Chain-Length Dependence of Relaxation and Dynamics in Poly

Article Cite This: Macromolecules XXXX, XXX, XXX−XXX

Chain-Length Dependence of Relaxation and Dynamics in Poly(methyl methacrylate) from Oligomers to Polymers Fabio Zulli, Marco Giordano, and Laura Andreozzi* Department of Physics “E. Fermi”, University of Pisa and IPCF-CNR, Pisa, Italy

ABSTRACT: A study of local relaxation is presented in a series of poly(methyl methacrylate)s (PMMAs) of varying molar mass from oligomers to polymers. Decoupling phenomena from the viscous flow in the rotational relaxation at temperature TC and violations of the Debye−Stokes−Einstein relation characterize the rotational dynamics of the series, also evidencing the influence exerted by the chain length on segmental relaxation. We discuss different dynamics regimes, coupling degrees of local relaxation to viscosity, and step-like behaviors with mass. The latter is a recurrent result in rotational relaxation and its dynamic parameters, discriminating the molecular-like behavior of the oligomers from a more specific polymeric response. Particular attention has been devoted to the onset of the Rouse dynamics in polymer samples. Scaling laws of the relaxation times with the mass are considered in the different temperature regions. Above TC the results agree with the forecasts of the Rouse theory. Below TC, the scaling exponent could be accounted for by considering the Guenza theory of the cooperative interchain dynamics.

1. INTRODUCTION The phenomenon of the glass transition governs the dynamics of liquids at temperatures below the melting point. The associated dynamic arrest drives the changes in the physical properties of the systems. It still is a topic of great interest in the condensed matter physics, having an impact on many areas of science.1−5 Significant differences between polymers and other glass-forming materials come from the polymer chain connectivity because the presence of covalently bonded chains leads to dynamic processes usually described in terms of relaxation processes over different lengths and time scales, which are slow with respect to the time scale of the segmental dynamics of the chain. The polymer dynamics was formerly described by the Rouse model for unentangled chain6 and by the de Gennes reptation motion7,8 for the topologically constrained chains of entangled polymers. Starting from the works by Rouse and de Gennes, many theoretical, molecular dynamics, and experimental studies have progressively focused on the peculiarities of the dynamics of the polymeric chain of entangled and unentangled polymer melts, including the crossover features between the two mass regimes. Only in more recent times the properties of materials with both increasing molecular weight and fixed chemistry have attracted attention, focusing on the transition from the small molecule behavior to the polymer one.9−11 Indeed it is expected that the © XXXX American Chemical Society

Rouse dynamics of the chain sets in at a certain mass, above which molecules exhibit Gaussian statistics and become polymers.12 Literature studies investigated how the crossover from simple glass-formers behavior to polymer may be associated with different dynamic features. Alba-Simionesco and co-workers investigated the role of chain length in the nonergodicity factor and fragility of polymers,13 the weight dependence of fragility in polymers was studied by Rössler and co-workers,14 and Casalini and co-workers dealt with the dielectrometry in oligomers of PMMA.11 Focusing the interest on the segmental dynamics, it is relevant to gain insight with studies as a function of mass and temperature on the time scale and the length scale characterizing dynamics and crossovers in glass-formers and polymers. In this respect, X-band ESR spectroscopy is a powerful technique to investigate molecular and segmental dynamics in simple and complex liquids.15−23 Most materials are diamagnetic and thus do not exhibit ESR signals. Therefore, small quantities of stable paramagnetic species such as nitroxide radicals are introduced in the host material,22 so that size and shape of the probe select the length scale of the experiment. The cholestane spin probe Received: November 1, 2017 Revised: January 15, 2018

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DOI: 10.1021/acs.macromol.7b02330 Macromolecules XXXX, XXX, XXX−XXX

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indium and zinc standards. Highly pure nitrogen was used as the purge gas. Thermograms were recorded on heating at 10 K min−1, after cooling at 40 K min−1. The glass-transition temperatures, determined according to the enthalpic definition,30 are reported in Table 1. In the table the characterization data of literature PMMA samples, with microstructure same as our samples and higher molecular weight, are also reported.27 2.2. Rheological Characterization. The viscoelastic properties were measured with a stress-controlled Haake RheoStress RS150H torsional rheometer by applying a torque with a parallel-plate sensor system (20 mm diameter). The loading gap between the sensor plates was 0.750 mm at T ∼ Tg + 60 K. The gap was chosen significantly larger than the chain lengths to ensure gap independent measurements. The rheometer automatically varied the parallel-plate gap with temperature to take into account thermal dilatation effects in the measuring system. The temperature control system is the Haake TC501 accessory, equipped with two platinum thermoresistances (PT100) and with a thermal bath with circulation of refrigerant. The temperature at the sample was stable within 0.1 K during all measurements, which were carried out under highly pure nitrogen flow. Steady-state viscosity, creep, and creep-recovery experiments were carried out in order to evaluate the zero-shear viscosity, η0.31 In creep and creep-recovery experiments, the viscosity of the materials was evaluated without breaking the linear viscoelastic limit, by imposing an upper limit to the strain; moreover, the procedure adopted in calculating the viscosity rules out any contribution except that from the terminal relaxation.32 Oscillatory tests were carried out in the linear viscoelastic regime and in the range of frequency 10−2 to 24.4 Hz; they provided the shear complex modulus G* = G′ + iG″, where G′ and G″ are the storage modulus and the loss modulus, respectively,29 and i2 = −1. From oscillatory experiments, an independent evaluation of the zero-shear viscosity was obtained according to the relationship η0 = limω→0G″/ω.33 2.3. ESR: Apparatus and Experimental Technique. ESR measurements were carried out with a continuous wave Bruker ER200D-SRC spectrometer, equipped with an X-band bridge (Bruker ER042-MRH) and with a NMR gaussmeter ER035M. The temperature was controlled with a gas-flow variable temperature unit (Bruker BVT100) with nominal accuracy of ±0.1 K. ESR studies were performed on PMMA samples in which the cholestane paramagnetic probe (Figure 1) was dissolved. The samples were prepared at room

was profitably employed to study reorientation dynamics of polymers and glass-formers.17,19,24,25 It locates length and time scales at nanometers and nanoseconds, characteristic of cooperative processes and of dynamic heterogeneity of the supercooled material.26 This peculiar feature could play a fundamental role in connecting and evidencing possible discrepancies of the global melt dynamics because of specific local chain properties. In previous studies by means of ESR spectroscopy, it was possible to reveal dynamics crossovers in polymers and soft matter [see for example refs 15 and 16] and decoupling from the macroscopic overall dynamics of the matrix and its dynamics at nanoseconds. A further main finding in probe ESR studies is the sensitivity to cooperative dynamics of the matrix associated with signatures and dynamic anomalies in the temperature dependence of the times of the molecular reorientation. The present contribution reports on a ESR study of the crossover from molecule behavior to polymer in PMMA, where the samples were chosen as monodisperse and with the same microstructure in order to reduce effects that can blur the mass dependence of the dynamic and relaxation properties. The presence of such effects was already reported in rheological studies on PMMA by Fuchs et al.27 and in fast dynamics and fragility studies on PIB and PS by Ding et al.28 In this work we highlight the effects of the onset of polymer chain connectivity by looking at the dynamic consequences over the features of the rotational relaxation in the PMMA series as well as the coupling of the local dynamics to the terminal relaxation of the material, detected by viscosity measurements.

2. EXPERIMENTAL SECTION 2.1. Physicochemical Characterization. Six different syntheses of PMMA were investigated in this work; the three samples with the highest molecular masses were purchased from LabService Analytica and the others from Polymer Source Inc. All of them were used as received without any further purification. Their average molar masses are reported in Table 1. Mw was evaluated with GPC and, in some samples, with light scattering. The

Table 1. Chemical and Physical Characterization of PMMA Seriesa sample PMMA PMMA PMMA PMMA PMMA PMMA PMMA PMMA PMMA

200 300 660 1200 2900 4900 22R 62R 108R

Mn (Da)

Mw (Da)

Mw/Mn

Tg (K)

200 300 520 1370 2280 4470 24000 58000 100000

200 300 660 1200 2900 4860 25000 62000 108000

1 1 1.28 1.06 1.09 1.1 1.04 1.06 1.08

163 215 259 317 345 366 392

Figure 1. Structure of the cholestane molecular tracer. temperature by mixing two chloroform solutions containing predetermined amounts of polymer and cholestane, respectively. The resulting solution (10−3 cholestane/repeat unit molar ratio) was evaporated to complete dryness under vacuum at a proper temperature. The sample was finally sealed in a standard ESR tube. The nitroxide cholestane (3β-doxyl-5α-cholestane, 98%, Aldrich), used as a paramagnetic molecular tracer, was chosen because of its good thermal stability, stiffness, and geometry.22 The cholestane molecule is quite asymmetric in shape and can be sketched as a prolate ellipsoid with semiaxes of about 0.99 and 0.29 nm.15,36 Its stiffness and cylindrical shape make it suitable to ESR studies in anisotropic media and polymers.17,19,22 Its size selects the length scale of the experiment, which delivers the relaxation time of the spontaneous fluctuation of the system. Regarding the nanometer length scale pertinent to cholestane, its suitability in ESR experiments on polymers and glass-formers comes from the finding26 that it

394

a

Mw in PMMA 1200, 2900, and 4900 was evaluated with light scattering. Data of samples 22R, 62R, 108R are also reported from the literature.27

PMMA samples are almost monodisperse (Mw/Mn ≤ 1.1), excluding PMMA 660, and certified by the sellers as GPC standards. It appears that the average molar masses of all the samples are smaller than the entanglement mass Me of PMMA [about 10.1 kg/mol29]. According to the supplier, the microstructure at 1200 and higher masses is 60% syndiotactic, 35% atactic, and 5% isotactic. Differential scanning calorimetry (DSC) measurements were carried out with a PerkinElmer DSC7 calorimeter, frequently calibrated with B


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Macromolecules Table 2. Values of the Magnetic Tensor Components in the Molecular Frame sample PMMA PMMA PMMA PMMA PMMA PMMA

200 300 660 1200 2900 4900

gxx

gyy

gzz

Axx (G)

Ayy (G)

Azz (G)

2.0026 2.0026 2.0026 2.0026 2.0026 2.0026

2.0091 2.0092 2.0091 2.0093 2.0093 2.0093

2.0068 2.0070 2.0067 2.0066 2.0068 2.0066

33.2 33.2 33.2 32.4 32.7 32.3

5.8 5.8 5.8 6.0 6.2 6.0

5.0 5.0 5.0 5.0 4.8 5.0

Figure 2. Temperature dependence of the zero shear viscosity in PMMA samples. The continuous lines are the best fit curves obtained with the VFT law.

⎛ Tb ⎞ η(T ) = η0 exp⎜ ⎟ ⎝ T − T0 ⎠

characterizes dynamic heterogeneity and cooperativity in polymers and supercooled liquids on approaching the glass transition.17,37−39 Moreover, very importantly to the study of complex systems, the relaxation times of cholestane fall in the nanosecond time scale, characteristic of crossover and cooperative phenomena of complex systems.25,40,41 The rotation of cholestane in PMMAs is described in terms of a diffusion stochastic model under cylindrical symmetry,42−44 characterized by a spinning diffusion coefficient D∥ and a tumbling diffusion coefficient D⊥.45,46 For comparison purposes with the results of rotational dynamics reported in the literature,17,19,25,37,47 in this work the reorientation of cholestane is discussed in terms of correlation times τ∥ and τ⊥ defined as τ∥ = 1/(6D∥) and τ⊥= 1/(6D⊥).48,49 More details regarding ESR spectroscopy, the theory, and the simulation program of ESR line shapes that provides dynamic information in slow motion regime are reported elsewhere.44,50−54 The principal components of the Zeeman g and hyperfine A magnetic tensors were set by the ESR line shape recorded in the ultraslow motion regime, according to a procedure detailed elsewhere.43 The values of magnetic parameters for the cholestane dissolved in the PMMA samples, expressed in the molecular frame,22 are reported in Table 2.

(1)

The values of the prefactor η0, the pseudoactivation temperature Tb, and the Vogel temperature T0 are given in Table 3. The best fits are superimposed to the experimental results in Figure 2. Table 3. VFT Parametersa sample PMMA PMMA PMMA PMMA PMMA PMMA PMMA PMMA PMMA

200 300 660 1200 2900 4900 22R 62R 108R

Tb (K) 850 1120 1750 2440 2800 3100 3500 3500 3550

± ± ± ± ± ± ± ± ±

60 50 40 60 50 50 50 50 50

η0 (Pa s)

T0 (K) 144 186 212 249 266 278 286 289 287

± ± ± ± ± ± ± ± ±

2 3 3 3 3 3 3 3 3

(1.0 (6.0 (2.2 (1.3 (1.4 (2.0

± ± ± ± ± ±

0.1) 0.6) 0.2) 0.1) 0.1) 0.2)

× × × × × ×

10−4 10−6 10−6 10−6 10−6 10−6

a The Tb values of 22R, 62R, and 108R samples are calculated from ref 27.

3. RESULTS AND DISCUSSION 3.1. Viscosity of PMMA Samples. The zero shear viscosity of PMMA samples was evaluated with flow, creep, and small-amplitude oscillatory experiments,29 carried out starting from few degrees above the pertinent glass transistion temperature. The resulting behaviors are collected in Figure 2. For each sample, the temperature dependence of the viscosity can be fitted by a Vogel−Fulcher−Tamman (VFT) law:55−57

3.2. ESR Line Shapes and Relaxation. ESR experiments on PMMAs spanned as a whole a temperature range of about 300 K. Temperatures greater than about 490 K were not accessed because of the thermal instability of cholestane molecule, while the ESR sensitivity to the mobility of the molecular probe in the ultraslow motion regime set the lower bound of temperatures. C


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Figure 3. ESR line shapes of cholestane dissolved in PMMA 200 (left) and PMMA 4900 melts. Comparison of experimental (continuous line) and calculated (dotted line) spectra at different temperatures is also shown.

Figure 4. Temperature dependence of ESR spinning times in PMMA samples.

As an example, some experimental ESR spectra recorded at

spectral details. Comparable agreements were obtained for the line shapes of all samples. In calculating the theoretical line shapes, the anisotropy ratio D∥/D⊥ was found independent of temperature, with a value ranging from 6.5 in the dymer to 6.0 in the hexamer. The value is in nice correspondence with the anisotropy ratio of cholestane diffusing in low molecular weight glass-formers, set

different temperatures and the corresponding calculated line shapes are reported in Figure 3 for PMMA 200 and PMMA 4900 samples. The strong temperature dependence of the ESR line shape should be remarked as well as the capability of the numerical algorithms in reproducing the main experimental D


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Macromolecules in the range 5−7.15,58 In the samples with the highest masses, the diffusion anisotropy ratio was found constant with temperature but with values increasing with the mass of the sample. The interval ranged from 15 to 18, with values similar to the ones reported in previous ESR studies in poly(alkyl acrylate)s (range of the ratio 12−20),37,47 and liquid crystalline polymers and copolymers exhibiting an anisotropy ratio of 15.17,19 The spinning correlation times τ∥ of the cholestane dissolved in PMMAs are shown in Figure 4 in the overall investigated temperature ranges. All the sets of data exhibit a similar qualitative behavior with three different dynamics regimes, that onset at specific temperatures. In order to better evidence the definition of crossover temperatures and dynamics regimes, in Figure 5 the case of

The presence of a thermally activated regime seems to indicate that the main-chain relaxation has become too slow to drive the faster rotational diffusion of the probe molecule and that, instead, at this stage a coupling with less cooperative and more local relaxation mechanisms could be expected. The activated regime starts at temperatures such as the values covered by the TA/Tg ratio (Table 4) allow the separation of the PMMA samples in two distinct groups. One of them includes the series of low molecular weight PMMA up to the hexamer while the other group includes the highest molecular weight PMMA samples. This changeover from molecular to polymeric behavior also reflects in the activation energies ΔE. From Table 5, it is seen that ΔE starts with the dimer values slightly higher than the ones, ranging from 10 to 22 kJ mol−1, characteristic of the cholestane or other molecular tracers in molecular glasses36,59−61 or oligomers,25,62 and increases as the mass increases up to almost reach typical values of polymers. As one example, at temperatures below Tg, ΔE was usually found to be 31−35 kJ mol−1 for cholestane reorientation in liquid crystalline homopolymers and copolymers63 and various poly(alkyl acrylate)s.37,42,50 These values were ascribed to coupling of the spin probe dynamics to very local secondary relaxation modes of the matrices.17,25 Literature studies provide for PMMA polymers values of the activation energy of the secondary relaxation of about 80 kJ mol−1,64,65 and it was recently recognized66 that this relaxation process is a Johari− Goldstein relaxation. It is the slowest secondary relaxation and has been interpreted as a precursor of the structural relaxation. Accordingly, it appears that cholestane is not coupled to it and the present findings suggest for a probe reorientation taking place in solid-like environment, with no relation to the structural relaxation. Let us focus now on the presence of other crossovers in the rotational dynamics of the polymer series, in particular with the one at high temperature. In the literature,67 the presence of crossovers of the transport or relaxation mechanisms were proposed at some temperature above Tg, where the molecules of the liquid pack more closely with decreasing temperature, and the molecular mobility is driven by activated mechanisms. The mode coupling theory (MCT) of the glass transition also focused on the temperature region above Tg, predicting the existence of a dynamic crossover from liquid-like to solid-like dynamics at a critical temperature Tcr.68 From the experimental point of view, dynamic changes above Tg of the relaxation of molecular and polymeric glass-formers have been observed and have been interpreted as a universal feature of the glass-forming systems; so, they appear to play a fundamental role in understanding the glass transition phenomenon.69−71 Dynamic crossovers have been detected by using different spectroscopic techniques40,72,73 and molecular dynamic approaches. More specifically, the amount of collected data

Figure 5. Temperature dependence of ESR spinning times in PMMA 1200. The definition of crossover temperatures is shown.

PMMA 1200 is reported as an example in a magnified view. In Table 4, the crossover temperatures are reported along with the ranges of temperature of the dynamic regions. In particular, in the high and intermediate temperature regions of each sample, namely the regions labeled as I and II in Figure 5, the rotational relaxation times are accounted for by two VFT behaviors. τ (T ) = τ

∞

⎛ Tb ⎞ exp⎜ ⎟ ⎝ T − T0 ⎠

(2)

The values of the fit parameters are reported in Table 5. Below TA, rotational diffusion follows activated dynamics according to the Arrhenius law (Table 5): τ (T ) = τ

∞

⎛ ΔE ⎞ exp⎜ ⎟ ⎝ kBT ⎠

(3)

Table 4. Temperature Details of Dynamic Regions sample PMMA PMMA PMMA PMMA PMMA PMMA

200 300 660 1200 2900 4900

TC (K)

TA (K)

TC/Tg

TA/Tg

Tg (K)

HT (I) (K)

IT (II) (K)

LT (III) (K)

262 307 373 404 430 451

206 262 332 366 398 405

1.6 1.42 1.44 1.27 1.25 1.23

1.26 1.22 1.28 1.15 1.15 1.11

163 215 259 317 345 366

320−262 332−307 463−373 468−404 474−430 488−451

262−206 307−262 373−332 404−366 430−398 451−405

206−190 262−226 332−295 366−334 398−367 405−382

E


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× × × × × × 0.1) 0.1) 0.1) 0.1) 0.1) 0.1)

τ∥∞ (s)

± ± ± ± ± ± (5.4 (7.7 (4.7 (2.2 (3.8 (6.5 0.7 0.7 0.7 0.9 0.9 0.9 ± ± ± ± ± ± 23.5 23.0 24.8 29.4 30.2 31.0 ± ± ± ± ± ± 5 6 8 8 9 9 ± ± ± ± ± ± 144 186 214 247 266 278 10 10−12 10−11 10−10 10−10 10−10 50 70 90 80 90 90 ± ± ± ± ± ± 600 650 890 610 730 800 ± ± ± ± ± ± 9 8 9 9 8 8 ± ± ± ± ± ± 143 186 212 249 266 278 10 10−13 10−13 10−13 10−13 10−13 PMMA PMMA PMMA PMMA PMMA PMMA

200 300 660 1200 2900 4900

850 1045 1660 1790 1650 1775

± ± ± ± ± ±

70 95 90 80 90 70

(3.1 (2.1 (1.9 (1.3 (5.2 (4.4

± ± ± ± ± ±

0.1) 0.1) 0.1) 0.1) 0.2) 0.2)

× × × × × ×

−13

τ∥∞ (s) Tb (K) sample

dynamic region I

Table 5. Dynamics Parameters of PMMA Series

T0 (K)

1.0 0.94 0.95 0.73 0.59 0.57

ξ

0.1 0.08 0.07 0.07 0.07 0.07

Tb (K)

(2.4 (5.6 (2.3 (2.7 (1.4 (1.2

± ± ± ± ± ±

0.1) 0.2) 0.1) 0.1) 0.1) 0.1)

× × × × × ×

−12

τ∥∞ (s)

dynamic region II

T0 (K)

0.70 0.58 0.51 0.25 0.26 0.26

ξ

0.07 0.06 0.05 0.03 0.03 0.03

ΔE (kJ mol−1)

dynamic region III

10−14 10−13 10−12 10−12 10−12 10−12

associate with TC the onset of a variety of phenomena such as spatially heterogeneous dynamics,25,74−76 fragile to strong transition,69,77 breakdown of the Stokes−Einstein (SE) and Debye−Stokes−Einstein (DSE) relations,15,77 a “universal” relaxation time τ(TC) ∼ 10−7 s,70,78 and so on. In this respect, the observation of crossover phenomena in our experiments by using ESR spectroscopy dates back to almost 20 years ago and is a recurrent result, on both molecular glass-formers and polymeric materials.15,16,19,25,47,50,79,80 The specific value of the crossover temperature depends on the material and may depend also on the guest molecule used [see ref 25 and references therein]; however, the range of obtained values by ESR experiments spans over the values that different spectroscopies provide for TC73 and for the critical temperature of the mode coupling theory Tcr, in the case of both small molecules and polymers.70 More specifically, in the case of the investigated PMMAs, the high-temperature crossover is located in the range 1.22Tg−1.6Tg (Table 4). The highest values are pertinent to the oligomers up to hexamer; the other extreme value pertains to high polymer masses and fits with the ones more frequently observed in the literature.70 In Figure 6, the crossover temperatures scaled to Tg are plotted as a function of the polymer mass; here, a transition from two regimes is evidenced by a step-like behavior. In the figure the crossover temperatures obtained with ESR experiments of cholestane probe in small molecules and in other polymer systems are superimposed to the PMMA data, showing a plateau region at masses greater than 1000 Da that according to this analysis appears as the watershed toward a polymeric behavior. This is also confirmed by the values of the correlation times at TC of the PMMA samples with the three highest masses (see Figure 4) that assume there the nearly universal relaxation time τ = 10−7±1 s, interpreted as one of the signatures of dynamic crossover from liquid-like to solid-like behavior.69,70,73 As far as the dynamics above and below TC is concerned, the T dependence of the correlation times is described by two VFT relations, whose parameters are reported in Table 5. T0 resulted in all cases virtually coincident with the corresponding Vogel temperature obtained for the viscosity by means of rheology, signaling that the local rotational relaxation is somewhat coupled to the structural relaxation of the oligomers. It must be pointed out that in simple liquids the classical hydrodynamics quantitatively describes the coupling between macroviscosity and microviscosity, at the level of segmental friction, by the classic SE law. It may become questionable in the case of a solute molecule in unentangled polymer melts because at least three length scales could affect its validity: the size of the probe nanoparticle, the gyration radius of polymers, and the correlation length of the polymer that corresponds in melts to the size of the monomer/polymer segment.81 However, what length scale dependent mechanism determines the observed breakdown of SE law is not well understood and is a topic of research interest.81−83 In the case of ESR spectroscopy, the rotational diffusion is coupled to the viscosity through the DSE law that under cylindrical symmetry can be written as D

,⊥

= C , ⊥/ η

τ , ⊥ = C′ , ⊥ · η F

(4) DOI: 10.1021/acs.macromol.7b02330 Macromolecules XXXX, XXX, XXX−XXX

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Figure 6. Values of the TC/Tg ratio as a function of the mass for PMMA of this work, molecular glass-formers,58 poly(alkyl acrylate)s,37,47 other polymers,25 and azobenzene methacrylates.19 The dashed area is a guide for the eye. The mean value of the TC/Tg ratio calculated over the plateau values is 1.19 ± 0.1.

Figure 7. Behavior of the fractional exponent ξ of the high (HT) and intermediate (IT) temperature dynamic regions as a function of the mass in the PMMA series and in poly(alkyl acrylate)s.37,47

τ (T ) ∝ [η(T )]ξ

In eq 4, C∥, C′∥, C⊥, and C′⊥ are constant and related to the volume of the diffusing particle.84 Therefore, in the case where viscosity and rotation follow VFT laws with identical Vogel temperature, the degree of coupling between rotation and viscosity can be evaluated by expressing the spinning correlation time as a power law of the viscosity:

(5)

The fractional exponent ξ is the ratio of the pseudoactivation temperature Tb of the VFT law relevant to the rotational dynamics over Tb pertaining to the viscosity. In particular, ξ = 1 signals a complete coupling of the molecular tracer dynamics to G


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Figure 8. Temperature dependence of the Debye−Stokes−Einstein ratio η/(Tτ∥) in PMMA 300. Validity of DSE equation at molecular level is proved for temperatures greater than 310 K.

PnBA where the value of exponent ξ was found to decrease by increasing the side chain length.37 On the other hand, important differences regarding the exponent ξ with respect to the polymer case are found if one takes in consideration low molecular weight inorganic glassformers. In fact, in the case of both o-therphenyl (OTP)15 and phenyl salicylate (salol),16 a complete coupling is found between tracer reorientation and shear viscosity (ξ ≈ 1 in eq 5) above TC. The results of the present investigation, involving PMMA samples going from dimer to oligomers, clarify the ESR ability to detect DSE law in low molecular weight materials and, at the same time, provide some insight regarding the crossover from molecular behavior to polymeric one in PMMA. In Figure 7 the values of the ξ exponent pertaining to the different PMMAs in the highest temperature dynamic region are reported as a function of the polymer mass (filled circles). By inspection, it is apparent a step-like molecular weight dependence also for the coupling parameter ξ signaling a crossover from a molecularliquid-like behavior to a polymeric-like dynamic behavior. In fact, up to the hexamer, a complete coupling with ξ ≈ 1 is present, whereas the two highest molecular weight systems PMMA 2900 and PMMA 4900 evidence the typical coupling value ξ ≈ 0.6 found in other non-nanostructured polymers.37,47 An intermediate value of about 0.74 characterizes the PMMA 1200 sample, that in the present case as well as for the crossover temperatures (Figure 6) marks the border between two regimes. It is interesting to note how the crossover is identified on the length scale of the probe molecule, being about 3 nm the length of the PMMA 1200 chain and about 2 nm the length of the cholestane long axis. However, other experiments also report that in PMMA behavior typical for oligomers turns into that of the polymer for a molecular size corresponding to about 10 monomer units.12 In Figure 7, the ξ values pertinent to the intermediate dynamical region below the crossover TC are also shown (open circles). The trend with the PMMA mass is the same as for the high temperature dynamic region, only with different values of

the terminal viscosity relaxation of the host matrix and the validity of DSE law. Decoupling phenomena with ξ < 1 correspond to violations of the DSE relation. The values of the ξ exponent for the PMMA samples in the two VLF regions are reported in Table 5. Focusing on the dynamic region above TC(HT), it is seen that the rotational correlation loss of the PMMAs up to the hexamer exhibits a complete coupling to the viscosity of the material, being the Tb values from ESR and viscosity virtually coincident for each sample (Tables 3 and 5). This result deserves much consideration, providing a proper frame and more insight on the customary findings regarding the partial decoupling of rotational relaxation from viscosity, usually observed above TC in ESR experiments in polymers.19,25,80 In fact, as far as the case of polymeric materials is concerned, we compared in ref 25 the behavior of rotational diffusion of different tracers in several polymers. A coherent description of the polymer data was proposed, where the reorientational dynamics of the probes is coupled, at high temperatures above T C, to relaxation modes of the polymer main chain characterized by proper length scales. However, in all the samples taken into account in that work, the value of the ξ exponent of eq 5 was lower than unity. This partial decoupling from the main relaxation was ascribed to steric hindrance effects or to a sort of segregation mechanisms that arise between probe molecule and polymer system and that are possibly also related to the polymer connectivity and/or to the presence of side groups mediating the interaction between tracer diffusion and chain dynamics. Indeed, the possible effect of the polymer chain connectivity was suggested by the results on a liquid crystalline oligomer,62 while the presence of effects due to side groups was supported by the finding that in the linear poly(propylene glycol) polymer, where side groups are almost inexistent, a complete coupling (ξ = 1 in eq 5) to the dielectric α relaxation was actually observed.25 Further support to this interpretation was given by the series of poly(alkyl acrylate)s PMA, PEA, and H


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Macromolecules the fractional exponent that in the region of high masses match with the ones found in other polymeric matrices. In this region the dynamics is characterized by a smoother temperature dependence of the correlation times for all the samples of the series. The dynamics yet develops according to a VFT with parameter values reported in Table 5. Looking at the table, the fit values of T0 still are virtually coincident with the ones of the viscosity, whereas a greater decoupling degree in all the PMMA samples was signaled by the decrease of Tb and ξ values. This decrease of the ξ value below TC has always been found in ESR studies in polymers and small molecules, and it has been ascribed to the onset of cooperative effects on the length scale probed by the tracer.15,17,19,25 At this stage, it has to be noted that, differently from the molecular glass-formers, the hydrodynamic limit has never been found in ESR studies on polymers, not only because ξ < 1 but also because the ratio τ∥/(η/T) was not compatible with the volume of the tracers. The ability of reaching the hydrodynamic limit for low PMMA masses could be tested for the cases of unitary value of the fractional exponent to confirm their nonpolymeric nature. In Figure 8 the temperature dependence of the quantity η/ (Tτ∥)85 for the PMMA 300 case is shown. It is evident that at temperature higher than about 310 K such a value was found quite constant, and also, it is compatible with the DSE scaling under cylindrical symmetry providing hydrodynamic axes of the cholestane tracer in agreement with the dimension of the cholestane molecule with semiaxes of about 0.99 and 0.29 nm. It is found apparent average radii r*∥ = 1.0 nm and r*⊥ = 0.3 nm for slip boundary conditions and r*∥ = 1.1 nm and r*⊥ = 0.4 nm for stick boundary conditions.46,86 Similar results were found for both PMMA 200 and PMMA 660, confirming that all these systems behave as molecular glass-formers as far as the tracer rotational diffusion is concerned. Referring to low molecular weight materials for which the ordinary hydrodynamics holds at high temperature, the crossover at TC and the next decoupling are likely related to the onset of dynamic heterogeneity,25 to be pictured in terms of clusters of molecules slower than the average. Indeed, we found that cholestane reorienting in supercooled OTP and salol exhibits a sort of universal behavior, because a unique master curve was obtained over the two VFT dynamic regions of the correlation loss after introducing the reduced temperature scale Tcr/T and a proper shift along the vertical axes.16 Note that this scaling usually does not work in polymers. Conforming to the idea that the behavior of the PMMA samples up to the hexamer follows the dynamics of molecular liquids, in Figure 9 the rotational correlation times are plotted for PMMA 300 and PMMA 660 as a function of a scaled temperature TS/T, being TS = 1.2Tg. [Note that this choice about coincides with the use of the reduced temperature TA/T (Table 5).] In figure, the correlation times pertaining to PMMA 300 have been shifted by multiplying by a factor 5, according to the fact that the value of the ratio between the pertinent shear viscosities at the same reduced temperatures is about 5. It is seen that the two curves collapse in a master curve holding in a large dynamic range, which only excludes the low-temperature Arrhenius-like region. The result parallels the one observed by comparing rotational diffusion of the cholestante tracer in OTP and salol and supports the idea that PMMA needs more than six monomeric units for undergoing Rouse dynamics.

Figure 9. Scaling of the rotational correlation times of PMMA 300 and PMMA 660.

It must be noted that the possibility of scaling does not include the PMMA 200 sample. This finding could be related to specific features of their shear viscosities that also did not rescale. The presence of steps in Figures 6−8 suggests the onset to a new domain in PMMA when its mass exceeds the value 1200 Da. This new domain can be definitely ascribed to the onset of the Rouse dynamics characteristic for polymers. The prediction of the Rouse model can be here tested on a quantitative level by recurring to the forecasts for the mass dependence of the relaxation times of the polymer chain.87 The Rouse theory provides for the relaxation times a scaling law according to τ ∝ M2, whose exponent originates both from the dimension of the chain, namely the gyration radius in the Rouse case, and from the diffusion coefficient of the center of mass D:87 D ∝ M −1 ,

τ ∝ M2

(6)

An agreement with this behavior could be expected for the highest PMMA polymer masses of the present study, considering that they fulfill the condition Mw < Me. We tested the scaling law of correlation times by considering the series samples with the highest masses, in order to avoid as much as possible the effects of the transition region from oligomer to polymer. Figure 10 provides an overview of the results obtained for the Rouse tests in the different dynamical region of the samples. In fact, the figure shows the ratio of the rotational correlation times pertaining to the samples PMMA 4900 to PMMA 2900 calculated for the temperatures for which both samples behave according either to a high temperature VFT or to an intermediate VFT law. In the high temperature region, labeled as PMMAs HT, three values are compared: the expected Rouse value, calculated according Table 1 and the scaling law of eq 6, the mean values of set of ratios of correlation times calculated according to the pertinent VF parameters reported in Table 5, and the mean value of the ratio of the experimental correlation times. In figure, it is seen that the x-coordinate value of the plot is quite nicely according to the forecast of the Rouse dynamics that, in agreement with eq 6, provides the value of about 2.8 for the squared value of the ratio of the polymer masses. I


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Figure 10. Plot category of PMMA in the HT and IT dynamics regions: the ratios of the correlation times of cholestane in PMMA 4900 and PMMA 2900 were raised to the scaling exponent 2 and 1.5, respectively, for the HT and IT dynamics temperature regions. The reported values are the mean values, pertaining to either couples of experimental data or of data calculated according to the values of Table 5. In the plot the expected values are also reported. For the categories of PEAs in HT and IT dynamic regions, the same quantities are reported by using data published in refs 37 and 47.

A second set of data in the figure, labeled as PMMAs IT, pertains to the ratio of the cholestane rotational times in PMMA in the temperature regions below TC. It is seen that the mean data, both experimental and calculated by using the pertinent parameters of Table 5, set to a value compatible with the ratio of the polymer masses to the scaling exponent 1.5. This finding could have some rationale resorting to the works of Guenza,88−91 going beyond the Rouse model. Such works, while maintaining the central idea of intrachain relaxation modes of different length scales, introduce intermolecular and intramolecular cooperative effects giving account of some of the typical shortcomings highlighted by the Rouse model. According to Guenza,92 the onset of many-chain cooperative diffusion affect the Rouse diffusion by a factor (M)0.5, so that D ∝ M−1.5. As a consequence, that would therefore extrapolate to an overall scaling of the correlation times according to τ ∝ M2.5. Coming back to the present experimental findings in IT region, the presence of a different scaling exponent with respect to the Rouse one would allow us to speculate that dynamic crossovers and anomalies are observed in the temperature dependence of the ESR rotational relaxation when cooperative intrachain modes become effective over the length scale of the probe. Also, in the IT dynamic region, the experimental value of the scaling exponent of the correlation times at the time scale of nanoseconds would turn out compatible with probe relaxation times driven only according to the mass dependence of a cooperative diffusion that includes intrachain dynamics. Another contribution to the mass dependence of correlation time, which in the case of Rouse dynamics (eq 6) is related to the gyration radius of the polymer, would appear in this temperature region somehow frozen, as if it could be related to a constant typical length, eventually connected to cooperativity of intrachain dynamics. Indeed, here forecasts on the scaling exponent of relaxation times have been extrapolated starting from the cooperative

diffusion coefficient of the Guenza model and, in principle, the length scale effective for evaluating the relaxation times in the presence of cooperative relaxation would be connected not to the dimensions of the polymer chain but rather to the ones of the cooperative regions, which are supposed to be identical at constant temperature in all the polymer of the series. A deeper insight of the present experimental results could be obtained by performing direct theoretical calculations of the correlation times and their scaling exponent according to the model. The above general interpretative scheme for the dependence of the relaxation of cholestane in unentangled polymers can also be tested with the rotational relaxation in PEA samples, with masses Mn = 7500 Da and Mn = 11 650 Da, investigated by ESR spectroscopy in previous studies.37,47 As seen in Figure 10, the ratios of correlation times in PEA provide the values 2.4 and 1.9, respectively for the VFT regions of high and intermediate temperature. These values agree with a power law of the masses with exponent 2.0 and 1.5, respectively. The finding confirms the presence of the same power laws as in PMMA and nicely recommends further investigation on more polymer series characterized by same chemistry and different masses.

4. CONCLUSIONS Viscosity and segmental dynamics of almost monodisperse PMMAs were studied as a function of the mass from oligomers to polymers and as a function of T by using rheology and electron spin resonance spectroscopy. The experimental data of zero-shear viscosity were collected and used to gain insight into the relaxation dynamics mechanisms and cooperative processes arising in the material series on cooling toward the glass transition. The structural relaxation time of the all samples, independent of their mass, exhibits monotonous change with temperature and shows a similar qualitative behavior with dynamic changes at a crossover temperature TC. The relaxation time corresponding to the crossovers set at the so-called “magic value” of about J


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Macromolecules 10−7 s. Two VFT relations account for data above and below TC, relating relaxation time to the viscosity with different power laws. The general observed trend for relaxation and structural parameters of the series is that they are strongly influenced by the chain length, as evidenced by various step-like variations and transitions to plateau regimes. In particular, the step-like behaviors with the sample mass found in the TC/Tg ratios and in the values of the fractional exponents of the two VFT regimes evidence that the mass of PMMA 1200 acts as a separation element between samples exhibiting a molecular behavior and samples that behave as polymers. In the molecular-like region of masses, the temperature dependence of correlation times can be suitably scaled to a single master curve, as we already found in ESR studies on simple molecular glass-formers, and the validity of DSE law was confirmed in the high temperature region of these samples. This is a nice result because usually ESR studies in polymers provide a fractional DSE behavior also above the crossover temperature. Accordingly, one can argue that as the length of the chain increases, the connectivity of the chain and the nature of the side groups influence the host−guest coupling and that different coupling mechanisms drive the probe dynamics in small molecules and polymers. Finally, in the polymeric domain of PMMA, we have tested the scaling law exponents of the correlation times above and below TC. In the high temperature region above TC, the forecasts of the Rouse theory are fulfilled; below TC, the scaling law exponent can be accounted for by recurring to cooperative intrachain dynamics.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] (L.A.). ORCID

Laura Andreozzi: 0000-0002-8235-275X Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the University of Pisa (fondi di Ateneo). Massimo Faetti is acknowledged for his support at the early stages of this work. The authors acknowledge Marina Guenza for helpful discussions with Laura Andreozzi at 8IDMRCS in Poland.

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