CC BY
28
UDC 678
G. E. Zaikov, G. V. Kozlov, A. K. Mikitaev, O. V. Stoyanov, Bob A. Howell
THE STRUCTURAL ANALYSIS OF PARTICULATE-FILLED POLYMER NANOCOMPOSITES
MECHANICA LPROPERTIES
Keywords: nanocomposite, ultrafine particles, mechanical properties, structure, fractal analysis.
A number of the main mechanical characteristics (yield stress, impact toughness, microhardness) of particulate-filled polymer nanocomposites was described quantitatively within the framework of general conception - fractal analysis. Such approach allows to study the main specific features of the indicated nanomaterials mechanical behavior. The influence of both nanofiller initial particles size and their aggregation degree on nanocomposites mechanical properties has been shown.
Ключевые слова: нанокомпозит, сверхтонкие частицы, механические свойства, структура, фрактальный анализ.
В рамках общей концепции - фрактального анализа количественно описан ряд основных механических характеристик (предела текучести, ударной вязкости, микротвердости) дисперсно-наполненных полимерных нанокомпозитов. Такой подход позволяет изучить основные специфические особенности механического поведения указанных наноматериалов. Показано влияние размера исходных частиц нанонаполнителя и степени их агрегации на механические свойства нанокомпозитов.
Introduction
Mechanical properties represent a very important part of polymer materials characteristics, particularly if the talk is about their application as engineering materials. Nevertheless, even if the indicated materials have another functional assignment, mechanical properties remain always practical application important factor in this case as well. Particulate-filled polymer nanocomposites mechanical properties have a specific features number, which will be considered below.
As it is well-known [1], the yield stress of polymeric materials is an important operating characteristic, restricting the range of their application as engineering materials from above. Therefore the theoretical treatment of yielding process was always paid special attention, to that resulted in the development of a large number of theoretical models, describing this process [2]. For particulate-filled polymer nanocomposites the specific feature of the dependence of yield stress cty on nanofiller contents is observed [3, 4]: unlike microcomposites of the same class [5], the value ct7 is not increase to some extent perceptibly at nanofiller contents growth and even can be reduced. It is obvious, that the indicated effect is a negative one from the point of view of these polymeric materials exploitation, since it restricts their using possibilities as engineering materials.
The authors of papers [6, 7] found out, that the introduction of particulate nanofiller (calcium carbonate (CaCO3)) into high density polyethylene (HDPE) results in nanocomposites HDPE/CaCO3 impact toughness Ap in comparison with the initial polymer by about 20 %. The authors [6, 7] performed this effect detailed fractographic analysis and explained the observed Ap increase by nanocomposites HDPE/CaCO3 plastic deformation mechanism change in comparison with the initial HDPE. Without going into details of the indicated analysis, one should note some reasons for doubts in its correctness. In Fig. 1 the schematic
diagrams load-time (P-t) for two cases of polymeric materials samples failure in impact tests are adduced: by instable (a) and stable (b) cracks. As it is known [8], the value Ap is characterized by the area under P-t diagram, which gives mechanical energy expended on samples failure.
P P
Fig. 1 - The schematic diagrams load-time (P-t) in instrumented impact tests. Failure by instable (a) and stable (b) crack
The polymeric materials macroscopic fracture process, defined by main crack propagation, begins at the greatest load P. From the schematic diagrams P-t it follows, that fracture process as a matter of fact practically does not influence on Ap value in case of crack instable propagation and influences only partly in case of stable crack. Although the authors [6, 7] performed impact tests on instrumented apparatus, allowing to obtain diagrams P-t, these diagrams were not adduced. Moreover, the structural aspect of fracture process has been considered in works [6, 7] with the usage of secondary structures (crazes, shear zones and so on). Their interconnection with the initial undeformed material structure is purely speculative. It is obvious, that it does not occur possible to obtain quantitative relationships structure properties at such method of analysis.
At present it is known [10-12], that microhardness Hv is the property sensitive to morphological and structural changes in polymeric materials. For composite materials the existence of the filler, whose microhardness exceeds by far polymer matrix corresponding characteristic, is an additional powerful factor [13]. The introduction of sharpened indentors in the form of a cone or a pyramid in polymeric material a stressed state is localized in small enough microvolume and it is supposed, that in such tests polymeric materials real structure is found [14]. In connection with the fact, that polymer nanocomposites are complex enough [15], the question arises, which structure component reacts on indentor forcing and how far this reaction alters with particulate nanofiller introduction.
The interconnection of microhardness, determined according to the results of the tests in a very localized microvolume, with such macroscopic properties of polymeric materials as elasticity modulus E and yield stress cty is another problem aspect. At present a large enough number of derived theoretically and obtained empirically relationships between Hv, E and cty exists [16].
Proceeding from the said above, the purpose of the present work is the indicated mechanical properties of particulate-filled polymer nanocomposites treatment within the framework of general structural approach, namely, the fractal analysis.
Experimental
Polypropylene (PP) "Kaplen" of mark 01030 with average weight molecular mass of ~ (2-3)x103 and polydispersity index 4.5 was used as a matrix polymer. Nanodimensional calcium carbonate (CaCO3) in a compound form of mark Nano-Cal P-1014 (production of China) with particles size of 80 nm and mass contents of 1-7 mass % and globular nanocarbon (GNC) (production of corporations group "United Systems", Moscow, Russian Federation) with particles size of 5-6 nm, specific surface of 1400 m2/g and mass contents of 0.25-3.0 mass % were applied as nanofiller.
Nanocomposites PP/CaCO3 and PP/GNC were prepared by components mixing in melt on twin screw extruder Thermo Haake, model Reomex RTW 25/42, production of German Federal Republic. Mixing was performed at temperature 463-503 K and screw speed of 50 rpm during 5 min. Testing samples were prepared by casting under pressure method on a casting machine Test Sample Molding Apparate RR/TS MP of firm Ray-Ran (Taiwan) at temperature 483 K and pressure 43 MPa.
Uniaxial tension mechanical tests have been performed on the samples in the shape of a two-sided spade with the sizes according to GOST 112 62-80. The tests have been conducted on a universal testing apparatus Gotech Testing Machine CT-TCS 2000, production of German Federal Republic, at temperature 293 K and strain rate ~ 2x 10-3 s-1.
The impact tests have been conducted by Sharpy method on samples by sizes of 80x10x4 mm. Samples have V-like notch with length of 0.8 mm. Tests have been performed on pendulum apparate model
Gotech Testing Machine GT-7045-MD, production of Taiwan, with the energy dial of 1 J so that no less than 10 % and no more than 80 % of energy reserve was consumed on sample failure, with distance between supports (span) of 60 mm. No less than 5 samples were used for each test.
The microhardness Hv measurements by Shore (scale D) were performed according to Gost 24621-91 on scleroscope HD-3000, model 05-2 of form "Hildebrand", production of German Federal Republic. The samples have cylindrical shape with diameter of 40 mm and height of 3 mm.
Results and Discussion
Yield stress
For the dependence of yield stress cty on particulate nanofiller contents theoretical analysis the dispersive theory of strength was used, where nanocomposite yield stress at shear xn is determined as follows [17]:
Gn bB
X
(1)
where Xm is shear yield stress of polymer matrix, Gn is shear modulus of nanocomposite, bB is Burgers vector, X is distance between nanofiller initial particles in nanocomposite.
In case of nanofiller particles aggregation the equation (1) has the look [17]:
GnbB
Xn = Tn
(2)
k(r)X'
where k(r) is aggregation parameter.
It is easy to see, that the equation (2) describes the initial nanoparticles aggregation influence on nanocomposite yield stress. This effect is important from both theoretical and practical points of view in virtue of well-known nanoparticles tendency to aggregation, which is expressed by the following relationship [15]:
k{r) = 7.5 x 10~3St,, (3)
where Su is nanofiller specific surface, which is determined as follows [18]:
(4)
PnDp
where pn is nanofiller density, Dp is its particles diameter.
From the equations (3) and (4) it follows, that the nanofiller particles size decreasing results in Su enhancement, that intensifies nanofiller initial particles tendency to aggregation.
Let us consider determination methods of the parameters, included in the equation (2). The general relationship between normal stress ct and shear stress x has the look [19]:
CT (5)
x =
S'
The stress Xm is determined according to the equation [17]:
xm =rm(1-^/3), (6)
where xm is shear yield stress of matrix polymer, pn is nanofiller volume content, determined according to the well-known formula [1]:
Vn ='
Pn
(7)
Gn
the
is connected with following simple
(9)
where Wn is nanofiller mass content and the value pn for nanoparticles is determined as follows [15]:
pn= , kg/m3, (8)
where Dp is given in nm.
The shear modulus Young's modulus En by relationship [20]:
C -En
df
where df is fractal dimension of nanocomposite structure, which is determined according to the equation [20]:
df -(d-1)1 + v), (10)
where d is dimension of Euclidean space, in which a fractal is considered (it is obvious, that in our case d=3), v is Poisson's ratio, estimated according to the mechanical test results with the aid of the relationship [14]:
En
1 - 2v
(11)
6(1 + v)'
where En is nanocomposite elasticity modulus.
The value of Burgers vector bB for polymeric materials is determined according to the equation [2]:
bR =
/ N 1n ' 60.
C„
Â,
where Cm is characteristic ratio, dimension df by the equation [2]:
(12)
connected with
C„, = ■
2df
4
(13)
d(d-i)(d -df) 3'
It is obvious, that for the value xn theoretical estimation according to the equation (2) an independent method of parameter k(r)X determination is necessary. The following equation gives such method [3]: k(r)X - 2.09 x 10-2£>p(Sul<pn)112, (14)
where Dp is given in nm, Su - in m2/g.
The value Su estimation according to the equation (4) gave the following results: Su=3280 and 93 m2/g for GNC and CaCO3, respectively.
In Fig. 2 the comparison of the received experimentally uy and calculated according to the described above method UY yield stress values for
nanocomposites PP/GNC is adduced. As one can see, the theory and experiment good correspondence is observed (the average discrepancy between uy and uty
makes up 5.5 %). Besides, the value cty for nanocomposites does not differ to some extent significantly from the corresponding parameter for matrix PP: for nanocomposites cty=36.0-32.9 MPa, for PP cty=31.5 MPa, i.e. cty enhancement at nanofiller introduction does not exceed 15 %.
MPa
60
40
A-l
/
/
/
0
1
W„, mass. %
Fig. 2 - The dependences of yield stress cty on nanofiller mass contents Wn for nanocomposites PP/GNC. 1 - experimental data; 2, 3 - calculation according to the equations (2) and (1), respectively
The causes of such effect can be elucidated by the equation (1) using, where the value X is calculated as follows [17]:
2 =
/ N 1/3
3V
- 2
n /
2
(15)
The equation (1) supposes nanofiller initial particles aggregation absence (k(r)=1.0) and the
Y
dependence GY (Wn), calculated according to the
indicated equation, is also adduced in Fig. 2. The absence of GNC initial particles aggregation results in nanocomposites PP/GNC yield stress strong increasing within the range of Wn=0.25-3.0 mass % - from 44 up to 86 MPa.
In Fig. 3 the similar dependences of yield stress on I/I/,, for nanocomposites PP/CaC03 are adduced, try, MPa
34
30
26
A-l
0
8
W,,. mass. %
Fig. 3 - The dependences of yield stress cty on nanofiller mass contents Wn for nanocomposites PP/CaCO3. 1 - experimental data; 2, 3 - calculation according to the equations (2) and (1), respectively
As one can see, Uy estimation according to the equation (2) gives an excellent correspondence to
the experiment - the average discrepancy between cty
t
and CTY makes up 0.7 % only.
Besides, for nanocomposites PP/CaCO3 cty reduction within the range of Wn=1-7 mass % is observed. And at last, CaCO3 initial nanoparticles aggregation suppression does not give positive effect for these nanocomposites. It is obvious, that the cause of the indicated cty reduction for non-aggregated CaCO3 is a relatively large diameter of its initial nanoparticles, approaching to upper limit of nanoparticles dimensional range, which is equal to ~ 100 nm [21]. Owing to that X value for nanocomposites PP/CaCO3 varies within the limits of 200-66 nm within the range of Wn=1-7 mass %, whereas for nanocomposites PP/GNC the value X is essentially smaller: 17-4 nm within the range of Wn=0.25-3.0 mass %. Thus, in particulate-filled polymer nanocomposites yield stress value definition two competeting factors played critical role: nanofiller initial particles size and their aggregation level. It is important to note, that weak dependence of yield stress on nanofiller contents is typical not only for particulate-filled polymer nanocomposites, but also for other classes of these nanomaterials: polymer/organoclay [22] and polymer/carbon nanotubes [23].
Let us consider alternative, specific for nanocomposites with semicrystalline matrix, treatment of yield stress change. The equations (10) and (11) combination at the condition d=3 allow to obtain the following dependence of ratio EJcty on the main structural characteristic df:
— = 7Tf>. (16)
In Fig. 4 the dependence of ratio EJcty on dimension df is adduced, which demonstrates strong nonlinear growth of the indicated ratio at df increasing,
specifically at df>2.7.
E/ny
200
100
Fig. 4 - The dependence of elasticity modulus and yield stress ratio E/cty on structure fractal dimension df for polymeric materials
Thus, the postulated in work [24] cty and En proportionality is true in special case only, namely, in case of polymeric material structure invariability, i.e. in case df=const. This rule is fulfilled for particulate-filled
polymer nanocomposites with amorphous matrix: phenylone/p-sialone and phenylone/oxynitride silicium-yttrium [15]. For the indicated nanocomposites df=const=2.416 and then En/CTy=12.4, that is confirmed experimentally [15].
Let us consider further yield stress cty behavior as a function of nanofiller mass contents Wn for the considered nanocomposites. The value df for them can be estimated by an independent mode, using the following equation [2]:
, 1/2
df = 3 - 6
Pd
(17)
where pci is local order domains (nanoclusters) relative fraction, S is macromolecule cross-sectional area, which is equal to 27.2 A2 for PP [2].
The value pci can be estimated according to the following percolation relationship [2]:
pd= 0.03(1 -K \Tm-T)055, where K is crystallinity degree,
(18)
Tm and T are melting and testing temperatures, respectively. For the considered nanocomposites the value K according to DSC data varies within the limits of 0.637-0.694 for PP/GNC and 0.637-0.668 for PP/CaCO3 and the value Tm for PP was accepted equal to 445 K [25].
Since the value pcl is considered for nanocomposites, where nanoclusters are concentrated in polymer phase only, then one should use its reduced
value Pci ,
which is equal to [15]:
(19)
Pcf =PC/(1-Pj.
In Fig. 5 the comparison of the received experimentally and calculated according to the described above method dependences of yield stress cty on nanofiller mass contents Wn for the considered nanocomposites is adduced.
or, MPa
40
35
30
25
A -1
O -2
0
8
mass. %
Fig. 5 - The dependences of yield stress cty on nanofiller mass contents Wn for nanocomposites PP/GNC (1, 3) and PP/CaCO3 (2, 4). 1, 2 - the theoretical calculation; 3, 4 - experimental data
As one can see, a good correspondence of theory and experiment is obtained (their mean discrepancy makes up 2.5 %). This circumstance allows to explain the cause of insignificant increasing and in case of nanocomposites PP/CaCO3 even reduction of
yield stress at nanofiller contents growth. As it is known [15], the fractal dimension df of crystallizing polymeric materials structure depends on their crystallinity degree K as follows:
df - 2 + K + pif , (20)
where pif is a relative fraction of interfacial (crystallizing also) regions.
Therefore the values df for the considered nanocomposites are within the range of ~ 2.75-2.80, i.e. within the range, where the ratio En/uY strong increase begins (see Fig. 4). So, for nanocomposites PP/GNC the ratio En/uY value varies within the range of 31.0-41.8 and for PP/CaCO3 - within the limits of 31.0-37.2. This increase corresponds by absolute value to nanocomposites elasticity modulus enhancement, which makes up ~ 40 % for PP/GNC and ~ 13 % for PP/CaCO3. Hence, the ratio En/uY increasing compensates En growth owing to nanofiller introduction and in case of nanocomposites PP/CaCO3 En small increase results in uy reduction. Let us note, that the proposed model is true for nanocomposites with amorphous matrix as well, having high enough df values (for example, for rubbers) [4].
Impact toughness
In Fig. 6 the dependences of impact toughness Ap on nanofiller volume contents pn are adduced for the considered nanocomposites.
Fig. 6 - The dependences of impact toughness Ap on nanofiller volume contents pn for nanocomposites PP/GNC (1) and PP/CaCO3 (2)
As it follows from the data of this figure, for both nanocomposites the dependence Ap(pn) has an extreme character, whose maximum is reached at pn«0.03. Ap increasing for nanocomposites in comparison with the corresponding parameter for matrix polymer can be significant: so, for nanocomposite PP/CaCO3 at pn=0.03 Ap value exceeds impact toughness for PP in 1.5 times. Nanocomposites PP/GNC and PP/CaCO3 mechanical properties study has shown that a similar extreme dependence of property on nanofiller contents has yield stress uy only (see Figs. 2 and 4). As it has been shown above, such dependence
uY(pn) shape is due to nanofiller initial particles aggregation, which is intensified at pn growth. This interconnection is not accidental: as it has been noted above, Ap value is proportional to area under P-t diagram or curve stress-strain (u-s). In its turn, for plastic polymeric materials, which are investigated nanocomposites, a stress is restricted from above by yield stress uy and limiting strain is equal to failure strain sf. Therefore it is to be expected that impact toughness Ap is proportional to product uYsf. Within the framework of fractal analysis the value sf is determined as follows [2]:
sf- C^-1 -1, (21)
where Dch is fractal dimension of a polymer chain part between its fixation points (chemical cross-linking nodes, physical entanglements, nanoclusters etc).
The parameters Cm and Dch characterize polymer chain statistical flexibility and molecular mobility level, respectively [2]. The dimension Dch can be determined with the aid of the following equation [2]:
— - . (22)
Pd
In Fig. 7 the dependence Ap(uYsf) for the considered nanocomposites is adduced, which proves to be linear and passing through coordinates origin, that allows to describe it analytically by the following empirical equation:
Ap - (0.4 x 10-3 m)uysf . (23)
Fig. 7 - The dependence of impact toughness Ap on parameter CTySf for nanocomposites PP/GNC (1) and PP/CaCO3 (2)
The adduced above analysis allows to elucidate the cause of higher values Ap for nanocomposites PP/CaCO3 in comparison with PP/GNC. This cause is higher Dch values: for PP/CaCO3 Dch=1.33-1.34, for PP/GNC Dch=1.13-1.29, i.e. higher molecular mobility level for nanocomposites PP/CaCO3, although uy values are somewhat higher for PP/GNC.
Microhardness
Let us consider the interconnection of microhardness Hv and other mechanical characteristics, among their number yield stress uy for nanocomposites PP/GNC and PP/CaCO3. The following relationship
was received by Tabor [26] for metals, which were considered as rigid perfectly plastic solids, between Hv and cy:
— »c, (24)
crY
where c is constant, which is approximately equal to 3.
The relationship (24) implies, that the exerted in microhardness tests pressure under indentor is higher than yield stress in quasistatic tests owing to restriction, imposed by undeformed polymer, surrounding indentor. However, in works [12, 16, 22, 27, 28] it has been shown that the value c can differ essentially from 3 and varied in wide enough limits: ~ 1.5-30. In the work [28]
it has been found out, that for the composites
•
HDPE/CaCO3 depending on strain rate s and type of quasistatic tests, in which the value cy was determined (tensile or compression) c magnitude varies within the
limits of 1.80-5.83. To c=3 the ratio Hv/cy approaches
•
only at minimum value S and at using cy values, received by compression tests. Therefore in the work [28] the conclusion has been obtained, that the value c=3 can be received only at comparable strain rates in microhardness and quasistatic tests and at interfacial boundaries polymer-filler failure absence.
An elasticity role in indentation process was proposed to consider for the analysis spreading on a wider interval of solids. For the solid, having elasticity modulus E and Poisson's ratio v, Hill has obtained the following equation [16]:
"v=7 i1+ln IFVarK • (25)
and empirical Marsh equation has the look [16]:
f E 1 Hv = 0.07 + 0.6ln- a . (26)
I CJ
The equations (25) and (26) allow the microhardness Hv theoretical estimation for particulate-filled polymer nanocomposites at the known E and cy condition and the value v can be calculated according to the equation (11).
In table 1 the comparison of experimental Hv and calculated according to the equation (26) Hi
microhardness values for the considered nanocomposites is adduced. The equation (26) was chosen according to a simple reason that it gives better correspondence to experiment than the equation (25) for all classes of nanocomposites [4, 22, 29]. As it follows from the data of this table, a good enough correspondence of theory and experiment is obtained (the mean discrepancy between Hv and Hi makes up ~
8 %). This correspondence indicates, that Hv value for the considered nanocomposites is controlled by their macroscopic mechanical properties to the same extent, as for other materials.
Let us consider the physical nature of the ratio Hv/ay deviation from the constant c»3 in the equation (24), using for this purpose the relationships (25) and (26). The value df can be determined according to the
equation (10) and then the relationships combination allows to obtain the following equations [4, 22]:
cY
and
H
crY
for case d=3.
2df
(4-OVX3-OV).
0.07 + 0.6ln
f 3df 1 3-df
(27)
(28)
Table 1 - The comparison of obtained experimentally Hv and calculated according to the equation (26) Hi microhardness values for nanocomposites PP/GNC and PP/CaCO3
Nano Nanofiller Hv, h T, MPa A,
composite mass content, mass % MPa %
0 68 66.3 2.5
0.25 75 71.7 4.4
0.50 73 75.8 9.4
1.0 74 78.7 6.4
PP/GNC 1.50 72 76.4 6.1
2.0 72 75.5 4.9
2.50 72 75.2 4.4
3.0 72 75.9 4.8
1.0 72 67.0 10.7
2.0 75 66.8 10.9
2.5 76 66.8 12.1
3.0 75 66.5 10.9
PP/CaCOs 3.5 75 66.7 11.3
4.0 75 66.4 11.1
5.0 75 66.7 11.5
6.0 75 66.2 11.7
7.0 75 66.5 11.3
Footnote: A is relative discrepancy between parameter Hv and Hi
From the equations (27) and (28) it follows, that the ratio Hv/cty is defined by structural state of nanocomposite (matrix polymer) only, which is characterized by its fractal dimension df. In Fig. 8 the dependences of the ratio Hv/cty on df is adduced, calculated according to the equations (27) and (28), which found out complete similarity, but absolute values Hv/cty, calculated by the equation (27), proved to be on about 15 % higher than the analogous magnitudes, calculated according to the equation (28).
The identical results for extrudates of polymerization-filled compositions on the basis of ultra-high-molecular polyethylene were obtained in work [12] and for polymer nanocomposites of different classes -in works [4, 22, 29]. In Fig. 8 the condition Hv/cty=c=3 according to the equation (24) is given by a horizontal stroke line. As it follows from this figure data, the indicated condition is reached at df«2.85 according to the equation (27) and df&2.93 according to the formula (28). As it is known [20], for real solids the limiting greatest value df is equal to 2.95. Thus, these df values
at c=3 indicate again, that the empirical Marsh equation (the formula (28)) gives more precise estimation of ratio Hv/uY, than more strictly derived Hill relationship (the equation (27)). Hence, the adduced above results show, that the ratio Hv/uY value is defined by polymer nanocomposites structural state only and Tabor criterion c=3 is realized for Euclidean solids only.
HJar
Fig. 8 - The dependences of ratio Hv/aY on structure fractal dimension df. 1, 2 - calculation according to the equations (27) (1) and (28) (2); 3, 4 - experimental data for nanocomposites PP/GNC (3) and PP/CaCO3 (4). The horizontal stroked line indicates Tabor criterion Hv/ay=c=3
In Fig. 8 the dependence of the obtained experimentally values Hv/uY on nanocomposites structure fractal dimension df is shown by points. As one can see, the obtained experimentally dependence HJaY(df) corresponds well to the theoretical curve, calculated according to the equation (28) (the mean discrepancy between theory and experiment makes up 8.5 %), whereas calculation according to the equation (27) gives overstated absolute values of the indicated ratio.
Conclusions
The yield stress value of particulate-filled polymer nanocomposites is controlled by two competeting factors: diameter of nanofiller initial particles and their aggregation degree. The cause of weak increasing (and even reduction) of the indicated nanocomposites yield stress at nanofiller contents growth is the initial nanoparticles strong aggregation. The application of nanoparticles disaggregation artificial methods might be worth-while only for nanocomposites with a small diameter of the initial nanoparticles. High enough values of nanocomposites structure fractal dimension are an additional factor, influencing on this effect.
The impact toughness of particulate-filled polymer nanocomposites is defined by a number of factors on various structural levels: molecular, topological and suprasegmental ones. The indicated
levels characteristics are interconnected and changed at nanofiller introduction. The molecular mobility level is the main parameter, defining the considered nanocomposites impact toughness.
The ratio of microhardness and yield stress for particulate-filled polymer nanocomposites is defined by these polymer nanomaterials structural state only. Tabor criterion is correct for Euclidean (or close to them) solids only.
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© G. E. Zaikov - Doctor of Chemistry, Full Professor, Plastics Technology Department, Kazan National Research Technological University, Kazan, Russia, G. V. Kozlov - Senior Researcher, Kh.M. Berbekov Kabardino-Balkarian State University, Nal'chik, Russia, i_dolbin@mail.ru; A. K. Mikitaev - Doctor of Chemistry, Full Professor, Head of Organic Chemistry and Macromolecular Compounds Department, Kh.M. Berbekov Kabardino-Balkarian State University, Nal'chik, Russia; O. V. Stoyanov - Doctor of Engineering, Full Professor, Dean of the Faculty Of Plastics and Composite Materials Technology, Processing and Certification, Head of Plastics Technology Department, Kazan National Research Technological University, Kazan, Russia, ov_stoyanov@mail.ru; Bob A. Howell - Doctor of Chemistry, Professor, Center for Applications in Polymer Science, Central Michigan University, Mount Pleasent, Michigan, USA.
© Г. Е. Заиков - доктор химических наук, профессор кафедры Технологии пластических масс, Казанский национальный исследовательский технологический университет, Казань, Россия, Г. В. Козлов - старший научный сотрудник, Кабардино-Балкарский государственный университет им. Х.М. Бербекова, Нальчик, Россия, i_dolbin@mail.ru; А. К. Микитаев - доктор химических наук, профессор, заведующий кафедрой Органической химии и высокомолекулярных соединений, Кабардино-Балкарский государственный университет им. Х.М. Бербекова, Нальчик, Россия, О. В. Стоянов - доктор технических наук, профессор, декан факультет Технологии, переработки и сертификации пластмасс и композитов, заведующий кафедрой Технологии пластических масс, Казанский национальный исследовательский технологический университет, Казань, Россия, ov_stoyanov@mail.ru; Боб А. Хауэлл - доктор химии, профессор, Центр прикладной полимерной науки, Центральный университет Мичигана, Маунт-Плезант, Мичиган, США.
