Evidence of Deformation-Dependent Heat Capacity and Energetic

Article Cite This: Macromolecules XXXX, XXX, XXX−XXX

Evidence of Deformation-Dependent Heat Capacity and Energetic Elasticity in a Cross-Linked Elastomer Subjected to Uniaxial Elongation David Nieto Simavilla,† Jay D. Schieber,†,‡ and David C. Venerus*,† †

Department of Chemical & Biological Engineering and Center for Molecular Study of Condensed Soft Matter and ‡Department of Physics and Department of Applied Mathematics, Illinois Institute of Technology, Chicago, Illinois 60616, United States S Supporting Information *

ABSTRACT: We present a novel infrared thermography technique to investigate the dependence of heat capacity on deformation in cross-linked polymers. This phenomenon is directly related with the longstanding question of whether or not there is an energetic contribution to the stress in deformed polymers and, in general, to the nonisothermal, viscoelastic behavior of polymer materials. By tracking the temperature evolution of samples heated by a laser, we are able to measure heat capacity changes relative to the equilibrium value in an elastomer subjected to uniaxial extension. We find that the heat capacity increases with elongation in lightly cross-linked cis-1,4polyisoprene. Remarkably, the onset of heat capacity dependence on deformation is observed at strains similar to those required to achieve finite extensibility. The deviation from the equilibrium value of heat capacity is consistent with an independent set of experiments comparing anisotropy in thermal diffusivity from forced Rayleigh scattering and thermal conductivity from steadystate infrared thermography. Finally, we propose a straightforward thermodynamic analysis of the results based on classical rubber elasticity.

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INTRODUCTION Deformation during the processing of polymeric components induces molecular orientation, which in turn induces changes in thermophysical properties such as thermal conductivity or, as this study shows, heat capacity. Polymeric materials are ubiquitous, and understanding the connection between microstructural orientation and macroscopic physical properties is important not only in optimizing fabrication processes but also in assessing the performance of polymeric materials during use. Elastomeric materials, or rubbers, are polymers that possess permanent cross-links. In contrast to polymer liquids, the presence of cross-links in elastomers prevents the complete relaxation of the chain segments back to the isotropic state as long as the macroscopic deformation is held.1 The ability to achieve equilibrium in the deformed state offers important experimental advantages in the study of the effect that deformation has on thermophysical properties, since the anisotropic orientation of the polymer chains is maintained during the measurements.2−4 Nevertheless, similar effects are expected in polymer melts. The thermodynamics of rubber elasticity has been the focus of extensive research for more than half a century.1,5−8 Since most experiments are carried out under controlled temperature and deformation, we consider the Helmholtz free energy density f(T,λ) = u − Ts, where λ is the stretch ratio in uniaxial elongation and u and s are the internal energy and entropy densities, respectively. The engineering stress σeng = - /A 0 , where - is the © XXXX American Chemical Society

tension and A0 is the cross-sectional area of the undeformed sample, is the thermodynamic variable conjugate to λ ⎛ ∂σeng ⎞ ⎛ ∂f ⎞ ⎛ ∂u ⎞ ⎛ ∂s ⎞ ⎛ ∂u ⎞ σeng = ⎜ ⎟ = ⎜ ⎟ − T ⎜ ⎟ = ⎜ ⎟ + T ⎜ ⎟ ⎝ ∂λ ⎠T ⎝ ∂λ ⎠T ⎝ ∂λ ⎠T ⎝ ∂λ ⎠T ⎝ ∂T ⎠ λ (1)

From the second equality in (1), we can separate energetic σeeng = (∂u/∂λ)T and entropic σseng = −T(∂s/∂λ)T contributions to σeng. Neglecting the energetic contribution, which is usually much smaller than the entropic contribution, is referred to as the assumption of purely entropic elasticity (i.e., the internal energy density u is considered a function of temperature only). The third equality in (1) is obtained using standard thermodynamic manipulations and gives (∂u/∂λ)T = σeng − T(∂σeng/∂T)λ. Invoking the assumption of purely entropic elasticity, we have σeng = T(∂σeng/∂T)λ, which implies σeng is linear in T or that (∂2σeng/∂T2)λ = 0. This result is consistent with well-established experimental data for elastomers subjected to uniaxial elongations with 1.1 ≳ λ ≳ 3.1,5,7 The The specific heat capacity per mass at constant elongation ĉλ is defined as Received: October 5, 2017 Revised: November 29, 2017

A

DOI: 10.1021/acs.macromol.7b02139 Macromolecules XXXX, XXX, XXX−XXX

Macromolecules ⎛ ∂ 2f ⎞ ⎛ ∂s ⎞ ρcλ̂ = T ⎜ ⎟ = −T ⎜ 2 ⎟ ⎝ ∂T ⎠ λ ⎝ ∂T ⎠ λ

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INFRARED THERMOGRAPHY We consider a rectangular sheet of material having thickness d in the z3-direction. The sample is subjected to simple elongation in the z1-direction so that the length L = λL0, where L0 is the undeformed length. For an isochoric deformation the width, W = W0/ λ and d = d0/ λ , where W0 and d0 are the initial width and thickness of the sheet. A laser beam with Gaussian intensity distribution propagating in the z3-direction is applied at t = 0 and passes through the center of the sheet (see Figure 1). The beam

(2)

where ρ is the mass density, which we assume is constant.5,9 Since σeng = (∂f/∂λ)T, if purely entropic elasticity is assumed, then from (2) one finds (∂ĉλ/∂λ)T = 0, or ĉλ is independent of deformation λ. Some experiments on cross-linked polyisoprene,5−7 however, indicate that the energetic contribution (∂u/∂λ)T accounts for 10−20% of σeng and shows a deformation dependence at moderate values of λ. Hence, the assumption of purely entropic elasticity does not appear to be generally valid and this leaves open the possibility of a deformation-dependent heat capacity. A number of studies have examined the assumption of purely entropic elasticity in flowing polymer liquids. Astarita et al.10−12 measured the temperature changes in polyisobutylene (PIB) and poly(vinyl acetate) (PVA) melts induced by viscous dissipation during shear flow. By assuming adiabatic conditions, the heat capacity was obtained as the ratio between the applied work and the change in temperature. Astarita et al. found the heat capacity to be independent of deformation for PIB, as suggested by the assumption of purely entropic elasticity. On the other hand, they observed an increase in heat capacity for PVA subjected to uniaxial extension. This deviation from the theory was attributed to an energetic contribution to stress. However, their analysis assumes that these experiments were carried out under adiabatic conditionsan assumption whose violation would also explain the observed deviations from the equilibrium heat capacity. Also, as Ionescu et al.13 pointed out, the deformation rates employed were too low to obtain significant orientation of the microstructural conformation. These arguments suggest that the experiments of Astarita et al.10−12 are ambiguous and not a clear test of the purely entropic elasticity assumption. More rigorous thermodynamic analysis of nonisothermal flows in viscoelastic liquids have been considered by several researchers.13−17 In these studies, the internal energy density is allowed to depend on a conformation tensor leading to a modified form of the temperature equation. The internal energy contribution to the temperature equation due to local conformation changes was argued to be essential in the description of nonisothermal viscoelastic flows. However, the temperature differences resulting from the changes in heat capacity induced by the flow of polyethylene melt through a die carried out by Ionescu et al.13,16 experimental setup were too small to be measured. It is fairly well-known that deformed polymers exhibit anisotropy in thermal conductivity,18,19 which has been investigated by our group using an optical technique known as Forced Rayleigh Scattering (FRS)2,20−22 and, more recently, with a complementary method based on Infrared Thermography (IRT).3 In the present study, we use IRT to measure the changes in heat capacity relative to the equilibrium value in lightly crosslinked cis-1,4-polyisoprene subjected to uniaxial elongation. First, we present a model for the analysis of the transient IRT experiments. Then, we give the details on sample preparation, experimental setup, and procedures. Next, the results of our experiments are presented and compared with those given by classical rubber elasticity. Finally, we summarize our findings and discuss the remaining questions for this fundamental research topic.

Figure 1. Schematic of rectangular sheet heated by a laser beam propagating in the z3-direction.

has a waist w and maximum intensity I0. Only a small fraction of the beam’s energy is absorbed by the sample Kd < 1, where K is the absorption coefficient of the sample at the laser beam wavelength. The diffusive heat flux q in anisotropic materials is described by a generalized form of Fourier’s law: q = −k·∇T, where k is the thermal conductivity tensor and ∇T the temperature gradient. For an isotropic solid, symmetry implies k = keqδ, where keq is the equilibrium thermal conductivity and δ is the identity tensor. Using the coordinate system described above, the thermal conductivity tensor of a material subjected to uniaxial extension is given by a diagonal matrix with k22 = k33.23 Hence, the temperature equation can be written as ∂T ∂ 2T ∂ 2T ∂ 2T = k11 2 + k 22 2 + k 22 2 + KI0 ∂t ∂z1 ∂z 2 ∂z 3 2 2 ⎛ z +z ⎞ × exp⎜ −2 1 2 2 ⎟ exp( −Kz 3) w ⎠ ⎝

ρcλ̂

(3)

where the kii are taken as constants. Initially, and far from the source, the sample is at temperature T0, the temperature in the gas surrounding the sheet, so that T (z1, z 2 , z 3 , t ≤ 0) = T (±∞ , z 2 , z 3 , t ) = T (z1, ±∞ , z 3 , t ) = T0

(4)

The boundary conditions at the sample faces are given by Newton’s law of cooling B


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Figure 2. Temperature evolution at x1 = x2 = 0 for the exact solution given in (10) (solid) and approximate solution in (11) (dashed) for two different stretch ratios: λ = 1 (α1 = α2 = 1) black curves and λ = 4 (α1 = 1.3, α2 = 0.95) red curves. The insets show the fractional residuals for λ = 1 and λ = 4. Comparison for Bi0 = 0.035 (a) and Bi0 = 0.35 (b).

− k 22

∂T (z1, z 2 , ± d /2, t ) = ± h[T (z1, z 2 , ± d /2, t ) − T0] ∂z 3

characteristic time for diffusive heat transfer. We note that the fin approximation is expected to be valid when Bi ≤ Bi0 ≪ 1.24 The solution of (7) subject to (8) can be found using Green’s function (see Appendix A in the Supporting Information) and is given by

(5)

where h is the heat transfer coefficient. A detailed analysis on the validity of treating the finite sheet as infinite in the z1- and z2directions, as indicated by the boundary conditions in (4), can be found elsewhere.3 Since L ≫ d and W ≫ d, we assume that temperature variations in the z3-direction are small so that the fin approximation can be used. Hence, we define the average temperature: 1 ⟨T ⟩(z1 , z 2 , t ) = d

θ(x1, x 2 , τ ) =

where τ′ = τ − τ0. To obtain τD from the experiments described in the following section, we consider only temperature measurements at the origin, that is, at the center of the laser heating spot (x1 = x2 = 0). Hence, from (9) we obtain

(6)

Introducing ⟨T⟩ and using the approximations T(z1, z2, ±d/2,t) ≈ ⟨T⟩ and Kd ≪ 1 allows us to write (3)−(5) as ⎛w⎞ ∂θ ∂ 2θ ∂ 2θ = α1 2 + α2 2 − 2Bi⎜ ⎟ θ + exp[− 2(x12 + x 2 2)] ⎝d⎠ ∂τ ∂x1 ∂x 2 2

(7)

θ( ±∞ , x 2 , τ ) = θ(x1 , ±∞ , τ ) = 0,

θ(0, 0, τ ) =

θ(x1 , x 2 , 0) = 0 (8)

where xi = zi/w (i = 1,2), θ =

⟨T ⟩ − T0 , 2KP0 / πkeq

∫0

τ

⎡ exp⎢ −2Bi ⎣

2

⎤

( wd ) τ′⎥⎦

(1 + 8α1τ′)(1 + 8α2τ′)

dτ ′ (10)

which requires numerical integration. From steady-state IRT experiments,3 we find the Biot Number Bi ∼ 0.035, which is consistent with the values given by correlations.25 Since our estimated beam waist to sample thickness ratio is w/d ∼ 0.05, we have Bi(w/d)2 ∼ 10−4, which suggests that the exponential term can be neglected at sufficiently small times. For the case Bi(w/ d)2τ ≪ 1, (10) can be integrated analytically26 to obtain

and P0 = I0πw /2 is the 2

total power in the laser beam. In (7) we have introduced the thermal conductivity ratios αi = kii/keq, and the Biot number Bi = Bi0/ λ with Bi0 = hd0/keq. Additionally, we have introduced the dimensionless time τ = t/τD, where τD = w2/(keq/ρĉλ) is the θ(0, 0, τ ) ≅

⎡ ⎡ 2x 2 ⎤ 2x 2 ⎤ exp − 1 exp⎢ − 1 + 82α τ ′ ⎥ ⎡ ⎣ ⎛ w ⎞2 ⎤ ⎣⎢ 1 + 8α1τ ′ ⎦⎥ 2 ⎦ exp⎢ −2Bi⎜ ⎟ τ′⎥ dτ ′ ⎝ d ⎠ ⎦ 1 + 8α1τ′ 1 + 8α2τ′ ⎣

(9)

d /2

∫−d/2 T(z1, z 2 , z3 , t ) dz3

∫0

τ

⎡ 2 α α (1 + 8α τ )(1 + 8α τ ) + 16α α τ + α + α ⎤ 1 1 2 1 2 1 2 1 2⎥ ln⎢ 2 ⎥⎦ 8 α1α2 ⎢⎣ ( α1 + α2 )

In order to establish the range of time for which the approximate expression in (11) can be used to analyze experiments, we compare it to (10) for the cases λ = 1 and λ =

(11)

solution in (10) and the approximate solution in (11) are in good agreement for τ ≲ 60. We conclude that for a dimensionless time τ = 50, with w ∼ 50 μm and keq/ρĉλ ∼ 10−3 cm2/s, the expression given by (11) can be used to model our experiments for times up to t ∼ 1 s. In Figure 2b, we show a similar comparison for a larger Biot number Bi0 = 0.35. In this case, (11) deviates from (10) at much shorter times (τ ≲ 20), which significantly reduces the

4 for different values of Bi = Bi0/ λ . For the λ = 4 case, we use typical values2,3 for the thermal conductivity ratios α1 = 1.30 and α2 = 0.95. Using the estimated Bi0 = 0.035, we see that the exact C


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Macromolecules number of data points available for the fit of the experimental data to (11). These results are used in the design and analysis of the experiments described in the following sections.

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EXPERIMENTAL SECTION

Sample Preparation. Natural rubber samples were prepared by cross-linking deproteinized cis-1,4-polyisoprene (SMR-S from H.A. Astlett Co.) with dicumyl peroxide (Luperox DCP). This is the same polymer and cross-linking agent used in the study by Mott et al.27 and Nieto et al.2 For un-cross-linked polyisoprene (PI), we find28 the molecular weight between entanglements Me = 6.5 kDa, the plateau modulus GN = 372 kPa, and density ρ = 864 kg/m3. Both FRS and IRT techniques require the samples having certain optical absorption properties3,20 at the heating beam wavelength (∼514 nm). Addition of a dye known as Oil Red O (Sigma-Aldrich) at a concentration of 0.015% (relative to the polymer mass) results in an absorption coefficient K = 13 cm−1 (see Figure S1 in the Supporting Information). Samples were prepared by a solvent-casting procedure in which dye, cross-linker DCP (2% relative to the polymer mass) and polymer (nominally 10% by mass) were dissolved in toluene. The solution was poured into a Teflon mold assembly with a removable glass bottom and then was allowed to dry for 1 week in ambient conditions and four additional weeks in a vacuum chamber. Confirmation that the residual solvent had been removed was made by comparing small-amplitude oscillatory shear measurements of the as-received material with the dried material. Frequency-dependent dynamic modulus measurements in small-amplitude oscillatory shear indicate that the as-received material is partially cross-linked (see Figure S2). After removal of the solvent, a second glass plate was pressed on the exposed surface of the polymer film. The glass−sample−glass sandwich was placed in a temperaturecontrolled press for cross-linking at 160 °C for 30 min, after which the sample was quenched in an ice bath.2,27 The cross-linked sheet was removed from the glass plates, and a dog-bone-shaped specimen was punched from it. Test specimens with a thickness of 1.2 mm, center width of 18.0 mm, and overall length of 60 mm were used for subsequent testing. In order to obtain reproducible mechanical results, all samples were prestretched and allowed to relax for at least 24 h. On the basis of work by Mott et al.,27 who designate this material as NR-2, we estimate a cross-link density of 100 mol/cm3, which is close to the entanglement density ρ/Me ≅ 133 mol/cm3. Mechanical Testing. Dog-bone-shaped samples were uniaxially stretched (in the z1-direction) using a custom-made stretching frame in which the sample was clamped between fixed and movable clamps. A weight with known mass was attached to the movable clamp causing the sample to elongate. The distance L between two ink lines (L0 = 10.0 mm) marked on the sample was measured periodically using a cathetometer to determine the stretch ratio λ = L/L0. A minimum of 3 h was allowed for the sample to reach equilibrium (constant λ), after which time the movable clamp was locked in position. The extra stress tensor τ in uniaxial elongation is diagonal and for symmetry reasons τ33 = τ22. The applied force - and cross-sectional area A = A0/λ determine the true stress σ = σ engλ = τ11 − τ22 = - /A . The equilibrium stress−strain results from these experiments are presented in Figure 3 in the form of a Mooney−Rivlin plot, which is based on the expression

σ = 2(C1 + C2/λ) λ 2 − 1/λ

Figure 3. Mooney−Rivlin representation of the mechanical behavior of cross-linked cis-1,4-polyisoprene (blue squares) at 21 °C. The solid line corresponds to a least-squares fit to the modified Langevin equation in (13). where n is the number of statistical units in the chain and 3(x) = coth(x) − 1/x is the Langevin function. The fit of (13) shown in Figure 3 uses 2C1 = 271 ± 43 kPa, 2C2 = 111 ± 20 kPa, and n = 83 ± 22. For lightly cross-linked materials in the limit λ → 1 the plateau modulus is approximated by GN ≃ 2(C1 + C2) = 382 ± 63 kPa, which is in agreement with the literature value of GN = 372 kPa.28 Infrared Thermography (IRT). The details on the IRT experimental setup have been given elsewhere,3 and here we give a brief description. An Ar+ laser passing through an optical train consisting of three mirrors, an attenuator (OD = 2.0), and a focusing lens is used as heating source. The stretching frame is placed horizontally after the last mirror, so that the sample center is at the minimum waist of the laser (w ≃ 50 μm). An infrared camera (FLIR A320) measures the surface temperature of the sample from the side opposite to the incoming laser. The camera has a spatial resolution of 320 × 240 pixels and a sensitivity of about 0.1 °C, and it is equipped with an 18 mm focal length lens, which results in a pixel pitch of ∼50 μm. During the experiments, the power P0 of the laser beam is adjusted (∼0.750−0.900 W) so that the amplitude of the change in temperature once steady state is reached (≲10 °C) is small enough so as to not alter the sample properties but large enough to give a reasonable signal-to-noise ratio. Note, however, that the temperature change in the data used for the analysis of the transient experiments is ≲5 °C. All experiments were carried out at room temperature, which was nominally 21 °C. After aligning the sample in the optical train, recording of the sample temperature is initiated, and after approximately 1 s the sample is exposed to the Ar+ laser. Data collected before exposure to the laser are used to determine the initial temperature T0. Figure 4 shows the temperature evolution for a stretched sample (λ = 3.86). The first five images (t = 0 → 1 s) are in the time range relevant for transient IRT experiments using the temperature evolution given by (11). The last image (t = 200 s) shows the steady-state temperature distribution, which can be used to determine the thermal conductivity ratios α1 and α2.3 To improve the accuracy of the method, several runs were performed and averaged. Sufficient time was allowed between experiments for the sample to recover a uniform temperature (T = T0). As discussed in the preceding section, our analysis of the transient IRT experiments is based on monitoring the time dependence of the sample temperature at the origin using the expression in (11). To fit the dimensionless expression in (11) to experimental data, parameters used to make temperature and time dimensionless must be determined. The parameter for temperature is Cθ = 2KP0/πkeq and the parameter for time is τD = w2/(keq/ρĉλ). For stretched samples (λ > 1), the expression in (11) contains the parameters α1 and α2, which are determined from steady-state IRT measurements.3 For each experiment, a least-squares

(12)

where C1 and C2 are phenomenological constants. It is clear from Figure 3 that (12) does not capture the upturn observed at large strains, which is commonly associated with finite extensibility of the chain segments. The solid line in Figure 3 corresponds to the modified Langevin equation,5 which accounts for non-Gaussian statistics of the chain segments at large deformations σ = 2(C1 + C2/λ)

λn1/2 −1 −1/2 [3 (λn ) − λ−3/2 3−1(λ−1/2n−1/2)] 3 (13) D


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Figure 4. Typical time sequence of a transient IRT experiment in a sample stretched in the z1-direction to λ = 3.86.

Figure 5. (a) Measured temperature evolution ΔT = T(0,0,d/2) − T0 for an unstretched sample (λ = 1, blue circles) and stretched samples (λ = 1.98, green circles; λ = 3.06, red circles). Fitting of (11) to the different data sets (black lines) results in the parameter values given in Table 1. (b) Residuals cumulative distributions (symbols) compared to normal cumulative distributions (solid lines). method is used to determine the parameters Cθ and τD. An example of this fitting procedure is given in Figure 5a, which shows data for one unstretched and two stretched samples. In Figure 5b, we see that the residuals cumulative distribution P is in good agreement with a normal distribution with standard deviation of 0.0008 ± 0.0002. The results from fits to the data in Figure 5a are presented in Table 1. Here, we have neglected the deformation-induced changes in density for the stretch ratios of interest in this study (λ = 1−4) since for rubbers ΔV/V < 10−4.5,9 Therefore, ρ is independent of λ, and we can write cλ̂ τD(λ) = ceq̂ τD(λ = 1)

Table 1. Results from the Fit of (11) to the Data in Figure 5a

a

λ

Cθ (°C)

τD (s)

α1a

α2a

1 1.98 3.06

11.6 ± 0.3 12.1 ± 0.3 12.5 ± 0.2

1.61 ± 0.10 1.60 ± 0.11 1.86 ± 0.08

1 1.04 1.10

1 0.97 0.97

From steady-state IRT.3

Forced Rayleigh Scattering. Forced Rayleigh scattering (FRS) is a sensitive and noninvasive optical technique for the study of diffusive transport in transparent condensed matter.20,29,30 Only a short description of the technique is given here. The intersection of two beams from a coherent (Ar+ 514.5 nm) laser creates a sinusoidal modulation of intensity with period Λ within a sample. The sample contains a dye that absorbs a small fraction of the laser energy, which leads to a sinusoidal temperature field with modulation amplitude δT ∼ 0.01 K. The temperature modulation leads to a modulation of refractive index, which is detected using a second, low-power, reading laser (HeNe

(14)

where ĉeq = ĉλ=1 is the equilibrium heat capacity of the unstretched sample. Because of variations in T0 and w from experiment to experiment, we use the ratio of τD’s from unstretched and stretched samples obtained from consecutive experiments. By doing so, the unstretched sample measurement is used to obtain the relative changes of heat capacity due to deformation. E


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Figure 6. IRT and FRS results for cross-linked PI 100k subjected to uniaxial elongation. (a) Comparison of thermal conductivity (IRT) and thermal diffusivity (FRS)2 in the parallel and perpendicular directions as a function of the stretch ratio λ. (b) Test of the stress-thermal rule using thermal conductivity ratios from steady-state IRT (CtGN = 0.018 ± 0.009) and thermal diffusivity from FRS (CtGN = 0.014 ± 0.005) measurements.2 632.8 nm) that passes through the sample at the Bragg angle. The intensity of the first-order diffracted beam is measured by a photodetector. Following a pulse of the writing laser, δT, for times large compared to the time scale for sound wave propagation, decays exponentially in time: δT ∝ exp(−t/τg). The grating relaxation time is τg =

Λ2 4π 2Dii

examine the effect deformation on heat capacity is to compare FRS (Dii) to steady-state IRT3 (kii) measurements on the same system. Figure 6a shows a comparison of the thermal conductivity ratios (red) and thermal diffusivity ratios (black) in the parallel (squares) and perpendicular (circles) directions to the stretch as a function of the stretch ratio λ. From this figure it appears that results from IRT and FRS techniques are, within the experimental uncertainty, in agreement. However, when the same data are plotted as a test of the stress-thermal rule in (15), as shown in Figure 6b, small but systematic differences between the IRT and FRS measurements can be observed (i.e., different slopes). These results put into question the assumption that Dii ≃ kii/ρĉeq and suggest that to test the stress-thermal rule, one needs to account for a deformation-dependent heat capacity (i.e., Dii = kii/ρĉλ) and therefore apply the correction

, where Dii = kii/ρĉλ is the component of the thermal diffusivity

tensor in the direction of the grating. By changing the relative orientation of the samples with respect to the grating, the thermal diffusivity in different directions is obtained. The uncertainty in Dii measurements for FRS technique is roughly 1−2%.20 For an unstretched sample, kii = keq and ĉλ = ĉeq so that Dii = keq/ρĉeq = Deq. Hence, if both ρ and ĉλ are assumed to be deformation independent, we have Dii/Deq = kii/keq = αi, which allows evaluation of the anisotropy in thermal conductivity with measurements of thermal diffusivity. Note that both unstretched and stretched samples are considered to be at thermodynamic equilibrium during our measurements, and throughout the text we have loosely used the subscript “eq” to refer to the properties of unstretched samples.

c ̂ D − D22 k11 − k 22 = λ 11 keq ceq̂ Deq

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RESULTS AND DISCUSSION Over the past decade, our group has investigated the anisotropy in thermal conductivity induced by deformation in elastomers2,21 polymer melts31−33 and thermoplastics34,35 using the FRS technique. These studies have been the first, and so far, only experimental evidence of the linear relationship between thermal conductivity and stress known as the stress-thermal rule36 k−

⎛ ⎞ 1 1 δ tr k = keqCt ⎜τ − δ tr τ ⎟ ⎝ ⎠ 3 3

(16)

Note that either side in (16) corresponds to the left-hand side of (15) for the specific case of uniaxial extension. In Figure 7, we present the results for the heat capacity ratio ĉλ/ ĉeq as a function of the stretch ratio λ measured via transient IRT experiments. For λ ≲ 2.5, the heat capacity appears to be independent of deformation; however, for larger λ, ĉλ increases by up to 20%. It should be noted that this is the range of λ for which an upturn in the Mooney−Rivlin plot due to finite extensibility is observed (see Figure 3). In steady-state IRT,3 a large section of the temperature field of the sample surface is used to obtain accurate measurements of the thermal conductivity ratios. This approach, which compensates for the relatively large uncertainty in the measurements of temperature (ΔT = ±0.1 °C), is not plausible with the current analysis given for the transient IRT method. In contrast, the transient IRT method uses a single point to study the temperature evolution of samples, and as a result, these measurements present larger uncertainty. We wish to interpret our observations making use of classical rubber elasticity results and a simple thermodynamic analysis. As discussed in the Introduction, the specific heat capacity at constant elongation ĉλ defined in (2) is independent of deformation if one assumes purely entropic elasticity. A more

(15)

where Ct is the stress-thermal coefficient. From our previous studies, we find (15) is valid in both shear and elongational deformations and that GNCt is relatively insensitive to polymer chemistry, where GN is the plateau modulus of the material. As noted in the previous section, FRS provides measurements of thermal diffusivity Dii = kii/ρĉλ. This means that it is not possible to separate the effects of deformation on thermal conductivity and heat capacity from FRS experiments alone. In a previous study, we performed FRS measurements on the crosslinked cis-1,4-polyisoprene used in this study to obtain components of the thermal diffusivity tensor in directions parallel (D11) and perpendicular (D22) to the direction of elongation at several values of λ.2 A less direct approach to F


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behavior of our samples (GN = 382 kPa and n = 83) and those in the study by Smith et al.7 (GN = 227 kPa and n = 144) make it difficult to directly infer the stress−temperature dependence by extrapolating from their results. Therefore, we opt for generating an empirical ∂2σeng/∂T2 that reduces to the observations by Treloar et al. at small strains5 and can be integrated analytically using (18) to obtain an expression that captures the trend in our observations for heat capacity. These criteria are satisfied by ∂ 2σeng ∂T

2

⎛ k = −k1 exp⎜ − 2 ⎝ λ−

⎞ ⎟ 1⎠

(19)

where k1 and k2 are fitting parameters that determine the critical stretch ratio λc after which ∂2σeng/∂T2 deviates from zero and the rate of decline thereafter (see Figure S3a). The analytic integration26 of (18) using (19) gives Figure 7. Specific heat capacity ratio from transient IRT measurements (symbols) as a function of the stretch ratio λ. The blue solid line is a fit to (20) with k1 = 10.1 ± 1.8 kPa/K2 and k2 = 5.2 ± 0.2. The black dashed line represents equilibrium heat capacity ratio ĉλ/ĉeq = 1.

⎛ cλ̂ k2 T ⎡ =1− ⎢k1 exp⎜ − ⎝ ⎣ ceq̂ ρceq̂ λ−

∫T

T

0

+

∫1

which can be fitted to the heat capacity data (solid blue line in Figure 7) to obtain k1 = 10.1 ± 1.8 kPa/K2 and k2 = 5.2 ± 0.2 using typical values for ρ = 930 kg/m3 and ĉeq = 1900 J/(kg K).40 Equation 19 is somewhat arbitrary but has been selected to satisfy a number of conditions. First, according to stress−temperature data5,7 and our measurements, the contribution of ∂2σeng/∂T2 must vanish at small deformations. Second, in agreement with the characterization given by Treloar,1,5,41 after a transition at small deformations the energetic contribution to stress becomes independent of λ, which results in ∂2σeng/∂T2 ∝ λ for moderate to large deformations. Finally, (19) is a continuous function and can be integrated analytically, making (20) amenable for fitting k1 and k2 to heat capacity measurements. Using (13) and (19) with the mechanical properties of our specimens, we are able to predict a stress−temperature behavior for our specimens that show qualitative agreement with the stress−temperature measurements by Smith et al.7 (see Figure S3b). However, the magnitude of ∂2σeng/∂T2 necessary to explain the heat capacity measurements presented in Figure 7 at any given λ is roughly 10 times larger than the one found in stress− temperature measurements by Smith et al.7 The discrepancy in the stress−temperature results and our predictions based on heat capacity measurements could be explained by the different mechanical behavior of our samples and/or the possible failure to achieve true equilibrium in the Smith et al. experiments.7 Note that in the creeplike experiments used to obtain the stress−strain behavior of our specimens, we allowed 3 h for the samples to reach equilibrium, whereas Smith et al.7 used 15 min intervals in the stress−temperature measurements. In their study, longer experiments toward true equilibrium were prevented because of the chemical degradation of samples above room temperature. In view of (16) and Figure 7, an increase in heat capacity with deformation implies that the anisotropy in thermal conductivity (k11 − k22)/keq will be larger than the anisotropy in thermal diffusivity (D11 − D22)/Deq. In Figure 8, we have applied the correction in (16) making use of (20) to the FRS data in Figure 6b. The improved agreement between IRT and the corrected FRS anisotropy measurements to test the stress-thermal rule (red squares and green circles in Figure 8, respectively) provides further evidence of the heat capacity dependence on deformation.

T − T′ ρceq̂ (T ′) dT ′ T′

λ

σeng(T , λ′) dλ′

(17)

where f 0 and s0 are constants. Using the definition for specific heat capacity in (2) and the expression in (17), we obtain ρcλ̂ = ρceq̂ − T

∫1

λ

⎛ ∂ 2σ ⎞ ⎜ eng ⎟ dλ′ ⎜ ∂T 2 ⎟ ⎝ ⎠ λ′

⎞⎤ ⎟⎥ 1 ⎠⎦ (20)

general form for the Helmholtz free energy density can be derived under the assumption of incompressibility37 as f (T , λ) = f0 − s0(T − T0) −

⎞ ⎛ k ⎟(λ − 1) + k 2 Ei⎜− 2 ⎠ ⎝ 1 λ−

(18)

Hence, for the heat capacity ĉλ to be deformation-dependent, it is required that the engineering stress σeng is a nonlinear function of T. As noted earlier, experiments on natural rubber1,5−7 indicate deviations from the assumption of purely entropic elasticity with energetic contributions to σeng of 10−20% that result on a nonlinear dependence of the stress−temperature data. Deviations from the linear approximation (i.e., purely entropic stress) become more apparent at large deformations5,7 and have been attributed to (1) finite extensibility increasing the energetic contribution to stress and (2) strain-induced crystallization. In the natural rubber (DC2) specimens used in a study by Smith et al.,7 which constitute one of the most significant examples of a strong nonlinear dependence of stress on temperature, deviations from linear behavior are attributed to crystallization. Therefore, we might be inclined to assign crystallization as the source of the stress dependence on temperature suggested by the results in Figure 7. However, the samples in our study do not display a downturn at high strains in the Mooney−Rivlin plot that is characteristic of strain-induced crystallization (see Figure 3), and we do not observe an increase in the scattering baseline due to the formation of crystalline domains during FRS measurements (see Figure S4). The absence of crystallization in our measurements can be attributed to the use of deproteinized natural rubber in the preparation of samples,38 differences in the cis/trans content, and the presence of impurities,39 which have been shown to affect the onset and degree of crystallization. Furthermore, the different mechanical G


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Solution of the temperature equation; rheological characterization and absorption spectra of as-received and cross-linked cis-1,4-polyisoprene; baseline for the scattered light during the FRS experiments supporting the absence of strain-induced crystallization; stress−temperature predictions (PDF)

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] (D.C.V.). ORCID

David Nieto Simavilla: 0000-0001-5389-4827 Notes

The authors declare no competing financial interest. Figure 8. Test of the stress-thermal rule using thermal conductivity ratios from IRT (red squares) and the thermal conductivity ratios (green circles) obtained by correcting the FRS data in Figure 6b using (16) and the heat capacity measurements in Figure 7 for cross-linked PI 100k subjected to uniaxial elongation.

ACKNOWLEDGMENTS

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REFERENCES

The authors thank the National Science Foundation for the support of this study under Grants DMR-0706582 and CBET1336442.

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(1) Treloar, L. R. G. The Physics of Rubber Elasticity; Oxford University Press: London, 1958. (2) Nieto Simavilla, D.; Schieber, J. D.; Venerus, D. C. Anisotropic Thermal Transport in a Crosslinked Polyisoprene Rubber Subjected to Uniaxial Elongation. J. Polym. Sci., Part B: Polym. Phys. 2012, 50, 1638− 1644. (3) Nieto Simavilla, D.; Venerus, D. C. Investigation of anisotropic thermal conductivity in polymers using infrared thermography. J. Heat Transfer 2014, 136 (11), 111303. (4) Kloczkowski, A.; Mark, J. E.; Erman, B. A Diffused-Constraint Theory for the Elasticity of Amorphous Polymer Networks. 1. Fundamentals and Stress-Strain Isotherms in Elongation. Macromolecules 1995, 28, 5089−5096. (5) Treloar, L. R. G. The Elasticity and Related Properties of Rubbers. Rep. Prog. Phys. 1973, 36, 755−826. (6) Ciferri, A.; Hoeve, C. A. J.; Flory, P. J. Stress-Temperature Coefficients of Polymer Networks and the Conformational Energy of Polymer Chains1. J. Am. Chem. Soc. 1961, 83, 1015−1022. (7) Smith, K. J.; Greene, A.; Ciferri, A. Crystallization under Stress and Non-Gaussian Behavior of Macromolecular Networks. Kolloid Z. Z. Polym. 1964, 194, 49−67. (8) Roland, C. M. Viscoelastic Behavior of Rubbery Materials; Oxford University Press: New York, 2011. (9) Gee, G.; Stern, J.; Treloar, L. R. G. Volume changes in the stretching of vulcanized natural rubber. Trans. Faraday Soc. 1950, 46, 1101−1106. (10) Sarti, G. C.; Astarita, G. A Thermomechanical Theory for Structured Materials. Trans. Soc. Rheol. 1975, 19, 215−228. (11) Sarti, G. C.; Esposito, N. Testing Thermodynamic Constitutive Equations for Polymers by Adiabatic Deformation Experiments. J. NonNewtonian Fluid Mech. 1977, 3, 65−76. (12) Astarita, G.; Sarti, G. C. The Dissipative Mechanism in Flowing Polymers: Theory and Experiments. J. Non-Newtonian Fluid Mech. 1976, 1, 39−50. (13) Ionescu, T. C.; Edwards, B. J.; Keffer, D.; Mavrantzas, V. G. Energetic and Entropic Elasticity of Nonisothermal Flowing Polymers: Experiment, Theory and Simulation. J. Rheol. 2008, 52, 105−140. (14) Dressler, M.; Edwards, B. J.; Ö ttinger, H. C. Macroscopic Thermodynamics of Flowing Polymeric Liquids. Rheol. Acta 1999, 38, 117−136. (15) Hütter, M.; Luap, C.; Ö ttinger, H. C. Energy elastic effects and the concept of temperature in flowing polymeric liquids. Rheol. Acta 2009, 48, 301−316.

CONCLUSIONS We present a novel transient method based on infrared thermography (IRT) to directly measure the effect of deformation on heat capacity by studying the temperature evolution of samples heated by a laser beam while subjected to uniaxial extension. Applying transient IRT to cross-linked cis-1,4polyisoprene samples, we observe that after a critical stretch ratio λc ∼ 2.5 heat capacity increases monotonically up to 20% for the highest strain λ ∼ 4. The deviations from the heat capacity of unstretched samples are confirmed indirectly with an independent set of experiments comparing anisotropy in thermal diffusivity from FRS and thermal conductivity from steadystate IRT. These results reveal that a correction is necessary to test the stress-thermal rule using thermal diffusivity measurements. From mechanical and scattering experiments, we conclude that strain-induced crystallization is not responsible for the observed deformation-dependent heat capacity. As a result, we are able to use a straightforward thermodynamic analysis to identify an energetic contribution to stress, which manifests as a nonlinear dependence of stress on temperature, as the remaining explanation for our observations. Finally, we propose a function for the second derivative of stress with respect to temperature that explains the heat capacity dependence on deformation and satisfies the main features of well-known stress−temperature data.5,7 Our results are in direct contradiction with the assumption of purely entropic elasticity, since a deformationdependent heat capacity implies energetic contribution to elasticity. From a more practical point of view, our study suggests that this small, yet significant, energetic contribution to stress needs to be accounted for in the study of nonisothermal deformation of polymers.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.macromol.7b02139. H


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Evidence of Deformation-Dependent Heat Capacity and Energetic

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