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0277-The Adhesion Quality and the Extent of the Mesophase 1984

0277-The Adhesion Quality and the Extent of the Mesophase 1984

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Unformatted text preview: VT The Adhesion Quality and the Extent of the Mesophase in Particulates P. S. THEOCARis Department of Theoretical and Applied Mechanics The National Technical University of Athvn: 5, Heroes of Palytechnion Avenue. GR—l57 73 Athens, Greece SUMMARY The three-term and two-term unfolding models, which were developed. to describc mesophase phenomena in fiber-rcinforced composites, are extended in this paper for particulate composites. It has been already shown that the thin boundary layer, developed during casting and curing of the matrix around the inclusions in the com- posite, influences considerably the overall properties of the composite working as a crust between the two main phases of a bimaten'al composite. ' _ However, in a fiberxtomposite the elastic modulus of the composite (E,) consrsts of three components contributing parallelly in a meridional section, that is the modulus of the fiber, mesophase and matrix, which then contribute additiver to the Efmodulus, multiplied each one by a weight parameter equal to the particular volume content of each phase. In a particulate the situation is inversed. Here the compliances of the typical phases contribute in an equatorial plane superimposrng their contnbutron and therefore the respective compliances are additive. Based on this fact the three-term and two-term unfolding models, introduced for thc fiber-composites, were established also for the particulates. it was shown that, by measuring the heat capacity jumps of the matrix and the composites, the thickness or the mesophase layer was experimentally established. The experimental value. of DC- compliancc, combined together with the Df-filler compliance and the Dm«matrix corn, pliance could define completely the model and gave, furthermore, important indica- tions about the adhesion quality of the phases. by defining the so—called adhesron we} flcient of the composite. INTRODUCTION RIGOROUS MATHEMATICAL REPRESENTATION OF A COMPOSITE SYSTEM consisting of a polymer matrix, in which particulate fillers have been dispersed, presents unsurmountable difficulties even for the cases where both constituent phases are considered as elastic. _ Although many geometric, topological, mechanical and physrcal parameters interfere for the statistic evaluation of the average properties of a particulate, it has been shown that the representative volume element, con» sisting of a series of concentric spheres, adequately describes the mechanical 204 Journal of REINFORCED PLASTICS AND COMPOSITES, Vol. 3—July I984 0731-6844/84/03/0204—28 $04.50/0 @1984 Technomic Publishing Co., Inc. The Adhesion Quality and the Extent of the Mesophase in Particulates 205 and physical behavior of the composite. In the two-sphere model introduced by Hashin [l] for the particulates, a gradation of sizes of spheres is an- ticipated, corresponding to the particular filling volume content of the com- posite considered. Hashin gave an exoression for the bulk modulus K, of the composite, considered as the effective homogeneous bulk modulus for the particulate, in terms of the bulk moduli K, and Km of the filler spheres and the matrix material enveloping the spheres, as well as of the filler and matrix volume contents v, and vm (v, + 9', = 1). On the other hand. the homogeneous shear modulus was determined by bounded values. This, and all other two-phase models, assumed ideally smooth and mathematically described interfaces between phases and perfect adhesion be- tween them, thus accommodating the different mechanical propertics between phases. Thus, analytic solutions are in general based on variational principles of mechanics and yield upper and lower bounds of approximations for the effec- tive moduli of the composites. These solutions are valid only for rather low filler contents, since they ignore, for reasons of efficiency, all mechanical in- teractions between neighboring fillers, as well as physico-chemical influences on either phase from the other one. A great number of empirical and semi— cmpirical expressions for the effective moduli exist, expressing some kind of law of mixtures. or trying to match theoretical expressions to experimental data by appropriately defining the existing constants in these expressions. In all these models perfect adhesion is assumed holding between phases. The most important of these models were described in References [2] and [3]. They are characterized by the fact that they express the ratio of the moduli of the composite and the matrix in terms of constants and the filler volume fraction v,, ignoring completely the influence of the value of the elastic modulus E( of the tiller. Only the models presented by Takahashi and co-workers [4] and by Takayanagi [5] introduced explicitly in the expression of the Ec-modulus the filler modulus E,. Indeed, in Takayanagi’s models A and B the elements of the composite, weighed by their respective volume fractions, were arranged in units in series or in parallel, with one of the units having its elements disposed in parallel or in series. In this way a combination of units in parallel and in series was devised, suitable for each case studied. The characteristic of this model is that the matrix was divided into two parts, the one belonging in the one unit, and the second contributing together with the inclusion to the other unit. The relations of these models, however, may be shown [6] to derive from the exact expressions for the moduli of the components of the composite, by introducing simplifications based on the comparison of the values of the mechanical constants of the constituents (for instance 13{ >> Em > Gm, vm = v, etc.). Another type of model is referred to multiphase particulates. It assumes three consecutive phases as concentric spheres and the external one is ex- tended to infinity. The spherical filler is surrounded by a concentric spherical 206 RS. THEOCARIS layer having the properties of the matrix, which in turn is embedded in the in- finite medium with the properties of the composite. The model was intro- duced by Kerner [7]. Van der Peel [8] used a model similar to the Kerner model, but with different boundary conditions. Finally, Maurer, in his disser- tation, used extenstvely the Van der Poel model and derived interesting results [6]. Although the Kerner and Van der Poel models may be adapted for the study of the mesophase phenomena developed at the boundary layers be— tween phases, only Lipatov [9] considered extensively the phenomenon of creation of the mesophase between main phases in particulates and gave rela- tions interconnecting the heat-capacity jumps at the glass transition tem— peratures of the matrix polymer and the composites with the thickness of this boundary layer. In all these models the boundaries of the phases were idealized as smooth surfaces described by exact mathematical expressions. In reality, around an inclusion a complex state develops, which consists of areas of imperfect bonding, permanent stresses, due to shrinkage of the polymer phases during the curing period and the change of the thermal conditions there, high stress» gradients and stress-singularities, due to the complicated geometry of the in- terfaces, voids, impurities and microcracks, appearing at the vicinity of these boundaries. Moreover, the interaction of the matrix polymer during its curing period with the surface of the solid inclusion is always a complicated procedure. In- deed. the presence of the filler restricts the free segmental and molecular mobility of the polymeric matrix, as adsorption interaction between phases occurs. This phenomenon influences considerably the quality of adhesion between phases, contributing to the development of a hybrid phase between main phases, which is called interphase, or better, mesophase. The existence of mesophase was proved experimentally and its extent was evaluated by the theory developed by Lipatov [9], or it may be calculated by dynamic measurements of the storage moduli and the loss factors at the vicinity of Tg’s of the matrix and the composites [10]. Although the extent of mesophase may be determined from ther- modynamic measurements, the influence of this pseudophase to the mechanical behavior of the composite was not extensively studied. For the ease of fiber-reinforced and particulate composites the multi- cylinder model was introduced in References [l l] and [12], which studied the influence of the mesophase on the properties of the composite. Another model was afterwards presented, where the variable with polar distance elastic modulus of the mesophase was expressed as sum of a constant term and two variable terms, expressed as modifications of the moduli of the filler and the matrix by negative power laws [13,14]. The two—term improvement of the previous model [15], where the variable mesophase modulus was ex- pressed by a negative power term of variation of the E,—modulus and a linear term for the variation of the Em-modulus, gave better and much more stable results [16]. The Adhesion Quality and the Extent of the Mnophm in Particulate: 207 All these models were based on an improved law of mixtures, where the composite modulus was assumed as the sum of the filler, mesophase and matrix moduli, multiplied, each one of them, by their respective volume frac- tions in the composite. Whereas this law is simple for fiber reinforced com- posites, where the components or the moduli are connected in parallel, and they add themselves, for the case of particulates a new form of improved law of mixtures should be sought, since, in this case, the contributing moduli may not any more be assumed as acting parallelly. The models of Kerner and Van der Poel, which are valid for the study of three-phase models, may be adapted to derive an improved law of mixtures for particulates, if one assumes the intermediate phase representing the matrix in the Kerner model to occupy the place of the mesophase, whereas the infinite medium possessing the unknown properties of the composite is restricted to an external phase having the shape of a sphere and the properties ofthe matrix [16,17]. Some attempts have already been made to derive in a very sittiplified manner the expression for the elastic modulus Ec of a particulate composite, by taking into consideration the existence of mesophase. One of these is the expression given by Lipatov [18], who attributed this simple model to Sagalaev and Simonov-Emeljanov [19]. According to this model, when the composite attains its critical content of filler, that is the content above which no changes in density of the composite appear, the matrix phase of the Kerner model becomes a mesophase. In this case the compliance of the composite is expressed by the sum of compliance; of the constituent phases, multiplied by their respective volume contents. Although this model was not tested in the praxis, it seems conceptually in« compatible with the real behavior of composites, since it is impossible to assume in the praxis a composite with an overall compliance (Dc) larger than the compliance attributed to the matrix. Another approximate model, which considers the influence of mesophase for the evaluation of the elastic modulus of the composite, was introduced by Spathis, Sideridis and Theocaris [20], according to which the elastic modulus of the composite is expressed as the sum of the filler, mesophase and matrix moduli, multiplied with their respective volume fractions. The term express- ing the contribution of the filler-modulus was further multiplied by a factor k, taking care of the quality of adhesion. This factor was derived from the ex- perimental determination of the mesophase volume fraction vi and its modulus by measuring the Ec-values for different volume contents. Measurements at least at two different filler contents suffice for the evalua- tion of the factor k. Further tests with different v,s were used to check the constancy of the values of k. This modulus gave a lower bound for the com, posite modulus EC. ln this paper a new model was introduced. relating the composite com- pliance (Dc = l/Ec) to the compliances of the matrix, mesophase and filler. Then, by measuring the compliance of the composite and the matrix materials and knowing the compliance of the tiller, the compliance of the mesophase 208 P.S. THEOCARKS can be determined. Introducing the two-term unfolding model for the com- Dliances. as it has been developed for the fiber-reinforced composites. the adhesion coefficient between phases can be determined, by evaluating the mode of variation of the elastic compliance or modulus of the mesophase, to match the bounds of D’s and E’s of the filler and matrix. The new model gave reasonable results, as it did the same model for fiber composites and thus characterized the quality of adhesion of the composite. Determination of the Elastic Modulus of the Composite In order to determine the Ec-modulus of the composite, we consider that the representative volume element consists of three concentric spheres, having radii r,, ri and rm respectively. The external radius of the RVE was taken such that this model corresponds to the average properties of the respective par- ticulate. All quantities are normalized to the volume of the outer sphere 4/31rr,’n to be equal to unity. We assume lurther that all phases are elastic, homogeneous and isotropic, with the exception of the mesophase layer, which has a variable modulus, changing with the polar distance between the values of Efand E,“ at its bound- aries. However, this layer is also averaged, so that Ei corresponds to its E,“- mean value. Moreover, it was assumed that the fillers are perfect spheres, their distribu tion is uniform and their intercenter distance large enough to ascertain a negligible interaction between neighbouring fillers. Finally, we assumed that the RVE is submitted to a radial pressure of magnitude pm, so that, at the other boundaries, equal radial pressures of magnitudes pi and pr are exerted, which exhibit the interaction between successive phases. The representative volume element, as descrlbed above, is presented in Figure 1. Because of the spherical symmetry of the problem we introduce spherical (r,0,¢p)—coordinates with the simplification that, from all the components of the spherical displacement vector, the ur—radial displacements are different than zero, because of the nature of geometry and loading of the problem. (u) (bl (r) ngm 1. Principal sections of the three concentric spheres forming the representative volume element lRVé‘I of a typical particulate and the forces applied to their boundaries. The Adhesion Qualin and the Extent of the Mesophase in Particulates 209 The solution of the problem may be derived by the stress functions in] f(r) (with j = f,i,m for the filler, mesophase and matrix respectively), which are expressed by: it, =A,r"+19.r2 (1) with the constants A, and .3, defined by the boundary conditions between phases, With A, = 0 for fin1teness of stresses at r = 0. The components of displacements are given by: u.,- = (~A,-r*2 + 28,.r)/ZG,- (2) with the other components: “or = "av = 0 (3) The component u,, is expressed by: Mr - 7, (4) since A J = 0. The components of stresses are given by: =fi+2r1+w 2(1+v ,3 m3” aoj=ow=—A,r3+H’) (1 — 2v.) 3] (5) a,,- The boundary conditions yield: 17A! r = r,.' 0,, = 0., = ‘1), ii) At r = r,: o..- = 0,... = - p. (6) 170A! r = r,,: 0,... = ‘11.. The solution of the system derived from the boundary conditions (6) yields the values of the constants A,, 3,: AI=(PJ_P1—i)’7"3'-i (’73P1_’}1—1P1—1)”_2Vj) I 2(’f"7‘-i) B, = — 2m ~ r; . .i (1 + v.) (7’ Where the index (j + 1) means the next layer of the layer j considered, with the successron from the filler outwards, so that j = 1 corresponds to the filler, 1:.2 to the mesophase, and j = 3 to the matrix of the particulate, and n» — r,- _ 1 = 0 forj = l, r, = r,,r.r,r, forj = 1,2,3 respectively. 210 P.S. THEOCARIS The equations for the radial displacements are: _ _ P!” _ 2V1), 8 “r! ‘ E, ( ) 03”: (PI ‘ Pa) (1 + Vi) (1 " 21h) (1730/ ‘ P9P.) u . = + r (9) " 2(r.-‘ - r;‘)E.r2 E. n‘ - I? __ if: ([7: ‘17...) (1 + V...) (1 ' 2V-) (n’pr — lip...) 10 “'“‘ 2(I3.-n‘)Ef + E. e—e ’ ( ’ The components of strains in the three layers are given by: _du,,_i _ _u,,___fi c"_ dr_G,'£"-£W_r G, du - A- 8- ud A,- B, = "=4 V: ,= .=_=_ +i 1 "" dr Ho: + G: ' “" ‘e r 2r’G.- G: (1 ’ dun, A," + B... _ _ um _ (A. + Bin "'4 ' dr ‘ r’G... 0,. ’ ‘W W" r 2140,, 6,. The boundary conditions, implying continuity of displacements at the boundary between filler and mesophase, yield: For r = r, 14,, = u,, then: -p,(l - 2w) _ (I - 2vt)_ (P13 -p.-) + (l + v.»)(pf “pd (12) E, ‘ E; 1 ~ B 2(1 ~ B)E_. in which the ratio r,:‘/r.3 is replaced by the quantity B where: U! = 3 B u, + ul. (1 ) Solving Equation (12) with respect to the ratio A,. = p,/p.- we have: 3(1 — v,)(u, + u.-)E, = 14 2u.E,.(1 — 2v,) + [3u,(l — v.) + u,-(1 + v.)]E, ( ) 1,, We apply now the second boundary condition concerning the compatibility of the radial displacements at the second interface between mesophase and matrix. In this boundary it is valid that u,, = u". and this yields: = 3(1 — v”) — 2u,,.(1 — 2v”) _ Emu, (1 + vi) 3(1 — v") E, 3(1 — 11..) (15) Air" The Adhesion Quality and the Extent of the Mesaphase in Particulates 21 1 In order to evaluate the elastic modulus Ea of the composite, we define the elastic strain-energy balance on the representative volume element. Then we have: 1 V25 Il’gdu‘=I/z§ W,du,+%s W,dul.+%5 Mary" H c U! U: V». e which yields: I‘m 171 r! rl. " r’dr = W,r‘dr + W,-r‘dr + o Kc o r, where Kc is the bulk modulus of the composite, related to the elastic modulus E: by the well known relationship: r». er‘dr (16) r.‘ E K“ = 3(1 — 2m (17) The elastic strain energies of each one of the layers of the particulate may be defined by the relation: "/1 = "'1‘?! + "9150/ + aw‘w = firm) lntroducing the appropriate values for the components of stresses and strains for each phase, we may readily derive that: 93% W/ = TE; ’ (18) 3A2 933k, W" = r‘G: + 0,? (19) and . ,5. B2 m Wm _ 34 + 9 ,K (20) ' PG... 0}, Introducing now the values for W’s into relation (16) we obtain, after some algebra, that: 1_u, 1,:1 31;; I—A,.-]‘ u; i,._ 1 2 K: ‘ K, (it) + 4BG.u. L- + Kiu. .. mm + 3n, _ L ‘ 1 u, _ 2 + 430,21. [ hm] +K_u... [mm ] (2” 212 PS. THEOCARIS where: _ a _ I: a = h 2 Aft _ p; ; Ann " p; and a“ ll“ ( and: . v, + u,- rE/ri, = u, + 1),, ri/rfi, = u, and fi/r}= u (23) 1' Equation (21) yields the bulk modulus Kc of the composite in terms of the bulk and shear moduli of the phases and the stress-ratios Al,- and A“, which, on the other hand, depend on the elastic moduli and Poisson’s ratios of the phases. _ Since the mesophase is derived from the matrix material, it is reasonable to accept that: v.- E v" (24) whereas for the Poisson ratio 1/, of the composite we use a modification of the relationship given by Reference [21] interconnecting the values of Porsson’s ratios. This relationship is given by: L = a + a + U_~ (25) 1/6 1/, v, v, This equation completes the number of relationships necessary for evaluating the characteristic properties of the composite. Comparison with Exlstlng Models for Plrticulntes Relation (21), interconnecting the inverses of the bulk- and shear-moduli of the phases and the composite, constitutes an improved law of mixtures for particulates. It indicates that some relation between compliances, instead of moduli, should hold for particulates. Indeed, Equation (21) shows that the composite bulk-compliance (or simply any compliance, since the material is assumed at a thermal equilibrium, where the reduced elastic relations between moduli should hold, provided the actual values of Poisson ratios at this temperature level are known, together with the values of any modulus at the same temperature or ti...
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