Simultaneous Slowdown of Segmental and Terminal Relaxation of

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Simultaneous Slowdown of Segmental and Terminal Relaxation of Both Components in Dynamically Asymmetric Poly(ε-caprolactone)/ Poly(styrene-co-acrylonitrile) Blends Yafang Xu, Wei Yu,* and Chixing Zhou

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Advanced Rheology Institute, Department of Polymer Science and Engineering, Shanghai Key Laboratory of Electrical Insulation and Thermal Ageing, State Key Laboratory for Metal Matrix Composite Materials, Shanghai Jiao Tong University, Shanghai 200240, P. R. China S Supporting Information *

ABSTRACT: The molecular relaxation of the dynamically asymmetric miscible polymer blend poly(styrene-co-acrylonitrile) (SAN)/poly(ε-caprolactone) (PCL) was investigated by differential scanning calorimetry, broadband dielectric spectroscopy, and rheological measurements. The relaxation times and the broadness of relaxation spectrum at both segmental scale and chain scale increase when the temperature decreases. The relaxation times at two length scales are linearly correlated, and the broadness of the terminal relaxation is wider than that of the segmental relaxation. Moreover, it is found that both the segmental relaxation time and the terminal relaxation time increase with the content of the counterpart at the same temperature distance from the effective glass transition temperature. It implies that segmental relaxation time at the effective glass transition temperature is composition-dependent due to the specific interactions between PCL and SAN. By combining the time-dependent diffusion double reptation (TDD-DR) model and the self-concentration and concentration-fluctuation (SCCF) model with the composition-dependent segmental relaxation time at the effective glass transition temperature, the linear viscoelasticity of SAN/ PCL blends can be well predicted over a wide range of compositions and temperatures. fluctuations and chain connectivity should be taken into account for the understanding of segmental dynamics.7,8,12 The dynamic heterogeneity was also observed in the chain dynamics of miscible polymer blends.1,4,13,16,17 Watanabe et al.9 found that the thermorheological complexity appeared in a range of temperature when the composition of one component was above an overlapping concentration in blends polyisoprene (PI)/poly(vinyl ethylene) (PVE) and PI/poly(4-tert-butylstyrene) (PtBS). It was ascribed to the different temperature dependence of monomer friction coefficients ζ; i.e., the frictional nonuniformity increased when the test temperature approaches the respective glass transition temperatures of components. As the thermorheological complexity was attributed to the difference of the monomer friction coefficients ζ of the Rouse segments (the motional unit for terminal relaxation), the natural question is whether the global dynamics is directly coupled with the segmental dynamics. Although in PI/polystyrene (PS) blends Harmandaris et al.18 reported that the ratio of the terminal to the segmental relaxation time presents a clear qualitative difference for the constituents due to the different sizes of segment in relation to the length scale of motion, Haley et al.19 investigated PI/PVE blends and found

1. INTRODUCTION Understanding the molecular dynamics of miscible polymer blend is one of the most important subjects of polymer physics. Extensive experimental studies1−4 and simulations5−8 have elucidated the existence of dynamic heterogeneities in miscible polymer blends, which manifest itself as broadening of the glass transition (even two separate transitions), a broad distribution of the segmental relaxation modes, and the thermorheological complexity (failure of the time−temperature superposition, TTS).9 The local molecular dynamics in segmental scale is known to govern the glass transition. It had been found experimentally that the broad glass transition arises both from the wide distribution of segmental motional rates for each species and from intrinsic differences in the motional rate between the two species.2,3,10,11 Theoretical models based on the local environment have also been proposed to explain the segmental dynamics. The thermally driven concentration fluctuations12,13 considers intermolecular composition surrounding the specific segment in a “cooperative volume”, which can explain the dramatic broadening relaxation observed in experiments. The intramolecular chain connectivity, which is the core of the selfconcentration model,2,3,14 can explain the presence of different dynamic identities in miscible blends, although the selfconcentration experienced by a test segment is not a fixed value.8,15 Now, it is believed that both thermal concentration © XXXX American Chemical Society

Received: June 7, 2018 Revised: September 7, 2018

A

DOI: 10.1021/acs.macromol.8b01214 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules Table 1. Characteristic Parameters of SAN and PCL SAN PCL

Tg (°C)

GNd (MPa)

density (g/cm3)

Me (g/mol)

p (Å)

b (nm)

C∞

φs

a (nm)

C1

C2 (°C)

104.4 −63.9

0.217 0.467

1.07 1.14

17347 8626

4.07 3.15

1.5a 0.7b

9.6a 5.9c

0.18 0.41

8.13 6.30

29.9 27.8

84.0 52.0

From ref 14. bFrom ref 27. cFrom ref 28. dAt 150 °C.

a

Figure 1. Heat capacity (a) and its differential curve (b) for SAN, PCL, and their blends. The glass transition temperature in (c) is determined from the peak in the differential curves. The dashed lines are the predictions of the self-concentration model. The shadow area denotes the temperature span of the glass transition.

that the segmental and terminal dynamics of two components exhibit equivalent dependences on temperature and composition. Based on these understandings on the correlation between segmental and terminal relaxation, different models had been suggested to describe the linear viscoelasticity of miscible polymer blends. Several reptation-based models, such as the double reptation (DR) model,16,20,21 the time-dependent diffusion double reptation (TDD DR) model,5 and double reptation with tube dilation,17 have been combined with the self-concentration model and can give satisfactory predictions on the linear viscoelasticity of miscible blends. However, in contrast to the focus on the width of segmental relaxation spectrum, the broadness of terminal relaxation in miscible blends was seldom considered. Most studies on miscible polymer blends focused on model systems without component interaction, such as PI/PS,18 PI/ PVE,9 PI/PtBS,9 and PI/PB.13 In fact, the intermolecular and intramolecular interaction also play important roles in compatibility, especially when at least one component is a copolymer. For example, Chiu et al.22 and Fernandes et al.23 found that the compatibility of PCL/SAN is determined by AN content and also the blend composition. The binary interaction models23,24 and the mean-field theory25 suggest that the miscibility of PCL/SAN is caused by a dilution of the intramolecular repulsive interaction between styrene and acrylonitrile segments. In addition, the SAN/PCL blend has significant dynamic asymmetry as a big difference in their glass transition temperatures (ΔTg ≈ 170 K). The molecular dynamics in such a dynamically asymmetric system with specific interactions is expected to have different features from those of the athermal system. In this work, we will reveal the segment and chain dynamics over the entire composition in

wide temperature range. Both the relaxation times and the broadness of the relaxation spectrum will be studied. Specifically, we will illustrate that the segmental relaxation time at the effective glass transition temperature of the component is different from the pure polymer and depends on the blend composition. Such a result will be incorporated with the TDD DR model and the self-concentration concentration fluctuation model to predict the linear rheological behavior of miscible blends.

2. EXPERIMENTAL SECTION 2.1. Materials. PCL (Capa 6800, Perstrop UK Limited) has a weight-average molecular weight (Mw) 80 kg/mol and a polydispersity (=Mw/Mn, with Mn the number-average molecular weight) of 1.62. Poly(styrene-co-acrylonitrile) (SAN, 81HF, LG Chemical) contains 28.4 wt % acrylonitrile with Mw = 141 kg/mol and Mw/Mn = 2.08. PCL and SAN were dried at 50 °C in a vacuum oven for 48 h before processing. The blends of PCL/SAN with different compositions were melt blended in a torque rheometer (XSS-300, Shanghai Kechuang Rubber & Plastic Equipment Co, China) at 150 °C under 60 rpm for 15 min. The sample for rheological measurements were made by compressing molding at 150 °C under a pressure of 10 MPa. No sign of phase separation of all blends was observed when investigated by phase contrast microscopy up to 200 °C. This is consistent with previous report about the effect of AN content in SAN on the phase behavior of SAN/PCL blends.26 Characteristic molecular parameters of SAN and PCL are listed in Table 1. 2.2. Dielectric Spectroscopy. The dielectric properties were investigated in the frequency range 10−1−107 Hz under different temperatures by using a broadband dielectric spectrometer (Novocontrol GmbH Concept 40 system with Quatro Cryosystem temperature control). The samples for dielectric measurement were made through compressing the molten blended mixtures in a disk-like mold with 15 mm diameter and 1 mm thickness at 150 °C under a pressure of 10 MPa. Then they were carefully deposited with gold films B


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Figure 2. Dielectric loss ε″ for pure SAN (a) and SAN/PCL 70/30 blend (b) at various temperature. The corresponding normalized data are shown in (c) and (d) for SAN and SAN/PCL 70/30 blend, respectively. Solid and hollow symbols are experimental results; solid lines are fitting curves using the HN equation (eq 4). on both sides and then placed between two round plate gold electrodes. 2.3. Differential scanning calorimetry (DSC). DSC measurements were performed on a Q-2000 (TA Instruments). For each measurement, about 10 mg of sample was equilibrated at 150 °C for 5 min, then quickly quenched to −80 °C, and held for another 5 min. Then the sample was heated again to 150 °C with a heating rate of 10 °C/min, where the glass transition temperature was determined. 2.4. Rheology. Rheological measurements were performed on a rotational rheometer (Bohlin Gemini HR, Malvern Instruments) with 25 mm parallel plate geometry and a gap of 1 mm. Oscillation frequency sweeps were conducted between 0.01 and 100 rad/s with a strain amplitude of 5% to ensure the linear responses of all samples. 2.5. Gel Permeation Chromatography (GPC). Molecular weight and molecular weight distribution were measured by GPC (Viscotek 270Max, Malvern Instruments, UK), and HPLC grade tetrahydrofuran (THF) was used as the solvent at room temperature. The concentration of solution was set at about 7 mg/mL. Viscometry, light scattering, and refractive index signals were collected for analysis.

connectivity. The effective local concentration ϕeff,i of i species (i = A or B) can be expressed as

3. RESULTS 3.1. Segmental Relaxation. 3.1.1. Glass Transition from DSC. The glass transitions of SAN, PCL, and their blends were studied by thermal analysis. The heat capacity and its first derivative are shown in Figures 1a and 1b, respectively. The glass transition temperature was determined as the peak temperature in the curve of the first derivative of heat capacity.28 It is seen that two glass transition temperatures appear clearly in the 50/50 blend, and a single but wide transition is observed in other blends. The plot of the glass transition temperature against the content of SAN is shown in Figure 1c. The Lodge−McLeish (LM) self-concentration (SC) model has been frequently adopted to describe the glass transition temperature of the miscible polymer blends.14,29,30 In this model, the local concentration of A monomer is higher than its macroscopic volume average due to the chain

The self-concentration of SAN and PCL can be estimated from the characteristic parameters in Table 1 to be 0.18 and 0.41, respectively. Compared with SAN, PCL is more flexible and thus has a larger ϕs. The composition dependence of the calculated Tig,eff of the two components is shown as dashed lines in Figure 1c. The predictions from SC model are quite consistent with the glass transition temperatures from DSC measurements. 3.1.2. Segmental Relaxation from Dielectric Spectrum. Broadband dielectric spectroscopy has been frequently used to study the segmental dynamics of miscible polymer blends.31−33 The dielectric relaxation behavior of pure SAN and SAN/PCL 70/30 blend at various temperatures is shown in Figure 2. SAN contains type B dipoles perpendicular to the chain backbone, and the main loss signal is attributed to the segmental relaxation (α-relaxation) under electric field excitation.34 The α-relaxation shifts to lower frequency with the decrease of temperature in

ϕeff, i = ϕs, i + (1 − ϕs, i)Φi

(1)

where Φi is the bulk composition and ϕs,i is the selfconcentration, which can be estimated through ϕs = C∞M0/ kρNAVVref, where C∞ is the characteristic ratio of the polymer, M0 is the molar mass of repeating unit, k is the number of backbone bonds per polymer repeating unit, ρ is the polymer density, NAV is Avogadro’s number, and Vref ∼ b3 is the reference volume occupied by the segment with Kuhn length b. The effective glass transition of i component Tig,eff(ϕeff,i) can be evaluated through effective concentration ϕeff,i using the Fox equation14 1 i Tg,eff

C

=

ϕeff, i i Tg,eff

+

1 − ϕeff, i j Tg,eff

(i , j = A or B) (2)


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components.15 These assumptions and treatments indicate that τi depends only on ΔT (ΔT = T − Tig,eff), and τi ∼ ΔT curves should be superposed for different blends. However, when the effective glass transition temperature of SAN (TSAN g,eff ) is taken ε from the DSC measurement (TDSC ) and τ is plotted against g,SAN peak the relative temperature ΔT (ΔT = T − TDSC g,SAN), it is found that the segmental relaxation time of SAN increases with the PCL content (Figure 3b). It means that the relaxation of SAN segment becomes slower as the PCL content increases at the same ΔT. One possible reason is the improper choice of the effective glass transition temperature due to the different length scales in the dielectric α-relaxation and the glass transition in DSC experiment. The other possible reason is the improper assumption that τg,i is independent of the blend composition. Although the broadening of segmental relaxation has been observed and reported widely, a quantitative analysis on the width of segmental relaxation has not been paid much attention. To describe the broadness of segmental relaxation, the Havriliak−Negami (HN) equation41 is adopted here to fit the experiment data

both pure SAN and SAN/PCL blends, which means the slowdown of SAN segmental relaxation at lower temperature. Moreover, the time−temperature superposition (TTS) works quite well in pure SAN, but it fails in SAN/PCL blends. The failure of TTS in SAN/PCL blends apparently can be ascribed to the evident broadening of dielectric relaxation when the temperature decreases. Such a broadening phenomenon of segmental relaxation in miscible polymer blends has been reported in 1,2-PBd/1,4-PI,35 PVME/PS,36,37 and PVE/PI38 and was regarded as a sign of dynamic heterogeneity.39 It can be explained by the thermal concentration fluctuation model,32 which attributes the enhanced broadness of dielectric relaxation with the decrease of temperature to the growing size of the “cooperatively rearranging domain”. A recent study further combined the intermolecular contribution (i.e., the concentration fluctuation model) with the intramolecular contribution (i.e., the self-concentration model) to interpret microheterogeneity for miscible polymer blend as the temperature approaches the apparent glass transition temperature.7 The plot of the segmental relaxation time (τεpeak = 1/2πf max with f max the peak frequency in the dielectric relaxation spectrum) of SAN against temperature is shown in Figure 3a.

ε* = ε∞ +

Δε [1 + (iωτ0)1 − αε ]βε

(4)

where ε* is the complex dielectric permittivity, ε∞ is the high frequency limiting value of the real part, Δε is the relaxation strength, ω is the frequency, and τ0 is the relaxation time. αε and βε are shape parameters of the HN function, with αε being the width parameter and βε being the symmetry of the relaxation peak. When βε = 1, the HN model becomes the Cole−Cole model.42 The best fitting curves are shown as solid lines in Figure 2, and the obtained αε values are presented in Figure 4.

ε Figure 3. Dependence of segmental relaxation time τpeak on temperature (a) and on the temperature difference ΔT = T − Tig,eff (b). The effective glass transition temperatures Tig,eff of SAN are determined by DSC (TDSC g,SAN) as shown in Figure 1. The solid lines are the fitting results for pure SAN using eq 3 with τεg,SAN = 0.05 s, C1,SAN = 29.9, and C2,SAN = 84 K.

It is clear that τεpeak increases as the temperature decreases and decreases as the PCL content increases. The latter is often ascribed to the decrease of the apparent glass transition temperature of SAN after been blended with the lower Tg component PCL. Actually, it has been suggested to use the Williams−Landel−Ferry (WLF) equation (or Vogel equation) to describe the component segmental relaxation time of blends with the effective glass transition temperature18

Figure 4. Dependence of the width parameter αε of dielectric αrelaxation for SAN/PCL blends on temperature (a) and on relative temperature ΔT = T − Tig,eff (b). Effective glass transition temperatures Tig,eff of SAN are determined by DSC (TDSC g,SAN) as shown in Figure 1.

i ij τ yz ) C1, i(T − Tg,eff logjjjj i zzzz = − i j τg, i z C2, i + T − Tg,eff (3) k { where τg,i is the segmental relaxation time for pure component i at the glass transition temperature; C1,i and C2,i are coefficients. τg,i and C1,i have always been assumed to be constant and independent of blend composition, while C2,i has been regarded as a constant40 or a linear combination of those of pure

βε is about 0.505 in SAN and blends, denoting that the shape of dielectric relaxation does not change in the blends. Because the dielectric loss of SAN can be well superposed in the experimental range of temperature (Figure 2c), its broadness is only determined from the shifted master curve. Therefore, only one value is determined for SAN and plotted as a horizontal line in Figure 4. The width of dielectric relaxation of all blends is larger than that of pure SAN, and αε of blends exhibits remarkable increase as temperature decreases. It is D


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Figure 5. Normalized dynamic viscosities η′/η0 (hollow symbols) and η″/η0 (solid symbols) for SAN (a) and SAN/PCL 50/50 blend (b) at different temperatures. Solid lines in (a) denote the fitting curves of the master curve of SAN using eq 5, and solid lines in (b) denote the fitting curves at 60 and 180 °C using eq 5.

found that the width parameter αε of SAN/PCL 90/10 blend is the largest at the same temperature. This unusual phenomenon can be visualized from direct comparison between the normalized dielectric spectra at the same temperature (Figure S1). Such behavior is ascribed to the change of effective glass transition temperature of SAN in blends. When αε is plotted with ΔT (= T − TDSC g,SAN), it is apparent that the width of segmental relaxation increases with PCL content at the same ΔT. 3.2. Chain Relaxation. The chain relaxation of polymer can be inferred from the linear viscoelastic properties. In almost all previous studies, the frequency dependencies of dynamic moduli have been adopted to determine the terminal relaxation time τw of polymer chains,43 while the broadness of the terminal relaxation time was seldom discussed. In contrast with those previous works, we choose dynamic viscosities (η′ and η″) to determine the terminal relaxation time. This approach is based on the consideration that the viscosity η″ and the dielectric loss ε″ have similar dependence on the frequency, which allows us to directly compare, in a consistent way, the characteristic relaxation time and the broadness of the relaxation processes from rheology (terminal) and dielectric spectrum (segmental). Such an approach is critical since different definitions of relaxation may lead to inconsistent comparison because of the wide distribution of different relaxation processes. The normalized η′/η0 and η″/η0 versus the normalized angular frequency ω/ωpeak for the SAN and SAN/PCL 50/50 blend are shown in Figure 5, where η0 is the zero shear viscosity and ωpeak is the angular frequency at the peak of η″/η0. The oscillation frequency sweeps were determined at temperatures as wide as possible; the lower limit of testing temperature should avoid the crystallization of PCL in PCL-dominant blends and be controlled within the torque range of instrument especially for SAN-dominant blends, while the upper limit of testing temperature should consider the thermal stability of samples. For SAN, the excellent superposition of η′/η0 and η″/η0 from different temperatures indicates the success of the TTS principle. For SAN/PCL 50/50 blend, the failure of superposition can be clearly seen in both η″/η0 and η″/η0 curves. The failure of TTS has also been observed using the frequently

adopted methods like the shifted dynamic modulus,44 the Cole−Cole plot,45 and the vGP plot.46 In the linear viscoelastic region, the failure of TTS has been used to identify phase separation47 or dynamic heterogeneity in polymer blends.44 In SAN/PCL blends, the failure of TTS in rheological functions also manifests the dynamic heterogeneity at the length scale of chain size, which results from the broadening of the terminal relaxation process as the temperature decreases. It might be related to the dynamic heterogeneity at the length scale of segment size. Similar to that in dielectric spectrum, the characteristic relaxation time in the terminal region of dynamic viscosities is taken as the reciprocal of the peak frequency, τηpeak = 1/ωpeak. The dependence of the characteristic relaxation time τηpeak on temperature in pure polymers (SAN, PCL) and their blends is shown in Figure 6. At the same temperature, it is seen that SAN (PCL) has the longest (shortest) characteristic relaxation time τηpeak, and in the blends τηpeak decreases monotonically as the PCL content increases. In contrast to the dielectric loss as shown in Figure 2, where the segmental relaxation is ascribed only to SAN, both components (SAN and PCL) contribute to the terminal relaxation process. Although it is difficult to separate the two contributions, SAN chains will dominate the terminal relaxation in SAN-dominant blends, and PCL chains play a major role in PCL-dominant blends. Therefore, we may plot τηpeak with the relative temperature ΔT = T − Tig,eff in Figure 6b, where Tig,eff is taken as TDSC g,SAN for SAN dominated blends (wSAN ≥ 0.5) and TDSC for PCL dominated blends (wSAN ≤ 0.3). It is g,PCL found that the characteristic relaxation time τηpeak increases as PCL (or SAN) content increases in SAN dominated blends (wSAN ≥ 0.5) (or PCL dominated blends (wSAN ≤ 0.3)) at the same ΔT. It implies that the chain relaxations of SAN and PCL are simultaneously slowed down after blending with the counter component. Such behavior in terminal relaxation is similar to that observed in segmental relaxation (Figure 3). Usually, the broadness of the terminal relaxation is described by using different relaxation times, for example, the ratio of the terminal relaxation time τw (= J0s η0 with J0s the steady shear compliance and η0 the zero shear viscosity) and the relaxation time τn (= η0/G0N with G0N the plateau modulus). Because the E


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Figure 6. Chain relaxation time τηpeak at different temperatures (a) and relative temperatures (b) for pure components and SAN/PCL blends. The relative temperature is defined as T − TDSC g,SAN for SAN dominated blends (wSAN ≥ 0.5) and T − TDSC g,PCL for PCL dominated blends (wSAN ≤ 0.3). The dashed lines are the fitting curves for SAN and PCL using the WLF equation (eq 4) with C1,SAN = 29.9, C2,SAN = 84 K, and C1,PCL = 27.8, and C2,PCL = 52.0 K. The solid lines in (a) are results predicted by the TDD-DR-SCCF model.

Figure 7. Width parameters of the terminal relaxation for pure SAN, PCL, and their blends. The solid lines are predictions of the TDD-DRSCCF model for SAN/PCL blends.

4. DISCUSSION 4.1. Segmental Relaxation versus Chain Relaxation. The relaxation times of SAN segmental motion and chain motion are compared in Figure 8. It should be stressed that the

plateau modulus cannot be detected for all samples in the range of temperatures in this study, the ratio of τw to τc (= 1/ωc with ωc the crossover angular frequency of G′ and G″) is adopted. The terminal relaxation time τw and the crossover relaxation time τc of SAN, PCL, and blends are shown in Figure S3, and their ratios are shown in Figure S4. The ratio τw/τc, ranging from 15 to 50, is much larger than 1, which is observed in monodispersed pure polymer. The ratio τw/τc increases as temperature decreases, manifesting the broadening of chain relaxation. However, such parameter (τw/τc) for the broadness of chain relaxation cannot (and should not) be directly compared with αε for segmental relaxation because τw/τc and αε are differently defined “measure” of the broadness. Because of the similarity of the frequency dependence of the dielectric loss, the frequency dependence of the dynamic viscosity is analyzed by the Havriliak−Negami equation to evaluate the broadness of the terminal relaxation process η0 = η′ − iη″ η* = [1 + (iωτ )1 − αη ]βη (5) where η* is the complex viscosity, αη is the width parameter of the terminal relaxation, and βη denotes the symmetry of the viscoelastic relaxation peak. An example of fitting result is shown in Figure 5a for SAN. Both η′ and η″ are fitted simultaneously using the same set of parameters (η0, τ, αη, and βη). The best fitting values of αη for all samples are shown in Figure 7, and βη values for all samples are shown in Figure S2. αη and βη for pure components (SAN and PCL) are similar and almost keep constant over a wide range of temperature. In SAN/PCL blends, αη changes slightly at high temperature but increased remarkably as temperature decreases. A similar trend is also seen for βη. Such a trend is similar to the temperature dependence of τw/τc (see Figure S3). Actually, when we plot αη versus τw/τc (Figure S4), a clear linear relationship between these two broadening parameters can be found, which indicates the equivalence of αη and τw/τc in describing the width of terminal relaxation process.

Figure 8. Comparison of chain relaxation time with segmental relaxation time in pure SAN and SAN-dominant blends.

characteristic time τηpeak from dynamic viscosity represents the apparent chain relaxation, which is contributed by the motion of SAN chain as well as the motion of PCL chain. In SANdominant blends, τηpeak is mainly determined by the terminal relaxation of SAN chains, and PCL has a minor influence on the apparent terminal dynamics. It is found that in these blends τηpeak is linearly proportional to the segmental relaxation time τεpeak of SAN, and the dependency is almost not affected by the PCL content and close to that of pure SAN. Whether the segmental relaxation and the terminal relaxation have the same temperature dependence (or have the same local frictions) has been a controversial topic for a long time.48−50 The assumption that friction is the same for segmental and terminal mode results in F


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Macromolecules the thermorheological simplicity,51 while failure of TTS especially when approaching the glass transition temperature has been found using different techniques.52−56 For pure polymer, Ding et al.57 illustrated that the segmental and terminal relaxations have the same friction when the segmental relaxation time is below 10−7−10−5 s, which could be explained as the decoupling of translational and rotational motions in polymer melts52,53,57 or could be linked with a dynamic heterogeneity.58,59 In miscible polymer blends, Harmandaris et al.18 investigated the dynamics of polyisoprene (PI)/polystyrene (PS) by atomistic simulations; the ratio of the terminal to the segmental relaxation time for PS remains approximately constant, which is independent of the composition of the blend and very similar to the ratio of the bulk pure PS. Their simulation results are consistent with our experimental data. In fact, it is seen that the linear correlation between the terminal relaxation time and segmental relaxation time extends up to ∼10−4 s of the segmental relaxation time (Figure 8), which covers most experimental data in Figure 6. The reason might be the experimental temperatures for rheological measurements are at least 50 °C above the apparent Tg. In addition to the relaxation time, the width of relaxation spectrum is also an important issue in polydispersed polymer blends. Here we quantify it as the broadness of segmental relaxation αε and the broadness of terminal relaxation αη. It can be seen that the width of relaxation process broadens as the temperature decreases for both the segmental relaxation (Figure 4) and the chain relaxation (Figure 7). It is instructive that the widths of relaxation of both processes are analyzed by the same width parameter α of the HN model (eqs 4 and 5), which allows direct comparison of the segmental relaxation and the terminal relaxation (Figure 9). As the α-relaxation of

of the segmental relaxation process. However, such a conjecture is inconsistent with the experimental data. In fact, the wider relaxation spectrum of chain motion than that of segment motion in Figure 9 can be ascribed to the polydispersity in chain length, i.e., the wide molecular weight distribution, because the polydispersity is not a factor in segmental scale. Second, when the temperature decreases, the width of both relaxation processes increases, and the difference between the broadness of two relaxation processes becomes larger. Such behavior can be ascribed to the mixing effect of dynamically asymmetric components. Actually, the experimentally determined terminal relaxation in blends is an apparent one, which includes the contributions from SAN chains and PCL chains. In the simplest case, even if the width of chain relaxation of either component is the same as that in the pure polymer and does not change in the blends, the width of the apparent terminal relaxation can still increase at lower temperature. This is because of the intrinsic different temperature dependence of the mean chain relaxation times of SAN and PCL, especially when it is approaching the effective glass transition temperature. Such a mechanism is clearly justified by the predictions (hollow symbols in Figure 9) of the TDD-DR-SCCF model using this idea (for details, see section 4.3), which are consistent with the experimental data and illustrate the same trend. 4.2. Component Dependency of Relaxation Time at Tig,eff. It has been shown in Figure 3 that the segmental relaxation time of SAN in pure polymer and blends cannot be described by the WLF equation using the glass transition temperature from DSC. It is known that the glass transition from DSC actually covers a wide range of temperature (see the shadow region in Figure 1c). It may be explained by the fluctuation of the effective concentration. Kumar et al.7 combined the intrachain effect (self-concentration) with the interchain effect (concentration fluctuation). In contrast to the LM self-concentration model, the effective concentration of i segment (i = A or B) in such SCCF (self-concentration− concentration fluctuation) model7 is expressed as ϕeff, i = ϕself, i + (1 − ϕself, i)ϕi

(6)

The self-concentration ϕself,i is evaluated as 3νi/(2πbiRc2) with νi = lpibi2 the volume of a single i segment, lpi the packing length of the polymer, bi the Kuhn length of the polymer, and Rc the radius of the reference sphere. ϕi in eq 6 represents the intermolecular concentration experienced by a test segment. When it takes the mean blend composition Φi, the effective concentration ϕself,i becomes the mean effective concentration ϕ̅ self,i. The key idea to show the effect of concentration fluctuation on glass transition is the assumed correlation between the probability distribution function of the effective concentration around a segment and of the relaxation time experienced by that segment

Figure 9. Comparison of the width parameter for the chain relaxation with the segmental relaxation in pure SAN and SAN-dominant blends. The solid symbols are from experiments, and the corresponding hollow symbols for blends are from the TDD-DR-SCCF model.

p(ϕeff, i)dϕeff, i = pτ (ln τ ) d ln τ

(7)

By assuming a standard Gaussian form for the distribution of intermolecular concentration (see eqs S1−S4), it is possible to obtain the distribution function of relaxation time from eq 7. Therefore, the effective concentration corresponding to the most probable time (the time corresponding to the maximum in pτ(ln τ)d ln τ) and the mean relaxation time could be expressed as15

dielectric spectrum is only determined for SAN segment, the width of two processes is discussed only in pure SAN and SANdominant blends. First, the width of the terminal relaxation is always larger than that of the segmental relaxation both in pure SAN and in SAN/PCL blends. If the chain relaxation time can be scaled with the segmental relaxation time (Figure 8), the width of chain relaxation process should be at least equal to that G


Macromolecules ϕpeak, i − ϕeff, ̅ i = −⟨δϕeff 2⟩

ϕavg, i − ϕeff, ̅ i=

ij d ln τ yz d zz lnjjjj z dϕeff, i j dϕeff, i zz k {

ij d ln τ yz 1 d zz ⟨δϕeff 2⟩ lnjjjj z 2 dϕeff, i j dϕeff, i zz k {

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that such kind of correction cannot eliminate the influence of composition unless the composition dependent relaxation time at the reference temperature is considered in PCL/CAN blends. As shown in Figure 10, the segmental relaxation times

(8)

(9)

If ϕpeak,i is taken as the effective concentration of a segment, we can determine the effective glass transition temperature corresponding to the most probable segmental relaxation time from i ,peak i Tg,eff = Tg,eff |ϕeff,i = ϕpeak,i

(10)

where the dependence of segmental relaxation time on the effective concentration can be determined by using the WLF equation (eq 3) and the Fox equation (eq 2). Similarly, if ϕavg,i is taken as the effective concentration of a segment, we can determine the effective glass transition temperature corresponding to the average segmental relaxation time from i ,avg i Tg,eff = Tg,eff |ϕeff,i = ϕavg,i

(11)

Ti,peak g,eff

Figure 10. Dependence of the relative segmental relaxation time on . The dependence of vertical shift factor on the ΔT = T − TSAN,avg g,eff content of PCL is shown in the inset.

Ti,avg g,eff

The details of the calculation of and can be found in eqs S5−S6 and eqs S7−S8, respectively. When the effective concentration is taken as the mean effective concentration ϕ̅ eff,i, the effective glass transition temperature Tig,eff(ϕ̅ eff,i) can be directly determined from the Fox equation (eq 2), which is identical to the LM self-concentration model. In the SCCF model, the radius of the reference sphere Rc is determined by fitting the DSC glass transition temperature using Tig,eff(ϕ̅ eff,i). Comparisons among different effective glass transition temperatures are shown in Figure S6. It is clear that the effective glass transition temperatures corresponding to the different segmental relaxation times are quite different. Specifically, the effective glass transition temperature corresponding to the average segmental relaxation time is higher than Tig,eff(ϕ̅ eff,i), while that corresponding to the probable segmental relaxation time is lower than Tig,eff(ϕ̅ eff,i). Such a phenomenon is a natural result of concentration fluctuation, and it is not odd to see that the glass transition temperature from DSC (or Tig,eff(ϕ̅ eff,i)) may not be suitable to describe the universal dependence of relaxation time in pure polymers and polymer blends. However, even the effective glass transition temperatures from SCCF model are unable to give a satisfied universal dependence of relaxation time on ΔT = T − Ti,SCCF (see Figure g,eff S7). The failure of superposition can also be seen when the relaxation times (segmental and terminal) are plotted against the relative temperature, Ti,SCCF g,eff /T (see Figure S8). Furthermore, if TSAN g,eff is adopted as a fitting parameter to superpose the segmental relaxation time (Figure S9), the fitted glass transition temperature of SAN are much higher than TSAN,avg from the g,eff SCCF model (see Figure S6 for the comparison). More importantly, the obvious failure of superposition and deviation from the WLF fitting of pure SAN at large ΔT can be found. All these attempts indicate that shifting the segmental relaxation time along the ΔT axis using TSAN,avg only from the SCCF g,eff model is insufficient to get the master curve. Actually, our approaches to shift the segmental and terminal relaxation times by choosing proper reference temperature (Figures S7−S9) are numerically equivalent to the correction to the isofrictional state,3,4 although conceptually different. These efforts indicate

are shifted vertically to form a master curve after proper choice of the effective glass transition time from the SCCF model, which implies that the segmental relaxation time at the effective glass transition temperature may depend on the blend composition. It is seen that the shifting factor (b) is a parabolic function of the composition (inset of Figure 10). Such behavior has not been observed for nearly athermal blends and could be explained from the specific interaction between SAN and PCL.60 Actually, in a simple binary interaction model of homopolymer−copolymer mixtures, the heat of mixing is also a parabolic expression of the compositions of two components,60 where the binary interaction energy density is a function of the monomeric binary interaction parameters and the composition of copolymer. Addition of PCL to the random copolymer SAN dilutes the most unfavorable interactions between styrene and acrylonitrile, which leads to a net exothermic mixing condition. Because the interactions between CL and styrene as well as CL and acrylonitrile are mutual, and will affect the motion of both components, it is natural to expect that relaxation of both components in blends would be slower than those in pure polymers. Such an effect is ascribed to the higher friction coefficient of components in blends than that in pure polymer. Experimental results on dielectric spectrum support the slowdown relaxation of the SAN segment (Figure 3), while those on the viscoelastic spectrum support slowdown relaxation of both SAN chain and PCL chain in blends (Figure 6). The slowdown of segmental relaxation of PCL cannot be determined directly from dielectric experiments. However, from the linear relation between the segmental and chain relaxation time (Figure 8), it is expected that the segmental relaxation of PCL will also be slower in blends as that of chain relaxation. Therefore, the dynamics of both components are slowed down for segmental relaxation and chain relaxation. 4.3. Comparison with the TDD-DR-SCCF Model. In the above sections, we have discussed the character of chain dynamics and its relation to segmental dynamics in respect of H


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Macromolecules

Figure 11. Linear viscoelastic properties of pure SAN (a) and PCL (b) at various temperatures. The solid lines are fitting results of TDD-DR model with molecular weight distribution described by the generalized exponential function (GEX).

number of Kuhn segment between entanglement points), and τ0 is the relaxation time of Kuhn segment. The generalized exponential function (GEX)62−64 is used to describe the molecular weight distribution É ÄÅ a+1 ÅÅ i M yc ÑÑÑ ij M yz c j z Ñ Å wGEX(M ) = jj zz expÅÅÅ−jjj zzz ÑÑÑ ÅÅ j Ms z ÑÑ Γ((a + 1)/c) jjk Ms zz{ (15) ÇÅ k { ÖÑ

relaxation time and its broadness. Moreover, we have illustrated the important role of the effective concentration fluctuation and the composition dependent segmental relaxation time at Tig,eff. In this part, we will demonstrate if such understanding in segmental dynamics can be used to describe the linear viscoelasticity of dynamically asymmetric miscible blends. The time-dependent diffusion double reptation model (TDDDR)5 will be adopted to describe the rheological property of polydispersed entangled linear polymer blends, which has been established in our previous work5 to predict the linear viscoelasticity and concentration fluctuation for athermal dynamically asymmetric polymer blends. The mechanisms like the constraint release and/or the dynamic tube dilation will not be considered, although Haley et al.17 had shown that combination of the double reptation with tube dilation may improve the predictions on viscosity. This attempt was not made here because the TDD-DR model can give sufficiently satisfactory predictions. In the model, the relaxation modulus can be calculated by integrating the molecular weight distribution w(M) with proper “weight” function Fk(t,M)5,61 G(t ) = Greptation(t ) + G Rouse(t ) ÄÅ ÅÅ Gk (t ) = GN ÅÅÅÅ ÅÅÇ

ÉÑβK ÑÑ [FK (t , M )]1/ βK w(M ) d log M ÑÑÑÑ , ÑÑÖ log Me

∫

where Γ is the Gamma function and a, c, and Ms are model parameters. These parameters together with the Kuhn segmental relaxation time τ0 are adopted as adjusting parameters when fitting the experimental dynamic moduli. Figure 11 displays the experimental data and the best leastsquares fitting on the master curves of pure SAN and PCL at reference temperature (Tref = 150 °C). The agreement between the TDD-DR model and the experimental data for both polymers is very good. The fitted Mw is very close to those from GPC, but the polydispersity (Mw/Mn) is slightly higher (2.18 for PCL and 2.76 for SAN). In the TDD-DR model, we assume that the molecular weight distribution of blend can be expressed as a linear combination of two components5

(12)

∞

k = reptation or Rouse

SAN PCL wblend(M ) = w0wGEX (M ) + (1 − w0)wGEX (M )

with w0 the content of SAN in the blend. To account for the dynamic heterogeneity, the self-concentration model14 is introduced into the TDD-DR model5 and in short named as the TDD-DR-SC model. The idea is to express the Kuhn segmental relaxation time τ0(Φ,T) as functions of blend composition and temperature. Therefore, the reptation time for SAN in blend becomes65

(13)

where Greptation(t) and GRouse(t) denote the contribution from reptation and Rouse motion, respectively. GN is the plateau modulus, and Me is the average molecular weight between entanglements. The detailed definition of the kernel relaxation function FK(t,M) of monodisperse polymer with molecular weight M can be found elsewhere.5 The time scales for reptation motion (τw) and Rouse motion (τRouse) are 2

3

bN τw(M ) = τ0 2 , a

τw,SAN(Φ, T ) =

2

N τRouse(M ) = τ0 6

(16)

bSAN 2NSAN 3 [a(Φ)]2

τ0,SAN(Φ, T )

(17)

For the dependence of tube diameter a(Φ) on composition, the harmonic mixing rule suggested by Pathak et al.37 is adopted here, which gives a good description on the plateau modulus of blends (Figure S10). Through some straightforward calculations, we can have

(14)

where b is the Kuhn length, N is the number of Kuhn segment per chain (= M/Mk with Mk the molecular weight of Kuhn segment), a is the tube diameter (= Ne1/2b with Ne = Me/Mk the I


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Macromolecules

Figure 12. Comparison of the predictions by the TDD-DR-SCCF model (solid lines for G′ and dashed lines for G″) with the experimental data (solid symbols for G′ and hollow symbols for G″) for SAN/PCL 30/70 (a), 50/50 (b), and 70/30 (c) blends at different temperatures. The data are shifted vertically to avoid overlap.

τw,SAN(Φ, T ) =

bSAN 2NSAN 3 2

a ji zy × jjjΦ + SAN (1 − Φ)zzz j z aPCL k { aSAN

for f(Φ) (1 + 18.3Φ(1 − Φ)) are adopted for both components, although only that for SAN has been determined from experiments. Such an assumption indicates that the segmental relaxation time of both components in SAN/PCL blends slows down as the content of the other component increases. Comparisons between the predictions from TDD-DR-SCCF model and the experimental data are presented in Figure 12 for SAN/PCL blends over a wide range of temperatures. It is seen that the TDD-DR-SCCF model can give quite good predictions on the linear viscoelastic properties. Both the crossover frequency of G′ and G″ and the frequency dependency are well captured by the TDD-DR-SCCF model. The characteristic relaxation time and the broadness of the terminal relaxation are also extracted from the model predictions and compared with experimental data in Figures 6a and 7, respectively. The model predictions on the terminal relaxation time are well consistent with the experiments for all blends. We would stress that incorporation of the function f(Φ) is critical here. If such a function f(Φ) is taken as 1 (no interaction included), we found that the terminal relaxation of the model would be much faster than the experiments (Figure S11). The predicted broadness of terminal relaxation also follows the same trend as the experiments; i.e., the relaxation becomes broader as temperature decreases. It should be stressed here that broadening of the segmental relaxation is not included in the present model, and the broadening of terminal relaxation is merely ascribed to the enlarged difference in the segmental relaxation time of two components as the temperature approaches the effective glass transition temperature and the polydispersity of molecular weight distribution.

τ0,SAN(Φ = 1, Tr)f (Φ)g (T )

2

(18)

ÄÅ ÉÑ g ÅÅ C g (T − T SAN(Φ)) C1,SAN (Tr − Tg,SAN) ÑÑÑÑ ÅÅ 1,SAN g,eff Å ÑÑ + g g (T ) = expÅÅ− g SAN ÅÅ C 2,SAN + T − Tg,eff C 2,SAN + Tr − Tg,SAN ÑÑÑÑ (Φ) ÅÇ Ö

with

(19)

where the harmonic mixing rule for tube diameter (eq S10) is adopted, and Tr is the reference temperature to determine the Kuhn segmental relaxation time τ0,SAN(Φ = 1, Tr) of pure polymer by the TDD-DR model. A similar expression can be written down for PCL. When the self-concentration model is adopted, the effective glass transition temperature can be calculated from eqs 1−3. Two modifications will be made here. First, we have illustrated that concentration fluctuation will cause a distribution of segmental relaxation time, which results in different definitions of glass transition temperature corresponding to the different characteristic segmental relaxation times. Here, we suggest to replace the effective glass transition temperature Tig,eff(Φ) of LM SC model with the effective glass transition temperature Ti,avg g,eff corresponding to the mean segmental relaxation time from the SCCF model, and the model will be named as the TDDDR-SCCF model. In fact, more strict calculation concerning the concentration fluctuation may require the introduction of the probability distribution function of segmental relaxation into the TDD-DR model (eq 13). However, as will be shown below, our approach is sufficient to capture the temperature and composition dependence of the time and broadness of chain relaxation. Second, we introduced the coefficient f(Φ) in eq 18 to represent the composition dependency of the segmental relaxation time at the effective glass transition temperature. Such a dependency is found to be a parabolic function of composition in the SAN/PCL blend and has been shown clearly in the inset of Figure 10. Moreover, similar expressions

5. CONCLUSIONS The segmental relaxation and chain relaxation of PCL/SAN blends were investigated by the dielectric spectrum and rheology, respectively. A unified approach was adopted to extract the relaxation time and the broadness of the relaxation process from the HN model for both the dielectric spectrum and linear viscoelastic spectrum, which allows direct comparison of two relaxation processes. It is found that both the J


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(2) Chung, G. C.; Kornfield, J. A.; Smith, S. D. Component dynamics in miscible polymer blends: A two-dimensional deuteron NMR investigation. Macromolecules 1994, 27, 964−973. (3) Chung, G. C.; Kornfield, J. A.; Smith, S. D. Compositional dependence of segmental dynamics in a miscible polymer blend. Macromolecules 1994, 27, 5729−5741. (4) Chen, Q.; Matsumiya, Y.; Masubuchi, Y.; Watanabe, H.; Inoue, T. Component Dynamics in Polyisoprene/Poly(4-tert-butylstyrene) Miscible Blends. Macromolecules 2008, 41, 8694−8711. (5) Yu, W.; Zhou, C. Rheology of miscible polymer blends with viscoelastic asymmetry and concentration fluctuation. Polymer 2012, 53, 881−890. (6) Kamath, S. Y.; Colby, R. H.; Kumar, S. K. Evidence for dynamic heterogeneities in computer simulations of miscible polymer blends. Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 2003, 67, 108011−108014. (7) Kumar, S. K.; Shenogin, S.; Colby, R. H. Dynamics of miscible polymer blends: Role of concentration fluctuations on characteristic segmental relaxation times. Macromolecules 2007, 40, 5759−5766. (8) Liu, W.; Bedrov, D.; Kumar, S. K.; Veytsman, B.; Colby, R. H. Role of distributions of intramolecular concentrations on the dynamics of miscible polymer blends probed by molecular dynamics simulation. Phys. Rev. Lett. 2009, 103, 037801. (9) Watanabe, H.; Urakawa, O. Component dynamics in miscible polymer blends: A review of recent findings. Korea-Australia Rheol. J. 2009, 21, 235−244. (10) Adam, G.; Gibbs, J. H. On the temperature dependence of cooperative relaxation properties in glass-forming liquids. J. Chem. Phys. 1965, 43, 139−146. (11) Hoffmann, S.; Willner, L.; Richter, D.; Arbe, A.; Colmenero, J.; Farago, B. Origin of dynamic heterogeneities in miscible polymer blends: a quasielastic neutron scattering study. Phys. Rev. Lett. 2000, 85, 772−775. (12) Kant, R.; Kumar, S. K.; Colby, R. H. What length scales control the dynamics of miscible polymer blends? Macromolecules 2003, 36, 10087−10094. (13) Zawada, J. A.; Fuller, G. G.; Colby, R. H.; Fetters, L. J.; Roovers, J. Component dynamics in miscible blends of 1,4-polyisoprene and 1,2-polybutadiene. Macromolecules 1994, 27, 6861−6870. (14) Lodge, T. P.; McLeish, T. C. B. Self-concentrations and effective glass transition temperatures in polymer blends. Macromolecules 2000, 33, 5278−5284. (15) Shenogin, S.; Kant, R.; Colby, R. H.; Kumar, S. K. Dynamics of miscible polymer blends: Predicting the dielectric response. Macromolecules 2007, 40, 5767−5775. (16) Pathak, J. A.; Kumar, S. K.; Colby, R. H. Miscible polymer blend dynamics: Double reptation predictions of linear viscoelasticity in model blends of polyisoprene and poly(vinyl ethylene). Macromolecules 2004, 37, 6994−7000. (17) Haley, J. C.; Lodge, T. P. Viscosity predictions for model miscible polymer blends: Including self-concentration, double reptation, and tube dilation. J. Rheol. 2005, 49, 1277−1302. (18) Harmandaris, V.; Doxastakis, M. Molecular dynamics of polyisoprene/polystyrene oligomer blends: The role of self-concentration and fluctuations on blend dynamics. J. Chem. Phys. 2013, 139, 034904. (19) Haley, J. C.; Lodge, T. P.; He, Y.; Ediger, M. D.; von Meerwall, E. D.; Mijovic, J. Composition and temperature dependence of terminal and segmental dynamics in polyisoprene/poly(vinylethylene) blends. Macromolecules 2003, 36, 6142−6151. (20) des Cloizeaux, J. Double reptation vs. simple reptation in polymer melts. EPL (Europhys. Lett.) 1988, 5, 437−442. (21) des Cloizeaux, J. Relaxation and viscosity anomaly of melts made of long entangled polymers: time-dependent reptation. Macromolecules 1990, 23, 4678−4687. (22) Chiu, S. C.; Smith, T. G. Compatibility of poly(ε-caprolactone) (PCL) and poly(styrene-co-acrylonitrile) (SAN) blends. II. the influence of the AN content in SAN copolymer upon blend compatibility. J. Appl. Polym. Sci. 1984, 29, 1797−1814.

segmental relaxation time and the terminal relaxation time increase as temperature decreases, and two relaxation times are linearly correlated in blends. It is also found that the broadness of the segmental relaxation and the terminal relaxation increases as the temperature decreases, while the terminal relaxation becomes broader than the segmental relaxation as the temperature gets close to the effective glass transition temperature. We found that the LM self-concentration model can quantitatively describe the composition-dependent effective glass transition temperature from DSC; however, it underpredicts the segmental relaxation from the dielectric spectrum. Considering the possible distribution of segmental relaxation time, the effective glass transition temperature corresponding to the mean effective concentration may not be suitable in describing the segmental relaxation behavior. Then, we adopted the SCCF model to calculate the effective glass transition temperatures corresponding to the probable segmental relaxation time and the mean segmental relaxation time. With such revision, it is found that a composition dependent interaction parameter is necessary to account for the slowdown of the segmental relaxation time and the chain relaxation time in blends, which implies the segmental relaxation time at the effective glass transition is not the same as that in the pure polymer but depends on the surrounding environment of the segments. By combining such idea with the TDD-DR model, the linear viscoelastic properties of SAN/PCL blends can be well described over wide range of compositions and temperatures.

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ASSOCIATED CONTENT

* Supporting Information S

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.macromol.8b01214.

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Comparisons of dielectric spectrum, different definitions of chain relaxation times, various definitions of effective glass transition temperatures, segmental and terminal relaxations versus effective glass transition temperatures, plateau modulus, and TDD-DC-SCCF model prediction on dynamic moduli (PDF)

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] (W.Y.). ORCID

Wei Yu: 0000-0002-5615-9198 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work is supported by the National Natural Science Foundation of China (No. 21474063, 21790344, and 51625303). We also thanks CAS Interdisciplinary Innovation Team.

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REFERENCES

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Simultaneous Slowdown of Segmental and Terminal Relaxation of

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