VISCOELASTIC PROPERTIES OF INHOMOGENEOUS NANOCOMPOSITES

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1 VISCOELASTIC PROPERTIES OF INHOMOGENEOUS NANOCOMPOSITES V. V. Novikov ), K.W. Wojciechowski ) ) Odessa National Polytechnical University, Shevchenko Prosekt, 6544 Odessa, Ukraine; novikov@te.net.ua ) Institute of Molecular Physics, Polish Academy of Sciences, M. Smoluchowskiego 7, 6-79 Poznań, Poland; ABSTRACT Hierarchic model of structure [V. V. Novikov, K. W. Wojciechowski, D. V. Belov and V. P. Privalko, Phys. Rev. E 63, 36 ()] is generalized and alied to study viscoelastic roerties of a two-comonent inhomogeneous medium with chaotic, fractal structure and with one of the comonents exhibiting a negative shear modulus. It is shown that similarly to the results obtained recently in frames of the Hashin-Strikman model [R. S. Lakes, Phys. Rev. Let. 86, 897 ()], the resent model redicts ossibility to obtain comosites of the effective shear and duming coefficient much higher than those characterizing both the comonent hases. The viscoelastic roerties of the fractal medium are, however, qualitatively different from the roerties of the Hashin-Strikman medium.. INTRODUCTION Recent studies of inhomogeneous materials with inclusions of negative stiffness indicated that such comosites show very interesting roerties, e.g. they can exhibit much higher stiffness and higher duming coefficient than the hases constituting them [-3]. Analysis of influence of inclusions with negative shear modulus on the effective shear modulus of a comosite was erformed on the basis of the Hashin-Strikman formulae [4, 5] which were obtained in aroach that roerties don t deend on scale. Recently, however, increasing attention has been aid to hysical roerties of media which can be thought of as fractal structures in some size range. Examles of such structures are article aggregates in colloids [6-8], olymer molecules, ercolationg clusters, structures of some binary solutions and olymers [9-], and structures roduced in rocesses of diffusion-controlled aggregation (olymerization) [3-5]. One should add that media of fractal-like structures show qualitatively different roerties than those characteristic for gases, liquids and solids. Let us consider elastic roerties of inhomogeneous medium of chaotic, fractal structure which one of the hases shows negative shear modulus. (One should stress here that any ure hase of such a roerty is, by definition, unstable. It can be stabilized, however, when inclusions of it are ut into a stabilizing matrix comosed of a hase exhibiting ositive shear modulus [-3].) The analysis is carried on by the method described in [6-].. FRACTAL MODEL OF VISCOELASTIC PROPERTIES OF CHAOTIC STRUCTURE The resent analysis is erformed on the basis of a hierarchical model of twocomonent structure and is a generalization of the method of elastic roerties calculations described in detail in [8-]. In the modeling of chaotic structures resented there, hierarchical lattices of random distribution of arameters are alied. The main set of bond configurations, Ω, is obtained by an iterative rocedure. In the initial ste of the rocedure, a finite lattice is considered with bonds of length l and robability that a given bond belongs to the first hase. In the following stes (k =,,...) each bond of the initial lattice is substituted by an examle of a lattice obtained in the revious ste. In contrast to the case considered in [], where the bonds of the lattice reresented urely elastic roerties, the bonds considered in the resent aer may reresent more general, viscoelastic case (see Fig. ). The limiting set of bonds, Ω( ), which deends on the size of the initial lattice and the l,

2 robability, is self-similar, i.e. it constitutes a fractal [6,8-]. (In ractical calculations, the iterative rocedure is finished when the roerties of the lattice become indeendent of the index k.) Two ensembles of structures were introduced in [8-]. The first one is the bonded ensemble (BE), being the set of all configurations in which bonds reresenting the first hase connect oosite surfaces of the lattice. The second one, the non-bonded ensemble (NBE) constitutes of the remaining configurations, for which bonds of the first hase do not connect the oosite surfaces of the lattice. According to [], in calculations of the elastic roerties one exloits simle analytic formulae, in articular the robability function, Rl (, ), and deendencies of the elastic roerties of the BE and NBE configurations on roerties and concentrations of hases forming the considered inhomogeneous medium. Figure. Illustration of the self-similarity of the fractal system Ω n (L, =3/8 ) (where n ) of viscoelastic bonds for L =. The robability function, Rl (, ), is defined as the ratio of the number of the BE configurations to the number of all configurations for a cubic initial lattice. The numerical analysis [] of initial lattices of various sizes showed that good results can be obtained for the lattice as small as xx for which [] R ( ) = ( , () )

3 According to () the ercolation threshold, i.e. the transition from the NBE to the BE, occurs at c = The deendencies of the elastic roerties of the BE and NBE configurations on roerties and concentrations of hases forming the considered inhomogeneous medium were modeled by the blob model [8- ]. In this aroach each BE configuration is treated as a continuum of the first hase with a sherical inclusion of the second hase (see Fig.a), and each NBE configuration is treated as a sherical inclusion of the first hase in a continuum of the second hase (see Fig.b). The elastic roerties of configurations corresonding to the BE and NBE were calculated by alying the Hashin-Shtrikman formulae [4,5,,] ). Fig.. Simulation of (a) a connected and (b) a disconnected set Hashin-Strikman formulae were obtained on the basis of the rincile of minimum of addition energy by using variational calculation method for an inhomogeneous medium [Z. Hashin, The elastic moduli of heterogeneous materials, J. Alied Mechanics,9,43-5 (96).] to determine the uer К с, µ с and the lower К n, µ n bounds of effective elastic roerties. The uer bound К с, µ с corresonds to the comosite structure in which sherical inclusions of the elastic constants К, µ. are laced into a matrix of the elastic constants К, µ ; in the following, it is assumed that К >K, µ > µ. The lower bound К n, µ n is obtained in the when the comonents are relaced, i.e. when the matrix is described by К, µ and the sherical inclusions by К, µ. From the Hashin-Strikman sheres (inside a shere of one material a shere of the other material is laced centrally) one can form a comosite as follows [, ]: sheres of various sizes, down to infinitely small, are taken and the sace V is filled by them without emty saces. Only one condition is required: in each Hashin-Strikman shere the volume concentrations of both comonents are the same, i.e. all the Hashin- Strikman sheres exhibit the same elastic roerties []. Such a comosite will be further referred to as the Hashin-Strikman comosite. Elastic roerties of the Hashin-Strikman comosite are described by the formulae which are obtained basing on the exactly solvable model consisted of a single sherical inclusion of one hase in an infinite matrix of the second hase []. The elastic roerties of the Hashin-Strikman comosite deend only on the volume concentrations and elastic roerties of the hases forming it; they do not deend on the scale. Inthis sense the Hashin-Strikman comosite can be thought of as uniform. In contrast to the Hashin-Strikman comosite real comosites are not uniform in this sense as their roerties do deend on the scale on which they are measured. One exects that the macroscoic roerties of such non-uniform comosites should be obtained from the microscoic ones by a roer averaging. In [8-] a simle averaging scheme, based on the idea of the renormalization grou and the blob model, was roosed. The non-uniform comosites for which this scheme works will be further referred to as fractal comosites. Results obtained for static elastic roerties corresond to results for viscoelastic roerties (for arising harmonic vibrations) obtained by relacing the real elastic modules, (the bulk modulus) and (the shear modulus), by the comlex modules K *, µ * [-4]. Alying µ

4 this corresondence one obtains the following formulae for the comlex bulk modulus, K*, and c the comlex shear modulus, µ *, for configurations belonging to the BE at the (k+)-th iteration c ste [4,5]: ( k) ( k) *( k+ ) ( k) ( k )( K * * ) * n K K c c = K c +, () ( k) ( k) ( k) + a k c ( K* n K* c ) () k () k *( k+ ) ( i) ( k )( µ * * ) * n µ µ c c = µ c +, (3) () k () k () k + b k c ( µ * n µ * c ) where 6( * () k * () k () k 3 () ) ; k K c + µ c (4) ac = b 3 * () 4 * () c = K k k 5 K* () k (3 K* () k 4 * () k с + µ c c c + µ c ) and () () K* = K*, µ * = µ * denote the comlex bulk modulus and the comlex shear modulus of c c the first hase of the inhomogeneous medium, and () () K* = K*, µ * * n n = µ denote the comlex bulk modulus and the comlex shear modulus of the second hase, resectively. ( k+ ) ( k+ ) For non-bonded configurations the viscoelastic modules K * n, µ * n are described by the formulae which come from (), (3) after the following relacements c n and (- ). k k 3. RESULTS & DISCUSSION Calculations were erformed for a two-comonent, inhomogeneous medium. For simlicity, it has been assumed that both the hases are isotroic and the first hase is urely elastic whereas the second hase is elastic from the oint of view of volume deformations and viscoelastic from the oint of view of shear deformations. The concentration of the urely elastic hase is denoted by. It is convenient to write the shear modulus of the second hase, µ *, in the form µ * = x( + iy) µ ', (5) where y= tgϕ ( ) = µ " / µ ', µ ' / µ ' = x, and µ * = µ ' is the (real) shear modulus of the first hase. As it has been mentioned before, fractal structures can show roerties different from uniform structures. To illustrate the oint let us comare the effective shear modulus and the duming coefficient of an inhomogeneous medium with fractal structure (further referred to as a fractal comosite ) with a comosite material corresonding to the Hashin-Strikman formulae (further referred to as the Hashin-Strikman comosite ).

5 mm mm mm HeL.88.9 HaL.6.8.4x HcL HdL mm mm mm HbL.8 HfL.8.4x Figure 3. Comarison of the ratio of the effective shear modulus to the shear modulus of the elastic hase at y=. as a function of the concentration of the elastic hase and x :in the fractal comosite (a), (b), (c) and (d), (e), (f) in the Hashin-Strikman comosite. In Fig.3, the ratio of the effective shear modulus to the shear modulus of the elastic hase is shown as a function of the concentration of the elastic hase and x for y= -3. It is assumed that the viscoelestic hase in Fig. 4 has a negative shear modulus (i.e. negative real art of the comlex shear modulus) and is characterized by y= tgϕ =. and the Poisson s ratios of both hases (calculated from real arts of the elastic moduli) are equal to (The latter

6 assumtion means that the ratio of the real art of the shear moduls to the bulk modulus is equal to µ i /K i =.8, where i =, numerates the hases; in consequence the ratio of the bulk modulus of the second hase to the bulk modulus of the first hase is equal to K '/ K ' = x.) It can be seen there that for the inhomogeneous fractal medium the shear modulus shows in some ranges of concentration a resonance-like behavior similar to that discussed in [-3] whereas in the Hashin-Strikman comosite there exists only one such a resonance in the vicinity of the concentration =. HaL HbL tg j.6.4. t g j.6.8.4x HcL HdL HeL.4. t g j tgj HfL x. 6 6 t g j 4 t g j Figure 4. The ratio of the imaginary to the real art of the effective shear modulus (a), (b), (c) in the fractal comosite and (d), (e), (f) in the Hashin-Strikman comosite; the other details are the same as in Fig. 3.

7 As it has been mentioned before, fractal structures can show roerties different from uniform structures. To illustrate the oint let us comare the effective shear modulus and the duming coefficient of an inhomogeneous medium with fractal structure (further referred to as a fractal comosite ) with a comosite material corresonding to the Hashin-Strikman formulae (further referred to as the Hashin-Strikman comosite ). In Fig.3, the ratio of the effective shear modulus to the shear modulus of the elastic hase ' ' is shown as a function of the concentration of the elastic hase and x = µ / µ for y= -3. It is assumed that the viscoelestic hase in Fig. 3 has a negative shear modulus (i.e. negative real -3 art of the comlex shear modulus) and is characterized by y= tgϕ = and the Poisson s ratios of both hases (calculated from real arts of the elastic moduli) are equal to (The latter assumtion means that the ratio of the real art of the shear moduls to the bulk modulus is equal to µ i /K i =.8, where i =, numerates the hases; in consequence the ratio of the bulk modulus of the second hase to the bulk modulus of the first hase is equal to x = K '/ K '.) It can be seen there that for the inhomogeneous fractal medium the shear modulus shows in some ranges of concentration a resonance-like behavior similar to that discussed in [-3] whereas in the Hashin-Strikman comosite there exists only one such a resonance in the vicinity of the concentration =. In Fig.4 the ratio of the imaginary to real art of the effective shear modulus, i.e. the tangent of loses tg ϕ, is shown as a function of the concentration of the elastic hase and x for y= -3. The arameters of the hases are the same as in Fig.3. It can be seen again that, deending on x, tg ϕ of the fractal comosite shows very large values in some ranges of concentration whereas the Hashin-Strikman comosite exhibits only one large value, in the vicinity of the concentration =, for x in the range considered. The above comarisons show that the viscoelastic roerties of the fractal comosite differ qualitatively from the roerties of the Hashin-Strikman comosite. The observed differences can be understood taking into account that the hierarchical model considered takes into account clusters of various length scales and different arameters of their resonances which are formed in the fractal comosite whereas the Hashin-Strikman comosite is uniform in this asect. The above comarisons show that the viscoelastic roerties of the fractal comosite differ qualitatively from the roerties of the Hashin-Strikman comosite. The observed differences can be understood taking into account that the hierarchical model considered takes into account clusters of various length scales and different arameters of their resonances which are formed in the fractal comosite whereas the Hashin-Strikman comosite is uniform in this asect. It follows from the calculations of the viscoelastic roerties (Figs.3, 4) that in a fractal comosite eaks ( resonances ) are obtained in a broad range of concentrations of hases. The nature of the eaks and the way in which they are formed deending on the effective shear modulus µ *, the frequency ω, and the concentration one can understand considering a single inclusion with the negative shear modulus ( µ * = µ ' ( µ ' = xk) ) which is immersed in a medium (stabilizing matrix) of the ositive shear modulus µ * = µ '. The shear modulus µ * of the comosite with a single inclusion can be determined by the formula [4, 5, ] where µ * µ * ( µ * µ * ) = + + ( b ) ( µ *, (6) µ * )

8 b 6( K * + µ * ) = 5 µ * (3 K * + 4 µ * ). It follows from (6) that when µ * = µ ' ( µ ' = xk), then µ * µ ' From the equation xk ( + µ ' ) = ( b ) ( xk +. (7) µ ' ) ( b ) ( xk + µ ' ) =, (8) one can determine the resonance arameters of the comosite for which the eaks arise, i.e. the arameters for which the external disturbance is in the resonance with the inclusion arameters (being equal to its characteristic frequency). As the comosite constitutes self-similar, chaotic system, comosed of clusters of various sizes, on the k-th scale-level each cluster will have its own resonance arameters, i.e. its own characteristic frequency, which roduces the system of eaks (characteristic frequencies) in the deendence of the effective shear modulus on the arameters of the comosite. At this oint let us notice that a material of fractal structure which should show the viscoelastic roerties similar to those obtained by solving the model described can be manufactured in reality by the following scheme. At the first ste (the lowest size level) one roduces tablets, e.g. of a olymer with required inclusions. At the next ste, the tablets obtained at the receding level are used as inclusions to larger tablets. The rocess is continued and a hierarchy shown in Fig.5 is obtained. Figure 5. The idea of construction of a material with fractal structure. It has been shown that a hierarchic blob model used to study viscoelastic roerties of an inhomogeneous fractal medium (the fractal comosite) gives results, which differ qualitatively from the results obtained by alying to an inhomogeneous medium the Hashin-Shrikman aroximation (the Hashin- Strikman comosite). In articular, studies of the fractal model comosed of an elastic hase (which can be seen as a stabilizing matrix) and a viscoelastic hase with a negative shear modulus roved that the effective shear modulus and the effective tangent of loses calculated in this model exhibit much more comlex behavior (more singularities ) than those redicted by the standard Hashin-Strikman model. The new singularities observed in the fractal comosite are interreted as resonances coming from ( mesoscoic ) clusters of various length scales which are described by different (mesoscoic) arameters of resonances. Such clusters are taken into account by the hierarchical model whereas they are neglected comletely in the standard Hashin-Strikman aroximation. At the end of this section let us notice that a material of fractal structure which should show the viscoelastic roerties similar to those obtained by solving the model described can

9 be manufactured in reality by the following scheme. At the first ste (the lowest size level) one roduces tablets, e.g. of a olymer with required inclusions. At the next ste, the tablets obtained at the receding level are used as inclusions to larger tablets. The rocess is continued and a hierarchy shown in Fig.8 is obtained. CONCLUSIONS It has been shown that a hierarchic blob model used to study viscoelastic roerties of an inhomogeneous fractal medium (the fractal comosite) gives results, which differ qualitatively from the results obtained by alying to an inhomogeneous medium the Hashin-Shrikman aroximation (the Hashin-Strikman comosite). In articular, studies of the fractal model comosed of an elastic hase (which can be seen as a stabilizing matrix) and a viscoelastic hase with a negative shear modulus roved that the effective shear modulus and the effective tangent of loses calculated in this model exhibit much more comlex behavior (more singularities ) than those redicted by the standard Hashin-Strikman model. The new singularities observed in the fractal comosite are interreted as resonances coming from ( mesoscoic ) clusters of various length scales which are described by different (mesoscoic) arameters of resonances. Such clusters are taken into account by the hierarchical model whereas they are neglected comletely in the standard Hashin-Strikman aroximation. ACKNOWLEDGEMENTS This work was suorted by the KBN grant 4TF3.and by the National Academy of Science of Ukraine. References. Lakes R. S., «Extreme daming in comosite materials with a negative stiffness hase.» Phys. Rev. Let. 86,. () Lakes RS, Lee T, Bersie A, Wang YC. «Extreme daming in comosite materials with negative-stiffness inclusions.» Nature, 4, (), Lakes RS. «Extreme daming in comliant comosites with a negative-stiffness hase» Philosohical Magazine Letters, 8, (), Hashin Z., Shtrikman S., «A variational aroach to the theory of the elastic behavior of multihase materials.» J. Mech. Phys. Solids (963)., Hashin Z, J. «Viscoelastic behavior of heterogeneous media.» J. Al. Mech. 3 (965), Weitz DA, Oliveria M. «Fractal structures formed by kinetic aggregation of aqueous gold colloids.» Physical Review Letters,.5,( 984), Feder J, Jossang T, Rosenqvist E. «Scaling behavior and cluster fractal dimension determined by light scattering from aggregating roteins.» Physical Review Letters 53, (984), Lin MY, Lindsay HM, Weitz DA, Ball RC, Klein R, Meakin P. «Universality of fractal aggregates as robed by light scattering.» Proceedings of the Royal Society of London Series A-Mathematical &Physical Sciences 43, (989), Stauffer D., Introduction to the Percolation Theory, (Taylor & Francis, London-Philadelhia, 985).. Sahimi M., Alications of Percolation Theory, (Taylor and Francis, London, 994).. Privalko V. P., Novikov V. V., The Science of Heterogeneous Polymers. Structure and Thermohysical Proerties, (J. Wiley, Chichester-New York-Brisbane-Toronto-Singaore, 995) Elam WT, Wolf SA, Srague J, Gubser DU, Van Vechten D, Barz GL Jr, Meakin P. «Fractal aggregates in sutter-deosited NbGe/sub / films.» Physical Review Letters 54, (985), Witten TA, Sander LM. «Diffusion-limited aggregation.» Physical Review B-Condensed Matter 7( 983), Meakin P. «Formation of fractal clusters and networks by irreversible diffusion-limited aggregation». Physical Review Letters 5, (983), Vicsek T. «Pattern formation in diffusion-limited aggregation.» Physical Review Letter 53, ( 984), Novikov V. V., Wojciechowski K. W., Belov D.V. and PrivalkoV. P. «Elastic roerties of inhomogeneous media with chaotic structure» Phys. Rev. E63, 36 (). 7. Novikov V. V, V.P. Privalko. «Temoral fractal model for the anomalous dielectric relaxation of inhomogeneous media with chaotic structure» Phys. Rev. E, 64, 354, () 8. V. V. Novikov, K. W. Wojciechowski, «Frequency deendences of dielectric roerties of metal-insulator comosites» Solid State Physics 44, 55 ()

10 9. Novikov V. V., Wojciechowski K. W., «Viscoelastic roerties of media with a fractal structure» J.Exerimental and Theoretical Physics 95, () Novikov V. V, Belov V. P., «Inverse renormalization - grou transformation in the bond ercolation roblem» J.Exerimental and Theoretical Physics 79, (994) Bernasconi J., «Real-sace renormalization of bond-disordered conductance lattices» Phys. Rev. B8, (978), Shermergor Т. D., Teoriya urugosti mikroneodnorodnykh sred (Nauka, Мoscow, 977).399 (in Russian). 3. Christensen R., Mechanics of comosite materials (New York, Wiley, 979). 4. See e.g. Nonnenmacher T. F., Meltzer R. in Alications of Fractional calculus in hysics, ed. Hilfer R. (World Scientific, Singaore, ),.395.