CC BY
19
UDC 678
G. E. Zaikov, G. V. Kozlov, A. K. Mikitaev, O. V. Stoyanov, Bob A. Howell
THE REINFORCEMENT OF PARTICULATE-FILLED POLYMER NANOCOMPOSITES
BY NANOPARTICLES AGGREGATES
Keywords: nanocomposite, globular nanocarbon, calcium carbonate, aggregation, interfacial effects, reinforcement.
The applicability of irreversible aggregation model for theoretical description of nanofiller particles aggregation process in polymer nanocomposites has been shown. The main factors, influencing on nanoparticles aggregation process, were revealed. It has been shown that strongly expressed particulate nanofiller particles aggregation results in sharp (in about 4 times) formed fractal aggregates real elasticity modulus reduction. Nanofiller particles aggregation is realized by cluster-cluster mechanism and results in the formed fractal aggregates density essential reduction that is the cause of their elasticity modulus decreasing. As distinct from microcomposites nanocomposites require consideration of interfacial effects for elasticity modulus correct description in virtue of a well-known large fraction of phase's division surfaces for them.
Ключевые слова: нанокомпозит, глобулярный наноуглерод, карбонат кальция, агрегация, межфазные эффекты, усиление.
Показана применимость модели необратимой агрегации для теоретического описания процесса агрегирования частиц нанонаполнителя в полимерных нанокомпозитах. Выявлены основные факторы, влияющие на процесс агрегации наночастиц. Показано, что сильно выраженная агрегация частиц дисперсного нанонаполнителя приводит к значительному (примерно в 4 раза), уменьшению реальных значений модуля упругости образованных фрактальных агрегатов. Агрегация частиц нанонаполнителя реализуется по механизму кластер-кластер и приводит к существенному снижению плотности образовавшихся фрактальных агрегатов, что является причиной уменьшения их модуля упругости. В отличие от микрокомпозитов, для корректного описания модуля упругости нанокомпозитов необходимо учесть межфазные эффекты в силу хорошо известных для них больших долей поверхностей раздела фаз.
Introduction
In the course of technological process of par-ticulate-filled polymer composites in general [1] and nanocomposites [2-4] in particular preparation of the initial filler powder particles aggregation in more or less large particles aggregates always occurs. The aggregation process exercises essential influence on composites (nanocomposites) macroscopic properties [1-5]. For nanocomposites the aggregation process gains special significance, since its intensity can be such, that nanofiller particles aggregates size exceeds 100 nm -the value, which is assumed (although conditionally enough [6]) as upper dimensional limit for a nanoparti-cle. In other words, the aggregation process can be resulted in the situation, when initially supposed nanocomposite ceases to be as such. Therefore at present a methods number exists, allowing to suppress na-noparticles aggregation process [4, 7].
Analytically this process is treated as follows. The authors [5] obtain the relationship:
k(r) = 7.5 x 10~3 Su, (1)
where k(r) is aggregation parameter, Su is specific surface of nanofiller initial particles, which is given in m2/g.
In its turn, the value Su is determined as follows [8]:
(2)
PnDp
where pn is nanofiller density, Dp is diameter of its initial particles.
From the equations (1) and (2) it follows, that Dp reduction results in Su growth, that in its turn reflects in the aggregation intensification, characterized by the parameter k(r) increasing. Therefore in polymer
nanocomposites strengthening (reinforcing) element are not nanofiller initial particles themselves, but their aggregates [9]. This results in essential changes of nanofiller elasticity modulus, the value of which is determined with the aid of the equation [9]: f \3+d'
■ agr
= En
CD
\nagr у
(3)
where Eagr is nanofiller particles aggregate elasticity modulus, Enan is elasticity modulus of material, from which the nanofiller was obtained, a is an initial nano-particles size, Ragr is a nanoparticles aggregate radius, d| is chemical dimension of the indicated aggregate, which is equal to ~ 1.1 [9].
As it follows from the equation (3), the initial nanoparticles aggregation degree enhancement, expressed by Ragr growth, results in Eagr decrease (the rest of parameters in the equation (3) are constant) and, as consequence, in nanocomposite elasticity modulus reduction.
Very often the elasticity modulus (or reinforcement degree) of polymer composites (nanocomposites) is described within the frameworks of numerous micromechanical models, which proceed from elasticity modulus of matrix polymer and filler (nanofiller) and the latter volume contents [10]. Additionally it is supposed, that the indicated above characteristics of filler are approximately equal to the corresponding parameters of compact material, from which filler is prepared. This practice is inapplicable absolutely in case of polymer nanocomposites with fine-grained nanofiller, since in this case a polymer is reinforced by nanofiller fractal aggregates, whose elasticity modulus and density differ essentially from compact material
characteristics (see the equation (3)) [5, 9]. Therefore the microcomposite models application, as a rule, gives a large error at polymer composites elasticity modulus evaluation that in its turn results in the appearance of an indicated models modifications large number [10].
Proceeding from the said above, the present work purpose is the theoretical treatment of particulate nanofiller aggregation process and elasticity modulus (reinforcement degree) particulate-filled polymer nanocomposites with due regard for the indicated effect within the framework of irreversible aggregation models and 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 matrix polymer. Nanodimensional calcium carbonate (CaCO3) in 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 0.253.0 mass % were applied as nanofiller.
Nanocomposites PP/CaCO3 and PP/GNC were prepared by components mixing in melt on a 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 Apparatus RR/TS MP of firm Ray-Ran (Taiwan) at temperature 483 K and pressure 43 MPa.
The nanocomposites melt viscosity was characterized by a melt flow index (MFI). MFI measurements were performed on an extrusion-type plastometer Noselab ATS A-MeP (production of Italy) with capillary diameter of 2.095+0.005 mm at temperature 513 K and load of 2.16 kg. The sample was maintained at the indicated temperature during 4.5+0.5 min.
Uniaxial tension mechanical tests have been performed on the samples in the shape of a two-sided spade with 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.
Results and Discussion
The particulate nanofiller aggregation degree can be evaluated and aggregates diameter Dagr quantitative estimation can be performed within the framework of strength dispersive theory [11], where shear yield stress of nanocomposite xn is determined as follows: x ' , GnbB
I
(4)
where xm is shear yield stress of polymer matrix, bB is Burgers vector, Gn is nanocomposite shear modulus, X is distance between nanofiller particles.
In case of nanofiller particles aggregation the equation (4) has the look [11]:
T = r (5)
X X k(r)X' K )
where k(r) is aggregation parameter.
The parameters, included in the equations (4) and (5) are determined as follows. The general relationship between normal stress ct and shear stress x has the look [12]:
The intercommunication of matrix polymer xm and nanocomposite polymer matrix Xm shear yield stresses is given as follows [5]: xm =xm(1-^/3), (7)
where <pn is nanofiller volume content, which can be determined according to the well-known formula [5]:
Pn =■
Pn
(8)
where Wn is nanofiller mass contents, pn is its density, which for nanoparticles is determined according to the equation [5]:
pn =188^ )1/3, kg/m3, (9)
where Dp is given in nm.
The value of Burgers vector bB for polymeric materials is determined as follows [13]:
bB —
/ \1/2 ' 60.
Â,
(10)
v œ /
where CM is characteristic ratio, connected with nanocomposite structure dimension df by the equation [13]:
C„, — ■
2d,
f
4
(11)
d(d- 1)d-df) 3' where d is dimension of Euclidean space, in which a fractal is considered (it is obvious, that in our case d=3).
The value df can be calculated according to the equation [14]:
df =(d-1)(1 + v), (12)
where v is Poisson's ratio, estimated according to the mechanical tests results with the aid of the relationship [15]:
<Jv
1 - 2v
(13)
En 6(1 +
where cty and En are yield stress and elasticity modulus of nanocomposite, respectively.
Nanocomposite moduli En and Gn are connected between themselves by the relationship [14]:
(14)
And at last, the distance X between nanofiller nonaggregated particles is determined according to the equation [11]:
2 —
4n 3Pn
1/3
-2
(15)
From the equations (5) and (15) k(r) growth from 5.65 up to 43.70 within the range of Wn=0.25-3.0 mass % for nanocomposites PP/GNC and from 1.0 up to 2.87 within the range of Wn=1-7 mass % for nanocomposites PP/CaCO3 follows. Let us note, that the indicated variation k(r) for the considered nanocomposites corresponds completely to the equations (1) and (2). Let us consider, how such k(r) growth is reflected on nanofiller particles aggregates diameter Dagr. The equations (8), (9) and (15) combination gives the following relationship:
k(r)l =
f 0.25br^1/3 Wn
- 2
agr
2
, (16)
allowing at Dp replacement on Dagr to determine real, i.e. with accounting of nanofiller particles aggregation, nanoparticles aggregates diameter of the used nanofiller. Calculation according to the equation (16) shows Dagr increasing (corresponding to k(r) growth) from 25 up to 125 nm within the range of Wn=0.25-3.0 mass % for GNC and from 80 up to 190 nm within the range of 1-7 mass % for CaCO3. Further nanofiller particles aggregates density can be calculated according to the equation (9) at the condition of Dp replacement by Dagr.
Within the framework of irreversible aggregation model Dagr value is given by the following relationship [16]:
( Ac0kT ^
Hd*gr
3^/77,
tHdf9r
(17)
0 7
where c0 is nanoparticles initial concentration, k is Boltzmann constant, T is temperature, q is medium viscosity, m0 is mass of initial nanoparticle, df is fractal
dimension of particles aggregate, t is aggregation process duration.
Let us consider estimation methods of the parameters, included in the relationship (17). In the simplest case it can be accepted that all particles of nanofiller initial powder have the same size and mass. In this case c0«9n, where <pn value is determined according to the equation (8) with using nanofiller particles aggregates diameter Dagr. q value is accepted equal to reciprocal of MFI value and m0 magnitude was calculated as follows. In supposition of nanofiller initial particles spherical shape the nanoparticle volume was calculated according to the known values of their diameter Dp and then, using pn value, calculated according to the equation (8), their mass m0 can be estimated. T value is accepted as constant and equal to nanocomposites processing duration, i.e. 300 s.
The fractal dimension of nanofiller particles
aggregates structure of the equation [17]:
sd^-d
d f was calculated with the aid
pn Pdens
agr
2a
(18)
where pdens is density of compact material of nanofiller particles, a is self-similarity (fractality) lower scale of nanofiller particles aggregates.
pdens value for carbon is accepted equal to 2700 kg/m2, for CaCO3 - 2000 kg/m2 [5] and a value is accepted equal to the initial GNC particle radius, i.e. 2.5 nm f. values, calculated according to the equation (18), are equal to 2.09-2.67 and 2.47-2.75 for GNC and CaCO3 nanoparticles aggregates, respectively.
In Fig. 1 the dependences Dagr(Wn), plotted according to the equations (16) and (17), comparison is adduced.
Dagr, nm
Dagr, nm
W„, mass. %
Fig. 1 - The dependences of nanofiller particles aggregates diameter Dagr on nanofiller mass contents Wn for nanocomposites PP/GNC (1, 3) and PP/CaCO3 (2, 4): 1, 2 - calculation according to the equation (16); 3, 4 - calculation according to therelationship (17)
As one can see, the good enough correspondence of estimations according to both indicated methods was obtained (the average discrepancy of Dagr values, calculated with the usage of these relationships, makes up ~ 16 %). This circumstance indicates, that irreversible aggregation models can be used for the theoretical description of particulate nanofiller particles aggregation processes. Besides, the equation (17) analysis demonstrates various factors influence on nanofiller particles aggregates size (or their aggregation degree). So, c0, T and t increasing results in aggregation processes intensification and q, m0 and d^9r enhancement - to
their weakening.
Let us note in conclusion, that proportionality coefficient in the relationship (17) for GNC and CaCO3
(CgnC and CСаСОз, respectively) can be approximated by the following relationship:
'CaCO3
'GNC
f /770CaC°3
. Udf
m,
GNC
(19)
where m:
CaCO3
and /77q are masses of the initial par-
#0 aiiu ,,,q
ticles of CaCO3 and GNC, respectively , dfv is average fractal dimension of the indicated nanoparticles aggregates.
Further elasticity modulus Eagr of nanofiller particles aggregates according to the equation (3) can be determined. Let us consider the concrete conditions of this equation usage in reference to nanocomposites PP/GNC. Two possible variants exist at parameter a
choice in the indicated equation. The first from them supposes, that the value a is equal to GNC initial particles diameter [9], i.e. 5.5 nm. Such supposition means, that GNC nanoparticles aggregates are formed by particle-cluster (P-Cl) mechanism, i.e. by separate particles GNC joining to a growing aggregate [18]. However, such supposition gives unreal high Eagr values of order of 5x105 GPa. The other variant assumes, that nanofiller aggregation is realized by a cluster-cluster (Cl-Cl) mechanism, i.e. small clusters association in larger ones [18]. In such model aggregate radius R^ on the previous (i-1 )th aggregation stage is accepted as a and then the equation (3) can be rewritten as follows:
En =E.
agr
( R-1 A 41
agr
R'
(20)
The elasticity modulus Eagr real values within the range of 21.3-5.0 GPa were obtained at such calculation method. Further the simplest microcomposite models can be used for nanocomposite elasticity modulus En estimation. For the case of uniform strain in nanocomposite phases the theoretical value En (ETn ) is given by a parallel model [10]:
ETn=Eagr(Pn+ Emd-(Pn), (21)
where Em is elasticity modulus of matrix polymer.
For the case of uniform stress in nanocomposite phases the lower theoretical boundary E^ is determined according to the serial model [10]:
(22)
El =■
E agr En
mPn '
In Fig. 2 the comparison of the received experimentally En and calculated according to the equations (21) and (22) ETn elasticity modulus values of the considered nanocomposites PP/GNC is adduced. As one can see, the experimental data correspond better to the determined according to the equation (21) E^ upper boundary (in this case average discrepancy of En and E^ makes up ~ 8%). The indicated discrepancy is due to objective causes. As it is known [10], at the equations (21) and (22) derivation the equality of Poisson's ratio for nanocomposite both phases were supposed. In practice this condition non-fulfillment defines discrepancy between experimental and theoretical data.
In Fig. 2 the dependence E^ (Wn), calculated according to the equation (21) in supposition Eagr=const=21.3 GPa, is also adduced. As one can see, in this case the theoretical values of elasticity modulus E^ exceed essentially experimentally received ones En. Hence, the good correspondence of experiment and calculation according to the equation (21) is due to real values Eagr usage only.
It is obvious, that nanoparticles aggregates elasticity modulus reduction is due to the indicated aggregates diameter growth and, as consequence, their density p„ reduction, which can be calculated according to the equation (18). In Fig. 3 the dependence Eagr(pn) is adduced, which, as was expected, proves to be linear,
passing through coordinates origin and is described ana lytically by the following empirical equation: E = 12.6 x 10-3pn , GPa,
■agr
where pn is given in kg/m3.
E„, GPa
3 -
(23)
W„, mass. %
Fig. 2 - The dependences of elasticity modulus En on nanofiller mass contents Wn for nanocomposites PP/GNC. 1 - calculation according to the equation (22); 2 - according to the equation (21) at Eagr=variant; 3 - according to the equation (21) at Eagr=const=21.3 GPa; 4 - experimental data
05 10 ^¿.„xl^kg/m3 Fig. 3 - The dependence of GNC nanoparticles fractal aggregates elasticity modulus Eagr on their density pn for nanocomposites PP/GNC
The limiting magnitude pn=pdens allows to obtain the greatest value Eagi&34 GPa for GNC aggregates, that is the real value of this parameter [1].
The authors [19] proposed to use for nanocomposites elasticity modulus En determination a modified mixtures rule, which in original variant gives upper limiting value of composites elasticity modulus [10]:
En = Em (1 - cpn) ■+ bEnancpn , (24)
where b< 1 is coefficient, reflected nanofiller properties realization degree in polymer nanocomposite. In the present work context the parameter bEnan as a matter of fact presents nanofiller effective modulus or, more precisely, its aggregates modulus Eagr (compare with the equation (21)).
In Fig. 4 the dependence of parameter b in the equation (24) on nanofiller particles aggregates diameter
Dagr, calculated according to the equation (16), for the studied nanocomposites is adduced.
b
Fig. 4 - The dependence of parameter b on nanofiller
particles aggregates diameter
D;
agr
for
nanocomposites PP/GNC (1) and PP/CaCO3 (2). The shaded region indicates transition of nanofiller particles aggregates from nano- to microbehavior
As one can see, this dependence disintegrates on two linear parts: at small Dagr fast decay of b at Dagr growth is observed and at large enough Dag the value b«const«0.175. Let us note, that dimensional interval of the indicated transition, showed in Fig. 4 by a shaded area, makes up Dag^70-100 nm, i.e. it coincides approximately with upper dimensional boundary of nanoparticles interval (although and conditional enough [6]), which is equal to about 100 nm. As a matter of fact, the indicated dimensional interval defines the transition from nanocomposites to microcomposites, the dependence b(Dag ) for which differs actually qualitatively. The adduced in Fig. 4 dependence b(Dag ) can be described analytically by the following integrated equation:
b = 0.67 - 6.7 x10-3 b = const = 0.175 for Dagr>70 nm
D for Dagr<70 nm,
agr y '
(25)
In Fig. 5 the comparison of experimentally obtained and calculated according to the equation (25) dependences En(<n) is adduced for the studied nanocomposites. In this case the parameter b value was estimated according to the equation (25) and values Enan were accepted equal to 30 GPa for GNC and 15 GPa for CaC03.
E„, GPa
Fig. 5 - The comparison of experimentally received (1, 2) and calculated according to the equations (24) and (25) (3, 4) dependences of elasticity modulus En
on nanofiller volume contents <n for nanocomposites PP/GNC (1, 3) and PP/CaCO3 (2, 4)
As one can see, the good correspondence of theory and experiment is obtained (their mean discrepancy makes up 3 %, that approximately equal to the experimental error of En determination). Higher values En for nanocomposites PP/GNC in comparison with PP/CaCO3 even at Dag> 100 nm are due to two factors: the initial nanoparticles smaller size, that gives higher values <n at the same Wn values (see the equations (8) and (9)) and higher value Enan. It is important to note close values Eagr for nanocomposites PP/GNC, determined according to the equation (20) and as bEnan.
The authors [8] proposed the following percolation relationship for polymer microcomposites reinforcement degree EJEm description:
= 1 + 11(^)L
(26)
where Ec is elasticity modulus of microcomposite.
Later the relationship (26) was modified in reference to the polymer nanocomposites case [20]:
-f- = 1 +11(„+^)
1.7
(27)
where <if is relative fraction of interfacial regions.
It is easy to see, that the modified relationship (27) takes into consideration a factor of sharp increase of division surfaces polymer matrix-nanofiller [21]. In Fig. 6 the comparison of experimentally obtained and calculated according to the equation (26) dependences En(<n) for the considered nanocomposites is adduced. As it follows from this figure data, the equation (26) describes well the experimental data for nanocomposites PP/CaCO3, but the corresponding data for nanocomposites PP/GNC set essentially higher than theoretical curve. This discrepancy cause is obvious from the equations (26) and (27) comparison - for nanocomposites PP/GNC interfacial effects accounting is necessary, i.e. parameter accounting. Hence, in the considered case only compositions PP/GNC are true nanocomposites.
Fig. 6 - The comparison of experimentally received (1, 2) and calculated according to the equations (26) (3) dependences of elasticity modulus En on
nanofiller volume contents фп for nanocomposites PP/GNC (1, 3) and PP/CaCO3 (2, 4)
Conclusions
The applicability of irreversible aggregation models for theoretical description of particulate nanofiller particles aggregation processes in polymer nanocomposites has been shown. Analysis within the framework of the indicated models allows to reveal either factors influence on aggregation degree.
Strongly expressed aggregation of particulate nanofiller particles results in sharp (in about 4 times) formed fractal aggregates real elasticity modulus reduction. In its turn, this process defines nanocomposites as the whole elasticity modulus reduction. Nanofiller particles aggregation is realized by a cluster-cluster mechanism and results in the formed fractal aggregates density essential reduction, that is the cause of their elasticity modulus decreasing.
A nanofiller elastic properties realization degree is defined by the aggregation of its initial particles level. Unlike microcomposites nanocomposites require interfacial effects accounting for elasticity modulus correct description in virtue of well-known large fraction of phases division surfaces for them.
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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, 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, 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, Ма-унт-Плезант, Мичиган, США.
