Exponential Power Law Model for Predicting the Dielectric Constant of New Nanocomposite Industrial Materials A.Thabet1, Y. A. Mobarak2, and S. Abozeid3 Nano-Technology Research Centre, High Institute of Energy, South Valley University, Aswan, Egypt 1
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Abstract— This research presents systematic theoretical investigation of the effective dielectric constant for polymer/filler nano-composites and its dependence “filler concentration, the interphase interactions, polymer filler dielectric constant, and interphase dielectric constant”. Exponential Power Law model is a good fitting, and enabled us to control in rate of change in interphase properties, interphase volume constant, and filler particle shape and orientation thereby giving us wider class of functions. This paper investigated the prediction of the dielectric constant of new nano-composite materials based on Exponential Power Law model. Thus, this research has been improved the dielectric properties of polymer matrix and predicted the dielectric properties of new nano-composite materials for using in high voltage modern insulation and conducting materials by adding specified nano-fillers to polymer matrix. Keywords: Interphase, EPL, Dielectric Properties, Nanocomposite Materials
I. INTRODUCTION
NE of the important properties of nanocomposite materials in general is their dielectric properties which have been studied extensively [1-6]. The dielectric properties of composites play an important role in areas such as microelectronic and optoelectronic packaging materials [1]. The more recently, researches into physical and electrical properties of polymer/filler nanocomposite materials have supported the existence of the interphase region between polymer and filler [2-4]. With the existence of the interphase region, some researchers have gone on to consider this region as being a homogeneous region with constant properties. However, current researches tend to suggest that the properties of the interphase would best be modeled not by a constant but by a smooth variation in properties. Also, recently research tends to suggest that the dielectric properties of the interphase are also inhomogeneous, varying with respect to the radial distance from the centre of the spherical inclusion [4-6].
O
Such an inhomogeneous transition is due to the bonding mechanisms occurring in the space between the inclusion and matrix. Results for the dielectric constant have been published for two-phase composites in which perfect bonding is assumed to exist between the inclusion and the matrix. The first of these results are from Maxwell-Garnett theory [5]. With the same results derived later by Hashin using the composite spheres assemblage model [7-11]. There has been a first principles approach developed by Dong et al [7] to find the effective dielectric response of composites with a dilute suspension of graded spherical particles. Use of this approach however is difficult for any arbitrary graded profile since the solution procedure is dependent on finding the exact solution to the governing differential equations. Vo and Shi [1] measured the dielectric properties of composites as a function of inclusion
concentration using a proposed theoretical model based on effective medium theory. The dielectric property of composites and its dependence on the filler concentration is taken into account in their model. Therefore, the model is valid overall volume fractions and showed good agreement with experimental results. Their model however cannot account for a variation in the properties of the interphase region but instead measures the property of the interphase by a single constant. One of the difficulties in modeling the interphase region of composite materials is to know what properties of the interphase best model reality. If the properties of the interphase are being modeled by a single constant, then how that constant is chosen becomes an important factor. If the properties of the interphase are modeled by a smooth variation then choosing an appropriate function becomes an issue of importance. The more parameters that are included in such a function, the greater is the freedom of choice the dielectric profile, however it then becomes a harder problem to solve. There are different profiles which model the inhomogeniety of the interphase like Power Low Profile PLP, Exponential Profile and Exponential Power Law EPL Profile. All this models equation derived using the replacement method on the Maxwell-Garnett mixing rule [7]. This paper presents EPL profile as an analytical model to explain and predict the dielectric properties of the composite interphase region and dielectric constant of polymer/filler composite and its dependence on the filler concentration and the interphase interactions. The results of EPL model investigated using specific polymers and fillers in different cases to show the effect of interphase dielectric constant in effective dielectric constant of the composite. EPL model is used due to its good fitting and enables us to control three different parameters, thereby giving us wider class of functions. The present results have important implications to the targeted formulation of optoelectronic packaging materials [1]. The rest of this paper is documented in following headings. Section II provides analytical model studied, section III in provide industrial materials used in this study, section IV provides analytical model studied, and the results obtained from this model and observations are described in section V, and concluding remarks have been focused in section VI. II. ANALYTICAL MODEL STUDIED A. Inhomogeneous Interphase A polymer/filler composite material can be considered by the arrangement shown in Fig. 1(a), which spherical particle of radius a, surrounded by an annular interphase region of radius b, embedded in a surrounding matrix. Maxwell demonstrated that its always possible to find a value of composite dielectric constant so that the field outside the sphere of radius c is precisely what one would expected of the composite system. If
replaced by a single sphere of relative permittivity ε c [8-9]. The Maxwell-Garnett approximation of the dielectric constant of a composite εc consisting of isotropic spherical inclusions embedded in an isotropic matrix is given by [11]: (1) Where: εm is the dielectric constant of the matrix, εp is the dielectric constant of the inclusions, and c is their volume fraction c=do(b3/a3), where do is the volume fraction of inclusion relative to all phases. Interphase
r+∆r r
Thus, for an inhomogeneous interphase, the Vo and Shi method gives us a way of finding the equivalent homogeneous property of the interphase. Since now know what the equivalent homogenous property of the interphase is, this result may be incorporated into other existing three-phase models, which assume a homogeneous interphase surrounding each inclusion. Therefore propose here a way of fusing two different models together, an inhomogeneous and a homogeneous interphase model. This joining process is called the fused model. D. Composite Dielectric Constant For instance, the value εi calculated from expression (8) can be used in the Vo and Shi model [10]. According to the Vo and Shi model, the dielectric constant of a particulate composite is given by: (9) Where:
Fig. 1. Schematic for filler/polymer composite materials consisting of filler, matrix and interphase
B. Exponential Power Law EPL Model An EPL model suppose that the dielectric properties of the interphase region vary according to the function given by: (2) Where: c, p, and β, is constants, and x is radial distance from the center of sphere inclusions. Such a function is useful in that it enables us to control in three different parameters, that is c, p and , thereby giving us a wider class of functions than power low profile and exponential profile this function seem to have been first considered by [10]. Substitution of EPL function (5) into following differential equations: (3) )
(4)
The subscripts m, i and p stand for matrix, interphase and particle inclusion, respectively. The parameters a and b are as defined in inhomogeneous interphase model while the parameter c represents the radial distance from the centre of the inclusion to the outer boundary of the matrix phase of a representative composite sphere. The parameters a, b and c are related to each other through the parameters k and do where k is the interphase volume constant and do is the volume fraction of filler, that is:
(5)
(10)
(6) With initial condition S(a)=1, U(a)=0, T(a)=0, V(a)=1, and xЄ[a, b] the equations of S(x), U(x), T(x), V(x), can be derived in ref. [16]. Then the effective dielectric constant of inclusion and interphase determined by: (7) Where:
is the inclusions dielectric constant
C. Interphase Dielectric Constant Dielectric constant value εi of the interphase can be determined by doing a reverse mapping as shown in Fig. 2 that is the mapping of a homogeneous sphere of radius b onto a two-phase sphere of identical size to determine the value of εi by solving:
(11) (12) The value of k reflect the matrix /filler interaction strength, k is equal zero when the interaction (bonding) between the matrix and filler negligible, and the large positive k value indicates a strong polymer/filler interaction [12]. By rearranging Eqs. (10) to (13) the parameter b may expressed by: (13) The bavg used in our inhomogeneous interphase model which gives the value of interphase dielectric constant over the domain 0≤do≤1 given by: (14)
(8) III. SELECTED INDUSTRIAL MATERIALS
2a 2b 2b
Fig. 2. A mapping of a homogeneous particle consisting of inclusion and interphase onto a two-phase composite
In this paper two polymer and five fillers used to form five nanocomposites. The polymers are Epoxy and Polypropylene with dielectric constant 2.3, and 5.0 respectively. Epoxy-based materials include coatings, adhesives, industrial tooling, civil engineering materials, electrical and electronic materials especially in generator ground wall insulation for medium and high voltage insulations. Polypropylene(PP) is a widely used polymer as insulation of modern high voltage capacitors HVC due to its good properties. polypropylene has excellent resistance to chemical and good thermal endurance. The fillers are Clay, Fumed Silica SiO2, Zinc Oxide ZnO, Magnesium
IV. RESULTS AND DISCUSSION EPL profile enables us to control in interphase properties by changing three different factors c, p, and β, where these constants are chosen to agree this condition ɛ(a)=ɛP and ɛ(b)=ɛm. Effects of rate of change in interphase properties with respect to radial distance x “p” and interphase volume constant on the effective dielectric constant are discussed in this paper. A. Effect of “Rate of Change in Interphase Properties with Respect to Radial Distance x”on the Effective Dielectric Constant Choosing the parameter c, and β such that ɛ(a)=ɛP and ɛ(b)=ɛm and allow parameter p to vary, then it’s possible to plot a several results for specific values of interphase volume constant k. Fig. 3 shows the effects on the effective dielectric constant ɛc with the inclusion concentration do at different values of interphase dielectric constant. The value of ɛi can be controlled by changing the interphase parameter p which represents the rate at which dielectric properties change with respect to radial distance x. Three different nanocomposites used to show the effect of interphase dielectric constant in composite dielectric constant. Fig. 3(a) shows the case of the matrix dielectric constant equal to the filler dielectric constant “PP with Clay composite”. Also this fig shows at constant value of k=2.8, interphase dielectric constant is higher than matrix dielectric constant at values of p=10, p=500 but interphase dielectric constant is lower than matrix dielectric constant at values of p=-500, p=-350.
B. Effect of Interphase Volume Constant on the Effective Dielectric Constant The effective dielectric constant of nanocomposite also depend on interphase volume constant k which reflects the interaction between filler and polymer. Fig. 4 shows the variation of the effective dielectric constant ɛc with the inclusion concentration do at different values of interphase volume constant (k=1, k=2, k=3 and k=7) and at constant value of p=350. Three different nanocomposites used to show the effect of interphase dielectric constant in composite dielectric constant. Fig. 4(a) shows the matrix dielectric constant equal to filler dielectric constant“PP with Clay composite. Fig. 4(b) shows the matrix dielectric constant higher than filler dielectric constant “PP with SiO2 composite”. Fig. 4(c) shows the matrix dielectric constant lower than filler dielectric constant “Epoxy with SiO2 composite”.
ɛi>ɛm
2.5
k=1 k=2 k=3 k=7
ɛi>ɛm
2.6
2
P=-500 P=-350 P=10 P=500
1.5 0
0.2
0.4
0.6
Concentration of Filler do
0.8
0
1 (a)
PP with Clay composite, and k=2.8 2.7
P=-500 P=-350 P=-250 P=500
ɛi>ɛm
2.5
2.2
2.3
0.2
0.4
0.6
Concentration of filler do
0.8
1
(a) PP with Clay composite, and p=350 4.9
Effective Dielectric Constant
ɛi>ɛm
k=1 k=2 k=3 k=7
ɛi>ɛm
4.4
ɛi
