is not yet fully understood.1−7 One of the most striking features of the glass transition is the increase of the viscosity or the α-relaxation time with a decrease of the temperature approaching the glass transition. This dramatic increase can be interpreted in several ways depending on the theoretical approach used. The dynamic nature of the glass transition is related to dynamic heterogeneities,8 considered in spatial and temporal correlations of molecular motions, as well as a distribution of the relaxation time. The distribution of the relaxation time is related to the breadth of the segmental function or dynamic susceptibility.9 Dynamic heterogeneities contain information on both spatial and temporal correlations and are associated with the cooperative dynamics of the structural relaxation.10 The size of dynamic heterogeneities corresponds to a dynamic length scale, ξα, in which molecular rearrangements occur coordinately with other molecules within their local environment. With temperature decrease ξα becomes larger, leading ultimately to vitrification as the number of molecules moving in the same time becomes overly large.11,12 Classical theories of the glass transition such as the well-known entropic model of Adam and Gibbs,13 or the free volume approach,14,15 invoke the growth of the dynamic correlation size as the cause of the slowed © XXXX American Chemical Society

Received: May 28, 2015 Revised: September 17, 2015

A

DOI: 10.1021/acs.macromol.5b01152 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules

Figure 1. Chemical structure of the polymers studied in this work.

⎡ k T2 ⎤ ⎥{max tχ (t )}2 Nc(T ) ≈ ⎢ B T ⎢⎣ ΔCp(T ) ⎥⎦

to Donth et al.16,17 the cooperativity length ξα is associated with the volume of a CRR, Vα = ξα3, and Nα, the number of relaxing structural units per CRR, also called the cooperativity degree, is estimated by using the following relations:

(1)

where ΔCp is the change in heat capacity between liquid and glassy states and kB is the Boltzmann constant. The spatial variations lead to a dynamic correlation volume, defined from the maximum of the four-point dynamic susceptibility χ4(t) for a time scale of the order of the α-relaxation.23,25 As is wellknown,23,26 direct experimental investigations that would result in the determination of the four-point dynamic susceptibility χ4(t) function are difficult. However, a simpler method is to measure the nonlinear response of polymers, as an example, by measuring the nonlinear dielectric constant, which contains information similar to χ4(t).23 Assuming a stretched exponential form for the relaxation function, i.e. a Kohlrausch−Williams− Watts relationship exp[−(t/τα)βKWW,27,28 a simpler form for eq 1 has been suggested:11,26,29−33 Nc =

2 2 R ⎛ βKWW ⎞ ⎛ d[ln τα] ⎞ ⎟ ⎜ ⎟⎜ ΔCpm0 ⎝ e ⎠ ⎝ d[ln T ] ⎠

Vα = ξα 3 =

Nα =

ΔCp−1kBTα 2 ρ(δT )2

(3)

NAkBTα 2ΔCp−1 m0(δT )2

(4)

ΔCp−1

where is the difference in the inverse of the isobaric heat capacity between the liquid and glass at Tα (for details see ref 34), Tα the dynamic glass transition temperature, ρ the density at Tα, kB the Boltzmann constant, NA Avogadro’s number, m0 the molar mass of the relaxing structural unit, and δT the temperature fluctuation in a CRR. The evolution of the cooperativity length in glass-forming liquids with time and temperature always attracts the interest of researchers.12,35−42 As an example, Pieruccini et al.43,44 proposed a canonical model allowing for the estimation of monomeric units cooperatively involved in the α-relaxation process. Classically, Donth’s approach allows for the estimation of the cooperativity length at the glass transition temperature using calorimetric techniques.45 Over the past few years, researchers have developed new experimental techniques46−48 in order to extend the CRR analysis range and then to study the whole spectrum of structural relaxations. Recently, Saiter et

(2)

where ΔCp is the change in heat capacity between the liquid and glassy states, m0 is the molar mass of the relaxing structural unit, τα is the α-relaxation time, R is the gas constant, βKWW is the stretch exponent, and e is Euler’s number. In Donth’s approach,34 the CRR size is estimated using the temperature fluctuation of the amorphous medium. According B


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Table 1. Sample Name, Molecular Weight Mw, and Glass Transition Temperature Tg Taken at the Middle Point of the ΔCp Step Obtained from Classical DSC Measurements, Density, and Source for Each Sample

al.36,49 proposed an extended form of the Donth model by combining two experimental techniques (MT-DSC and BDS), enabling the calculation of the temperature dependence of the CRR size in a wide temperature range, starting from the onset of cooperativity in the crossover region down to the calorimetric glass transition temperature Tg. Indeed, there are several common and/or complementary features for the glass transition phenomenon studied by these two techniques. The assumptions made for the extended Donth approach are detailed in ref 49. In recent years, the evolution of the cooperativity length upon varying external parameters as well as molecular characteristics has been experimentally studied to understand how cooperativity length correlates with other relaxation parameters such as fragility,50−53 glass transition temperature,54 and relaxation time.55 Roland et al.,11 and Paluch et al.56,57 also used pressure dependence studies for different glass-formers in order to quantify the respective contribution of free volume, thermal energy and configurational entropy on the relaxation properties. Casalini et al.30−32 studied the variations of the number of dynamically correlated segments (Nc) with varying pressure, and compared these variations with the cooperativity degree Nα according to Donth’s approach. The question arises “why is Donth’s approach used as comparative studies?” The main answer lies in the length calculation without any assumption on the relaxing unit. Indeed, for Nc and Nα, the hypothesis on the relaxing unit has a large impact on the result making this assumption an issue. The determination of the relaxing unit is quite difficult and theoretically complex. As an example, Kuhn developed a model assuming a flexible polymer chain consisting of connected segments called “Kuhn’s segments”, forming a long random walk.58,59 In this work, we propose an experimental investigation of relaxation parameters like the fragility and the degree of cooperativity at the dynamic glass transition. The structural relaxation parameters are studied using three different classes of polymers: (a) polymers with a stiff backbone such as poly(ether imide) (PEI), poly(bisphenol A carbonate) PBAC and poly(ethylene 1,4-cyclohexylenedimethylene teretphate glycol) PETg, (b) polymers with flexible backbone and flexible pendent groups such as poly(lactide acid) PLA and poly(vinyl acetate) PVAc, and (c) polymers with polar pendent groups and a saturated aliphatic backbone such as poly(vinyl chloride) PVC. This work aims in particular at understanding the experimental comparison of Nc and Nα with variations of time and temperature and the application of the models to the different amorphous polymers mentioned above having very different calorimetric glass transition temperatures.

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Mw [g/mol]

Tg,mid [K]

density [ρ (g/cm3)]

PEI

55 000

488.5

1.27

PBAC PETg PVC

47 400 26 000 N/A

418.3 350.7 356.6

1.20 1.27 1.40

PLA PVAc

188 000 500 000

329.5 315.5

1.25 1.19

polymers

source Sabic Innovative Plastics General Electrics Eastman Chem. Co. EVIPOL SH6030, INEOS Nature Works LLC Aldrich Chem. Co.

than heating, i.e. 10 K.s−1 and for a constant heating rate of 10 K.min−1. The MT-DSC measurements have been performed in two different modes: (a) Quasi-isothermal mode: The measurements have been performed with modulation amplitude of ±1K and an oscillation period of 100 s (for detail see ref 60). The instrument was calibrated for heat flow, temperature, and baseline using the standard Tzero technology. Calibration in temperature is carried out using the standard values of indium and zinc, calibration in energy is carried out using indium. Calibration in specific heat capacity was carried out using sapphire as a reference. (b) Heat−cool mode: By using heat−cool measurements, the real and imaginary parts of the heat capacity were obtained with a modulation amplitude of ±1 K, an oscillation period of 60 s and an underlying cooling rate of 0.5 K·min−1 except for the PLA and PVC samples. For the PLA and PVC samples, a heating ramp (0.5 K·min−1) has been applied for both modes of MT-DSC measurements. In the case of the PLA, this is due to its ability to crystallize, and for the PVC it is due to its low thermal stability. More details are given in ref 60. Broadband Dielectric Spectroscopy (BDS). BDS is a very powerful tool widely used to study the molecular mobility.61−64 In this work, the measurements were carried out with a Novocontrol Alpha analyzer. Samples were placed between parallel electrodes, using 30 mm diameter gold plated electrodes. The broadband dielectric converter (alpha analyzer interface) allows the measurement of the complex dielectric permittivity (real and imaginary parts) in the frequency range 10−1 Hz to 106 Hz. To remove thermo-mechanical previous history and improve the material/electrode contact, the samples were annealed at a temperature 10 K upper than the glass transition during 5 min, for each sample. The temperature was varied between 553 and 480 K for the PEI, between 470 and 410 K for the PBAC, between 410 and 340 K for the PETg, and between 373 and 310 K for the PVAc, in consecutive decreasing steps of 1 K. For the PVC and PLA samples, the temperature was varied in consecutive increasing steps of 1 K between 340 and 403 K and between 323 and 363 K, respectively. The temperature was controlled by a Quatro Novocontrol Cryosystem with stability better than 0.2 K. During the whole period of measurement, the sample was kept in a pure nitrogen atmosphere. For quantitative analysis of the dielectric spectra in the frequency domain, the complex permittivity signals associated with the α-peaks were fitted with the empirical Havrilik−Negami (HN) function:65

EXPERIMENTAL SECTION

Materials. The six polymers studied herein are poly(ether imide) (PEI), poly(bisphenol A carbonate) (PBAC), poly(vinyl chloride) (PVC), PETg, an amorphous copolymer consisting of cyclohexanedimethanol, ethylene glycol and terepththalic acid with a molar ratio of approximately 1:2:3, poly(lactide acid) (PLA) having 95.7% L and 4.3% D isomers, and poly(vinyl acetate) (PVAc). The chemical structure of each sample is presented in Figure 1. The fully amorphous character of the samples has been confirmed by calorimetric investigations. Details of the molecular weight, the glass transition temperature, the density and the source are presented in Table 1. Calorimetry. The classical DSC (Q100 TA Instruments) has been used to estimate the calorimetric glass transition temperature Tg taken at the inflection point of the ΔCp step after cooling at the same rate

ε*(ω) = ε∞ +

Δε [1 + (iωτHN)αHN ]βHN

(5)

where ε∞ is the unrelaxed dielectric permittivity, Δε is the relaxation strength, τHN is a characteristic relaxation time and αHN and βHN are shape parameters describing the symmetric and asymmetric broadening factor of the dielectric spectra. The conduction effects were analyzed by adding a contribution σ″cond = σ0/[ωsε0] to the dielectric C


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Macromolecules loss, where σ0 accounts for the Ohmic conduction related to the mobile charge carriers, s is a fitting parameter and ε0 the dielectric permittivity of vacuum. From the estimated value of τHN, αHN, βHN parameters, the relaxation time, τmax = (2π f max)−1, was calculated according to66

τmax = τHN

⎡ ⎢ sin ×⎢ ⎢ sin ⎢⎣

( (

αHNβHNπ 2 + 2βHN αHNπ 2 + 2βHN

) )

⎤1/ αHN ⎥ ⎥ ⎥ ⎥⎦

symbols and solid lines respectively in Figure 2). From the Cp measurements, extrapolated baselines are defined for the liquid and the glassy states represented by dashed dot lines (see in Figure 2). To estimate the degree of cooperativity (Nα) and the cooperativity length (ξα) from calorimetric investigations, the parameters δT and Tα were extracted from heat−cool protocol, and ΔCp−1 was calculated from quasi-isothermal protocol. The method proposed by Donth16 assumes that the mean temperature fluctuation δT (δT = fwhm/2.35) is associated with the standard deviation σ obtained from a Gaussian fit of the imaginary part of the complex heat capacity Cp″ (as presented in Figure 2). Dielectric Spectroscopy Analysis (α-Relaxation). Representative dielectric spectra are shown as a function of frequency for different temperatures in Figure 3. Some general trends can be noted, for example, the decrease of the relaxation peak amplitude. At low frequencies, a strong increase of ε″ is observed corresponding to the dc-conductivity contribution. The dielectric relaxation process is characterized by a steplike decrease of ε′ and a peak in ε″ versus frequency at isothermal condition. Figure 4 presents an example of the HN fitting procedure for the real and imaginary parts of the complex permittivity as a function of frequency for the PBAC at 433 K. At high frequency a well-defined loss peak is observed (see also in Figure 3) which can be fitted by the single HN function. Similarly, at lower frequency the electrode polarization and the conductivity phenomena are observed for the real and the imaginary parts. A similar procedure was followed to analyze the relaxation data for all the samples in the study. Figure 5 shows the normalized plots of the dielectric loss versus frequency scaled to f max for each isothermal curve of the PBAC sample. It can be seen that there is very good overlap of all the curves forming a single master curve. However, systematic deviation from the master curve is noted on the high frequency side of the α-Relaxation. This deviation is mainly due to the contribution of noncooperative localized molecular mobility, i.e., β-Relaxation, associated with the molecular mobility of localized molecular entities. As the measurement temperature is increased, the increased influence of the β-Relaxation process is observed on the master curve. Figure 6 shows isochronal spectra (in the frequency range 10−1 to 5 × 105 Hz) of the dielectric loss (ε″) as a function of the temperature for the PBAC. These spectra are obtained from subtracting the contribution of the conductivity and the secondary relaxation (β-Relaxation). The relaxation peak corresponding to the structural or the main relaxation process is observed. Relaxation Map for α-Relaxation. Figure 7 shows τmax as a function of the inverse temperature (Arrhenius diagram) for all of the samples. The temperature dependence of the relaxation time for the α-relaxation usually presents a superArrhenius behavior (i.e., deviation from the Arrhenius behavior) and is well described by the empirical Vogel− Fulcher−Tamman (VFT) equation:68−70

(6)

Fitting procedure has been conducted on both the real and imaginary parts of the signals for all the samples in order to improve the consistency of the fit results. From the dielectric data analysis, a relaxation map can be constructed providing a picture of the dynamical behavior of the probed system in a broad range of temperature and frequency. The correlation function is calculated using the HN relationship:67 βHNαHN

( ) sin(β φ) 1 f(τ ) = π⎡ + 2( ) cos(πα ⎢1 + ( ) ⎣ τ τHN

2αHN

τ

τHN

HN

τ

αHN

τHN

⎤ βHN /2

HN)⎥

⎦

(7)

with ⎛ π −1⎜ φ = − tan ⎜ ⎜ 2 ⎝

HN

⎞ + cos(παHN) ⎟ ⎟⎟ sin(παHN) ⎠

αHN

(ττ )

(8)

where f(τ) is the distribution function, τHN is the HN relaxation time, αHN and βHN are the HN shape parameters. In term of the distribution of relaxation time, the correlation function Φ(t) is expressed as67

Φ(t ) =

∫0

∞

f (τ ) exp−t / τ dt

(9)

Each function has then been fitted with the KWW relationship in order to obtain KWW stretching parameter βKWW.

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RESULTS AND DISCUSSION MT-DSC Measurements. Figure 2 shows an example of the heat capacity (Cp) as a function of temperature for the PEI, for quasi-isothermal and heat−cool measurements (see hollow

⎛ B ⎞ τ(T) = τ0 exp⎜ ⎟ ⎝ T − T0 ⎠

Figure 2. PEI sample: hollow symbols represent the heat capacity (Cp) as a function of temperature under quasi-isothermal measurement. The solid lines represent the real part (Cp′) and the imaginary part (Cp″) of the complex heat capacity versus temperature obtained from heat−cool measurement. δT, the mean temperature fluctuation in a CRR, is associated with the standard deviation, σ, obtained from a Gaussian fit of the imaginary part of the complex heat capacity Cp″.

(10)

where τ0 is the relaxation time at infinite temperature, B is a fitting parameter, and T0 is the so-called Vogel temperature. In general, for many glass-forming liquids it has been found that a single VFT law describes the temperature dependence of D


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Figure 3. Dielectric loss ε″ vs frequency on the whole temperature range investigated for the PEI, PBAC, PETg, PVC, PLA, and PVAc samples.

the relaxation time up to a characteristic temperature TB. Ngai71 indicates that most of the glass-forming liquids, including polymers, present a crossover between two differentiated temperature dependences of the relaxation time at a temperature TB. To provide a good fit for relaxation data in the whole temperature range, two VFT equations instead of one are sometimes required.72−74 In recent years, this phenomenon has been extensively investigated by analyzing the temperature dependence of the structural relaxation time with the support of the analysis originally proposed by Stickel,75 confirming the presence of such crossover regions. Stickel et al.75,76 indicated that the validity of VFT description in supercooled glassforming liquids is associated with the linearity at ΦT vs T plot (see in Figure 8). ⎡⎛ d[log τ ] ⎞⎤−1/2 10 max ⎥ ⎟ ΦT = ⎢⎜ ⎢⎣⎝ d(1000/T ) ⎠⎥⎦

It appears to be a key tool for estimating the dynamic crossover temperature between two dynamic domains.72,77−80 Figure 8 presents the derivative analysis of the dielectric relaxation time for the PEI, PBAC, PETg, PVC, PLA, and PVAc samples. For all the samples, except the PBAC, a single VFT fit is useful to describe the behavior on the temperature range explored. For the PBAC a change in dynamics at a temperature T = 446 ± 3 K corresponding to log (τmax) = −5.2 ± 0.3 is determined from the intersection of the two linear fits. Table 2 gives all the parameters and values obtained from the derivative-based analysis for each polymer. The following procedure was applied to estimate the VFT parameters and the fragility index value for quantitative comparison. B and T0 were taken from the derivative technique using eq 12 by linear regression.81,82

(11) E


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Figure 7. Dielectric relaxation time (τmax) for α-relaxation as a function of the inverse temperature for the PEI, PBAC, PETg, PVC, PLA ,and PVAc samples. Hollow symbols are from BDS experiments and filled symbols from MT-DSC experiments (period = 60 s, τ ∼ 10 s). Solid red lines and dashed red lines represent the VFT fits to the corresponding data for each polymer.

Figure 4. Real and imaginary parts of the complex permittivity vs frequency for PBAC at T = 433 K. The blue dashed dot lines are HN fits and the red line corresponds to the conductivity contribution.

Figure 8. Linearized, derivative-based analysis (Stickel derivative plot) of the temperature evolution of dielectric relaxation data for the validity of the VFT equation. Red lines (solid and dashed dots) correspond to the linear fits to the relaxation data for all the samples. The arrow corresponds to the intersection of the fitted lines and the deviation from the single VFT behavior for the PBAC. T0 is the Vogel temperature.

Figure 5. Log−log plot of the dielectric loss normalized to the peak maximum vs the normalized frequency at various temperatures for the PBAC. Blue dashed dot line corresponds to the HN fit at 441 K for the α-relaxation and the blue dot line corresponds to the HN fit at 423 K for the β-relaxation.

The degree of deviation from Arrhenius-type temperature dependence near Tg provides a useful classification of glassformers. Angell et al.83,84 defined the so-called “fragility concept”. Materials called “fragile”, exhibit markedly nonArrhenius temperature dependence of the relaxation time τα close to Tg. The systems called “strong”, exhibit linear temperature dependence of the relaxation time in the Arrhenius diagram. Fragility or steepness index is quantitatively defined by the following equation:85,86 m=

Figure 6. Isochronal spectra of the dielectric loss (ε″) as a function of the temperature for the PBAC.

⎡⎛ d[log 1/τ ] ⎞⎤−1/2 ⎛ B ⎞−1/2 max 10 ⎢⎜ ⎟ ⎟⎥ (T − T0) =⎜ ⎝ 2.303 ⎠ dT ⎢⎣⎝ ⎠⎥⎦

d[log10τ(T )] d(Tg /T )

T = Tg

(13)

Using the VFT expression eq 10 and eq 13, the fragility index m is calculated according to (12)

m= F

BTg (Tg − T0)2 (ln 10)

(14) DOI: 10.1021/acs.macromol.5b01152 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules

Table 2. VFT Parameters DT, T0 (K), and B (K) obtained from Stickel’s analysis (eq 12), Glass Transition Temperature Tg(τ=100s) and Fragility Index Value m (VFT) Obtained from eq 10, and the Fragility Index Value m (Stickel’s) from eq 12 for the PBAC, PVC, PETg, PLA, PVAc, and PEI Samples sample

DT

T0 [K]

B [K]

Tg(τ=100s) [K]

m [VFT]

m [Stickel’s]

PBAC (low T) PBAC (high T) PVC PETg PLA PVAc PEI

5.04 1.20 2.25 3.25 4.76 6.86 3.12

370 406 331 315 292 255 425

1866 490 746 1024 1392 1750 1362

415 424 355 348 327 307 485.5

159 258 191 151 144 89 162

148 278 199 142 137 86 156

Dielectric Relaxation Strength. The dielectric strength (Δεα) is obtained in addition to the relaxation data from the fit of the HN-function eq 5. The starting point for the analysis of Δεα is the generalized form of the Debye theory by Onsager, Fröhlich, and Kirkwood:66

Figure 9 shows Angell’s plots for all the samples studied. A large range of fragility indexes is observed. Earlier studies on a

Δεα =

μ2 N 1 gK F 3ε 0 kBT V

(15)

Here, ε0 is the dielectric permittivity of vacuum, μ the mean dipole moment of moving unit in vacuum, N/V the density of dipoles involved in the relaxation process, and gK the Kirkwood correlation factor taking into account the short-range intermolecular interactions that lead to specific static dipole− dipole orientations. The Onsager factor F is equal to 1 for the sake of simplicity. Figure 10 presents the temperature dependence of the dielectric strength (Δεα) of the α-relaxation versus temperature Figure 9. Relaxation time vs Tg/T (Angell’s plot) for the polymers studied herein: The symbols correspond to the experimental data, and the red lines represent the VFT fits for each sample. Tg corresponds to τ = 100 s.

large number of glass formers, including polymers,87,88 show that the fragility of the polymer depends on its molecular structure as well as the glass transition temperature Tg: High Tg polymers exhibit high fragilities. However, this trend is no longer valid in many glass-forming systems.52 In our study, polymers having relatively more rigid backbones such as PEI, PBAC, and PETg are more fragile and exhibit also higher values of Tg compared to PLA and the PVAc. For PVC, the glass transition temperature is in the intermediate range and it is the most fragile polymer of our samples. PVC is composed of a flexible backbone and bulky side chain (−Cl) group. This results in a poor packing efficiency and leads to a significant increase of the fragility.89 As defined by Kunal et al.,90 the packing efficiency of the amorphous chain in the glassy state and the chain flexibility are the main parameters influencing the fragility index.89,91 The lower the packing efficiency, the higher the fragility index. For example in the case of stiff backboned polymers such as PEI, PBAC, PETg, etc., while those with flexible backbones and no side group such As poly(tetrahydrofuran) PTHF and poly(ethylene) PE are the strongest.88 Recently Kunal et al.90 presented a large review of fragility index values in several polymers with different molecular structures. Their conclusions are very similar to those proposed by Dudowicz et al.91 for linear backbone polymers. According to Dudowicz et al.,92 the rigidity/flexibility of the backbone and the side groups are essential parameters controlling the fragility in polymers.

Figure 10. Temperature dependence of the dielectric strength for αrelaxation (Δεα) versus temperature for all the samples. The inset shows the ratio TΔεα/(TΔεα)max at lowest measured temperature versus temperature for each polymer.

for the PEI, PBAC, PETg, PVC, PLA, and PVAc samples. As a general feature, the dielectric relaxation strength Δεα decreases on increasing temperature for each sample.66,81 Similar temperature dependencies of Δεα are also found for other kinds of glass-forming systems for example, polymers and thin polymer films.66,81,93,94 As shown in Figure 10, the dielectric strength of PVAc is higher than the other samples. These results are supported by previously reported results of Urakawa et al.95 on PVAc (∼6.2), Pluta et al.94 on PLA (∼2.5), and Huajie et al.81 on PBAC (∼0.5). The inset in Figure 10 shows the values of TΔεα normalized to the maximum value at the lowest measured temperature as a function of the temperature for each sample studied: TΔεα G


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Macromolecules decreases with increasing temperature for each polymer. As mentioned by Schönhals,96 a stronger temperature dependency of Δε α than predicted by eq 15, indicates stronger intermolecular interactions with decreasing temperature. Thus, a recognized characteristic of the α-relaxation can be explained by the cooperative character of the underlying molecular motion responsible for the glassy dynamics. With decreasing the temperature, the size of the cooperative rearranging region (and therefore effective dipole moment) increases.96 Dynamic Heterogeneity. The concepts of dynamic heterogeneity and/or cooperativity are invoked for providing explanations for drastic variations of different experimental parameters like relaxation times and viscosity of supercooled liquids approaching the glass transition.97−103 In the past 20 years, different experimental techniques have been developed to study the dynamic heterogeneity. The most commonly used technique are dielectric spectroscopy101,102 and techniques like single-molecule spectroscopy,103,104 solvation dynamics,46,105 atomic force microscopy106 and fluorescence recovery after photobleaching.98,107 In recent years, some indirect methods (model dependent) such as multipoint dynamic susceptibility function20,21,23,24and thermodynamic method based on the fluctuation−dissipation theory97,108 have been used extensively to study the dynamic heterogeneity. Dynamic heterogeneity has also been observed through molecular dynamic simulation.109−111 All of these approaches provide a consistent estimation of the average length scale of the dynamically correlated heterogeneities, which is approximately 1−4 nm20,98,46,97 in different glass-forming materials at the dynamic glass transition. Despite a large number of theoretical, computational and experimental advances, there are many questions concerning dynamic heterogeneities that remain unanswered. The existence of a length scale associated with dynamical heterogeneity allows for the coexistence of molecules with substantially different relaxation times.112 These spatially correlated heterogeneities are organized into fast and slow dynamic regions with a characteristic size increasing with cooling and with a dramatic slowing down of the α-relaxation process. In the case of the α-relaxation, the length scale of dynamic correlations has been invoked as a measure of a cooperativity size.26 Recent works based on computer simulation using a model glass-forming polymer melt19,113 demonstrated that the fourpoint function does not probe a CRR scale envisioned by Adam and Gibbs. In the current study, our motivation is to check this theoretical result by using experimental data obtained on different amorphous polymers scaling a wide range of Tg. The two different models used in this work based on the higher order correlation function and the thermodynamic fluctuation approach, have been widely used to analyze the dynamic heterogeneities in different glass-formers at the dynamic glass transition.29,31,37,49,54,114 Figure 11 presents the number of dynamically correlated segments Nc (estimated by using eq 2) as a function of τ/τ0 (where τ0 is a microscopic time scale fixed to 1 fs) for each sample. Data taken from different publications are also reported in Figure 11 for 39 glass-formers in order to give a broader view of Nc parameter on a wide range of relaxation times and glassformers. Interestingly, the Nc values calculated for our samples and the data reported from literature for small molecular glass-formers,

Figure 11. Number of dynamically correlated segments Nc for the PBAC, PETg, PLA, PVAc, PVC, PEI samples, and data from refs 11, 26, 30, and 31 (square dot symbols) as a function of τ/τ0, (where, τ0 = 10−15s). The data in refs 11, 30, and 31 have been studied under different pressures ranging from 0.1 to 500 MPa. The red dashed line is log(τ/τ0) = (Nc/N0)ψ + z ln(Nc/N0) + ln A, with A = 4.07, N0 = 4.0, z = 1.0, and ψ = 0.5.

low and high molecular weight polymers11,23,26,30,31are in the same range. We clearly notice similar behavior for relaxation time dependence of Nc whatever the chemical structure of the glass formers. The increase of Nc with increasing relaxation time can be described by two regimes, one fast increase at short relaxation times and one slower increase for relaxation times approaching the glass transition. Thus, the extent of spatial correlation in the dynamics grows as the glass transition is approached. This result is supported by several studies23,26 and is predicted by the mode coupling theory.115 Two power laws can be used to numerically describe these two separated regimes.23,26 The red dashed curve represents the fitting of the experimental data using an empirical equation proposed in literature23,26 to fit the evolution of the number of correlated unit during this two-regime evolution: ⎛ τ ⎞ ⎛ Nc ⎞ψ ⎛N ⎞ log⎜ ⎟ = ⎜ ⎟ + z ln⎜ c ⎟ + ln A ⎝ τ0 ⎠ ⎝ N0 ⎠ ⎝ N0 ⎠

(16)

It is noticed that the fitting parameters of the average evolution of Nc for our samples are quite similar to the parameters presented in the literature for the other glass formers. This result supports the idea of a common behavior for a large family of glass formers regarding the evolution of dynamic fluctuation with temperature and/or relaxation times. In addition, some effects of pressure and molecular weight on dynamically correlated segments have been reported for different glass-formers.11,30,31 It has been shown that Nc depends only on relaxation time but not on pressure, and Nc increases with increasing molecular weight. In fact, these results seem to provide a common picture of dynamic heterogeneity relaxation time dependences for glass-formers with molecular mobility ranging from the onset of the cooperative process down to the glass transition temperature Tg. Figure 12a presents the cooperativity length (ξα) associated with structural relaxation as a function of the relaxation time for the PEI, PBAC, PETg, PVC, PLA, and PVAc samples. For each sample, the MT-DSC results are reported in Table 3. As expected, the cooperativity length (ξα) increases with increasing relaxation time. It is noted that the extrapolation of H


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observed. However, this nonlinear temperature dependence of ξα is related to the temperature dependence of the relaxation time. Independently from the glass transition temperature or from the flexibility of the macromolecular backbone, the cooperativity lengths ξα at the glass transition temperature for all the samples are quite similar, i.e., 3.0 ± 0.2 nm. In the literature, different experimental techniques have been used to estimate the cooperativity length: For example, thermodynamic fluctuation techniques associated with Donth’s approach16,97,34 or 4-D NMR.51,98,116,117 Recently, Schick et al.114 used dielectric spectroscopy, MT-DSC, and Ac-chips calorimetry to study the cooperativity size from Donth’s approach on PS and PMMA samples. They found that the different experimental probes provide coherent values of cooperativity length clearly associated with the structural relaxation temperature and time dependences. These studies report typical length scales of cooperativity ranging from 1 to 3.5 nm at the glass transition temperature for really different glass formers. As reported in Table 4 and in Figure 12b (main figure and inset), the dispersion of ξα at the glass transition is relatively narrow and the values obtained for the six amorphous polymers studied here seem to be representative of a very widespread behavior not only for linear backbone polymers but also for a wide range of glass formers. The similarity between values of cooperativity length at Tg is in agreement with the fact that the cooperativity length can be explained in terms of interaction volume around the relaxing unit.37,52,118 Figure 13 shows the cooperativity degree Nα in a CRR plotted as a function of the relaxation time. It can be seen that, the evolution of Nα measured from MT-DSC and BDS investigations are consistent for each sample. It has been reported that the cooperativity degree measured by different probes seems to give similar values.36,49,123 As clearly shown in Figure 13, the values of Nα for each polymer shows similar behavior for relaxation time dependence with a shift in the Nα absolute values. The Nα(τ) variations can be separated in three groups of polymers (see Figure 1): the highest values are observed for PLA, PVAc and PVC, then PBAC and PETg, and finally PEI for the smallest values. A very basic observation can be made for these polymers: the highest the glass transition temperature has the lowest Nα(τ). The Nα calculation from Donth’s formula eq 4 indicates this assumption is wrong because (Tα)2 and Nα are mathematically correlated. In fact, the differences between the three groups mentioned above can be attributed to the chemical structure of the monomeric unit and to its molar mass m0. As an example, comparing m0 values of the PEI (m0 = 592 g mol−1) and the PVC (m0 = 63 g mol−1) gives a ratio close to 100, which could explain the shift put in evidence in Figure 13.

Figure 12. (a) Cooperativity length (ξα) versus relaxation time for each polymer as deduced from BDS (hollow symbols) and MT-DSC (period = 60 s, τ ∼ 10 s) measurements (filled symbols). (b) Cooperativity length (ξα) versus temperature normalized at Tg. The inset shows the cooperativity length evolution as a function of the glass transition temperature.

the cooperativity length (ξα) estimated from BDS over a wide range of relaxation times, fits very well with the values estimated at Tα with MT-DSC at τ ≈ 10s. This result validates the use of the extended Donth approach in the case of these six polymeric systems. As previously noticed for Nc, the cooperativity lengths for each sample are quite similar and have similar behavior regarding relaxation time dependence. This result is interesting taking into account that this set of samples has quite different glass transition temperatures and VFT behaviors. In order to quantify the influence of the difference in temperature dependence of ξα, we plotted in Figure 12b, the cooperativity length (ξα) as a function of temperature normalized at Tg (corresponding to Tα at τ = 10s in our case) for each studied sample. A nonlinear increase of the cooperativity length with decreasing temperature is

Table 3. MT-DSC Results on All Investigated Samples: m0, the Molar Mass of the Relaxing Unit; Tα, the Glass Transition Temperature (τ ∼ 10 s); δT, the Mean Temperature Fluctuation in a CRR; ξα, the Cooperativity Length at Tα; and Nα, the Number of Structural Units in a CRR sample

m0 [g/mol]

PEI PBAC PETg PLA PVC PVAc

592 254 218 72 63 86

δT [K]

Tα,τ=10s [K] 489.1 419.7 351.8 330.3 356.4 315.6

± ± ± ± ± ±

0.3 0.3 0.3 0.3 0.3 0.3

3.50 2.92 2.81 3.08 3.37 3.06 I

± ± ± ± ± ±

0.05 0.05 0.05 0.05 0.05 0.05

ξα,Tα [nm] 2.8 2.9 2.8 2.9 2.8 3.0

± ± ± ± ± ±

0.2 0.2 0.2 0.2 0.2 0.2

Nα 20 72 77 266 258 246

± ± ± ± ± ±

2 7 7 10 10 10


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Table 4. Glass Transition Temperature and the Cooperativity Length Estimated from Different Experimental Techniques: DSC, MT-DSC, 4-D NMR, Boson Peak Spectra (BP) for All of the Systems Presented in Figure 12b with Corresponding References sample

Tg [K]

ξTα[DSC] [nm]

ξTα[MT‑DSC] [nm]

Polymeric Glass Formers 3.0,a 3.23d 2.50a d 3.13 3.10a 3.20a 2.30a 2.80i 2.60i 2.50i 3.00e 3.10j

PS P4MS PVC PVAc EVA90 EVA80 EVA70 PLA PBAC PIP PMPS PEK PtBS PPX C

373a,b 395d 353a 313a 292i 276i 259i 330e 413b 200c 243c 426a 419f 363g

salol OTP DBP m-toluidine

219a 249a 179b 182b

3.10a 3.00a

sorbitol glycerol PG 1-propanol

267c 189a 168b 95b

3.60a 2.90a

As2S3 Se As10Se90 As20Se80 As30Se70 As55Se45 B2O3 SiO2 CKN

455b 413k 348k 382k 412k 428k 547a 749−818a 337a

3.20a 3.20f 2.90g

ξ[4‑D NMR] [nm]

ξ[BP] [nm]

3.30b

3.30b,c

3.40b

2.88,b 2.80c

3.18b 2.23e 3.36e

3.10c

2.48h 2.50b 1.96h

2.48b 2.50h 1.47b 2.29b

3.00a

Molecular Glass Formers 3.30a

H-Bonding Glasses 2.60

a

2.50h 1.30c 1.38b

1.42c 1.30c 1.40b

Inorganic Glasses 1.78b 2.50k 2.43k 1.58k 1.57k 1.25k 1.50a 1.20a 3.20a

1.84 2.60a

h

2.00b 2.55b 1.70h

a j

Reference 97. bReference 52. cReference 117. dReference 35. eReference 37. fReference 119. gReference 120. hReference 51. iReference 121. Reference 49. kReference 122.

(Figure 11), the Nα increase with increasing the relaxation time can be described by two regimes, at short relaxation time a single power law and another one for the long relaxation times up to Tg (τ ≈ 100 s). Close to Tg, similar values of Nα and Nc are obtained for “low” m0 polymers, but these quantities seem to diverge while the relaxation time decreases. Concerning the “high” m0 polymers, these two quantities diverge in the relaxation time domain studied. In order to compare more accurately the number of dynamically correlated segments Nc (Figure 11) with the degree of cooperativity Nα calculated from Donth’s approach (Figure 13), the ratio Nc/Nα is plotted as a function of the relaxation time (Figure 14). As depicted in Figure 14, the Nc/Nα ratio shows two clear trends for our samples. This observation can be related to the molecular weight of the monomeric units (m0). As shown in Figure 13 for “low” m0 polymers (PVC, PLA and PVAc) and for τ values close to τTg, the values of Nc and Nα are similar (Nc/ Nα ∼ 1). From this observation, we may assume that for these polymers, similarities can be found in the physical definition of Nα and Nc. This is particularly interesting because the

The term cooperativity can be discussed in terms of monomeric unit number in a CRR (Nα) or in terms of length scale (volume). However, the specific choice of the elementary structural unit to estimate Nα from Donth’s approach is debatable. As an example, one PEI repeating unit consists of five benzene rings covalently bounded with imide groups. Therefore, it is hard to determine whether one benzene ring, other units or the whole molecule should be included in the relaxing unit of Donth’s approach. In fact, it can be found that for more rigid backbone polymers, cooperative motions involve some statistical segments of the macromolecule rather than the whole repeating units.51,59 These segment sizes could be associated with Kuhn’s segment length.124 This assumption is in agreement with the work of Shogo et al.,119 showing that the cooperative motion in stiff backbone polymers may be correlated to Kuhn’s segment length. Even if a direct comparison between Nα and Nc is not trivial due to the shift in the Nα absolute values, similar trends are observed. In order to compare the evolution of the parameters from the two approaches, the fitted curve from the Nc values has been reported in Figure 13. Just like for the Nc variations J


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asymptotic behaviors are different between these two models. These new and original results indicate the origin of the correlated and cooperative motions Nc and Nα respectively: The correlated motions appear at a time scale lower than the cooperative motions, implying that the correlated motions are more sensitive to the localized motions than the cooperative motions envisioned by Adam and Gibbs, as computed by Lacevic et al.125

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CONCLUSION In this work, the dynamic heterogeneity and the cooperative length scale at the dynamic glass transition have been determined for six different amorphous polymers from two different theoretical approaches by using BDS and MT-DSC investigations. As expected, from both approaches, the number of dynamically correlated segments increases on increasing the relaxation time. We compared Nc calculated data of our samples with literature data on a set of 39 glass-forming materials in a wide range of relaxation time. For this large set of glass-formers (more than 40), the number of dynamically correlated segments is comparable and seems to be characteristic of similar dynamic heterogeneity features. The cooperativity length estimated using Donth’s approach for the six polymeric systems studied here are found to be very close at the glass transition (≈3 nm) and no effect of chemical structure is reported. From a comparison of the cooperativity degree as a function of the normalized relaxation time, three tendencies can be observed that seem to be related to the molar mass of the relaxing unit. Finally, by plotting the Nc/Nα ratio, we show that the asymptotic behaviors are different between the two models used in this work: The correlated motions Nc appear at a time scale lower than the cooperative motions Nα.

Figure 13. Nα as a function of τ/τ0 (where, τ0 = 10−15s) for the PEI, PBAC, PETg, PVC, PLA, and PVAc samples. Hollow symbols are from BDS experiments and filled symbols from MT-DSC experiments (period = 60 s, τ ∼ 10 s). The red dashed line corresponds to the same fitting law shown in Figure 11, i.e., log(τ/τ0) = (Nα/N0)ψ + z ln(Nα/ N0) + ln A, with A = 4.07, N0 = 4.0, z = 1.0, and ψ = 0.5. The red dot line corresponds to the Nα fit for the PVAc with A = 5.9, N0 = 4.2, z = 1.0, and ψ = 0.45.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Telephone: 33 (0) 2.32.95.50.86. Fax: 33 (0)2.32.95.50.82. Figure 14. Nc/Nα ratio as a function of the relaxation time for each polymer.

Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS The authors are grateful for the financial support from the French Ministry for Higher Education and Research, Simone Capaccioli is gratefully acknowledged for fruitful discussions and for providing data for Nc (Figure 11). Andreas Schönhals and Eric Dantras are gratefully acknowledged for fruitful discussions, and Peter E. Mallon is acknowledged for having carefully read the paper.

calculation of Nc based on the calculation of the number of dynamically correlated segments from the 4-point correlation approach, differs from Donth’s approach, since it allows for the calculation of the number of molecules whose dynamics are correlated assuming thermodynamic fluctuations. However, with a relaxation time decrease the ratio of Nc/Nα increases. For “high” m0 polymers (PEI, PBAC, PETg) the value of Nc/ Nα is close to 10 whatever the relaxation time. The divergences between “low” and “high” m0 polymers can also be observed comparing the fitting curves for Nc and Nα (Figures 11 and 13). For these two approaches, the temperature or the relaxation time associated with the value Nc or Nα = 1 corresponds to the limit of the cooperativity, and defines a “crossover” point in Arrhenius diagram. It is noticed that Nc = 1 at the relaxation time τ ∼ 10−12 s for all the systems according to the fitting curve presented in Figure 11, but Nα = 1 in the range of τ ∼ 10−10−10−9 s for “low” m0 polymers according to the fitting curve presented in Figure 13. Concerning the “high” m0 polymers, Nα = 1 for τ ≫ 10−9s. This suggests that Nα decreases faster than Nc with a decreasing relaxation times, achieving a value Nα = 1 at higher relaxation time for all the samples. From this observation, we may conclude that the

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REFERENCES

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