Nanoindentation Study of Yielding and Plasticity of Poly(methyl

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Nanoindentation Study of Yielding and Plasticity of Poly(methyl methacrylate) Leila Malekmotiei, Aref Samadi-Dooki, and George Z. Voyiadjis* Computational Solid Mechanics Laboratory, Department of Civil and Environmental Engineering, Louisiana State University, Baton Rouge, Louisiana 70803, United States ABSTRACT: In this study, an experimental approach is used to characterize the geometrical and micromechanical properties of the shear transformation zones (STZs) in glassy polymers. Nanoindentation experiments have been conducted on both ascast and annealed poly(methyl methacrylate) (PMMA) at different strain rates and temperatures, utilizing continuous stiffness measurement (CSM) technique at room temperature indentations, and conventional loading rate control method for nanoindentations at elevated temperatures. Employing a homogeneous flow theory for analyzing the experimental data, the geometrical properties of the STZs are observed to be almost independent of the thermal history of the samples. While the transformation shear strain of the STZ in PMMA is found to be slightly smaller than that in glassy metals, the size of the STZ in this polymer is shown to be at least 10 times of that in metallic glasses. On the other hand, the activation energy of a single STZ is found to change drastically with annealing. In addition, analysis of the rate sensitivity of the shear flow stress reveals a remarkable transition at a certain strain rate which is believed to pertain to the β-transition. This transition is well-matched with a jump in the STZ activation energy at the same strain rate range; hence, the jump is referred to as the β-transition activation energy, which is found to be about 10% of the STZ nucleation energy for PMMA.

1. INTRODUCTION The broad use of polymers as transparent, low density, and high impact resistant materials has drawn the interest of researchers to investigate the mechanical properties of this class of materials for a long time. In general, polymers are composed of disordered and entangled long molecular chains. Amorphous (glassy) polymers are one of the main groups of polymeric materials that show no considerable chain alignment in their intra- and intermolecular structures. In the excessive loading condition, the polymeric glasses (PGs) undergo plastic deformation which is basically different from that in the crystalline solids. The post yield behavior of the PGs starts with a softening at the onset of the yielding, accompanied by a hardening (due to the chain alignment) which continues to the break point.1−3 Because of the absence of the long-range structural order in glassy solids, their flow and plastic response mechanism in the microstructures level are different from crystal plasticity where dislocations, as line defects, are the main carriers of plasticity. Many efforts have been made to characterize the mechanism of nonlinear elasto-plastic deformation in disordered polymeric solids, which resulted in different physical and phenomenological models (see, for instance, refs 3−12). The current widely accepted mechanism for the plastic deformation in glassy solids is the cooperative localized rearrangement of atomic or molecular clusters in small distinct regions which are called shear transformation zones (STZs). Although this theory was first developed for metallic glasses (MGs),13,14 currently © XXXX American Chemical Society

STZs are recognized as the carriers of plasticity in other forms of noncrystalline solids such as covalent glasses and PGs.15,16 In particular, Oleinik has experimentally observed the presence of discrete plastic deformation units in PGs referred to the formation of STZs.17,18 Shear transformation zones are viewed as discrete irreversible processes for stress relaxation which initiate around free volume regions under an applied shear stress with the assistance of thermal fluctuations.14 Since in comparison with crystals there is no structural order in glassy solids, the STZs are immobile and fixed in their nucleation sites,19 and upon formation, they do not broaden by translational movements of their boundaries;16 they might propagate only by joining to the just formed neighboring STZs.20 Therefore, the kinetics of the STZs mediated plastic deformation in glassy solids is nucleation controlled.19,21 To evaluate the stress and strain fields generated in the continuous medium caused by localized disturbance due to STZs, one can use the homogenization methods based on micromechanical approach. The pioneering work of Eshelby22 has been used as an advanced homogenization technique for treating a wide range of inhomogeneous media like composites,23 cracks,24 and even problems in biomechanics.25 This method has been used successfully in addressing the nucleation energy of STZs in glassy solids, which Received: May 18, 2015 Revised: July 13, 2015

A

DOI: 10.1021/acs.macromol.5b01064 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules in turn can be used for relating the shear flow strain rate to the shear flow stress by means of an Arrhenius function.16 The nucleation energy of an STZ is a function of its geometrical and micromechanical properties. These characteristic properties have been massively worked out for MGs; the average nucleation energy of an individual STZ in MGs has been found to be about 1.5 eV (see Table SIV in the Supporting Information of ref 26), with the average size of ≤10 nm3 which includes less than 500 atoms,27 and the average transformation shear strain of 0.07.16 For the case of the PGs, however, there still exists a big gap in the data for evaluating these parameters. Argon and Bessonov28 obtained the geometrical properties of plastic deformation units in some glassy polymers. However, their investigations were based on the Argon’s double kink theory6 which was abandoned later in favor of the STZs mediated plasticity59. Mott et al.29 developed a set of molecular dynamics simulations to investigate the kinematics of plastic deformation in amorphous atactic polypropylene. They found the spherical plastic flow units to possess shear strain of about 0.015 and an average diameter of 10 nm. Ho et al.30 also obtained a power law correlation between the entanglement density and the shear activation volume by means of performing uniaxial and plane stress compression tests on miscible polystyrene−poly(2,6-dimethyl-1,4-phenylene oxide) (PS−PPO) blends at different PS/PPO ratios. In the current study, the nanoindentation technique is used to evaluate the flow stress of poly(methyl methacrylate) (PMMA) at different strain rates and temperatures. Demonstrating that the flow in this polymer is homogeneous, the experimental results are used to obtain the geometrical and micromechanical properties of the STZs in PMMA. This paper is organized as follows: in section 2, the experimental procedure is explained with the dedication of part 2.1 to sample preparation and part 2.2 to the nanoindentation technique. In section 3, it is first shown that the flow in PMMA is homogeneous at tested temperatures and strain rates. The homogeneous flow theory is then elaborated based on the Eshelby’s solution for nonelastic strain in an embedded inclusion in the representative volume element (RVE) and the Arrhenius function relating the shear flow stress to the rate of the shear flow strain. In section 4, the experimentally obtained results for annealed and as-cast samples are presented. Incorporating the flow theory, the characteristic properties of STZs are obtained which include the nucleation energy barrier, size and shape of an STZ, and the shear activation volume. In addition, by attributing the observed jump in the STZ activation energy to the β-transition, the energy barrier for this transition is found to be about 10% of the STZ nucleation energy. Finally, the concluding remarks are summarized in section 5.

remove any moisture due to the washing process. The glass transition temperature (Tg) of the samples was measured to be about 110 °C, using the differential scanning calorimetry (DSC). The calorimetric measurements were carried out on a TA Instruments 2920 DSC device, which was operating under the nitrogen flow, using standard aluminum pans. The DSC cycles were performed with the rate of 10 °C/min from room temperature to 200 °C. Since one of the main purposes of this investigation is studying the effect of thermal history on the microstructural and micromechanical properties of STZs in PMMA, one-half of the specimens were annealed at 120 °C for 4 h and then cooled down to room temperature with the rate of 10 °C/h. This thermal process was performed in a vacuum oven to avoid the oxidation of the specimen surfaces at temperatures above the PMMA glass transition.31 The samples surface roughness, which is a key factor affecting the nanoindentation results,32,33 was measured by both an Agilent 5500 atomic force microscope (AFM) and a Wyko Optical Profiler. For 3 × 3 μm scan regions (Figure 1), the average surface

Figure 1. Sample of AFM scanning of the PMMA specimen surface. roughness was Ra = 0.372 ± 0.013 nm, which allows the assumption of the flat surface for samples;32 therefore, no modification is required for the obtained experimental data. All the samples were mounted on aluminum stubs using super glue for ambient temperature tests and thermoresistant epoxy putty (Drummond Nu-Doh epoxy repair compound titanium reinforced) for high temperature indentation tests on hot stage apparatus. 2.2. Nanoindentation. The nanoindentation technique has been employed to measure the mechanical properties of the PMMA samples. The indentation load-hold-unload cycles have been conducted using an MTS Nanoindenter XP equipped with a threesided pyramidal Berkovich diamond tip. The nanoindentation technique evaluates the mechanical properties of the material including the hardness, H, and the elastic modulus, E. The analysis of nanoindentation load-penetration curves is mainly based on the procedure developed by Oliver and Pharr,34 and will be briefly described in the following. The hardness of the sample can be computed from the definition

H=

2. EXPERIMENTAL PROCEDURE

Pmax Ac

(1)

where Pmax is the peak indentation load and Ac is the projected area of the tip−sample contact at the maximum load. The contact area can be described as a function of contact depth, hc, as follows

2.1. Sample Preparation. Commercially available amorphous poly(methyl methacrylate) (PMMA) 2.0 mm-thick sheets, Goodfellow Catalog No. ME303020, Cambridge, U.K., were selected for the investigation in this study. The PMMA sheets have been produced through the traditional cell cast method; therefore, no preferred molecular chain orientation is likely to exist in the as-cast sheets. Since the sheets were casted as100 × 100 mm plates, they were cut into 20 × 20 mm squares suitable for conducting nanoindentation tests. To remove the residues of the protective film layer covering the sheets, the samples were washed using 30% isopropyl alcohol (IPA) and thoroughly rinsed by pure distilled water. All specimens were stored in a desiccator cabinet for 10 days prior to any test or measurement to

Ac = 24.56hc 2 + C1hc1 + C 2hc1/2 + C3hc1/4 + ... + C8hc1/128 (2) The first term on the right-hand side of eq 2 describes a perfect Berkovich indenter and the other terms account for deviations from ideal geometry due to the blunting of the tip. The parameters C1 through C8 are constant coefficients determined by using the indentation results on a standard fused silica sample and curve fitting performed on the Analyst software. The area function of the diamond tip was calibrated prior to experiments and the coefficients were B


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Figure 2. Load-indentation depth curves for as-cast and annealed PMMA samples at different temperatures and the load rates of (a) 10 mN s−1, and (b) 300 mN s‑1. No pop-in events were observed in any of the tests. obtained as C1 = 297.54, C2 = 144.08, C3 = 27.90, C4 = 116.55, C5 = 98.11, C6 = 85.79, C7 = 78.97, and C8 = 75.41. Since the indentations are deep in our study, the fitting coefficients are allowed to be positive only. The elastic modulus of the sample, E, is calculated from eq 3 as follows 1 − νi 2 1 1 − ν2 = + Er E Ei

ε̇ =

S 2λ

π Ac

(5)

and since the Ṗ /P is kept constant during the CSM test, the indentation strain rate also remains constant. The indentation effective shear strain rate can then be calculated incorporating the relation γ̇ =

(3)

3 Cεi̇

(6) 40−42

where the constant C = 0.09. The tests were carried out in the following sequence: first, after the tip reached the surface of the sample, the loading segment was carried out with constant value of Ṗ /P until a maximum depth of about 10 μm; the load was then kept at the maximum value for 10 s to account for the material creep behavior of the polymer surface; and finally, the unloading segment was performed until 10% of the maximum load was reached. In order to investigate the response of the mechanical behavior of the PMMA to the indentation strain rate, a series of experiments were conducted with the values of Ṗ /P varying from 0.001 to 0.2 s‑1. A total of 25 indents were performed for each strain rate with a minimum distance of 150 μm between adjacent indents to avoid the interaction of the plastic zones formed around the indents. 2.2.2. High-Temperature Nanoindentation Procedure. The Nanoindenter XP is equipped with temperature control system to perform high-temperature nanoindentation tests. This system consists of a hot-stage, a heat shield to isolate the indenter transducer from the heat source, and a coolant apparatus to transfer the extra heat to the outside of the instrument. Employing the basic method, the load control experiments were performed with a maximum load of 300 mN to ensure the elimination of the indentation size effect and making the data comparable to those of the CSM tests. Loading rates of 4, 10, 50, 100, and 300 mN s−1, which are constant during the tests, were applied for temperatures varying from room temperature to 100 °C (slightly lower than the PMMA glassy temperature). While the loading rate Ṗ /P is constant in this method, by examining the value of Ṗ /P it can be easily obtained that the indentation strain rates ε̇i varies as 1/hξ during the test, owing to the fact that the load− depth curve follows a Hertzian relation according to the statements in section 2.2.1. Following the approach of Schuh et al.,42 an average value of Ṗ /P within the indentation depth beyond a certain value (5

In the above equation, Ei and νi are the elastic modulus and Poisson’s ratio of the indenter, respectively, ν is the Poisson’s ratio of the sample, and Er is the reduced elastic modulus which accounts for elastic deformation in both the indenter and the sample. The reduced modulus can be obtained from the relation developed by Sneddon35 Er =

ḣ 1 Ṗ = h ξP

(4)

where S is the measured contact stiffness and λ is a constant that depends on the geometry of the indenter equal to 1.034 for the Berkovich tip. 2.2.1. Room-Temperature CSM Nanoindentation. A continuous stiffness measurement (CSM) technique36,37 was utilized for experiments at room temperature. With this technique, the elastic contact stiffness is measured as a continuous function of depth during the loading path, and not just at the point of unloading as in the conventional measurements. The CSM is accomplished by imposing a small sinusoidally varying load on top of the main applied load that drives the motion of the indenter. The frequency and displacement amplitude of the superimposed oscillating force are set as 45 Hz and 2 nm, respectively, which are optimum values for this MTS nanoindenter XP. In a deep indentation test, where the indentation size effect is negligible (i.e., the hardness value is almost constant), the load− displacement curve obeys the Hertzian contact relation as P = ηhξ, where P represents the indentation load, h is the penetration depth, η is a material dependent parameter, and ξ is a curve fitting parameter close to 2 for the Berkovich tip.38,39 Accordingly, the indentation strain rate ε̇i can be obtained by C


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Figure 3. Variation of the PMMA hardness with temperature at different load rates for (a) as-cast and (b) annealed samples. The data points at the load rate of 4 mN s‑1 and 361 K were not shown due to their high standard deviation values. μm in this study) was used to evaluate the effective shear strain rate using eqs 5 and 6. Accordingly, the corresponding effective shear strain rates for the load rates given above respectively are 0.0014, 0.0035, 0.0175, 0.035, and 0.105 s−1. Before recording any measurement, each sample was heated to the set temperature inside the indenter; and then it was left to equilibrate for 2−3 h. Once the test was started, the tip was kept at the distance of about 1 μm from the sample surface for about 10 min to adjust the allowable thermal drift. It was concluded from the observations on thermal tests that the delayed contact helped the contact area of the tip and the specimen to reach a thermal equilibrium, since no fluctuations of the load and displacement signals were seen during the test. It should be mentioned that since the polymer sample has low thermal conductivity and the adhering thermoresistant putty layer between the sample and aluminum stub is thick, the set temperature and the temperature of the sample surface may not be exactly equal. To evaluate the precise temperature of the sample surface, an Omega SA1K-SRTC thermocouple was attached to a separate PMMA sample at the end of the tests. The temperature was recorded by using an Omega HH74K hand-held monitoring device. The discrepancy between the measured and set temperatures were significant for high temperatures.

were performed on both annealed and as-cast samples. The loading segment-displacement curves of the samples at loading rates of 10 and 300 mN s‑1 and at different temperatures are depicted in Figure 2. All the curves are represented with the origin offset of 2 μm except the first one at room temperature. No pop-in events are observed in any of the nanoindentation tests. In addition, the hardness-temperature curves under various strain rates (Figure 3) show that the hardness is greatly strain rate sensitive in a way that at a given temperature, a higher hardness is obtained at a higher strain rate. The strain rate sensitivity of the flow and absence of pop-in events in the P−h curves indicate a homogeneous flow in amorphous PMMA polymer at temperatures below its glass transition temperature Θg, which is consistent with the observation of Casas and coworkers.45 The flow in glassy solids is mediated by the irreversible regional disturbances in their initial organization which form rearranged atomic (in MGs) or molecular (in PGs) domains. These cooperative rearrangements result in isolated unit increments of shear, and are known as shear transformation zones (STZs). In the homogeneous flow regime each volume element has contribution to the total plastic strain, whereas in the inhomogeneous flow regime the strain is localized in distinct shear bands.13 According to the statement made in the preceding paragraph, the thermally activated flow in glassy polymers can be described based on the homogeneous flow mechanism developed by Spaepen13 and Argon.14 For the RVE shown in Figure 4, the free energy of the nucleation of a single STZ as an inclusion problem is given by Eshelby22 as follows:

3. THEORY OF HOMOGENEOUS FLOW FOR GLASSY POLYMERS As stated in the introduction, despite the inhomogeneous flow of glassy solids, their homogeneous flow is strain rate sensitive.41,43,44 Another significant difference between inhomogeneous and homogeneous flows of glassy solids is the generation of several pop-in events in the load−displacement curves obtained by nanoindentation tests during the inhomogeneous flow.39,44 It has been previously shown that the flow behavior of metallic glasses is temperature dependent, in a way that an inhomogeneous to homogeneous transition happens at a certain temperature. The transition temperature is also strain rate dependent: the higher the applied strain rate, the higher the transition temperature.44 In order to investigate the flow nature in glassy PMMA polymer, several nanoindentation tests

ΔF0 = [Ξ(ν) + Ψ(ν)β 2]μ(γ T )2 Ω

(7)

in which Ω and γ are the volume and transformation shear strain of the STZ, respectively, and μ is the shear modulus. The parameters Ξ(ν) and Ψ(ν) are functions of the Poisson’s ratio T

D


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energy of the STZ have been experimentally obtained for different MGs (see Chapter 7 of ref 16 for details), their numerical evaluation for PGs have been limited to some theoretical modeling and simulations,16,29 and a few experimental studies.28,30 For the conventional polymeric materials where γTΩτ ≫ 2kBΘ at temperatures below their glass transition temperatures, eq 8 can be reduced to ln γ ̇ = Figure 4. Representative volume element (RVE) of PMMA matrix containing a single shear transformation zone (STZ).

(9)

with C1 = ln((γ̇0)/(2)) − ((ΔF0)/(kBΘ)) representing a temperature-dependent parameter. According to eq 9, the important characteristic parameter of glassy polymers known as shear activation volume, which is proportional to γTΩ, can be determined from the derivative of the natural logarithm of the strain rate with respect to the shear flow stress. Considering the effect of hydrostatic pressure on the shear yield stress of polymers, the shear activation volume is obtained from eq 9 as follows:30,47

ν, and pertain to the shear and dilatational components of the transformation strain tensor, respectively. The coefficient β is the dilatancy parameter and can be obtained from the pressure sensitivity of yielding. It is worth noting that Ψ(ν) = ((2(1 + ν))/(9(1 − ν))) is independent of the inclusion shape, whereas Ξ(ν) is shape dependent.46 The kinetics relation for the shear strain rate γ̇ due to the applied shear stress τ is well expressed by an Arrhenius relation as follows:13,14 ⎛ γ T Ωτ ⎞ ⎛ ΔF ⎞ γ ̇ = γ0̇ exp⎜ − 0 ⎟ sinh⎜ ⎟ ⎝ kB Θ ⎠ ⎝ 2kBΘ ⎠

γTΩ τ + C1 2kBΘ

V * = (1 + βα)γ T Ω = 2kBΘ(1 + βα)

∂[ln γ ]̇ ∂τ

(10)

in which β is the yield stress sensitivity to pressure as defined before, and α is the loading condition related constant which is between −0.7 and −0.6 for different compressive loading conditions.47,48 In fact, eq 10 implies that the shear activation volume is a modified STZ volume with considering the dilatation effect. In addition, since γT is believed to be a universal constant for glassy polymers which is about 0.02,29,30 the size of the shear transformation zone Ω can then be calculated. At a fixed shear strain rate, eq 8 can be rearranged as follows:

(8)

in which kB is the Boltzmann constant, Θ is the absolute temperature, and γ̇0 is the pre-exponential factor proportional to the attempt frequency. The factor 2 in the denominator of the argument of the hyperbolic function is due to the reverse transformation probability.13 The quantity γTΩ is proportional to the size scale of plastic relaxation units in PGs known as shear activation volume V*.47 Although the individual material parameters in eq 9, such as size, shear strain, and activation

Figure 5. Variation of the hardness versus the indentation depth for three different Ṗ/P values for (a) as-cast and (b) annealed PMMA samples. The legend numbers represent the Ṗ /P values. E


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Figure 6. Variation of the elastic modulus versus the indentation depth for three different Ṗ /P values for (a) as-cast and (b) annealed PMMA samples. The legend numbers represent the Ṗ /P values.

Figure 7. Variation of the shear flow stress with the shear strain rate for both as-cast and annealed PMMA samples.

τ = ΘC2 +

2ΔF0 γTΩ

and elastic contact stiffness of the sample during the loading segment. Figures 5 and 6 represent the variation of these mechanical properties versus the indentation depth for some selected Ṗ/P values. As can be seen in these figures, the elastic modulus is almost constant for the indentation depth beyond 100 nm for both as-cast and annealed samples, while it takes about 2 μm for the hardness to reach a stable value; this is known as the indentation size effect in polymers which has been previously addressed in the literature (see ref 10 and references cited therein). More importantly, it is only the hardness which shows the strain rate dependency, and the elastic modulus does not change significantly with the strain rate, as apparently seen.

(11)

with C2 = ((2kB)/(γTΩ))(ln γ̇ − ln((γ̇0)/(2))) represents a strain rate dependent constant. Consequently, the activation energy of a single STZ, ΔF0, can be obtained from the linear interpolation of the variation of the flow shear stress with temperature. Determining ΔF0 and incorporating eq 7, the numerical value of Ξ(ν) can be calculated which leads to the evaluation of an approximate shape of the STZ in PMMA.

4. RESULTS AND DISCUSSION 4.1. Shear Activation Volume. As mentioned in section 2.2.1, the CSM nanoindentation method records the hardness F


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Figure 8. Variation of the shear flow stress with temperature at different load rates for (a) as-cast and (b) annealed PMMA samples. The data points at the load rate of 4 mN s‑1 and 361 K were not shown due to their high standard deviation values.

Figure 9. STZ activation energy for both as-cast and annealed samples at different shear strain rates.

annealed ones have slightly bigger flow stresses at the same shear strain rate.50 Another interesting feature of Figure 7 is the existence of a significant transition at a certain value of strain rate beyond which the sensitivity of the flow stress to the strain rate increases. This transition is believed to be a result of strain rate shift of the β-relaxation process in the storage modulus of the PMMA, which is attributed to the restriction of the ester side group rotations at high strain rates, besides the intermolecular and local backbone motion restrictions.9,16,51 The transition shear strain rate is about 0.005 s‑1 for both annealed and as-cast samples, as shown in Figure 7, which well matches with the experiments of Mulliken and Boyce.9 Following the descriptions presented in Mulliken and Boyce,9 the flow stress regimes below and above the transition strain rate might be referred to as α and β regimes, respectively.

The hardness data obtained from the nanoindentation test can be converted to the yield strength σy of the specimen using Tabor’s relation H = κσy

(12)

in which H is the hardness and κ is Tabor’s factor. The constraint factor κ can be taken equal to 3.3 for PMMA at high indentation strains.49 In addition, since the shear flow stress is about half of the yield stress in plane stress condition for monotonic loading,44 the ratio of the hardness to the shear flow stress is about 6.6. Figure 7 shows the variation of the shear flow stress with the shear strain rate for both as-cast and annealed PMMA samples at room temperature. It should be noted that the shear flow stress is obtained based on the average hardness values over the indentation depth beyond 5 μm where indentation size effect is absent. As it is expected, the G


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system containing a diglycidyl ether of bisphenol A (DGEBA) and 1,3-bisaminomethylcyclohexane (1,3-BAC). This ratio is different for MGs where the β-transition activation energy has been shown to be almost equal to the STZ activation energy.26 It is also noticeable that the activation energy has an increasing trend with increasing the strain rate. In addition, as it was stated before, the activation energy is a function of the shear modulus which varies with temperature. Figure 10 shows the variation of

According to eqs 9 and 10, the shear activation volume for an amorphous polymer can be calculated from the shear flow stress-shear strain rate curves with logarithmic interpolation. Using eq 9 and the data points of Figure 7, the factor γTΩ for as-cast and annealed samples is obtained about 3.66 and 3.69 nm3, respectively. Consequently, using eq 10 and assuming β = 0.20447 and α = −0.65, the shear activation volume is found to be 3.17 and 3.20 nm3 in the α-regime for as-cast and annealed samples, respectively, which are in good agreement with the data obtained from molecular dynamics simulations.16 The shear transformation zones form around the free volumes in the glassy solids;16 and it has been previously shown that the average free volume size for annealed and quenched PMMA samples is almost the same for plastically deformed specimens.52 Hence, it is expected that the shear activation volume for samples with different thermal histories to be identical, which confirms the obtained results from the experiments. In addition, the thermal history of the sample was shown to be immaterial for the segmental mobility of the plastically deformed PMMA.53 Since the segmental mobility is closely related to the molecular rearrangement of the PGs, it is expected that the size of the activation volume to be independent of the thermal history as well, which is experimentally confirmed here. 4.2. Activation Energy. In light of eq 11, linear interpolation of the flow stress as a function of the temperature provides the data for calculation of the thermal activation energy of a single STZ. As it is defined in eq 7, the Helmholtz free energy is a function of the shear modulus (μ) of the material which itself is temperature dependent. However, the shear modulus of the PMMA does not vary significantly for the temperatures below around 100 K.54 Hence, eq 11 remains valid, and consequently, ΔF0 might be referred to as the activation energy at 0−100 K. Figure 8 shows the variance of the shear flow stress with temperature for different strain rates. Assuming the parameter γTΩ not to change with temperature,16 the STZ activation energy for both as-cast and annealed samples can be calculated as shown in Figure 9. Interestingly, there exist jumps in the activation energy of different samples at the strain rate range 0.0035−0.0175 s‑1, which is consistent with the β-transition strain rate obtained from room-temperature CSM nanoindentation. Therefore, this jump might be referred to as the β-transition activation energy. Although the difference in the activation energy for the annealed and as-cast samples is small for strain rates above the β-transition strain rate, the discrepancy is considerable for strain rates below this transition. Since the annealed PMMA sample is expected to have more ordered chains compared to the as-cast one, the rotation and slip of these chains are more restricted in this sample; consequently, one expects the activation energy of the STZ to increase. In contrast, beyond the β-transition strain rate the rapid loading does not allow the chains to slip or rotate smoothly which puts the annealed and as-cast samples in the same deformation condition; and as a consequence, the activation energy of STZs for high strain rates is approximately equal for these two samples with different thermal histories. Considering the jumps in Figure 9 as the β-transition activation energy, it turns out that this activation energy is much bigger for the as-cast polymer than the annealed one (almost 3 times). Moreover, the obtained β-transition energy is about 1 order of magnitude smaller than the thermal activation energy of an STZ for PMMA, which is in consonance with the findings of Barral et al.55 who found almost the same ratio for a

Figure 10. Variation of the shear modulus of the PMMA with temperature for both as-cast and annealed samples at two different load rates.

the shear modulus with temperature for both as-cast and annealed samples at two different strain rates. These data points are excellently interpolated by linear trend lines. Assuming the variation to be linear for temperatures down to 100 K, and a constant shear modulus for temperatures below 100 K (which is a valid assumption based on the experiments of Gall and McCrum54), one can obtain the variation of the STZ activation energy with temperature as shown in Figure 11. As it is seen in this Figure, the STZ activation energy is about 0.6 eV for PMMA at room temperature which is about one-third of that for MGs.26 4.3. Geometry of STZs. Taking eqs 11 and 7 into consideration, the intercept of τ−Θ linear interpolation with τ axis is equal to 2[Ξ(ν) + Ψ(ν)β2]μγT. As was mentioned in section 3, Ψ(ν) does not change with the shape of the STZ and is about 0.5 for PMMA with the Poisson’s ratio of 0.38. But, the value of Ξ(ν) is determined by the shape of the disturbance

Figure 11. Variation of the activation energy of a single STZ of the PMMA with temperature for annealed and as-cast samples at two different load rates. H


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Macromolecules zone which varies between ((7 − 5ν)/(30(1 − ν))) and 0.5 for spherical and flat ellipsoidal shapes, respectively.46 Using μ0 = 3.1 GPa from Figure 10, and β = 0.204, the value of γT changes between 0.03 and 0.05, with the upper and lower bounds pertaining to the spherical and flat ellipsoidal transformation zones, respectively. While the shear strain (γT) of 0.05 is a large value for PGs according to the molecular dynamics simulation,29 the shape of the transformation zone is more likely to resemble a flat ellipsoid for PMMA instead of a sphere. Since the factor γTΩ has already been obtained in section 4.1, and assuming γT = 0.03 for PMMA, the volume of an individual plastic deformation unit, Ω, can be calculated easily. This volume, which is almost the same for as-cast and annealed samples, is obtained about 123 nm3 which is at least 1 order of magnitude bigger than that of the metallic glasses.26,27 Assuming the PMMA monomers as cylinders with radius of 2.85 Å and length of 1.55 Å,28 the single STZ is found to contain about 3000 monomers, which is in excellent agreement with the simulations.16

Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors gratefully acknowledge Prof. Ali S. Argon of MIT for fruitful discussion on the mechanism of plasticity in polymers. The authors also thank Dr. Rafael Cueto of LSU for technical assistance and performing DSC experiments.

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5. CONCLUSIONS In summary, the mechanical and geometrical characteristics of shear transformation zones, the plastic deformation units in glassy polymers, have been investigated in the molecular level for the PMMA in this study. Nanoindentation experiments, which have been previously used for evaluation of the mechanical properties of the polymeric materials in the nanoscale,56−58 have been conducted on both as-cast and annealed specimens to explore the temperature and strain rate sensitivity of the shear flow stress. Incorporating the homogeneous flow theory, the nucleation energy of the shear transportation zones as well as the β-transition energy barriers have been obtained for this glassy solid with different thermal histories. The accuracy of the interpolation of the experimental results using the homogeneous flow assumption further supports the idea that the yield and plastic deformation in PMMA is homogeneous and mediated by the STZs. The procedure used for obtaining the β-transition energy based on the nanoindentation technique is an innovative approach which can be used for other glassy solids. In addition, the analysis of the geometrical properties of the shear transformation zones suggests that in PMMA these discrete local molecular disturbances take place in regions with the shape close to a flat ellipsoid and volume of 123 nm3 possessing transformation shear strain of about 0.03. The obtained transformation shear strain γT for PMMA is slightly bigger than what is believed to be the universal value for PGs. This finding suggests the requirement of further investigations on determining this parameter, which might be unique to a particular polymer depending on the molecular size, weight, and structure. The main advantage of using the nanoindentation is the accuracy and repeatability of the tests, as well as its nondestructive experimentation technique from macroscopic point of view. The experimentally evaluated parameters offer unequivocal numerical values for the characteristics of the STZs as the main carriers of plasticity in PMMA, which can be used as dependable input data for numerical and analytical studies.

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Nanoindentation Study of Yielding and Plasticity of Poly(methyl

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