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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 threeterm and twoterm unfolding models, which were developed. to describc
mesophase phenomena in ﬁberrcinforced 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 threeterm and twoterm unfolding models, introduced for thc
ﬁbercomposites, 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 Dfﬁller 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 07316844/84/03/0204—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 twosphere 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 twophase 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 physicochemical 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 coworkers [4] and by
Takayanagi [5] introduced explicitly in the expression of the Ecmodulus 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 heatcapacity 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 stresssingularities, 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 ﬁberreinforced 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 Emmodulus, 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
threephase 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 SimonovEmeljanov [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 fillermodulus 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 Ecvalues 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 twoterm unfolding model for the com Dliances. as it has been developed for the fiberreinforced 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 Ecmodulus 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 strainenergy 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 stressratios 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 shearmoduli 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 bulkcompliance (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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