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

0277-The Adhesion - VT The Adhesion Quality and the Extent of the Mesophase in Particulates P S THEOCARis Department of Theoretical and Applied

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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 time level) equals the sum of the bulk compliance of the fiber, the bulk and shear compliances of the mesophase and the bulk and shear compliances ot the matrix material, all these terms multiplied by con» venient factors depending on the volume fractions of phases and the ratios of the elastic moduli of filler and matrix, normalized to the elastic modulus of the mesophase through the convenient factors 1,, and l,,,.. It is obvious from the structure of relation (21) that this constitutes an im- The A dhesion Quality and the Extent of the Mesoplmse in Particulate: 2 l 3 provement of previous assumptions concerning the interrelations between moduli in particulates. indeed, while a simple improved law of mixtures for fiber-reinforced composites exists, which may be expressed as: E‘ = 15,11, + En), + Emu... (26) the same is not valid for the particulates. This is because, if we assume separated the contributions of the moduli in the RVE of the fiber—composite, these component moduli cooperate through their interfaces, which are assumed parallel to the direction of application of the external load. this cooperation achieved by the lateral surfaces of cylinders developing shears between phases. Thus, this model necessitates an addition of the moduli of the phases multiplied, all of them, by weight- factors which are simply their respective volume contents. In an electrical analogy scheme then, the weighted moduli constitute capacitances connected in parallel, which explains the validity of relation (26). In the case of particulates the situation is much different. The component phases are interconnected through consecutive spherical surfaces of the fiber, mesophase and matrix. The dominant transverse interconnection through shear, holding in the fiber-composites, is now insignificant, the adhesion is now achieved by a continuously varying combination of shear and normal forces at the interfaces, derived as variable components of the hydrostatic pressure mainly applied on these surfaces. It is now closer to the real situation of force-distribution, to assume that the contributions of phases are done by superposed layers of the phases, lying the one on the top of the other in the direction of the application of the external load. in this case the compliances, instead of the moduli, should be added to derive the compliance of the composite. since now the capacitances of the respective electrical analogy are connected in series. This argumentation ex- plains the necessity of adding in this case the compliances of phases, instead of moduli. Figure 2 presents a schematic of the difference of the two cases in a RVE of fiber (a) and particulate (b) composites. The drawn surfaces between phases explain the reasoning of the previous ideas. However, in the case of particulates, since the connecting surfaces are not flat, engendering almost equal distributions of shear forces (except at sin~ guloritics and discontinuities), but spherical, with variable contributions for the boundary conditions, relation (21) is much more complicated than the relation (26) holding for fiber-reinforced materials. There are several models expressing the E,-modulus of the particulate, which are based on simple relations interconnecting the matrix with the filler through a mesophase. One of the interesting models is the model introduced by Sagalaev and Simonov-Emiljanov [19], which is expressed by: EfEiEm E6 = mEIEiUM + EmEiUl + (27) a 14 ES. Tuuocmus Flynn 2. Schematic of the difference in the cooperation of the phases in a fing unidirectional composite (ai and a spherical particulate (bl. This expression may be reduced to: I), = u,D, + u,D; + u_D- (28) which interrelates the extension compliance of the composite (D;) to the com- plianees (D,) of the phases multiplied by their respective volume fractions (11,} (j = f,i,m). I _ This model is the analogy for particulates of the respective model for the elastic moduli of fiber reinforced composites, expressed by relation (26), However, as it is already stated here and proved by several authors, this simple relation is an oversimplification of the real situation in particulates Another approximate model was introduced by Spnthis, Sideridis and Theocaris [20] which constitutes an upper-bound for the value of the elastic modulus of the composite. This model is expressed by: E: = E,u,k + E111; + Emu, (29) where the adhesion factor k between filler and matrix is introduced to take care of the influence of the shape (approximately spherical) and number of TheAdhesion Quality and the Extent oft/u.- Metsophatm in Particulate: 215 fillers and their interaction, on the value of the average elastic modulus of the composite. In addition, the existence of the mesophasc layer, contributing to the quality of adhesion, was put with to take care of the proper influence of this pseudophase on the value of [Sc-modulus. Finally the study by Maui-er is worthwhile mentioning [6]. This study, based on the Van cler Poel model [8], has examined the influence of the existence of a mesophase on the viscoelastic behavior of a composite containing spherical inclusions. The influence of discrete amounts and qualities of mesophases on the overall mechanical prop- erties of the particulate composite was exhaustively studied, concerning the two principal simple modes of loading, that is hydrostatic pressure and pure shear, Besides other important contributions, this study has shown the in fluence of a third phase in-between the two others, be it a real phase, or pseudophase developed during polymerization of the composite. It was shown that, for the bulk modulus of a three-phase material. the following relation holds: K _ (KW! + KiUiR + KmUmS) ‘ u, + u,R#+#uM—S— (30) where the factors R and S are functions of the bulk and shear moduli of the three phases. For the shear modulus of the composite a (10 X 10)-matricial expression for this modulus is anticipated. It can be derived, by comparing relations (29) and (30), that relation (29) is a simplified expression of relation (30) incorporating into the factor k the in- fluence of the quantities R and S. Then, relation (29) may be considered as a reasonable upper bound for the K6, or E,»moduli. In completing this discussion about the expression for the moduli in com- posites, it is worth mentioning the Kerner-Kerner model [7], where the classical Kcrncr model was uscd twice, once for the filler-mesophasc material, and for the second time for the internal heterogeneous material and the matrix. In this model we have again the expression for the shear modulus G‘ (if the particulate in the form: GU +Guk+Gu1 :_J_I_%1_L___'LL 0‘ (u,+u.-k+u,,.l) (3” which is similar in conception with relations (29) and (30). The factors k and l are complicated expressions of the shear moduli G,- and Poisson‘s ratios 11,- of the phases. Similar expression is also given by the Kudykina and Pervak model [22), where the: shear modulus of the composite is given by: _ 0,114+ 0,115+ vat G“ _ ("If + ms + U...) (32) 216 RS. THEOCARIS All these expressions are more or less similar, displaying the correction fac- tnrs between two of the three phases. Quite different expression for the composite moduli is given by the Takano and Sakanishi model, which includes the concept of mesophase. The ratio (PE/Gm of the shear moduli is given by [6]: G, _[r—3/2(u,+u,)r' m] 0., _ [F+(u,+u,-)|"Q,,,] (32) where the quantities F,l' ' and PWQ, are complicated expressions of the bulk and shear moduli of the phases. Besides these models incorporating explicitly the existence of a mesophase, which may be a real one, or may be a pseudophnse developed by the matrix at its boundary with the fillers, there is a quantity of simple phenomenological models, based on the interaction of the two main phases. These models have been exhaustively described in the literature and they )n'eld more or less reliable results [3]. One important model, where a convenient combination of the two limiting arrangements, that is in parallel and in series, was introduced to express, in an effective way, and with a high approximation, the elastic moduli of par- ticulates is the Takayanagi model [5,23]. While the Takayanagi model is designed mainly for two—phase materials, it could be extended to incorporate the influence of mesophase [24]. It seems that a further extension of the concept of using mixed-mode connections, that is in series and in parallel, for the elements contained in the model represent- ing the mechanical properties of the particulate, is Very promising, since it yields high flexibility in the model to be adapted to the real behaviour of the substance. However, if one examines relation (21) expressing Kc, he may recognize that the terms in the right-hand side of this relation could not be strictly considered as connected in series, The Unfolding Model for the Mesophase A decisive factor for the physical behaviour of a particulate composite is the adhesion efficiency at the boundaries between phases. In all theoretical models this adhesion is considered as perfect, assuming that the interfaces en- sure continuity of stresses and displacements between phases, which should be different because of the proper nature of the constituents of composites. However, such conditions are hardly fulfilled in reality, leading to imperfect bonding between phases and variable adhesion between them. The introduc— tion of the mesophase layer has as function to bring together, in a smooth way, the differences on both sides of interfaces. The model for the representative volume element of a particulate consists of a unit of three concentric spheres with respective radii r,, r, and r,,,, whose volume contents are expressed by: The Adhesion Quality and the Extent of (he Mesaphase in Particulates 217 _ 51‘ _ 5:0“, _ [ti u,— I: ,u.— n," andum— ,7" Assuming the appropriate boundary conditions between the internal sphere and any number of layers surrounding it, in the RVE of the composite, which assure continuity of radial stresses and displacements, we may establish, by an energy balance between phases, a relationship, interconnecting the com- pliances, or the moduli between phases and composite. Such a relationship is given by Equation (21). In this relation the quantity E, corresponds to the average value of the modulus of the mesophase and, in the following, it will be denoted as E‘.-'. However, this effective or average value of the mesophase modulus, necessary for introducing the contribution of the mesophase to the value of the modulus of the composite, does not really exist, except in one very thin spherical layer. In order to approach closer to reality we assume that the mesophase layer consists of a material having progressively variable mechanical properties, in order to match the respective properties of the two main phases surrounding the mesophase. We can then define a variable elastic-modulus for the mesophase, which, for reasons of symmetry, depends only on the polar distance from the fiber-mesophase boundary. In other words, we assume that the mesophase layer consists of a series of elementary layers, whose constant mechanical properties differ to each other by a quantity (small enough), defined by the law of variation of E,(r) . In this way the elastic modulus of the mesophase is unfolding between two limits, from the E,- to [Em—modulus. The unfolding model for the mesophase may be expressed in two forms: In the first form, the E,(r)-modulus is expressed by the sum of three terms, i.e.: i) a constant one and equal to the Em-modulus. ii) a variable one, depending on the modulus of the filler (E,), which should be added to the first one, and iii) a third variable-one, which should depend on the Enrmodulus, and which should be antagonistic to the second term. The two variable terms should yield very abrupt variations in the E.(r)-modu1us, since the generally large differences between the moduli of the fillers and the matrix must be accommodated in very thin layers for the mesophases. The appropriate functions for such steep variations are power functions of r with large exponents. Then, the E,(r)vmodulus may be ex- pressed by: E,(r) = E," + E, ( ‘ —E,,, 1 (33) Expression (33) may be interpreted by the fact that the variable E,(r).modulus, which connects two phases with highly different mechanical 218 P.S. THEOCARIS properties and elastic moduli, must interconnect and span these differences. Indeed, for hard-core composites it is valid that EI >> Em, whereas for rubbery fillers we have the opposite relation E, >> E,. However. it never happens that E, = 5,. Relation (33) has been already established for expressing the mesophase variable modulus for fiber-reinforced composites with excellent results [14 to 17]. In relation (33) the second right-hand term expresses the contribution of the Ermodulus to the variation of EMU-modulus, whereas the third right- hand term defines the counterbalancing contribution of the Em—modulus, to correct the contribution of the E,-modulus, and insert the influence of the matrix to the outer layers of the mesophase-annulus. From the compatibility conditions for the moduli at the boundaries be- tween filler, mesophase and matrix it may be readily derived that for r = r, relation (33) yields E,(r) = E, and therefore satisfies automatically the bound- ary condition at the filler-mesophase interface. On the other hand, at the other boundary between mesophase and matrix, that is for r = n, we obtain E.(r) = E... if the following relation holds [16]: logE,/E,,, A = (m — "1) = logr/r (34) The constant A = (n, - m), which depends on the ratio of the moduli of the two phases and the ratio of the radii of the mesophase and the filler, characterizes the adhesion quality of the composite, because for large values of A the extent of the mesophase layer is reduced and the adhesion is more ef- fective. The constant A is called the adhesion parameter. Higher values offiiindicatg ibegeLa’dhgsign‘for a particular composite, because’irnilysmalldifferencesinthe valuescfrt'anfl r',". For’haid-core materials, where E, >> 5,, the radius r, must be always larger than r,, and E, >> E," , therefore the logarithm of the ratio r,/r, is a positive number and since log(E,/E,..) > 0 the values of A are always positive. This means for a hard-core composite it is always valid that n, > 7);. On the contrary, for rubber-core composites log(E,/E,.) is now negative and therefore it is valid that Y], < m. A simple and more stable version of the previous model for the variation of the E.(r)-modulus is given by the two-term form of the previously described model. Since the third term of the right-hand side of Equation (33) takes care of the influence of the matrix modulus E... to the variation of E, (r)-modulus and since always for strong-core composites this contributiOn is secondary, rela- tion (33) may be somehow relaxed, by assuming that this third term varies linearly with the radius r along the interphase. This means, in other words, that the exponent n; may be assumed equal to unity without loosing gene erality, and imposing to the first right-hand side term to take care of the totality of the change of slope of the E,(r)-modulus. The Adhesion Quality and the Extent of the Mesaphase in Particulates 219 Then, relation (33) may be written by: ’1 2" _ fl 1" (V _ 7'1) Ex”) - E! (r) +{E— El<rl) }(ri _ r!) (35) Now relation (35) contains only two terms in its right-hand side, since the contribution of the En-constant term in the Equation (33) is now incor— porated into the second linear right-hand side term of Equation (35). It is easy to show that the boundary conditions for this equation are automatically satisfied. Indeed, for r = r, we have the second right-hand term of Equation (35) equal to zero and the first term equal to the E,-modulus, as it should be. Moreover, for r = r,, Equation (35) yields automatically E,(r) = E, and this satisfies the exterior boundary condition. Relation (35) has the advantage to contain only one unknown exponent and therefore simplifies considerably the evaluation of the unknown quan- tities in the definition of the variable E,(r)-modulus. Moreover, it was established during the numerical evaluation of the unknown quantities in relations (33) and (35) that the definition of the two ex— ponents m and m in relation (3 3) is rather unstable, depending fraily on small variations of the value of the E,—modulus. 0n the contrary, the single unknOWn Zn-exponent, defining relation (35), yields rather stable and reliable results. In the two-term unfolding model the 2n-exponent is the characteristic parameter defining the quality of adhesion and therefore this exponent may be called the adhesion coefficient. This quantity depends solely on the ratios of the main-phase moduli (E,/E,,.), as well as on the ratio of the radii of the filler and the mesophase. The Thickness of the Mesophase It has been observed that, for the same volume fraction 11, of the filler, an increase of T, indicates an increase of the total surface of the filler [18]. This is because an increase in T, may be interpreted as a further formation of molecular bonds and grafting between secondary chains of molecules of the matrix and the solid surface of inclusions, thus restricting significantly the mobility of neighbour chains. This increase leads to a change of the overall viscoelastic behaviour of the composite, by increasing the volume fraction of the strong phase of inclusions. A considerable amount of experimental work indicates an increase of T, in composites with an increase of the filler content [18]. The degree, however, of this variation and the character of its change may differ from composrte to composite and, also, for the same composite, is depending on the method used for its measurement [25,26]. Moreover, in many cases, a shift of T, to lower values of temperature has been detected, but in these cases the quality of adhesion between phases may be the main reason for the reversing of this attitude [18,27]. If calorimetric 220 RS. THEOCARIS measurements are executed in the neighbourhood of the glass transition zone, it is easy to show that lumps of energies appear in this neighbourhood. These jumps are very sensitive to the amount of filler added to the matrix polymer, and they were used for the evaluation of the boundary layers developed around fillers, The experimental data Show that the magnitude of the heat capacity (or similarly of the specific heat), under adiabatic conditions, decreases regularly with the increase of filler content This phenomenon was explained by the fact that the macromolecules appertaining to the mesophase layers are totally or partly excluded to participate in the cooperative process taking place in the glass-transition zone, due to their interactions with the surfaces of the solid inclusions. It was shown [28,29] that, aflhe‘ fuller-7vplume’fraption“ismincreasedtthe proportiongt macromgggggspartigpaghg‘n’mis boundary layers with reagentmobmmathenymjflifi} Bmwules participating in the T,-process is reduced. This is eguivalepttgamlatlye in- crease :9f 11,-. ‘ " W patov [18] has indicated that the following relation holds between a weight constant A, defining the mesophase volume—fraction 11,-, and the jumps of the heat capacity AC], of the filled-composite and AC}, of the unfilled polymer for particulate composites: Act i=1—Aq (36) where A is a real constant, which multiplies the filler volume fraction u,, in order to take into consideration the contribution of the mesophase volume fraction u,- to the mechanical behaviour of the composite. In order to define the volume—fraction u,- of the mesophase for the par- ticular composite studied, which was an iron—epoxy particulate, a series of dilatometric measurements were executed in a differential scanning calorimeter (DSC) over a range of temperatures including the glass transition of a pure epoxy polymer used as matrix, and a series of samples of com- posites containing different amounts of iron—particles of three different diameters d, = 150,300 and 400m” varying between u, = 5 percent to u. = 25 percent. The graphs of all the data, which were similar to the heat-capacity graphs for iron-epoxy particulates, shown in Reference [27], presented shapes, which were qualitatively similar to one another. They consisted of two linearly in— creasing regions separated by the glass transition zone. While the glassy linear regions presented a positive and significant slope, the rubbery linear regions were, all of them, almost horizontal. The AC,‘s were calculated by ignoring the smooth protrusions B 'BC ap- pearing in the CP = f(T) curves, as indicated in Figure 3 where the variation of the specific heat, and its jumps at the region of glass-transition temperatures Tgc, versus temperature was plotted for an iron-epoxy par- Cp Ital, gr‘, "[4] —> 50 75 100 125 150 T (“C l -—> Figum 3. The variation of heat capacity, C,, 0/ immepaxy particulates plotted against tempera fare, [or Iaur different lid/er volume Iraot‘ime, e,, and for a particle diamator d, = 0.40 x 103m, ticulate with d, = 400nm and for various u,’s. Then, we measured the distance of the intersection A of the tangents AA ’ of the glassy curve and AB of the transition curve from the horizontal tangent CC' of the rubbery part of the curve AC, = f(T). The values of chs for the different composites are in- dicated, among others, in Table l. in order. now. to define the radius r. of the spherical layer corresponding to the mesophase, we express it as r. = (r, + Ar) and we use the respective rela- tion, given by Lipatov [18] for particulates, which is given by: (r,+Ar)3 _1_ Au, 7; l-Ul (37) which yields: hfi‘fi‘T—VF—T) (38) The real constant B depends only on the filler-volume fraction and the coef f i- cient A, and it is critical for evaluating the exponents n, and m. Introducing the values of A from Table l we can define the values of Ar for the various filler-volume contents u, for the iron-epoxy particulates. Then, it is easy to evaluate the volume fraction u.- for the mesophase layer. Figure 4 presents the variation of the differences, Ari, of the radii of the mesophases and inclusions, (Ar, = (r.- — r,)), versus the filler-volume content u, for the three different types of iron—epoxy particulates with diameters of fillers d, = 150,300 and 400m, as they have been derived from relation (37), Figure 5 presents the variation of the heat-capacity jumps, AC1, at the respective glass-transition temperatures of the particulates, versus the filler 15 Zn 11114 1520 424 158 74 654 1N2 N2 124 64 744 m6 2% 118 58 o df=150pm x macaw“ / E c. d,=L00,un E < $3” 88888 88888 a E “19" E" 'gBEN ) g 6 g?“ " Eg" : '” g *1 -§E§§~§§EEB $5328 3 —' ' & ° n E_ 8.8.888. 88.888. 88.8.8.8. Wm,» Iv: > :9 “at! 89 :tfisg eana '0 Figure 4. The variation of the differences Ar. 0! the radii of mesaphases and inclusions ° ‘3 i Ar = l r. — r,ll, versus the inclusion volume content u, for three different diameters of inclusions 8 _ of iranlepaxy particulates. E is a; SNK’EB 93mm: 8838?. ‘2; “no castva mvvmo v‘rvmw c ' 1w 5 55‘ a 3:8 '* F g: .-e fi§§§5 “$313 :8 1 88888 §§§E§ §§§§E 1 E g 75 E 2‘s a; m g g 2 : 3:5 a a , g .a g « Eaéfiawzssaéaeaas :_ 000°C OOOOO 00000 2 :1 g :4 "8 9 m o L .5 ‘ n 9 3 — F E -n ° 2: $83.32 @5538. §8§8§ D 8 E 8 oooeo ocooo cocoa 125 T: > E 5 8.8.2.28. 8.8.888. 88.8.8.8. “ ’ EBSRB 388R$ 3$8R$ U Q - Q 5 n ’5 =- § 3§§§§ §8§§§ @Nifii unvn—e fl- oo—Nm oo—Nm DDFNlfl Figure 5. The venetian of the specific heat jumps at glass-transition temperature of iron-epoxy Particulate composites, versus the filler volume content u,. The values for the factor A, the - .n o m 8 a m o m a g: m c, m a a mesophaseluil, and the matrix (u..)-volume fractions versus 1),, as derived from the values of the " " " respective AC,’s are also plotted. 222 223 214 P.S. Tneocmus volume content u,, for the three different diameters of the fillers (d, = 150,300 and 400ym). In the same figure the variation of the coefficient A and the volume fractions for the mesophase and the matrix, versus u, were plotted, as they have been derived from relation (38). It is apparent from these graphs, that the niesuphase-volume content 11, for the three different diameters of inclusions were varied only insignificantly and, therefore, they may be assumed as independent of the diameters of the fillers. It may be concluded from Figure 5 that the variation of the heat—capacity- jump curves are smoothly decreasing curves, as the filler-volume content is increased. This behaviour is logical, since addition of iron particles makes the composite more rigid [18]. It is also interesting noting that, for various diameters of the fillers, the ACT, = f(u,)-curves differ only slightly, with the respective curves for different filler-diameters. mutually intertwining. This fact indicates that the size of diameter of the filler plays only a secondary role on the influence of the heat-capacity jumps, which are primarily influenced by the filler-volume content. A similar behaviour is expected for the variation of the coefficient A, as indeed it is indicated in Figure S. It is worth indicating that the values for u,-'s fitted excellently a third degree curve expressed by: u,- = Cu,J (39) where the constant C was evaluated experimentally to be equal to C = 3.5. The values of ms, together with the respective values for EC ’5, were intro- duced in the respective models and gave excellent coincidence with their ex- perimental values. based on the cubic relation between it.» and u,. The values of the characteristic quantities for the three-term unfolding mode], as derived by this procedure, were included in Table i. In order now to evaluate the one exponent n. of the model. we make recourse to the law of mixtures given for the spherical particulates by relation (2]). Evaluation of Adhesion Coefficient A for the Three-Term Unfolding Model As soon as the Ar’s were determined and the values of as are found, the values of the adhesion parameter A may he readily defined, by using relation (34). The values of A's for the different filler-volume contents are given in Table l for the iron-epoxy particulates with different amounts of fillers, up to 25 percent [27]. In order to define the m- and nI-exponents it is necessary to dispose a sec- ond equation, besides relation (38), for the evaluation of n-radius and rela- tion (34) for the definition of the difference (qr-In). For this purpose we use relation (33), in which the value for E}, as it has been derived from relation (21) was used. Indeed, in relation (21) every other quantity is either known, or measured, as it is the bulk modulus K. of the composite and its Poisson's ratio v. from very meticulous experiments. However, the values for the t The Adhesion Quality and the Extent of the Mesophase in Particulates 225 Poisson’s ratios of the various composites may be found by using relation (25) and applying the values of v’s for the known phases and composites. It has been derived that the values of Poisson's ratios for the mesophase v.- are close to v... and they may be found when the Poisson ratio for the composite vc is evaluated. For known values of E: derived from Equation (21) the variation of the mesophase modulus along its thickness may be defined by using relation (33) for the three-tenn undolding model. It is valid that: « —ir'E +E '4 "—E 1“ r‘dr (40) [Vi—r:- rl m l I. m r integrating relation (40) from r = r, to r = r.- we find: . .— jE’i _(fl.+_L an” ~ 1) Ely-[BIL]: -ll.'l >11: _ E,u.-—(2n_3){l 2n_3)3 }+ 4 {1+3 +3 } _ Ezv/ {1+B-l/J + 3—2/3 _,3B»I} (4]) Relation (41) uses the value of E7, derived from Equation (2|), and it con- stitutes one equation interrelating the two unknown exponents n, and n,. Then, relations (34) and (41) form a system of two equations and two unknowns, which can be solved and yields the values of the exponents n, and q. and their difference A, which expruses the quality of adhesion and it was already called the adhesion parameter. The values of the exponents n, and n2, as well as of their difference are given in Table l and plotted in Figure 6 for the iron—epoxy particulate composites. Figure 7 presents the variation of the terms E,(r,/r)"I and E,(r,./r)"2 in the mesophase layer for a 25 percent iron-epoxy particulate, as they have been derived from Equation (40). It is worthwhile indicating the smooth transition of the E.—modulus to the En-modulus at the region r E r.-. Similar behaviours present all other compositions. It is interesting plotting the variation of the E‘_(r)-morlnlus, versus polar distance around a typical filler. Figure 8 presents this transition of the moduli from the fibers to the matrices, exemplifying the important role played by the mesophasc layer to the overall mechanical behaviour of the composite. Figure 9 presents the variation of the various moduli of the composite and its constituents for various volume fractions of the series of iron—epoxy par— ticulates. It is of interest to point out the small variation of the average value of the variable E,(r)—modulus of the mesophase, which reflects the uniformity of the adhesion quality of these series of composites, which is also indicated by the almost linear variation of the longitudinal composite modulus, versus the filler—volume content. 226 PS. THEOCARIS E. (r) =EmoE, (r, Ir)“‘ -E,,.lr,lrl"i uf=015 115% 111:12 E;(r .10" NM" —-> 2.0 Arxlo‘m —’ Figure 6. The mode of evolution a! the variable terms l:',(/,/rl"1 and Emlrflrl'”, contributing to the definition of the mesophese modulus, versus the polar distance r from the filler boundary It” a 25% iron~epoxy particulate composite, Evaluation of the Adhesion Coefficient 2:1 for the Two-Term Unfolding Model For the case of the two-term unfolding model we have to replace the in- tegral in the right-hand side of Equation (40) by the integral corresponding to the two-term model derived from relation (35). In this case we have: i r. i in L In r-” Ems r; SUEC) +{E, E,(n) }ri_r, r‘dr (42) In this relation the only unknown is the exponent 2n, since the value for E: is derived from relation (21). Integrating relation (42) we find: EN! = _ anls— ]_ % (EflI _ Esznn)‘:1 + B-i/a + B-zn _ 33-] (43) Relation (43) yields the value of 2n, which is also tabulated in Table l for all iron-epoxy particulates studied. Figure 6 contains also the plot of the adhe- The A dhesion Quality and the Extent of the Mesaphase in Particulate: 227 2000 0 0.05 0,10 0.15 0.20 025 u) l°/o) —-> Noun 1. The van'nhbn a! 2;» adhesion pmmm A for the thmnrm unfolding model, and the adhesion coefficient 2r, for the two-mm modal. versus the We! volume fraction v,. sion coefficient 2n, versus filler volume fraction for all iron-epoxy par- ticulates studied. It is worth mentioning here that the three 2r] = f(u,) curves for the three different diameters of the fillers are almost coincident. Furthermore, there is an equivalence between the mode of variation of the two adhesion coeffi- cients A and 21, corresponding to the two versions of the unfolding model. Figure 10 presents the variation of the interphase moduli E,(r) for the various inclusion—volume fractions, versus the extent of the interphase Ar, normalized to the highest inclusion-volume fraction of 25 percent. This was done in order to show the similarity of variation of the Ermodulus for the various values of u, for this series of composites, possessing the same adhe- sion properties between them. These normalized patterns are equivalent with those presented in Figure 9 for the three—term unfolding model, since the dif— ferences between corresponding values of the two versions of the model are insignificant. CONCLUSIONS All theoretical models, describing the physical and mechanical properties of composites, consider the surfaces of the inclusions as perfect mathematical surfaces. In this way the transition of the mechanical properties from the one phase to the other is done by jumps in the characteristic properties of either L0 [1,104 Nn‘l —> m c N o D DDS 010 015 0,20 025 u, (°/ol —> Figure 8. The variation of the moduli of the particulate composite Er, versus the filler-volume fraction, u,, and the mode of variation of the average mesophose modulus, E, as derived from the models. l i ‘1 2L0 180 T A 11,:150 um + d.=300 um as u u,:noo um f 120 l '9 .1 L:— 60 050,05 U,:0‘lO' . 0.3 0a 05 10 L0 60 3.0 100 2 a 5.6 6 I. 11.2 1L0 Ar x 105 m —-—> figure 9. The variation of the mesophase moduli, versus polar distance r for different filler volume contents u, for the throeterm unfolding model. 228 The Adhesion Quality and the Extent of the Mesophase in Particulates 229 2L0 l , z o nV :rm A u,=o,2o 15° + u,=015 «it u,=010 A u,=0.05 120 E,x10‘9 "m" -—> 60 0 0 10 20 10 L1) 50 Arxloim ——> Figure 10. The variation of the mesophase moduli, normalized to the mesophase thickness Al for the 25 percent filler~volume content composite, versus polar distance I for the two-term unfo/ding model phase. This fact introduces high shear straining at the boundaries, which is an unrealistic fact. in order to alleviate this singular and unrealistic situation, two versions of models were presented, in which a third pseudo-phase was considered, as developed along a thin boundary layer between phases during the polymeriza- tion of the matrix, and whose properties depend on the individual properties of the phases and the quality of adhesion between them. The two versions of this unfolding model consider that the mesophase layer, lying between the main phases, possesses varying physical and mechanical properties, assuring a smooth transition from the properties of the fillch to the properties of the matrix. Then, it is achieved in a very short distance, corresponding to the thickness of this boundary layer, the pro~ grcssive change of its mechanical properties from the filler to the matrix. By using Lipatov’s theorv. interrelating the abrupt iumps in the specific heat of composites at their respective glass transition temperatures with the values of the extents of these boundary layers, the thickness of the mcsophase was accurately calculated Assuming in the one version of the model a three—term representation of the unfolding value for the elastic modulus of the mesophasc. where each term is expressed as a negative power function of the polar distance from the inclusion, the variation of the elastic modulus of the mesophasc was ac- curately determined. The second version of the unfolding model uses, instead of three, two terms, one of which is a negative power function ofthe polar distance, acting upon the inclusion modulus, and the other~one, acting upon the matrix modulus, is expressed as a linear function of r. 230 P.S. THEOCAius Both versions of the unfolding model gave excellent results for the evalua- tion of the variation of modulus of mesophase, with the second one (the two- term model) yielding always more stable results derived in an easy way. The difference in the exponents n, and n1 of the two terms in the three-term model (the third term was a constant), and the exponent Zn in the single power-term or the two-term model, gave a measure of the quality 01’ adheslon between phases and they were called the adhesion parameters or coefficients. Higher values of these coefficients characterized better adhesion, whereas, as these values were reducing, the adhesion became less and less effective. Both types of the unfolding model describe satisfactorily the state of transi- tion of the mechanical and physical properties in the particulate composite from its fillers to the matrix. with the two-term version yielding more stable and therefore more reliable results than the three-term model. REFERENCES I. Hashin, Z., Intern. I. 50!. and Slrucl, 6, p. 539 and p. 797 (1970). 2. Theocan's, P.S., and Sideridis, E., J. Appl. Polymer Science, 29 (1984). 3. Theocaris, P.S., Papanicolaou, G., and Kontou, E., J. Reinf. Plastics and Comp., 1, p. 207 (1982). 4. Takahashi, K.,1keda, M., Hunk-we, K., Tlnalrn. K., and Sakai, T., J. Pol. Sci., 16, p. 415 (1973). Takayanagi, M., Harima, E., and lwlll, Y., Jnl. Soc. Mat. Sal, (Japan) 12, p. 389 (1963). Maurer, F., Ph.D., Thesis at the University of Duisburg FRG (I983). Kemer, E.H., Prat. Phys. Soc. Lond, 869, p. 808 (1956). Van der Poel, C.. Rheologica Acts, 1, 2—3, p. 198 (1958). Lipalov, Yu. "Physical Chemistry of Filled Polymers,” Eng]. Trans. by RJ. Moseley, Intern. Polymer Sci. and Techn., Monograph No. 2 (1977). 10. Thoocan's. P.S., Kefalas, E., and Spathis, G., Jnl. App]. Pol. Sal, 28, 12, p. 3641 (1983). 11. Papanicolaou. G., Paipetis, S., and Theocaris, P.S., Colloid and Polymer Science, 256, p. 625 (1978). 12. ‘I'heocan's, P.S., and Papanicolaou, 6., Fib. Sci. Tech, 12, p. 421 (1979). 13. Papanicolaou, 0.. Theocaris, P.S., and Spathis, (1.. Call. Polymer Sell, 258, p. 1231 (1930). 14. Theocaris, P.S., Proc. Nat. Acad, Athens 591, p. 327 (1984). 15. Theocaris, P.S., Coll. and Polym. Sci, 260, (1984). 16. Theocaris, P.S., “New Developments in the Characterization of Polymers in the Solid State." Advances in Polymer Science, H.H. Kausch Jr H.G. Zachmann Editors, Springer Verlag (1984). 17. Theocaris, P.S., Proc. intern. Cont. lnlerface—lnterphase in Composite Materials, J. Liegeois and S. Okuda Editors, SPE publication. p. Tl-TZB (1983). 18. Lipatov, Yu, "Advances in Polymer Science," 22, p. 31 (1977). 19. Sagnlnev GA, and Simonov-Emeljanov. 1., Plastinassy N2, p. 48 (1973). 20. Spathis, G., Sideridis, E., and Theocaris, P.S., Intern. J. Adhesion and Adhesives, l, p. 195 (1931). 21. Jones, R., Mechanics of Composite Materials, McCrraw»Hill Publ., New York (1975). 22. Kudykinn. T.A., and Pervak, 1.G., UkraiIery Flzicherky Zhurnal, 20, p. 1664 (1975). 23. Takayanagi, M., Uemura, S., and Minami, 8., Jill Polymer Sclenne, Part C, No. 5, p. 113. PP‘HPMV' The Adhesion Quality and the Extent of (he Mesophase in Particulates 231 7A. 25. 16. 27. as. 29. Spathia. 6.. Komou, E., and Theocu'ls, P.S., Jnl. of Rheology, 28 (2), p. 161 (1984). Landel, E., Rubber Chem. and Technology, 35 (2). p. 291 (1962). Droste. 13.11., and Dchnedmo, A.T., Jnl. Appl. Poly. Soil. 13, p. 2149 (1969). Theoclris, P.S., Plplnioolaou, 0., and Sideridis, E., Jnl. Reinfar. Plastics and Comp.. 1 (l), 93 (1982). Price, E., French, D.M., and Tampa. A.S., Jul. Appl. Poly. Sal, 16 (z), 151 (1972). Kwei, T.K., Jul. Polym. 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0277-The Adhesion - VT The Adhesion Quality and the Extent of the Mesophase in Particulates P S THEOCARis Department of Theoretical and Applied

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