Elastic and Viscoelastic Properties of Non-bulk Polymer Interphases in Nanotube-reinforced Polymers

Transcription

1 Elastic and Viscoelastic Properties o Non-bulk Polyer Interphases in Nanotube-reinorced Polyers Frank T. Fisher 1, Kon-Chol Lee 1, and L. Catherine Brinson 2 1 Departent o Mechanical Enineerin, Stevens Institute o Technoloy Castle Point on Hudson, Hoboken, NJ Departents o Mechanical Enineerin and Materials Science and Enineerin Northwestern University, 2145 Sheridan Road, Evanston, IL ABSTRACT There is considerable interest in usin carbon nanotubes (NTs) to create ultiunctional polyer coposite aterials with outstandin echanical, electrical, and theral properties. A hurdle in odelin the behavior o these systes is the non-bulk interphase reion in these systes that ors due to nanoscale interactions between the ebedded NTs and adjacent polyer chains. This interphase reion coprises a substantial portion o the volue raction o the coposite due to the treendous NT surace area per unit volue available or interaction with local polyer chains and results in siniicant chanes in the viscoelastic properties o the nanotube-polyer coposite copared to those o the bulk polyer. However, the echanical properties o this interphase reion are unknown and very diicult to easure directly ro experiental testin due to the size scale o this interphase reion. Thus a three-phase (iber/nanotube annular interphase atrix) Mori- Tanaka icroechanical odel has been developed such that the properties o this interphase reion can be inerred ro acroscale viscoelastic data obtained usin dynaic echanical analysis. Such analysis will be particularly useul as a eans to assess chanes in the obility and echanical behavior o this interphase reion and as a eans to characterize the ipact o cheical unctionalization on interphase oration in these systes. BACKGROUND Due to the inherent strenth o the carbon-carbon bond and the potential o a deect-ree structure, it has been suested that nanotubes ay approach the theoretical liits or any iportant echanical properties, includin axial stiness and tensile strenth. Lare increases in racture strain and touhness, and superior electrical/theral properties, are other potential beneits o usin NTs as the iller aterial in a polyer-based coposite. Detailed discussions o the echanical properties o nanotubes and their use in coposites are provided in a nuber o places in the literature; see or exaple [1, 2] and [3, 4], respectively. Eorts to odel the echanical behavior o nanotube-polyer coposites are coplicated by the presence o a polyeric interphase reion surroundin the nanotubes with properties dierent ro that o the atrix polyer. While irst postulated based on chanes in the viscoelastic behavior o nanotube-polyer coposites in coparison to the behavior o the pure atrix polyer aterial, a nuber o recent coputational atoistic odelin results are consistent with the presence o this interphase reion [5-7]. For at least soe nanotube-polyer systes, this interphase polyer has been characterized as havin restricted olecular obility due to the ipeded olecular otion o those polyer chains interactin with the ebedded nanotubes. Such restricted olecular obility within the interphase reion has been inerred experientally in a nuber o ways, includin: shitin o the nanocoposite lass transition teperature to hiher teperatures [8, 9], broadenin o the loss tanent peak [10], chanes in the relaxation spectra [11], and oration o a stronly adhered polyer layer on the surace o the nanotubes [12]. Cheical unctionalization o the nanotubes and subsequent covalent tetherin o short-chain polyers to the nanotube surace has been shown to increase the interphase eect in nanotube-polyer coposites [13, 14]. INTRODUCTION TO THE MORI-TANAKA METHOD FOR MULTIPHASE COMPOSITES The Mori-Tanaka ethod is a popular icroechanics tool or the study o ultiphase aterials; the reader is reerred to the literature or an in-depth derivation and discussion o the ethod [15-17]. A critical coponent o the Mori-Tanaka ethod is

2 the deterination o the ute strain concentration tensor A r, which relates the averae strain o the rth inclusion to the arield strain in the atrix ε 0 such that ε r = A r ε 0. (1) I such a tensor can be deterined, one can show that the eective stiness o a coposite with randoly oriented inclusions can be written as [17] 1 r r C = 0 C 0 + r { C r A r } 0 I + r{ A r }, (2) n=1 n=1 where C is the eective stiness tensor o the coposite, r and C r are the volue raction and stiness o inclusion phase r, 0, and C 0 are the volue raction and stiness o the atrix aterial, I is the identity tensor, and brackets {} denote the appropriate orientational averae (see [18]). The ability to handle spatial orientation distributions o the ebedded nanotubes within the polyer is critical, as the nanotubes are rarely alined within polyer nanocoposites. For inclusions which are ellipsoidal in shape, the solution or the ute strain concentration tensor A r can be written explicitly in ters o the Eshelby tensor S r, which or an isotropic atrix is only a unction o the inclusion shape and the Poisson ratio o the atrix, such that A r = [ I + S r C 1 0 ( C r C 0 )] 1. (3) General ors or the Eshelby tensor or ellipsoidal inclusions [19], includin the liit case o ininitely lon cylindrical inclusions [20], are provided in the literature. As shown in Fiure 1A, the standard ultiphase Mori-Tanaka approach requires separate, physically distinct ellipsoidal inclusion phases or the calculation o the Eshelby tensor o each phase or substitution into (3). Thus, while one could odel a three phase nanocoposite (bulk polyer as the atrix phase and interphase polyer and nanotube as the inclusion phases), such an approach represents a physical syste as in Fiure 1A where the interphase is a separate ellipsoidal inclusion ro the nanotube inclusions. However, the actual physical eoetry o nanotube-reinorced polyer systes is such that the interphase reion is an annular reion, continuous with and surroundin the nanotube (see Fiure 1B). Thus, the developent o the ute strain concentration tensor or coated ibers is needed, as presented in the next section. Note that the sipliied odel shown in Fiure 1A has been used previously by the authors to odel the echanical behavior o viscoelastic interphases [21] and the echanical behavior o a nanotube-reinorced polyers [14, 22]. Thereore the results o this work will also clariy the deree o error in this approxiation and help to deine conditions under which accurate representation o the annular interphase is essential. In the next section o the paper, an analytical ethod is used to derive the ute strain concentration tensors or the nanotube and its continuous annular interphase, respectively, via the deinition in (1). Subsequently, these tensors can be substituted directly into (2) to deterine the eective stiness o the nanotubereinorced polyer. Such exaple calculations are then provided in the Results section. (A) (B) iber interphase Fiure 1. Scheatic o icroechanics approaches. (A) Standard ultiphase Mori-Tanaka approach. (B) Coated ibrous inclusion approach.

3 DILUTE STRAIN CONCENTRATION FOR INFINITELY LONG COATED INCLUSIONS Several icroechanics-based odels have been developed to deterine the eective oduli o coposites with coated inclusions. A nuber o these ethods (includin the present) are based on the analytical solution or the stress ields in an ininitely lon coated iber irst developed by Benveniste and co-workers or alined inclusions [23]. For exaple, Wen and co-workers developed an iterative approach where a two-phase coposite is odeled usin a three-phase inclusion-atrixeective coposite eoetry in ters o a odiied Eshelby tensor to account or particle interactions at hih concentrations o inclusions [20, 24]. Alternative icroechanics approaches, such as one developed speciically or thickly coated inclusions based on a proressively illed coposite eleent assebly [25], have also been developed. The approach presented here is a re-orulation o the oriinal Benveniste ethod, where or reasons discussed previously we solve explicitly or the ute strain concentration tensors or the iber and interphase reions usin the deinition in (1). Such an approach is related our previous work, where the coponents o the ute strain concentration tensor were calculated directly usin the inite eleent ethod to odel the ipact o nanotube waviness on the eective odulus o a two phase nanotube-polyer coposite [26]. Fiure 2. (A) Deterination o ute strain concentration based on a sinle coated ibrous inclusion in an ininite atrix aterial. (B) Axial loadin case. (C) Hydrostatic loadin case. (D) Transverse shear loadin. (E) Lonitudinal shear loadin. Usin contracted notation, the ute strain concentration tensors iven in (1) can be written explicitly or the iber 1 and interphase, respectively, as 1 Because the icroechanics approach presented here assues a continuu description or the echanical behavior o the nanotube, the ters nanotube and iber are used interchaneably.

4 A A A A A A A A = A A A A A, A A A A A A A A = A A (4) A A A The oal o the current work is to develop an analytical solution or each o the individual coponents o the ute strain concentration tensors iven in (4). First, usin (1) with (4) one inds that ε 33 = A 31 ε 11 + A 32 ε 22 + A 33 ε 33, ε 33 = A 31 ε 11 + A 32 ε 22 + A 33 ε 33. (5) By inspection, or the case o an ininitely lon coated inclusion ebedded in an ininite atrix considered here, the averae r axial strain in the iber and interphase reions (ε 33 ; r =,) will be independent o the transverse strains o the atrix and equal to the axial coponent o the atrix strain. Thus A 31 = A 32 = A 31 = A 32 = 0, (6) A 33 = A 33 = 1 such that the ute strain concentration tensors can be urther sipliied to A A A A A A A = A A A, A A A A A A A =. (7) A A A In addition, as shown in Fiure 2A, one can take advantae o the syetry o the 1-2 plane to urther sipliy the ute strain concentration tensors iven in (7), i.e. A A A A A A A = A A A, A A A A A A A =, (8) A A A such that there are now ive independent coponents ( A r 11, A r 12, A r 13, A r, A r ; r =,) o the ute strain concentration tensor or each inclusion phase. The reainin coponents o the ute strain concentration tensors are ound via the analytical solution o the averae strains in the iber and interphase reions or the loadin cases shown in Fiure 2B-2E ollowin the analysis o Benveniste [23]. The eneral solution procedure is as ollows: or each o these loadin cases, one can write the appropriate ors o the displaceents o each phase as a unction o position in ters o unknown constants. Given these displaceents, straindisplaceent relationships and Hooke s Law are invoked to deterine the strains and stresses in each phase. Appropriate boundary conditions are then invoked, which allows one to deterine the unknown constants in the displaceent (and hence strain and stress) ields, thus providin the analytical solution or each loadin case. Fro these analytical solutions, the phase-averaed strains or the iber and the interphase can be deterined, at which point the coponents o the ute strain concentration tensor or each phase as deined in (1) can be deterined. To illustrate the procedure, we will consider the solution o the lonitudinal shear loadin case shown in Fiure 2E, which will provide the analytical solution or A and A in (8). A ore detailed presentation o this derivation, as well as the derivation

5 or the other loadins cases shown in Fiure 2 will be presented elsewhere [27]. For this loadin case, the non-zero displaceents can be written as = A 4 r sinθ u z = A r + B r sinθ u z = A 4 r + B 4 sinθ r u z (9) where A 4, A, B, A 4, and B 4 are unknown constants to be deterined. Non-zero strain coponents in cylindrical coordinates are iven as ε rz ε zθ = A 4 sinθ, ε rz = A B r 2 sinθ, ε rz = A 4 B 4 sinθ r 2 = A 4 cosθ, ε zθ = A + B r 2 cosθ, ε zθ = A 4 + B 4 cosθ r 2 (10) Usin Hooke s Law, the non-zero stress coponents can be written as σ rz = 2µ A 4 sinθ, σ rz = 2µ A B r 2 sinθ, σ zθ = 2µ A 4 cosθ, σ zθ = 2µ A + B r 2 cosθ, = 2µ A 4 B 4 sinθ r 2. (11) σ zθ = 2µ A 4 + B 4 cosθ r 2 σ rz r The above stress ields can be converted to Cartesian coordinates usin the relationship σ yz Appropriate boundary conditions or the iven case are r = σ rz r sinθ +σ zθ cosθ (r =,,). u z = u z, σ rz u z = u z, σ rz ε yz 0 = ε yz = σ rz = σ rz at r = a at r = b, (12) as r the application o which can be shown to yield the ollowin syste o equations (in atrix or) which can be solved to ind the unknown ield constants in (9)-(11). a a a 1 0 b b 1 b b µ yz µ yz µ yz 0 0 a 2 0 µ yz µ yz µ yz µ a 2 yz b A 4 A B A 4 B = ε yz (13) Once the known constants A 4, A, B, A 4, B 4 have been deterined via (13), their substitution into (10) provides analytical expressions or the strain ields or each phase in the coposite. The phase averaed strain coponents in the iber and inclusion phases can then be deterined usin ε yz = 2π a ε yz (r,θ ) rdrdθ 0 0, ε π a 2 yz = 2π b ε yz (r,θ ) rdrdθ 0 a π ( b 2 a 2 ). (14)

6 Likewise, the ar-ield strain in the atrix or the case o a sinle ininitely lon iber in an ininite atrix can be deterined ro (10) as ε 0 = ε yz = A 4. (15) r Thus, by the deinition in (1) the ollowin coponents o the ute strain concentration tensors or the iber and interphase can be deterined based on exaination o the lonitudinal shear case presented above: where the values o A 4, A, and A 4 are ound ro the solution to (13). A = A 4 A 4, A = A A 4, (16) The reainin ute strain concentration ters (or both the iber and the interphase) in (8) can be ound via analysis o the loadin cases iven in Fiure 2 as ollows: - A ro the transverse shear loadin case (Fiure 2D) - A 11 and A 12 ro the transverse hydrostatic loadin case iven in Fiure 2C, with the additional constraint that A 11 A 12 = A - A 13 ro a superposition o the axial loadin case (Fiure 2B) and the transverse hydrostatic case (Fiure 2C) While the solution to the reainin ute strain concentration ters iven in (8) ollows the outline o the procedure outlined above, the alebra is unwieldy and will be presented elsewhere [27]. RESULTS One can show that the resultin ute strain concentration tensors or the coated ibrous coposite deterined in this ashion are identical to those calculated usin the Eshelby tensor via (2) or a two phase coposite when the interphase properties atch those o the iber or the atrix, respectively. Preliinary results coparin the eective Youn s oduli and shear oduli predictions or a three phase coposite usin the standard Mori-Tanaka ipleentation (assuin physically distinct iber and interphase reions; see Fiure 1A) and the coated ibrous inclusion odel presented here are shown in Fiures 3 and 4, respectively. Constant diensionless Youn s oduli o 100, 10, and 1 were chosen or the iber, interphase, and atrix, respectively. Here the volue raction o the iber was kept constant at 1%, while the volue raction o the interphase varied ro 0 to 50%. For all phases a Poisson ratio o 0.3 was assued. Two types o coposites were considered: coposites where the iber and interphase reions are alined, and a coposite with the iber and interphase reions randoly orientated in 3D space. Fiure 3 shows that there is relatively little dierence in the coposite oduli predictions between the two ethods or the values o elastic oduli shown here. This is not surprisin or the alined coposite case, where or ininitely lon ibers icroechanical ethods approach the siple rule o ixture approxiations. More surprisinly, or the case o randoly orientated inclusions in 3D space there is relatively little dierence between the standard Mori-Tanaka ultiphase analysis and the coated inclusion odel presented here or the phase properties assued here. As shown in Fiure 4, there appears to be a larer dierence between the ethods when evaluatin the eective shear odulus o the coposites. Further analysis o the dierences between the two ethods is onoin. Note also that, or nanotube-polyer coposites with one o the viscoelastic phases close to the lass-rubber transition, it ay be possible or the dierences in the phase oduli to be uch larer than those assued here. Thus, via ipleentation o the dynaic correspondence principle to the elastic odel developed here, dierences in eective viscoelastic oduli predictions between the standard Mori-Tanaka ultiphase odel and the coated inclusion odel will be investiated in detail in uture work. Ultiately, iven the viscoelastic properties o the bulk atrix aterial and the eective behavior o the nanocoposite, it will be possible to iner the viscoelastic properties o the interphase reion necessary to atch the results o the icroechanical odelin with experiental data.

7 Fiure 3. Eective Youn s odulus as a unction o interphase volue raction or dierent iber orientations. Fiber: E = 100, v = 1%. Interphase: E = 10. Matrix: E = 1. All phases considered isotropic with Poisson s ratio = 0.3. Fiure 4. Eective shear odulus as a unction o interphase volue raction or dierent iber orientations. Fiber: E = 100, v = 1%. Interphase: E = 10. Matrix: E = 1. All phases considered isotropic with Poisson s ratio = 0.3. CONCLUSIONS A hurdle in odelin the behavior o nanotube-polyer coposite systes is the non-bulk interphase reion in these systes that ors due to nanoscale interactions between the ebedded NTs and adjacent polyer chains. This interphase reion coprises a substantial portion o the volue raction o the coposite due to the treendous NT surace area per unit volue and results in siniicant chanes in the viscoelastic properties o the nanotube-polyer coposite copared to those o the bulk polyer. The icroechanical odel developed here was otivated by the desire to odel the eective echanical behavior o these systes while aintainin an accurate physical representation o the annular interphase reion surroundin the nanotubes, and particularly, in a anner suitable to odel dierent orientational distributions o the coated nanotubes within the coposite. While the derivation and results presented here are or an elastic analysis, one can reay use the dynaic correspondence principle to extend this odel or viscoelastic aterials. In this reard the current odel will be useul or inerrin the chane in echanical properties o the interphase polyer based on acroscale experiental data or a particular nanotube-polyer syste; such an approach ay also be useul in characterizin the ipact o various cheical unctionalization strateies on interphase oration in nanocoposite systes. The present odel ay also ind use or bridin atoistic siulations o nanotube-polyer systes with continuu-level predictions o the eective echanical behavior o these aterials.

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