The phase behavior of polydisperse multiblock copolymer melts : (a theoretical study)

Summary The main theme of this thesis is the influence of polydispersity on the phase behavior of copolymer melts. With “polydispersity” we do not only refer to polydispersity in overall chain length, but also to polydispersity in the composition and the monomer sequence of the chains. Study of the influence of polydispersity is important because synthesizing purely monodisperse copolymers is very difficult, and for most polymerization techniques the occurrence of a certain degree of polydispersity is inevitable. We start with a short discussion about phase separation. A homopolymer is a chain molecule consisting of only one sort of link (monomer). For many homopolymer blends, i.e. mixtures of different homopolymers, the homogeneous state becomes unstable on lowering the temperature, and the different molecule species tend to separate from each other. The result is a splitting of the system into coexisting phases. Each of these phases separately is homogeneous, but they differ in composition. The separation of a polymer blend into coexisting homogeneous phases is called macrophase separation. In a copolymer, on the other hand, different monomer types are chemically linked together. Therefore, a complete separation of the system into the different monomer types is impossible. Instead, on lowering the temperature the phase separation occurs on a microscopic length scale. Small domains rich in one monomer type are alternated by small domains rich in the other. Usually, these domains are arranged in a regular pattern. When in a copolymer system such domains arise, we talk about microphase separation. The research described in this thesis was restricted to copolymers consisting of two monomer types, henceforth denoted by A and B. Most attention was paid to the so-called random copolymers. In random copolymer chains, the correlation in chemical identity between two monomers decays exponentially with their mutual distance along the chain. It has been assumed that within the chains, like monomers tend to aggregate to form long sequences of identical monomers. Such sequences are called blocks. The block length distribution in random copolymers is very broad: the variation in the block lengths is of the same order of magnitude as the block lengths themselves. Homopolymers having such a length distribution can be formed by a polycondensation reaction, after which they can be linked together to form a multiblock copolymer chain. With the phase behavior of a polymer system we mean the phase of the system as a function of temperature. The phase contains information about the volumes and compositions of the coexisting phases in case of macrophase separation, and the size and the spatial arrangement of the microscopic domains in case of microphase separation. In chapter 1 we describe the theory which enables the calculation of the phase behavior of a large class of polydisperse copolymer melts. In chapter 2 we describe how the regular periodic spatial arrangement of the domains in a microphase separated copolymer melt can be described mathematically. In chapter 3 the phase behavior of the correlated random copolymer melt is calculated in the so-called meanfield approximation, which means that it is assumed that the concentration profile is static (the concentration profile describes the spatial dependence of the A-monomer fraction). This approximation becomes more accurate if the block lengths in the system increase. In chapter 3 we derive an expression for the free energy of a random copolymer melt, and using this expression it is shown that the system tends to microphase separate. The A-rich and B-rich domains appear to have a regular spatial arrangement despite the intrinsic disorder present in the sequence distribution along the chains. In chapter 4 the study of the correlated random copolymer is continued by taking into account the possibility of macrophase separation. It is shown that for certain values of the composition and temperature the melt can indeed separate into coexisting phases, but at least one of these phases has to be microphase separated. Nevertheless, it is very doubtful whether the system will ever reach this two-phase state under experimental conditions, because macrophase separation requires a complete spatial rearrangement of the molecules, which is a very slow process due to the restricted mobility of the chains. In chapter 5 we go beyond the mean-field approximation. As indicated above, in the mean-field approximation it is assumed that the profile is regular, smooth, and static. In reality, however, irregular, time-dependent disturbances are present. These disturbances are called fluctuations. It is to be expected that due to the intrinsic disorder in the monomer distribution along random copolymer chains, fluctuations will be rather important in systems consisting of these molecules. This expectation is confirmed by the analysis in chapter 5. It is shown that the regular structures predicted in chapter 3 are strongly distorted, giving the concentration profile a disordered appearance. In chapter 6 a more general class of copolymers is considered, namely polydisperse multiblock copolymers for which the average number of blocks per chain, and the average number of momomers per block are very large. The length distribution of the A-blocks is arbitrary, and may differ from the arbitrary length distribution of the B-blocks. The correlated random copolymer studied in the previous chapter belongs to this general class, if we choose a Flory distribution both for the lengths of the A-blocks, and for the lengths of the B-blocks. In chapter 6 we calculate and compare the mean-field phase diagrams for various realizations of the block length distributions. By changing continuously the degree of polydispersity, it is possible to study its influence on the phase behavior.

Microphase separation in thin block copolymer films: a weak segregation mean-field approach

In this paper we consider thin films of AB block copolymer melts confined between two parallel plates. The plates are identical and may have a preference for one of the monomer types over the other. The system is characterized by four parameters: the Flory-Huggins chi-parameter, the fraction f of A-monomers in the block copolymer molecules, the film thickness d, and a parameter h quantifying the preference of the plates for the monomers of type A. In certain regions of parameter space, the film will be microphase separated. Various structures have been observed experimentally, each of them characterized by a certain symmetry, orientation, and periodicity. We study the system theoretically using the weak segregation approximation to mean field theory. We restrict our analysis to the region of the parameter space where the film thickness d is close to a small multiple of the natural periodicity. We will present our results in the form of phase diagrams in which the absolute value of the deviation of the film thickness from a multiple of the bulk periodicity is placed along the horizontal axis, and the chi-parameter is placed along the vertical axis; both axes are rescaled with a factor which depends on the A-monomer fraction f. We present a series of such phase diagrams for increasing values of the surface affinity for the A-monomers. We find that if the film thickness is almost commensurate with the bulk periodicity, parallel orientations of the structures are favoured over perpendicular orientations. We also predict that on increasing the surface affinity, the region of stability of the bcc phase shrinks.Comment: 35 pages, 20 figure

Freezing in polyampholyte globules: Influence of the long-range nature of the interaction

In random heteropolymer globules with short-range interactions between the monomers, freezing takes place at the microscopic length scale only, and can be described by a 1-step replica symmetry breaking. The fact that the long-range Coulomb interaction has no intrinsic length scale suggests that freezing in random polyampholyte globules might take place at all length scales, corresponding to an overlap parameter q(x) that increases continuously from zero to its maximum value. Study of the polyampholyte globule within the independent interaction approximation seems to confirm this scenario. However, the independent interaction model has an important deficiency: it cannot account for self-screening, and we show that the model is only reliable at length scales shorter than the self-screening length. Using the more realistic sequence model we prove that in the general case of a random heteropolymer globule containing two types of monomers such that unlike monomers attract each other, freezing at arbitrarily large length scales is not possible. For polyampholyte globules this implies that beyond the self-screening length, the freezing behavior is qualitatively the same as in the case of short-range interactions. We find that if the polyampholyte globule is not maximally compact, the degree of frustration is insufficient to obtain freezing.Comment: 28 pages, 5 figures, submitted to J.Chem.Phy

Dynamic charge density correlation function in weakly charged polyampholyte globules

We study solutions of statistically neutral polyampholyte chains containing a large fraction of neutral monomers. It is known that, even if the quality of the solvent with respect to the neutral monomers is good, a long chain will collapse into a globule. For weakly charged chains, the interior of this globule is semi-dilute. This paper considers mainly theta-solvents, and we calculate the dynamic charge density correlation function g(k,t) in the interior of the globules, using the quadratic approximation to the Martin-Siggia-Rose generating functional. It is convenient to express the results in terms of dimensionless space and time variables. Let R be the blob size, and let T be the characteristic time scale at the blob level. Define the dimensionless wave vector q = R k, and the dimensionless time s = t/T. We find that for q<1, corresponding to length scales larger than the blob size, the charge density fluctuations relax according to g(q,s) = q^2(1-s^(1/2)) at short times s < 1, and according to g(q,s) = q^2 s^(-1/2) at intermediate times 1 < s 0.1, where entanglements are unimportant.Comment: 12 pages RevTex, 1 figure ps, PACS 61.25.Hq, reason replacement: Expression for dynamic corr. function g(k,t) in old version was incorrect (though expression for Fourier transform g(k,w) was correct, so the major part of the calculation remains.) Also major textual chang

Weak Segregation Theory of Microphase Separation in Associating Binary Homopolymer Blends

In this paper we study the phase behavior of blends of associating homopolymers A and B in the weak segregation regime. The homopolymers are “associating” in the sense that hydrogen bonds are possible between the A chains and the B chains. Hydrogen bonds between two A chains, or between two B chains, are not possible. Each B chain can form at most one hydrogen bond, whereas each A chain might form bonds with several B chains, leading to the formation of block copolymer-like clusters. If the hydrogen bonds are strong enough, the system might undergo a microphase separation transition. However, due to the reversible nature of the hydrogen bonds, the system is in dynamic equilibrium, enabling it to adapt its cluster composition to changing conditions. Therefore, to construct the phase diagram, the free energy should be minimized simultaneously with respect to the cluster composition and the parameters describing the microstructure. We show that in the weak segregation regime this minimization can be split into two independent steps. In the first step, one determines what the cluster composition would have been if the system were homogeneous. In the second step, this composition is inserted into the expression for the Landau free energy without the nonlocal term. We show that the error made in the first step (neglecting the change in cluster composition due to the presence of the microstructure) exactly cancels the error made in the second step (omission of the nonlocal term from the Landau free energy). For the simplest associating homopolymer blend the phase diagram is presented.