Q.H.Zeng et al Prog.Polym.SeL 33(2008)191-269 20 (iv)conservation of energy.based on the first law of such as aspect ratio:(iii)well-bonded filler-polymer thermodynamics:and (v)conservation of entrony interface and the ignorance of interfacial slip. based on the second law of thermodynamics.These filler-polymer debonding or matrix cracking. laws provide the basis for the continuum model and The first concept is the linear elasticity,i.e.. must be coupled with the appropriate constitutive linear relationship between the tota stres equations d the equatio state to provide al infinitesimal st nsors for the filler and s necessary for aimias'cpiedbyheoioSingconsimte equations mediun to the forces acting on the medium and the resulting For filler, =C8 (35) internal stress and strain.Computational an For matrix,o'm= (36) proaches range from simple closed-form analytical expressions to micromechanics and complex struc where Cis the stiffness tensor The second concent is the average stress and mechanics calculations based on beam and ory ods we int strain.Since the pointwise stress field a(x)and the corresponding strain field ax)are usually t have been use d in poly includ non-uniform in polymer composites,the volume- s mo average stress 6 and strain E are then defined ove ntin model,self-co the representative averaging volume V,respectively, and finite element analysis =1/. (37) 2.3.1.Micromechanics Since the assumption of uniformity in continuum (38) mechani at the microscale leve :-dv microme to exp Therefore the average filler and matrix stresses nu i- nfi are the averages over the corresponding volumes V and V.respectively. of th nt theme of micromechanics models is the develon (39) ment of a representative volume element (rve)to 1 ties. to ensure that the c(x)dv (40) length scale is consistent with the smallest constitu ent that has The average strains for the fillers and matrix are defined,respectively,as 61 1 (41) nstituents discontinuities and coupled mechanical and non-mechanical properties. purpose is to review the micromechanics methods 1 a(x)dv. (42) used for polymer nanocomposites.Thus,we only discuss here some important concepts of micro Based on the above definitions,the relationships ferred to mod re G=y可+miw (43) Basic concepts.When applied to ticle E=r十mnm (44) reinforced polvmer comnosites models usually follow such basic assumptions as where p (i)linear elasticity of fillers and polymer matrix: (ii)the fillers are axisymmetric,identical in shape erage properties 01 and size,and can be characterized by parameters
(iv) conservation of energy, based on the first law of thermodynamics; and (v) conservation of entropy, based on the second law of thermodynamics. These laws provide the basis for the continuum model and must be coupled with the appropriate constitutive equations and the equations of state to provide all the equations necessary for solving a continuum problem. The continuum method relates the deformation of a continuous medium to the external forces acting on the medium and the resulting internal stress and strain. Computational approaches range from simple closed-form analytical expressions to micromechanics and complex structural mechanics calculations based on beam and shell theory. In this section, we introduce some continuum methods that have been used in polymer nanocomposites, including micromechanics models (e.g., Halpin–Tsai model, Mori–Tanaka model), equivalent-continuum model, self-consistent model and finite element analysis. 2.3.1. Micromechanics Since the assumption of uniformity in continuum mechanics may not hold at the microscale level, micromechanics methods are used to express the continuum quantities associated with an infinitesimal material element in terms of structure and properties of the microconstituents. Thus, a central theme of micromechanics models is the development of a representative volume element (RVE) to statistically represent the local continuum properties. The RVE is constructed to ensure that the length scale is consistent with the smallest constituent that has a first-order effect on the macroscopic behavior. The RVE is then used in a repeating or periodic nature in the full-scale model. The micromechanics method can account for interfaces between constituents, discontinuities, and coupled mechanical and non-mechanical properties. Our purpose is to review the micromechanics methods used for polymer nanocomposites. Thus, we only discuss here some important concepts of micromechanics as well as the Halpin–Tsai model and Mori–Tanaka model. Interesting readers are referred to the literature [41] for details. 2.3.1.1. Basic concepts. When applied to particlereinforced polymer composites, micromechanics models usually follow such basic assumptions as (i) linear elasticity of fillers and polymer matrix; (ii) the fillers are axisymmetric, identical in shape and size, and can be characterized by parameters such as aspect ratio; (iii) well-bonded filler–polymer interface and the ignorance of interfacial slip, filler–polymer debonding or matrix cracking. The first concept is the linear elasticity, i.e., the linear relationship between the total stress and infinitesimal strain tensors for the filler and matrix as expressed by the following constitutive equations: For filler; sf ¼ Cf f , (35) For matrix; sm ¼ Cm m, (36) where C is the stiffness tensor. The second concept is the average stress and strain. Since the pointwise stress field s(x) and the corresponding strain field e(x) are usually non-uniform in polymer composites, the volume– average stress s¯ and strain ¯ are then defined over the representative averaging volume V, respectively, s¯ ¼ 1 V Z V sðxÞ dV, (37) ¯ ¼ 1 V Z V ðxÞ dV. (38) Therefore, the average filler and matrix stresses are the averages over the corresponding volumes Vf and Vm, respectively, s¯f ¼ 1 Vf Z Vf sðxÞ dV, (39) s¯ m ¼ 1 Vm Z Vm sðxÞ dV. (40) The average strains for the fillers and matrix are defined, respectively, as ¯f ¼ 1 Vf Z Vf ðxÞ dV, (41) ¯m ¼ 1 Vm Z Vm ðxÞ dV. (42) Based on the above definitions, the relationships between the filler and matrix averages and the overall averages can be derived as follows: s¯ ¼ vf s¯f þ vms¯ m, (43) ¯ ¼ vf¯f þ vm¯m, (44) where vf, vm are the volume fractions of the fillers and matrix, respectively. The third concept is the average properties of composites which are actually the main goal of a ARTICLE IN PRESS Q.H. Zeng et al. / Prog. Polym. Sci. 33 (2008) 191–269 201
202 O.H.Zeng et al.Prog.Polvm.Sci.33 (2008)191-269 micromecha of th (lower bound). (52) E C8. (45) Conversely,.ifξ→o,the theory reduces to the The average compliance S is defined in the same rule of mixtures (upper bound). way: E=vEy+(1-v)Em (53) 龙=Sa (46) Another important concept is the strain-concen tration and stress-concentration tensors A and B 23.1.3.MoriTanaka model The Mori-Tanaka model is derived based on the princinles of which are basically the ratios between the average filler strain Eshelby's inclusion model for predicting an elastic (or stress)and the corresponding average of the composites stress field in and around an ellipsoidal filler in an infinite matrix.The complete analytical solutions 可=A (47) or d with aligned =Ba. (48) spherica inclusion are [42.43] Using the concepts and e E e above E A0+y(1+204 (54) and the filler and E22 Γ2A0+y(-2A3+(1-o)A4+(1+o)AsA0 C=Cm Dy(Cf-Cm)A. (49 (55) where Em represents the Young's modulus of the 10 l2HirTaimTeHepo-Taimod isson' el stiffness of unidirectionalc as a functional and thes ties of the filler and the matrix of aspect ratio.In this model.the longitudinal E including Young's modulus.Poisson's ratio.filler concentration and filler aspect ratio 1421. and transverse E22 engineering moduli are expressed in the following general form 2.3.2.Equicalent-continuum and self-similar (50) approaches Numerous micromechanical models have been where E and E represent the Young's modulus of successfully used to predict the macroscopic beha the composite and orc Howe matrix,respectively,t is the direct useof volume fraction of filler.and n is given by E/Em-1 n nanotuhe and t ical carbo = (51) fiher Recently two methods have b Er/Em for modeling the mechanical behavior of single where E,represents the Young's modulus of the walled carbon nanotube (SWCN)composites: filler and the shape p rameter depending on the equivalent-continuum approach and self-similar filler geometry and loading direction.Wher n calcu approach [44] he alent nwas pr en ca sverse mod by C 43 this appr equ H n wa ole h ne reinford theory converges to the inverse rule of mixture SWCN surrounded by a cylindrical volume
micromechanics model. The average stiffness of the composite is the tensor C that maps the uniform strain to the average stress s¯ ¼ C¯. (45) The average compliance S is defined in the same way: ¯ ¼ Ss¯. (46) Another important concept is the strain–concentration and stress–concentration tensors A and B which are basically the ratios between the average filler strain (or stress) and the corresponding average of the composites. ¯f ¼ A¯, (47) s¯f ¼ Bs¯. (48) Using the above concepts and equations, the average composite stiffness can be obtained from the strain–concentration tensor A and the filler and matrix properties: C ¼ Cm þ vf ðCf CmÞA. (49) 2.3.1.2. Halpin– Tsai model. The Halpin–Tsai model is a well-known composite theory to predict the stiffness of unidirectional composites as a functional of aspect ratio. In this model, the longitudinal E11 and transverse E22 engineering moduli are expressed in the following general form: E Em ¼ 1 þ xZnf 1 Znf , (50) where E and Em represent the Young’s modulus of the composite and matrix, respectively, vf is the volume fraction of filler, and Z is given by Z ¼ E=Em 1 Ef =Em þ xf , (51) where Ef represents the Young’s modulus of the filler and xf the shape parameter depending on the filler geometry and loading direction. When calculating longitudinal modulus E11, xf is equal to l/t, and when calculating transverse modulus E22, xf is equal to w/t. Here, the parameters of l, w and t are the length, width and thickness of the dispersed fillers, respectively. If xf-0, the Halpin–Tsai theory converges to the inverse rule of mixture (lower bound), 1 E ¼ nf Ef þ 1 nf Em . (52) Conversely, if x-N, the theory reduces to the rule of mixtures (upper bound), E ¼ nf Ef þ ð1 nf ÞEm. (53) 2.3.1.3. Mori– Tanaka model. The Mori–Tanaka model is derived based on the principles of Eshelby’s inclusion model for predicting an elastic stress field in and around an ellipsoidal filler in an infinite matrix. The complete analytical solutions for longitudinal E11 and transverse E22 elastic moduli of an isotropic matrix filled with aligned spherical inclusion are [42,43]: E11 Em ¼ A0 A0 þ nf ðA1 þ 2u0A2Þ , (54) E22 Em ¼ 2A0 2A0 þ nf ð2A3 þ ð1 u0ÞA4 þ ð1 þ u0ÞA5A0Þ , (55) where Em represents the Young’s modulus of the matrix, nf the volume fraction of filler, n0 the Poisson’s ratio of the matrix, parameters, A0, A1,y, A5 are functions of the Eshelby’s tensor and the properties of the filler and the matrix, including Young’s modulus, Poisson’s ratio, filler concentration and filler aspect ratio [42]. 2.3.2. Equivalent-continuum and self-similar approaches Numerous micromechanical models have been successfully used to predict the macroscopic behavior of fiber-reinforced composites. However, the direct use of these models for nanotube-reinforced composites is doubtful due to the significant scale difference between nanotube and typical carbon fiber. Recently, two methods have been proposed for modeling the mechanical behavior of singlewalled carbon nanotube (SWCN) composites: equivalent-continuum approach and self-similar approach [44]. The equivalent-continuum approach was proposed by Odegard et al. [45]. In this approach, MD was used to model the molecular interactions between SWCN–polymer and a homogeneous equivalent-continuum reinforcing element (e.g., a SWCN surrounded by a cylindrical volume of ARTICLE IN PRESS 202 Q.H. Zeng et al. / Prog. Polym. Sci. 33 (2008) 191–269
Q.H.Zeng et al Prog.Polym.SeL 33(2008)191-269 203 Molecular model 2RsHnewoctoriar,iakatnsandoquiakaommumodhsReprotacodiomotkeard polymer)was constructed as shown in Fig.2.Then. equivalent-truss model of the RVe is used as micromechanics are used to determine the effective an intermediate step to link the molecular and bulk properties of the equivalent-continuum rein- equivalent-continuum models.Each atom in the forcing element embedded in a continuous polymer. molecular model is represented by a pin-joint,and The equivalent-continuum approach consists of each truss element represents an atomic bonded or iefly described below ed to generate the The ocntaleey of the U'=∑U(E)+∑U(E)+∑U(E),(57 model and the equivalent-continuum model. Step 2:The potential energies of deformation for where and U are the energies associated the molecular model and effective fiber are derived with truss elements that repres ovalent hond and equated for identical loading conditions.The stretching,bond-angle bending.and van der Waals bonded and non-bonded interactions within a interactions,respectively.The energies of each truss element are a function of the Young's modulus the er system. tota E.The potential energy (or strain energy)of the homogeneous,linear-elastic.effective fiber is Um=∑Urk,)+∑Uka)+∑U(k), U=U(C). (58) (56 where Cis the elastic stiffness tensor of the effective where U,U and U are the energies associated fiber.Equating Eqs.(56)(58)yields with covalent bond stretching,bond-angle bending. and van der Waals interactions,respectively.An Ul=U!=Um. (59
polymer) was constructed as shown in Fig. 2. Then, micromechanics are used to determine the effective bulk properties of the equivalent-continuum reinforcing element embedded in a continuous polymer. The equivalent-continuum approach consists of four major steps, as briefly described below. Step 1: MD simulation is used to generate the equilibrium structure of a SWCN–polymer composite and then to establish the RVE of the molecular model and the equivalent-continuum model. Step 2: The potential energies of deformation for the molecular model and effective fiber are derived and equated for identical loading conditions. The bonded and non-bonded interactions within a polymer molecule are quantitatively described by MM. For the SWCN/polymer system, the total potential energy Um of the molecular model is Um ¼ XUr ðkrÞ þXUy ðkyÞ þXUvdwðkvdwÞ, (56) where Ur , Uy and Uvdw are the energies associated with covalent bond stretching, bond-angle bending, and van der Waals interactions, respectively. An equivalent-truss model of the RVE is used as an intermediate step to link the molecular and equivalent-continuum models. Each atom in the molecular model is represented by a pin-joint, and each truss element represents an atomic bonded or non-bonded interaction. The potential energy of the truss model is Ut ¼ XUa ðEa Þ þXUb ðEb Þ þXUc ðEc Þ, (57) where Ua , Ub and Uc are the energies associated with truss elements that represent covalent bond stretching, bond-angle bending, and van der Waals interactions, respectively. The energies of each truss element are a function of the Young’s modulus, E. The potential energy (or strain energy) of the homogeneous, linear-elastic, effective fiber is Uf ¼ Uf ðCf Þ, (58) where Cf is the elastic stiffness tensor of the effective fiber. Equating Eqs. (56)–(58) yields Uf ¼ Ut ¼ Um. (59) ARTICLE IN PRESS Fig. 2. Representative volume elements of molecular, equivalent-truss and equivalent-continuum models [44]. Reproduced from Odegard, Ripes and Hubert by permission of Elsevier Science Ltd. Q.H. Zeng et al. / Prog. Polym. Sci. 33 (2008) 191–269 203
204 Q.H.Zeng et al Prog.Polym.Sci.33(2008)191-269 1.93x105m mear16xi0 SWCN SWCN micro-fiber 169x107m SWCN nano-wire SWCN 1.48x108m SWCN nano-array 1.38×10°m SWCN gwSnmy19x1o SWCN Eq.(59)relates the elastic stiffness tensor of the wire.Finally,the SWCN nanowires are further effective fiber to the force constants of the molecular impregnated with a polymer matrix and assembled model. the SWCN microfiber 10a1 priori.a set of loading condi tion s are che all three geometries [461. such that each component is uniquely determined from Eq.(59). 233 finite element method Step 4:Overall constitutive properties of the FEM is a general numerical method for obtaining dilute and unidirectional SWCN/polymer compo approximate solutions in space to initial-value anc site are determined with Mor -Tanaka mode boundary-value problems including time-dependent anica me or po yme ully capture the spata (Fig.2)is included in thee ffective fiber and it is It also allows com nonlinear tensile relation assumed that the matrix olvmer surrounding the shins to he inco rated into the analysis.Thus.it effective fiber has mechanical properties equal to has been widely used in mechanical,biological and those of the bulk polymer. geological systems. The self-similar In FEM,the entire domain of interest is spatially discretized into an assembly of simply shaped First,a helical array Thi xahedra or tetr ane ano nens in two tho or t gaps an the SWCN r ma trix and ass mbled into a b des The i on of FEM includes the second twisted array,termed as the SWCN nano- important steps 、shown in Fig..4
Eq. (59) relates the elastic stiffness tensor of the effective fiber to the force constants of the molecular model. Step 3: A constitutive equation for the effective fiber is established. Since the values of the elastic stiffness tensor components are not known a priori, a set of loading conditions are chosen such that each component is uniquely determined from Eq. (59). Step 4: Overall constitutive properties of the dilute and unidirectional SWCN/polymer composite are determined with Mori–Tanaka model with the mechanical properties of the effective fiber and the bulk polymer. The layer of polymer molecules that are near the polymer/nanotube interface (Fig. 2) is included in the effective fiber, and it is assumed that the matrix polymer surrounding the effective fiber has mechanical properties equal to those of the bulk polymer. The self-similar approach was proposed by Pipes and Hubert [46] which consists of three major steps. First, a helical array of SWCNs is assembled. This array is termed as the SWCN nanoarray where 91 SWCNs make up the cross-section of the helical nanoarray. Then, the SWCN nanoarrays is surrounded by a polymer matrix and assembled into a second twisted array, termed as the SWCN nanowire. Finally, the SWCN nanowires are further impregnated with a polymer matrix and assembled into the final helical array—the SWCN microfiber. The self-similar geometries described in the nanoarray, nanowire and microfiber (Fig. 3) allow the use of the same mathematical and geometric model for all three geometries [46]. 2.3.3. Finite element method FEM is a general numerical method for obtaining approximate solutions in space to initial-value and boundary-value problems including time-dependent processes. It employs preprocessed mesh generation, which enables the model to fully capture the spatial discontinuities of highly inhomogeneous materials. It also allows complex, nonlinear tensile relationships to be incorporated into the analysis. Thus, it has been widely used in mechanical, biological and geological systems. In FEM, the entire domain of interest is spatially discretized into an assembly of simply shaped subdomains (e.g., hexahedra or tetrahedral in three dimensions, and rectangles or triangles in two dimensions) without gaps and without overlaps. The subdomains are interconnected at joints (i.e., nodes). The implementation of FEM includes the important steps shown in Fig. 4. ARTICLE IN PRESS Fig. 3. Self-similar scales (left) and number of single-walled carbon nanotube per meter length [44]. Reproduced from Odegard, Ripes and Hubert by permission of Elsevier Science Ltd. 204 Q.H. Zeng et al. / Prog. Polym. Sci. 33 (2008) 191–269
Q.H.Zeng et al Prog.Polym.SeL 33(2008)191-269 205 rate of change of the displacement field i,and the mass density p.The strains are related to the displacements according to + (63) Describe the problem under investig ation by using variational principle These are fields defined throughout space in the continuum theory Thus the total enerey of constitutive laws the system is an integral of these quantities over the volume of the sample de.The FEM has been Divide the system eover individu nents Replace the coeraby interpolation proper te ha oly the FEM ap ment sti sites.In order to capture the multiscale material behaviors,efforts are also underway to combine the multiscale models spanning from molecular to macroscopic levels [48,491. 3.Modeling and simulation of polymer aking into count the boundar nanocomposites tion 3.1.Nanocomposite thermodynamics Calculate the state equation values from state variables Fig.4.Important steps in the finite element mcthod The formation of stable nanocomposites depends on the thermodynamics of the multicomponent mixture concerned.In the case of polymer-nano- The energy in FEM is taken from the theory of particle mixtures,their final structures are strongly linear elasticity and thus the input parameters are influenced by the characteristics of nanoparticles (e.g.,size, shape, aspect ratio),polymer (e.g.. P .the nd bol and nts compa by M observed from the s ructure in whichn rticle are uniformly dispersed in polymer matrix.There. given by [47] fore,in order to achieve such polymer nanocompo- sites and the maximal property improvement.it is U=Uv+Uk (60) very important to determine thoroughly the effects of various factors on the final struct (61) ate the thermodynamic con ditions fo the st Exper at a nan oscale -fordoup (62) Computer modeling and simulations have shown great success in colloid-polymer solutions.How- where Ur is the Hookian potential energy term ever,the colloid particles in such systems are which is quadratic in the symmetric strain tensor s. much larger than the gyration radius of a polymer. contracted with the elastic constant tensor C.The These methods may not directly be extendable to
The energy in FEM is taken from the theory of linear elasticity and thus the input parameters are simply the elastic moduli and the density of the material. Since these parameters are in agreement with the values computed by MD, the simulation is consistent across the scales. More specifically, the total elastic energy in the absence of tractions and body forces within the continuum model is given by [47] U ¼ UV þ UK , (60) UV ¼ 1 2 Z dr X 3 m;n;l;s¼1 mnðrÞCmnlslsðrÞ, (61) UK ¼ 1 2 Z drrðrÞ u_ðrÞ 2 , (62) where UV is the Hookian potential energy term which is quadratic in the symmetric strain tensor e, contracted with the elastic constant tensor C. The Greek indices (i.e., m, n, l, s) denote Cartesian directions. The kinetic energy UK involves the time rate of change of the displacement field u_, and the mass density r. The strains are related to the displacements according to mn ¼ qum qrn þ qun qrm . (63) These are fields defined throughout space in the continuum theory. Thus, the total energy of the system is an integral of these quantities over the volume of the sample dv. The FEM has been incorporated in some commercial software packages and open source codes (e.g., ABAQUS, ANSYS, Palmyra and OOF) and widely used to evaluate the mechanical properties of polymer composites. Some attempts have recently been made to apply the FEM to nanoparticle-reinforced polymer nanocomposites. In order to capture the multiscale material behaviors, efforts are also underway to combine the multiscale models spanning from molecular to macroscopic levels [48,49]. 3. Modeling and simulation of polymer nanocomposites 3.1. Nanocomposite thermodynamics The formation of stable nanocomposites depends on the thermodynamics of the multicomponent mixture concerned. In the case of polymer–nanoparticle mixtures, their final structures are strongly influenced by the characteristics of nanoparticles (e.g., size, shape, aspect ratio), polymer (e.g., molecular weight, structure, polarity and its compatibility with the particle), and surfactant if employed. The remarkably improved properties are usually observed from the structure in which nanoparticles are uniformly dispersed in polymer matrix. Therefore, in order to achieve such polymer nanocomposites and the maximal property improvement, it is very important to determine thoroughly the effects of various factors on the final structure and then isolate the thermodynamic conditions for the stable and uniform dispersion of nanoparticles. Experimentally, it is very difficult to ascertain these issues at a nanoscale. Computer modeling and simulations have shown great success in colloid–polymer solutions. However, the colloid particles in such systems are much larger than the gyration radius of a polymer. These methods may not directly be extendable to ARTICLE IN PRESS Replace the continuum domain with an assemblage of subdomains (i.e., finite elements) Select the appropriate constitutive laws Select interpolation functions necessary to map the element topology and to approximate the course of state variable in the elements Describe the problem under investigation by using variational principle Select the appropriate constitutive laws Divide the system level integral into subintegrals over individual elements Replace the continuum state variables in integral by interpolation functions Assemble system of element equations containing element stiffness, load, and unknown variable Assemble system of global equations from element equations containing system stiffness, load, and unknown variable Solve global system of equations, taking into account the boundary conditions Calculate the state equation values from state variables Fig. 4. Important steps in the finite element method. Q.H. Zeng et al. / Prog. Polym. Sci. 33 (2008) 191–269 205