Linear homopolymers

Random copolymers and homopolymer blends have some fundamental properties in common with ordinary homopolymers, hence it is useful to start with a model for simple homopolymers as a basis for the description of random copolymers in section 2.2.2, and of crosslinked homopolymer blends in section 3.2.

R

Figure 2.1: Coarse-graining of a polymer chain. The orig-inal contour (thick line) is approximated by a chain of straight segments (solid ar-rows). The dashed arrow in-dicates the end-to-end vec-torR_ee.

Chain connectivity

The most prominent feature of polymers is their connectivity, i.e.the fact that the monomer units are linked into macromolecules. Out of the various possible topolo-gies like trees or stars we pick the simplest one, namely linear chains, consisting of bifunctional monomers binding to (at most) two other monomer units.

On a small length-scale, the geometrical and other properties of the polymer strongly depend on the chemical details of the monomer species. The positions and orientations of neighbouring units in the chain are correlated due to usually quite restricted bond angles and bond lengths, and less rigid, but still constrained bond rotation angles. Typically, the amplitudes of the thermal oscillations of bond lengths and angles are of the order of 3% and 3-5%, respectively [34], and can often be neglected. The potentials for bond rotation angles generally exhibit multiple minima, three in the case of single bonds between tetravalent carbon atoms. The energy barriers in-between are typically larger thankBT, yet small enough to allow for frequent transitions between the possible states [34], giving rise to a diversity of possible chain configurations that grows large with the number of rotating bonds.

In turn, the correlations between two monomers decay with their distance along the chain contour [1, 34–36], so that the details of the monomer properties are effectively masked on a larger length-scale. The decay length, the persistence length, is a measure for the stiffness of the polymer.

A coarse-grained picture allowing for the calculation of distribution functions of, e.g., the end-to-end distance of the chain, can be obtained as follows [1, 14, 35]:

The polymer chain is divided into L pieces of equal number of subunits or arc-lengthl, and their endpoints are connected with straight segments having a mean square length a²; this procedure is illustrated in fig. 2.1. Provided l is much larger than the persistence length, the orientations of the segments are practically uncorrelated, so that the contour of the segment chain formally equals the path of a Brownian particle or a random walk¹. Consequently, the distance vector of two segments approximately follows a Gaussian distribution, with a mean square dis-tance proportional to the number of segments in-between. For insdis-tance, the mean square end-to-end distance of the segment chain is given byR²_ee :=hR²i=La². In the construction, the segment size a is chosen such that the mean square

end-to-1 Strictly speaking, this only holds true if the volume occupied by the monomers can be ne-glected, since the polymer chains cannot overlap. This issue will be discussed in section 2.2.1.

end distances of the segmented and the real chain coincide. This also holds for the radius of gyrationRg, an alternative measure for the chain extension appearing in the low momentum expansion of the scattering factor of single polymer chains. It is defined as the mean square distance between the monomers,i.e.

R²_g := 1 2n²

Xn i,j=1

(ri−r_j)²

(2.1) for a linear chain of n monomers located at r_i, where i= 1, . . . , n.

Up to this point, the choice of the number of segmentslis arbitrary, in the range from the persistence length to the arclength of the polymer, and the segment sizea is determined depending on l. This arbitrariness can be removed by requiring not only the average coil size, but also the length of the fully extended chain (roughly speaking the arclength) to be the same for the real polymer (R_max) and the segmented one (La). (The average size of the individual segments is assumed to be unaffected by the stretching of the chain.) It follows that a is given by the Kuhn lengthaK:=hR²i/Rmax, which, like the persistence length, is a measure for the stiffness of the polymer. In fact, both quantities take similar values [1].

The Gaussian approximation made above can be tested explicitly for simple models like the freely jointed chain, in which adjacent monomers are linked by bonds having fixed length and random orientation. It turns out that the approx-imate distribution of the end-to-end distance, even for a chain as short as five units, is in good agreement with the exact result [1]. Hence it seems well justified to employ the Gaussian chain model, which offers invaluable advantages in the analytical treatment.

Gaussian model

To parametrise the model for Gaussian chains, we consider a melt of N poly-mer chains containing L segments each, thus neglecting effects of polydispersity.

The position of segment s on chain i in the d-dimensional space is denoted as R_i(s) ∈ R^d where i = 1, . . . , N and s = 1, . . . , L. The distance vector between neighbouring segments on the chain, in thermal equilibrium, is taken to follow a Gaussian distribution with zero average and a root mean square length ofb, which can be interpreted as the Kuhn length of the polymer. The chain configuration is described by the Wiener Hamiltonian

H^W :=kBT · d 2b²

XN i=1

L−1

X

s=1

(Ri(s+ 1)−R_i(s))². (2.2) The pre-factor of kBT in eq. (2.2), which cancels out with the factor of 1/ kBT in the Boltzmann weight and yields a temperature-independent partition function, reflects the entropic nature of the distribution of chain configurations.

In the case of Gaussian chains, the radius of gyration defined in eq. (2.1) (withL segments instead ofnmonomers) can be calculated analytically. The mean square

distance of two monomers i and j of the same chain is given by h(Ri−R_j)²i =

|i−j|a², henceRg =La²/6 (up to corrections of the orderL⁻¹, which are due to not fully accounting for the segments at the chain ends). To indicate the dependence on the chain length, the radius of gyration will be also referred to asRL. Obviously, the average end-to-end distance and the radius of gyration of the Gaussian chain are related viaRee =√

6RL.

Excluded volume

The analogy of the polymer contour with a random walk, used to derive the ideal chain model in the previous section, must be taken with a pinch of salt. It had been obtained considering only short-range interactions of subunits being neighbours on the polymer chain, disregarding the long-range intramolecular interactions of subunits that are close in space, yet distant along the chain contour.

In the case of homopolymers, the most important example is the steric hindrance or interference: In contrast to the path of a Brownian particle, a polymer chain may not cross itself, since the subunits occupy a certain volume, the excluded vol-ume. For long chains, many of the configurations of the ideal chain predicted by the Gaussian model are ruled out. Since extended configurations are more likely to “survive”, it appears that the mean extension of real chains must be larger than predicted. This is indeed the case for dilute polymer solutions, especially in a good solvent, which interacts more favourably with the polymer than the poly-mer itself, thus amplifying the interference effect and leading to polypoly-mer swelling.

Conversely, the effect is counteracted by poor solvents. In some systems, the ef-fective attraction due to the poor solvent is comparable to the self-repulsion of the polymer, depending on temperature. At the θ temperature, in an ideal solution, the two interactions just cancel, and the polymer chains adopt the configurations of the ideal Gaussian chain.

Measurements and theoretical considerations show that in polymer melts the chain configurations are also nearly ideal, with only small distortions [1, 14]. This is because of the intermolecular forces not taken into account so far: for the in-terference of two portions of polymer, it is irrelevant whether they belong to the same chain or not, and there is no reason to discriminate between the according intermolecular and long-range intramolecular interactions. In a polymer melt, the chains could avoid self-interference by assuming a more extended shape, but only at the cost of more frequent collisions with other chains. Therefore, the shape and size of coils effectively remain unchanged despite steric hindrance.

Nevertheless, it is necessary to account for the excluded volume effect in the model as a means of limiting compressibility, since the copolymer systems discussed in the present thesis could break down in the absence of repulsive forces: In a copolymer melt, the attraction between monomers of the same kind could induce the collapse onto one point in space per species. In a crosslinked melt, the network favour the contraction onto a single point. To model steric repulsion we shall use

the usual excluded volume interaction [14], adapted to polymer melts, H^EV:=kBT · µ