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Glass Transition Distribution in Miscible Polymer Blends: From Calorimetry to Rheology Peiluo Shi,*,† Régis Schach,‡ Etienne Munch,‡ Hélène Montes,† and François Lequeux† †

Laboratoire de Physico-Chimie des Polymères et Milieux Dispersés, ESPCI ParisTech - CNRS, UMR 7615, 10 rue Vauquelin, 75005 Paris, France ‡ Manufacture Française des Pneumatiques Michelin, Centre de Technologies, 63040 Clermont Ferrand Cedex 9, France ABSTRACT: At their glass transition, simple liquids and polymers exhibit a broad spectrum of relaxationwith a width of more than 4 decades in time. Blends of miscible polymers exhibit an even broader relaxation spectrum. Here, we assume that a blend can be considered as an ensemble of domains of various local glass transition temperatures. Using calorimetric data, with or without aging, we probe the distribution of glass transition temperature (Tg) of miscible polymer blends of polybutadiene (PB) and styrene butadiene rubber (SBR). Using the self-consistent averaging method inspired by the Olroyd−Palierne model, we predict quantitatively, with no adjustable parameter, the viscoelastic spectrum of our blends from the Tg distribution obtained by calorimetry. This quantitative prediction confirms thus the assumption that mechanically a blend can be considered as an ensemble of domains, each of which have a different glass transition temperature.

1. INTRODUCTION Blending is a powerful and convenient way to tune mechanical properties of materials. A great amount of effort has been made during the past 15 years in order to understand the effect of blending on the local dynamics in polymers. In fact, the local dynamics are heterogeneous because of the local arrangement of the two types of monomers in space. Some models have been developed to describe these effects, where the chain connectivity is taken into account (self-concentration effect)1 and later the thermally driven concentration fluctuations. 2 In these approaches, at the nanometric scale, a mixture of miscible polymers is considered as a random distribution of polymer domains exhibiting a wide distribution of glass transition temperatures. In this frame, dielectric or DSC data measured on various miscible blends have been quantitatively described through detailed analysis of the dynamical heterogeneity.2−11 On the contrary, there are in the literature only few results that related the structure and the dynamic of the system at the scale of the dynamic fluctuation to the macroscopic viscoelastic properties of miscible blends. The object of this article is to discuss the consequences of the heterogeneous dynamics on the linear viscoelastic properties of blends. In this paper, we will not discuss the relevance of the concentration fluctuation or self-concentration. We will simply assume that the blends can be considered as an ensemble of domains as shown in Figure 1, each of those exhibiting its own glass transition temperature. This idea has been suggested more that 10 years ago.12 Here we will prove experimentally for the first time that the assumption of a distribution of glass transition temperature gives a quantitative relation between calorimetry © XXXX American Chemical Society

and rheology. In this picture, the elastic modulus of each domain varies with frequency from the rubbery elastic plateau to the glassy one, i.e., from 106 to 109 Pa typically, and each domain may exhibit a very different characteristic relaxation time. So at a given frequency some domains may appear as glassy, with a modulus of about 109 Pa, and others as rubber-like, with a modulus of 106 Pa. Hence, to interpret quantitatively the viscoelastic data, one must be able to derive the macroscopic viscoelastic spectrum from local modulus values of randomly arranged domains varying over 3 orders of magnitude. It is in general quite difficult to predict from the knowledge of the mechanical properties of each constituent the macroscopic mechanical behavior of the blend.13 In the mechanical engineering community, the most efficient way to solve this question appears to be the homogenization technique.14 This latter is a sophisticated mean-field approach predicting the relation between stress and deformation in heterogeneous systems. The homogenization technique consists of dividing the materials in periodic cells and of building a constitutive equation at the scale of a cell. Then the local constitutive equation is extended at the macroscopic level. To do so, however, a simple averaging either of the stress or of the strain of the various constituents is performed at one step. This approach fails for a very broad distribution of mechanical properties because neither the stress nor the strain averaging is satisfactory: the stress averaging overestimates the contribution of the rigid domains, Received: February 28, 2013 Revised: April 9, 2013

A

dx.doi.org/10.1021/ma400417f | Macromolecules XXXX, XXX, XXX−XXX

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2. SAMPLES AND EXPERIMENTAL TECHNIQUES We study blends of polybutadiene (PB) and styrene−butadiene random copolymer chains (SBR) provided by Michelin. The PB contains 93% cis-1,4-PB, 3% trans-1,4-PB, and 4% 1,2-PB, with TPB g = 173 K. The SBR contains 16% cis-1,4-PB, 16% trans-1,4-PB, 43% 1,2-PB, and 25% of = 258 K. These two polymers are miscible in all styrene, with TSBR g proportions. As commonly observed in other miscible polymer blends, the SBR/PB mixtures exhibit a great broadening of the glass transition. Blends with various concentrations are prepared in chloroform solution with cross-linkers, which are 1.2 PCR (per hundred rubbers) of sulfur and 1.9 PCR of accelerator, CBS, and then cross-linked after evaporation of solvent. Cross-linking allows avoiding the crystallization of raw PB material. However, we have carefully checked that the cross-linking changes neither DSC nor rheology measurements in the glass transition domain, except a shift of Tg position of about 8 K. So all the results presented here are not at all modified by the presence of cross-linkers. DSC measurements were conducted with a TA Instruments Co. 2920 DSC, at a heating rate of 10 K/min (conventional DSC measurement, not modulated) with sample weights of 10 mg or less. Mechanical properties are measured by an Anton Paar Physica MCR 501 rheometer.

Figure 1. Schematic view of nanometric domains relevant for the glass transition. Blue chains and red chains have very different Tg. In the glass transition regime each of the domains has its own dynamic (because of either self-concentration effect or concentration fluctuations). We will assume that, irrespective of any microscopic model, each domain exhibitsdepending on its local glass transition Tgan elastic modulus between 106 and 109 Pa.

3. FROM CLASSICAL DSC MEASUREMENT TO Tg DISTRIBUTION In the following two sections we will present three methods that allow measuring the distribution of glass transition in the blends. The first method, which we call here the “fit” method, consists just of fitting the raw response of the blends as a linear combination of different polymer domains, similar to the pure components, with a Tg distribution P(Tg). Figure 2a presents the differential calorimetric scanning response measured with a TA Instruments Co. 2920 DSC setup at a heating rate of 10 K/min for different macroscopic PB volume fraction Φ. All the DSC curves have been shifted vertically for a better visualization of the results. One immediately sees that, as observed classically in the literature, the calorimetric signature of the glass transition spread over tens of kelvin for our blends. Here we assume that polymer blend are arrangements of independent domains of different local Tg, with a distribution P(Tg). As a consequence, the calorimetric spectrum of a blend is a linear superposition of all the contributions of domains of various local Tg. For each domain of composition Tg, we assume that the calorimetric response corresponding to such a domain that we call f(T − Tg) can be linearly extrapolated from the ones of the pure PB and SBR polymers:

and the strain averaging overestimates that of the softer domains. Here we propose and test a mechanical approximation that accounts for a 3D average of the viscoelastic modulus. We show that this averaging method of the linear viscoelastic modulus, based on the self-consistent approach of the Olroyd−Palierne model,15,16 gives excellent quantitative results even in the case of a wide distribution of viscoelastic moduli. In this paper, first we propose a novel method to probe the Tg distribution of miscible polymer blends by using DSC data, and this distribution is found to be in good agreement with more intuitive results obtained form special physical aging measurements. Second, we show that it is possible to predict quantitatively from the Tg distribution the viscoelastic spectrum of our blends by using a self-consistent mechanical model of viscoelastic inclusion, i.e., a self-consistent Olroyd−Palierne model. We use miscible polymer blends composed of two polymers having very different glass transition temperatures, namely polybutadiene (PB) of TPB g = 173 K and polystyrene−butadiene copolymer (SBR) of TSBR = 258 K. g

Figure 2. (a) DSC curves for different blends with experimental data (circles) and curve fitting (solid lines); curves are shifted vertically for clarity. (b) Tg distributions P(Tg) for macroscopic blends Φ = 0.25, 0.5, and 0.75. The inset is the calorimetric spectrum of pure PB (left), pure SBR (right), and a local domain of Tg = −77 °C (middle). B


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f (T − Tg) = kPBfPB (T − Tg) + k SBR fSBR (T − Tg)

(1)

with kPB = (Tg − TgSBR )/(TgPB − TgSBR ) k SBR = (TgPB − Tg)/(TgPB − TgSBR ) = 1 − kPB

where kPB and kSBR are the linearly extrapolation coefficients. TPB g and TSBR are the glass transition temperatures of respectively g pure PB and SBR polymers. f PB(T − Tg) and f SBR(T − Tg) are the calorimetric response of respectively pure PB and SBR. In fact, the relaxation spectrum of SBR and that of PB have different widths because SBR is a copolymer. Expression 1 assumes that the width of the glass transition, which is slightly broader for SBR than for PB, varies linearly with the glass transition of the domains. We will see later that this approximation does not modify in practice the obtained Tg distribution. Hence, the total calorimetric response of a blend, F(T), can be calculated by summing over all the domains of Tg varying from SBR TPB g to Tg . F (T ) =

∫T

TgSBR

PB g

f (T − Tg)Pfit(Tg) dTg

Figure 3. Tg distributions P(Tg) by the derivative method. The inset is an example of the temperature derivative of the heat flow curve of PB50%.

these methods in the following paragraph, and before that we will now introduce a third calorimetric method for determining P(Tg) based on annealing.

4. FROM PHYSICAL AGING DSC MEASUREMENT TO Tg DISTRIBUTION Traditional DSC measurement consists of a “three-step” thermal cycle: cooling, annealing, and heating. Some studies focused on the influence of annealing time at a fixed annealing temperature Ta, which is typically 15−20 K below Tg;19,20 others focused on the influence of heating rate.21 There is in the literature very little information on the influence of annealing temperature, especially for polymer blends. Here we choose to anneal the sample at a given temperature, to “print” by a memory effect the response of the domains at this given temperature. More precisely, annealing leads to a modification of a tiny domain’s relaxation spectrum, the domain being the one that has a relaxation time about the annealing time at the temperature of annealing. When heating back the sample, this relaxation spectrum modification printed by annealing can be read. This effect is well-known as the memory effect previously studied on dielectric measurements22,23 and mechanical measurements.24 The procedure is illustrated in Figure 4a. The scan S1 includes an annealing step, while the scan S2 has no annealing step (this is the case of the curves in Figure 2a). To put into evidence the “printing” effect of annealing, we calculate the difference between the curves S1 and S2. Figure 5a,c shows the responses for pure polymers. They are concentrated around the position of Tg of each and are observable only for limited range of Ta. On the contrary, those of the blend Φ = 0.5 are distributed over a very wide range of Ta from −110 to −60 °C, as shown in Figure 5b. We thus see clearly from Figure 5a that the annealing is able to select the contribution of domains of a given Tg, with a width of less than 10 K that corresponds indeed to the natural width of the PB glass transition observed in calorimetry. In the inset of Figure 5a, the amplitude of the peak is plotted as a function of Ta, which confirms this selectivity of the annealing process. More precisely, a certain Ta will “select” domain with a Tg of about Ta + 15 K as shown in Figure 5d where the peak temperature is plotted both at its maximum for the pure components and for the blends as a function of Ta. The temperature at the overshot maximum that is systematically equal to about Ta + 15 K. The contributions of those domains will then be revealed in a peak at their Tg. If this assumption of “selectivity” is correct, it infers that two

(2)

where Pfit(Tg) is the distribution of glass transition temperature we are seeking for. For the sake of simplicity for the determination of Pfit(Tg), we take the Tg distribution Pfit(Tg) as a sum of two Gaussian distributions, corresponding to the contribution of PB or SBR. It appears that it is similar to the method of Shenogin et al.2 The amplitude of each is thus fixed by the macroscopic composition Φ or (1 − Φ). These conditions impose to consider only four free parameters that are the position Tg and the width w of the two Gaussian-like contributions. This writes indeed PB 2 ⎫ ⎧ ⎪ (Tg − Tg ) ⎪ 1 ⎬ Pfit(Tg) = Φ exp⎨− ⎪ ⎪ 2πwPB 2wPB ⎭ ⎩ SBR 2 ⎫ ⎧ ) ⎪ ⎪ (Tg − Tg 1 ⎬ + (1 − Φ) exp⎨− ⎪ ⎪ 2wSBR 2πwSBR ⎩ ⎭

(3)

We vary the four free parametersby the best chi-square fitting methodsuch that we obtain a good description of the experimental DSC curves (see Figure 2a). As we can see in Figure 2b, our procedure allows to extract Pfit(Tg) properly from the DSC data, and the Pfit(Tg) has two peaks; this is in fact required by the self-concentration model. Another method to extract P(Tg) consists of simply doing the temperature derivative of the heat flow curve as shown in Figure 3, using P(Tg) ≅ Pderiv(Tg) =

d F (T ) dT

(4)

which is in principle similar to eq 2 only if f(T − Tg) can be approximated by an Heaviside step function H(T − Tg). This method is already used in the literature17,18 and gives results that are quite similar to the previous one. In fact, the main difference between the derivative method and the fit one is that one takes into account the natural broadness of the pure component glass transition with the derivative method, while the other does not. This difference is thus given by the broadness of the homopolymer’s Tg width observed by calorimetry. We will compare C


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Figure 4. (a) Physical aging procedure: cooling at 10 K/min from equilibrium, annealing of 5 h at different Ta, and heating scans at 10 K/min (S1), followed by a second cooling−heating scan without annealing (S2). (b) Physical aging measurement for pure SBR with different annealing temperatures of 5 h and without annealing.

Figure 5. Curves of calorimetric overshoots (S2−S1) at various Ta for pure PB (a), Φ = 0.5 (b), and pure SBR (c). Insets (a−c) are the amplitudes of peaks as a function of aging temperature. In (d) the temperature at the overshot maximum as a function of the aging temperature. Inset (d) is the comparison with different annealing time 5 h or 12 min for the PB25% blend.

consecutive annealing processes with very different values of Ta will give two distinct peaks; each of them will be the same as it is in a single annealing. This is exactly the case as it is shown in Figure 6, where two consecutive annealing processes at Ta = −70 °C and Ta = −90 °C give two separate peaks; each of them superpose very well with the peak obtained by a single annealing. The “selectivity” of local Tg by different Ta is therefore confirmed, in agreement with the assumption that different domains are nearly independent. As explained above, we have assumed the frequency domain being selected by annealing is the one that has a relaxation time about the annealing time at the temperature of annealing. Another test of this statement is to use shorter annealing time. So we have performed similar experiments in PB25% blend but with a smaller aging time, 12 min. In that case, the overshoot peak shift toward slightly higher temperature with ΔT = 10 K as indicated

in the inset of Figure 5d, while the whole picture is quantitatively very similar to the one obtained with a 5 h annealing time, but with a weaker precision. In addition, the ΔT given by WLF for the annealing time of 12 min is 10 K and that of 5 h is 15.6 K, being extremely similar to the ones observed, i.e., 10 and 15 K, respectively. This confirms the statement that the domain being selected by annealing is the one that has a relaxation time about the annealing time at the temperature of annealing. It can be seen that this selectivity effect is not very evident for pure polymers with narrow Tg distribution, since only the narrow domain around the maximum of P(Tg) that can be selected, while this selectivity effect is more evident for blends, where this maximum domain is broad (see insets of Figure 5a−c). To sum up, the calorimetric signal after annealing at Ta exhibits a peak that reveals the contribution of domains of Tg ≈ Ta+ ΔT K, the value of ΔT depending on the annealing time. D


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Figure 6. (a) Physical aging procedure for two consecutive annealings. (b) Comparison of two consecutive annealings and two separate annealings.

constituent’s glass transition. More precisely, we expect that the convolution between Pfit(Tg) by the response of the single components would approach Pderiv(Tg) and Paging(Tg). In Figure 8, on the left column (a), (b), and (c) we have plotted the raw value of Pderiv(Tg), Paging(Tg), and Pfit(Tg) for three PB/SBR mixtures, 25/75, 50/50, and 75/25, respectively. On the right column (d), (e), and (f) are plotted the same Pderiv(Tg) and Paging(Tg), while Pfit(Tg) is convoluted response of the pure components obtained with the aging techniques. We see that there is a very good agreement between the aging and the convoluted “fit” in (d), (e), and (f), at least better than that in the (a), (b), and (c). On the contrary, the amplitude of the maximum of the derivative technique, compared to other techniques, is respectively smaller and higher for PB25% and PB75%. We observe in addition that the agreement between Paging(Tg) and the convoluted Pfit(Tg) is excellent, while the value of Pderiv(Tg) seems slightly different from the two previous. To sum up, the aging technique is the more accurate one, but it is time-consuming and has to be deconvoluted from the intrinsic broadness of the homopolymers to get the blend information. The fit technique is extremely powerful, since it does not require deconvoluting the data. Finally, the derivative technique gives a fast result but is less precise than the others. Therefore, we prefer to use either aging or fit techniques, taking care of the convolution by the natural shape of the glass transition of each of the components of the blend.

The envelop of all this contribution provides an approximation for P(Tg) that we will call Paging(Tg). The physical aging measurements in annealed DSC curves give us a direct and intuitive insight into the broad distribution of Tg in polymer blends. Thus, we can approximate P(Tg) by the normalized envelope of these aging peaks, as shown in the inset of Figure 7.

Figure 7. P(Tg) by envelop of physical aging DSC measurements for two pure polymers and three blends. The inset is an example of envelop of the blend PB50%.

6. FROM Tg DISTRIBUTION TO RHEOLOGY 6.1. Estimation of the Viscoelastic Spectra of a Local Domain of Tg. As we now have precise data of the glass transition temperature distribution, our next step consists in predicting the viscoelastic spectra from the Tg distribution obtained by DSC measurements. First, we estimate the local viscoelastic spectrum Gloc * (Tg) corresponding to each local domain of Tg. We can deduce it from the viscoelastic spectrum of both pure PB and pure SBR. Then Gloc * (Tg) is combined with the Tg distribution function P(Tg) extracted from the previous calorimetric methods. According to previous dielectric studies,6,13 the viscoelastic spectra of the pure polymers can be described using a modified Havriliak−Negami function:25

5. DISCUSSION OF THE CALORIMETRIC RESPONSE TECHNIQUE USED TO EXTRACT P(Tg) We have shown that it is possible to obtain by three different methods an approximated distribution of the glass transition temperature in polymer blends. However, the three methods differ also by the fact that they may exhibit or not an intrinsic broadness for the glass transition of the pure components themselves. Indeed, Pfit(Tg) has by definition a zero width for the glass transition of each of the pure components, which is not the case for Pderiv(Tg) and Paging(Tg). One of the difficulties of our blend is that the width of the glass transition as measured by calorimetry is different for the two componentsbecause they are different copolymers. Our objective is to relate the width of the glass transition of the blends to its viscoelastic spectrum. But it is clearly out of the scope of this paper to discuss the relation between the width of the calorimetric response and the viscoelastic one for the pure components. So in the next section we will assign a single value of Tg to the viscoelastic response of a pure component PB or SBR. However, to compare quantitatively the three methods, one have to take into account the broadness of the single

G* = Gglass −

τHN E

Gglass − Grub (1 + (jωτHN)α )β

1/ α ⎡ ⎛ ⎞⎤ π ⎢ ⎥ = τ tan⎜ ⎟ ⎣ ⎝ 2(β + 1) ⎠⎦

(5)


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Figure 8. (a−c) Comparison of Paging(Tg), Pderiv(Tg), and Pfit(Tg) for Φ = 0.25, 0.5, and 0.75. (d−f) Convolution of these three blends’ Pfit(Tg) with pure polymers’ Paging(Tg) and comparison.

Figure 9. Master curves for storage modulus (circles) and loss modulus (triangles) as a function of frequency and their H−N function fittings (solid SBR = 258 K. lines): (a) PB with Tref = TPB g = 173 K; (b) SBR with Tref = Tg

The parameters α, β, τ, Gglass, and Grub were calculated from the fits to the experimental data for pure PB and pure SBR (see Figure 9 and Table 1). Experimental data in Figure 9 are master curves obtained from time−temperature superposition. The

reference temperature is the Tg of each, chosen as the temperature at which the maximum of G″ is located at frequency = 1 Hz. * (Tg) of a local We suppose that the viscoelastic modulus Gloc domain of Tg is given by a modified Havriliak−Negami function, F


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6.2. Description of the Macroscopic Mechanical Response of the Blends. 6.2.1. First Observations on the Parallel and Series Averaging Methods. It is interesting to start by analyzing the results that would be obtained from classicalbut approximatedmean of averaging. There have been many discussions in the literature on the accuracy of averaging the stress at a given strain or the strain at a given stress in polymer blends. In Figure 11, we have plotted for the blends Φ = 0.25, 0.5, and 0.75 the results of the stress averagethe representative domains being in parallel spring/dashpot representation:

Table 1. Glass Transition Temperatures, Best Chi-Square Fitting Havriliak−Negami Parameters for Pure Polymers and Any Local Domain of Tg log(τ) α β Gglass (Pa) Grub (Pa)

PB

SBR

local Tg

−0.947 0.705 0.284 9.5 × 108 6.3 × 105

−0.871 0.752 0.254 6.5 × 108 7.4 × 105

fixed by WLF equation kPBαPB + kSBRαSBR kPBβPB + kSBRβSBR SBR kPBGPB glass + kSBRGglass SBR kPBGPB + k G rub SBR rub

whose parameters are linearly extrapolated from the one of the two pure polymers. * (Tg) = Gglass(Tg) − G loc

G* =

Gglass(Tg) − Grub(Tg)

1 = * Gser

P(T )

∫ G*(Tg ) dTg

(9)

g

(6)

Clearly neither the parallel nor the series averaging is able to give a correct description, and there are very different. The parallel estimation overestimates the elastic modulus. A glassy domain with a modulus of 109 Pa will have a contribution equivalent to the one of 103 rubber-like domains, according to eq 8, while the same glassy domain, as a single rigid inclusion with a volume fraction of ϕ = 10−3, is known to increase the elastic modulus by a factor of 1 + (5φ/2) ≅ 1.0025. The series estimation eq 9, for symmetrical reasons, underestimated the modulus even for a volumefraction of rubber-like domains of only 10−3. Consequently, in the highfrequency domain, the stress averaging (parallel) gives the most efficient approximation of measured ones, while the strain averaging (series) is better at lower frequency. However, none of the models are able to give a good prediction of the data in the glass transition domain. For that we need to take into account that there are inclusions of domains in a matrix self-consistently. This can be done thanks to the Olroyd−Palierne model. 6.2.2. Application of the Olroyd −Palierne Model. The Olroyd−Palierne model15 is a continuum mechanics calculation of the macroscopic viscoelastic complex modulus Gm * of a viscoelastic matrix of complex modulus Gmat * embedding various spherical viscoelastic inclusions G*i , each one with a small volume fraction f i. Originally the authors of this model have included effects of surface tension that will not be considered in this specific study. The model gives a rigorous expression for the modulus in the limit of infinitely dilute inclusions, in the absence of interfacial stress that can be written as follows:

The same approach is used for the glassy and rubbery modulus, Gglass and Grub, as well as the H−N exponentsall these quantities being in fact hardly different for PB and SBR. The value of τ(Tg) can be derived from the WLF equation.26 ⎛ τ(Tg) ⎞ C1g(T − Tg) ⎜ ⎟ log⎜ ⎟ = −Cg + T − T ⎝ τg ⎠ 2 g

(8)

and the one for a compliance averagethe representative domains being in series in a spring/dashpot representation.

(1 + (jωτHN(Tg))α(Tg))β(Tg)

1/ α(Tg) ⎡ ⎛ ⎞⎤ π ⎟⎥ τHN(Tg) = τ(Tg)⎢tan⎜⎜ ⎢⎣ ⎝ 2(β(Tg) + 1) ⎟⎠⎥⎦

∫ G*(Tg)P(Tg) dTg

(7)

with τg = 1/2π. The Cg1 and Cg2 coefficients are determined from WLF fitting of pure polymers. In Figure 10, we have the WLF plot of the segmental

Figure 10. Data of segmental relaxation time of PB (rectangles) and SBR (circles). Classic WLF law (dotted line) and a modified WLF law (solid line): the derivation occurs at T − Tg = 10 °C.

⎛ 5 ** ⎜ Gm* = Gmat ⎜1 + 2 ⎝

relaxation time log τ = log(aTτg), where aT is shift factor of master curves of Figure 9. We can see that these two polymers have similar WLF law, and the fitting gives Cg1 = 11.9 and Cg2 = 37.7. We note that at T − Tg < −10 °C the data derivate from the WLF behavior and should be replaced by an Arrhenius law because it is in the nonequilibrium glass regime.27−30 It appears that this replacement is indeed important for the quality of our prediction. Hence, in this section we explain how we deduce the local viscoelastic modulus Gloc * (Tg) of a local domain of Tg, from the mechanical property of the pure PB and SBR polymers. We now have to explain how we can deduce, from the distribution function P(Tg) of local viscoelastic modulus Gloc * (Tg) extracted from the DCS technique, the mechanical spectrum of the whole blend.

⎞

∑ fi Hi⎟⎟ i

⎠

(10)

where Hi =

* 2Gi* − 2Gmat * 2Gi* + 3Gmat

(11)

This expression can be seen as a generalization of the Einstein relation for the viscosity of solid suspensions. Indeed, if the inclusions’ modulus tends toward infinity, Hi tends toward 1, and eq 10 simply becomes G*m = G*mat(1 + 5/2∑i f i), which transforms into the Einstein relation for the complex modulus of a purely viscous system, i.e., Gmat * = iηω. G


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Figure 11. Experimental data of master curves for storage modulus (circles) and loss modulus (triangles) as a function of frequency. Comparison between the prediction of the Olroyd−Palierne model (solid lines) and the series (dotted lines) and parallel (dashed-dotted lines) averagings for three blends, all of them using Pfit(Tg). Inset (b): three predictions of G″ corresponding to the use of the Pfit(Tg), Pderiv(Tg), and Paging(Tg) of Figure 8a.

the blend Gblend * . The assumption used implicitly here is that the matrix can be considered as exhibiting a homogeneous viscoelastic modulus around each domain. We will see that this approximation is highly satisfactory in the present case. So we take a distribution of representative inclusions of modulus G*(Tg) with a volume distribution λP(Tg) and a total volume fraction λ ≪ 1. The fact that the system’s modulus is exactly the same with and without the representative inclusions leads to the integral equation that can be deduced from eqs 10 and 11:

In our case, the system is entirely constituted of domains, where each of them has a peculiar viscoelastic modulus. However, it can be considered homogeneous at a macroscopic scale with a modulus G*blend.16,31 In order to determine this modulus, one can consider the system as a homogeneous matrix containing a small fraction of viscoelastic inclusions. Let us choose the inclusions that are representative of the whole domains moduli distribution, i.e., that they have the same moduli distribution P(Tg) as the system itself. Thus, as the inclusions are representative of the matrix itself, they must not modify the system modulus. Thanks to eq 11, the modulus of the matrix of modulus G*blend is modified by the presence of the distribution of inclusions, but as the inclusions have the same P(Tg) as the matrix, this modification must vanish. This gives us a self-consistent approximation for the modulus of

⎛ * = G blend * ⎜1 + 5 λ G blend ⎜ 2 ⎝

2G*(T ) − 2G *

⎞

P(Tg) dTg ⎟⎟ ∫ 2G*(Tg) + 3G blend * g

blend

⎠

(12) H


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One can see that whatever the value of λ, this relation holds if

in addition, the width of SBR is still narrow compared to that of the blends, so the width of SBR should not be of great importance. On the contrary, it is worthy to testify other blend systems where the miscibility is worse, or with greater difference between the two components’ Tg, or with very different flexibility. This is an orientation of a future study.

2G*(T ) − 2G *

P(Tg) dTg = 0 ∫ 2G*(Tg) + 3G blend * g

blend

(13)

This gives an integral equation for G*blend as a function of P(Tg) and G*(Tg). Indeed, this is an integral equation that is not easy to solve, and we choose a recursive method that appears to be quite efficient. It consists in iterating the following suitestarting from any value of G*blend,1, for instance the value given by eq 8 or 9: * n+1 = G blend,

7. CONCLUSION In conclusion, compatible polymer blends exhibit drastic fluctuations of modulus values in the vicinity of their glass transition temperature. The glass transition distribution P(Tg) can be extracted by different methods from DSC measurements with or without physical aging. From this distribution, we can predict quantitativelythanks to a self-consistent Olroyd− Palierne modelthe viscoelastic spectrum of blends in the glass transition domain. The model assumes simply that glassy domains are nearly spherical and randomly distributed. The experimental data confirm the assumption. This means that regarding at least the viscoelastic spectrum of blends, a polymer blend can be considered in its glass transition regime as an arrangement of independent viscoelastic domains, each one with a specific glass transition.

1

∫

5P(Tg) dTg * t ,n 2G * (Tg) + 3G blend

(14)

In practice, this suite converges in a few tens of steps toward the solution of eq 13. As the pure components viscoelasticity are represented by a Dirac distribution of Tg, the more convenient choice for P(Tg) is the fit one. The results are shown in Figure 11. Clearly the series and parallel approach fail to describe the glass transition domain. The discrepancy between both the series and parallel prediction and the data is indeed of the order of 4 decades in frequency. At opposite, the agreement between the data and the self-consistent approach is always good with a discrepancy smaller than a decade in the glass transition regime. Of course, the prediction is not satisfactory at lower frequency. In these domains, the Rouse mode becomes predominant and cannot be accounted simply by the self-consistent method, as segment of chain that relaxes in this regime belongs to more than one glassy domain. Indeed, taking into account the Rouse modes that may extend over various glassy domains is clearly out of the scope of our approach, which is focused on the glass transition. Lastly, we can compare the various values of P(Tg). In the inset of Figure 11b, we have zoomed the three predictions of G″ corresponding to the Pfit(Tg), Pderiv(Tg), and Paging(Tg) of Figure 8a. We see that the difference is clearly not significant. This originates indeed from the fact that the mechanical averaging decreases the broadness of the viscoelastic spectrum compare to the one obtained with an additive averaging. As soon as part of the domains just consists of independent inclusion with a volume fraction less than 10%, they do not influence significantly the viscoelastic response any longer. At opposite, calorimetry measures the sum of the contribution of the relaxations of each domain and is thus sensitive to the volume fraction of the domains. So it is perfectly comprehensive that the calorimetry relaxation spectrum is broader than the viscoelastic technique. To sum up, the use of this method to predict the viscoelastic spectrum of our polymer blends appears to be extremely efficient. In Figure 11, the viscoelastic data and the prediction from Olroyd−Palierne self-consistent model agree nearly perfectly. We recall that there is no adjustable parameter in the mechanical prediction, all the data being deduced from the calorimetric measurements and the viscoelastic spectrum of the pure polymers. An open question is about the universality of this proposed approach. In fact, our blend system has the following characteristics: very good miscibility, similar flexibility for the two pure components, and the width of the glass transition as measured by calorimetry of SBR is broader than that of PB. As already discussed in section 5, our objective is to relate the width of the glass transition of the blends to its viscoelastic spectrum;

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] (P.S.). Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We are indebted to the Michelin-ESPCI chair program for this work. REFERENCES

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