7/29/2019 2012D33

1/6

SISOM 2012 and Session of the Commission of Acoustics, Bucharest 30-31 May

BASIC SEQUENCES OF DYNAMIC DISSIPATIVE QUANTITIES IN CASE OF LINEAR

VISCOELASTIC BEHAVIOR OF POLYMER-LIKE MATERIALS. II. ISOCHRONALCIRCUMSTANCES.

Horia PAVEN

National Institute of Research and Development for Chemistry and Petrochemistry - ICECHIM, Bucharest, Splaiul Independentei,202, ROMANIA; email: htopaven@netscape.net.

Typical linear viscoelastic features in case of polymer-like materials, regarding the frequency

and temperature dependence, are inspected from the view point of temperature dependence at given

frequency. Furthermore, given the effects of strain- and stress-controlled conditions, the loss modulus,and the corresponding loss factor, as well as the attached loss compliance, or the loss compliance, thecorresponding loss factor, and the attached loss modulus, are considered. Consequently, the generalframe of isochronal characterization circumstances is concerned, the natural restrictions of

temperature variation of storage modulus and storage compliance being the used for obtaining therelative locations of characteristic temperatures.

Key words: dynamic viscoelasticity, dissipative quantities, polymers, isochronal circumstances,characteristic temperatures.

1. INTRODUCTION

The results established in the analysis of linear viscoelastic behavior in case of strain- and stress-

controlled conditions and isothermal circumstances need to be also considered for the temperature

dependence at given frequency.

In case of dynamic strain-controlled conditions, in isochronal circumstances, the linear viscoelastic

behavior is described by means of temperature, T, and frequency, , dependences.

On the hand, well defined properties regard the primary quantities, including

- Storage modulus, ),;(TM

- Loss modulus, ),;('' TM

- Absolute modulus, | );(* TM |,

- Loss factor, );( TM ,

as well as the attached, secondary ones,

- Storage compliance, ),;( TJM - Loss compliance, );(TJ

M,

- Absolute compliance, | |,);(* TJM

- Loss factor, );( TJ , which is similar to );( TM ,

where the semi-colon, ;, point out a variable quantity, at left, and a parameter, at right.

On the other hand, in case of stress-controlled conditions, well defined properties arise as the

primary ones, containing

- Storage compliance, ),;(TJ

- Loss compliance, ),;(TJ

- Absolute compliance, | |,);(* TJ

- Loss factor, );( TJ ,

7/29/2019 2012D33

2/6

Basic sequences of dynamic dissipative quantities in case of linear viscoelastic behavior. II. Isochronal circumstances.147

as well as the attached, secondary ones,

- Storage modulus, ),;(TJM

- Loss modulus, );(TJM ,

- Absolute modulus, | |,);;(* TTJM

- Loss factor, );( TJ , which is identical to );( TM .

2. METHOD AND RESULTS

Aiming to establish in a phenomenological, model-free way the sequence of typical peaks arising

from the temperature, T, dependence, at given frequency, , of linear viscoelastic losses quantities, both the

set of loss modulus, );(TM , the corresponding loss factor, );( TM , and the attached loss compliance,

);(TJM , and the well as the set of loss compliance, corresponding loss factor and attached loss modulus

are considered [1 - 6].

Strain-controlled temperature-dependent primary-like quantities

From the experimental data, obtained in isochronal circumstances, there is a temperature dependence

of the storage modulus, expressing a decrease of the stiffness with the temperature increasing, i. e.,

0);(

);(T

TMTMDT (1)

Taking into account the standard definition of the loss factor as the ratio of loss to storage modulus,

the corresponding temperature derivative is given as

);(

);();();();(

);(

);(

);(

2

TM

TMDTMTMDTM

T

TM

TM

TD

TT

MT(2)

In order to obtain the needed information about the locations of loss modulus and loss factor, the

variation trend of this loss factor is evaluated at the temperature

)};({ TMmTT , i. e., at the temperature where there is a maximum of the loss modulus,

0|);()};({

TMmTTT

TMD (3)

Thus, one obtains

)};({)};({|);(

);(

);(|);(

2

TMmTMm TTTTTMTTMD

TM

TMTD (4)

and, if the relation (1) is used, it results

0|);()};({

TMmTTMT

TD (5)

7/29/2019 2012D33

3/6

Horia PAVEN 148

i. e., for the value of temperature at which there is a maximum of the ''M peak in the range of temperature

dependence. Consequently, a positive slope results, ad the peak of );( TM

is located at a frequency which

is higher than that of the ''M peak, as

)};({)};({ TmTMm MTT (6)

Strain-controlled temperature-dependent secondary-like quantities

The above presented result needs to be also done from the view point of the attached loss compliance

location in isochronal circumstances. This needs the use of the storage compliance restriction and

establishing of the trend of loss compliance variation with the temperature.

Accordingly, by using the experimental data

0);(

);(T

TJTJD MMT

(7)

and taking into consideration that

);(

);();();();(

);(

);(

);(

2

TJ

TJDTJTJDTJ

T

TJ

TJ

TD

M

MTMMTM

M

M

JT(8)

the appropriate temperature sequence, which provides the relative location of compliance-like quantities,

results by evaluating the variation of the loss factor with the temperature increases, at the temperaturewhere the maximum of the loss compliance is positioned, i. e., when)};({ TJm MTT

0|);()};({ TMJm

TTMT TJD (9)

Accordingly, at the peakMJ

)};({)};({|);(

);(

);(|);(

2

TMJmTMJmMTTMT

M

MTTJT TJD

TJ

TJTD (10)

and given the restriction (7), it results

0|);()};({

TMJmM

TTJT TD (11)

This means that at the characteristic temperature )};({ TJm MTT , where the maximum peak of loss

compliance is located, the slope of loss factor is negative, resulting that theMJ

peak temperature is lower

than that corresponding to peak. The basic temperature sequence arises asMJ ''

)};({)};({ TJmTm MMJTT (12)

7/29/2019 2012D33

4/6

Basic sequences of dynamic dissipative quantities in case of linear viscoelastic behavior. II. Isochronal circumstances.149

If the relations (6) and (12) are taken into account, the complete sequence of characteristic

temperatures is obtained in the form

)};({)};({)};({ TJmTmTMm MTTT (13)

Stress-controlled temperature-dependent primary-like quantities

The above presented result needs to be also done from the view point of the loss compliance location

in isochronal circumstances. This means to use the storage compliance and to establish the trend of variation

with the temperature increase for storage compliance.

Accordingly, by using the experimental data

0);(

);(T

TJTJDT (14)

and taking into consideration that

);(

);();();();(

);(

);(

);(

2

TJ

TJDTJTJDTJ

T

TJ

TJ

TD

TT

JT(15)

the appropriate temperature sequence which provides the relative location of compliance-like quantities

results by evaluating the variation of the loss factor with the temperature increases, at the temperature

where the maximum of the loss compliance is positioned, i. e., when)};({ TJmTT

0|);()};({ TJmTTT

TJD (16)

Accordingly, at the ''J peak

)};({)};({|);(

);(

);(|);(

2

TJmTJm TTTTTJTTJD

TJ

TJTD (17)

and given the restriction (17), it results

0|);()};({

TJmTTJT

TD (18)

This means that at the characteristic temperature )};({ TJmTT , where the maximum peak of loss

compliance is located, the slope of loss factor is negative, resulting that the J peak temperature is lower

than that corresponding to peak. The basic temperature sequence arises as''J

)};({)};({ TJmTm TT J (19)

7/29/2019 2012D33

5/6

Horia PAVEN 150

Stress-controlled temperature-dependent secondary-like quantities

Given the experimental data, in isochronal circumstances, there is a temperature dependence of the

storage modulus, expressing a decrease of the stiffness with the temperature increase, i. e.,

0);(

);(T

TMTMD JJT (20)

Taking into account the standard definition of the loss factor as the ratio of loss to storage modulus,

the corresponding temperature derivative is given as

);(

);();();();(

);(

);(

);(

2

TM

TMDTMTMDTM

T

TM

TM

TD

J

JTJJTJ

J

J

MT J (21)

In order to obtain the needed information about the locations of loss modulus and loss factor, the

variation trend of loss factor being evaluated at the temperature )};({ TMm JTT , i. e., at the temperature

where there is a maximum of the loss modulus,

defined as

0|);()};({ TJMm

TTT TMD (22)

results in

)};({)};({|);(

);(

);(|);(

2

TJMmTJMmJTTJT

J

JTTMT TMD

TM

TMTD (23)

and, if the relation (1) is used, one obtains

0|);()};({

TJMmJ

TTMT TD (24)

i. e., for the value of temperature at which there is a maximum of the JM peak in the corresponding

temperature dependence. Consequently, since a positive slope resulted, the peak of );( TJM

is located at a

frequency which is higher than for the peak, respectivelyJM

)};({)};({ TmTMmJMJ

TT (25)

and taking into account the relations , it results the sequence

)};({)};({)};({ TJmTmTMm TTT J (26)

7/29/2019 2012D33

6/6

Basic sequences of dynamic dissipative quantities in case of linear viscoelastic behavior. II. Isochronal circumstances.151

3. CONCLUSIONS

As a direct consequence of typical temperature dependence of storage modulus, and the standard

definition of loss factor, in isothermal circumstances, the complete sequence of characteristic temperatures

corresponding to maximum peaks of losses quantities is presented.

The temperature at which the loss factor peak resultes is lower than that obtained in case of the losscompliance one, but higher than the temperature at which the is located the peak of loss modulus.

The dynamic linear viscoelastic quantities obtained in isochronal circumstances are suitable for

identifying extensions and locations of dissipative relaxation/ retardation mechanisms.

The approach is a general, model-free one.

REFERENCES

1. BRINSON, H. F., BRINSON, L. C., Polymer Engineering Science and Viscoelasticity. An Introduction, Springer, New York,

2008.2. van KREVELEN, D. M., te NIJEHUIS, K., Properties of Polymers, Elsevier, Amsterdam, 2009.3. LAKES, R., Viscoelastic Materials, Cambridge University Press, Cambridge, 2009.

4. PAVEN, H., Dual Thermorheodynamical Approaches of DMTA Data in the Framework of Standard Viscoelastic Behavior, IstCentral and Eastern European Conference on Thermal Analysis and Calorimetry, Craiova, 7 - 10, Sept., 2011.

5. PAVEN, H., VULUGA, Z., Characteristic Frequencies and Temperatures Underlying 2D - 3D DMTA Data in Case of Strain-and Stress-Controlled Conditions, NANOTOUGH, PC - NMP - 21346, Copenhagen, 21 - 24, Sept., 2011.

6. PAVEN, H., VULUGA, Z., NICOLAE, C. A., IORGA, M., GABOR, R.,Localizations of Dissipative Energetical Effects inPolymer Maerials.II. Characteristic Temperatures in Isochronal Circumstances, PRIOCHEM- ICECHIM, 27 - 28 Oct., 2011.