Effect of Free-Volume Holes on Dynamic Mechanical Properties of

Article pubs.acs.org/Macromolecules

Effect of Free-Volume Holes on Dynamic Mechanical Properties of Epoxy Resins for Carbon-Fiber-Reinforced Polymers H. J. Zhang,*,† S. Sellaiyan,† T. Kakizaki,† A. Uedono,*,† Y. Taniguchi,‡ and K. Hayashi‡ †

Division of Applied Physics, Faculty of Pure and Applied Science, University of Tsukuba, Tsukuba, Ibaraki 305-8573, Japan Epoxy Resin Materials Center, Nippon Steel & Sumikin Chemical Co. Ltd., 11-5 Kitasode, Sodegaura, Chiba 299-0266, Japan

‡

ABSTRACT: Six types of matrices of carbon-fiber-reinforced polymers were prepared from different epoxies, amines, and thermoplastics at different curing conditions. Dynamic mechanical analysis measurements were performed to investigate the dynamic mechanical properties of storage modulus E′, loss modulus E″, damping factor tan δ, and complex viscosity |η*|. Positron annihilation lifetime (PAL) spectroscopy was carried out to evaluate each size and fraction of free-volume holes in the sample. The correlations between the dynamic mechanical properties and relative free-volume fraction were studied by using the Williams−Landel−Ferry equation. With increasing relative free-volume fraction, regular changes of dynamic mechanical properties are revealed: log[E′(T)] and log[|η*|(T)] decrease linearly in the temperature range of Tg(PAL) < T < Trub(E′) (Tg(PAL) is the glass transition temperature determined by PAL measurements; Trub(E′) is the lowest temperature where E′(T) curve coincides with its fitting line in the rubbery status stage) and then remain nearly unchanged at T > Trub(E′); log[E″(T)] and log[tan δ(T)] increase linearly at Tg(PAL) < T < Tg(tan δmax) (Tg(tan δmax) is the temperature of maximum tan δ) and then decrease linearly at T > Tg(tan δmax). In the present work, PAL spectroscopy is demonstrated to be a reliable experimental technique to provide precious quantitative information on free-volume holes in polymers.

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INTRODUCTION Carbon-fiber-reinforced polymers (CFRPs) have been widely used as structural materials of aerospaces, aeroplanes, transportation, and sports equipment due to the excellent mechanical properties and low density.1,2 The epoxy resins are usually used as the polymer matrices for CFRPs. In order to improve the mechanical properties of epoxy resin matrices, it is important to understand the effect of microstructure on dynamic mechanical properties. Positron annihilation spectroscopy has long been considered as a unique method to study the nanoscaled free-volume holes in polymers which are rather difficult for the traditional experimental techniques.3,4 In polymers, a positron could trap an electron prior to annihilation to form a hydrogen-like bound state which is usually called as a positronium (Ps) atom. A Ps atom exists in one of two spin states: p-Ps (spin-singlet parapositronium, |S,m⟩ = |0,0⟩) and o-Ps (spin-triplet orthopositronium, |S,m⟩ = |1,1⟩, |1,−1⟩, and |1,0⟩). A p-Ps atom soon annihilates into two γ-rays with a lifetime (the time interval between the producing of a positron and its annihilation) of 125 ps in a vacuum. The vacuum lifetime of o-Ps self-annihilation into three γ-rays is much longer than that of p-Ps atom. Therefore, in polymers the o-Ps atoms experience a large number of collisions in free-volume holes. During the collisions with the walls of free-volume holes, the positron of an o-Ps atom picks up an electron to annihilate into two γ-rays which is called o-Ps pick-off annihilation. In polymers, the lifetime of o-Ps pick-off annihilation is usually in the range from © XXXX American Chemical Society

1 to 10 ns. The o-Ps lifetime is determined by the size of the free-volume holes (free space between the polymer chains), and its intensity is proportional to the fraction of the free-volume holes. The correlations between the dynamic mechanical properties and free-volume holes were investigated by using the Williams−Landel−Ferry (WLF) equation for several thermoplastic polymers such as chlorinated isoprene−isobutylene rubber (CIIR), polycarbonate (PC), and multiwalled-carbonnanotube (MWCNT) modified PC.5−8 According to the literature, the WLF equation is applicable in the temperature range between Tg and Tg + 100 °C.9 But in these studies of thermoplastic polymers, the temperature ranges were very limited. In the study of CIIR, log[E′(T)] and log[E″(T)] were found to decrease linearly with increasing relative free-volume fraction in the temperature range between Tg and Tg + 80 °C.5 In the study of PC and MWCNT-modified PC, log[tan δ(T)] increased linearly with increasing relative free-volume fraction in the temperature range between Tg and Tg + 30 °C.6 In another study of PC and MWCNT-modified PC, a linear decrease of log[|η*|(T)] with increasing relative free-volume fraction was found in the temperature range between Tg and Tg + 30 °C.7 As far as we know, whether these monotonous changes of dynamic mechanical properties could be valid for Received: March 3, 2017 Revised: April 20, 2017

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DOI: 10.1021/acs.macromol.7b00472 Macromolecules XXXX, XXX, XXX−XXX

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Figure 1. Chemical structures of the epoxies, thermoplastics, and curing agents: (a) diglycidyl ether of bisphenol A (DGEBA), (b) bis(paminocyclohexyl)methane (PACM), (c) tetraglycidyl diaminodiphenylmethane (TGDDM), (d) epoxy phenolnovolac (EPN), (e) triglycidyl paminophenol (TGAP), (f) poly(ether sulfone) (PES), (g) diaminodiphenyl sulfone (DDS), (h) dicyandiamide (DCD), (i) dichlorophenyldimethylurea (DCMU), (j) diglycidyl ether of bisphenol F/diglycidyl ether of bisphenol A (DGEBF/DGEBA), (k) diglycidyl ether of bisphenol F (DGEBF), and (l) tris(dimethylamino)phenol (TDAP).

Table 1. Constituents and Curing Condition of Each Sample curing condition sample

epoxies (wt %)

amines (wt %)

E1 E2

DGEBA liquid (76) TGDDM (30) EPN (18) TGAP (12) TGDDM (30) EPN (18) TGAP (12) DGEBA liquid (25) DGEBA solid (25) EPN (30) DGEBA liquid (25) DGEBA solid (25) EPN (30) DGEBA liquid (45) DGEBF liquid (30)

PACM (24) DDS (28)

E2P

E3

E3P

E4

thermoplastic (wt %)

DDS (28)

PES (12)

DCD (4) DCMU (3) DCD (4) DCMU (3)

DGEBF/DGEBA (13)

second step

100 °C (1 h) 160 °C (1 h)

150 °C (3 h) 200 °C (3 h)

160 °C (1 h)

200 °C (3 h)

100 °C (1 h)

140 °C (3 h)

100 °C (1 h)

140 °C (3 h)

130 °C (5 min)

PACM (20) TDAP (5)

constituents of each sample was carried out in a planetary centrifugal mixer (THINKY ARV-310). The degassing was performed during the mixing process by setting the vacuum of the mixer to 4 kPa. The mixed composite was poured into a mold and then cured in air at the curing condition in Table 1. The prepared epoxy resins are denoted as E1, E2, E2P, E3, E3P, and E4, respectively. In these six epoxy resins, E1 was prepared from only one type of epoxy and one type of amine, while the other epoxy resins were synthesized from at least four constituents. To study the effect of additive of thermoplastic into amine-cured epoxy resin, thermoplastics were added into E2 and E3. The resulted polymers were denoted as

epoxy resins in a wider temperature range is still unknown. In this study, we made an attempt to study six different types of epoxy resins in a wider temperature range and gained a further understanding on the relationship between dynamic mechanical properties and free-volume holes.

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first step

EXPERIMENTAL SECTION

Samples Preparation. The chemical structures of all epoxies, amines, and thermoplastics are shown in Figure 1. The constituents were mixed at the weight ratio as shown in Table 1. The mixing of the B


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Macromolecules E2P and E3P, respectively. The curing time of E4 was much shorter than the other five epoxy resins. PAL Measurements. The PAL spectra were recorded by a conventional fast−fast coincidence system with a time resolution of ∼210 ps. Each positron source was prepared by depositing and drying of aqueous 22NaCl (activity of ∼500 kBq) onto the central zone (diameter of less than 2 mm) of a Kapton polyimide foil (10 mm × 10 mm × 7.5 μm, Nilaco) and then covering by another Kapton polyimide foil of the same size. The positron source was sandwiched between two identical samples with a dimension of 10 mm × 10 mm × 1 mm. The sample−source−sample set was sealed in a vacuum chamber which was evacuated by a rotary pump and a turbo molecular pump. To prevent the backscattering of γ-rays by the PAL scintillators, the two PAL detectors were perpendicularly positioned. The distance between sample−source−sample set and each lifetime detector was ∼20 mm. The measurements were performed under vacuum by heating the sample−source−sample set from 29.0 to 206.3 °C. The vacuum inside the sample chamber was better than 1 × 10−6 Pa. Each spectrum was collected for 4096 channels with a channel width of 10.0 ps/channel. The total count of each PAL spectrum is 1 × 106, with a counting rate of around 50 cps. For each sample, 45 PAL spectra were collected in approximately 10 days. TMA Measurements. The TMA measurements were carried out by using a thermal mechanical analyzer (TMA6100, SII Nanotechnology Inc.). The measurements were performed in a compression mode in a nitrogen atmosphere. A force of 9.8 mN was applied to each cylinder-shaped sample (around ϕ 5 mm × 4 mm). The first heating scan was performed in the temperature range from room temperature to 310 °C at a heating rate of 10 °C/min. After cooling to room temperature, the second heating scan was performed from room temperature to ∼250 °C at a heating rate of 5 °C/min. DMA Measurements. The dynamic mechanical properties were investigated by using a dynamic mechanical analyzer (DMA6100, SII Nanotechnology Inc.). Bar-shaped samples in the size of 20 mm × 10 mm × 4 mm (length × width × thickness) were tested in dualcantilever flexural mode at a frequency of 10 Hz. The samples were heated from room temperature up to ∼280 °C in a nitrogen atmosphere at a heating rate of 2 °C/min. The storage modulus E′, loss modulus E″, and damping factor tan δ were collected.

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Figure 2. Temperature dependences of (a) o-Ps lifetime, (b) o-Ps intensity, and (c) relative fractional free volume of each sample. The arrow denotes Tg(PAL) which corresponds to the intersection point of the fitting lines of o-Ps lifetime (or free-volume hole size) of each sample.

RESULTS AND DISCUSSION PAL Measurements. By using the PATFIT program, all PAL spectra were decomposed into three lifetime components of τ1, τ2, and τ3 (τ1 < τ2 < τ3), with the corresponding intensities of I1, I2, and I3, respectively.10 Among the three lifetime components, the longest component corresponds to the pick-off annihilation of o-Ps in the free-volume holes. The temperature dependence of o-Ps lifetime (τ3) of each sample is shown in Figure 2a. The intersection point of the two fitting lines in the temperature ranges of glassy state and rubbery state (it must be mentioned here, dτ3/dT in the rubbery state is higher than that in the glassy state) is the glass transition temperature determined by PAL measurements (Tg(PAL)). For both E2 and E2P, no clear increase of dτ3/dT could be found. This indicates that the glass transition temperatures (Tg) of E2 and E2P are not in the temperature range (from 29.0 to 206.3 °C) of PAL measurements. For E1, E3, and E3P, the temperature dependence of o-Ps lifetime (τ3) could be well fitted by two lines. While for E4, two intersection points are given by the three fitting lines. The first intersection point at ∼108 °C is its Tg(PAL), and the second intersection point is at around 160 °C. The different variation of τ3 of E4 is most probably due to its insufficient curing process (130 °C for 5 min) before PAL measurement.11 The curing time of E4 (5 min) was much shorter than the other five samples (4 h). The temperature dependence of o-Ps intensity (I3) of each sample is shown in Figure 2b. Among the six samples, the o-Ps

intensities of E3 and E3P remain nearly unchanged with increasing temperature. For E1, E2, and E2P, the o-Ps intensity initially increases in the temperature range between 29.0 and ∼170 °C and then decreases gradually. For E4 which is not well-cured (cured at 130 °C for only 5 min) prior to PAL measurements, in the temperature range between 120 and 140 °C (higher than its Tg of 108 °C), I3 drops abruptly. The dropping and quick rising of I3 of E4 in the temperature range between 120 and 160 °C is probably caused by the enhancement of three-dimensional cross-linking due to the heating of sample in PAL measurement.11 By using a semiempirical equation based on a spherical infinite potential well model (Tao−Eldrup model),12 the average radius (R) of free-volume holes could be estimated from the o-Ps lifetime τ3: ⎡ ⎛ R 1 R ⎞⎤ ⎟ sin⎜2π + τ3−1 = 2⎢1 − ⎝ R + ΔR ⎠⎥⎦ ⎣ R + ΔR 2π

(1)

where ΔR is the thickness of the electron layer on the surface of free-volume holes, which is an empirical parameter of 0.1656 nm.13 The average volume (Vf) of free-volume holes, which is usually called as free-volume hole size, could be estimated from C


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Figure 3. Temperature dependences of TMA dimension change (shown in red circles) and free-volume hole size (shown in black squares) for (a) E1, (b) E2, (c) E2P, (d) E3, (e) E3P, and (f) E4. The black and red vertical double-headed lines correspond to Tg(PAL) and Tg(TMA), respectively.

Figure 4. Temperature dependences of storage modulus (shown in red circles) and free-volume hole size (shown in black squares) for (a) E1, (b) E2, (c) E2P, (d) E3, (e) E3P, and (f) E4. The black, red, and green vertical double-headed arrows correspond to Tg(PAL), Tg(E′onset), and Tg(E′offline), respectively. The violet single-headed arrow denotes the temperature Trub(E′) of each sample.

Vf = 4πR3/3

each sample, except E3, the black double-headed arrow is very close to the red double-headed arrow. As shown in Figure 3f, at the temperature of the second transition point of Vf (∼160 °C), no clear transition was found in the TMA curve of E4. This is probably due to the significant difference in the measuring time of TMA and PAL experiments. For the PAL measurements of each sample, 45 PAL spectra were collected in ∼10 days in the temperature range from 29.0 to 206.3 °C. Therefore, the unsufficiently cured sample E4 has the chance to be cured in PAL measurements. But for the TMA experiments, the second heating scan (from 30 to ∼250 °C at a heating rate of 5 °C/min) was completed in ∼44 min. Therefore, in this short measuring time, the curing degree of E4 was not enhanced significantly to be clearly observed by its TMA curve. Tg from DMA Measurements. From DMA data, the glass transition temperatures (Tg) are traditionally determined by the ′ ), maximum loss modulus onset of storage modulus drop (Eonset ″ ), and maximum damping factor (tan δmax).16 Among the (Emax three types of Tg, Tg(tan δmax) is the most widely accepted. The

(2)

The fractional free volume (in %) is defined as f = CVfI3, where C is a constant.14 For the epoxy resins with different chemical structures, the constant C must be different. To evaluate the relative change of free-volume holes due to the temperature for each epoxy resin, it is sufficient to utilize a simpler definition of relative fractional free volume:5−8 fr = Vf I3

(3)

The temperature dependence of relative fractional free volume (f r) of each sample is shown in Figure 2c. Tg from TMA Measurements. The TMA dimension changes as a function of temperature derived from the second heating scans of all six samples are shown in Figure 3. To compare with Tg(PAL), the temperature dependence of Vf is also plotted for each sample. Two tangent lines are drawn in a conventional way along discontinuities in the TMA curves.15 The Tg determined by the PAL technique and TMA are denoted by black and red vertical double-headed arrows, respectively. At the first glance of Figure 3, we notice that for D


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Figure 5. Temperature dependences of loss modulus (shown in red circles) and free-volume hole size (shown in black squares) for (a) E1, (b) E2, (c) E2P, (d) E3, (e) E3P, and (f) E4. The black and red vertical double-headed arrows correspond to Tg(PAL) and Tg(Emax ″ ), respectively.

Figure 6. Temperature dependences of tan δ (shown in red circles) and free-volume hole size (shown in black squares) for (a) E1, (b) E2, (c) E2P, (d) E3, (e) E3P, and (f) E4. The black, red, and green vertical double-headed lines correspond to Tg(PAL), Tg(tan δmax), and Tg(tan δonline), respectively.

Table 2. Glass Transition Temperatures (Tg) and Trub(E′) Determined by Different Measurements and Analyses sample Tg(PAL) (°C) Tg(TMA) (°C) Tg(Eoffline ′ ) (°C)

E1 160 ± 8 161 ± 8 156 ± 8

E2 231 ± 8 226 ± 8

E2P

E3

E3P

E4

229 ± 8 216 ± 8

139 ± 8 106 ± 8 137 ± 8

123 ± 8 120 ± 8 119 ± 8

108 ± 8 102 ± 8 110 ± 8

Tg(tan δonline) (°C)

155 ± 8

236 ± 8

220 ± 8

140 ± 8

125 ± 8

110 ± 8

Tg(E′onset) (°C)

164 ± 5

250 ± 5

243 ± 5

149 ± 5

138 ± 5

117 ± 5

Tg(Emax ″ ) (°C)

170 ± 1

257 ± 1

249 ± 1

156 ± 1

143 ± 1

122 ± 1

Tg(tan δmax) (°C)

181 ± 1

264 ± 1

258 ± 1

164 ± 1

157 ± 1

132 ± 1

Trub(E′) (°C)

195 ± 2

182 ± 2

176 ± 2

146 ± 2

E′ starts to deviate from the fitting tangent line of the slight decrease stage (corresponds to the glassy state). We denote the corresponding temperature of this deviation point as Tg(Eoffline ′ ). It is easy to observe in Figure 4, rather than Tg(Eonset ′ ), Tg(Eoffline ′ ) is closer to Tg(PAL). The temperature dependences of loss modulus E″ and damping factor tan δ of all samples are shown in Figures 5 and 6, respectively. From Figure 5 we notice that the Tg(Emax ″ ) (shown

temperature dependence of storage modulus E′ of all samples are shown in Figure 4. The Tg(Eonset ′ ) of each sample is determined by plotting two tangent lines in the slight decrease stage (in glassy state) and drastic decrease stage (change from glassy state to rubbery state). To compare with Tg(PAL), the temperature dependence of Vf is also plotted for each sample. It is clear that for E3, E3P, and E4, the Tg(PAL) is about 10−20 °C lower than Tg(Eonset ′ ). Interestingly, at the temperature of Tg(PAL), E


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Macromolecules as the red vertical line) is around 10−20 °C higher than that of Tg(PAL) (shown as the black vertical line). In Figure 6, we can find Tg(tan δmax) (shown as the red vertical line) of each sample is around 20−30 °C higher than Tg(PAL) (shown as the black vertical line). We plot a tangent line in the left side of tan δ peak for each sample in Figure 6. Then it is easy to find the lowest temperature where the left half tan δ peak coincides with its tangent line, which is denoted as Tg(tan δonline) (indicated by the green vertical line), is very close to Tg(PAL) for each sample. It is well-known that Tg(PAL) is the onset temperature of main-chain movement.3,4 Therefore, this suggests that at the onset temperature of main-chain movement the damping factor tan δ starts to increase drastically. Correlation between Tg and Free-Volume Holes. In Table 2, seven types of glass transition temperatures derived from PAL, TMA, and DMA techniques are listed. It is obvious that Tg(TMA) of all samples except E3 are very close to Tg(PAL). Instead of the three traditional glass transition temperatures (Tg(Eonset ′ ), Tg(Emax ″ ), and Tg(tan δmax)) from DMA data, Tg(Eoffline ′ ) and Tg(tan δonline) are closer to Tg(PAL). This indicates that for aminecured epoxy resins Tg(PAL) is closely correlated with the dynamic mechanical properties. The PAL spectroscopy is demonstrated to be a reliable experimental method to characterize the structure transition of polymers. Table 2 and Figure 2 reveal that the six epoxy resins have different Tg and different temperature dependences of Vf and f r. Thereafter, we could make a preliminary attempt to correlate Tg and the changes of free-volume holes. The change of freevolume holes with increasing temperature, may be roughly expressed by Vf (206.3 °C)/Vf(29.0 °C) (change of Vf from the lowest to the highest measuring temperature of PAL measurements) and f r(206.3 °C)/f r(29.0 °C) (change of f r from the lowest to the highest measuring temperature of PAL measurements). In Figure 7, the variations of four types of Tg (Tg(PAL), Tg(TMA), Tg(Eoffline ′ ), and Tg(tan δonline)) as a function of Vf change and f r change are plotted. Surprisingly, the variations of Tg vs both Vf and f r changes could be roughly fitted by a single line. This linear relationship may not be applicable to all epoxy resins. But at least, this indicates that the epoxy resin which has a higher Tg, tends to exhibit smaller changes of Vf and f r with increasing temperature. Effect of Free-Volume Holes on Storage Modulus. The storage modulus E′ is a parameter to characterize the elastic response of a material. In the temperature range from Tg to Tg + 100 °C, the temperature dependence of E′ could be well described by the Williams−Landel−Ferry (WLF) equation: log

C1′(T − Tr) E′(T ) = E′(Tr) C2 + (T − Tr)

Figure 7. Variations of Tg as a function of (a) Vf(206.3 °C)/Vf(29.0 °C) and (b) f r(206.3 °C)/f r(29.0 °C) for all samples. The black lines are plotted to guide the eyes.

and E4. For the discussion of each sample, the lowest PAL measuring temperature (of the 45 PAL spectra) which satisfies Tr ⩾Tg(PAL) is selected as the reference temperature. The selected Tr of E1, E3, E3P, and E4 are 164.0, 139.4, 127.3, and 111.3 °C, respectively. Figure 8 shows the variation of log[E′(T)] as a function of relative free-volume fraction (1 − f rr/f r) for E1, E3, E3P, and E4. It is obvious that for all four samples the log[E′(T)] curve could be divided into two distinct stages. With increasing relative free-volume fraction (1 − f rr/f r), log[E′(T)] first decreases linearly and then keeps as a constant. In Figure 4, a tangent fitting line is plotted in the rubbery state stage of each sample. The lowest temperature where E′(T) curve coincides with the tangent line of E′(T) in the rubbery state stage, is denoted as Trub(E′). The temperature Trub(E′) of each sample is indicated by a violet vertical line in Figure 8. Therefore, a simple conclusion could be made on the variation of log[E′(T)] with increasing (1 − f rr/f r): decreases linearly in the temperature range between Tg(PAL) and Trub(E′) and remains nearly unchanged in the temperatures above Trub(E′). Effect of Free-Volume Holes on Loss Modulus and Damping Factor. The loss modulus E″, which describes the viscous property, is correlated with the energy loss in the oscillation. Damping factor is the delayed response between E′ and E″:

(4)

where C′1 and C2 are constants and Tr is the reference temperature (Tr ⩾Tg(PAL)).9 Since C2 = frr /αf

(5)

we have log[E′(T )] = log[E′(Tr)] + C1′(1 − frr /fr )

(6)

tan[δ(T )] =

where f rr is the relative fractional free volume ( f r) at Tr and αf is the thermal expansion coefficient of free-volume holes.5−7,17 Because the Tg of E2 and E2P are higher than 206.3 °C, the discussion of the WLF equation is only valid for E1, E3, E3P,

E ″ (T ) E′(T )

(7)

According to the WLF equation, in the temperature range between Tg and Tg + 100 °C, the temperature dependence of E″ is expressed as9 F


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Figure 9. Variation of log[E″(T)] as a function of relative free-volume fraction (1 − f rr/f r) for (a) E1, (b) E3, (c) E3P, and (d) E4. The red vertical line denotes the Tg(tan δmax) of each sample.

Figure 8. Variation of log[E′(T)] as a function of relative free-volume fraction (1 − f rr/f r) for (a) E1, (b) E3, (c) E3P, and (d) E4. The violet vertical line denotes the Trub(E′) of each sample.

log

C1″(T − Tr) E ″ (T ) = E″(Tr) C2 + (T − Tr)

(8)

where C1″ is a constant. From eqs 5 and 8, we obtain log[E″(T )] = log[E″(Tr)] + C1″(1 − frr /fr )

(9)

From eqs 6−9, we have log[tan δ(T )] = log[tan δ(Tr)] + C1d(1 − frr /fr )

(10)

where Cd1 equates to C1″ − C1′ .5−7 The variation of log[E″(T)] as a function of relative freevolume fraction (1 − f rr/f r) for E1, E3, E3P, and E4 is shown in Figure 9. In this Figure, Tg(tan δmax) of each sample is indicated by a red vertical line. In the temperature range between Tg(PAL) and Tg(tan δmax), log[E″(T)] increases linearly with increasing (1 − f rr/f r). While at the temperatures above Tg(tan δmax), log[E″(T)] decreases linearly with increasing (1 − f rr/f r). From Figure 9 we can clearly observe that at the temperatures above Tg(tan δmax) the log[E″(T)] change of E4 is different from the other three samples. For the other three samples the decrease of log[E″(T)] could be well fitted by a single line, while for E4 the decrease of log[E″(T)] should be fitted by two lines which gives an intersection temperature of 159.8 °C (the second transition temperature of Vf and f r). The different log[E″(T)] variation of E4 is induced by the aforementioned curing effect of the PAL measurement. Damping factor tan δ is a material property that indicates the degree of vibration energy loss to a system. By using the tan δ values of DMA measurement and f r values of PAL results, the relationship between log[tan δ(T)] and (1 − f rr/f r) of each sample is calculated and shown in Figure 10. In this figure, Tg(tan δmax) of each sample is also indicated by a red vertical line. We can clearly observe that for E4 the log[tan δ(T)] variation could be divided into three stages, while for the other three

Figure 10. Variation of log[tan δ(T)] as a function of relative freevolume fraction (1 − f rr/f r) for (a) E1, (b) E3, (c) E3P, and (d) E4. The red vertical line indicates Tg(tan δmax) of each sample.

samples the variation could be divided into two stages. The different log[tan δ(T)] variation of E4 is also induced by the aforementioned curing effect of the PAL measurement. In the temperature range between Tg(PAL) and Tg(tan δmax), log[tan δ(T)] increases linearly with increasing (1 − f rr/f r). While at the temperatures above Tg(tan δmax), log[tan δ(T)] decreases linearly with increasing (1 − f rr/f r). The variations of log[E″(T)] and log[tan δ(T)] with increasing (1 − f rr/f r) must be induced by the change of movement of the chains. Below the temperature of Tg(PAL), the G


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not be applicable to a wider temperature range between Tg and Tg + 100 °C.7 Correlation between C1 Constants and Cross-Linking Density. The variations of log[E′(T)], log[E″(T)], log[tan δ(T)], and log[|η*|(T)] with increasing relative free-volume fraction (1 − f rr/f r) are discussed by using the WLF equation. The constants of C1′ , C1″, Cd1, and C1*, which are calculated from the slopes of the least-squares fitting lines in Figures 8−11, are listed in Table 3. As already discussed in Figures 9 and 10, the different fittings (by two straight lines) of log[E″(T)] and log[tan δ(T)] variations of E4 in the temperature range of T > Tg(tan δmax) are caused by the significant change in curing degree during the PAL experiments. Therefore, for E4, the slopes of fitting lines of log[E″(T)] (T > Tg(tan δmax)), log[tan δ(T)] (T > Tg(tan δmax)) are calculated only in the temperature range of Tg(tan δmax) < T ⩽ 159.8 °C. The different slopes of fitting lines indicate that for different amine-cured epoxy resins the effect of free-volume holes on the four dynamic mechanical parameters (log[E′(T)], log[E″(T)], log[tan δ(T)], and log[|η*|(T)]) are quantitatively different. To reveal the correlation between the four C1 constants and their microstructure, we could study the cross-linking density (Lc) of each sample which is usually considered as the most important microstructure parameter of epoxy/amine systems. According to the rubbery elasticity theory, the cross-linking density could be derived from the storage modulus E′ of DMA measurements:

movement of main chains is restricted, only side-chain movement is allowed. At the temperature of Tg(PAL), the main chains start to move. In the temperature range from Tg(PAL) to Tg(tan δmax), due to the increase of damping factor with increasing temperature, the enhancement of movement of main chains results in the increase of internal friction energy dissipation between chains. While at the temperatures above Tg(tan δmax), with increase of f r due to increasing temperature, the enhancement of segmental chain movements induces the decrease of internal friction energy dissipation which eventually results in the decreases of E″ and tan δ. Effect of Free-Volume Holes on Complex Viscosity. Complex viscosity is a function of E′ and E″ in the form of |η*| = (E′ 2 + E″ 2)1/2 /ω

(11)

where ω is the frequency (10 Hz) of DMA measurements. According to the WLF equation, we have log

C1*(T − Tr) |η*|(T ) = |η*|(Tr) C2 + (T − Tr)

(12)

7

From eqs 5 and 12

log[|η*|(T )] = log[|η*|(Tr)] + C1*(1 − frr /fr )

(13)

The variation of log[|η*|(T)] as a function of (1 − f rr/f r) of the four samples are shown in Figure 11. The temperature Trub(E′)

′ /(ϕRT ) Lc = Erub

(14)

where E′rub is the E′ at the temperature of Tg(tan δmax) + 30 °C, ϕ is the front factor (equates approximately to 1 according to Flory theory), R is the gas constant, and T is the absolute temperature of Tg(tan δmax) + 30 °C.18−20 In Table 3, the E′ at Tg(tan δmax) + 30 °C and the deduced Lc of each sample are listed. The variations of C″1 (T > Tg(tan δmax)) and Cd1 (T > Tg(tan δmax)) as a function of Lc are shown in Figure 12. It is clear that both the magnitudes of C″1 (T > Tg(tan δmax)) and Cd1 (T > Tg(tan δmax)) increase with increasing Lc. This indicates that for the amine-cured epoxy resin which has a higher Lc, the loss modulus E″ and damping factor tan δ are more heavily influenced by the free-volume holes.

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CONCLUSIONS In this study, six types of amine-cured epoxy resins, which were prepared from different epoxies, amines, and thermoplastics at different curing conditions, were designed as the CFRP matrices. The dynamic mechanical properties of these six matrices were investigated by DMA measurements. The temperature dependences of free-volume hole size (Vf) and relative fractional free volume (f r) of each sample were revealed by PAL measurements. The glass transition temperature which was determined by PAL technique, Tg(PAL), was compared with those determined by TMA and DMA measurements. Rather than the three traditional Tg of DMA measurements (Tg(Eonset ′ ), Tg(Emax ″ ), and Tg(tan δmax)), the values of Tg(E′offline) and Tg(tan δonline) are closer to Tg(PAL). The variation of Tg was found to decrease with increasing Vf(206.3 °C)/Vf(29.0 °C) and f r(206.3 °C)/f r(29.0 °C). This indicates that with increase in temperature from 29.0

Figure 11. Variation of log[|η*|(T)] as a function of relative freevolume fraction (1 − f rr/f r) for (a) E1, (b) E3, (c) E3P, and (d) E4. The violet vertical line denotes Trub(E′) of each sample.

of each sample is indicated by a violet vertical line. It is obvious that all the log[|η*|(T)] curves could be divided into two distinct stages with a transition temperature of Trub(E′). We could summarize the log[|η*|(T)] variation with increasing (1 − f rr/f r): decreases linearly in the temperature range between Tg(PAL) and Trub(E′) and then remains nearly unchanged at the temperatures above Trub(E′). The linear anticorrelation between log[|η*|(T)] and (1 − f rr/f r) for PC and MWCNT-modified PC in the temperature range between Tg and Tg + 30 °C could H


Article

Macromolecules

Table 3. C1 Constants Derived from the Slopes of the Least-Squares Fitting Lines in Figures 8−11 and the Cross-Linking Density Lc Derived from the Storage Modulus E′a sample C1′ (T < Trub(E′)) C1′ (T > Trub(E′)) C″1 (T < Tg(tan δmax))

E3

E3P

E4

−8.78 ± 0.30 −0.12 ± 0.03 3.18 ± 2.11

−7.35 ± 0.39 −0.11 ± 0.06 0.75 ± 0.79

−10.01 ± 0.78 0.05 ± 0.04 0.64 ± 1.25

C1″ (T > Tg(tan δmax))

−26.56 ± 1.90

−17.90 ± 1.11

−16.48 ± 0.74

−10.21 ± 0.46

Cd1 (T < Tg(tan δmax))

6.55 ± 0.54

11.67 ± 1.98

5.92 ± 0.44

8.38 ± 0.89

Cd1 (T > Tg(tan δmax))

−26.47 ± 2.05

−15.44 ± 1.12

−17.03 ± 0.99

−8.09 ± 0.53

C*1 (T < Trub(E′)) C*1 (T > Trub(E′)) E′ at Tg(tan δmax) + 30 °C (MPa)

−11.68 ± 0.80 0.50 ± 0.28 47.56

−8.66 ± 0.30 −0.13 ± 0.04 37.24

−7.31 ± 0.37 −0.12 ± 0.06 25.80

−9.73 ± 0.63 0.01 ± 0.04 19.80

11.83

9.60

6.75

5.47

Lc (mol/dm3) a

E1 −11.74 ± 0.74 0.61 ± 0.17 0.21 ± 0.71

For E4, C1″ (T > Tg(tan δmax)) and

Cd1

(T > Tg(tan δmax)) are calculated from the fitting lines in the temperature range of Tg(tan δmax) < T ⩽ 159.8 °C.

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected] (H.J.Z.). *E-mail: [email protected] (A.U.) Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was financially supported by Cross-Ministerial Strategic Innovation Promotion Program - Unit D66 Innovative Measurement and Analysis for Structural Materials (SIP-IMASM) operated by the Cabinet Office of Japan.

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Figure 12. Variations of C″1 (T > Tg(tan δmax)) and Cd1 (T > Tg(tan δmax)) as a function of cross-linking density Lc. The solid line is plotted to guide the eyes.

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to 206.3 °C, the sample which has a higher glass transition temperature, tends to exhibit a smaller change in both Vf and f r. By using the WLF equation, the dynamic mechanical properties were observed to change regularly with increasing relative free-volume fraction (1 − f rr/f r): The values of log[E′(T)] and log[|η*|(T)] initially decrease linearly in the temperature range between Tg(PAL) and Trub(E′) and then remain nearly unchanged at the temperatures above Trub(E′). The values of log[E″(T)] and log[tan δ(T)] initially increase linearly in the temperature range between Tg(PAL) and Tg(tan δmax) and then decrease linearly at the temperatures above Tg(tan δmax). The constants of C′1, C″1 , Cd1, and C*1 were derived from the slopes of the fitting lines in Figures 8−11. Additionally, the cross-linking density Lc of each epoxy resin was calculated from the storage modulus E′ by using the rubbery elasticity theory. Thereafter, the magnitudes of both C″1 (T > Tg(tan δmax)) and Cd1 (T > Tg(tan δmax)) were found to increase with increasing Lc. This indicates that for the amine-cured epoxy resin which has a higher Lc, the loss modulus E″ and damping factor tan δ are more heavily influenced by the free-volume holes. The PAL spectroscopy is therein demonstrated to be a reliable experimental technique to provide precious quantitative information on nanoscaled free-volume holes in polymers. I


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Effect of Free-Volume Holes on Dynamic Mechanical Properties of

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