The comparison of calculated data from proposed mathematic model and experimentally obtained data of PP/clay nanocomposites was done with the focus on the layered shape of MMT platelets. Based on the well-known Kerner's model and the Halpin-Tsai' equation with the use of some described presumption, the mathematic model for PP/clay nanocomposite was proposed. Data from the measurement of prepared PP/clay samples were taken and compared with the calculated ones from the proposed model. The good agreement was found.
Firstly, the mathematical description of mechanical and physical properties of composites based on a polymer matrix has been studied in 1950s [
These models mainly describe the dependence of shear modulus of elasticity on the content of the filler and spherical or fibre type of filler is considered. They are named self-consistent, it means that homogenous spherical particles are placed in the homogenous continuous polymer matrix. With the increasing use of composites as construction materials in the industry models with cylindrical dispersed phase has been proposed. This was used in order to simulate short or long fibres.
The classical Kerner’s model is usually presented as:
The equation of Halpin-Tsai is very often cited and properly it is the conversion of Hill’s equation [
In the literature one can find a lot of attempts of simplification and specification in earlier models [
Nanocomposite materials based on the polymer matrix have been studied more than twenty years. The main reason is the unusual combination of properties. They are unique in comparison with conventional composites because of their large interphase surface (connecting with aspect ratio) and very small distances between reinforcing particles [
The behaviour of polymer chains in the neighbourhood of nanoparticles and the influence of the nanoparticles shape on the nanocomposite properties have been studied in the several next papers [
The proper model derivation (modelling).
At the majority of above-mentioned model is based on the mixing principle when the slope of the curve increases with the increasing filler content. Slope of curve represents the elasticity modulus—and it is the function of the filler content. This is valid till the composite modulus reaches for
More complicated models have a disadvantage—the dependence on stress or deformational fields. In the case of our studied system PP/MMT the tensile modulus initially grows very quickly, but with the increasing filler content the slope of curve falls down. Figure
The dependence of reduced elasticity modulus on the filler content.
Based on these observations mentioned above we tried to deduce such an equation which at the use of the easiest mathematical apparatus would correspond with obtained experimental data and would be suitable for prognosis of future research in the use of layered nanoclays. The interval of filling was chosen between 0–10 wt. %.
The outgoing assumptions are as follows: the filler has an unlimited modulus → it has zero deformability; the primary deformation is observed in case of torque and strain modulus, after the created stress is measured; particles are fixed in matrix with the perfect adhesion (index
When the particles are freely inserted into the vacancies in the matrix, without any adhesion, the system behaves like an expanded material. With the growing number of ratio of
The simple scheme of the deformation gradient at the liquid flow along fixed plates.
A complete spectrum of deformation matrix from zero at the filler particle to the maximum somewhere along particles is simplified into two values. After the layer at the thickness
The consequence of this thought is the increasing of the origin undeformable volume
The next consequence based on the thought discussed above, is that the created stress and by this the rigidity of the composite depends on the size of filler particles.
At the observations it was found that the size of filler particles
The constant
After the summarization of the above-mentioned thoughts (
With the growing content of filler at the same initial deformation the stress inside composite is created and increases according to (
Now we have all mathematical apparatus needed for the calculation of the module of elasticity: firstly, we calculate the initial stress from the previous equation we can get the ratio of no delaminated particles is (according to ( next we calculate the stress equal to free particles (according to ( after we calculate the rest of fixed particles: next the stress equal to bonded particles (according to ( and the resulting stress: if the deformation at measurement is unit and it is small enough (in order to be responding to Hook’s principle), after
This system of equations contains only three unknown parameters of physical merit which it is necessary to estimate for the next calculation:
In order to obtain the real result of this calculation, the particular parameters must be chosen in the values fulfilling following conditions:
From the influence of the mentioned three parameters it is possible to see from the following three-dimensional graphs (Figures
The simple scheme of the deformation gradient at the liquid flow along fixed plates—with the filler in polymer melt.
Reduced modulus of elasticity as a function of parameters
Reduced modulus of elasticity as a function of parameters
The low value of the adhesion
In case of the microscopy observations of nanocomposite samples it was found that with the growing content of nanofiller the number of agglomerates increases; thus, the average size of particles increases too. The velocity of this increase is included in the constant
Reduced modulus of elasticity as a function of parameters
Parameter
Time and conditions of the kneading, the technology of the nanocomposite preparation are projected in the parameters
It is necessary to underline that the presented model (like all models) is valid for the limited interval of layered nanocomposite filling only.
In order to verify the validity of the proposed model for the layered type of nanofiller nanocomposite samples of PP/modified MMT were prepared.
Polypropylene (PP) Mosten GB 003 produced by Chemopetrol Litvínov, Czech Republic, was used as a polymer matrix. The density of chosen PP was 907 kg/m3 and melt flow index (MFI) was 3.2 g/10 min at 230°C.
The density of maleated polypropylene (PP-Ma) EXXELOR PO 1015 was 900 kg/m3, MFI was 22, and contents of maleic groups were 1 wt. % (used as a compatibilizator). Maleated polypropylene was supplied by ExxonMobil Chemical Europe, Belgium. Two types of nanofiller Dellite were used (Dellite 72T and Dellite 67G); their concentration was 2, 4, 6, and 10 wt. %. The nanofiller were supplied by Laviosa Chemical Mineraria S.p.A., Italy.
All nanocomposites in this work were prepared by melt blending on a Brabender Plasticorder compounder at 40, 60, and 80 rpm and 220°C for 10, 20, 30, and 40 min. The samples were prepared by pressuring at 220°C at 9 min and cooling was 7 min.
The measurement of E modulus was done by DMA analysis on the equipment DMA DX04T (company RMI) on FT TBU in Zlin. Values presented in this work are values at 30°C.
Resulting data of measurement of dynamic and strength modulus of elasticity for both used nanofiller types are listed in the Tables
Dynamic modulus as a function of speed of rotation of kneader and volume ratio of nanofiller D72T.
D 72T | Dyn. modulus | ||||
rpm. ¥ | 0 | 0,008 | 0,016 | 0,024 | 0,040 |
400 | 1626 | 1827 | 1796 | 1864 | 1843 |
600 | 1626 | 1864 | 1905 | 1911 | 1912 |
800 | 1626 | 1816 | 1843 | 1846 | 1905 |
1200 | 1626 | 1852 | 1827 | 1910 | 1912 |
1600 | 1626 | 1733 | 1844 | 1878 | 1950 |
1800 | 1626 | 1882 | 1854 | 1912 | 1915 |
2000 | 1626 | 1806 | 1806 | 1870 | 1874 |
2400 | 1626 | 1831 | 1884 | 1896 | 1958 |
2800 | 1626 | 1851 | 1860 | 1907 | 1901 |
3000 | 1626 | 1766 | 1824 | 1893 | 1880 |
3400 | 1626 | 1931 | 1971 | 2006 | 1991 |
4200 | 1626 | 1811 | 1885 | 1869 | 1911 |
Average | 1626 | 1831 | 1858 | 1897 | 1913 |
Summary of results of dynamic modulus and strength modulus for both nanofiller types (data calculated by the same way as in the case of Table
rpm/V | Dynamic modulus | Shear modulus | ||
---|---|---|---|---|
D 72T | D 67G | D 72T | D 67G | |
0 | 1626 | 1626 | 733 | 733 |
0,008 | 1831 | 1807 | 1180 | 1099 |
0,016 | 1858 | 1830 | 1183 | 1055 |
0,024 | 1897 | 1862 | 1195 | 1009 |
0,040 | 1913 | 1888 | 1286 | 1221 |
The summary of results from Tables
D72T-d | D67G-d | D67G-s | M67dy | Kerner | |||||
yn | yn | D72T-str | tr | M72dyn | M72str | M67str | |||
0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
0,001 | 1,024 | 1,019 | 1,054 | 1,043 | 0,002 | ||||
0,005 | 1,086 | 1,071 | 1,243 | 1,189 | 0,009 | ||||
0,008 | 1,126 | 1,111 | 1,610 | 1,499 | 1,114 | 1,094 | 1,358 | 1,276 | 1,015 |
0,016 | 1,143 | 1,125 | 1,614 | 1,440 | 1,152 | 1,128 | 1,572 | 1,441 | 1,03 |
0,024 | 1,167 | 1,145 | 1,630 | 1,377 | 1,170 | 1,145 | 1,687 | 1,546 | 1,046 |
0,040 | 1,176 | 1,161 | 1,754 | 1,666 | 1,182 | 1,160 | 1,745 | 1,662 | 1,077 |
38,1 | 31,1 | 62,3 | 50 | ||||||
104 | 104 | 75 | 75 | ||||||
3,02 | 3,02 | 8,5 | 8,5 |
Table
Based on Figure
Reduced elasticity modulus as a function of nanofiller content of prepared samples.
The growing of the particle agglomeration with their rising content makes it possible to observe qualitatively in TEM pictures. The lowering of the parameter
Models and mathematical descriptions of the various composite behaviours have been created for more than 50 years. Nanofillers especially due to their relatively short time of application and the specific shape and size, unique interface surface, small distance of particles, and the tendency to the agglomeration do not fit to these models.
In this work the authors tried to create a relatively simple model which with the variation of three parameters allows very nice well description of the dependency of the elasticity modulus on the filler content for the layered nanofiller type with the platelet shape of particles. This model is valid in the interval of the clay nanofiller content at least to the 10 wt. %.
Model brings the better look at the material containing particles and technological parameters influenced by the rigidity of prepared composites and the more optimal planning of the next experimental work in this field.
This project was supported by the Academy of Sciences of the Czech Republic (Project KAN 100400701), Board of Trade in the frame of Project FI-IM3/085 and Internal Grant Agency IGA/FT/2012/040, and by Operational Programme Research and Development for Innovations cofunded by the European Regional Development Fund (ERDF) and national budget of the Czech Republic within the framework of the Centre of Polymer Systéme Project (reg. number: CZ.1.05/2.1.00/03.0111). Some of these data were presented in ANTEC 2009.