Journal of Computer-Aided Materials Design, 3 (1996) 403-408 403 ESCOM

J-CAMATD 055

Hierarchical nano-/micromaterials based on electrospun polymer fibers: Predictive models for

thermomechanical behavior

Yuris A. Dzenis

Department of Engineering Mechanics, Center for Materials Research and Analysis, University of Nebraska-Lincoln, 221 Bancroft Hall, Lincoln, NE 68588-0347, US.A.

Received 8 January 1996 Accepted 15 January 1996

Keywords: Polymer nanofibers; Hierarchical composites; Elastic and thermal properties

SUMMARY

Applications of small electrospun polymer fibers in hierarchical composite materials are discussed. Micromechanics models for effective elastic, thermal, and thermoelastic behavior of these materials are developed. The principle of effective homogeneity is applied to connect scales in the materials. Effective thermoelastic characteristics of nano- and microfiber composite are analyzed to illustrate the developed approach. Strong hybrid effects are observed in the dependence of effective modulus and thermal expansion coefficient on fractional content of fibers of different diameters. The extrema are located at the higher fractions of larger reinforcing elements. The methodology developed can be utilized for connecting scales in modeling other hierarchical materials.

Nanocomposites based on ceramic [1-3] and metal [3,4] matrices, as well as cermets [5,6], were extensively studied during the past decade. The polymer-based nanocomposites are much less investigated and their studies are mostly limited to layered and particulate systems [7,8]. At the same time, it is well-known that advanced composites possessing outstanding mechanical prop- erties for structural applications are usually composed of strong fibers dispersed in a matrix material. The absence of strong and tough fibers in the submicron diameter range prevented the development of advanced polymer nanocomposites until recently.

A method of electrospinning of thin polymer fibers was recently developed [9]. Fibers with diameters ranging from 50 nm to several microns were prepared from over 30 different synthetic and natural polymers, including high-temperature high-modulus polyimide and polyaramid (Kevlar) fibers [10]. These submicron polymer fibers are candidates for reinforcing agents in advanced nanocomposites due to their continuity, orientation, inherent flexibility, and potential high compatibility with polymer matrices.

0928-1045l$ 6.00 + 1.00 © 1996 ESCOM Science Publishers B.V.

404

.... : I

Fig, 1. Schematic representation of a hierarchy of nano- and microfiber composites.

Even though electrospinning is not an expensive process, the amount of submicron fibers required for a composite structural part is presently expensive to produce. A combination of nanofibers with conventional reinforcement at the micron scale seems to be a feasible solution. The objective of this paper is to study the possible applications of high-performance submicron fibers in hierarchical polymer composites reinforced at both the nano- and microscales.

The main mechanism of modification of properties in nanocomposites is interfacial changes, In ceramic or polycrystalline nanocomposites, the relative content of the grain-boundary zones may be comparable to the content of grain bulk material. In polymer nanocomposites the prop- erties of components can also change due to strong interfacial interactions. Classical micromecha- nics models for two-component composites assume perfect bonding and absence of physical or chemical interactions of constituents at the interface. These models show no effect of reinforce- ment size on effective properties, but only the total volume fraction of reinforcement matters. However, in case of three-component composites, if the reinforcement consists of elements of significantly different size, both the total volume fraction and the fractional content of reinforcing elements of different sizes are important.

It is suggested here that micromechanics models for two-component materials can be used in conjunction with stage-by-stage analysis to evaluate properties of hierarchical materials. Firstly, properties of the matrix reinforced with smaller elements can be computed. Then, this material reinforced with the elements of a larger size can be analyzed. This approach is consistent with the principle of effective homogeneity [11]. Note that conventional advanced reinforcing fibers have diameters in the range from 5 to 12 microns. Assuming that a significant difference in size means at least an order of magnitude, submicron (or nano-) reinforcing elements at smaller scale are required in order for a hybrid composite to be considered hierarchical. Such a hybrid nano- and microfiber composite is analyzed below as an illustrative example.

A schematic representation of a hierarchical structure of a random three-dimensional nanofiber and microfiber composite is shown in Fig. 1. Let v= and vl be the absolute volume fractions of small and large fibers, respectively. The volume fraction of the matrix is then 1 - v=- v 1. The elastic bulk and shear moduli of a matrix reinforced with nanofibers can be calculated using the following correlations [12]:

K = [Ell +4(1 + v12)2K2s] / 9

G ~-~ [gi1 + (1 - 2vlg)2K23 + 6(G12 + G23)] / 15

(1)

405

where:

t u = Efvf + Em(l -v f )

V12 = VfVf + Vm(1 - Vf)

K 2 3 = K m + a m + vr 3 1/(Kf - K m + (Gf - G m ) /3) + (1 - vf ) / (U m + 4G m / 3)

G12 ~ G m G f ( l + vf) + G i n ( l - vf )

G r ( 1 - vf) + Gin(l+ vf)

(2)

G23 = G m + VfGm

G m (Km + 7 G ~ / 3 ) ( 1 - v r ) / 2

Gf - G m K m + 4 G m/3

where vf = Vs/(1-vl) is a volume fraction of small fibers in the polymer matrix; and indices f and m denote characteristics of fibers and matrix, respectively. The effective Young's modulus E and Poisson's coefficient v can be calculated from K and G using isotropic relations. In the second stage, the properties of this nanocomposite reinforced with conventional microfibers can be calculated by repeated use of Eqs. 1 and 2, where the volume fraction of fibers vf = v v

A similar approach can be used for effective transport characteristics determination. The conductivity of a random three-dimensional fiber composite can be calculated from [12]:

£ = )Vu/3 + 2~.22/3 (3)

where:

~"11 = ~,fVf "t- )k,m(l -- Vf)

Vf~. m

~Vm/(£f --)Vm) + (1--Vf)/2

(4)

Equations 3 and 4 hold for the thermal and electric conductivity, and the dielectric and magnetic permeability.

The following correlation between effective elastic and expansion characteristics of isotropic inhomogeneous media is used to calculate the coefficient of thermal expansion [12]:

0~'=~1 "[ 1/~2 ~l~-K1 • I~ 1 (5)

406

.= 10

0.4

0.2

s 0.1

o~s i v,/(vs+v,)

Fig. 2. Variation of the effective Young's modulus of nano- and microfiber composites. Numbers indicate the total volume fraction of reinforcement.

where K is the effective bulk modulus (1); subscripts are phase indices. Equation 5 holds for both thermal and moisture expansion coefficients.

In order to perform the parametric study of effective behavior of hybrid composites it is useful to firstly simplify the problem. The hybrid composites under consideration consist of three ma- terials and are described by two structural parameters, v s and v v Thus, the number of variables in the transport problem is 5, in the elastic problem -8, and in the expansion problem -11. Clearly, such a large number of variables makes the analysis difficult. Let us assume incompres- sibility of composite constituents. This implies that their bulk moduli are infinite, the Poisson's coefficients are equal, and the following simple relationship holds for Young's and shear moduli: E = 3G. In this case, only one elastic characteristic is independent and Eqs. 1 and 2 can be reduced to:

1 [ E l ( l+ vf) + Em(1- vf) ] E= Efvf + Era(l- v f ) + 4 E m E l ( l - vf) + E~(I+ vf)J (6)

Equation 6 is used in the following analysis of effective elastic modulus of polymer composite. However, the assumption of incompressibility cannot be applied in the expansion problem. Therefore, the full solution (Eqs. 1, 2 and 5) is used in the analysis. It is well-known that an incompressible elastic solution is a good approximation for the solution of transport problems. Thus, the results of analysis of effective stiffness given below can be qualitatively applied to transport characteristics.

The calculated variation of the effective Young's modulus of hybrid composites with relative content of different reinforcing elements is shown in Fig. 2. The variation of the thermal expan- sion coefficient is presented in Fig. 3. Material properties of reinforcing elements, i.e., nano- and microfibers, are assumed to be equal. The effects observed can be attributed solely to hierarchical composite morphology. The Young's modulus of reinforcing elements is 100 times higher than that of the matrix. The following Poisson's coefficients of constituents are utilized in the analysis of thermal expansion: Vm = 0.33 and vf=0.22. The coefficients of thermal expansion of the matrix and reinforcements are o~ m = 10 x 10 s K -1 and off= 1 x 10 ~ K -I.

407

7

×

0.1

0.2

0.3

o o:s v~/(v~+v~)

Fig. 3. Variation of the effective coefficient of the thermal expansion of nano- and microfiber composites. Numbers indicate the total volume fraction of reinforcement.

For all composites, the effective stiffness increases and the thermal expansion decreases with increasing total volume fraction of reinforcement. Calculated properties of two-component fiber composites (vl/(v~+vl)= 0 and 1 in Figs. 2 and 3) are equal due to the assumed equality of prop- erties of nano- and micro fibers. In real material, however, one can expect differences in effective properties of nanocomposites and microcomposites due to the different interfacial interactions on different scales. Nevertheless, even without taking into account interfacial phenomena, the hierarchical nano-/microcomposite reinforced with a mixture of nano- and microfibers exhibit extremal properties. The stiffness of hybrid composite is higher, whereas expansion coefficients are lower compared to properties of two-component composites. The location of extrema shifts toward a higher content of larger fibers and the improvement becomes more pronounced with increasing total volume fraction of reinforcement. Thus, hybrid reinforcement on different scales can be utilized to increase stiffness and suppress the thermal expansion of fiber composite. The effects observed can be used for composite microdesign and optimization of properties. Similar- ities of composite microstructures having synergistic 'extremal' properties with some biological composites [13] can be noted. The methodology developed can be utilized for connecting scales in modeling other hierarchical materials.

ACKNOWLEDGEMENTS

This work was funded by NSF, Grant DMI-9523022, and the Center for Materials Research and Analysis, UNL.

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