The following post is a modified version of a short paper I
wrote for MatSE 542 last semester. It is essentially a summary of the paper “Self-assembly of block copolymers” by Yiyong Mai and Adi Eisenberg published in

*The Royal Society of Chemistry*in 2012.

**Introduction**

Block copolymers (BCPs) are a fascinating class of materials that have recently attracted significant attention due to their ability to self-assemble into a variety of morphologies, such as spheres, cylinders, gyroids, and lamellae. This post will discuss what block copolymers are, the thermodynamics behind microphase separation, and their theoretical and experimental phase diagrams.

Simply put, block copolymers consist of two or more
chemically dissimilar polymer blocks that are thermodynamically immiscible yet
covalently bonded. Figure 1 below illustrates the most popularly studied
structures of block copolymers which can be formed with two types of blocks, A
and B. Such a material is referred to as a diblock copolymer, while structures
consisting of three blocks are called triblock copolymers, and so on. The
chemically distinct blocks will separate into different domains while the
covalent bonds restrict this demixing to local length scales, resulting in what
is called microphase separation and giving rise to the aforementioned myriad of
morphologies.

Figure 1: Schematic of different structures of diblock copolymers. Reproduced with permission from The Royal Society of Chemistry. |

**Microphase separation**

Microphase separation in BCPs is driven by a
combination of the unfavorable mixing enthalpy and a small mixing entropy, with
the covalent bonds holding blocks together to prevent macroscopic phase
separation. For a diblock copolymer consisting of blocks A
and B, microphase separation is influenced by three parameters: the volume
fractions of the A and B blocks (f

_{A}and f_{B}), the total degree of polymerization of the two blocks (N = N_{A}+ N_{B}), and the Flory-Huggins interaction parameter (χ_{AB}). The Flory-Huggins parameter is dependent upon several factors and can be described with the equation below:
where

*z*is the number of nearest neighbors per repeat unit,*k*is the Boltzmann constant,_{B}*T*is the temperature, and ε is the interaction energy of the respective block pairs. The degree of microphase separation is determined by the product of the interaction parameter χ_{AB}and the total degree of polymerization*N*. Thus, as χ*N*decreases (or as temperature increases), the blocks become increasingly miscible while the combinatorial entropy increases and the copolymers become disordered. This behavior is referred to as an order-to-disorder transition (ODT).

Phase diagrams

Phase diagrams

We will now use phase diagrams to discuss the morphologies formed through microphase separation. As mentioned earlier and as depicted in Figure 2 below, diblock copolymers can form a variety of morphologies, including spheres (S), cylinders (C), gyroids (G), and lamellae (L). The morphology which occurs is dependent upon the composition

*f*and the combination parameter χN.

Figure 2: The range of morphologies corresponding to diblock copolymers. Reproduced with permission from The Royal Society of Chemistry. |

In an effort to better understand the phase behavior of diblock copolymers, self-consistent mean-field (SCFM) theory has been used to predict their phase diagrams. In block copolymer phase diagrams, the χ

*N*< 10 region is called the weak segregation limit (WSL) and the χ

*N*>> 10 region is called the strong segregation limit (SSL). SCMF theory has shown that when χ

*N*> ~10.5, increasing f

_{A}corresponds with a transition that goes as follows: closely packed spheres, body-centered cubic spheres, hexagonally packed cylinders, bicontinuous gyroids, and lamellae. These transitions occur in reverse when the composition is inverted. Using the model system polyisoprene-

*block*-polystrene (PI-

*b-*PS), Bates and co-workers report an experimental phase diagram for diblock copolymers. Figure 3 shows both the theoretical and experimental phase diagrams of diblock copolymers; the two agree remarkably.

Figure 3: Theoretical (left) and experimental (right) phase diagrams for PI-b-PS. Reproduced with permission from The Royal Society of Chemistry. |

The
formation of these morphologies occurs through the cone-column mechanism.
Basically, there are two competing factors at play: an enthalpic contribution
from the interfacial energy between the two blocks and an entropic contribution
from chain stretching. The two blocks separate from each other in order to
minimize the interfacial area and thus lower the total interfacial energy.
Meanwhile, this separation causes the chains to stretch and it is the volume
fractions of the blocks which determines the degree of stretching. As depicted
in Figure 4, when f

_{A}is small, the A blocks bunch together to form spherical microdromains while the B blocks surround them. As f_{A}is increased, the surrounding blocks decrease, resulting in less curved interfaces between the two domains and forcing the polymer chains to rearrange in a way that minimizes their stretching. This behavior corresponds with the transition from a spherical morphology to a lamellar morphology.Figure 4: Schematic of the cone-column mechanism. Reproduced with permission from The Royal Society of Chemistry. |

**Triblock copolymers**

We
can now consider linear ABC triblock copolymers. The addition of a third block
comes with increased parameters to consider (such as block sequence,
composition, and block molecular weights) which correspond to an increased
number of possible morphologies. For example, ABC triblocks, BAC triblocks, and
ACB triblocks would all exhibit different morphologies and phase transitions. Figure
5 below shows some examples of the many morphologies that could occur in
triblock systems. While AB diblocks required only χ

_{AB}*N*and f_{A}to specify a particular phase, an ABC triblock would require three interaction parameters (χ_{AB, }χ_{AC, }χ_{BC}) and two independent composition variables (for example, f_{A}and f_{B}). With this increase in variables comes both experimental and theoretical challenges. On the experimental side, synthesis, isolation, and characterization must be done for each change in composition, molecular weight, or architecture. On the theoretical side, SCMF theory calculations become very computationally expensive.

**Summary**

Block
copolymers consist of chemically distinct polymer blocks which would like to
phase separate, but can only do so locally because the blocks are covalently
bonded. This phenomenon results in a range of morphologies that arise from a
competition between minimizing interfacial energies and minimizing chain
stretching and for diblock systems are dependent upon χ

_{AB}N and f_{A}. As more blocks are introduced, such as in triblock copolymers, the parameter space greatly increases and so too does the range of possible morphologies.

**References**

Mai, Y.
& Eisenberg, A. Self-assembly of block copolymers.

*Chem. Soc. Rev.***41,**5969 (2012).
## Post a Comment