J Mater Sci(2008)43:6747-6757 6755 b2N2 relaxation( Fig. 6)(46]. In the case of the non-local normal khTN ISo(N-Na)+ saNa] mode of relaxation, its characteristic volume is the upper In the case that Na=N, the reptation time takes the size. Below this upper limiting characteristic volume V aro N2. where r is the chain end-to-end distance b2N5/2 and n is the number of monomer units in the chain. The =2k.[wN2+5 (6) characteristic time, te, for each particular relaxation pro- cess varies from 10 s for bond vibrations above Tsto To describe the change in reptation dynamics of the chains the infinitely long times below Tg. Thus, the macroscopic as a function of nanoparticle volume fraction, percolation viscoelastic response of a polymer is a manifestation of a model was used. At the percolation threshold, physical range of molecular relaxations localized in some charac network formed by interconnection of immobilized chains teristic volume and the rate of the relaxation mode on individual nanoparticles penetrates the entire sample indirectly proportional to its locality (Figs. 7 and 8) volume. In this case, only physical"cross-links" are The physical reasons for the expected breakdown of considered and the terminal relaxation time reaches the ontinuum elasticity on the nano-scale include increasing value characteristic for the life time of the physical filler- importance of surface energy due to appreciable surface to polymer bond. Thus, the relaxation time near the percolation volume ratio [47], the discrete molecular nature of the hreshold is expressed in the form [44] polymer matrix resulting in non-local behavior in contrary to local character of classical elasticity [48], the presence of nano-scale particles with the length scale similar to the here v* is critical effective filler volume fraction (veff=0.04 for PVAc-HAP at 90C)and b is the perco- lation exponent(b= 4 for the same system). The veff is a sum of the filler volume fraction and the volume fraction of immobilized Snt=42m2/g the percolation, random clustering of effective hard spheres 8 o.249 immobilized chains and was shown to equal 0.04 for a PVAc-HAP nanocomposites at 90C. In order to simpli E was considered only in the way similar to that originally outlined by Jancar et al. [45] for micro-scale composites Percolation threshold Percolation threshold at approximately 2 m of the filler- were immobilized at the internal contact area of 42 mper B LER SuP G polymer contact area per l g of the composite was found in AREA PER 1g OF NANOCOMPOSITE(m) PVAC-HA ( Fig. 6). all the chains 1 g of the Characteristic length scale for transition between continuum and discrete elasticity in polymer composites layer pproximately R, The classical continuum mechanics is designed to be size- independent. For nano-composites, however, size-depen surface dent elastic properties have been observed which cannot be readilly explained using continuum mechanics and, thus, prevent simple scalling down the existing continuum elasticity models [23]. Polymers are unique systems with macroscopic viscoelastic response driven by the relaxation processes on the molecular level [46]. These relaxation processes represent particular molecular motions occurring in some characteristic volume, Ve. The V depends on the type of the relaxation process and temperature. The char Fig. 7 Simple approach combining the reptation dynamics and percolation model to describe the retarded reptation of chains in the acteristic volumes vary from 10-3 nm for localized bond vicinity of solid nano-sized inclusions representing the nano-scale ibrations to 10 nm for the non-local normal mode of 2 Springersads rep ¼ b2N2 p2kbTNe ½ ð f0ð Þþ N Na faNa 5Þ In the case that Na = N1/2, the reptation time takes the form: sads rep ¼ b2N5=2 2p2kbTNe f0N1=2 þ fa h i ð6Þ To describe the change in reptation dynamics of the chains as a function of nanoparticle volume fraction, percolation model was used. At the percolation threshold, physical network formed by interconnection of immobilized chains on individual nanoparticles penetrates the entire sample volume. In this case, only physical ‘‘cross-links’’ are considered and the terminal relaxation time reaches the value characteristic for the life time of the physical filler￾polymer bond. Thus, the relaxation time near the percolation threshold is expressed in the form [44]: srec composite ¼ sads rep veff veff v eff 1 v eff
b ð7Þ where v eff is critical effective filler volume fraction (v eff = 0.04 for PVAc-HAP at 90 C) and b is the perco￾lation exponent (b = 4 for the same system). The v eff is a sum of the filler volume fraction and the volume fraction of immobilized chains and was shown to equal 0.04 for PVAc-HAP nanocomposites at 90 C. In order to simplify the percolation, random clustering of effective hard spheres was considered only in the way similar to that originally outlined by Jancar et al. [45] for micro-scale composites. Percolation threshold at approximately 2 m2 of the filler￾polymer contact area per 1 g of the composite was found in PVAc-HA nanocomposite system (Fig. 6). All the chains were immobilized at the internal contact area of 42 m2 per 1 g of the nanocomposite. Characteristic length scale for transition between continuum and discrete elasticity in polymer composites The classical continuum mechanics is designed to be size￾independent. For nano-composites, however, size-depen￾dent elastic properties have been observed which cannot be readilly explained using continuum mechanics and, thus, prevent simple scalling down the existing continuum elasticity models [23]. Polymers are unique systems with macroscopic viscoelastic response driven by the relaxation processes on the molecular level [46]. These relaxation processes represent particular molecular motions occurring in some characteristic volume, Vc. The Vc depends on the type of the relaxation process and temperature. The char￾acteristic volumes vary from 10-3 nm3 for localized bond vibrations to 106 nm for the non-local normal mode of relaxation (Fig. 6) [46]. In the case of the non-local normal mode of relaxation, its characteristic volume is the upper limit for Vc displaying strong dependence on the chain size. Below this upper limiting characteristic volume, Vc * R3 N3/2, where R is the chain end-to-end distance and N is the number of monomer units in the chain. The characteristic time, sc, for each particular relaxation pro￾cess varies from 10-14 s for bond vibrations above Tg to the infinitely long times below Tg. Thus, the macroscopic viscoelastic response of a polymer is a manifestation of a range of molecular relaxations localized in some charac￾teristic volume and the rate of the relaxation mode is indirectly proportional to its locality (Figs. 7 and 8). The physical reasons for the expected breakdown of continuum elasticity on the nano-scale include increasing importance of surface energy due to appreciable surface to volume ratio [47], the discrete molecular nature of the polymer matrix resulting in non-local behavior in contrary to local character of classical elasticity [48], the presence of nano-scale particles with the length scale similar to the Fig. 7 Simple approach combining the reptation dynamics and percolation model to describe the retarded reptation of chains in the vicinity of solid nano-sized inclusions representing the nano-scale ‘‘interphase’’ J Mater Sci (2008) 43:6747–6757 6755 123