Volume preserving and non-preserving strains
2 Rheology of linear polymer chains
2.1 Entropic elasticity
First we recap briefly the statistical physics of a polymer chain (as covered in the lectures by Khokhlov, this volume). Each chain is a random walk in space modelled by some local rule for spatial links; an example is the freely jointed chain. The step length of the chain corresponds to the Kuhn length of the polymer which we denote b. (This is the shortest independently orientable segment length, usually 4 or 5 monomers long.) Suppose the whole walk has N links and end-to-end vector R(N). From the theory of random walks, ( R ( N ) 2 ) = Nb2 where R = IRI. Also P(R) must have a Gaussian form (since R is a sum
90 Tom McLeish
8 '
-4 -2 0 2 4 8
I / L
log
0(rads)
8 - n
4 -2 0 2 4 6
log
0( d s )
Figure 9. Comparison of G' and G" for a linear (top) and star polymer melt of similar molecular weight polyisoprene [3]. Note the much broader range of relaxation times for the star polymer.
of many independent random vectors). So
P ( R ) d 3 R =
(-)
2.irNb2 3 3'2 exp(=)
-3R2 d 3 R .Now define a macrostate by the end-to-end vector R. The microstates are the different random walks of given R. In a freely flexible chain, each has the same energy, so the number of microstates obeys n(R) = OtotdP(R). Since the entropy of the walk is given by
S
= k~ In Q we have S ( R ) = S ( 0 ) - 3ksR2/2Nb2. The free energy of the chain is then F ( R ) = U - T S where U , the internal energy, is a constant: therefore we have F ( R ) = const.+
3ksTR2/2Nb2. Finally the entropic force ('Brownian tension') on thechain is
Rheology of linear and branched polymers 91
Thus a random walk polymer, or ‘Gaussian coil’ is like a Hookean spring with stiffness
K proportional to T I N . In dilute solutions, polymer chains are not gaussian (Khokhlov, this volume), except at the Theta temperature. However, it transpires [7] that in con- centrated solutions and polymer melts, the excluded volume interaction responsible for chain swelling is screened. (Although such chains remain selfavoiding at short distances, the driving force for chain swelling, is to decrease the probability of contacts within the chain. This is removed at high density: most collisions are with other chains, and swelling does not reduce the probability of these.) Chains in concentrated solutions and melts are gaussian at large enough distances, and Equation 14 applies to them.
2.1.1 Stress tensor
Equation 14 will enable us to calculate the stress tensor in any polymeric fluid provided the following conditions are met: (i) we know the instantaneous configuration of the chains at scales above some characteristic number
fi
of links; (ii) the configurations have achieved a local equilibrium for chain segments at smaller scales than this; (iii) we may average over many subchains (offi
links) within a local volume large enough to define a macroscopic stress, but small enough to define uniform physical conditions for the polymer chains within it.is the i-th component of total force per unit area transmitted across a plane whose normal lies in the j-th direction. Now consider a small cubic volume in a polymeric fluid of side L (Figure 10). It contains @ / N
Recall that component U;, of the stress tensor
L
L
.
Figure 10. Contrabutzon of a szngle subchazn to the stess tensor.
subchains of length N , where C is the monomer concentration (we drop the tilde on
R ) .
The probability that one subchain of end-to-end vector R cuts a given 3-plane across the volume is just R,/L (the fraction of the sample length L in the 3 direction spanned by its end-to-end vector). The z-th component of force transmitted by this chain across the j-plane is, from Equation 14, K&. So its contribution to the mean local stress utJ is n&R3/L3. The sum over all subchains may be replaced by the average (...) over the ensemble multiplied by the number of subchains, @ L 3 / N :
92 Tom McLeish
We will find it convenient to work with a continuous representation of the chains R(n) that maps the arclength position of the n-th monomer onto its spatial position €2.. Then we may identify R / N for a (small) subchain with dR/an. The formula for the stress tensor becomes pleasingly simple:
The second moment average (...) that governs the stress now needs to be calculated under various different assumptions for the dynamics. For example, it is sometimes pos- sible to identify subchains containing N monomers that have end-to-end distributions P(R) fixed by external constraints (as in a network, when N is the number of monomers between cross links) or by dynamics at a particular time scale (as in an entangled melt, when N is the number of monomers between entanglements), but which are equilibrated at all smaller length scales. In this case the natural unit of arc length is the coarse-grained step length of the segments f i b . Writing nb = s ’ f i the stress may be calculated from any known distribution of (coarse-grained) chain tangent vectors as:
aij = 3 k ~ T -
--
Nas’ as! .
Each sub-chain thus contributes kBT of stress, distributed tensorially via the second moment of its orientation distribution.
2.2
Dynamics
In polymer solutions and melts, the stress formula (Equation 16 or 17 above) is always appropriate given the validity of the three criteria listed at the start of Section 2.1.1, and the applicability of the Gaussian chain approximation. But there are important physical regimes in which the dynamics themselves differ qualitatively.
(i) Unentangled Chains. In the first regime, topological interactions between chains are not important because the chains are not sufficiently overlapped. Note that entanglement is only achieved at remarkable degrees of (spatial) overlap: even in the melt, chains must be several hundred monomers long, in order to see entanglement. The unentangled regime divides into two classes depending on whether long-range hydrodynamic interactions are important for the drag on the chains. If not, there is just local dissipation due to frictional forces as the chains slide past one another. Rouse [8] proposed this simplest case as a model for dilute solution, but it actually finds its realisation in low molecular weight melts and concentrated solutions. In dilute solution the more complex issue of hydrodynamic interaction dominates. We will not deal with this subject here, but the relevant model was devised by Zimm [9].
(ii) Entangled Chains. In this case the dissipation is local on the scale of the entangle- ment spacing (whether in melt or concentrated solution) but the chains’ motion is severely restricted by the topological constraints of their surroundings-two chains may not cross each other. Rouse’s formulation of the local drag needs to be supplemented by a model of these topological restrictions. The most powerful approach has proved to be the tube model of Doi, Edwards and de Gennes (see below and [7]). This entangled regime also
Rheology of linear and branched polymers 93
divides into two classes, but now depending on the topological structure of the chains themselves, that is, whether they are linear or branched. The branched case offers a nice example of hierarchical dynamics in soft condensed matter (see Section 3 and [2]).
Starting below for unentangled chains, we will outline the calculation of two aspects of the polymer dynamics, one microscopic and one macroscopic. The first is the mean- square monomer displacement as a function of time $(t) = (IR(n, t) - R(n, O)lz) averaged over all chains and monomer positions. This may be measured directly via NMR in some circumstances, and by scattering experiments indirectly [7]. The second aspect is the linear rheological response G(t) and its frequency-dependent representations, G’(w) and G”(w). In each case, we first use a formal approach in which the Brownian motion of chains is handled using a random thermal force on the monomers (a ‘Langevin’ equation). Then we discuss the result using simple physical arguments.