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Quantification of different contributions to dissipation in elastomer nanoparticle composites
Polymer 111 (2017) 48e52
Contents lists available at ScienceDirect
Polymer journal homepage: www.elsevier.com/locate/polymer
Short communication
Quantification of different contributions to dissipation in elastomer nanoparticle composites Sriharish M. Nagaraja a, Anas Mujtaba a, Mario Beiner a, b, * a b
Fraunhofer Institut für Mikrostruktur von Werkstoffen und Systemen IMWS, Walter-Hülse-Str. 1, 06120 Halle (Saale), Germany €t Halle-Wittenberg, Naturwissenschaftliche Fakulta €t II, 06120 Halle (Saale), Germany Martin-Luther-Universita
a r t i c l e i n f o
a b s t r a c t
Article history: Received 19 November 2016 Received in revised form 6 January 2017 Accepted 8 January 2017 Available online 10 January 2017
We present an approach to quantify different contributions to dissipation in elastomer nanoparticle composites based on strain sweeps. A modified Kraus equation is successfully used to approximate the 00 loss modulus depending on strain amplitude G g measured at different temperatures. For natural rubber composites containing > 10 vol% carbon black or > 3 vol% carbon nanotubes two different contributions to dissipation due to (i) breaking and (ii) deformation of glassy rubber bridges in the filler network are identified. Filler fraction and temperature-dependent trends support physical pictures considering that glassy rubber bridges are responsible for the visco-elasticity of the filler network. Constant extra contributions to dissipation are associated with the bulk-like fraction of the elastomer matrix or filler network independent effects. The achieved understanding should be very important for the optimization of elastomer nanoparticle composites for special applications like tire treads. © 2017 Elsevier Ltd. All rights reserved.
Keywords: Polymer composites Dissipation Reinforcement Polymer dynamics Mechanical properties
Developments towards a rational design of reinforcement and dissipation of elastomer nanoparticle composites are of extraordinary importance for various fields of applications [1e3]. A famous example are materials for tire treads where high reinforcement has to be combined with an optimized dissipation spectrum in order to combine long tire life with low rolling resistance and good wet grip [4e6]. In the recent years some progress has been made in understanding different contributions to reinforcement [7]. In particular, it has been shown based on strain-amplitude sweeps that the load carrying capacity of the filler network in highly filled composites depends significantly on temperature and frequency. One can conclude that the filler network shows visco-elastic properties what has been subsequently also confirmed by independent solid-state NMR investigations [8]. These findings have been interpreted as a strong indication for the existence of immobilized rubber bridges which are glassy under ambient conditions but soften sequentially at high temperatures and low measurement frequencies (Fig. 1a). This molecular interpretation predicts a relaxation spectrum Hu of elastomer nanoparticle composites which is broadened at the low frequency end of the a
* Corresponding author. Fraunhofer Institut für Mikrostruktur von Werkstoffen und Systemen IMWS, Walter-Hülse-Str. 1, 06120 Halle (Saale), Germany. E-mail address: [email protected] (M. Beiner). URL: http://www.imws.fraunhofer.de http://dx.doi.org/10.1016/j.polymer.2017.01.011 0032-3861/© 2017 Elsevier Ltd. All rights reserved.
relaxation peak as compared to that of unfilled elastomers due to interaction between rubber matrix and filler surface (Fig. 1b). The main peak position of the relaxation spectrum is, however, not changed since the major fraction of the elastomer matrix is unaffected by the filler particles even for high filler fractions above the percolation threshold fC indicating the transition from isolated filler aggregates to a percolated filler network. A very interesting point of this molecular picture is that dissipation should be also systematically affected if the relaxation spectrum of the elastomer is changed by the incorporation of fillers. In this letter, we present to our knowledge for the first time an approach that is able (i) to quantify different contributions to dissipation in elastomer nanoparticle composites based on simple strain-amplitude sweeps and (ii) to identify their molecular origin. This is very interesting since there is limited knowledge about the molecular origin of dissipation in elastomer nanoparticles composites and the factors that influence dissipation on the microscopic scale, which can be of very different nature [9e11]. Dissipation in the rubber matrix [12], friction between nanoparticles [13] and fracture work needed for breaking the filler network [9,14] are effects that have been considered in the literature. Starting point for the discussion in this work is the wellestablished phenomenological model by Kraus [9] describing the so-called Payne effect [15] in highly filled elastomer nanoparticle composites, which is commonly attributed to a breakdown of the
S.M. Nagaraja et al. / Polymer 111 (2017) 48e52
00
Gg¼
49
m 00 00 2 G m G ∞ gg 2m g gc
1þ
c
þG
00
(2)
∞
00
with G m being the peak height introduced as intensity measure. 00 Eq. (2) predicts that G g shows a peak at gC and identical values at small (g/ 0) and large (g/∞) strain amplitudes what is obviously not the case for differently filled NR composites (Fig. 2) where the 00 G g values at high and small amplitudes are significantly different. In particular, it is found that the loss modulus values at small strain 00 amplitudes (g < 101 %) are commonly much higher than G ∞ in contradiction to that what is predicted by Eq. (2). This clearly in00 dicates that the original Kraus equation for G g is insufficient to describe experimental data taken from strain sweeps. The necessary modification is obviously an additional sigmoidal contribution (blue lines in Fig. 2) with a g dependence similar to that obtained for the storage modulus G0 in Eq. (1). If such an additional term is introduced, one gets the modified equation
G
00
Gg¼
00
00
0
G ∞ 2m þ
1 þ gg c 00
00
m 00 00 2 G m G ∞ gg 1 þ gg c
00 ~ ~ ¼ G g;D þ G g;F þ G ∞
Fig. 1. (a) Sketch showing the filler network in highly filled elastomer nanoparticle composites containing glassy rubber bridges with internal mobility gradients related to different a relaxation times and glass temperatures Tg . (b) Comparison of the relaxation spectra Hu in the a relaxation range for NR and a related NR composite filled with 17.5 vol% carbon black. The color code for the mobility used in part (a) corresponds to that in part (b).
with G
00
0
and G
00
∞
2m
c
00
þG
∞
(3)
being the loss moduli at small (g/0 ) and large
filler network. Experimental evidence is a sigmoidal decrease in the storage modulus as function of strain amplitude G0 g as shown for a natural rubber (NR) composite in Fig. 2 (open symbols). This behavior is successfully modeled by Kraus based on rate equations describing the number of bridges in the filler network as function of strain amplitude Ng . For the storage modulus G0 g one gets
G0 g ¼
G0 0 G0 ∞ 0 2m þ G ∞ g 1þ g
(1)
c
with G0 0 and G0 ∞ being the storage moduli at small (g/ 0) and large (g/∞) strain amplitudes, respectively. Their difference defines the load carrying capacity of the filler network DG0 ¼ G0 0 G0 ∞ . Further parameters are the critical strain amplitude gC and the exponent m describing the shape of the sigmoidal decrease in G0 g . Eq. (1) fits the experimental G0 g data in Fig. 2 obviously quite well. In the following, we will focus on the quantification of different contributions to dissipation based on data for the loss modulus 00 depending on strain amplitude G g as obtained from strain sweeps 00 (Fig. 2, full symbols). Characteristic for G g is a peak occurring 00 concurrently with a sigmoidal decrease in G g . Kraus associated the 00 peak in G g with the heat released by breaking cohesive bonds between filler particles. Based on the rate equations which are 00 already used for the derivation of Eq. (1), he proposed that G g can be described by
00
Fig. 2. Strain-dependent storage modulus G0 (open symbols) and loss moduluas G (full symbols) for NR composites filled with (a) 20.4 vol% carbon black (CB) and (b) 6.5 vol% carbon nanotubes (CNT), respectively. Kraus approximations to real and imaginary parts are shown as a black lines. The modified approximation for the loss modulus according to Eq. (3) is shown as red line. The blue line indicates the additionally introduced dissipative contribution.
50
S.M. Nagaraja et al. / Polymer 111 (2017) 48e52 00
(g/∞) strain amplitudes, respectively. G 0 accounts here for the amplified dissipation at small strain amplitude where the filler network is intact. These contributions are obviously absent if the 00 filler network is totally broken at G ∞ . Fits based on Eq. (3) (red lines) approximate the experimental data for the overall dissipation 00 G g for structurally different fillers like carbon black (CB, Fig. 2a) or carbon nanotubes (CNT, Fig. 2b) obviously very well. Moreover, contributions to dissipation with physically distinct origins can be discriminated what becomes more clear if a generalized form of Eq. 00 ~ (3) is used, where G g;D is the 00dissipation due to oscillatory defor~ mation of the filler network, G g;F is related to the heat released by breaking bridges in the filler network originally introduced by 00 Kraus, while G ∞ represents contributions to dissipation in composites, which are not related to the filler network. Note that an equation similar to Eq. (3) has been proposed by Ulmer [16] some time ago on a phenomenological basis without physical interpretation of the individual parameters and clear connection to the molecular situation. The interpretation of the terms in Eq. (3) given above fits well to results from recent studies demonstrating that the filler network in highly filled rubbers shows viscoelastic properties arising from glassy rubber bridges connecting individual nanoparticles or aggregates (Fig. 1a) [8]. These glassy bridges are formed by a tiny fraction of immobilized rubber having Tg values, which can be several 10 K higher than the bulk rubber [17,18] due to a strong interaction between elastomer segments and filler surface [19e22]. 00 ~ From that point of view G g;D quantifies dissipative contributions caused by the deformation of intact glassy rubber bridges being the weakest elements in the filler network [11,23,24]. If the strain 00 ~ amplitude g increases, G g;D should also decrease since the number of intact glassy rubber bridges Ng decreases as predicted already by the classical Kraus model for G0 g (Eq. (1), for details see ESI). The 00 ~ second strain-dependent term in Eq. (3), G g;F , is due to the heat released during breaking of glassy rubber bridges and physically of 00 quite different nature. Its contributions to G g are largest in the range where most of the00 glassy rubber bridges break indicated by 00 m ~ the peak in G g . Hence, G g;F is assumed to be proportional to Ng g as already done by Kraus in the derivation of Eq. (2) (cf. ESI). Most interesting consequence of the proposed physical picture is that the 00 00 00 00 00 dissipation measures DG D ¼ ðG 0 G ∞ Þ as well as DG F ¼ ðG m R ~00 00 G ∞ Þf G g;F dg should be under idealized conditions both proportional to the initial number of bridges N0 characterizing the undisturbed filler network since their extrapolated number at infinite strain amplitude N∞ is commonly assumed to be 0 (for details see ESI). The dissipation remaining even at very high strain 00 amplitudes, G ∞ , is probably due to contributions of relaxators in the bulk-like fraction of the elastomer matrix or other processes without relation to the filler network. Obviously, this approach allows to discriminate different contributions to dissipation in elastomer nanoparticle composites with filler contents above the percolation threshold ðf > fC Þ based on classical strain sweeps. The proposed physical picture can be further checked using the results of strain sweeps on NR-CB composites with variable filler fractions f performed at different temperatures T, which are presented in Fig. 3. The commonly high quality of the fits (solid lines) 00 underlines that Eq. (3) is precisely predicting the behavior of G g for different temperatures T and filler fractions f. In addition, several non-trivial trends predicted by the discussed physical picture are confirmed. For all temperatures (0, 25, 60 C) the characteristic peak at intermediate strain amplitudes is only observed for higher filler loading above the percolation threshold (f > fC ). The peak intensity is, however, much higher for the lowest temperature (0oC) and goes systematically down if temperature increases. This is expected considering the results for the storage modulus G0 g indicating that the total number of bridges N0 (or analogously the filler
network strength DG0 ) is reducing commonly with increasing temperature [7,8]. Accordingly, the total number of glassy rubber bridges that break in a strain sweep (and/or the glassy bridge volume) should also decrease with increasing temperature and decreasing filler fraction in percolated systems. Most interesting is 00 now that the loss modulus values at small strain amplitudes G 0 00 and the step heights DG D behave qualitatively like or the peak 00 00 intensity G m or DG F . Both quantities decrease obviously also with increasing T and decreasing f (above fC ) while the limiting dissi00 pation G ∞ is only slightly changing. Within the proposed physical picture these trends are understood by differences regarding the initial number of bonds in the intact filler network N0 (and/or the glassy bridge volume), which is lowered if the temperature increases or the filler fraction decreases. If N0 decreases, the dissipation due to oscillatory deformation of glassy bridges in the intact filler network decreases accordingly. Note that the rate equations behind Eqs. (2) and (3) are not only applicable to count the actual number of bridges Ng as proposed by Kraus but should be analogously also able to quantify the volume in glassy bridges, which is possibly a more relevant quantity for measuring strength and dissipation of a filler network. Summarizing this part, one can conclude that there is a parallel influence of temperature T and filler 00 fraction f on both, dissipation at low strain amplitudes G 0 and 00 peak intensity measures like G m00 . This indicates strongly that both 00 ~ ~ contributions to dissipation, G g;D and G g;F , are due to glassy rubber bridges being part of the filler network. In order prove further correlations between the two strain00 00 dependent terms in Eq. (3) the fit parameters, DG D and DG F , are
00
Fig. 3. Loss modulus (G ) as function of strain amplitudes for different filler fractions of carbon black at (a) 0 C, (b) 25 C and (c) 60 C. The fits to the measured data are based on the modified Kraus equation for the loss modulus (Eq.(3)).
S.M. Nagaraja et al. / Polymer 111 (2017) 48e52
compared in Fig. 4 for different NR-CB composites measured under various conditions. It is quite clear from Fig. 4a that both quantities do increase for a given temperature with f as expected owing to the trend in the number of glassy rubber bridges ðN0 Þ in the undeformed filler network. Interestingly, the dependence on filler fraction is also similar for both quantities. For a given filler fraction 00 00 DG D and DG F decrease with increasing temperature. This indicates that at low temperature the number of glassy rubber bridges (and/or their volume) is higher since partial softening occurs as the temperature is increased. Accordingly, the dissipation caused by glassy rubber bridges should decrease with T. In Fig. 4b, 00 00 DG D is plotted vs DG F for samples with f > fC . A linear dependence is commonly observed for different temperatures. This strongly supports the already discussed prediction that both quantities are proportional to N0 . Obviously, the glassy rubber bridges influence the dissipation in the investigated natural rubber composites most. This is a striking finding underlining the importance of the physical picture used in this letter (Fig. 1). Note that m is basically constant for the investigated NR-CB composites and that the observed proportionality may also indicate that other important quantities like the heat needed for breaking a glassy rubber bridge qF or the dissipation occurring during a full cycle of oscillatory deformation of a glassy rubber bridge qD are mainly independent on filler volume fraction f and total number of bridges N0 in highly filled composites (f > fC ). Differences regarding the temperature dependence of qD and qF may be relevant for the temperature dependence of the slopes in Fig. 4b. Finally, we should also highlighted that the overall dissipation behavior in NR-CNT composites (Fig. 2b) is qualitatively similar and well described by Eq. (3). However, the situation is characterized by 00 smaller DG D values as compared to NR-CB composites although
DG
51
00
0 F and reinforcement related quantities (G 0 ) are quite similar. This may be partly explained by a smaller number of glassy rubber bridges but demonstrates that there are filler-specific effects and probably differences in the ratio qD =qF which definitively need further investigations comparing such quantities more systematically. In summary, we have compiled in this work experimental evidences for the existence of at least three different contributions to 00 dissipation G g in strain sweeps for highly filled elastomer nanoparticle composites. These dissipative contributions are obviously of physically quite different origin. A physical picture is proposed that allows to quantify these contributions and to associate them with distinct molecular mechanisms. For natural rubber composites containing large amounts of carbon black (f > 10 vol%) dissipation contributions due to oscillatory deformation of glassy 00 ~ rubber bridges G g;D are found to be most relevant at low strain amplitudes (g < 0:1%). The heat caused by breaking glassy rubber 00 ~ bridges G g;F is usually dominating at intermediate strain amplitudes close to gC z2%. Filler network independent dissipation ef00 fects G ∞ , appearing for example in the bulk-like rubber matrix, are obviously most important for extremely large strain amplitudes (g > 50%). The situation is shown to vary with temperature as expected due to a sequential softening of the glassy rubber bridges connecting neighbored filler particles or aggregates. In the framework of the proposed physical picture this is related to a reduction of the number (and/or the volume) of the glassy rubber bridges N0 in the undeformed filler network. We strongly believe that the predictions of this physical picture can help to optimize elastomer nanoparticle composites for special applications like tire treads.
Acknowledgments Financial support by the state Sachsen-Anhalt in the framework of projects 1304/00022 and 1604/00028 is greatly acknowledged. Appendix A. Supplementary data Supplementary data related to this article can be found at http:// dx.doi.org/10.1016/j.polymer.2017.01.011. References
00
00
Fig. 4. (a) DG D and DG F (insert) vs filler volume fraction f. The parameters are ob00 00 00 tained from fits to G g curves shown in Fig. 3 using Eq. (3). (b) DG F vs DG D for various NR-CB composites with high filler fractions (f > fC ) measured at three different temperatures.
[1] E.M. Dannenberg, The effects of surface chemical interactions on the properties of filler-reinforced rubbers, Rubber Chem. Technol. 48 (1975) 410e444. [2] J.E. Mark, B. Erman, F.R. Eirich (Eds.), Science and Technology of Rubber, third ed., Elsevier, Amsterdam, 2005. [3] A. Balazs, T. Emrick, T. Russell, Nanoparticle polymer composites: where two small worlds meet, Science 314 (2006) 1107e1110. [4] M.K.H. Nordsiek, The integral rubber concept - an approach to an ideal tire tread rubber, Kautsch. Gummi Kunstst. 38 (1984) 178e185. [5] A. Lechtenboehmer, H.G. Moneypenny, F. Mersch, A review of polymer interfaces in tyre technology, J. Brit Polym. 22 (1990) 265e301. [6] G. Heinrich, The dynamics of tire tread compounds and their relationship to wet skid behavior, Prog. Colloid Polym. Sci. 90 (1992) 16e26. [7] A. Mujtaba, M. Keller, S. Ilisch, H.-J. Radusch, T. Thurn-Albrecht, K. Saalw€ achter, M. Beiner, Mechanical properties and cross-link density of styrene-butadiene model composites containing fillers with bimodal particle size distribution, Macromolecules 45 (2012) 6504e6515. [8] A. Mujtaba, M. Keller, S. Ilisch, H.-J. Radusch, T. Thurn-Albrecht, €chter, M. Beiner, Detection of surface-immobilized components and K. Saalwa their role in viscoelastic reinforcement of rubber-silica nanocomposites, ACS Macro Lett. 3 (2014) 481e485. [9] G. Kraus, Mechanical losses carbon-black filled rubbers, Appl. Polym. Sci. Appl. Polym. Symp. 39 (1984) 75e92. [10] G. Heinrich, M. Klüppel, Recent advances in the theory of filler networking in elastomers, Adv. Polym. Sci. 160 (2002) 1e44. [11] A.A. Gusev, S.A. Lurie, Loss amplification effect in multiphase materials with viscoelastic interfaces, Macromolecules 42 (2009) 5372e5377. [12] J. Busfield, A. Muhr (Eds.), Constitutive Models for Rubber III, Swets & Zeitlinger, Lisse, 2003. [13] A. Medalia, Effect of carbon black on dynamic properties of rubber vulcanizates, Rubber Chem. Technol. 51 (1978) 437e523.
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[14] L. Mullins, Softening of rubber by deformation, Rubber Chem. Technol. 42 (1969) 339e362. [15] A.R. Payne, The dynamic properties of carbon black-loaded natural rubber vulcanizates, J. Appl. Polym. Sci. 6 (1962) 57e63. [16] J.D. Ulmer, Strain dependence of dynamic mechanical properties of carbon black filled rubber compounds, Rubber Chem. Technol. 69 (1996) 15e47. [17] S. Merabia, P. Sotta, D.R. Long, A microscopic model for the reinforcement and the nonlinear behavior of filled elastomers and thermoplastic elastomers (payne and mullins effects), Macromolecules 41 (2008) 8252e8266. [18] G.B. McKenna, Ten (or more) years of dynamics in confinement: perspectives for 2010, Eur. Phys. J. Spec. Top. 189 (2010) 285e302. [19] M.J. Wang, Effect of polymer-filler and filler-filler interactions on dynamic properties of filled vulcanizates, Rubber Chem. Technol. 71 (1998) 520e589.
[20] J. Berriot, H. Montes, F. Lequeux, D. Long, P. Sotta, Gradient of glass transition temperature in filled elastomers, Europhys. Lett. 64 (2003) 50e56. [21] N. Jouault, P. Vallat, F. Dalmas, S. Said, J. Jestin, F. Boue, Well-dispersed fractal aggregates as filler in polymer-silica nanocomposites: long-range effects in rheology, Macromolecules 42 (2009) 2031e2040. [22] N. Jouault, J.F. Moll, D. Meng, K. Windsor, S. Ramcharan, C. Kearney, S.K. Kumar, Bound polymer layer in nanocomposites, ACS Macro Lett. 2 (2013) 371e374. [23] Y. Kantor, I. Webman, Elastic properties of random percolating systems, Phys. Rev. Lett. 52 (1984) 1891e1894. [24] A.A. Gusev, Micromechanical mechanism of reinforcement and losses in filled rubbers, Macromolecules 39 (2006) 5960e5962.
Contents lists available at ScienceDirect
Polymer journal homepage: www.elsevier.com/locate/polymer
Short communication
Quantification of different contributions to dissipation in elastomer nanoparticle composites Sriharish M. Nagaraja a, Anas Mujtaba a, Mario Beiner a, b, * a b
Fraunhofer Institut für Mikrostruktur von Werkstoffen und Systemen IMWS, Walter-Hülse-Str. 1, 06120 Halle (Saale), Germany €t Halle-Wittenberg, Naturwissenschaftliche Fakulta €t II, 06120 Halle (Saale), Germany Martin-Luther-Universita
a r t i c l e i n f o
a b s t r a c t
Article history: Received 19 November 2016 Received in revised form 6 January 2017 Accepted 8 January 2017 Available online 10 January 2017
We present an approach to quantify different contributions to dissipation in elastomer nanoparticle composites based on strain sweeps. A modified Kraus equation is successfully used to approximate the 00 loss modulus depending on strain amplitude G g measured at different temperatures. For natural rubber composites containing > 10 vol% carbon black or > 3 vol% carbon nanotubes two different contributions to dissipation due to (i) breaking and (ii) deformation of glassy rubber bridges in the filler network are identified. Filler fraction and temperature-dependent trends support physical pictures considering that glassy rubber bridges are responsible for the visco-elasticity of the filler network. Constant extra contributions to dissipation are associated with the bulk-like fraction of the elastomer matrix or filler network independent effects. The achieved understanding should be very important for the optimization of elastomer nanoparticle composites for special applications like tire treads. © 2017 Elsevier Ltd. All rights reserved.
Keywords: Polymer composites Dissipation Reinforcement Polymer dynamics Mechanical properties
Developments towards a rational design of reinforcement and dissipation of elastomer nanoparticle composites are of extraordinary importance for various fields of applications [1e3]. A famous example are materials for tire treads where high reinforcement has to be combined with an optimized dissipation spectrum in order to combine long tire life with low rolling resistance and good wet grip [4e6]. In the recent years some progress has been made in understanding different contributions to reinforcement [7]. In particular, it has been shown based on strain-amplitude sweeps that the load carrying capacity of the filler network in highly filled composites depends significantly on temperature and frequency. One can conclude that the filler network shows visco-elastic properties what has been subsequently also confirmed by independent solid-state NMR investigations [8]. These findings have been interpreted as a strong indication for the existence of immobilized rubber bridges which are glassy under ambient conditions but soften sequentially at high temperatures and low measurement frequencies (Fig. 1a). This molecular interpretation predicts a relaxation spectrum Hu of elastomer nanoparticle composites which is broadened at the low frequency end of the a
* Corresponding author. Fraunhofer Institut für Mikrostruktur von Werkstoffen und Systemen IMWS, Walter-Hülse-Str. 1, 06120 Halle (Saale), Germany. E-mail address: [email protected] (M. Beiner). URL: http://www.imws.fraunhofer.de http://dx.doi.org/10.1016/j.polymer.2017.01.011 0032-3861/© 2017 Elsevier Ltd. All rights reserved.
relaxation peak as compared to that of unfilled elastomers due to interaction between rubber matrix and filler surface (Fig. 1b). The main peak position of the relaxation spectrum is, however, not changed since the major fraction of the elastomer matrix is unaffected by the filler particles even for high filler fractions above the percolation threshold fC indicating the transition from isolated filler aggregates to a percolated filler network. A very interesting point of this molecular picture is that dissipation should be also systematically affected if the relaxation spectrum of the elastomer is changed by the incorporation of fillers. In this letter, we present to our knowledge for the first time an approach that is able (i) to quantify different contributions to dissipation in elastomer nanoparticle composites based on simple strain-amplitude sweeps and (ii) to identify their molecular origin. This is very interesting since there is limited knowledge about the molecular origin of dissipation in elastomer nanoparticles composites and the factors that influence dissipation on the microscopic scale, which can be of very different nature [9e11]. Dissipation in the rubber matrix [12], friction between nanoparticles [13] and fracture work needed for breaking the filler network [9,14] are effects that have been considered in the literature. Starting point for the discussion in this work is the wellestablished phenomenological model by Kraus [9] describing the so-called Payne effect [15] in highly filled elastomer nanoparticle composites, which is commonly attributed to a breakdown of the
S.M. Nagaraja et al. / Polymer 111 (2017) 48e52
00
Gg¼
49
m 00 00 2 G m G ∞ gg 2m g gc
1þ
c
þG
00
(2)
∞
00
with G m being the peak height introduced as intensity measure. 00 Eq. (2) predicts that G g shows a peak at gC and identical values at small (g/ 0) and large (g/∞) strain amplitudes what is obviously not the case for differently filled NR composites (Fig. 2) where the 00 G g values at high and small amplitudes are significantly different. In particular, it is found that the loss modulus values at small strain 00 amplitudes (g < 101 %) are commonly much higher than G ∞ in contradiction to that what is predicted by Eq. (2). This clearly in00 dicates that the original Kraus equation for G g is insufficient to describe experimental data taken from strain sweeps. The necessary modification is obviously an additional sigmoidal contribution (blue lines in Fig. 2) with a g dependence similar to that obtained for the storage modulus G0 in Eq. (1). If such an additional term is introduced, one gets the modified equation
G
00
Gg¼
00
00
0
G ∞ 2m þ
1 þ gg c 00
00
m 00 00 2 G m G ∞ gg 1 þ gg c
00 ~ ~ ¼ G g;D þ G g;F þ G ∞
Fig. 1. (a) Sketch showing the filler network in highly filled elastomer nanoparticle composites containing glassy rubber bridges with internal mobility gradients related to different a relaxation times and glass temperatures Tg . (b) Comparison of the relaxation spectra Hu in the a relaxation range for NR and a related NR composite filled with 17.5 vol% carbon black. The color code for the mobility used in part (a) corresponds to that in part (b).
with G
00
0
and G
00
∞
2m
c
00
þG
∞
(3)
being the loss moduli at small (g/0 ) and large
filler network. Experimental evidence is a sigmoidal decrease in the storage modulus as function of strain amplitude G0 g as shown for a natural rubber (NR) composite in Fig. 2 (open symbols). This behavior is successfully modeled by Kraus based on rate equations describing the number of bridges in the filler network as function of strain amplitude Ng . For the storage modulus G0 g one gets
G0 g ¼
G0 0 G0 ∞ 0 2m þ G ∞ g 1þ g
(1)
c
with G0 0 and G0 ∞ being the storage moduli at small (g/ 0) and large (g/∞) strain amplitudes, respectively. Their difference defines the load carrying capacity of the filler network DG0 ¼ G0 0 G0 ∞ . Further parameters are the critical strain amplitude gC and the exponent m describing the shape of the sigmoidal decrease in G0 g . Eq. (1) fits the experimental G0 g data in Fig. 2 obviously quite well. In the following, we will focus on the quantification of different contributions to dissipation based on data for the loss modulus 00 depending on strain amplitude G g as obtained from strain sweeps 00 (Fig. 2, full symbols). Characteristic for G g is a peak occurring 00 concurrently with a sigmoidal decrease in G g . Kraus associated the 00 peak in G g with the heat released by breaking cohesive bonds between filler particles. Based on the rate equations which are 00 already used for the derivation of Eq. (1), he proposed that G g can be described by
00
Fig. 2. Strain-dependent storage modulus G0 (open symbols) and loss moduluas G (full symbols) for NR composites filled with (a) 20.4 vol% carbon black (CB) and (b) 6.5 vol% carbon nanotubes (CNT), respectively. Kraus approximations to real and imaginary parts are shown as a black lines. The modified approximation for the loss modulus according to Eq. (3) is shown as red line. The blue line indicates the additionally introduced dissipative contribution.
50
S.M. Nagaraja et al. / Polymer 111 (2017) 48e52 00
(g/∞) strain amplitudes, respectively. G 0 accounts here for the amplified dissipation at small strain amplitude where the filler network is intact. These contributions are obviously absent if the 00 filler network is totally broken at G ∞ . Fits based on Eq. (3) (red lines) approximate the experimental data for the overall dissipation 00 G g for structurally different fillers like carbon black (CB, Fig. 2a) or carbon nanotubes (CNT, Fig. 2b) obviously very well. Moreover, contributions to dissipation with physically distinct origins can be discriminated what becomes more clear if a generalized form of Eq. 00 ~ (3) is used, where G g;D is the 00dissipation due to oscillatory defor~ mation of the filler network, G g;F is related to the heat released by breaking bridges in the filler network originally introduced by 00 Kraus, while G ∞ represents contributions to dissipation in composites, which are not related to the filler network. Note that an equation similar to Eq. (3) has been proposed by Ulmer [16] some time ago on a phenomenological basis without physical interpretation of the individual parameters and clear connection to the molecular situation. The interpretation of the terms in Eq. (3) given above fits well to results from recent studies demonstrating that the filler network in highly filled rubbers shows viscoelastic properties arising from glassy rubber bridges connecting individual nanoparticles or aggregates (Fig. 1a) [8]. These glassy bridges are formed by a tiny fraction of immobilized rubber having Tg values, which can be several 10 K higher than the bulk rubber [17,18] due to a strong interaction between elastomer segments and filler surface [19e22]. 00 ~ From that point of view G g;D quantifies dissipative contributions caused by the deformation of intact glassy rubber bridges being the weakest elements in the filler network [11,23,24]. If the strain 00 ~ amplitude g increases, G g;D should also decrease since the number of intact glassy rubber bridges Ng decreases as predicted already by the classical Kraus model for G0 g (Eq. (1), for details see ESI). The 00 ~ second strain-dependent term in Eq. (3), G g;F , is due to the heat released during breaking of glassy rubber bridges and physically of 00 quite different nature. Its contributions to G g are largest in the range where most of the00 glassy rubber bridges break indicated by 00 m ~ the peak in G g . Hence, G g;F is assumed to be proportional to Ng g as already done by Kraus in the derivation of Eq. (2) (cf. ESI). Most interesting consequence of the proposed physical picture is that the 00 00 00 00 00 dissipation measures DG D ¼ ðG 0 G ∞ Þ as well as DG F ¼ ðG m R ~00 00 G ∞ Þf G g;F dg should be under idealized conditions both proportional to the initial number of bridges N0 characterizing the undisturbed filler network since their extrapolated number at infinite strain amplitude N∞ is commonly assumed to be 0 (for details see ESI). The dissipation remaining even at very high strain 00 amplitudes, G ∞ , is probably due to contributions of relaxators in the bulk-like fraction of the elastomer matrix or other processes without relation to the filler network. Obviously, this approach allows to discriminate different contributions to dissipation in elastomer nanoparticle composites with filler contents above the percolation threshold ðf > fC Þ based on classical strain sweeps. The proposed physical picture can be further checked using the results of strain sweeps on NR-CB composites with variable filler fractions f performed at different temperatures T, which are presented in Fig. 3. The commonly high quality of the fits (solid lines) 00 underlines that Eq. (3) is precisely predicting the behavior of G g for different temperatures T and filler fractions f. In addition, several non-trivial trends predicted by the discussed physical picture are confirmed. For all temperatures (0, 25, 60 C) the characteristic peak at intermediate strain amplitudes is only observed for higher filler loading above the percolation threshold (f > fC ). The peak intensity is, however, much higher for the lowest temperature (0oC) and goes systematically down if temperature increases. This is expected considering the results for the storage modulus G0 g indicating that the total number of bridges N0 (or analogously the filler
network strength DG0 ) is reducing commonly with increasing temperature [7,8]. Accordingly, the total number of glassy rubber bridges that break in a strain sweep (and/or the glassy bridge volume) should also decrease with increasing temperature and decreasing filler fraction in percolated systems. Most interesting is 00 now that the loss modulus values at small strain amplitudes G 0 00 and the step heights DG D behave qualitatively like or the peak 00 00 intensity G m or DG F . Both quantities decrease obviously also with increasing T and decreasing f (above fC ) while the limiting dissi00 pation G ∞ is only slightly changing. Within the proposed physical picture these trends are understood by differences regarding the initial number of bonds in the intact filler network N0 (and/or the glassy bridge volume), which is lowered if the temperature increases or the filler fraction decreases. If N0 decreases, the dissipation due to oscillatory deformation of glassy bridges in the intact filler network decreases accordingly. Note that the rate equations behind Eqs. (2) and (3) are not only applicable to count the actual number of bridges Ng as proposed by Kraus but should be analogously also able to quantify the volume in glassy bridges, which is possibly a more relevant quantity for measuring strength and dissipation of a filler network. Summarizing this part, one can conclude that there is a parallel influence of temperature T and filler 00 fraction f on both, dissipation at low strain amplitudes G 0 and 00 peak intensity measures like G m00 . This indicates strongly that both 00 ~ ~ contributions to dissipation, G g;D and G g;F , are due to glassy rubber bridges being part of the filler network. In order prove further correlations between the two strain00 00 dependent terms in Eq. (3) the fit parameters, DG D and DG F , are
00
Fig. 3. Loss modulus (G ) as function of strain amplitudes for different filler fractions of carbon black at (a) 0 C, (b) 25 C and (c) 60 C. The fits to the measured data are based on the modified Kraus equation for the loss modulus (Eq.(3)).
S.M. Nagaraja et al. / Polymer 111 (2017) 48e52
compared in Fig. 4 for different NR-CB composites measured under various conditions. It is quite clear from Fig. 4a that both quantities do increase for a given temperature with f as expected owing to the trend in the number of glassy rubber bridges ðN0 Þ in the undeformed filler network. Interestingly, the dependence on filler fraction is also similar for both quantities. For a given filler fraction 00 00 DG D and DG F decrease with increasing temperature. This indicates that at low temperature the number of glassy rubber bridges (and/or their volume) is higher since partial softening occurs as the temperature is increased. Accordingly, the dissipation caused by glassy rubber bridges should decrease with T. In Fig. 4b, 00 00 DG D is plotted vs DG F for samples with f > fC . A linear dependence is commonly observed for different temperatures. This strongly supports the already discussed prediction that both quantities are proportional to N0 . Obviously, the glassy rubber bridges influence the dissipation in the investigated natural rubber composites most. This is a striking finding underlining the importance of the physical picture used in this letter (Fig. 1). Note that m is basically constant for the investigated NR-CB composites and that the observed proportionality may also indicate that other important quantities like the heat needed for breaking a glassy rubber bridge qF or the dissipation occurring during a full cycle of oscillatory deformation of a glassy rubber bridge qD are mainly independent on filler volume fraction f and total number of bridges N0 in highly filled composites (f > fC ). Differences regarding the temperature dependence of qD and qF may be relevant for the temperature dependence of the slopes in Fig. 4b. Finally, we should also highlighted that the overall dissipation behavior in NR-CNT composites (Fig. 2b) is qualitatively similar and well described by Eq. (3). However, the situation is characterized by 00 smaller DG D values as compared to NR-CB composites although
DG
51
00
0 F and reinforcement related quantities (G 0 ) are quite similar. This may be partly explained by a smaller number of glassy rubber bridges but demonstrates that there are filler-specific effects and probably differences in the ratio qD =qF which definitively need further investigations comparing such quantities more systematically. In summary, we have compiled in this work experimental evidences for the existence of at least three different contributions to 00 dissipation G g in strain sweeps for highly filled elastomer nanoparticle composites. These dissipative contributions are obviously of physically quite different origin. A physical picture is proposed that allows to quantify these contributions and to associate them with distinct molecular mechanisms. For natural rubber composites containing large amounts of carbon black (f > 10 vol%) dissipation contributions due to oscillatory deformation of glassy 00 ~ rubber bridges G g;D are found to be most relevant at low strain amplitudes (g < 0:1%). The heat caused by breaking glassy rubber 00 ~ bridges G g;F is usually dominating at intermediate strain amplitudes close to gC z2%. Filler network independent dissipation ef00 fects G ∞ , appearing for example in the bulk-like rubber matrix, are obviously most important for extremely large strain amplitudes (g > 50%). The situation is shown to vary with temperature as expected due to a sequential softening of the glassy rubber bridges connecting neighbored filler particles or aggregates. In the framework of the proposed physical picture this is related to a reduction of the number (and/or the volume) of the glassy rubber bridges N0 in the undeformed filler network. We strongly believe that the predictions of this physical picture can help to optimize elastomer nanoparticle composites for special applications like tire treads.
Acknowledgments Financial support by the state Sachsen-Anhalt in the framework of projects 1304/00022 and 1604/00028 is greatly acknowledged. Appendix A. Supplementary data Supplementary data related to this article can be found at http:// dx.doi.org/10.1016/j.polymer.2017.01.011. References
00
00
Fig. 4. (a) DG D and DG F (insert) vs filler volume fraction f. The parameters are ob00 00 00 tained from fits to G g curves shown in Fig. 3 using Eq. (3). (b) DG F vs DG D for various NR-CB composites with high filler fractions (f > fC ) measured at three different temperatures.
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