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SISOM 2004, BUCHAREST, 20-21 May

BASIC MIXING RULES AND RHEODYNAMIC EFFECTS IN TERNARY

PHASE-IN-PHASE IN PHASE COMPOSITE SYSTEMS WITH LINEAR

VISCOELASTIC COMPONENTS. II. THE LOSS MODULUS

Horia PAVEN*, Sandor POPOVICS*** Research-Development National Institute for Chemistry and Petrochemistry-ICECHIM,

Spl. Independentei, 202, Bucharest-060021, ROMANIA, htopaven@netscape.net** Drexel University, Philadelphia, PA-19401, USA

Abstract: The peculiarities of frequency-controlled mixing rules for the primary rheodynamic

quantity represented by the loss modulus are established in the case of ternary phase-in-phase inphase composite systems with linear viscoelastic components on the basis of corresponding

composite behaviour laws. Aiming at to identify relevant features of different VV, VR, RV and RRmorpho-rheological interactions, the mono-relaxant (retardant)-like rheological model is considered

for components. The basic rheodynamic effects are evidenced in the realm of 3D representation.

1. INTRODUCTION

In the case of viscoelastic behaviour the loss modulus is a natural measure of the vibration damping

availability, and if the composite systems are considered, it is reasonable to suppose that the composite

properties express the individual and cumulate, direct and cross-over contribution of components, thecomposite structural peculiarities at different levels, the amount of componens as well as the intrinsic

interface physical-chemical interactions /1/.

Consequently, in order to complete the phenomenological analysis of rheodynamic data for phase-in-

phase in phase composite systems with linear viscoelastic components, the loss modulus of ternary systems

is also of significant interest /2-4/

2. PHASE-IN-PHASE IN PHASE MORPHO-RHEOLOGICAL INTERRELATIONSHIPS

Taking into account that for the a//b/c ternaryphase-in-phase in phase composite system with linear

viscoelastic components, in the case of the basic [VV] morpho-rheological coupling, the corresponding

rheological behaviour law stands(II.1.1) ][][ VVVV QP =

or,

(II.1.2) )()( cbacbcbcabbcacbacba QPPvVQPPvVQPPVPPP ++=

from the mixing rule for the complex modulus

(II.1.3)****

][ ccbcbbbcaaVV MvVMvVMVM ++=

thespecificmixing rule for the primary rheodynamic quantity - the loss modulus, , is obtained as][VVM

(II.1.4)ccbcbbbcaaVVVV MvVMvVMVMM ++== }Im{*

][][

If the basic [VR] morpho-rheological couplingoperates, the corresponding rheological equation which

express the rheological behaviour law is:

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Basic mixing rules and rheodynamic effects in ternary phase-in-phase in phase composite systems. II. The loss modulus171

][][ VRVR QP = (II.2.1)

i.e., in an explicit form,

(II.2.2) )()( cbabcbaccacabbabcaccbab QQPVQQPvVQQPvVQPPvQPPv ++=+

Then the mixing rule for the complex modulus is

(II.2.3))/()( ********* ][ bccbcbbcbacacabaVR MvMvMMVMMvVMMvVM +++=

and for the primary rheodynamic quantity, the loss modulus, , results thespecific mixing rule][VRM

2

][,

2*

2**

][][ /)//(}Im{ RbccccbbbbcaaVRVR xMMvMMvVMVMM ++== (II.2.4)

where

2/122

*2

*22

*2

*

][, ])//()//[( cccbbbcccbbbRbc MMvMMvMMvMMvx +++= (II.2.5)

If the basic [RV] morpho-rheological couplingworks, the resulting behaviour law is

(II.3.1) ][][ RVRV QP =

and the explicit expression of behaviour law takes the form( (II.3.2) )() cabcbacbacbbccbacabcaba QQPvQQPvQPPVQPPvVQPPvV +=++

the resulting mixing rule for the complex modulus being

(II.3.3))/()( ******** ][ abcccabbacacbabRV MVMvVMvVMMvMMvM +++=

and thespecific mixing rule for the loss modulus one obtains

2

][

2

][,

2**

][][ /])(/[}Im{ RVVbcccbbbcaaaRVRV XxMvMvVMMVMM ++== (II.3.4)

2/122

][,

2*22

][,

2*

][ }]/[]/{[ VbcbcbcaaaVbcbcbcaaaRV xMVMMVxMVMMVX +++= (II.3.5)

(II.3.6)2/122

][, ])()/[(1 ccbbccbbVbc MvMvMvMvx +++=

Finally, the [RR] morpho-rheologicalcoupling is defined by the behaviour law(II.4.1) ][][ RRRR QP =

i.e.,

(II.4.2) )()( cbabaccbccabbbccbaa QQQQQPvVQQPvVQQPV =++

and for the complex modulus the corresponding mixing rule is

(II.4.3))/( ********** ][ bacbccabbccbacbaRR MMvVMMvVMMVMMMM ++=

thespecific mixing rule for the loss modulus, , resulting as][RRM

2

][

2*

2*

2**

][][ /)]//(/[}Im{ RRcccbbbbcaaaRRRR XMMvMMvVMMVMM ++== (II.4.4)

where

2/122][,][,

2*22][,][,

2*][ }]/[]/{[ RbcRbcbcaaaRbcRbcbcaaaRR xMVMMVxMVMMVX +++= (II.4.5)

2/122

*2

*22

*2

*

][, }]//[}]//{[ cccbbbcccbbbRbc MMvMMvMMvMMvx +++= (II.4.6)

3. 3D-REPRESENTATION OF MIXING RULES

Aiming at to point out the consequences of different rheodynamic effects from the standpoint of

specific mixing rules for the loss modulus, the case of (1, 1)- rheological model described by the rheological

behaviour law,

(II.5)p0+ p1Dt= q0+ q1Dt

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Horia PAVEN, Sandor POPOVICS 172

is considered, where are the nominal rheological parameters, and the corresponding

characteristic rheological parameters are

1010 ,;, qqpp

BMPa a,1 1= radsMPaAradsA aa /*1000,/1 10 ==

radsMPaAradsBMPaA bbb /*100,/1.0,1 110 ===

radsMPaAradsBMPaA ccc /*10,/01.0,1 110 ===

The results are summarized in Fig. II.x.1 to II.x.5, for the loss modulus at given frequencies.

The frequency dependence of (1, 1)loss modulus reveals the existence of the characteristic loss peak

corresponding to the value of component characteristic frequency of dynamic relaxation.

In principle, in the case of the considered composite system, as result of intrinsic morpho-rheological

interactions, the component contribution is controlled by v and V volume fractions, and as result of

cumulation of loss and storage moduli, it is rather difficult to establish analytical meaningful criteria of

evaluation and comparison. However, some qualitative guidance accounts can be evidenced from the

viewpoint of frequency control of specific mixing rules.

At low frequencies (=0.1 and 1 rad/s), the v and V dependences are linear for M2VV, and non-

linear for M2VR, M2RV and M2RR.

At intermediate values (=10 and 100 rad/s), there is an increase of non-linear trend of mixing rules forM2VV and M2VR, while for M2Rv and M2RR, M2(V)-maximum effects occur.

At high frequencies (=1000 rad/s), the v and/or V composition effects are increasingly non-linear

for morphologies from VV to RR.

4. CONCLUSIONS

The loss modulus of phase-in-phase in phase composite systems with components showing linear

viscoelastic behaviour results solely in terms of similar quantities of components in case of VV morphology.

The morpho-rheological interactions corresponding to VR, RV and RR basic models lead to relatively

complicated expressions for the composite loss modulus, when there is a well defined contribution of

different terms including both the loss- and storage-modulus.It is useful, both from scientific and application end use of approach, to continue the development of

the interactive knowledge data base for rational underlying of 2D-3D representation of rheodynamic

morpho-rheological interrelationships.

ACKNOWLEDGEMENT

The research was supported in part by the Grant no.6158/2000-2002 from ANSTI/MEC and by

Contract CERES no. 144/2001-2004 from MEC, respectively.

REFERENCES

1. Paven H., Dobrescu V., Model Reological Equations of State in the Linear Viscoelasticity of Polymeric Composites, Polymer

Bulletin (Berlin), 1980, 2, 727-730.2. Paven H., Dobrescu V., Viscoelastic Models in the Rheology of Hybrid Polymeric Composites of Phase-in-Phase in Phase Type,

in Rheology - Applications, vol. III, Astarita G., Marrucci G., Nicolais L. (eds.), Plenum Press, New York, 1980, 229-233.3. Paven H., Basic Interactions in Phenomenological Rheology of Ternary Hybrid Composites with Linear Viscoelastic Solid

Components, Proc. Ann. Symp, Inst Solid Mech., Romanian Academy, 1992, 77-82.4. Paven H., Dobrescu V., Rheology of Polymer Composites. VI. Linear Viscoelasticity of Hybrid Composites of Phase-in-Phase in

Phase Type, Intern. Chem. Eng., 1994, 34, 143-155.

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Basic mixing rules and rheodynamic effects in ternary phase-in-phase in phase composite systems. II. The loss modulus173

Fig. II.x.1. Phase-in-phase in phase rheodynamic mixing rules for the loss modulus for

different morphological architectures - VV, VR, RV, RR - and at = 0.1 rad/s.

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Horia PAVEN, Sandor POPOVICS 174

Fig. II.x.2. Phase-in-phase in phase rheodynamic mixing rules for the loss modulus fordifferent morphological architectures - VV, VR, RV, RR - and at = 1 rad/s.

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Basic mixing rules and rheodynamic effects in ternary phase-in-phase in phase composite systems. II. The loss modulus175

Fig. II.x.3. Phase-in-phase in phase rheodynamic mixing rules for the loss modulus for

different morphological architectures - VV, VR, RV, RR - and at = 10 rad/s.

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Fig. II.x.4. Phase-in-phase in phase rheodynamic mixing rules for the loss modulus for

different morphological architectures - VV, VR, RV, RR - and at = 100 rad/s.

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Fig. II.x.5. Phase-in-phase in phase rheodynamic mixing rules for the loss modulus for

different morphological architectures - VV, VR, RV, RR - and at = 1000 rad/s.