Stress effects on the elastic properties of amorphous polymeric materials S. Caponi, S. Corezzi, M. Mattarelli, and D. Fioretto Citation: The Journal of Chemical Physics 141, 214901 (2014); doi: 10.1063/1.4902060 View online: http://dx.doi.org/10.1063/1.4902060 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/141/21?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Competition between crystallization and vitrification of the rigid amorphous fraction in poly(3-hydroxybutyrate) AIP Conf. Proc. 1459, 36 (2012); 10.1063/1.4738390 Communication: Correlation of the instantaneous and the intermediate-time elasticity with the structural relaxation in glassforming systems J. Chem. Phys. 136, 041104 (2012); 10.1063/1.3681291 HighResolution Brillouin Scattering Study of Intermediate GlassForming Materials AIP Conf. Proc. 708, 693 (2004); 10.1063/1.1764265 Structural, morphological, and mechanical properties of plasma deposited hydrogenated amorphous carbon thin films: Ar gas dilution effects J. Vac. Sci. Technol. A 19, 1611 (2001); 10.1116/1.1336829 The sorption induced glass transition in amorphous glassy polymers J. Chem. Phys. 110, 11061 (1999); 10.1063/1.479042
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THE JOURNAL OF CHEMICAL PHYSICS 141, 214901 (2014)
Stress effects on the elastic properties of amorphous polymeric materials S. Caponi,1,a) S. Corezzi,2,3,a) M. Mattarelli,4 and D. Fioretto2 1
Istituto Officina dei Materiali del CNR (CNR-IOM) - Unità di Perugia, c/o Dipartimento di Fisica e Geologia, Perugia I-06100, Italy 2 Dipartimento di Fisica e Geologia, Università di Perugia, Via A. Pascoli, I-06100 Perugia, Italy 3 CNR-ISC (Istituto dei Sistemi Complessi), c/o Università di Roma “LaSapienza,” Piazzale A. Moro 2, I-00185 Roma, Italy 4 NiPS Laboratory, Dipartimento di Fisica e Geologia, Università di Perugia, Via A. Pascoli, I-06100 Perugia, Italy
(Received 12 August 2014; accepted 5 November 2014; published online 1 December 2014) Brillouin light scattering measurements have been used to study the stress induced modification in the elastic properties of two glass forming polymers: polybutadiene and epoxy-amine resin, prototypes of linear and network polymers, respectively. Following the usual thermodynamic path to the glass transition, polybutadiene has been studied as a function of temperature from the liquid well into the glassy phase. In the epoxy resin, the experiments took advantage of the system ability to reach the glass both via the chemical vitrification route, i.e., by increasing the number of covalent bonds among the constituent molecules, as well as via the physical thermal route, i.e., by decreasing the temperature. Independently from the particular way chosen to reach the glassy phase, the measurements reveal the signature of long range tensile stresses development in the glass. The stress presence modifies both the value of the sound velocities and their mutual relationship, so as to break the generalized Cauchy-like relation. In particular, when long range stresses, by improvise sample cracking, are released, the frequency of longitudinal acoustic modes increases more than 10% in polybutadiene and ∼4% in the epoxy resin. The data analysis suggests the presence of at least two different mechanisms acting on different length scales which strongly affect the overall elastic behaviour of the systems: (i) the development of tensile stress acting as a negative pressure and (ii) the development of anisotropy which increases its importance deeper and deeper in the glassy state. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4902060] I. INTRODUCTION
Vibrational properties of disordered materials are one of the currently lively topics of modern condensed matter physics.1–3 The evolution of elastic constants and of acoustic attenuation, their response to aging, relaxation, applied pressure and temperature are largely investigated in literature.4–10 The importance of these studies is twofold: on the one hand, they contribute to obtain a comprehensive scenario of the fundamental properties of amorphous materials; on the other hand, they can give important hints for designing materials with innovative properties. In fact, while the external conditions are clearly important, other factors are proved to be of critical importance for the final elastic properties including material microstructure, heterogeneities, pre-existing defects and residual stresses.11–15 The presence of internal stress, in particular, strongly affects the final macroscopic properties especially when the materials are produced by solidification from the melt, via quenches or crunches, which push the system in the final state through non-equilibrium configurations. As a matter of fact, the glass is typically a stressed state of matter since the reduced ability of the system to flow under changes of thermodynamic conditions prevents further adjustments into more stable and long range stress-free states. The a) S. Caponi and S. Corezzi contributed equally to this work. Au-
thors to whom correspondence should be addressed. Electronic mail: [email protected] and [email protected] 0021-9606/2014/141(21)/214901/7/$30.00
glass properties can be changed by the thermal and pressure history which is able to drive the system into states characterized by different amounts of residual stress.14, 16, 17 The role of stress has been recently demonstrated, for example, in the modification of the vibrational density of states18 or of the sound velocities, with effects on the Cauchy relation, i.e., the mutual relationship between the elastic moduli.19 The understanding of the mechanism by which the stress affects the elastic properties is important also for nanostructured glasses. As a matter of fact, the fabrication process can induce high tensile or compressive stresses, which have to be thoroughly controlled for technological applications. For instance, a strengthening higher than 10% was observed in nano-spheroids obtained by elongation of polystyrene spheres. Such particles can be used as building blocks of phononic crystals, whose final sound transmission properties will strongly depend on the shape, the constituent material, and also the different amount of stress in the spheroids.20 Moreover, the induced stress can beneficially be exploited in technological applications: it allows to increase the quality factor of nano-resonators21 or to strengthen a material, like in safety glasses or in the so called “gorilla glass,” deliberately pre-stressed during their production. Even if already exploited in the applications, the stress effect on the elastic properties is an aspect often overlooked in the fundamental physics of glasses, but it is, however, of
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clear importance for a correct analysis of the behaviour of amorphous materials. The amount of literature data and experimental studies in this field is poor and a complete experimental characterization and theoretical understanding of residual stresses, including their microscopic origin and their role on different length scales, have not yet been achieved. Internal stresses in a glass, depending on the length scale over which they average to zero, can be classified as stresses at the atomic level, micro-stresses at the grain scale, and macrostresses at the continuum scale which may involve the whole sample.12 The present work focuses on the effect of macroand micro-stresses on the elastic properties of glasses. In particular, we report on the stress-induced modification of the elastic properties of two different glassy polymers, used as model systems: polybutadiene and an epoxy resin.
II. EXPERIMENTAL
By means of Brillouin light scattering (BLS) experiments we measured the evolution of the frequency position of Brillouin peaks originating from longitudinal and shear acoustic modes, following different vitrification paths. The samples used and the chosen conditions in the different experiments are described in the next paragraphs. All the thermal treatments have been performed with cooling/heating rates between 0.5 and 2 K min−1 .
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stops, governed by a diffusion-controlled mechanism,26, 27 not all the possible bonds have been formed, and the reaction can resume if the diffusion increases by heating the sample. Combining the system’s ability to achieve the glassy phase both by decreasing the temperature (physical vitrification) and by increasing the number of bonds among the constituent monomers (chemical vitrification), the elastic properties have been investigated following five different thermodynamic pathways: (i) by cooling the unreacted mixture from 275 to 73 K, i.e., from above to well below its glass transition temperature; (ii) during the isothermal polymerization at T = 275 K. The reaction runs out in about 2.5 days, during which the glass transition temperature of the system reaches the value Tg polymerized = 300 K; (iii) by cooling from 298 to 171 K the isothermally polymerized glassy sample, i.e., always below its glass transition; (iv) by cooling from 298 to 73 K the glass obtained by annealing (post-curing) for 10 min at 415 K the isothermally polymerized sample. In the post-cured sample all the possible bonds compatible with the stoichiometry and microstructure of the system have been formed; and (v) by performing quenches at very low temperature during the isothermal polymerization of the reactive mixture: as the reaction proceeds at 275 K, the partially polymerized mixture is quenched to 73 K and then brought back to the reaction temperature. The quenched states are glasses at the same T with different extent of polymerization, and therefore they differently combine the effects of cooling and polymerization.
A. Samples and vitrification paths
Two different polymer systems have been studied: (1) The first system is the synthetic rubber 1,4polybutadiene (PB), a glass forming linear-chain homopolymer. The sample used is a deuterated one, with a molecular weight of 25.000 and a glass transition temperature Tg = 178 K, the same used in Refs. 22 and 23. PB has been investigated following four different thermal treatments: (i) First, it has been slowly cooled down from 328 K to 138 K, i.e., from well above to well below the glass transition temperature. In the other experiments, instead, the sample was cooled down fast below the glass transition using different cooling rates, and then measured as a function of temperature; in particular, the cooling rates were about (ii) 0.5 K min−1 , (iii) 2 K min−1 and (iv) 1 K min−1 . (2) The second system is a reactive epoxy-amine mixture, made of diglycidyl ether of bisphenol-A (DGEBA) and diethylenetriamine (DETA) in the stoichiometric ratio 5:2. The sample was prepared by mixing and stirring the liquid reagents for ∼2 min at room temperature, then transferred into the measurement cell. The unreacted mixture can be easily supercooled. Under cooling, the reaction is prevented and the system behaves as a usual molecular glass-former with Tg unreacted = 231 K. On the contrary, if maintained at a constant temperature, the initially liquid mixture undergoes a step polymerization reaction, by addition of the amino hydrogen to the epoxy group, and it spontaneously vitrifies.24–26 The final product is a glassy network-polymer with glass transition temperature some twenty of degrees above the reaction temperature (Tg polymerized ≈ 300 K). When the isothermal reaction
B. Brillouin light scattering measurements
BLS spectra have been acquired using a Sandercock type (3+3)-pass tandem Fabry-Perot interferometer28 with a finesse of ∼100 and a contrast ratio greater than 1010 . As exciting line, the single mode from a Coherent compass 532/400 laser operating at 532 nm has been used. PB has been measured by using two different sample environments. In the first one, the sample was transferred into a cylindrical Pyrex cell with inner diameter of 10 mm and outer diameter of 11 mm. The cell was sealed in a dry nitrogen environment and mounted into a copper sample holder cooled in vacuum by a nitrogen flux. In the second one, the sample was transferred into a cylindrical Pyrex cell with 3 mm outer and 1 mm inner diameter, sealed in a dry nitrogen environment, and temperature controlled by a helium cryostat Thor Cryogenics 3010 II. BLS measurements have been performed in the back-scattering geometry. DGEBA-DETA in all the experiments was loaded into a cylindrical Pyrex cell of 11 mm inner diameter and 12 mm outer diameter, and temperature controlled by the helium cryostat. Only the post-cured sample was a cracked, freestanding glass. In fact, during the post-curing treatment the Pyrex cell was broken due to the high tensile stress produced by the advancement of the polymerization reaction. Unpolarized spectra have been acquired in the 90◦ scattering geometry. This configuration allowed us to detect both longitudinal acoustic modes, which scatter light without change of polarization, and transverse modes, which scatter light with change of polarization.
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FIG. 1. Longitudinal Brillouin peak of polybutadiene at different temperatures. (Upper panel) Temperature evolution before and (lower panel) after the cracks development which occurs acquiring the spectrum at T = 105 K (full dots). The spectrum at T = 105 K (full circles) presents two different Brillouin peaks: the first before the stress release centered at about 16.5 GHz and the second one after the stress release centered at about 17.3 GHz.
FIG. 2. Longitudinal Brillouin peak of the epoxy resins glass (called reacted mixture) obtained by the polymerization reaction of DGEBA-DETA 5:2 at T = 275 K. Temperature evolution before (upper panel) and after (lower panel) the cracks development which occurs at T = 233 K.
Typical BLS spectra acquired in PB and DGEBA-DETA are shown in Figures 1 and 2, respectively. In order to obtain the angular frequency ω0 of the acoustic waves, each Brillouin peak of line-width , has been fitted by a damped harmonic oscillator function,29
III. RESULTS AND DISCUSSION
I (q, ω) =
ω0 2 I0 , π ω2 − ω 2 2 + (ω)2
(1)
0
convoluted with the instrumental resolution function. The sound velocity v has then been calculated by ω v = 0, (2) q where q is the exchanged wavevector, related to the scattering angle ϑ, the refractive index n of the medium, and the wavelength λi of the incident light through the relation q = 4π n/λi · sin(ϑ/2). In determining the sound velocity, it was considered the temperature dependence of the refractive index of PB, while for the epoxy resin it was considered also the dependence on the extent of polymerization, as previously described in Ref. 30. The real part of the longitudinal modulus M at the frequency of the longitudinal mode ω0L and the real part of the shear modulus G at the frequency of the transverse mode ω0T have been calculated through the relations M (ω0L ) = ρ ρ being the density.30
2 ω0L , q2
G (ω0T ) = ρ
2 ω0T , q2
(3)
A. Sound velocities
Figures 1 and 2 report the temperature evolution below Tg of the longitudinal Brillouin peak in PB and in the polymerized epoxy resin. The elastic behavior in both samples presents clear analogies and in the following the two cases are presented in parallel. The PB data reported in Fig. 1 show that, by decreasing the temperature, the continuous shift of the Brillouin peak towards higher frequency is abruptly interrupted. At T = 105 K an improvise transition from one to another elastic state occurs. In fact, in the same acquisition two different longitudinal Brillouin peaks are present: the peak centered at about 16.5 GHz is suddenly replaced by another one centered at about 17.3 GHz. The improvise modification in the elastic condition appears in correspondence of the development of internal cracks in the sample. The lower panel of Fig. 1 shows the temperature evolution of the Brillouin peak after the cracks development: starting from the new elastic state, the samples stiffness now evolves with a smooth temperature dependence. The temperature location of the jump and the frequency position of the Brillouin peak is strongly dependent on the cooling rate and on the sample environment. For the sake of comparison, all the acquired data on PB are summarized in terms of longitudinal sound velocity in Fig. 3. The data obtained following the temperature evolution from the liquid well into the glassy phase are reported as circles.
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FIG. 3. Temperature dependence of the longitudinal sound velocity in PB measured using different cooling rates and choosing different samples environments. The circles (line A) represent the values obtained following the evolution from the liquid state; the sample is contained in a cylindrical cell with the inner diameter of 10 mm. Stars (line D), triangles (line C), and dots (line B) are obtained quenching the system from room temperature to the glassy phase using increasing cooling rates; the samples are contained in a cylindrical cell with inner diameter of about 1 mm.
Decreasing the temperature below the glass transition, two effects are found: (i) increasing the cooling rate, a gradual departure towards lower sound velocity values progressively occurs; (ii) abrupt changes of the sound velocity are evident in correspondence of cracks developments in the glassy structure. The cracks occur at different temperatures in the two performed experiments (T = 35 K line B and T = 105 K line C). Starting from different elastic values, the sound velocity variation is also different. It reaches considerable values, up to ∼10%. However, it is worth noting that after cracks developments the system is brought towards a macroscopic elastic state which, within the experimental error, is independent from the thermal history and is maintained in the successive cooling (line B) or heating procedures (line C). A similar scenario is found in the fully polymerized sample of epoxy resin. The sample, which exploited the chemical vetrification process, is always in the glassy phase. In this case the chosen experimental setup let us to simultaneously collect the longitudinal and transversal Brillouin peaks. The temperature evolution of the spectra are shown in Figure 2 while the values of the sound velocities acquired following different cooling histories are summarized in Figure 4. By cooling the glass from room temperature, history dependent sound velocities are found. At T ∼ 240 K, in correspondence of cracks development, an upward variation is measured in the Brillouin frequencies (Fig. 2), and a smoother temperature dependence in the lower temperature range is found. Further information, in this case, can be also achieved: (i) the first cooling experiment evidences how the elastic transition at T ∼ 240 K simultaneously occurs in both longitudinal and transversal sound velocities and (ii) a similar relative variation of more than 4% has been found in both longitudinal and transversal sound velocities. Once cracks developed in the structure, the glass was heated at room temperature and the measurements were re-
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FIG. 4. Temperature dependence of the longitudinal (full symbols) and transversal (open symbols) sound velocities of the reacted mixture DGEBADETA 5:2. The dots (line A + B) represent the values obtained during the first cooling where the jump to higher values occurs at T = 240 K. The squares (line C) are obtained on the same sample reheated at room temperature and cooled down again.
peated. The data now show a smooth and continuous temperature dependence (line C in Fig. 4). It is worth noting that the two investigated samples are microscopically very different: the first is a linear polymer that by cooling reaches the glassy phase, while the second is arranged in a more complex branched structure and the glassy phase is obtained by a step-polymerization process. Nevertheless, both the obtained glasses develop strong internal stresses. In fact, on crossing the glass transition and deep in the glassy phase, the inability to flow put the system into a potentially stressed and out of equilibrium condition where a non-unique elastic behaviour can be expected. Moreover, because of the interaction with the environment (the cuvette, characterized by a strongly positive wettability and lower thermal expansion coefficient) the thermal contraction is hindered and the negative pressure consequently developed, increases during the cooling down of the system. The pressure release can occur either by the development of internal cracks in the glassy sample or by the breakage of the cell’s wall, depending on the relative fragility of the sample and sample holder. Indeed, the breakage of the Pyrex cell always occurs in the PB sample using the largest cuvettes. In this case, the measurements below ∼130 K are prevented (circles data), while the development of internal cracks occur using the smallest cuvette characterized, cause of its geometrical configuration, by a higher breaking load. The measurements in this case were possible down to 35 K. For both samples, the results suggest that, by cracks development, the glass is able to reduce the stresses trapped inside its structure, so inducing strong modifications of its elastic response. This process gives rise to the development of a permanent modification in the elastic properties of the glass, so that the same values of acoustic wave velocities are reproduced by further cycles of heating and cooling. Finally, it is noteworthy that in both samples the sound velocity increases after the crack formation. This result is in
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agreement with the previously observed behavior in the temperature evolution of DGEBA19 where, however, the jump in the sound velocities was only ∼1%. This is not an obvious effect since the release of the tensile stresses should be associated with an increase of the density and therefore, in the linear elasticity regime, with a reduction of the sound velocity. On the contrary, the observed increase indicates that, because of the tensile stress, the systems experienced a softening of the elastic moduli. This behavior is in agreement with the positive value of the Grüeneisen parameters for polybutadiene and epoxy resins31, 32 implying markedly nonlinear features, and in particular an increase of the moduli with the increase of density which overcompensates the pure density effect on the sound velocities. In Sec. III B we will show how finely monitoring the stress induced modification of the elastic constants, by studying the relative behavior of the elastic moduli. Our data provide new insights into this phenomenon as we study it in different materials and as function of different thermal and strain histories. B. The generalized Cauchy relation and its deviations
It is well known that the elastic behavior of a homogeneous, isotropic solid at equilibrium is fully described by the longitudinal and the shear modulus. If, in addition, the atoms interact through a central potential, the Cauchy identity, M = 3G , holds and only one independent elastic constant exists.33 In glass forming systems, the real parts of the longitudinal, M (ω), and transverse, G (ω), elastic moduli measured by BLS, satisfy a generalized Cauchy-like relation, M (ω) = A + BG (ω),
(4)
where B has been found close to 3 and A is a systemdependent constant which, at ambient pressure, does not appreciably depend on temperature. The validity of Eq. (4) has been verified by BLS both in the unrelaxed (high frequency) limit of glass forming systems34 and even in relaxing liquids.30, 35 Now the question we face is the following: Does the stress field modify the mutual relation between the elastic moduli? And if yes, how? To answer, we report in Figure 5(a) the data acquired at different temperatures and at different polymerization degrees in the epoxy-resins. In the low frequency region, the Cauchy like relation is well fulfilled, and the same line, M (ω) = A + BG (ω) with A = (3.09 ± 0.03) GPa and B = 3.00 ± 0.01, which connects the elastic moduli during the whole polymerization process, describes the data at the highest investigated temperatures for the unreacted mixture (red circles) and for the polymerized sample (green squares). A progressive departure from the expected values is instead well visible entering deeper and deeper in the glassy phase for the different investigated systems. In Figure 5(b) the difference between the experimental data and the estimated Cauchy line is reported. This data representation visually enhances the deviations that occur below as well as above the expected values shown as dashed line. The behavior of the polymerized sample, which presents a jump in the sound velocity (Line A + B in
FIG. 5. Panel (a) Real part of the longitudinal modulus, M , versus real part of the transverse modulus, G , of the DGEBA−DETA 5:2 mixture during the isothermal polymerization reaction at 275 K (gray ♦), by cooling the reacted mixture (black : line A + B of Fig. 4 and green : line C of the previous figure), the post cured sample (blue ), the unreacted mixture (red ●), and quenching at 73 K the sample at different polymerization extents (open triangles). Error bars are comparable to the symbol size. The solid line is the Cauchy-like relation M = (3.13 ± 0.03) + (3.00 ± 0.01)G . Panel (b) Difference between the experimental data and the expected Cauchy-like behaviour. The data are shown with the same symbols as in panel (a).
Fig. 4), will drive our discussion. Indeed, the data acquired before the sample cracking (line A, T > 240 K) deviate below the Cauchy line, while after the cracks development (line B, T < 240 K) the data exceed the expected values, suggesting that cracks development plays a role in the deviation from the Cauchy law. Repeating the measurements in the cracked glass where the tensile stresses are mostly released (green squares – line C showed in Fig. 4), the Cauchy line is recovered over a wider temperature range, up to T = 253 K. At this temperature a deviation again occurs but only above the expected values. The temperature evolution of the unreacted mixture (red dots) and the post-cured sample (stars) show both the deviations below and, at the lowest temperatures, above the Cauchy-line. In the quenched states at T = 73 K, at different polymerization degrees, cracks always occur in the samples, and the data reveal only upwards deviations.
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As a preliminary consideration, we would notice that in different samples, we obtain both the upwards or the downwards deviations evidencing that, independently from the thermal history, the mechanisms at the origin of the Cauchy breaking are different and at least two: both of them are always active and they play competitive roles causing opposite deviations. Changing the sample conditions, for example after the cracks development, their relative weight drastically changes making one dominant over the other. However, the deviations always occur when the samples are below their glass transition and hence under a condition in which a stressfield may extend over long range scales. By adding the cooling effect to the polymerization process, we are able to push the system deeper and deeper in the glassy phase increasing the amount of unrelaxed stresses. In fact, before the cracks development, the adhesion to the cell walls produces an effective negative pressure and the progressive development of tensile stress, which is mainly planar, as testified by the deep concave meniscus in the glassy states. In these conditions, also elastic anisotropy increases. The deviation of the M vs G relationship below and above the Cauchy line could be related to stresses on different length scales. In fact, while the long range stresses originate a negative pressure, stresses on lower length scales can be responsible for the permanence of anisotropy also after the cracks development. Both the negative-pressure and the microscopic residual anisotropy would produce a failure of the conditions for the Cauchy validity.33 It appears that their effects can produce deviations of different amount and can play a role in different experimental conditions. Before the cracks development, the effect of negative pressure seems to be the dominant mechanism responsible for the Cauchy law breaking. In fact, when the negative pressure is drastically reduced by cracks, the stress component on a macroscopic scale is released, the downwards deviation disappears and the Cauchy line is recovered over a wider temperature range in the cracked sample. The experimental data support this scenario also from a quantitative point of view. In fact, it is well documented that the applied pressure influences the longitudinal and the transversal phonon branches leading to a modification of the Cauchy relation with respect to that valid at P = 0.36–38 Considering the pressure dependent parameter predicted in the generalized Cauchy-like relation,34 the equation reads M (ω) = A + BG (ω) + 2P .
(5)
The last term is usually neglected and/or incorporated in the constant value A when the pressure variation is negligible; in fact, its value is usually very low compared with the material elastic constants (∼GPa). On the contrary, when its variation begins to be comparable with the elastic constants, its contribution cannot be any longer neglected. By cooling the sample under its glass transition, due to the adhesion to the cell walls, the glass can experience a negative pressure that becomes increasingly higher during cooling. A simple argument can lead to evaluate such pressure, assuming that it has to balance the contraction due to the decrease of temperature. From the stress-strain relation and considering the thermal expansion coefficients, we can estimate the tensile stress, σ , ex-
ercised by the cuvette surfaces as39 σ (T) =
2G (3M − 4G )α(T − Tg ), M
(6)
where α is the difference between the thermal expansion coefficient of the polymer (whose typical value is ∼1 × 10−440 ) and pyrex (3 × 10−6 ), and the term with the M and G combination is an effective elastic modulus for purely radial stress. On the basis of the elastic moduli values, the expected stress increase is ∼1 MPa K−1 . This prediction is well confirmed by the experimental data where, during cooling, the downward departure from the linear Cauchy relation, which we attributed to the term 2P of Eq. (5), is ∼200 MPa over a temperature range of 60 K. This maximum negative pressure is in agreement with the value of tensile strength of the epoxy resin, which is around 85 MPa:41, 42 when the negative pressure overcomes this limit, cracks develop in the material. The same arguments can be applied to PB, where using the data in Ref. 43 the induced pressure per Kelvin is estimated to be ∼1 MPa K−1 and therefore the total pressure increase that develops below the glass transition results in fairly good agreement with the PB tensile strength of the order of tens of MPa.44 Whereas the downward departure from the Cauchy behaviour seems to be well explained on the basis of a negative pressure term,45 the upward deviation at very low temperature is still a puzzling result, but a tentative explanation can be, however, provided. In fact, it should be considered that the violation of the Cauchy relation can be caused by different mechanisms other than pressure, including the development of non-central forces, many body interactions, heterogeneity or anisotropy in the sample.33 In particular, internal cracks are able to release most of the macroscopic stress, while stresses on smaller length scales may be not significantly modified. As a consequence, in a tentative interpretation the upward deviation may be ascribed to elastic anisotropy effects, which intensify by decreasing the temperature, and reach their maximum value in the quenched samples. Following this interpretation, the experimental data give also a hint on the characteristic size of the elastic anisotropy regions. In fact, due to the topologically disordered structure, microscopic stresses are always present in the glass but, in order to significantly affect the collective dynamics of the material, their characteristic size has to be comparable with the length scale probed by the investigated phonons. By changing the experimental technique the system is investigated on different length scales and, for example, the effect of the elastic heterogeneity at few nanometer scale well detectable by inelastic x-ray scattering,46 it is not revealed by BLS measurements. The wavelength of the phonons probed by BLS is λp ∼ 200 nm, and a stress field with characteristic dimension much smaller than λp /10 leaves unaffected the phonon propagation: the phonons in this case will travel as in an homogeneous and isotropic system.47 On the contrary, when the length scale of the stress field starts to be comparable with the phonon wavelength, its presence can strongly influence the propagation of the acoustic modes, modifying both their attenuation and velocity.
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IV. CONCLUSIONS
In conclusion, the paper reports how the stress field influences the acoustic properties of materials on the basis of original data for polybutadiene and epoxy resins, which are prototypes of linear and network polymers. The results show that the stress field affects on the one hand the absolute values of the Brillouin frequencies and on the other hand the mutual relationship between the elastic moduli. The results represent a systematic extension of existing experimental works. The rupture of the Cauchy relation seems to be a sensible marker of the stress presence in the sample: independently from the stiffening path, the states in which the system is able to release internal stresses follows the Cauchy law, otherwise the Cauchy relation fails. We point out how the existence of stress effects on the elastic properties is correlated with the presence of at least two different mechanisms acting on different length scales: the first one is interpreted in term of the development of tensile stress acting as a negative pressure, while the second one is tentatively interpreted with the persistence of anisotropy on the mesoscopic length scale. The two competitive mechanisms cause opposite deviations and their interplay is responsible of a complex elastic scenario. The picture that we provide is able to give a reasonable interpretation of the experimental data and may be a useful hint to a complete theoretical description, still missing. The study could also be useful for guiding the experimental work both in the application field and in the fundamental physics, where the role of the sample environment and the chosen cooling rate should be considered more carefully in the evaluation of the material properties. ACKNOWLEDGMENTS
S.C. and S.C. acknowledge support from MIUR-PRIN (Project 2012J8X57P), S. Caponi acknowledges support from CNR Flagship Project “Nanomax (N-CHEM),” PAT (Autonomous Province of Trento) “Grandi Progetti 2012” Project “MaDEleNA” and M.M. acknowledges support from EC founded Landauer project (G.A. No. 318287). 1 M.
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THE JOURNAL OF CHEMICAL PHYSICS 141, 214901 (2014)
Stress effects on the elastic properties of amorphous polymeric materials S. Caponi,1,a) S. Corezzi,2,3,a) M. Mattarelli,4 and D. Fioretto2 1
Istituto Officina dei Materiali del CNR (CNR-IOM) - Unità di Perugia, c/o Dipartimento di Fisica e Geologia, Perugia I-06100, Italy 2 Dipartimento di Fisica e Geologia, Università di Perugia, Via A. Pascoli, I-06100 Perugia, Italy 3 CNR-ISC (Istituto dei Sistemi Complessi), c/o Università di Roma “LaSapienza,” Piazzale A. Moro 2, I-00185 Roma, Italy 4 NiPS Laboratory, Dipartimento di Fisica e Geologia, Università di Perugia, Via A. Pascoli, I-06100 Perugia, Italy
(Received 12 August 2014; accepted 5 November 2014; published online 1 December 2014) Brillouin light scattering measurements have been used to study the stress induced modification in the elastic properties of two glass forming polymers: polybutadiene and epoxy-amine resin, prototypes of linear and network polymers, respectively. Following the usual thermodynamic path to the glass transition, polybutadiene has been studied as a function of temperature from the liquid well into the glassy phase. In the epoxy resin, the experiments took advantage of the system ability to reach the glass both via the chemical vitrification route, i.e., by increasing the number of covalent bonds among the constituent molecules, as well as via the physical thermal route, i.e., by decreasing the temperature. Independently from the particular way chosen to reach the glassy phase, the measurements reveal the signature of long range tensile stresses development in the glass. The stress presence modifies both the value of the sound velocities and their mutual relationship, so as to break the generalized Cauchy-like relation. In particular, when long range stresses, by improvise sample cracking, are released, the frequency of longitudinal acoustic modes increases more than 10% in polybutadiene and ∼4% in the epoxy resin. The data analysis suggests the presence of at least two different mechanisms acting on different length scales which strongly affect the overall elastic behaviour of the systems: (i) the development of tensile stress acting as a negative pressure and (ii) the development of anisotropy which increases its importance deeper and deeper in the glassy state. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4902060] I. INTRODUCTION
Vibrational properties of disordered materials are one of the currently lively topics of modern condensed matter physics.1–3 The evolution of elastic constants and of acoustic attenuation, their response to aging, relaxation, applied pressure and temperature are largely investigated in literature.4–10 The importance of these studies is twofold: on the one hand, they contribute to obtain a comprehensive scenario of the fundamental properties of amorphous materials; on the other hand, they can give important hints for designing materials with innovative properties. In fact, while the external conditions are clearly important, other factors are proved to be of critical importance for the final elastic properties including material microstructure, heterogeneities, pre-existing defects and residual stresses.11–15 The presence of internal stress, in particular, strongly affects the final macroscopic properties especially when the materials are produced by solidification from the melt, via quenches or crunches, which push the system in the final state through non-equilibrium configurations. As a matter of fact, the glass is typically a stressed state of matter since the reduced ability of the system to flow under changes of thermodynamic conditions prevents further adjustments into more stable and long range stress-free states. The a) S. Caponi and S. Corezzi contributed equally to this work. Au-
thors to whom correspondence should be addressed. Electronic mail: [email protected] and [email protected] 0021-9606/2014/141(21)/214901/7/$30.00
glass properties can be changed by the thermal and pressure history which is able to drive the system into states characterized by different amounts of residual stress.14, 16, 17 The role of stress has been recently demonstrated, for example, in the modification of the vibrational density of states18 or of the sound velocities, with effects on the Cauchy relation, i.e., the mutual relationship between the elastic moduli.19 The understanding of the mechanism by which the stress affects the elastic properties is important also for nanostructured glasses. As a matter of fact, the fabrication process can induce high tensile or compressive stresses, which have to be thoroughly controlled for technological applications. For instance, a strengthening higher than 10% was observed in nano-spheroids obtained by elongation of polystyrene spheres. Such particles can be used as building blocks of phononic crystals, whose final sound transmission properties will strongly depend on the shape, the constituent material, and also the different amount of stress in the spheroids.20 Moreover, the induced stress can beneficially be exploited in technological applications: it allows to increase the quality factor of nano-resonators21 or to strengthen a material, like in safety glasses or in the so called “gorilla glass,” deliberately pre-stressed during their production. Even if already exploited in the applications, the stress effect on the elastic properties is an aspect often overlooked in the fundamental physics of glasses, but it is, however, of
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© 2014 AIP Publishing LLC
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clear importance for a correct analysis of the behaviour of amorphous materials. The amount of literature data and experimental studies in this field is poor and a complete experimental characterization and theoretical understanding of residual stresses, including their microscopic origin and their role on different length scales, have not yet been achieved. Internal stresses in a glass, depending on the length scale over which they average to zero, can be classified as stresses at the atomic level, micro-stresses at the grain scale, and macrostresses at the continuum scale which may involve the whole sample.12 The present work focuses on the effect of macroand micro-stresses on the elastic properties of glasses. In particular, we report on the stress-induced modification of the elastic properties of two different glassy polymers, used as model systems: polybutadiene and an epoxy resin.
II. EXPERIMENTAL
By means of Brillouin light scattering (BLS) experiments we measured the evolution of the frequency position of Brillouin peaks originating from longitudinal and shear acoustic modes, following different vitrification paths. The samples used and the chosen conditions in the different experiments are described in the next paragraphs. All the thermal treatments have been performed with cooling/heating rates between 0.5 and 2 K min−1 .
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stops, governed by a diffusion-controlled mechanism,26, 27 not all the possible bonds have been formed, and the reaction can resume if the diffusion increases by heating the sample. Combining the system’s ability to achieve the glassy phase both by decreasing the temperature (physical vitrification) and by increasing the number of bonds among the constituent monomers (chemical vitrification), the elastic properties have been investigated following five different thermodynamic pathways: (i) by cooling the unreacted mixture from 275 to 73 K, i.e., from above to well below its glass transition temperature; (ii) during the isothermal polymerization at T = 275 K. The reaction runs out in about 2.5 days, during which the glass transition temperature of the system reaches the value Tg polymerized = 300 K; (iii) by cooling from 298 to 171 K the isothermally polymerized glassy sample, i.e., always below its glass transition; (iv) by cooling from 298 to 73 K the glass obtained by annealing (post-curing) for 10 min at 415 K the isothermally polymerized sample. In the post-cured sample all the possible bonds compatible with the stoichiometry and microstructure of the system have been formed; and (v) by performing quenches at very low temperature during the isothermal polymerization of the reactive mixture: as the reaction proceeds at 275 K, the partially polymerized mixture is quenched to 73 K and then brought back to the reaction temperature. The quenched states are glasses at the same T with different extent of polymerization, and therefore they differently combine the effects of cooling and polymerization.
A. Samples and vitrification paths
Two different polymer systems have been studied: (1) The first system is the synthetic rubber 1,4polybutadiene (PB), a glass forming linear-chain homopolymer. The sample used is a deuterated one, with a molecular weight of 25.000 and a glass transition temperature Tg = 178 K, the same used in Refs. 22 and 23. PB has been investigated following four different thermal treatments: (i) First, it has been slowly cooled down from 328 K to 138 K, i.e., from well above to well below the glass transition temperature. In the other experiments, instead, the sample was cooled down fast below the glass transition using different cooling rates, and then measured as a function of temperature; in particular, the cooling rates were about (ii) 0.5 K min−1 , (iii) 2 K min−1 and (iv) 1 K min−1 . (2) The second system is a reactive epoxy-amine mixture, made of diglycidyl ether of bisphenol-A (DGEBA) and diethylenetriamine (DETA) in the stoichiometric ratio 5:2. The sample was prepared by mixing and stirring the liquid reagents for ∼2 min at room temperature, then transferred into the measurement cell. The unreacted mixture can be easily supercooled. Under cooling, the reaction is prevented and the system behaves as a usual molecular glass-former with Tg unreacted = 231 K. On the contrary, if maintained at a constant temperature, the initially liquid mixture undergoes a step polymerization reaction, by addition of the amino hydrogen to the epoxy group, and it spontaneously vitrifies.24–26 The final product is a glassy network-polymer with glass transition temperature some twenty of degrees above the reaction temperature (Tg polymerized ≈ 300 K). When the isothermal reaction
B. Brillouin light scattering measurements
BLS spectra have been acquired using a Sandercock type (3+3)-pass tandem Fabry-Perot interferometer28 with a finesse of ∼100 and a contrast ratio greater than 1010 . As exciting line, the single mode from a Coherent compass 532/400 laser operating at 532 nm has been used. PB has been measured by using two different sample environments. In the first one, the sample was transferred into a cylindrical Pyrex cell with inner diameter of 10 mm and outer diameter of 11 mm. The cell was sealed in a dry nitrogen environment and mounted into a copper sample holder cooled in vacuum by a nitrogen flux. In the second one, the sample was transferred into a cylindrical Pyrex cell with 3 mm outer and 1 mm inner diameter, sealed in a dry nitrogen environment, and temperature controlled by a helium cryostat Thor Cryogenics 3010 II. BLS measurements have been performed in the back-scattering geometry. DGEBA-DETA in all the experiments was loaded into a cylindrical Pyrex cell of 11 mm inner diameter and 12 mm outer diameter, and temperature controlled by the helium cryostat. Only the post-cured sample was a cracked, freestanding glass. In fact, during the post-curing treatment the Pyrex cell was broken due to the high tensile stress produced by the advancement of the polymerization reaction. Unpolarized spectra have been acquired in the 90◦ scattering geometry. This configuration allowed us to detect both longitudinal acoustic modes, which scatter light without change of polarization, and transverse modes, which scatter light with change of polarization.
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FIG. 1. Longitudinal Brillouin peak of polybutadiene at different temperatures. (Upper panel) Temperature evolution before and (lower panel) after the cracks development which occurs acquiring the spectrum at T = 105 K (full dots). The spectrum at T = 105 K (full circles) presents two different Brillouin peaks: the first before the stress release centered at about 16.5 GHz and the second one after the stress release centered at about 17.3 GHz.
FIG. 2. Longitudinal Brillouin peak of the epoxy resins glass (called reacted mixture) obtained by the polymerization reaction of DGEBA-DETA 5:2 at T = 275 K. Temperature evolution before (upper panel) and after (lower panel) the cracks development which occurs at T = 233 K.
Typical BLS spectra acquired in PB and DGEBA-DETA are shown in Figures 1 and 2, respectively. In order to obtain the angular frequency ω0 of the acoustic waves, each Brillouin peak of line-width , has been fitted by a damped harmonic oscillator function,29
III. RESULTS AND DISCUSSION
I (q, ω) =
ω0 2 I0 , π ω2 − ω 2 2 + (ω)2
(1)
0
convoluted with the instrumental resolution function. The sound velocity v has then been calculated by ω v = 0, (2) q where q is the exchanged wavevector, related to the scattering angle ϑ, the refractive index n of the medium, and the wavelength λi of the incident light through the relation q = 4π n/λi · sin(ϑ/2). In determining the sound velocity, it was considered the temperature dependence of the refractive index of PB, while for the epoxy resin it was considered also the dependence on the extent of polymerization, as previously described in Ref. 30. The real part of the longitudinal modulus M at the frequency of the longitudinal mode ω0L and the real part of the shear modulus G at the frequency of the transverse mode ω0T have been calculated through the relations M (ω0L ) = ρ ρ being the density.30
2 ω0L , q2
G (ω0T ) = ρ
2 ω0T , q2
(3)
A. Sound velocities
Figures 1 and 2 report the temperature evolution below Tg of the longitudinal Brillouin peak in PB and in the polymerized epoxy resin. The elastic behavior in both samples presents clear analogies and in the following the two cases are presented in parallel. The PB data reported in Fig. 1 show that, by decreasing the temperature, the continuous shift of the Brillouin peak towards higher frequency is abruptly interrupted. At T = 105 K an improvise transition from one to another elastic state occurs. In fact, in the same acquisition two different longitudinal Brillouin peaks are present: the peak centered at about 16.5 GHz is suddenly replaced by another one centered at about 17.3 GHz. The improvise modification in the elastic condition appears in correspondence of the development of internal cracks in the sample. The lower panel of Fig. 1 shows the temperature evolution of the Brillouin peak after the cracks development: starting from the new elastic state, the samples stiffness now evolves with a smooth temperature dependence. The temperature location of the jump and the frequency position of the Brillouin peak is strongly dependent on the cooling rate and on the sample environment. For the sake of comparison, all the acquired data on PB are summarized in terms of longitudinal sound velocity in Fig. 3. The data obtained following the temperature evolution from the liquid well into the glassy phase are reported as circles.
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FIG. 3. Temperature dependence of the longitudinal sound velocity in PB measured using different cooling rates and choosing different samples environments. The circles (line A) represent the values obtained following the evolution from the liquid state; the sample is contained in a cylindrical cell with the inner diameter of 10 mm. Stars (line D), triangles (line C), and dots (line B) are obtained quenching the system from room temperature to the glassy phase using increasing cooling rates; the samples are contained in a cylindrical cell with inner diameter of about 1 mm.
Decreasing the temperature below the glass transition, two effects are found: (i) increasing the cooling rate, a gradual departure towards lower sound velocity values progressively occurs; (ii) abrupt changes of the sound velocity are evident in correspondence of cracks developments in the glassy structure. The cracks occur at different temperatures in the two performed experiments (T = 35 K line B and T = 105 K line C). Starting from different elastic values, the sound velocity variation is also different. It reaches considerable values, up to ∼10%. However, it is worth noting that after cracks developments the system is brought towards a macroscopic elastic state which, within the experimental error, is independent from the thermal history and is maintained in the successive cooling (line B) or heating procedures (line C). A similar scenario is found in the fully polymerized sample of epoxy resin. The sample, which exploited the chemical vetrification process, is always in the glassy phase. In this case the chosen experimental setup let us to simultaneously collect the longitudinal and transversal Brillouin peaks. The temperature evolution of the spectra are shown in Figure 2 while the values of the sound velocities acquired following different cooling histories are summarized in Figure 4. By cooling the glass from room temperature, history dependent sound velocities are found. At T ∼ 240 K, in correspondence of cracks development, an upward variation is measured in the Brillouin frequencies (Fig. 2), and a smoother temperature dependence in the lower temperature range is found. Further information, in this case, can be also achieved: (i) the first cooling experiment evidences how the elastic transition at T ∼ 240 K simultaneously occurs in both longitudinal and transversal sound velocities and (ii) a similar relative variation of more than 4% has been found in both longitudinal and transversal sound velocities. Once cracks developed in the structure, the glass was heated at room temperature and the measurements were re-
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FIG. 4. Temperature dependence of the longitudinal (full symbols) and transversal (open symbols) sound velocities of the reacted mixture DGEBADETA 5:2. The dots (line A + B) represent the values obtained during the first cooling where the jump to higher values occurs at T = 240 K. The squares (line C) are obtained on the same sample reheated at room temperature and cooled down again.
peated. The data now show a smooth and continuous temperature dependence (line C in Fig. 4). It is worth noting that the two investigated samples are microscopically very different: the first is a linear polymer that by cooling reaches the glassy phase, while the second is arranged in a more complex branched structure and the glassy phase is obtained by a step-polymerization process. Nevertheless, both the obtained glasses develop strong internal stresses. In fact, on crossing the glass transition and deep in the glassy phase, the inability to flow put the system into a potentially stressed and out of equilibrium condition where a non-unique elastic behaviour can be expected. Moreover, because of the interaction with the environment (the cuvette, characterized by a strongly positive wettability and lower thermal expansion coefficient) the thermal contraction is hindered and the negative pressure consequently developed, increases during the cooling down of the system. The pressure release can occur either by the development of internal cracks in the glassy sample or by the breakage of the cell’s wall, depending on the relative fragility of the sample and sample holder. Indeed, the breakage of the Pyrex cell always occurs in the PB sample using the largest cuvettes. In this case, the measurements below ∼130 K are prevented (circles data), while the development of internal cracks occur using the smallest cuvette characterized, cause of its geometrical configuration, by a higher breaking load. The measurements in this case were possible down to 35 K. For both samples, the results suggest that, by cracks development, the glass is able to reduce the stresses trapped inside its structure, so inducing strong modifications of its elastic response. This process gives rise to the development of a permanent modification in the elastic properties of the glass, so that the same values of acoustic wave velocities are reproduced by further cycles of heating and cooling. Finally, it is noteworthy that in both samples the sound velocity increases after the crack formation. This result is in
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agreement with the previously observed behavior in the temperature evolution of DGEBA19 where, however, the jump in the sound velocities was only ∼1%. This is not an obvious effect since the release of the tensile stresses should be associated with an increase of the density and therefore, in the linear elasticity regime, with a reduction of the sound velocity. On the contrary, the observed increase indicates that, because of the tensile stress, the systems experienced a softening of the elastic moduli. This behavior is in agreement with the positive value of the Grüeneisen parameters for polybutadiene and epoxy resins31, 32 implying markedly nonlinear features, and in particular an increase of the moduli with the increase of density which overcompensates the pure density effect on the sound velocities. In Sec. III B we will show how finely monitoring the stress induced modification of the elastic constants, by studying the relative behavior of the elastic moduli. Our data provide new insights into this phenomenon as we study it in different materials and as function of different thermal and strain histories. B. The generalized Cauchy relation and its deviations
It is well known that the elastic behavior of a homogeneous, isotropic solid at equilibrium is fully described by the longitudinal and the shear modulus. If, in addition, the atoms interact through a central potential, the Cauchy identity, M = 3G , holds and only one independent elastic constant exists.33 In glass forming systems, the real parts of the longitudinal, M (ω), and transverse, G (ω), elastic moduli measured by BLS, satisfy a generalized Cauchy-like relation, M (ω) = A + BG (ω),
(4)
where B has been found close to 3 and A is a systemdependent constant which, at ambient pressure, does not appreciably depend on temperature. The validity of Eq. (4) has been verified by BLS both in the unrelaxed (high frequency) limit of glass forming systems34 and even in relaxing liquids.30, 35 Now the question we face is the following: Does the stress field modify the mutual relation between the elastic moduli? And if yes, how? To answer, we report in Figure 5(a) the data acquired at different temperatures and at different polymerization degrees in the epoxy-resins. In the low frequency region, the Cauchy like relation is well fulfilled, and the same line, M (ω) = A + BG (ω) with A = (3.09 ± 0.03) GPa and B = 3.00 ± 0.01, which connects the elastic moduli during the whole polymerization process, describes the data at the highest investigated temperatures for the unreacted mixture (red circles) and for the polymerized sample (green squares). A progressive departure from the expected values is instead well visible entering deeper and deeper in the glassy phase for the different investigated systems. In Figure 5(b) the difference between the experimental data and the estimated Cauchy line is reported. This data representation visually enhances the deviations that occur below as well as above the expected values shown as dashed line. The behavior of the polymerized sample, which presents a jump in the sound velocity (Line A + B in
FIG. 5. Panel (a) Real part of the longitudinal modulus, M , versus real part of the transverse modulus, G , of the DGEBA−DETA 5:2 mixture during the isothermal polymerization reaction at 275 K (gray ♦), by cooling the reacted mixture (black : line A + B of Fig. 4 and green : line C of the previous figure), the post cured sample (blue ), the unreacted mixture (red ●), and quenching at 73 K the sample at different polymerization extents (open triangles). Error bars are comparable to the symbol size. The solid line is the Cauchy-like relation M = (3.13 ± 0.03) + (3.00 ± 0.01)G . Panel (b) Difference between the experimental data and the expected Cauchy-like behaviour. The data are shown with the same symbols as in panel (a).
Fig. 4), will drive our discussion. Indeed, the data acquired before the sample cracking (line A, T > 240 K) deviate below the Cauchy line, while after the cracks development (line B, T < 240 K) the data exceed the expected values, suggesting that cracks development plays a role in the deviation from the Cauchy law. Repeating the measurements in the cracked glass where the tensile stresses are mostly released (green squares – line C showed in Fig. 4), the Cauchy line is recovered over a wider temperature range, up to T = 253 K. At this temperature a deviation again occurs but only above the expected values. The temperature evolution of the unreacted mixture (red dots) and the post-cured sample (stars) show both the deviations below and, at the lowest temperatures, above the Cauchy-line. In the quenched states at T = 73 K, at different polymerization degrees, cracks always occur in the samples, and the data reveal only upwards deviations.
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As a preliminary consideration, we would notice that in different samples, we obtain both the upwards or the downwards deviations evidencing that, independently from the thermal history, the mechanisms at the origin of the Cauchy breaking are different and at least two: both of them are always active and they play competitive roles causing opposite deviations. Changing the sample conditions, for example after the cracks development, their relative weight drastically changes making one dominant over the other. However, the deviations always occur when the samples are below their glass transition and hence under a condition in which a stressfield may extend over long range scales. By adding the cooling effect to the polymerization process, we are able to push the system deeper and deeper in the glassy phase increasing the amount of unrelaxed stresses. In fact, before the cracks development, the adhesion to the cell walls produces an effective negative pressure and the progressive development of tensile stress, which is mainly planar, as testified by the deep concave meniscus in the glassy states. In these conditions, also elastic anisotropy increases. The deviation of the M vs G relationship below and above the Cauchy line could be related to stresses on different length scales. In fact, while the long range stresses originate a negative pressure, stresses on lower length scales can be responsible for the permanence of anisotropy also after the cracks development. Both the negative-pressure and the microscopic residual anisotropy would produce a failure of the conditions for the Cauchy validity.33 It appears that their effects can produce deviations of different amount and can play a role in different experimental conditions. Before the cracks development, the effect of negative pressure seems to be the dominant mechanism responsible for the Cauchy law breaking. In fact, when the negative pressure is drastically reduced by cracks, the stress component on a macroscopic scale is released, the downwards deviation disappears and the Cauchy line is recovered over a wider temperature range in the cracked sample. The experimental data support this scenario also from a quantitative point of view. In fact, it is well documented that the applied pressure influences the longitudinal and the transversal phonon branches leading to a modification of the Cauchy relation with respect to that valid at P = 0.36–38 Considering the pressure dependent parameter predicted in the generalized Cauchy-like relation,34 the equation reads M (ω) = A + BG (ω) + 2P .
(5)
The last term is usually neglected and/or incorporated in the constant value A when the pressure variation is negligible; in fact, its value is usually very low compared with the material elastic constants (∼GPa). On the contrary, when its variation begins to be comparable with the elastic constants, its contribution cannot be any longer neglected. By cooling the sample under its glass transition, due to the adhesion to the cell walls, the glass can experience a negative pressure that becomes increasingly higher during cooling. A simple argument can lead to evaluate such pressure, assuming that it has to balance the contraction due to the decrease of temperature. From the stress-strain relation and considering the thermal expansion coefficients, we can estimate the tensile stress, σ , ex-
ercised by the cuvette surfaces as39 σ (T) =
2G (3M − 4G )α(T − Tg ), M
(6)
where α is the difference between the thermal expansion coefficient of the polymer (whose typical value is ∼1 × 10−440 ) and pyrex (3 × 10−6 ), and the term with the M and G combination is an effective elastic modulus for purely radial stress. On the basis of the elastic moduli values, the expected stress increase is ∼1 MPa K−1 . This prediction is well confirmed by the experimental data where, during cooling, the downward departure from the linear Cauchy relation, which we attributed to the term 2P of Eq. (5), is ∼200 MPa over a temperature range of 60 K. This maximum negative pressure is in agreement with the value of tensile strength of the epoxy resin, which is around 85 MPa:41, 42 when the negative pressure overcomes this limit, cracks develop in the material. The same arguments can be applied to PB, where using the data in Ref. 43 the induced pressure per Kelvin is estimated to be ∼1 MPa K−1 and therefore the total pressure increase that develops below the glass transition results in fairly good agreement with the PB tensile strength of the order of tens of MPa.44 Whereas the downward departure from the Cauchy behaviour seems to be well explained on the basis of a negative pressure term,45 the upward deviation at very low temperature is still a puzzling result, but a tentative explanation can be, however, provided. In fact, it should be considered that the violation of the Cauchy relation can be caused by different mechanisms other than pressure, including the development of non-central forces, many body interactions, heterogeneity or anisotropy in the sample.33 In particular, internal cracks are able to release most of the macroscopic stress, while stresses on smaller length scales may be not significantly modified. As a consequence, in a tentative interpretation the upward deviation may be ascribed to elastic anisotropy effects, which intensify by decreasing the temperature, and reach their maximum value in the quenched samples. Following this interpretation, the experimental data give also a hint on the characteristic size of the elastic anisotropy regions. In fact, due to the topologically disordered structure, microscopic stresses are always present in the glass but, in order to significantly affect the collective dynamics of the material, their characteristic size has to be comparable with the length scale probed by the investigated phonons. By changing the experimental technique the system is investigated on different length scales and, for example, the effect of the elastic heterogeneity at few nanometer scale well detectable by inelastic x-ray scattering,46 it is not revealed by BLS measurements. The wavelength of the phonons probed by BLS is λp ∼ 200 nm, and a stress field with characteristic dimension much smaller than λp /10 leaves unaffected the phonon propagation: the phonons in this case will travel as in an homogeneous and isotropic system.47 On the contrary, when the length scale of the stress field starts to be comparable with the phonon wavelength, its presence can strongly influence the propagation of the acoustic modes, modifying both their attenuation and velocity.
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IV. CONCLUSIONS
In conclusion, the paper reports how the stress field influences the acoustic properties of materials on the basis of original data for polybutadiene and epoxy resins, which are prototypes of linear and network polymers. The results show that the stress field affects on the one hand the absolute values of the Brillouin frequencies and on the other hand the mutual relationship between the elastic moduli. The results represent a systematic extension of existing experimental works. The rupture of the Cauchy relation seems to be a sensible marker of the stress presence in the sample: independently from the stiffening path, the states in which the system is able to release internal stresses follows the Cauchy law, otherwise the Cauchy relation fails. We point out how the existence of stress effects on the elastic properties is correlated with the presence of at least two different mechanisms acting on different length scales: the first one is interpreted in term of the development of tensile stress acting as a negative pressure, while the second one is tentatively interpreted with the persistence of anisotropy on the mesoscopic length scale. The two competitive mechanisms cause opposite deviations and their interplay is responsible of a complex elastic scenario. The picture that we provide is able to give a reasonable interpretation of the experimental data and may be a useful hint to a complete theoretical description, still missing. The study could also be useful for guiding the experimental work both in the application field and in the fundamental physics, where the role of the sample environment and the chosen cooling rate should be considered more carefully in the evaluation of the material properties. ACKNOWLEDGMENTS
S.C. and S.C. acknowledge support from MIUR-PRIN (Project 2012J8X57P), S. Caponi acknowledges support from CNR Flagship Project “Nanomax (N-CHEM),” PAT (Autonomous Province of Trento) “Grandi Progetti 2012” Project “MaDEleNA” and M.M. acknowledges support from EC founded Landauer project (G.A. No. 318287). 1 M.
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