Volume preserving and non-preserving strains
1.6 Time dependence
Figure 6. Time dependent shear (lower curues) and extensional (upper curues) stresses normalised b y deformation rates ouer a range of rates. Lines are from a non-linear gen- eralisation of a model of branched polymers discussed in Section 3.
case the polymer melt is composed of monodisperse molecules of identical ('H-shaped') branched structure.
1.6 Time dependence
The stress-growth curves of Figure 6 indicate that viscoelastic materials do not achieve a steady state of stress in a steady flow (or vice versa) until a certain relaxation time has elapsed. This relaxation arises from dynamic processes intrinsic to the materials themselves and can be a very sensitive (if indirect) probe of structural dynamics in soft matter. The most common experiments measure the time dependence of materials in linear deformation.
1.6.1
In a step strain measurement, at time t = 0 a small strain y (usually shear) is suddenly imposed and sustained. The resulting (shear) stress component a ( t ) decays with time, and is measured. If the material is in a true linear response regime, one has o ( t ) = G ( t ) y . The function G ( t ) is the time dependent relaxation modulus, and is monotonically decreasing with time.
We will normally restrict ourself to isotropic materials, in which G ( t ) is a scalar func- tion of time. Lamellar, nematic and other ordered phases of surfactants and block co- polymers will have special directions in which measurements of G ( t ) may give very differ- ent results. However, polycrystalline samples of these materials recover isotropic rheology.
Very few materials exhibit a single-exponential relaxation modulus G ( t ) 0: exp[-t/~], Step-strain response and relaxation modulus
Rheology of linear and branched polymers 87
which is the linear response result of the Maxwell model (Equation 7). Many more can be described in terms of a sum of relaxation modes:
N
G ( t ) = z g i e - t / r t
.
(8)i = l
The relaxation modulus G ( t ) may be measured directly, but this suffers from two major drawbacks: (i) the initial step-strain is never quite instantaneous, degrading measurements of short relaxation times; (ii) the signal to noise ratio at long times is very weak, degrading measurements of long relaxation times.
The same information may instead be extracted from other flow histories, so long as the material properties are time-independent. That is, to each incremental strain d y ( t ’ ) applied prior to time t there is a corresponding incremental stress d o ( t ) = G(t - t’)dy(t’).
We say that the material then has Time Translation Invariance (TTI-see the lectures of Bouchaud, this volume). Exceptions to this class are materials that are not in equilibrium (even in the absence of a flow), but which ‘age’ towards it on time scales longer than the length of the experiment. Using TTI we may write (suppressing tensor indices)
a ( t ) = lim x G ( t - t’)by(t’) =
67-0 dt‘
67
(9) which is the linearised constitutive equation between shear strain y ( t ) and stress o ( t ) .
1.6.2 Frequency-dependent modulus
The most common strain history used to extract the equivalent of G ( t ) is the harmonic oscillation y(t) = Re (yOeawt). Then using Equation 9 we write
(10)
t
azy(t) = Re
([,
G(t - t’)yoiweiwt’dt’ = Re (YoG*(w)eiwt) , with the ‘complex modulus’ G*(w) defined byM
G*(w) = iw G ( t ) e - i w t d t . (11) The form of Equation 10 means that the stress will be simple harmonic at frequency U , but not in phase with the strain. If we write G * ( w ) = G ’ ( w ) + i G ” ( w ) , then we can identify the real part G‘ as the in-phase (elastic) part of the modulus and the imaginary part G” as the out-of-phase (dissipative) part. In general both will be frequency-dependent, crossing over from viscous (dissipative) behaviour at low frequencies to elastic behaviour at high frequencies. Before giving examples, let us summarise these two ideal limits:
Ideal newtonian fluid (viscosity 7 )
Ideal elastic solid (modulus G o )
88 Tom McLeish
1.6.3 Examples The Maxwell model
Now we can interpret what the frequency-dependent experiment will give us for the simplest model of a viscoelastic fluid, with a single relaxation time G ( t ) = GOe-t/T. The integral over t is readily done to yield
w2r2 w r
G’(w) = Go- G”(w) = Go-
1 +w*r2
’
1+
w2r2 ‘Note that the correct elastic and viscous behaviour are recovered at high and low frequency respectively. The characteristic time emerges as the inverse of the frequency at which the curves for Gf,Gff cross (or where G” is maximum, in this case). The result for the steady state viscosity is 9 = G r . More generally, Equation 9 gives the exact integral for the ratio of stress to strain rate in steady state as 9 =
JF
G(t)dt, so it is always true that 77 N Gr where G is an effective modulus and r a characteristic relaxation time.Polymeric matter
We finish this survey with a few examples of the elastic and loss modulus for polymeric materials. It is possible in many such cases to extract effective information on relaxations covering many decades of frequency, because of time-temperature superposition. For most polymers above both their melting point and glass temperature T,, the time scales of all viscoelastic relaxations shift with temperature by the same factor aT = exp[A/(T
-
TO)], with material-dependent values of A and To. (This is the Volgel-Fulcher, or WLF form;see e.g. the lectures by Kob, this volume). Up to 12 decades in frequency are then accessible for polymers with very low T,, by superposing data of different T .
1 1
I
I
A 8 P I 5Mk 100
Figure 7. Linear viscoelastic moduli G‘ and G” as functions of oscillation frequency w , of monodisperse melts of polystyrene, polyisoprene and polybutadaene of similar degree of entanglement (MIM,).
In Figure 7 we show results for three chemistries of near-monodisperse linear polymer melts. Note that the data are, as usual for such experiments, plotted on log-log axes in which the Maxwell model would have G“ with slopes of 1 and -1 each side of the
Rheology of linear and branched polymers 89
I I 1 I 1 I ) * ut 5''
I
10"
r k
lo" IO0Figure 8. Near-Maxwell behaviour of a wormlike surfactant solution.
maximum (see Equation 12). The slope in the data is much shallower on the right, indicating the presence of some shorter relaxation times (Equation 8) but there is still clearly a dominant time at the crossover from viscous to elastic behaviour.
There is one family of polymer-like systems with a near-Maxwell behaviour: the self- assembled wormlike surfactant micelles. These entangled polymers support an additional dynamics of breaking and reforming, that narrows the viscoelastic spectrum towards a single exponential (Figure 8); see [l].
Finally we examine the effect of a change of molecular topology on the linear rheology.
Figure 9 compares G*(w) for a linear and three-arm star architecture of polyisoprene melt.
As before, the linear polymer has a strong dominant relaxation time, but the branched variety is quite different: the maximum in G"(w) is no longer anywhere near the crossover point, indicating a much broader superposition of relaxation modes. The terminal time is also much longer in the case of the star polymer. We will examine the reasons for this critical effect of branching in Section 3.