Data supporting micromechanical models for the estimation of Young's modulus and coefficient of thermal expansion of titanate nanotube/Y2W3O12/HDPE ternary composites

This article presents several micromechanical models to predict the Young's modulus and the coefficient of thermal expansion of titanate nanotube/Y2W3O12/HDPE composites. The equations and assumptions of the selected micromechanical models are described in detail for this ternary system. Data of the elastic constants, coefficient of thermal expansion of composite components and other associated parameters, obtained either by literature survey or processing of literature information, are compiled in this work. For further interpretation of the data presented in this article, please see our research article entitled “The effect of titanate nanotube/Y2W3O12 hybrid fillers on mechanical and thermal properties of HDPE-based composites” (Pontón et al., 2019).


Data
The data presented in this article include a collection of four micromechanical models to predict the stiffness of polymer composites reinforced with particulate hybrid fillers comprising titanate nanotubes (TTNT) and yttrium tungstate (Y 2 W 3 O 12 ), which is a thermomiotic-like filler. These models are: the rule of mixtures (ROM), modified rule of mixtures (MROM), Halpin-Tsai and Hashin-Shtrikman. The equations of these models and the corresponding assumptions (see footnotes) are summarized in Table 1, along with the data of elastic constants and specific parameters for each composite phase. Table 2 presents a compendium of the micromechanical models chosen for the estimation of the coefficient of thermal expansion (CTE) of TTNT/Y 2 W 3 O 12 /HDPE composites, such as the rule of mixtures (ROM), Turner's and Schapery's models. The assumptions of these models (see footnotes) are also described, as well as, the bulk modulus and CTE for each composite phase.

Experimental design, materials, and methods
The data of micromechanical models presented in Tables 1 and 2 were selected from the literature to describe the TTNT/Y 2 W 3 O 12 /HDPE ternary system, depicted in Scheme 1. These ternary composites can be composed by different TTNT/Y 2 W 3 O 12 mass ratios: 1:2, 1:1 and 2:1 and prepared using three approaches whether the hybrid filler and the matrix are modified or not: i) pristine hybrid fillers < composites denoted as C1:2, C1:1 and C2:1>, ii) hybrid fillers modified with Specifications Table   Subject area Ceramics and composites More specific subject area

Polymer-based composites
Type of data The optimum titanate nanotube/Y 2 W 3 O 12 theoretical mass ratio for a specific application can be estimated from micromechanical analysis, depending on the properties expected for composites, as an initial criterion to define the amounts of fillers to be used in the manufacturing of composites. The approaches suggested in this work for the application of micromechanical models of binary composites to ternary ones, especially for the prediction of the coefficient of thermal expansion by Schapery's model could be considered as a benchmark for further modeling of other ternary polymer composites (reinforced with two different nanofillers), since articles dealing with the prediction of thermal expansion properties of three phase composites are still scarce. Table 1 Models for prediction of Young's modulus of HDPE-based composites reinforced with TTNT/Y 2 W 3 O 12 hybrid filler and data associated with the elastic constants and specific parameters for each composite phase.

Model Prediction
Eq.
composite, which is considered as a new matrix, calculated with Eq. 5. where, z f1 ¼ shape parameter of TTNT$20, calculated with Eq. 7.
z f2 ¼shape parameter of Y 2 W 3 O 12 $ 2, as the first approximation for spherical particulate fillers. Hashin-Shtrikman E l c , E u c ¼ lower and upper bounds of (continued on next page) K hyb ¼bulk modulus of the hybrid filler computed with Eq. 15.
TTNT within the hybrid filler calculated with Eq. 16.
G hyb ¼shear modulus of the hybrid filler calculated with Eq. 22.
n ¼ shear modulus of TTNT$ 106.06 GPa, calculated with Eq. 2, but using the corresponding data assumed for TTNT.  [12,13]. c G f2 is unavailable in the literature. Hence, G Y2 Mo3 O12 was used [14], since bulk moduli of both materials are similar (K Y2 Mo3 O12 ¼21 GPa). d The product between b and E f is termed as effective Young's modulus of filler (E eff ).
e Since the length of TTNT is much smaller than the specimen thickness, it is expected that TTNT are randomly oriented in 3D [7]. Thus, b 1 is assumed as 0.2, value used in the literature as a first approximation for 3D randomly oriented carbon nanotubes [7] and particulate fillers, such as TiO 2 [8]. anatase TiO 2 nanotubes, synthesized by a similar alkaline hydrothermal method followed by an acid washing and annealing treatment, was used [18]. k The volume fractions of TTNT and Y 2 W 3 O 12 within the hybrid filler are presented in Table 3. m These relationships are applicable when Km < K f and Gm < G f . n The shear modulus of TTNT was estimated considering them as isotropic materials as a first approximation.
f 0 f2 ¼volume fraction of Y 2 W 3 O 12 within the hybrid filler computed with Eq. 17 (see Table 1) K l c , K u c ¼ lower and upper bounds of bulk modulus of composite reinforced with the hybrid filler [16,17], calculated with Eq. 12 and Eq. 13. a Value measured by dilatometry in the temperature range of 30e70 C and during the second heating cycle [1]. b CTE of TTNT is unavailable in the literature. Therefore, corresponding value of TiO 2 was assumed. c Bulk modulus of hydrothermally synthesized TTNT is not reported in the literature. Hence, corresponding value of analogous anatase TiO 2 nanotubes, synthesized by a similar alkaline hydrothermal method followed by an acid washing and annealing treatment, was used [18]. d The bulk modulus of HDPE was calculated using the experimental Young's modulus of HDPE, assuming that this remains unchanged after heating samples up to 70 C, as a first approximation. e Since K l c and K u c for ternary phase composites are calculated in the literature using Eq. 12 and 13 [16,17], and Schapery's model takes into account these two values, it is reasonable to compute a l c and a u c defining K hyb and a hyb by the rule of mixture (see Eq. 15 and Eq. 27, respectively). cetyltrimethylammonium bromide (CTAB) <composites designated as C2:1-CTAB> and iii) HDPE modified with polyethylene-grafted maleic anhydride (PE-g-MA) as compatibilizer < composites called as C2:1-PE-g-MA>.
The computation of the volume fractions of TTNT and Y 2 W 3 O 12 (f f1 and f f2 , respectively) for the application of these models is presented in reference [1]. The mechanical and thermal properties of TTNT are assumed to be equal to those values of similar titania-based materials, as a first approximation, since they are not reported in the literature.
For the application of MROM for the TTNT/Y 2 W 3 O 12 /HDPE ternary system (see Eq. 3), the strengthening factor of Y 2 W 3 O 12 (b 2 ) was calculated from the experimental Young's moduli of HDPE/ Y 2 W 3 O 12 composites using data reported by Pont on et al. [14]. The MROM for these binary composites can be expressed as: Scheme. 1. Graphic illustration of the approaches used for the preparation of HDPE-based composites reinforced with (a) pristine hybrid filler, (b) CTAB modified hybrid filler, and (c) pristine hybrid filler with addition of PE-g-MA as compatibilizer ($$$$$representing hydrogen bonding).
Therefore, b 2 can be calculated from the slope of E c2 as a function of f f2 . The linear fitting of experimental Young's moduli of HDPE/Y 2 W 3 O 12 composites is presented in Fig. 1.
The volume fractions of TTNT and Y 2 W 3 O 12 within the hybrid filler, f 0 f1 and f 0 f2 , respectively, are presented in Table 3. These values were calculated with Eq. 16 and Eq. 17 from the volume fractions of both fillers (f f1 , f f2 ) inside the whole composite, and they are required for application of both Hashin-Shtrikman and Schapery's models.   There is a lack of information in the literature related to the application of Schapery's model to predict the CTE of polymer composites reinforced with a hybrid filler. The present approach of application of Schapery's model equation for binary composites to ternary ones (see Eq. 25 and Eq. 26) was based on the assumptions previously used in Hashin-Shtrikman model (see footnote "i" in Table 1), since both models depend on K l c and K u c . All micromechanical models presented in this article assumes perfect interfaces and homogenous dispersion of hybrid fillers. Experimental deviations from these models can be observed as a result of different dispersion states of fillers inside the matrix and/or the presence of the interfacial groups at hybrid filler-matrix interfaces.