B. Wessling:

Critical Shear Rate - the Instability Reason for the Creation of Dissipative Structures in Polymers

(Z. Phys. Chem., 191, (1995), S. 119-135)

Only very few of the countless different polymer systems we meet in our daily life are straight single-phase systems. At least most real polymers consist of two phases.

For most of them, many surprising and non-linear phenomena are well known and have been the subject of continuous work for decades. Especially the resulting properties, such as impact modification, viscosity and conductivity, display a non-linear dependence on a given parameter. The property/ parameter relationship can be described by an S-shaped curve. Theories which are in principle based on considerations of equilibrium thermodynamics have been developed and are currently being used to explain these phenomena. These theories include the "Flory-Huggins-Theory", the "Percolation Theory", the "Nearest-Neighbour-Model", but also the constitutive and related equations for the description of rheological phenomena.

In contrast to this, based on experimental results, we [1] have recently developed a new theory which is applicable to all heterogeneous polymer systems [6, 7]. Our main principle is to define the nonequilibrium character of colloidal dispersions in polymeric matrices and to interpret the experimental findings (phase separation, dispersion-flocculation phase transition, [3]) as "dissipative structures". This term was introduced by PRIGOGINE [2] for self-organising structures in non-linear systems far from equilibrium. In the case of dispersions in polymers, properties like conductivity, impact strength, etc., are measured as a result of the "frozen dissipative structure", whereby rheological phenomena (viscosity etc.) are to be considered as dynamic nonequilibrium phenomena. Dissipative structures can generally only survive under continuous entropy export; this is the case under dispersion conditions, in which a huge amount of high value energy is pumped through the system; it continues to be the case during melt flow in the molten stage, but not after the multiphase system has been quenched; then the high-viscosity energy barrier prevents the generated dissipative structures from falling apart.

Since colloidal dispersions in polymeric matrices are processes of isothermal nature, differences of temperature are neglected in the following; this means it is always assumed that T = 0.

In the sections below, the following abbreviations will be used:

G : Gibb's free enthalpy of mixing; G= U + pV - TS

H : dispersion or mixing enthalpy

S : entropy

Q : heat

d : distance

t : time

P : probability

Ji : flux, flow

Xi : force

: interaction parameter

: viscous strain

k : Boltzmann constant

: surface tension

T : temperature

: chemical or dispersion state potential

W : work, energy

A : area

: viscosity

a : rheological interaction parameter

n : number of particles

: volume fraction

1. Thermodynamic considerations concerning the present theory

The new "nonequilibrium thermodynamic theory of heterogeneous polymer systems" [1] is aimed at providing a basis for an integrated description for the dynamics of dispersion and blending processes, structure formation, phase transition and critical phenomena.

According to PRIGOGINE [2], dissipative structures are to be expected if in open systems the distance from thermodynamic equilibrium exceeds some critical value, di > di,crit. In that region, the relations between flows (fluxes) and forces are nonlinear and the standard Prigogine principle, which is valid only in the linear regime, is to be replaced by the Glansdorff-Prigogine evolution criterion


It is well known that none of the multiphase systems is spontaneously formed. The formation processes are all endergonic.

Therefore the entropy change of the (irreversible) dispersion process can be calculated according to irreversible thermodynamics of nonlinear processes. In principle, diffusion and dispersion can be treated in a similar way, they apparently lead to formally similar structures. Both processes are irreversible and nonlinear.

According to irreversible thermodynamics in the neighbourhood of equilibrium, the internal entropy change (production) would be the product of all fluxes and forces:


The entropy change (production) is then:


This is the difference between an "excess interfacial energy", 12eA12 , caused by the work needed for dispersing the particles, Wdisp , and the surface energy of component 2 before dispersion, (2A12 (the term (1A1 can be neglected), divided by the average dispersion path length.

This can be estimated empirically with a good theoretical basis by stating that (12eA12 is the energy which is necessary to provide 1 m2 of matrix polymer and force it to wet 1 m2 of carbon black or of an ICP, during which the interface A12 is formed. This is the minimum amount of work W , necessary to force the volume Vsyst , of the developing polymer dispersion with the viscosity at the shear rate sr , to flow:


whereby (see ref. [1], footnote [4d] therein):



(12 is the equilibrium surface tension (eq.(7) in [1]).

In reality, W is obviously only a minimum (ideal) value for 12eA12. A better experimental basis is the real dispersion work, Wdisp , which can be experimentally determined.

According to our experience, the total enthalpy needed for preparing such systems is about 2 MJ/kg, a value large enough to suggest that these systems are being driven far away from equilibrium.

A rough estimation can show that one of the conditions for order under nonequilibrium, a significant amount of negative entropy change, is fulfilled [1]. The next steps in analysing the nonequilibrium properties are to prove the nonlinearity of the process and to find out whether the distance to equilibrium is supercritical. Therefore we consider similarities to and differences from the irreversible diffusion process, which shows a positive entropy production.

To arrive at a solution for the dispersion law, entropy development during the diffusion process must be considered. This entropy develops over time as an e-function approaching a saturation value. This behaviour is characteristic of an irreversible process in the direction of thermodynamic equilibrium, and we are still in the thermodynamic branch.

The solution of a "dispersion law" is a function like


where ru , no = radius and number of the undispersed particles, respectively.

The "dispersion law" might therefore have a special solution like


Two new factors have been introduced. rd describes the particle radius after dispersion and c is a constant, describing dispersibility. It is used with the dimension [c] = [m].

The expression c/rd has the dimension 1, where [c] = [m] = [(m2*N)/ (N*m)] = [surface tension/pressure]. This correctly reflects the fact that dispersion is the result of a stress (N/m2), a "dispersion pressure" induced through the polymer by Wdisp, being applied against a surface tension (N/m) of the material to be dispersed, which works as a counter pressure induced by interfacial forces between the particles and the matrix and is directed against dispersion. The "dispersion stress" is transferred to the agglomerates to be dispersed via the shear force applied to the polymer matrix.

There is some evidence from the arguments given above that dS/dt <0, and also that dxP < 0 for the dispersion process itself. The entropy function is more complicated (visualised in the detail diagram in fig. 1.1

( and fig. 1.2 )), because in every dispersion, the extruder shear (leading to dispersion) is not continuous, but intermittent, followed by flow relaxation, with phase separation: a further small negative entropy change step (dS/dt <0).

So we see that dispersion provides enough negative entropy flow (entropy export) to force the system far away from equilibrium and allow it to build up "dissipative structures". The distance from equilibrium is very large, i.e. we are beyond the thermodynamic branch.

2. Critical shear rate at bifurcation point

This problem is related to the question of whether there is a minimum work input required before dispersion begins to take place. Or, in nonequilibrium terms: What is the critical parameter at which "bifurcation"

( fig. 1.3 ) occurs, and what is the value needed to make the system leave the thermodynamic branch?

The "dispersion law" and its possible solution, eq.(1.8), describing the dynamics of dispersion, could enable us to find the instability. With the values

c = 0.1 m, rd = 10-7 m, t = 100 s

we analysed the evolution of the particle numbers with changing sr ( shear rate):

tab. 1.1 . It can be seen that an appreciable initial degree of dispersion after a residence time of 100s will only be found if sr>1300 s-1 (dispersion degree > 0.1%). Therefore it can be concluded that the shear rate is the critical parameter, and its critical value above which dispersion takes place or dissipative dispersion structures are created is around 1000 s-1.

It seems helpful to reformulate the exponent in eq.(1.8), starting with the considerations about the dimension of c given there:

(1.9) with

and with

Empirically we know that there is no dispersion to be detected under pure pump extrusion conditions (pure melting and conveying screw design). It is known that a minimum shear stress has to be applied to obtain a significant degree of dispersion, e.g. of pigments (cf. fig. 1.4 ) .

Fig. 1.4 shows the development of the colour intensity (or: colour strength) of any kind of pigment in a polymer, which can be measured according to DIN 53234_. With increasing dispersion degree, the colour strength (represented by a increasing colour strength value [%]_) also increases. This can be achieved either by increasing dispersion time (at a given supercritical shear rate) or by increasing shear rate (for a given residence time). Also certain types of carbon black are used as pigments.

A recently published comparison [5] of such a carbon black dispersed in three low viscosity media (water, squalene, polydimethylsiloxane) showed that dispersion only takes place above a critical shear rate and: the lower the viscosity, the higher the necessary critical shear rate. Moreover, [5] is the only report available with a quantitative description of this qualitatively known dependence. But even [5] does not supply an answer to the question: "What is this critical shear rate in physical terms?"

Introducing the experimentally observed critical boundary for first occurrence of dispersion

, it follows that


This means that there is no mathematical information about n (in eq.(1.8)) for crit, and the equation (1.8) with the exponent as shown in (1.10) is not applicable. This is in accordance with our own experiments in low shear extrusion, and with the results published in [5]: if crit, no dispersion occurs.

Another approach to describing the observed behaviour is to introduce the above-mentioned definition of 12e: The value of 12eA12 is at least as large as the work

, necessary to overcome the viscous strain of the polymer before it is able to wet the dispersed phase (see above, eq.(1.4)). Introducing this in eq.(1.9), it follows,

( Vdisp = volume of the dispersed phase):


( Vsyst is the total volume of the matrix polymer/dispersed phase system).

At the present stage, in which we are now just entering a nonequilibrium thermodynamic description of multiphase polymer systems, an "ab initio" theoretical derivation of eq.(1.7) and (1.8)ff is still lacking. The foregoing thoughts and reformulations may at least lead to some important conclusions:

1) The critical shear rate above which eq.(1.8) results in a first physically appreciable degree of dispersion (> 0.1 %) is in the neighbourhood of what is known to cause "melt fracture" (sr 1000 s-1, 105 Nm). This leads to the hypothesis that dispersion can only occur and be observed under conditions of melt fracture, a widely known rheological instability [4][1.]

("Melt fracture" can be observed at the die of an extruder or melt rheometer as a sudden change in the surface aspect of the emerging melt beyond a critical point in the vicinity of  ~ 105 Nm or sr ~ 1000 s-1. Under given extrusion conditions it suddenly appears at a certain critical point during continuously increasing output. It can be viewed as the analogy of turbulent flow in low-viscosity media. Unlike there, "melt fracture" structures can be frozen by simply cooling the melt strand or the produced film. It exhibits irregular wave and/or fish-scale patterns. These patterns will again suddenly change to new patterns after a second critical point in response to a further increase in output.)

2) The non-linear behaviour of the melt is then well reflected in the exponent (eq.(3.10)), which leads to a definition of n = f(t) only for > crit (above "melt fracture").

3) Independently of this approach, eq.(1.11) tell us about two other non-linear phenomena:

a) the non-linear dependence of Vsyst on dispersed phase concentration, cf. the density non-linearity [8];

b) the relation of 12/12e (the "structure factor"), which behaves non-linearly according to fig. 1.5:

Fig. 1.5

12e is not defined for a dispersed phase concentration of zero; for low viscosity systems 12e will be identical to 12 for a certain concentration regime; for "easy-to-disperse" fillers 12e will only differ from 12 above a certain concentration; in general, two-phase systems cannot reach the 12e level from the 12 equilibrium level without experiencing a (non-linear) jump: there is no continuity between 12 and 12e.

These results, combining the widely known instability (and dissipative structure!) phenomenon of "melt fracture" with the new nonequilibrium description of multiphase polymer systems, will hopefully stimulate more experimental and theoretical work devoted to these (frozen) dissipative structures. It still remains an open question which property of the melt may be responsible for its suddenly occurring capability to disperse fillers (pigments, carbon black, etc.) or other incompatible polymers above melt fracture conditions. We can only speculate that especially the creation of microvoids (= inner surfaces) and a sudden increase in gas solubilisation capability at and above "melt fracture" allows the polymer melt to wet the surface of the material that is to be dispersed. This means that a polymer melt might have completely different (supercritical) properties above melt fracture than we usually observe.

This new theoretical view of dispersion in polymer systems and the important critical parameter can probably be used to describe any other colloidal systems [6] in an analogous way.

In the same way we can investigate microemulsion systems. The first attempts of Strey to explain microemulsion structures are still based on equilibrium thermodynamical considerations [9]. But structure generation is equivalent to an entropy decrease, which makes (-TS) a large positive term. Microemulsion scientists are thus forced to believe that the enthalpy of mixing is so large and negative that the total free energy of microemulsion formation is negative. However, it is legimate to doubt whether this is a general phenomenon or even whether it occurs at all. So, for some reasons, we can propose to consider microemulsions and their structure also to be the result of a supercritical energy input and entropy export, leading to self-organized dissipative structures.

Literature / References

1. B. Wessling, Synth. Met. 45 (1991) 119-149

2. a) I. Prigogine, Angew. Chem. 90 (1978) 704

b) G. Nicolis, I. Prigogine, Self-Organization in Non-Equilibrium Systems, J. Wiley, New York (1977)

3. B. Wessling, Synth. Met. 41-43 (1991) 1057-1062

4. J. Wiley & Sons, Encyclopedia of Polymer Science and Engineering 13 (1988) 441

5. S. Rwei, S. Horwatt, I. Manas-Zloczower, D. Feke, Intern. Pol. Proc. VI (1991) 98- 102

6. B. Wessling, Adv. Mat. 5 (4) (1993) 300-305

7. B. Wessling, Macromol. Symp. 78 (1994) 71-82

8. B. Wessling, Polymer Eng. Sci. 31 (16) (1991) 1200-1206

9. M. Kahlweit, R. Strey, G. Busse, J. Phys. Chem. 94 (1990) 3881