2.2. Thermodynamics of polymer melts
2.2.2. Phase separation in polymer melts
(nt−1
∑
i=0
ϕˆi(c)−1 )2
. (2.30)
The model parameter, κ0, is related to the inverse isothermal compressibility, compare toviiinEquation 2.26. The thermodynamic incompatibility between different particle types,i̸=j, is represented by the contribution
K0inter.[{ϕˆi}] =−∑
i̸=j
χ0ijN 4
(ϕˆi(c)−ϕˆj(c))2 or (2.31) K1inter.[{ϕˆi}] =∑
i̸=j
χ0ijNϕˆi(c) ˆϕj(c). (2.32) The first option K0inter. can be found in Ref. [33], the latter K1inter. in Ref. [58]. For systems with two particle types, both interaction potentials are equally suited. However, for systems with many different types the second optionK1inter. offers the advantage that it does not contain self-interaction terms, ˆϕ2i, for every type.
This type of potential is especially advantageous for melts with many particles interacting in a single grid cell. Because the Hamiltonian is a quadratic expansion in the densities, the potential can be written as a conventional pair-wise potential without approximations. Nevertheless, the translational invariance of a pairwise potential is lost, as the interaction refers to distinct grid cells,c. The calculation of energy differences is simple with these nonbonded interactions. Calculating forces, on the other hand, is a priori impossible. Hence, the interaction is best suited for Monte-Carlo (MC), but not optimal for MD simulation. In the context of this thesis, I am using this short-ranged interaction for SCMF simulations, seesection 2.3.3.
2.2.2. Phase separation in polymer melts
The repulsion of the nonbonded interaction between different particle species can lead to phase separations in polymer melts. The theoretical background of this macro- and microphase separation is the focus of this section.
The Flory-Huggins theory [59] is a lattice-based theory to understand the thermody-namics of binary homopolymer blends. The following description is well-known and the paragraph is based on the text of Rubinstein and Colby [35]. The original theory is based on a lattice where each lattice site is occupied by a single particle. Thus, monomers are hard and cannot overlap. Nonetheless, the results of this simple lattice-based theory are transferable to more realistic situations.
For this description uniform polymers,NA=NB =N, are assumed for both polymer species. Figure 1.1asketches such a blend schematically. The composition of the system can be described by the volume fraction of type A,φA:= nANnAA+nNABNB. nA denotes the number of polymers of type A andnthe total number of polymer chainsn=nA+nB. In this model perfect incompressibility is assumed. As a result, the volume fraction for type B is defined viaφA: φB:= 1−φA. Without the loss of generality, I describe the system composition with the volume fraction of type A:φ:=φA.
To calculate the phase diagram of this system the contributions to the free-energy difference per lattice site is separated in an energetic ∆Eand an entropic ∆Scontribution
∆F = ∆E−T∆S.
For a system of single particles, the entropic contribution can be described by the regular mixing theory [35]. The entropy reads as S ∝φlog(φ) + (1−φ) log(1−φ).
For a polymeric system, other entropic contributions exist, such as the change in conformational entropy [59]. However, a difference between the mixed and unmixed state is present but negligible. More important is the constraint of backbone bonds on the translational entropy. One monomer can only move with all its fellow monomers from the same polymer. The consequence is a reduction of the entropic contribution per polymer by the factor ofN. The dominant parts of the entropic difference between the mixed and the unmixed summarize as
∆S =kB ( φ
NAlog(φ) +1−φ
NB log(1−φ)). (2.33) Other entropic contributions, such as the temperature dependence of the fluid-like packing of the monomers in off-lattice situations, can be included in a temperature dependent Flory-Huggins parameter,χN.
For the energetic contribution to the free-energy two assumptions are important.
First, the interaction is evaluated on a lattice with zn interactions partners for each monomer. The relevant property of the grid is here the number of adjacent neighbors for each lattice site zn. The energetic contribution of an A monomer as a neighbor of B monomers isϵAB and for the other options ϵAA andϵBB respectively. In case of ϵAB > ϵAA and ϵAB > ϵBB the energetic contribution favors a demixing of the state opposing the entropic contribution. In all materials, I study in this thesis, unlike types repel each other. This is the common situation for polymerizable monomers. Even deuterating or protonating a molecule can be sufficient to induce a demixing in a polymer phase [60]. The second simplification is an MFA in the mixed state. The probability that the neighbor of a given monomer is of type A isφA and for type B respectively φB= 1−φ. On the other hand in a demixed state, the probability for an A monomer to have an A neighbor is one and for a B neighbor zero. With these simplifications, the energy difference per lattice site between the mix and the unmixed state reads as [35]
∆E = zn
2 φ(1−φ)(2ϵAB−ϵAA−ϵBB). (2.34) For this equation the Flory-Huggins parameterχ := 2kzBnT(2ϵAB−ϵAA−ϵBB) can be defined, simplifyingEquation 2.34 to
∆E =χφ(1−φ)kBT. (2.35)
Note that this equation can be used to estimate the χ parameter also for off-lattice simulations. In contrast, for off-lattice simulationsEquation 2.34 is not valid anymore.
The simplification of a fixed number of neighbors zn can be replaced by a mean-field
2.2. Thermodynamics of polymer melts
description via the radial distribution functiongαβ(r) of different typesα and β. χ≈ρ0
∫ dr [
gAB(r)VAB(r)−1 2
{gAA(r)VAA(r) +gBB(r)VBB(r)}] (2.36) Vαβ describes the nonbonded interaction. For a soft, coarse-grained model for a liquid polymer melt a good approximation can be achieved by settinggαβ(r) = 1.
However, as long as the total energy of a system can be measured and Equation 2.35 can be utilized to define a Flory-Huggins parameter χ. The mixing energy ∆E is measured as a function of the volume fraction φ. Simulations are performed with a mild repulsion of the particles to prevent demixing in order to comply with the MFA assumption. The opening of the parabola ∆E(φ) determines the Flory-Huggins parameterχ. Deviations from any of the assumption are automatically summarized in the finalχ parameter. In case of a single parameter pair potential, as it is discussed in section 2.2, a relation between the repulsion parameter vAB and χ is established.
Combining the entropic contribution (Equation 2.33) and energetic contribution (Equation 2.35) the free-energy reads
∆F kBT
√N¯ = V
R3eo(φlog(φ) + (1−φ) log(1−φ) +χN φ(1−φ)). (2.37) This equation makes use of the fact that both polymer species have an equal length N. Therefore, the analysis of the equation becomes easier and is sufficient for the scope of this thesis, but is not strictly necessary. Rubinstein and Colby [35] discuss the effects of chains of unequal length, especially the case ofNB= 1. Figure 2.10aplots this equation for three values ofχN. The critical value forχNcrit.= 2 is determined via the derivatives of the free-energy with respect to the compositionφ. For valuesχN <2 the free-energy is strictly convex and no phase transition can be observed, but at the criticalχNcrit. a nonconvex part establishes aroundφ= 1/2 indicating a phase separation. Figure 2.10 illustrates this phase transition.
Experimentally, the Flory-Huggins parameter χ is temperature-dependent. It is accepted in the literature [35] to empirically describe the relation as
χ= α
T +β. (2.38)
α and β are fit constants to be determined for a specific chemical species [35]. This empirical relation summarizes all missing energetic and entropic contributions into a single parameterχ. Consequently, a critical temperature, TODT, can be defined for the phase transition.
Interestingly, the phase transition is controlled by theχN parameter, the product of chain length and incompatibility. As a result, long polymer chains tend to phase separate as long as the Flory-Huggins parameter χ is bigger than zero. This is the reason why even small difference in the polymer species, as a change in the isotopes [60]
results in a phase separated system for sufficiently long macromolecules.
0.00 0.25 0.50 0.75 1.00
(a)Free-energy difference for the Flory-Huggins the-ory [35,59]. Above the criticalχNcrit. = 2 two minima emerge, indicating the phase separation.
0.00 0.25 0.50 0.75 1.00
(b)Spinodal and binodal of the Flory-Huggins system. The critical point is at χNcrit. = 2 andφcrit.= 1/2.
Fig. 2.10. Phase separation of a symmetric, binary blend of homopolymers in the Flory-Huggins theory [35,59].