Study of the Dynamic Heterogeneity in Poly (ethylene-ran-vinyl

Article pubs.acs.org/Macromolecules

Study of the Dynamic Heterogeneity in Poly(ethylene-ran-vinyl acetate) Copolymer by Using Broadband Dielectric Spectroscopy and Electrostatic Force Microscopy Mohammed M. Kummali,*,†,‡ Angel Alegría,†,‡ Luis A. Miccio,†,‡,§ and Juan Colmenero†,‡,§ †

Departamento de Física de Materiales, UPV/EHU, Fac. de Química, 20080 San Sebastián, Spain Centro de Física de Materiales, CSIC-UPV/EHU, Paseo Manuel de Lardizabal 5, 20018 San Sebastián, Spain § Donostia International Physics Center, Paseo Manuel de Lardizabal 4, 20018 San Sebastián, Spain ‡

ABSTRACT: Dynamic heterogeneities in ethylene−vinyl acetate (EVA) random copolymers were studied using broadband dielectric spectroscopy (BDS) over a broad frequency and temperature range. BDS data for EVA copolymers show relatively broad spectra extending over several frequency decades, which make interpretation of the data complicated. Thus, microscopic dielectric characterizations of the samples were done using single pass electrostatic force microscopy (SP-EFM) images. From these experiments two distinct dielectric contributions were found in the semicrystalline samples. Dielectric spectra of EVA copolymers were fitted using a model based on the SP-EFM images. According to this analysis, the broad dielectric responses of semicrystalline EVA copolymers were constituted from two dielectrically active phases, referred to as the constrained and nonconstrained phases, exhibiting different segmental relaxations. In the constrained phase, vinyl acetate (VA) motions were greatly restricted by crystalline microstructures, whereas in the nonconstrained phase, VA monomers were assumed to be completely free from crystalline effect. The glass transition temperature derived from the dynamics of the VA segments in the constrained phase was found to correspond well with that determined from DSC representing the overall amorphous phase in the more crystallized state. Moreover, crystallinity of the sample obtained using DSC was found to follow a similar variation with temperature to that of the dielectric relaxation strength of the constrained phase of the copolymers. However, at the earlier stage of crystallization, where crystallinity increases rapidly, the dielectric relaxation remains insensitive to it.

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copolymers.6,7 The effect of changes in the crystallinity and microstructures on the physical properties of EVA copolymers were actively investigated by probing the dynamics of the copolymer using DSC,4,9,10 13C NMR spectroscopy,4,11 dynamic mechanical analysis (DMA) and dielectric spectroscopy,5,7,11−13 and fluorescence spectroscopy,6,14 to name some. Morphological and microstructural complexities arising due to the complex crystallinity make the dielectric relaxation spectra of EVA copolymers complex; in particular, the structural origin of the relaxation process is not well-defined. One of the main reasons for this is the limited knowledge of the relative compositions of the amorphous and interface regions.6 Recent advances in atomic force microscopy (AFM) endue it to probe the structural and dynamic properties of polymer systems more locally. Because of its exceptionally high spatial resolution and sensitivity, AFM is widely being used to study the morphology of polymer samples and to probe their local dielectric properties.15−18 In this study we probe the relaxation

INTRODUCTION Copolymerization of polar vinyl acetate (VA) monomers with nonpolar and semicrystallizing ethylene monomers makes the structure of EVA copolymer complex. As reported in the literature,1,2 since the reactivity ratios of both ethylene and VA are close to one, EVA copolymers are statistically random in nature. Complexity of the structure of EVA copolymer, especially the crystallinity, has been a subject of intense research. While some research focused on the nature of change in crystallinity and microstructure as the VA content varies,1,3,4 others studied its effect on the mechanical and dielectric properties.5−8 Two types of crystalline structures, with different crystalline lamellae and morphologies, were attributed to EVA structures, with one type obeying Flory’s general theory of copolymer crystallization and the other disregarding the theory when VA content is increased.3,4 Because of the presence of crystalline microstructures, a three-phase model, comprising fully crystalline regions with only polyethylene units, interfacial regions with coexisting VA and crystallized polyethylene units, and complex amorphous regions with coexisting VA and noncrystallized polyethylene units, has been presented for the structure of EVA © XXXX American Chemical Society

Received: June 17, 2013 Revised: September 2, 2013

A

dx.doi.org/10.1021/ma4012522 | Macromolecules XXXX, XXX, XXX−XXX

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Table 1. Main Characteristics of EVA Samples Investigated in This Work

a

polymer nomenclature

VA content (wt %)

HDPE LDPEb EVA9 EVA18 EVA25 EVA33 EVA40 EVA50 EVA70 PVAc

0 0 9 18 25 33 40 50 70 100

Tg(DSC) (K)

crystallization onset tempa and melting end temp (K)

enthalpy of crystallization (J/g)

crystallinity percentage (% Xc) at 250 K

395−406 375−383 363−377 347−369 335−361 318−350 309−345 309−336

201 115 106 74.4 47.1 28.1 14.4 11.9

60 40 36.9 27.9 16.4 9.8 5.0 4.1 0 0

232 237 242 243 247 248 256 315

Obtained during cooling. bReferences 6 and 26.

Figure 1. (a) Heat flow (cooling part) of EVA9, EVA25, EVA40, and EVA70 samples measured at a cooling rate of 3 K/min. The corresponding reversing part of the heat flow signals is presented in (b). toluene with varying solution concentrations depending on the VA content. Samples for DSC and BDS measurements were obtained from this solution by the solution casting technique. For AFM measurements, 2 wt % solutions were prepared in toluene for EVA copolymers before being spin-coated over a gold sputtered disk. Sample thickness thus obtained was ca. 100 nm. In order to establish uniform thermal history, all EVA samples were subjected to a standard thermal treatment before undergoing any measurement. Samples were annealed at 373 K, well above their melting temperatures, under vacuum for a day. Modulated Differential Scanning Calorimetry (MDSC). MDSC measurements were performed using a DSC Q2000 from TA Instruments. Samples of about 5 mg were sealed in hermetic aluminum pans. Modulated heating−cooling cycles were performed under nitrogen flow in the temperature range from −173 to 448 K, with a heating/cooling rate of 3 K/min and a modulation amplitude of 0.5 K and a period of 60 s. The annealing time between cooling and heating runs was 5 min. MDSC allows separation of total heat flow signal to its reversing and nonreversing components23.24 Glass transition temperature (Tg) was determined at the inflection point of the reversing curve, and the heat of crystallization (ΔHc) was estimated from the total heat flow curve during cooling. Broadband Dielectric Spectroscopy (BDS). Low (f: 10−1−106 Hz) and high frequency ( f: 106−109 Hz) dielectric measurements were performed for this study. Low frequency measurements were conducted using a high-resolution broadband dielectric spectrometer (Novocontrol Alpha). In the case of high frequency measurements, an Agilent E4991A RF impedance analyzer was employed. Measurements were performed on disk-shaped samples using gold-plated electrodes with diameters of 20 mm and 10 mm, for low and high frequency measurements, respectively. Thicknesses of the samples in both cases

dynamics of ethylene−vinyl acetate (EVA) random copolymer, composed of VA units (polar) and ethylene units (apolar) using macroscopic and microscopic techniques. Macroscopic characteristics of EVA copolymers, including crystallinity and glass transition behavior, were obtained by using differential scanning calorimetry (DSC), and their bulk dielectric properties were probed by measuring its segmental relaxation using standard BDS. Nanoscopic characterization of the copolymers were obtained by using single pass electrostatic force microscopy (SP-EFM), a recently developed technique19−22 which enables one to map the local dielectric behavior using AFM. On the basis of the information obtained from imaging, we developed a two-component model for describing the dielectric relaxation data from the semicrystalline copolymers. This model allowed us obtaining the dynamical characteristics of the two VA containing phases, i.e., the interfacial and the fully amorphous ones. Thus, quantitative correlations between dynamic features of VA segments and the copolymer crystallinity are established.

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EXPERIMENTAL SECTION

Sample Preparation. All EVA random copolymers were purchased from Sigma-Aldrich Co., except EVA50 (with 50 wt % of VA) which was bought from Scientific Polymer Products Inc. Nomenclatures of the samples were done by suffixing their respective VA content (weight percentage, wt %, in bulk) to EVA. A list of the polymers used in this study along with the corresponding VA contents (wt %) is given in Table 1. EVA copolymers were first dissolved in B


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Figure 2. Dielectric images obtained through SP-EFM measurements are presented. AFM images were processed using WSxM.28 were typically of about 0.1 mm. For parallel-plate configuration, the sample capacitance is expressed as C = εε0A/d, where ε is the relative dielectric permittivity of the sample, ε0 is vacuum permittivity, A is the area of the sample electrode, and d is the sample thickness. The material properties are characterized by the complex dielectric permittivity, ε*, which is defined as ε*(ω) = C*(ω)/C0 = ε′(ω) − iε″(ω), where C0 is the capacitance of the empty capacitor and ω = 2πf. Isothermal frequency scans were performed every fifth degree over the temperature range 200−450 K. Sample temperature was controlled by nitrogen gas flow with temperature stability better than ±0.1 K. Single Pass EFM (SP-EFM) Imaging. Details of SP-EFM have been published elsewhere.19 In brief, in this method a sinusoidal voltage is applied to the conductive AFM probe during the main scan, where conventional tapping is performed. The motion of the cantilever is detected by the photodiode and analyzed by an external lock-in amplifier (LIA) in order to detect the electric force component. The cantilever responses at a frequency double of the electrical excitation gives information about the dielectric properties of the materials under investigation. The LIA analyzes this signal, and its amplitude and phase are mapped along with the topography of the sample. In this case dielectric permittivity of the sample is related to the measured displacement in the photodiode through the probe-sample capacitance16 according to the equation

A 2ωχkc = F2ω = −

1 ∂C 2 V0 cos(2ωt ) 4 ∂z

increase as VA content increases. These results are consistent with earlier studies.5 DSC exothermic curves presented in Figure 1a allow us to determine the crystallinity of EVA samples. The degree of crystallinity (χc) of EVA copolymers was calculated using the expression χc =

ΔHc × 100% ΔH0

(2)

where ΔHc is the measured enthalpy of crystallization of the copolymer and ΔH0 is the enthalpy of fusion of completely crystalline polyethylene taken to be 286.34 J/g.27 χc of EVA copolymers so obtained at low temperatures (T ≤ Tg) is presented in Table 1. Tg values for different EVA samples are also presented in Table 1. SP-EFM Measurements. Local dielectric characterization of the samples was done using SP-EFM. Single pass dielectric imaging was employed as it gives an enhanced contrast in the image compared to double pass imaging.19 SP-EFM images (shown in Figure 2) were obtained by applying an ac signal (10 kHz, 5 V) to the conductive tip at room temperature. EVA samples show continuous variation in their dielectric images, from dark to light color, as the VA content is increased. EVA18 shows a comparatively dark (red) dielectric image compared to the other samples, with darker spots (blue) in it. These blue spots, with lower dielectric signal, can be tentatively assigned due to PE crystallites. In EVA25, the dielectric image is mostly red in color with few blue spots in it. The red granular shapes in the image are separated by thin yellow boundaries, exhibiting increased dielectric values. In EVA40, the dielectric image is a mixture of dark and light (yellow) colors with negligible amount of spots in it. The dielectric image of EVA70 is very light, with an increased value of dielectric signal. A higher value of dielectric signal entails an increased permittivity of the sample according to eq 1. SP-EFM images in Figure 2 suggest the complex dielectric behavior of EVA copolymer13 and the corresponding change in the microstructure of the copolymer as VA content increases. Since the mobile VA monomers contribute more to the dielectric permittivity values, the blue spots with lower ε values in EVA18 and EVA25 could be attributed to crystalline structures from which VA monomers are excluded, whereas the relatively smooth, red background with intermediate ε values might be due to VA which would be strongly affected by the presence of polyethylene nanocrystallites (not resolved in the images). Dielectric images corresponding to EVA25 and EVA40 exhibits increased ε values, with mixture of red and yellow colors, compared to the previous case. The grain boundaries marked with higher dielectric values might be due to higher VA content and likely more mobile. Increased VA content as in the case of EVA70 increases the amplitude signal

(1)

where kc is the cantilever spring constant, χ is the sensitivity of the laser beam displacement on the photodiode, and A2ω is the amplitude of the 2ω component of the displacement. In order to obtain dielectric contrast images of the samples, 5 V sinusoidal voltage was applied to the probe using Stanford Research SR830 LIA. The conductive cantilevers used for the measurements were made of antimony (Sb) doped Si, coated with Pt/Ir (SCM-PIT tips, Bruker). The nominal values (manufacturer) for the natural frequency, tip radius, and cantilever spring constant were 75 kHz, 20 nm and 1.5−3 N/m, respectively. With this configuration, spatial resolution of the technique is about 20 nm.25

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RESULTS DSC Measurements. Figure 1a shows the cooling part of heat flow, plotted as a function of temperature for representative EVA samples. The figure shows crystallization peaks for all EVA samples except EVA70. In EVA70, the heat flow is relatively constant until the glass transition temperature. From the figure, an increase in the strength of the crystallization peak is apparent as VA content decreases. In addition, crystallization onset temperature (Tc) is found to decrease as VA content increases, and accordingly, the melting temperature (Tm) in the subsequent heating run (not shown) shifts toward lower values. MDSC allows one to separate the reversing contributions to heat flow, which highlights the glass transition range (see Figure 1b). Even though obtaining accurate values of glass transition temperatures (Tg) from the DSC curves are onerous, it can be found that the Tg tends to C


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Figure 3. ε″ is plotted as a function of frequency for all EVA samples investigated at 260 K (a) and at 290 K (b). ε″ is plotted as a function of frequency only for amorphous EVA samples at 340 K (c) and the same data in a normalized representation are shown in (d). In (e) the relaxation spectra of EVA40 at different temperatures (260, 270, 290, 310, and 330 K) through the crystallization range are plotted on a log−log scale. Lines correspond to data fitting as described in the text.

due to its high dielectric value. In summary, SP-EFM images gives three distinct features while varying degree of VA content viz. crystalline regions with no VA in it (blue color), as in the case of EVA18 and EVA25, regions with moderate dielectric

signal, red and a mixture of red and yellow color, where VA is affected by crystalline structures and regions with high dielectric signal, as in EVA70, which are richer in VA content and where VA is free from the influence of crystalline structures. D


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Figure 4. Calculated values of the maximum relaxation times for the nonconstrained components are plotted as a function of temperature. Filled circles were obtained from fully amorphous samples, whereas empty symbols were the result of a two-component fitting (see the text). Solid lines represent the corresponding VFT fit. The dashed-dotted line, corresponding to PVAc homopolymer,29 is included for comparison. Error bars at representative temperatures of 275 and 270 K for EVA25 and EVA18, respectively are shown; for other samples the errors are too small to be shown. In (b) α parameter from EVA70 is plotted as a function of temperature.

Therefore, these results are in agreement with the generally accepted multiphase character of EVA copolymers already mentioned in the Introduction. BDS Measurements. In Figure 3a−c, the relaxation spectra of the EVA samples investigated at three different temperatures are presented. Since only VA contributes to the dielectric relaxation, increasing its content in EVA samples cause corresponding increase in the dielectric loss. Plots in Figure 3a,b also show the general trend of a broadening of the dielectric spectra as VA content decreases. It can also be noticed that the frequencies corresponding to maximum loss shift to high frequency side as VA content decreases. These variations in the relaxation of the EVA samples are originated by both, changing VA concentration and changes in the crystallinity. At temperatures well above the melting temperatures of the polymers, the dielectric relaxation spectra, as in Figure 3c, display dramatic changes in peak frequency and intensity, but similar shapes. The latter is obvious in Figure 3d where the loss curves for EVA samples are compared in a normalized representation. At these temperatures, the samples are completely free from any crystalline effect, and thus, the spectra are governed by the mobility of the VA segments plasticized in a varying extent by the presence of neighboring ethylene units. Differences in the mobility of the VA segments in the samples alter the position of the spectra in time scale but somehow without changing its shape. On the other hand, Figure 3e presents segmental relaxation spectra of EVA40 at different temperatures where the direct influence of the crystallization process on the dielectric relaxation of a single copolymer is very clear. The figure shows that the dielectric relaxation is not just shifting over the measured temperature range; rather, it changes significantly in shape as it passes through the crystallization range, and consequently crystalline structures are developed in the VA poorer areas. At lower temperatures the spectra are broader and extend over a wide frequency range, whereas before crystallization the spectra remain relatively narrow. The characteristic broadening of the spectra at low temperatures due to the presence of crystalline microstructures in the

samples is evident not only within a given sample (Figure 3e) but also when comparing different copolymers (Figure 3a,b). Analysis of BDS Data. The dynamics of PVAc homopolymer has been previously studied using dielectric spectroscopy,29 and the results were interpreted using the phenomenological Kohlrausch−Williams−Watts (KWW)30 model. There it was possible to fit, in frequency domain, data obtained for PVAc relaxation using Havriliak−Negami (HN)31 function with a single independent shape parameter. The HN function can be written as ε*(ω) = ε∞ +

Δε [1 + (iωτHN)α ]γ

(3)

where the relaxation strength, Δε = (εs − ε∞), εs is the static, low frequency permittivity, ε∞ is the high frequency permittivity, and τHN is a characteristic relaxation time. The exponents α and γ (0 < α, γ ≤ 1) describe the broadening and asymmetry of the spectra, respectively. In this study, we follow the same procedure as followed by Tyagi et al.29 to fit the dielectric spectra of EVA70 copolymer, where the asymmetric shape parameter, γ, in HN function was restricted and written in terms of symmetric shape parameter, α, using the relation γ = 1 − 0.812(1 − α)0.387

(4)

as deduced from the analysis by Alvarez et al.32 for the correspondence between the HN equation and that corresponding to a stretched exponential time decay. In case of EVA70, the absence of crystalline structures has been confirmed by DSC and SP-EFM measurements. Thus, in this case the dielectric loss spectra can be fitted with this type of HN functions i.e., with only three fitting parameters. Some examples of fitting curves are included in Figure 3. Figures 4a,b show the temperature dependence of the fitting parameters (τ and α) so obtained. In the figure, the time corresponding to the maximum relaxation losses, τmax, calculated from the characteristic relaxation time of the sample τHN using the relation33−35 E


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⎡ ⎢ sin = τHN⎢ ⎢ sin ⎣

( ) ⎤⎥ ⎥ ( ) ⎥⎦ αγπ 2 + 2γ απ 2 + 2γ

Table 2. T0 and TgD Values of the Constrained and Nonconstrained Components for Different EVA Copolymers Obtained from VFT Fitting Are Shown; FWHM at 250 K for Different Samples Are Also Shown

1/ α

(5)

nonconstrained

is plotted against inverse temperature. Figure 4b presents the shape parameter α, which in turn represents the broadening of the spectra, with temperature. At higher temperatures α remains constant, within uncertainty, and decreases in a linear way below 320 K. As shown above in Figure 3d, the normalized relaxation spectrum of EVA70 at 340 K superimpose with spectra obtained for EVA50, EVA40, and EVA33 at the same temperature. At this temperature, all the three latter copolymers are well above their melting temperatures and thus are completely amorphous in nature. Being above the melting temperature, the VA monomer is completely free from crystallinity effects, and thus, the resulting dielectric signal will be referred to as nonconstrained relaxation. From Figure 3d similar shape parameters at each temperature are anticipated for the nonconstrained relaxations for different amorphous EVA copolymers. Thus, the analysis of the segmental relaxation spectra of these copolymers, at high temperatures, were done by fixing the shape parameter α from EVA70 at the same temperature, consequently reducing the number of fitting parameters for each spectrum from three (according to eqs 3 and 4) to only two. Relaxation spectra of EVA50, EVA40, and EVA33 at 340 K analyzed in this way are shown in Figure 3c. Accurate fittings of the spectra in the figure support the validity of this approach. The maximum relaxation times so obtained for VA content higher than 30 wt % are included in Figure 4 where the speed-up of the VA segment dynamics by decreasing the VA content is evident. Unfortunately, due to the high melting temperatures, for copolymers with VA content less than 30 wt % the loss peak in the melt state is out of experimental frequency window. Segmental relaxation times above the glass transition temperature use to exhibit a characteristic non-Arrhenius behavior33,36 that can be conveniently described by the well-known Vogel−Fulcher− Tamann (VFT).33 The latter can be written as ⎡ DT0 ⎤ τ = τ0 exp⎢ ⎥ ⎣ (T − T0) ⎦

samples

T0 (K)

TgD

PVAc EVA70 EVA50 EVA40 EVA33 EVA25 EVA18 EVA9

267.2 209.6 195.9 190 186.6 180.0 171.6

319.5 259.1 241.7 233.8 230.0 221.9 211.5

constrained T0 (K)

TgDc

fwhm at 255 K

204.3 200.6 197.9 193.4 190.4 186.1

250.6 245.7 242.3 237.2 233.6 228.3

2.6 3.5 4.0 6.6 7.8 10.2

Figure 5. Dielectric glass transition temperatures: squares represent nonconstrained and triangles the constrained components; filled obtained from τmax of single HN fits, empty obtained from double HN fits, and + from interpolation; circles: Tg as obtained by DSC.

the actual glass transition of most of the EVA copolymers is strongly influenced by the crystallization processes. While in EVA70, the dielectric relaxation of VA is completely free from crystalline effects on the whole temperature range, in EVA9 the VA segmental mobility must be strongly affected by the high crystallinity of the sample (see Table 1) due to the high dilution of the VA units. As can be seen in Figure 1a, the crystallization process in EVA9 seems to be completed already at room temperature where the dielectric relaxation in this sample is still too fast to be detectable in the available frequency range. So one can safely assume that the dielectric relaxation measured on EVA9 at lower temperatures correspond to a wellcrystallized sample. Selected dielectric relaxation spectra of EVA9 are presented in Figure 6a. It is noteworthy that EVA9 exhibits extremely broad relaxation loss at low temperatures, extending over more than 4 decades. According to refs 13, 36, 38, and 39, the restriction on the motion of amorphous chain segments near crystallites causes inhomogeneous broadening of the segmental relaxation dispersion. Therefore, this broadening of the segmental relaxation spectra can be considered as representative of the strong influence of the presents of crystallites in the mobility of VA segments, i.e., what we will refer to as constrained relaxation. Moreover, for the lowest

(6)

where τ0 is the reciprocal of a characteristic vibrational frequency,36 T0, the Vogel temperature, is the temperature at which the extrapolated relaxation time tends to diverge, and D is connected with the so-called fragility strength parameter.37 The temperature dependence so obtained for the maximum relaxation times for fully amorphous copolymers can be well described in this way assuming common values of τ0 = 1 × 10−12 s and D = 6.3 (see Figure 4), determined from the free VFT fitting of EVA70 data, and the value of T0 varying monotonously with VA content as expected by the plasticizing effect of ethylene units on the VA mobility. The corresponding parameter values are depicted in Table 2. From the VFT fitting lines we have calculated what is called dielectric glass transition temperature, TgD, determined from the temperature at which the maximum dielectric relaxation time equals ca. 1 s. The soobtained values are also included in Table 2 and are plotted as a function of VA content in Figure 5 in comparison with the calorimetric Tg data. Only for the fully amorphous samples, i.e., PVAc and EVA70, both values are close to each other. Thus, F


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Figure 7. Calculated values of the maximum relaxation times for the constrained components are plotted as a function of temperature. Filled symbols were obtained for EVA9 and correspond to a wellcrystallized sample, whereas empty symbols are the result of a twocomponent fitting (see text). Circles represent EVA50, squares for EVA40, and triangles for EVA33. Dashed lines are obtained assuming a linear dependence of T0 on VA content as depicted in the inset.

Figure 6. Dielectric relaxation spectra of EVA9 copolymer are shown at different temperatures. Solid lines are fits corresponding to an inverted HN function (see text).

temperatures the spectra seems to show a bimodal character with a relatively fast component that could be originated in the local mobility in VA segments typical of the beta relaxation of PVAc.29 Thus, by considering the relaxation spectra of EVA9 only above 250 K, it is clear that they appear as peaks skewed toward the low frequency part. This trend is opposite not only to the typical segmental relaxation loss of amorphous polymers but also to the rather symmetric shape commonly found in crystallized homopolymers. The inverted asymmetry of the loss peak from EVA 9 does not allow using the HN function, and consequently, instead of using a normal HN function, the loss peaks at these temperatures were conveniently fitted using an inverted HN function,40 using the equation ⎡ ⎤ −1 ε″(ω) = Δεc Im⎢ αc γc ⎥ ⎣ [1 + (1/iωτc) ] ⎦

The temperature dependence of maximum relaxation times from the EVA9 data can be well described using the VFT equation by maintaining again τ0 =1 × 10−12 s and D the same as for fully amorphous EVA samples (see Figure 7). From the resulting VFT fitting line we have calculated TgDc, i.e., the dielectric glass transition corresponding to the constrained VA mobility, which in this case compares rather well with the calorimetric Tg (see Figure 5) evidencing that in this copolymer the dynamics of the VA segments is well coupled with the (majority) remaining amorphous ethylene units.

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DISCUSSION The SP-EFM images shown in Figure 2 present rather uniform dielectric properties for EVA70, as expected for a single-phase copolymer (it cannot crystallize). However, for EVA40 and EVA25 the dielectric images evidence clear contrast indicative of the coexistence of phases with distinct dielectric properties. When decreasing the VA content the dielectric images again become more uniform, which suggests that a single phase dominates the dielectric properties in the samples with low VA content. When these results are analyzed in combination with those obtained from DSC, the following picture emerges. At room temperature the crystallization process of EVA18 is well developed and relatively high; therefore, the dielectric image would reflect the fact that there is a relatively low concentration of VA dipolar segments with a quite restricted mobility. On the other case, EVA70 is fully amorphous having a high concentration of VA units with no constraints. In EVA40 and EVA25 the situation seems to be somehow in between; the crystallization process is not completed, and regions with no constraints coexist with regions where crystallinity has been developed and VA segments mobility is restricted. Thus, a reasonable assumption is to consider that the BDS data obtained on EVA70, and also on EVA33, EVA40, and EVA50 before entering in the crystallization range, would be representative of the nonconstrained VA segmental dynamics,

(7)

Here again we found that γc can be written in terms of αc using relation 4, resulting in a good fitting (see Figure 6). The subscript c in eq 7 stands for the fact that EVA9 is assumed to be completely constrained. It is noteworthy that eq 7 can be written in terms of a superposition of Debye functions using a logarithmic distribution function,41 and therefore it obeys the Kramers−Kronig relation. Furthermore, the unusual shape of this component suggests that as far the crystallized phase is approached, the contribution of VA segments dramatically decreases, because of either the reduction in dipole reorientation ability or the decreasing VA local concentration. The time corresponding to the maximum relaxation loss of the constrained component was calculated using the equation

c τmax

⎡ ⎢ sin = τc⎢ ⎢ sin ⎣

( ) ( ) αcγcπ

2 + 2γc αcπ 2 + 2γc

⎤−1/ αc ⎥ ⎥ ⎥ ⎦

(8)

The power in eq 8 is inverted due to the inverted frequency in the HN function used in eq 7. The maximum relaxation times so obtained are plotted in Figure 7. G


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Figure 8. Relaxation spectra components of EVA40 (left) and EVA25 (right) at 270 K. Open symbols correspond to experimental data at 270 K, whereas lines represent the constrained (dashed) and nonconstrained (dotted) components.

are clearly separated in time scale, with the constrained component exhibiting a significantly slower relaxation compared to its nonconstrained counterpart. The coupling between fitting parameters give rise to typical uncertainties of about 10% for dielectric relaxation strength values, 5% for the shape parameter, and about 0.1 decade for the relaxation time. Note that using eq 9 is strictly valid only if a parallel model applies, which is obviously not the case in the present system where structural heterogeneity is very prominent and, subsequently, changes in the local field would be expected. However, as it will be confirmed below, eq 9 remains a rather good approximation for the EVA copolymers since the dielectric properties of the different phases are not drastically different. In such a situation the differences in the local field are small, justifying the general practice in the BDS literature of analyzing the losses resulting from multi phase systems using the parallel model. Of particular interest at this point are the values so obtained for the maximum relaxation times of the constrained components, which have been represented by empty points in Figure 7 in comparison to those obtained for EVA9. As can be seen, the temperature variation for this component obtained in the crystallizing range follows in good approximation the same VFT behavior as in EVA9 (but with a different T0), mainly at low temperatures where the constrained component gains more relevance on the whole relaxation process (see below). Note that this means that possible variations of parameter D as compared to that obtained for EVA9 cannot be addressed due to the limited accuracy of the data. Nevertheless, by fixing the value of D, we will reduce the uncertainties in the resulting values of the remaining single fitting parameter T0c. The so-obtained VFT behaviors (see solid lines in Figure 7) allows calculating the dielectric glass transition temperature corresponding to the VA constrained dynamics, TgDc, for these samples (see Table 2). When these values are compared with the corresponding calorimetric Tg values (see Figure 5), it becomes apparent that they are rather close to each other, and consequently one can conclude that the constrained component is that best reflecting at low temperatures the overall segmental mobility, involving also ethylene units. On the other hand, the values of T0c resulting from the VFT fittings of the constrained components vary quite linearly with VA

whereas those measured on EVA9 must correspond to the highly constrained VA segmental dynamics. However, in general, the dielectric relaxation of EVA copolymers, once in the crystallization range, would consist of two distinct contributions originating from two distinct VA containing regions as those inferred from the SP-EFM images of EVA25 and EVA40 at room temperature. Accordingly, the spectra of all EVA copolymers at temperatures below their onset crystallization temperatures, where two coexisting components can be expected, viz. EVA18, EVA25, EVA33, EVA40, and EVA50, were analyzed using the combination of eqs 3 and 7. The function can be written as ⎡ ⎤ −1 ε″(ω) = Δε Im⎢ α γ ⎥ + Δεc ⎣ [1 + (iωτ ) ] ⎦ ⎡ ⎤ −1 Im⎢ αc γc ⎥ ⎣ [1 + (1/iωτc) ] ⎦

(9)

where all symbols have the same meaning as above. Here, α and γ (αc and γc) are correlated as above. Note that fitting the data in this way without further constraints would involve a large number (six) of fitting parameters, so some restrictions (assumptions) are convenient to limit the uncertainties in the resulting fitting parameter values. According to the physical picture described above, a reasonable assumption is to consider that the characteristics of the nonconstrained component can be to some extent extrapolated from the results obtained in the fully amorphous state. So for the analysis of the samples EVA50, EVA40, and EVA33 the time scale of the nonconstrained component relaxations was fixed according to the VFT equations fitting the higher temperature data. Also, the value of the shape parameter for this component was fixed, except at very low temperatures where an extra broadening of this component was required for a good fitting quality. With this assumption the number of fitting parameters remains limited (four for most of the temperatures), and the obtained values of the remaining fitting parameters were rather stable. Some representative examples of the results of this fitting procedure are shown in Figure 8, where the description of the whole relaxation process and the two distinct components are shown. In the figure, the constrained and nonconstrained components H


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content as can be seen in the inset of Figure 7. Thus, it is possible to predict the VFT behavior of the constrained component for EVA18 and EVA25 and calculating the corresponding values of TgDc. Again, the agreement with calorimetric values is quite good. For samples EVA18 and EVA25, no reliable relaxation data can be obtained in the fully amorphous state, so the fitting of the dielectric losses in the crystallization range cannot be done using the same approach used for VA content higher than 30%. Note that for low VA content not only the dielectric losses are weaker but more important the crystallization takes place at relatively high temperature where the peak frequency is out of the accessible frequency window. Consequently, for these two low VA content copolymers the fitting procedure was based on fixing in eq 9 the maximum time scale of the constrained component according to the VFT equation described above. In this way, five parameters were allowed to vary in the fitting procedure, including the maximum relaxation time for the nonconstrained component. The obtained results for the latter are shown in Figure 4 as empty symbols. Again, these new set of data can be well described by the VFT equation varying only T0, and the corresponding values of TgD for eventually fully amorphous EVA samples with low VA content can be obtained (see Figure 5). It is clear that the new data points follow well the trend of those inferred using high temperature relaxation data on fully amorphous EVA samples. In addition to the time scales characteristic of the constrained and the nonconstrained dielectric relaxation components, the fitting procedure provides the corresponding relaxation strengths and shape parameters (we recall that for most temperatures a good fitting is obtained by fixing the shape parameter of the nonconstrained component from the behavior of EVA70). Figure 9 shows the values of the full width at halfmaximum (fwhm) of the dielectric loss peaks obtained for the constrained component. It is apparent that the fwhm increases with decrease in temperature, more dramatically as VA content decreases. Moreover, at low temperatures where crystallization

has been completed, there is a rather linear relationship between fwhm and crystallinity (see inset in Figure 9). An important parameter obtained from this phenomenological fitting of the dielectric data is the relaxation strength, not only the whole value (ΔεT = Δε + ΔεC) but more interestingly that of each component. In typical glass-formers, the relaxation strength of the segmental relaxation exhibits a monotonous and smooth decreasing with temperature.36 This behavior is what is approximately found for the whole dielectric relaxation in the different samples although some features likely related with the varying crystallinity are also apparent. However, the effects of the crystallinity set up in the dielectric relaxation at temperatures significantly lower than those detected by DSC. Note that due to the slower equivalent cooling rate corresponding to the isothermal dielectric experiments the effects of crystallization could start at temperatures slightly above that measured by DSC. Contrary, what we found is that during the crystallization process reflected as a rather sharp peak in DSC no effect on the dielectric relaxation of EVA can be detected. Only at lower temperatures the dielectric losses manifest the presence of the slower component (constrained component), i.e., at temperatures where the DSC curves present a rather broad (not so well-defined) peak associated with further EVA crystallization. In Figure 10 the comparison between the evolution of crystallinity and the relaxation strength of both components is presented for representative samples. There, it is clear that the increasing relevance of the constrained component varies monotonously by decreasing temperature in the range where also sample crystallinity increases smoothly. However, this component is not relevant at higher temperatures where DSC detects the sharp increasing of crystallinity. This result strongly support the idea that crystallization of EVA takes place first in highly segregated ethylene rich regions and only at lower temperatures the crystallization in VA containing regions starts. This “secondary” crystallization would imply the segregation of VA segments and would give rise to rather imperfect (likely nanometric scale) crystals that will influence the mobility of the VA neighboring segments. While the crystallinity curves presented in Figure 10 are purely empirical, the dielectric relaxation strength values were obtained from the fittings, which in turn involve many assumptions. Moreover, both quantities are obtained by measuring two different physical properties of the system. Thus, the visible correlation of the temperature dependence of the EVA crystallinity with that of the dielectric relaxation strength data from constrained component can be considered as an additional support of the assumptions involved in fitting the BDS data. Nevertheless, a direct correlation between Δε and crystallinity does not exist since dielectric relaxation is not sensitive to the first stages of crystallization, likely because only ethylene-rich regions would be involved in this early crystallization process. Another question that arises from the above analysis is whether the two components resolved in the crystallization range can be straightforwardly identified with the dielectric properties of the two phases. It is well established33 that when the measured dielectric losses arise from molecular motions in spatially separated phases, the resulting signal is not as simple as the superposition of the losses arising from each of the phases. Only if a parallel model applies the simple addition is fully correct. Despite that, it is a general practice in the BDS literature on polymers to analyze the losses resulting from multiphase systems using the parallel model as an approx-

Figure 9. Fwhm for the constrained components are plotted as a function of temperature. Symbols have their meaning as in Figure 7. Right pointed triangles represent EVA18 and left EVA25. The inset represents the linear correlation with crystallinity at low temperatures. I


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Figure 10. Dielectric relaxation strengths and crystallinity plotted together as a function of temperature for two different EVA copolymers, namely, EVA25 (left) and EVA40 (right).

should be taken as a semiquantitative estimation of the relative volume fraction of each of the phases but not an accurate one, not only because the model used to analyze the data but also because the uncertainties in the actual changes in the relaxation strength when transforming from the unconstrained to the constrained VA amorphous phase.

imation. This approximation is rather good as far the different phases in the material present similar dielectric properties, since in this case there are only minor differences in the local electric field. This means that when any of the phases present a dielectric relaxation, the parallel model approach would be still applicable only when the permittivity dispersion and consequently the associated dielectric losses are low enough. In the EVA samples analyzed in this work we have checked that the parallel model used still represents a good approximation. For doing so we have compared the results obtained above with that resulting from a more realistic physical model considering the coexistence of three different phases, namely, crystalline polyethylene regions surrounded by a constrained EVA phase all imbibed in a EVA amorphous matrix. The resulting permittivity for such a model will depend on the dielectric properties of the three phases and the corresponding volume fraction [see ref 33, p 502]. The volume fraction of the crystalline phase can be inferred from DSC experiments, but those of the VA containing phases are unknown. Moreover, applying this model would also require as an input the dielectric strength of both of the VA containing phases. So, at least one extra parameter would be needed for the fitting. In order to compare the fitting results using the same number of fitting parameters, one could make the assumption that the VA containing phases present a relaxation strength proportional to the VA content and that this remains the same in the two phases in a given copolymer. Under this extra assumptions, we have performed a fitting of the data collected form EVA40 at 260 K. We have selected this as the most unfavorable situation since the volume fraction of the two phases would be similar, and the VA content is relatively high. Furthermore, the measured loss peak is well centered in the explored frequency window, and therefore the loss peak is measured with the best accuracy. The obtained results are depicted in Figure 3e as a dashed line. As can be seen, the two fitting lines describe the experimental data with nearly the same accuracy. More importantly, the values of the parameters obtained by the two approaches differ less than typical uncertainties (0.1 decade for the time scale and 2% for the dielectric strength of the nonconstrained phase). Taking these results into account, the dielectric strength determined by the parallel model approach

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CONCLUSION The dielectric relaxation of EVA copolymers with varying amounts of VA content and crystallinity was analyzed. SP-EFM images revealed the presence of microstructures where three distinct dielectric behaviors can be identified and were associated with (i) relatively big ethylene crystallites, (ii) regions where vinyl acetate segments are highly constrained by neighboring ethylene nanocrystals, and (iii) amorphous regions where vinyl acetate segments are just plasticized by the presence of ethylene units. According to these results, the broad dielectric loss spectra of EVA copolymers would originate by the two vinyl-acetate containing (dielectrically active) phases, namely the constrained phase and nonconstrained phase, exhibiting different segmental relaxations. From this analysis, it was found that the temperature dependence of the relaxation strength arising from the constrained phase correlates with that of the sample crystallinity as determined from differential scanning calorimetry during cooling from the melt state. Moreover, the VA dynamics in the constrained phase was found to be very much coupled with the neighborhood amorphous ethylene units, since the dielectrically determined glass transition temperatures agree rather well with the overall one detected by DSC.

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AUTHOR INFORMATION

Corresponding Author

*E-mail [email protected] (M.M.K.). Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We acknowledge the financial support from Projects No. MAT2012-31088 (Spanish Government), No. IT654-13 J


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(35) Hansen, C.; Stickel, F.; Berger, T.; Richert, R.; Fischer, E. W. J. Chem. Phys. 1997, 107, 1086. (36) MaCrum, N. G.; Read, B. E.; Williams, G. Anelastic and Dielectric Effects in Polymeric Solids; Dover Publication: New York, 1991. (37) Angell, C. A. J. Non-Cryst. Solids 1991, 131−133 (1), 13. (38) Ngai, K. L.; Roland, C. M. Macromolecules 1993, 26, 6824− 6830. (39) Wetton, R. E.; Allen, G. Polymer 1966, 7, 331. (40) De La Fuente, M. R.; Pérez Jubindo, M. A.; Zubia, J.; Pérez Iglesias, T.; Seoane, A. Liq. Cryst. 1994, 16, 6. (41) Bottcher, C. J. F.; Bordewijk, P. Theory of Electric Polarization, 2nd ed.; Elsevier: Amsterdam, 1980.

(Basque Government), and the European Soft Matter Infrastructure (ESMI) Program (Grant Agreement number: 262348).

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Study of the Dynamic Heterogeneity in Poly (ethylene-ran-vinyl

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