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УДК 539.3
Влияние свойств межфазного слоя на модуль Юнга и предел текучести полимерных нанокомпозитов
Y. Zare, K.Y. Rhee
Университет Кёнхи, Йонъин, 446-701, Республика Корея В статье с использованием ряда моделей изучено влияние объемной доли, толщины, прочности и модуля упругости межфазного слоя между полимерной матрицей и наноразмерным наполнителем на модуль Юнга и предел текучести полимерных нанокомпозитов. На основе экспериментальных данных рассчитаны характеристики межфазного слоя для нескольких образцов. Установлено, что в некоторых образцах объемная доля промежуточной фазы выше, чем доля нанонаполнителя. Модуль Юнга нанокомпозитов в значительной степени зависит от концентрации наполнителя и промежуточной фазы. При этом при наибольших значениях объемной доли и прочности межфазного слоя наблюдается максимальный предел текучести для нанокомпозитов. Ключевые слова: полимерные нанокомпозиты, межфазные свойства, механическое поведение, модели микромеханики DOI 10.24411/1683-805X-2020-11009
How interphase properties control the Young's modulus and yield strength of polymer nanocomposites?
Y. Zare and K.Y. Rhee
Department of Mechanical Engineering, College of Engineering, Kyung Hee University, Yongin, 446-701, Republic of Korea
In this article, several models are applied to reveal the effects of volume fraction, thickness, strength and modulus of interphase region between polymer matrix and nanofiller on the Young's modulus and yield strength of polymer nanocomposites. The properties of interphase are calculated for several samples by experimental data of mechanical properties. It is found that the concentration of interphase is higher than that of nanofiller in some samples. The Young's modulus of nanocomposites largely depends on filler and interphase concentrations. In addition, the highest fraction and strength of interphase region produce the highest yield strength of nanocomposites. Keywords: polymer nanocomposites, interphase properties, mechanical behavior, micromechanics models
1. Introduction
Nanostructures suggest novel applications due to the excellent properties of materials at nanoscale [14]. Therefore, the structure and interaction at nano-scale cause the significant effects on the properties. The dramatic enhancement in the mechanical properties of polymers can be attained by incorporation of a low weight percentage (wt %) of various nano-fillers such as layered silicates [5, 6]. The large aspect ratio and stiffness of layered silicates may be the main reasons for the highly enhanced mechanical properties of polymer nanocomposites [7].
The Young's modulus of polymer nanocomposites increases by addition of nanoparticles, because they usually have a much higher modulus than polymer matrices [8-10]. However, the yield strength of nano-composites depends on the stress transferal between nanofiller and polymer matrix [11]. The stress applied to nanocomposites can be excellently transferred to nanoparticles in well-bonded nanoparticles to polymer matrix. In this condition, the yield strength of polymer nanocomposites noticeably improves in the tensile test. However, the yield strength reduces by adding of poorly bonded nanoparticles. As a result,
© Zare Y., Rhee K.Y., 2020
the properties of interface/interphase between polymer matrix and nanoparticles cause main effects on the mechanical properties of polymer nanocomposites and discounting of interphase characteristics results in wrong prediction of nanocomposites performances [12, 13].
The interphase dimension and stiffness have been determined by micromechanical models for mechanical behavior such as Young's modulus and tensile/ yield strength [14, 15]. It was also reported that shape memory polymer nanocomposites with a strong adhesion at polymer-nanofiller interface show pronounced shape memory properties [16]. However, there is not a model which directly expresses the effects of interphase properties such as interphase fraction on the Young's modulus and yield strength.
In this work, Ji and Pukanszky models are used to display the Young's modulus and yield strength of polymer nanocomposites containing different filler geometries as a function of the volume fraction, thickness, tensile strength and modulus of interphase region. The influences of interphase properties on the Young's modulus and yield strength of nanocomposites are discussed. Additionally, the mentioned equations are used to analyze the properties of interphase in various samples.
2. Background
Ji et al. [17] proposed a three-phase model for Young's modulus of nanocomposites taking into account matrix, nanofiller and interphase between polymer and nanoparticles. The Ji model for nanocompo-sites containing layered (1), spherical (2) and cylindrical (3) nanoparticles is expressed attributed to geometry of nanofillers as
" a-p
E = E„
1 -a + -
1 -a + a(k -1)/ln k
P
\-1
1 - a + 1/2(a - P)(k + 1) + PEf/En a1 =V (2?i/1 + 1)9f1 , a2 =4(ri2/r2 + 1)3 9f2 ' a3 =4(ra/ r3 +1)2 9f3 '
k=El,
where Em, Ef and E; are the Young's moduli of matrix, nanofiller and interphase, respectively, 9f is volume fraction of nanofiller, r and t are the radius and thickness of nanofillers, respectively, and r; and t; are the thickness of interphase.
The volume fractions of interphase 9; in different polymer nanocomposites are defined as
911 =9n2ti/1, (7)
912 = 9f 2[((r*2 + r2-Vr*2)3 -1], (8)
913 =9f3[((r3 + ri3)/r3)2-1]. (9) Therefore, all a parameters in Eqs. (2)-(4) can be related to 9; as
a = ^9i + 9f. (10)
As a result, Ji model for all nanocomposites is given by 9; as
Ec = Em[1 -^9i + 9f +
+ >i + 9f-4^f +
1 - +9f +4 9 +9f (k -1)/ln k
+ 49(1 -49i+9f + V2(V 9i+9f -^V97)(k+1) +V97 EjEm)-1]-1. (11)
According to Eqs. (7)-(9), 9f can be expressed as a function of 9; as
9i1t 9f1 = -i-i1-f1 2?-
9f2 =
9i2
9f3 =
((r2 + ri2)/r2)3-1
9i3
(12)
(13)
(14)
((r3 + ri3)/r3)2 -1' Accordingly, Ji model can be defined by the properties of interphase for polymer nanocomposites containing different nanoparticles as
E = Em[1 >/9i1 + 9i1?/(2 ?i) +
(1) +
(2) X
(3) +
+
(4)
+
(5)
(6) +
(49nt/(2?i) (1 + 9i1 V(2?i) ■
1/2 (^9i1 + 9i1 V(2?i) -(2?i) )(k +1) + V9iM2?i) eJ Em)-1]-1.
E = Em[(1 +M0-2 + ri2)Vr - 1)-1 + + (V9i2 +9i2((r-2 + ri2)Vr*2 -1)-1 -
+ ri2)Vr*2 -1)-1 ) X
X (1 -^9i2 +9i2((?2 + ri2)Vr*2 - 1)-1 +
W^i2 + M(r2 + ?l2)Vr*2 -1)-1 X
X (k - 1)ln-1 k )-1 +
+ VM(r2 + r|2)Vr*2 -1)-1 X
X (1 ^9i2 +9i2((r*2 + ri2)Vr*2 -1)-1 +
+ V2(4^ + ^i2 ((^2 + fh)3/r*2 - 1)-1 --^2«^2 + ri2)Vr*2 -1)-1 )(k + 1) + (16) W^i2((r2 + ri2)Vr*2 -1)-1 Ef/Em)-1]-1,
E = Em[(1 + 9i3 ((r3 + ri3 )Vr2 -1)-1 +
+ (V9i3 +9i3((r3 + r^2/r32 -1)-1 -
-^((r» + fi3)Vr32 -1)-1 )X
X((1 - ^3 + 9i3 ((r + ri3 )Vr32 -1)-1 +
W^i3 +9i3((r3 + ri3)Vr32 -1)-1 X X ( k -1) ln-1 k )-1 +
+ V^i3((r3 + ri3)Vr32 -1)-1 X
X (1 + 9i3 ((r + ri3 ) V?32 -1)-1 + +1/2 (^9i3 + 9i3 ((r3 + ri3)Vr2 -1)-1 -
-^((r» + r|3)Vr32 -1)-1 (* +1)) +
W^i3((^3 +13)7r2 -1)-1 W1]-1, (17)
which display the effects of interphase properties on the Young's modulus of polymer nanocomposites.
Pukanszky [18] suggested an equation based on the formation of interphase in composites, where the yield strength is determined as a function of filler content. Pukanszky model is presented as
_ 1 ~9f
g„ =-
-exp( Bf
(18)
1 + 2.59f
where ar is relative yield strength as ac/ am, ac and am are yield strengths of composite and matrix, respectively, B is an interfacial parameter which as-
sumes the capability of stress transfer between matrix and filler. This model was well applied for different polymer nanocomposites in the recent studies [19, 20]. Therefore, it is applied in this work to analyze the effects of interphase on yield strength of dissimilar nanocomposites. Parameter B depends to interphase characteristics as
B = (1 + AcPf ri)ln
"g )
(19)
where Ac is the specific surface area of filler, pf is density of filler, and is the tensile strength of interphase. To calculate parameter B, Pukanszky model can be rewritten as
ln
" 1 + 2.5fflf )
1 -9f
= Bf
(20)
where the linear plot of ln(ar(1 + 2.5%)/(1 -%)) against 9f shows the slope of B. Using Eqs. (12)-(14), Pukanszky model can be expressed as a function of interphase properties for polymer nanocom-posites as
Gr1 = 1 ^A2^ exp(B1 ^/(2^)), r1 1 + 2.5t(pj(2ti) 1 i
(21)
Gr2 =
= 1 -9i2((f2 + ri2)V -1)-1
G^, =
1 + 2.59i2((r2 + ri2)V- 1)-1 X exp[B29i2((r2 + f*i2)3/r23 -1)-1 L = (r + rj3)2/r32 -1 -9i3
X
X
r3 (r + ri3)Vr2 -1 + 2.59i3
X exp[B39i3 ((r3 + ri3)2/r3 -1)-1 ].
(22)
(23)
Additionally, Ac can be defined for layered, spherical and cylindrical nanoparticles as
c1
1
m,
2l2
Pf1v1
Pf1l 2t
Pnt
c2
2
2
4nr0
m2 Pf2v2 Pf2^3 ^2 Pf 2r2
2
c3
_ ^3 _ m
_
Inr^l
Pf3 nr32l
(24)
(25)
(26)
■'3 Pf3V3 Pf3nr l Pf 3r3 where A, m, v and l are the surface area, mass, volume and length of nanoparticles, respectively. As a result, B can be expressed for polymer nanocom-posites as
" m )
B1 =
1 +
9f1
ln Zl
(27)
Table
The samples and calculated interphase properties
No. Sample r or t, nm Pf> g cm-3 E GPa am> MPa ri or ti, nm Eqs. (2)-(4) E, GPa Eqs. (1)-(6) B Eq. (20) ai, MPa Eq. (19) Ac, m2 g-1 Eqs. (24)-(26)
1 PBTVnanoclay [21] 2 1.98 2.14 55.2 12.0 9.60 4.83 80.0 505.1
2 PA112/nanoclay [22] 4 1.90 0.61 32.8 12.0 2.50 8.40 108.9 526.3
3 LLDPE3/SiO2 [23] 8 2.20 3.70 51.0 12.0 11.10 24.60 4467.0 170.5
4 Epoxy/MWCNT4 [24] 15 1.90 1.90 45.0 22.5 133.00 7.66 305.4 70.2
5 PEI5/MWCNT [25] 9 2.10 2.96 102.0 11.0 8.80 5.94 573.5 117.0
1—poly(butylene terephthalate), 2—polyamide 11, 3—linear low density polyethylene, 4—multiwalled carbon nanotubes, 5—poly-etherimide.
b,
B3 =
+1
9f2
V3
- 2
9ü
9f3
V/2
+ 1
ln
ln Hü
(28) (29)
3. Results and discussion
3.1. The analysis of experimental data from the literature
In this section, the mentioned models are utilized to determine the properties of interphase in several samples from the literature. In addition, the effects of interphase characteristics on the modulus and strength of polymer nanocomposites are plotted. Table shows different samples from the literature as well as the properties of neat polymer and nanofiller. The experimental tensile moduli of samples are applied to Ji model (Eqs. (1)-(6)) and the average levels of ti or ri and Ei are calculated. The interphase thickness cannot be more than the gyration radius of polymer chains and Ei changes between the moduli of polymer matrix and nanofiller. Therefore, suitable ri or ti and Ei are chosen from the explained ranges and finally, the average values of interphase properties are given in the table. The presented data show the significant thickness of interphase in the reported samples. Additionally, a high interphase modulus is calculated in all reported samples, which is more than the stiffness of polymer matrix. Accordingly, the interphase can play an important role in the performances of polymer nanocomposites.
Moreover, the experimental yield strength of samples is fitted to Pukanszky model (Eq. (18)) to determine the B parameter. Parameter B is applied
into Eq. (19) to measure Gi values. Table gives the values of B and Gi data. The diverse B data show the dissimilar levels of interfacial adhesion in the
0.00 0.01 0.02 0.03 0.04 Nanoclay, vol. fraction
0.00
0.01
0.02
0.03
Si02, vol. fraction
0.00
0.01 0.02 0.03 MWCNT, vol. fraction
Fig. 1. ^i as a function of 9f in PBT/nanoclay [21] (No. 1) (a), LLDPE/SiO2 [23] (No. 3) (b) and epoxy/MWCNT [24] (No. 4) samples (c)
Fig. 2. Roles of and Ei in the Young's modulus of polymer nanocomposites containing spherical nanoparticles by Eq. (16) in r2 = = 15 nm, ri2 = 10 nm and Em = 2 GPa: (a) 3D and (b) contour plots (color online)
samples. Additionally, different data are obtained for reported samples. The highest is obtained for sample No. 3 as 4467 MPa and the least level is found for sample No. 1 as 80 MPa. The results are much higher than am demonstrating the significant strength of formed interphase in the reported samples.
The dimension and level of interphase are attributed to some parameters such as interfacial area and compatibility between polymer matrix and nanofiller which control the interfacial interaction. Some procedures such as treatment, modification and func-tionalization of nanofillers can encourage the compatibility and interfacial adhesion between the components of nanocomposites. Furthermore, Ac data are reported in the table using Eqs. (24)-(26). The nano-clay produces the highest level of Ac among the nanoparticles. This occurrence causes the highest level of interfacial area between polymer matrix and nanoclay layers, which finally creates the highest level of reinforcement in polymer nanocomposite. In fact, the high
value of Ac is the significant advantage of nanofiller, which makes the unexpected behavior in polymer nanocomposites. According to Eqs. (24)-(26), Ac is inversely related to r and t and the smallest nano-particles create the highest level of Ac as shown in the table.
Figure 1 illustrates the volume fraction in some reported samples by Eqs. (7)-(9). It is observed that the interphase occupies a large volume in polymer nanocomposites which is more than the nanofiller volume in some samples. Volume fraction increases with nanofiller content in all samples. The high level of interphase confirms the significant influence of this phase beside matrix and nanofiller phases. As a result, assuming the interphase is compulsory for estimation of mechanical properties in polymer nano-composites. In addition, is directly related to the thickness of interphase (according to Eqs. (7)-(9)), i.e. increases when the thickness of interphase enlarges.
Fig. 3. 3D (a) and contour plots (b) to show the effects of 9; and on parameter B in polymer nanocomposites containing spherical nanoparticles (Eq. (28)) in 9f3 = 0.1 and om = 50 MPa (color online)
r-n, nm
r-2, nm
5 10
25 r2, nm
Fig. 4. Volume fraction 9i2 as a function of r2 and ri2 (Eq. (8)) in f = 0.02: 3D (a) and contour plots (b) (color online)
3.2. The roles of interphase properties according to the models
Figure 2 demonstrates the effects of 9i and Ei on modulus of polymer nanocomposites containing spherical nanoparticles by Eq. (16). The Young's modulus more depends to than Ei.
The low generally results in a low modulus at all Ej levels. However, the best modulus is obtained by high levels of and Ei. It means that and Ei have optimistic effects on the modulus of nanocom-posites, where the effect of Ei becomes important at high values of indicating the important role of in Young's modulus of polymer nanocomposites.
Figure 3 exhibits the influences of and ai on parameter B in polymer nanocomposites containing spherical nanoparticles (Eq. (28)). Parameter B shows comparatively same levels at all values depended to the level of ai. In other words, ai more expressively affects the level of parameter B compared with 9j.
Parameter B shows negative values when ai is lower than am at all 9i. The best level of B is obtained in the highest values of and ai. As a result, B is more depended to ai value in polymer nanocomposites (especially at low ai), while formation of a high-volume interphase improves the magnitude of B interfacial parameter.
Figure 4 also shows the roles of nanoparticle radius and interphase thickness on the interphase fraction 9i2 of nanocomposites containing spherical nanoparticles by Eq. (8) in 9i2 = 0.02. The high values of nanoparticle size decrease the 9i2, but the small radius of nanoparticles increases it. On the other hand, a thick interphase grows the level of 9i2, while the
thinner one produces a smaller 9i2. Therefore, small nanoparticles and thick interphase show beneficial effects on the 9i2.
Since a higher volume fraction of interphase causes a better level for tensile modulus and strength (Figs. 2 and 3), it is concluded that small particles and thick interphase introduce the high mechanical properties in nanocomposites. Likewise, large particles and thin interphase result in low levels for mechanical performances. The small nanoparticles induce the high specific surface area between polymer and nanoparticles. A good interphase is gained by the high level of interfacial interaction/adhesion in nanocomposites [20, 26].
A micromechanical model was also proposed by Boutaleb et al. [27] to calculate the modulus and yield stress in polymer/SiO2 nanocomposites. It considers the interphase as the perturbed region of polymer matrix around the nanoparticles. The predicted effects of nanoparticle radius, interphase thickness and modulus on the modulus and yield strength of nano-composites in that work are similar to those suggested in the present study. All these remarks confirm the progressive roles of interphase properties in the mechanical behavior of nanocomposites.
4. Conclusions
Ji and Pukanszky models were used to show the Young's modulus and yield strength of different polymer nanocomposites as a function of interphase volume fraction, thickness, modulus and strength. The interphase properties of different samples were also studied based on Young's modulus and yield strength. The Young's modulus more depends on 9f and than
E^ A low generally results in a low modulus at all E; values. The poorest Young's modulus is found by the thinnest interphase. Besides, small nanoparticles and thick interphase increase the yield strength of nano-composites. Parameter g; more expressively affects the level of parameter B compared to 9 Nevertheless, the highest values of 9 and g; produce the highest level of interfacial adhesion expressed by parameter B. Since a high level of B increases the yield strength of nanocomposites, the high values of 9 and g; play positive roles in the yield strength.
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Received 12.11.2019, revised 14.12.2019, accepted 14.12.2019
Сведения об авторах
Yasser Zare, PhD, Dr., Islamic Azad University, Iran, y.zare@aut.ac.ir
Kyong Yop Rhee, PhD, Prof., Kyung Hee University, Republic of Korea, rheeky@khu.ac.kr
