# Polymer Chemistry. The Basic Concepts - Himenz P.C.

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A dynamic modulus can also be evaluated by following a procedure

similar to that used for the dynamic compliance above. The modulus is

most easily approached by considering a Maxwell element in which case the

differential equation analog to Eq. (3.77) is

ñîòî cos (cot) = - - + - a

(3.81)

G* dt ö*

Equations (3.77) and (3.81) both have the same general form dy/dt + Py =

Q, so the general solution-given in Example 3.5-is the same for both,

although the values of the constants are different. When the constants

are evaluated, the storage and loss components of the modulus are found

to be

G'(co) = co2r2 G*(l +co2t2)-1

(3.82)

and

G"(co) = gjtG*(1 + co2t2)-1

(3.83)

respectively.

The dynamic viscosity is related to the loss component of the shear

modulus through the result T?dyn = G"/co As ñî -> 0, the dynamic

viscosity approaches the zero shear viscosity of an ordinary liquid, t?n.

We commented above that the elastic and viscous effects are out of

phase with each other by some angle 5 in a viscoelastic material. Since

both vary periodically with the same frequency, stress and strain

oscillate with t, as shown in Fig. 3.14a. The phase angle 5 measures the

lag between the two waves. Another representation of this situation is

shown in Fig. 3.14b, where stress and strain are represented by arrows of

different lengths separated by an angle 5. Projections of either one onto

the other can be expressed in terms of the sine and cosine of the phase

angle. The bold arrows in Fig. 3.14b are the components of 7 parallel and

perpendicular to ñò. Thus we can say that ó cos 5 is the strain component

in phase with the stress and ó sin 6 is the component out of phase with

the stress. We have previously observed that the elastic response is in

phase with the stress and the viscous response is out of phase. Hence the

ratio of

178

The Elastic and Viscoelastic States

(a)

Çòã

Figure 3.14 General representations of stress and strain out of phase by

amount 5: (a) represented by oscillating functions and (b) represented by

vectors.

the two measures the relative contributions of these effects. These two

contributions are also measured by J'(co) and J"(co); therefore

7 sin 5

/Ýîë,

= tan 5 =

(3.84)

7 cos 5 J (ñî)

where the single parameter tan 5 is called the loss tangent. The loss

tangent is also proportional to the energy loss per cycle, so both J"(co)

and tan 5 measure viscous dissipation of mechanical energy in a sample.

Often the quantity n tan 6 is reported; it is called the logarithmic

decrement.

3.11 The Dynamic Components: Measurement and Interpretation

The wide range of variables involved makes experimental viscoelasticity

somewhat difficult to describe. In a particular study the range might be

narrowed because of the specific character of the material under

investigation. In more general terms, however, such wide ranges of

variation are involved that no single experimental design is suitable for

all purposes. Figure 3.12, for example, spans 9 orders of magnitude

variation in J(t) and 18 orders in t (no wonder log-log

The Dynamic Components: Measurement and Interpretation

179

coordinates are standard in this type of work!). In addition, samples can

vary from viscous liquids to brittle solids, and there may be

restrictions on the size or shape of available samples.

Both stress relaxation and creep experiments have their place in the

study of viscoelasticity, but measurement and interpretation of the

dynamic mechanical behavior of a polymer gives the most information on

the most convenient time scale. We shall describe the experimental

aspects of dynamic viscoelasticity only in broad conceptual terms. The

actual implementation of these concepts embraces many different kinds of

apparatus and experimental routines. In a conceptually simple apparatus

for measuring the dynamic components, the sample is sandwiched between

two surfaces, one of which is driven with a known periodic displacement.

The force at the other side of the gap is measured by a strain gauge. The

stress and strain at the two surfaces are proportional to the voltage

outputs of electromechanical transducers monitoring each plate. These can

be calibrated so that the maximum force F0 and maximum displacement x0

are read directly. The signals could be fed into a dual-beam oscilloscope

so that the sinusoidal variation of both the stress and strain appear on

the screen. The phase angle 6 can be measured from this trace, as shown

in Fig. 3.14a. Alternatively, the ratio of the two signals can be used to

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