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Explicit Analytical Relations for Strain-controlled

Rheodynamical Quantities in the Case

of Zener-Arrhenius Model

I. Underlying relations

HORIA PAVEN*

National Institute of Research and Development for Chemistry and Petrochemistry - ICECHIM, 202 Splaiul Independentei,060021, Bucharest, Romania

In order to point out the intrinsic peculiarities of thermorheodynamical behaviour of solid-like polymermaterials, the framework of a Zener - Arrhenius model is considered, the frequency or/and temperaturedependences being approached in the case of strain-controlled conditions. The complete set of the explicitgeneral relations providing the characteristic thermorheodynamical quantities (the storage, loss and absolutemoduli, as well as the loss factor, and the corresponding storage, loss and absolute compliances) is presentedin terms of meaningful rheological parameters including the low - and high frequency limit values of thestorage modulus, and the appropriate relaxation time. The underlying analytical relations are given in formssuitable for direct numerical simulation of both isothermal frequency - dependence, and well as of isochronaltemperature - dependence circumstances.

Keywords:analytical relations, characteristic thermorheodynamical quantities, strain-controlled conditions,Zener - Arrhenius model, isothermal and isochronal circumstances.

The ability of solid-like polymer materials to specificallydeform is a direct consequence of intrinsic mobility of theirmolecular entities to accomplish typical motions including,in the case of strain-controlled conditions, the relaxationones. Moreover, given the influence of the dynamicdeformation-driven level, as well as of those of frequency

and temperature dependences, the viscoelastic-like effectsare properly evidenced by the rheological quantities [1-5].In fact, linear viscoelasticity allows for time dependence

in the rheological stress-strain relationships by consideringa linear function of stress and its derivatives with respectto time to be equated to a linear function of strain and itstime derivatives [6-11].

Accordingly, the present article intend to clarify theessentials of different relationships expressing thefrequency and temperature variation trends ofrheodynamical quantities, and to provide an appropriatecharacterization of the linear viscoelastic peculiarities ofrheological behaviour by using a relaxation andcorresponding retardation description in the framework

of a coupled Zener - Arrhenius model. Thus, theconsequences of the explicit general and characteristicforms of the complete set of linear viscoelasticrheodynamical quantities point out the frequency or/andtemperature dependences in the case of a sinusoidal strain-controlled excitation, the underlying relations beingconsidered both in terms of phenomenological and physico-chemical meaningful parameters.

The proposed approach supposes the consideration ofthe typical rheological equation

(1)

in terms of and (the strain and stress, respectively),where po, p

1and q

0, q

1are the nominal temperature

dependent rheological parameters, the time contributionbeing stated by means of time derivative term, D

t=/t.

As a matter of fact, it results that this model is a three -parameter one, i. e.,

(2.1)

wi th th e corr espo ndin g char ac teri st ic rh eo logi ca l

parameters Ao(T), A1(T), B1(T) given by

(3)

Taking into account the intrinsic linearity of consideredrheological equation from the viewpoint of time-derivativeoperator, D

t, in case of a controlled sinusoidal dynamic

strain,(4)

where o

is the strain amplitude and - the angularfrequency of the given sinusoidal strain - the corresponding

dynamic form results as (Dt = i)

(2.2)

The involved stress - with the resulting amplitude o,

and of same frequency as that of the strain and, as result ofinternal friction, with a phase lag, , between the excitationand response - is

(5)

Accordingly, by definition

(6)

* email: [email protected]; Tel.: 0726394575

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is the complex modulus, M and M being the storage andloss moduli, respectively. On the other hand, for theabsolute (viscoelastic) modulus, |M*|, results

(7)

whereas M

= tan M

stands for the corresponding lossfactor, being given as

(8)

From a hierarchical viewpoint, it appears that in thecase of - controlled excitation, M, M are the directprimary rheological quantities, whereas |M*|,

Mare the

derivateprimary ones. Moreover, the complete set of rheological quantities also includes the well definedcompliance-like quantities - the corresponding storagecompliance, J

M,

(9)

loss compliance, JM,

(10)

and absolute compliance, |J*M|,

(11)

which are termed assecondary rheodynamical quantities.In order to include, besides the frequency contribution,

that of temperature one - it is suggested that, in principle,the deformation rate could be expressed by an Arrhenius-like relation [12-17], the most frequently used form, at leastas a preliminary evaluation tool, being an exponential one

defined for the controlled relaxation time, , as

(12)

where

= () is the prefactor (pre-exponential factor)

- denoting the value of the corresponding relaxation timeat infinite temperature (if

0), while

=A

/R - i. e., the

ratio of the Arrhenius activation energy, A, to the

universal gas constant, R = 8.314 J / (mol* K) - points avirtual temperature.

Consequently, the complete frequency andtemperature form of rheodynamic equation arises as

(2.3)where

(13)

the meaning of different rheological parameters beingexpressed in terms of the storage modulus at low (l) andhigh limit (h) frequency, respectively, and the characteristic relaxation time.

Results and discussionsAs results from the general form of the dynamic

rheological equation (2.3), there are distinct dependencesof characteristic rheodynamical quantities on frequencyand temperature, respectively. Thus, for the complete setof seven above quoted rheological quantities, twocircumstances are to be taken into account - on the hand,

that of frequency dependence at given temperature, whichcorresponds toisothermal circumstances, and, on the otherhand, that of temperature dependence, when the data areobtained inisochronal circumstances.

By using the general definition of temperature andfrequency dependences of rheodynamic quantities, thecharacteristic relations are indicated by a semicolon (;),where the first term denotes the continuous variable, whilethe second one appears as a parameter.

In the case of a sinusoidal strain-controlled excitation,the complete set of primary and secondary rheodynamicquantities includes the corresponding general andcharacteristic forms as follows.

Direct primary quantities- The general form of the storage modulus results, on

the basis of (2), (6) and (13), as

(14)

Accordingly, for the frequency dependence at giventemperature, T = T, one obtains

(14.1)

and

(14.2)

in the case of temperature dependence at given frequency,=.

- The general expression of theloss modulus is, by takinginto account (2), (6) and (13),

(15)

Correspondingly, for the - dependence at fixedtemperature, T=T,

(15.1)

(17.2)

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and

(15.2)

in the case of T - dependence at given frequency, =.

Derivate primary quantities- The general form of the absolute modulus results, by

using (2), (8) and (13), as

(16)

Accordingly, for the frequency dependence at giventemperature, T=T, one obtains

and

(16.2)


- The general expression of theloss factoris, given (2),(8), (13),

(17)

while for the - dependence at fixed temperature, T=T,

(17.1)

and


Coresponding secondary quantities- The general form of the corresponding storage

compliance results, on the basis of (2), (9) and (13), as

(18)

whereas fo r th e frequency de pe nden ce at gi ventemperature, T=T, one obtains

(18.1)

and

(18.2)


- The general expression of the corresponding losscompliance is, taking into account (2), (10) and(13), givenas

(19)

while for the - dependence at fixed temperature, T=T,

(19.1)

and

(16.1)

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(19.2)


- The general form of the corresponding absolutecompliance results, given (2), (11) and (13), as

(20)

wh ereas fo r the fr equenc y depe ndence at gi ve ntemperature, T=T, one obtains

and

(20.2)


The explicit analytical relations of the complete set ofseven rheodynamical quantities reveal both the potentialconsequences - from the viewpoint of different forms ofconsidered dependences - as well as the effectively ones -as a start up point for the identification of the independentvalues corresponding to characterist ic - extremum(maximum and minimum) and inflection points.

Thus, in the case ofisothermal circumstances, T=T, thefrequency dependence at strain-controlled conditionsshows:

- for the direct primary rheodynamical quantities -- that the - terms appearing in the

characteristic ratios are of same order (2 to 2) in the caseof M, whereas it is a dissimilar one, (1 to 2), for M;

- for the derivate primary rheodynamical quantities -- the ratio is (2 to 2), in the case of

|M*(,T), and a (1 to 2) one for M(, );

- for the corresponding secondary rheodynamicalquantities -, , - the valuesof characteristic ratios are (2 to 2) in the case of

while it is (1 to 2) for JM(, T).

On the other hand, in the case of is ochron alcircumstances, =, at strain-controlled conditions, the

temperature dependence points out:- for the direct primary rheodynamical quantities,, the T - exponentials appearing in the

characteristic ratios reveals that in the case of M the termsare of same order (2 to 2), but dissimilar ones (1 to 2) forM;

- for the derivate primary rheodynamical quantities -- the order of exponential terms which

appear in the characteristic ratios is (2 to 2) in the case of|M*(T; )|while the (1 to 2) result is obtained for

M(T; )

- for the corresponding secondary rheodynamicalquantities , - thecharacteristic ratios of exponential terms are given as (2to 2) in the case of and (1 to 2) for

ConclusionsThe explicit analytical relations obtained in the case of

dynamic strain-controlled processes for both the primaryrheological quantities as well as for the correspondingsecondary ones tell us well definite qualitative featuresconcerning the frequency or/and temperaturedependences.

Moreover, it is revealed the natural contribution offrequency dependence in isothermal circumstances,whereas for the temperature dependence in isochronalcircumstances well defined contribution of exponentialterms containing the activation energy (virtualtemperature) are present.

Furthermore, the direct thermorheodynamic quantities- the storage and absolute moduli - on the hand, as well as,on the other hand, the corresponding ones - the storageand absolute compliances - are typical in their frequencydependence, the situation being somewhat comparablefrom the standpoint of temperature dependence.

Henceforth, in the case of the other considered directthermodynamic quantities, including the loss modulus andthe loss facor, as well as for the corresponding losscompliance, the frequency or temperature trends appearto be similar.

Definite peculiarities of the full set of characteristicthermorheodynamic quantities are to be identified by taking

into account distinct frequency or/and temperaturedependences on the basis of numerical simulation route.

References1. LAKES, R. S., Viscoelastic Materials, Cambridge Univ. Press,Cambridge, 2009, p. 55-89, 207-270.2. BRINSON, H. F., BRINSON, L. C., Polymer Engineering Science and

Viscoelastici- ity, Springer, New Yok, 2008, p. 2213. TSCHOEGL, N. W., Phenomenological Theory of Linear ViscoelasticBehavior,Springer, New York, 1989, p. 69

4. FERRY, J. D., Viscoelastic Properties of Polymers, Wiley, NewYork,19805. ALFREY, T., Mechanical Behaviour of High Polymers, Interscience,New York, 1948, p. 103

6. MANLEY, T. R., Pure & Appl. Chem., 61, 1989, p. 13537. LI, Y., XU, M., Mech. Time-Depend. Mater., 10, 2006, p. 1138. LI, Y., XU, M., Mech. Time-Depend. Mater., 11, 2007, p. 19. SWAMINATHAN, G., SHIVAKUMAR, K., J. Reinf. Plast. Comp., 28,2009, p. 97910. PAVEN, H., Mat. Plast., 40,no. 4, 2003, p. 17111. PAVEN, H., Mat. Plast., 43, no. 4, 2006, p. 34512. KRAUSZ, A. S., EYRING, H., Deformation Kinetics, Wi le y-Interscience, New York, 1975.p. 6-28, 107-14213.] KARGIN, V. A., SLONIMSKY, G. L., BRIEF ESSAYS ON THE PHYSICS

AND CHEMISTRY OF POLYMERS, Himija, Moscow, 1967 (in Russian).14. ARRHENIUS, S., Z. Phys. Chem., 4, 1889, p. 22615. BECKER, Phys. Z., 26, 1925, p. 91916. ALEXANDROV, A. P., LAZURKIN, Yu. S., Acta Physicochim., 12,1940, p. 64717. PREVORSEK, D., BUTLER, R. H., Intern. J. Polymeric Mater., 1,1972, p. 251

Manuscript received: 10.06.2010

(20.1)

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