# Оценка модуля упругости нанокомпозитов на основе сшитого полиэтилена и глины с памятью формы

УДК 539.32

Оценка модуля упругости нанокомпозитов на основе сшитого полиэтилена и глины с памятью формы

Y. Zare1, K.Y. Rhee2

В статье с использованием ряда моделей проведен анализ модуля упругости полимерных нанокомпозитов из сшитого полиэтилена и глины с памятью формы. Традиционные модели, такие как модель модифицированного правила смесей, модели Guth, Paul, Counto, Kerner-Nielsen и другие, дают заниженные значения модуля упругости, показывая, что при его оценке для нанокомпозитов с памятью формы следует учитывать упрочняющий эффект нанонаполнителя. В некоторых моделях определены необходимые параметры для корректного расчета модуля упругости. В соответствии с моделью Halpin-Tsai для наполнителей с неравномерным трехмерным распределением и моделью Hui-Shia среднее значение относительного удлинения для слоев наноглины равно 56. Результаты, полученные с помощью модели Takayanagi, не согласуются с экспериментальными данными, что указывает на существенное влияние межфазной границы между полимерной матрицей и наноглиной. Предложена модификация моделей Guth, Halpin-Tsai и Kerner-Nielsen для корректировки расчета модуля упругости наноком-позитов на основе сшитого полиэтилена и глины с памятью формы.

DOI 10.24411/1683-805X-2020-14011

Estimation of tensile modulus for cross-linked polyethylene/clay shape memory nanocomposites

Y. Zare1 and K.Y. Rhee2

Many models are used for the analysis of tensile modulus in cross-linked polyethylene/clay shape memory polymer nanocomposites. The conventional models such as modified rule of mixtures, Guth, Paul, Counto, Kerner-Nielsen, etc. underestimate the modulus exhibiting that the reinforcing effect of nanofiller should be considered for the estimation of tensile modulus in the shape memory nanocomposites. In addition, the appropriate parameters in some models are indicated for proper prediction of tensile modulus. Several models such as Halpin-Tsai for fillers with random 3D distribution and Hui-Shia offer the average aspect ratio of 56 for nanoclay layers. The results obtained by the Takayanagi model are not fitted to the experimental results demonstrating the important effect of the interphase between polymer matrix and nanoclay. Some models such as Guth, Halpin-Tsai and Kerner-Nielsen are modified for better adjustment to tensile modulus of cross-linked polyethylene/clay shape memory nanocomposites.

Smart materials have attracted much attention in more studies. Shape memory materials are regarded by their potential to store a temporary deformed shape

and recover the originally permanent shape [1]. The shape memory behavior is introduced by a change in temperature, which is shown in metals, ceramics and polymers [2-4]. Shape memory polymers demonstrate

© Zare Y., Rhee K.Y., 2020

many advantages such as low density, high shape recovery, high recoverable strain, easy processing along with low cost. Especially, their recoverable strain is in order of 100%, while this term in shape memory metals and ceramics reaches to only about 10% and 1%, respectively [5].

All shape memory polymer structure retains two hard and soft phases. The hard segments keep the permanent shape and does not melt or soften in the transition temperature Tg of soft segments. These segments can be cross-linked as rigid local structures or entanglements, which will not separate in the recovery temperature. In another side, the soft part acts as a switch to remember the original shape of material. The shape memory polymers causing the large strains to recover in restricted environments are employed in many applications such as deployable aerospace structures, microsystems, heat-shrink tubing and biomedical tools [6-8].

A typical predeformation and recovery cycles for shape memory polymers were described as follows [9]. First, the shape memory polymer is heated to high temperature, which in this step, the sample is deformed to a desired strain. In the second step, shape memory polymer is cooled to fix the temporary shape. At the lower temperature, the shape memory polymer molecular segments are frozen in a temporary position and upon removing the restriction, the induced shape is recovered. In the last step, the shape memory polymer is heated up to transition temperature, which the original shape is recovered. The deformation, i.e., strain energy is amassed by a reversible morphological change or inhibition of molecular relaxation [10]. From the theoretical point of view, this stored energy can be converted to force, but to attain this, the rubbery modulus described as the elastic modulus above Tg should be increased [11]. Many types of fillers such as chopped carbon, glass or Kevlar fibers have been used to improve the rubbery modulus [11].

There is a growing attentiveness on the nanostruc-tures such as nanocomposites in recent years [12-26]. In the recent years, it has been found that the performance of shape memory polymers significantly improves using nanoparticles [2, 27, 28]. Koerner et al. [29] have revealed that the homogeneous dispersion of 1-5 vol % of carbon nanotubes can store and subsequently release more than 50% of recovery stress compared to the neat polymer. Most studies have made known that the addition of nanofillers to polymers affects the physical, mechanical, thermal and barrier properties of polymers [30-34]. In the case of shape memory polymer reinforced with traditional

microfillers, it frequently loses its shape memory effect due to the high weight portion of filler (20-30%). Nanofillers can overcome these disadvantages and the smart nanocomposites, even with small clay contents (0-8 wt %) show good mechanical properties and high physical properties such as high recovery force.

Although much investigation has focused on the experimental characterizations of shape memory polymer nanocomposites, their properties have not been analyzed so far. The theoretical analysis and modeling can offer valuable information, which facilitates the achievement of desired properties. Further, the evaluation of behavior makes possible to predict the behavior of shape memory polymer nanocomposites without requiring more experiments. The prediction of rubbery modulus is more helpful particularly to make an appropriate actuator with specified properties. In this article, the tensile modulus of cross-linked shape memory polymer/clay nanocomposites is analyzed using the composite theories. In addition, some equations are suggested for better prediction of tensile modulus in these samples.

The simplest models to predict the tensile modulus of composites are parallel and series models [35]. For the rule of mixtures, the equivalent strain is assumed in both matrix and filler phases. This model is given by

E _ EmK + E f (1)

where Em and Ef are the Young&s moduli of matrix and filler, and and are the volume fractions of the matrix and filler.

The inverse rule of mixtures presumes the same stress in the matrix and filler parts as

The rule of mixtures model was modified by Cox [36] as

E _ EmK +nEff (3)

, tanh m

n_1--, (4)

Ef ln(r/R)&

in which a is the aspect ratio of filler, a _ w/t, where w and t are the width and thickness of the filler, G is

the shear modulus of matrix, while R and r are the filler radius and the centre-to-centre distance of the fillers.

The rule of mixtures was modified as [37]:

E _ Em K + MRF^f Ef, (6)

MRF = 1 (7)

V E A (8)

a V Ef Am

Verbeek model [38] considers the porosity of the composite and the aspect ratio of the reinforcement as key factors influencing the Young&s modulus. This model assumes the perfect adhesion between the phases and stress is transferred through a shear mechanism. Verbeek model is given by

E _ EmAm + ^Ef f

À = 1 m

(9) (10)

(1 - X )3 G фf

фт(1 -у )

(1 -фm)f фmax

E = x( Em Фm + Ef фf) + (1 - x)Ef Em

Ef +

where x and (1 - x) represent the relative contributions of composite to the rule of mixtures and inverse rule of mixtures, respectively.

Authors of Ref. [41] considered that the consistent stress is created at the matrix-filler interface: 1 + (m - 1)f

In Ref. [42] under the assumption of the faultless adhesion in the matrix-filler boundary a model was suggested as

Em (1 -фГ)/фГ Em + Ef

Guth [43] believed the interaction between filler particles exploiting the Smallwood-Einstein approach. The Guth model is presented as

E _ Em(1 + 2.5Af + 14.1A2). (18)

The aspect ratio of filler was introduced to the Guth model in order to account the accelerated stiffening for rod-like filler [44] as

E _ Em[1 + 0.67aAf + 1.62(a^f)2]. (19)

Davies [45] suggested one of the most used model

E15 _ EmfAm + E15Af. (20)

Halpin and Tsai [46] proposed the most commonly used model as

E = Em ,

Ef/Em -1 Ef/ Em +Ç&

where ¥ is the voidage and Amax is the maximum volumetric packing fraction of the filler, Amax = true volume of the filler/apparent volume occupied by the filler, X is the modified void content as the modified porosity relative to the polymeric phase. The value of Amax was determined through the method proposed by McGeary [39] as 0.75.

Series and parallel models as we mixed [40]

= 2a. (23)

The tensile modulus for randomly oriented fillers can be calculated by laminate plate theory as

E = 0.49E1 + 0.51E2, (24)

E1 and E2 are the tensile moduli of the composite in the longitudinal and transverse directions, respectively. Both E1 and E2 are calculated from Eq. (9), but the aspect ratio of filler is assumed as 1 for E2 (a = 1).

Landel and Nielsen [47] modified the Halpin-Tsai model as

в = Ef/Em - 1

P = 1 + ф:

Ef/ Em + A& f 1 -Kax

where v is the Poisson ratio of matrix.

In addition, Lewis and Nielsen [48] took account the maximum volumetric packing fraction of the filler in the Halpin-Tsai model as

ф= 1 + фf

Kerner [49] proposed a model, which was originally developed to calculate the shear modulus of composites. The model is expressed as

Hui and Shia [50] developed a model assuming the perfect interfacial adhesion between the polymer matrix and particles as

-3(1 -fcf)(1 -g)a2 -g2, (33)

A _ (1 -f

g _ 2 a,

a 2 -1 &

Ji et al. [51] suggested a three-phase model (matrix, filler and interphase) as

where t and ti are the thickness of filler and interphase, respectively, k is the ratio of the interphase modulus on the surface of filler Ei and the Young&s modulus of matrix as k _ EilEm. If the effects of interface are neglected (ti = 0), the Ji model changes to the two phase model of Takayanagi [52] as

E _Em(1 -a)2 +a2 Em + Ef

Sato and Furukawa [53] developed a model for the tensile modulus of composites using an adhesion parameter Z as

f 1 (1 fcfv;

(1-fc13)fcf

The Z parameter of 0 indicates the good adhesion, while Z = 1 shows the poor adhesion.

The experimental measurements of tensile modulus were provided from Rezanejad et al. [6] study on the cross-linked polyethylene/clay shape memory na-nocomposites. The results showed that the tensile modulus, the recovery temperature, the recovery force and force recovery rate increased with the addition of nanoclay to the shape memory polymer, due to their improved mechanical properties, but final recovery strain decreases slightly about 12%. The XRD and TEM characterizations illustrated the intercalated structure. The presence of clay layers caused the cross-link density to decrease and crystallinity to increase.

When experimental data and theoretical measurements of tensile modulus are compared, except the rule of mixtures model, other models underestimate the tensile modulus showing the poor predictability of these models. It gives an idea that more parameters such as the aspect ratio of nanofiller should be assumed for estimation of tensile modulus in shape memory polymer nanocomposites. Modified rule of mixtures model suggests the lowest predictions in which the various values of aspect ratio of nanoclay do not change the predicted data, significantly. The obtained results by Davies model are found to be some better than others. The Kerner-Nielsen and Kerner models calculate the inappropriately identical results and their predictions do not depend on the maximum volumetric packing fraction of the filler fcmax and the Poisson ratio of matrix v. Furthermore, the Takayanagi model, which does not consider the interphase shows the much inferior results. It demonstrates that the effect of interphase should be taken into account in the Ji model for the cross-linked polyethylene/clay shape memory nanocomposites.

Using the Ji model, the values of nanofiller and interphase thickness (t and ti) and k should be established, which are very hard to characterize. However, in the previous studies, the value of k was randomly chosen and its role has not be determined, obviously [51]. Parameter k is defined as the ratio of the interphase modulus on the surface of the platelets Ei and the tensile modulus of matrix as k _ Ei/Em by the assumption of linear dependence of the modulus on space variable from the matrix to the surface of particles. As a result, the possible values of interphase modulus are Ei (min) = Em and Ei (max) = Ef. Accordingly, the parameter k can be selected in the range of 1 to 4450 for the present nanocomposites. The values of

k can be chosen from this range and the t/ti ratio is calculated from the fitting process to the experimental results. It should be indicated that the reduced thickness of the dispersed clay platelets t/ti expresses the level of exfoliation. The broken silicate layers in matrix with small thickness is indicative to smaller t/ti? which confirm the superior dispersion of silicate layers, i.e. higher degree of exfoliation. Figure 1 shows the values of t/ti as a function of k parameter. For k values ranged from 0 to 800, the t/ti increases to 0.25 rapidly, but for greater k values, the t/ti remains constantly at 0.25. The results of t/ti show clearly that the interphase thickness is at least four times higher than that of silicate layers, which verify the good exfoliation and reinforcing effect of nanoparticles in the polyethylene matrix. Evidently, further research by microscopic measurements is necessary for accurate determination of these parameters.

The Hirsch model [40] is properly fitted to the experimental data at x value of 0.01. It corroborates that the tensile modulus of cross-linked polyethylene/clay shape memory nanocomposites follows the rule of mixtures model more than the inverse rule of mixtures, due to the much higher modulus of nanoclay platelets. The developed Guth and Halpin-Tsai models calculate the tensile modulus at smaller aspect ratio of nanoclay compared to other models. These models usually over predict the tensile modulus [54] may be due to the minor contribution of two dimensional filler to tensile modulus in comparison to one dimensional fillers. Due to this reason, they are fitted to the experimental data at lower aspect ratios. Other models such as Halpin-Tsai for 3D random platelets, LewisNielsen, Hui-Shia and Verbeek suggest the various aspect ratios from 40 to 88 for nanoclay layers, which provide the average aspect ratio of 56. Moreover, Sa-to-Furukawa model predicts the tensile modulus accurately at Z of -14, while the variation range of Z is specified at 0 to 1. It is possibly due to the much slighter

level of nanoclay contents in the cross-linked shape memory nanocomposites compared to the conventional microcomposites.

For calculating the tensile modulus by the Cox model, two parameters including the aspect ratio of nano-filler a and the ratio of centre-to-centre distance of the filler and filler radius r/R should be measured. However, in the silicate layer nanocomposites containing randomly dispersed platelets, the accurate values of these parameters are difficult to determine. Therefore, the value of aspect ratio of nanoclay as a function of r/R is provided based on the fitting procedure to the experimental data as observed in Fig. 2. The aspect ratio of nanoclay increases quickly to 26 at r/R values of 1 to 35, but further values of r/R up to 300 do not affect the value of aspect ratio, noticeably. The obtained results show that the Cox model can predict the tensile modulus at the logical values of parameters.

The different aspect ratios of silicate layers have been indicated that do not affect the predicted results, considerably. In addition, the variation of MRF and u (Eqs. (7), (8)) at different nanoclay contents are negligible. As a result, modified rule of mixtures model for cross-linked polyethylene/clay shape memory nano-composites can be developed into

E = Em 4 m + MRF^f Ef. (42)

This equation is more similar to Cox and Verbeek models (Eqs. (3) and (9)) in which the MRF is the filler orientation efficiency factor [55]. MRF has been found to be 0.38 for PVA-nanotube thin films [56]. However, the best fitted results are obtained at MRF value of 0.01 for the current nanocomposites as illustrated in Fig. 3.

As mentioned, the Guth and developed Guth models do not estimate the suitable results for tensile

Nanoclay, wt %

Fig. 3. The predictions of tensile modulus by the modified rule of mixtures, Guth and Halpin-Tsai models

--1-1-1-1-1-1-1-r-1

Nanoclay, wt %

Fig. 4. The predicted modulus by the developed KernerNielsen, Kerner and Sato-Furukawa models

modulus of present shape memory polymer nanocom-posites. Therefore, because of overestimation of the developed Guth model, a modulus reduction factor (MRF) can be introduced as

E = Em[1 + 0.67MRFa4f +1.62(MRFa4f)2]. (43)

Figure 3 shows the predicted results by the proposed model with the MRF value of 0.35. Obviously, the best agreement between the experimental and the predicted data is demonstrated.

Like the developed Guth equation, the Halpin-Tsai model also overestimates the tensile modulus. So, a MRF can be taken into account in the model for improving the predictability. The Halpin-Tsai model after introducing the MRF is presented as

,= E 1 + MRF^f 1 "^f &

Figure 3 shows the calculations of the modified Halpin-Tsai model by MRF value of 0.4. The proposed results are extremely better fitted to the experimental data compared to the original model.

As mentioned, the Kerner-Nielsen model cannot predict the tensile modulus of present nanocomposite. It was noted that various 4max and v values do not alter the predicted results, due to the higher level of Young&s modulus of nanoclay (about 178 GPa) compared to the matrix modulus of 40 MPa. Accordingly, A, P and B parameters (Eqs. (26)-(28)) can be removed from the Kerner-Nielsen model and regarding the fitting process, the modified Kerner-Nielsen model can be shown as

The theoretical predictions by modified KernerNielsen model are shown in Fig. 4. The modified model presents a good ability for prediction of tensile modulus in the present shape memory polymer nano-composites. Most attention should be given to the simplicity of the suggested model in which more simple parameters are observed. Additionally, the Kerner model is not fitted to the experimental data properly. Furthermore, the different values of v in the long range of 0.33 to 0.5 do not play a key role in the calculations. As a result, v can be taken out from the Kerner model and the developed model is attained such as the modified Kerner-Nielsen model in Eq. (45).

The value Z = -14 in the Sato-Furukawa model suggests the accurate calculations for tensile modulus of current nanocomposite. But, the typical range of Z is determined as 0 to 1. By assuming Z = 0 for a perfect adhesion between nanoclay and matrix and using the fitiing procedure, the Sato-Furukawa model can be modified into

The predictions of the modified Sato-Furukawa model is observed in Fig. 4. More superior conformity between the experimental results and the predictions is provided using the modified model.

The tensile modulus of cross-linked polyethylene/clay shape memory nanocomposites was evaluated using many models. The conventional models such as Paul and Counto underestimate the modulus showing the requirement to more parameters such as the aspect ratio of nanofiller. For many models, the suitable values of parameters for proper prediction of tensile

modulus are indicated. The average aspect ratio of 56 for nanoclay layers is obtained according to several models such as the Halpin-Tsai model for randomly 3D fillers and the Hui-Shia model. Moreover, the Ta-kayanagi predictions demonstrate the significant effect of interphase on the present shape memory polymer nanocomposites. Lastly, the modified rule of mixtures, the Guth, Halpin-Tsai, Kerner-Nielsen, Kerner and Sato-Furukawa models are modified to better estimation of tensile modulus for cross-linked polyethylene/clay shape memory nanocomposites.

References

nanocomposites: Characterization and micro-macro modelling // Polymer Test. - 2010. - V. 29. -P. 245-250.

Received 21.07.2020, revised 21.07.2020, accepted 03.08.2020

Сведения об авторах

Yasser Zare, PhD, Dr., Motamed Cancer Institute, Iran, y.zare@aut.ac.ir

Kyong Yop Rhee, PhD, Prof., Kyung Hee University, Republic of Korea, rheeky@khu.ac.kr