Chapter 9: Self Assembly of Block Copolymers

Self Assembly in Soft Matter: Spontaneous Self Organization

Self-assembly refers to the spontaneous creation of order in a system from a previously disordered state. Self-assembly is driven or induced by a change in some environmental variable. This is similar to lowering temperature of a material that is initially disorder at higher temperatures can induce a orderdisorder transition (ODT) from a disordered to an ordered state. This is critically important for many systems, particularly magnetic materials. The key point is that self-assembly isn’t driven externally by applied stimuli like stress but instead occurs spontaneously driven by the thermodynamics of the system in response to some external change or stimuli.

Self-assembly is particularly prevalent and important for soft matter systems. A particularly important example of soft matter self-assembly occurs in proteins, which have an unfolded state at high temperatures, consisting of a single polymer chain with only primary structure that then folds and becomes ordered at lower temperatures, leading to secondary/tertiary structure (i.e. formation of helices, beta sheets, etc). Here the order-disorder transition is driven by the competition between favorable enthalpic interactions, like as hydrogen bonds, in the folded state, versus favorable entropic considerations at higher temperatures in the unfolded state. Are you sick of entropy vs enthalpy, the answer is no because it is still exciting and interesting. Now although we will talk a lot about temperature being the ODT or self-assembly can also be triggered by pH changes and other factors.

Previously we’ve pretty much only discussed disordered states, like the melt or completely mixed phases, and we only discussed a single macrophase separation process in the context of Flory-Huggins theory where two components either mix or demix. We will now discuss different ordering transitions, and the different morphologies that can exist in block co-polymer systems. We are particularly interested in understanding the key concepts behind the process of self-assembly, will construct a simple formalism to understand characteristic length scales in ordered copolymer microstructures, and will identify factors that enable us to tune the resulting morphology of diblock copolymers in the ordered state. Finally, self-assembly can lead to structures that are much more complicated than just a demixed or phase-separated state, leading to the rise of microphase separation.

Self-assembly is driven by the competition between enthalpy and entropy. Several order-forming processes are governed by this competition - macrosphase separation, which is what we discussed in the context of Flory-Huggins theory, microphase separation (our focus), mesophase separation, which is similar to microphase separation but involves separation over some larger length scale, adsorption/complexation, which involves the interaction of molecules with surfaces and boundaries, and crystallization, which involves the creation of order upon cooling below the melting point. The final state of a system after ordering will depend highly on the chemical affinities of the different components of the system and the architecture of the different species (e.g. star polymers in general will undergo different ordering processes than linear polymers). Finally, we will also have to consider the interfacial tension between any ordered phases that develop in the system, since in general a phase separation process will have some unfavorable enthalpic interaction between the two distinct phases. It is also important to consider kinetic implications of the self-assembly process, since many of these processes involve multiple energy minima which could lead to the formation of metastable phases that are not the desirable final state. These are all considerations that have to be taken into account in designing and studying self-assembled systems.

Soft material systems that undergo ordering processes are vast, and include but are not limited toLiquid crystals - anisotropic particles with some director vector

• Polymers and block copolymers
• Proteins via hydrogen bond complexes
• Nanoparticles - polymersomes or other particles formed from the assembly of polymers or other soft materials
• Many other biological systems
• Active Matter Systems (kind of but non-equilibrium so not really)

Structural order may vary across many length scales, from the atomistic level up to the level of grains (100s of µm) in metallic systems. When thinking about the strategic design of ordered, self-assembled structures, there are a wide variety of factors that must be considered to engineer order at multiple length scales. We need to think about the structure of our components, their interactions with each other, environmental constraints, the starting state of our system, any bias due to applied stresses, the influence of a substrate or other physical constraints involved in our ordering process, etc. There is a immense design space for self-assembly that could have implications in some of the most important industries, for example computer chip design.

Microphase Separation in Diblock Copolymers

We are going to tackle the task of develop a quantitative framework to understand microphase separation in diblock copolymers. To understand microphase separation, we will invoke the so-called min-max principle. The min-max principle states that ordered polymer systems will want to do 2 things

1. Minimize Interfacial Energy-remember back to nucleation and growth i.e. the energy penalty to create surfaces
2. Maximize conformational entropy of chains-energetically favorable to increase entropy.

Let’s keep this general and say that we are working with a diblock copolymer with one block of type A and one block of type B. As we discussed in Flory-Huggins theory, in general two different polymers will not prefer to mix, and hence A-B contacts will tend to be less enthalpicly preferable than A-A or B-B contacts. Above the order-disorder temperature (ODT), we expect a homogeneous, disordered mix of copolymers, where the A and B blocks freely interpenetrate

to maximize conformational entropy of the chains despite the unfavorable A-B contacts. Below the order-disorder temperature, the copolymers will enter an ordered state such that all the blocks of type A will segregate away from the blocks of type B, maximizing the number of A-A and B-B contacts while minimizing unfavorable A-B contacts.

Unlike in the case of polymer blends, however, there cannot be separation at the macroscale since the two blocks are chemically joined, enforcing a length scale constraint on the segregation of the two types. Hence, we establish ordered structures that depend on the relative proportions of the two blocks, and in general there will be a significant interface between A and B phases that leads to a large number of unfavorable A-B interactions. We call the dividing surface between the A phase and B phase the intermaterial dividing surface, (IMDS), where the IMDS can be thought of as the surface created by connecting all junction points where A and B blocks are joined. Since A and B blocks do not want to mix below the ODT, the interface defined by the IMDS wants to be minimized since there will be unfavorable A-B interactions along the IMDS. Note that in general there will be many IMDSs since the chemical linking of the diblocks limits the extent to which separation can occur this is in contrast to the case of homopolymers, where the amount of interface is effectively limited by the ability of polymers to move throughout the system without constraint. As we change the relative fraction of blocks A and B we will see different microphase separation behavior ranging form spheres, cylinders, double gyroids, double diamonds, lamellae phases, etc. The simplest case to study first is the lamellae so we will take this first.

50:50 A:B Mixture: Lamellae Microphase Separation

Let’s start with a quantitative model for a flat IMDS (corresponding to approximately a 50:50 mixture of A:B), with type A on one side and type B on the other side of the IMDS. The structure thus looks like just a flat interface between two phase-separated states. We want to calculate the change in free energy associated with forming this type of structure, since if we find this free energy change to be negative we can expect the formation of the ordered state to be favorable. We must first start with some definitions

• N = num of segments = N_A + N_B: i.e. total number of segments
• a = size of one segment, with a_a ≈ a_b. The volume of a segment is ≈ a³.
• λ = domain periodicity, i.e. size of lamellae in this case but will be different for different morphologies
• ^P = Interfacial area per chain. This can be thought of as the total area of the interface divided by the number of chains, which roughly corresponds

to the area of unfavorable overlaps between type A and type B polymers. The interface and IMDS is the same here.

• γ_AB = interfacial energy $\frac{kT}{a^2}\sqrt{\frac{\chi_{AB}}{6}}$ - note that this calculation to find this scaling is a bit involved, so just take it as fact and put your trust in me or if you are very motivated you can find the derivation in a book by Fredrickson.

We will assume we are in the strong segregation limit, meaning that the two blocks really do not want to associate, i.e. χ_ABN is very large and positive, so that both χ_AB > 0 and N is large. In this limit, there isn’t enough entropic drive force to mix and there is a strong enthalpic drive to segregate. Note that because of this limit, we assume that the ONLY type of phase-separation possible is where ALL type A goes to one phase and all of type B goes to another.

Practically, this means that looking at the change in free energy is sufficient to determine ordering, and we do not need to look at the chemical potentials of type A/B in either phase like we previously did with Flory Huggins.

Qualitative Understanding of Microphase Separation:

To do this let’s think about the free energy components present in this system. Well we know that we have a high positive value of χ between type A and type B, there is an unfavorable contribution to the enthalpy due to interactions along the interface. Thus the chains want to minimize their interfacial area as this scales with the number of A-B contacts. In order to do so, the chains will thus want to elongate away from the interface, and effectively increase their length to decrease their projected area onto the IMDS. What is the penalty in doing so? Remember that we have talked at length the consequences of elongating polymer chains right. To do this we have to pay an entropy penalty due to a decrease in the conformational entropy. Thus, we can see that equilibrium is determined by a competition between an enthalpic desire to minimize interfacial area and an entropic desire to maximize chain conformations/minimize stretching.

Quantitative Understanding of Microphase Separation:

Now let’s calculate the change in free energy. We can start with the disordered state approximated from Flory-Huggins theory:

$$H_{D} = N\chi_{AB}\phi_A\phi_BkT \nonumber$$

Notice that we’ve omitted the entropy of mixing because in general it will be a relatively small contribution.

This is because we have stated in this case that N is large and in Flory Huggins we saw that the mixing entropy of large polymers is small. Additionally it will not play a huge factor because we really have a single component system since the A and B blocks are not truly separate components since they are chemically linked. This observation helps justify our assumption of strong segregation since the entropy of mixing will in general be much smaller than in macrophase separated mixtures.

In the ordered state, the enthalpy of AA and BB interactions do not contribute as χ_AA = χ_BB = 0. Thus we only care about the interfacial energy, which is given by the interfacial energy per area times the interfacial area per chain, or $\gamma_{AB} \sum$.

$$\Delta H_{O} = \gamma_{AB} \sum \nonumber$$

We also need to account for the conformational entropy of the chains, which will tend to stretch out in order to minimize the interfacial energy and instead maximize same-block contacts. Luckily we already have this expression remembering back to previous lectures

$$S(r)_{stretching} = -\frac{3k}{2}\frac{r^2}{r_0^2} \nonumber$$

$$\Delta S_{stretching} = S_{O} - S_{D} = -\frac{3}{2}k\left [ \frac{r^2}{r_0^2} - 1 \right ] \nonumber$$

Note that the change in stretching free energy is relative to an unstretched state where $r^2 = r_0^2$ which is why we subtract 1. In the case for lamellae, the domain spacing is directly related to the stretching of these coils - specifically, the domain spacing λ/2 is equal to the total stretched length of both the A and B blocks, so that the total stretched length of any single chain is given by λ/2. The undeformed length is Na² from typical chain dimensions. We can thus replace r² in the entropy expression with (λ/2)² and $r_0^2$with Na², and note that in the initial state r = r₀ so the entropy is just $-\frac{3k_b T}{2}$. We can thus finally compute the total change in free energy

$$\Delta G = \Delta H - T\Delta S \nonumber$$

$$\Delta G = H_{O} - H_{D} - T(S_{O} - S_{D}) \nonumber$$

$$\Delta G = \gamma_{AB} \sum - N\chi_{AB}\phi_A\phi_BkT + \frac{3}{2}kT\left [ \frac{(\lambda/2)^2}{Na^2} - 1 \right ] \nonumber$$

The incompressible assumption allows us to write the volume conservation condition as

$$Na^3 = \frac{\lambda}{2}\sum \nonumber$$

where the left term is simply the volume of a segment times the number of segments, while the second term is the size of the resulting domain times the interfacial area, both of which are equivalent when the chain is stretched in the final state. Substituting in this relation and the relation for γ_AB given above allows us to eliminate $\gamma_{AB} \sum$

$$\Delta G = \frac{kT}{a^2}\sqrt{\frac{\chi_{AB}}{6}} \frac{Na^3}{(\lambda/2)} - N\chi_{AB}\phi_A\phi_BkT + \frac{3}{2}kT\left [ \frac{(\lambda/2)^2}{Na^2} - 1 \right ] \nonumber$$

From this expression, we can find the optimum period of the lamellae repeat unit, λ, by minimizing the free energy change with respect to λ.

$$\frac{\partial \Delta G}{\partial \lambda} = 0 \nonumber$$

$$\lambda_{optimal} \approx aN^{2/3}\chi_{AB}^{1/6} \nonumber$$

We can see the dimensions of the lamellae scale as λ ∝ N^2/3. It scales with a much larger exponent than expected in either the theta condition (r ≈ N^1/2) or in good solvent (r ≈ N^3/5). Hence, the chains are very extended when compared to the melt state.

Critical χ Parameter for Phase Separation: Nχ_AB

We have established the domain spacing dependence on molecular weight if we are in the phase-separated state; however, we have not actually determined when phase separation occurs. We can find the critical χ_AB parameter for the order-disorder transition by finding when

$$\Delta G(\lambda_{optimal}) = 0 \nonumber$$

All we have to do is substitute in the optimal value of λ which will yield the most negative possible free energy change; if this is less than 0 ordering should occur.

Simplifying the expression above for ∆G and substituting the expression for λ_opt gives us

$$1.2kTN^{1/3}\chi_{AB}^{1/3} \approx N\chi_{AB}\phi_A\phi_B kT \nonumber$$

where we ignore a factor of 3/2kT. Again this is small and we care about scaling in this class not prefactors. Solving for Nχ_AB for a 50/50 volume fraction thus gives a critical value of 10.5

$$N\chi_{AB} < 10.5 ~- \textbf{Homogeneous, mixed melt} \nonumber$$

$$N\chi_{AB} > 10.5 ~- \textbf{Lamellar, microdomains} \nonumber$$

Take a moment to appreciate that this is a much greater critical value than we derived for the macrophase separation of polymer blends (Nχ ≈ 2)! The major difference here is that stretching is required for microphase separation, while in the case of macrophase separation we assumed that chains were ideal and did not stretch. As a result, we need to decrease the temperature way more in the case of diblocks to overcome this additional entropy term opposing phase-separation, which is much greater than the mixing entropy present in the macrophase case. Again remember that decreasing temperature is equivalent to raising χN so this agrees with our quantitative analysis.

We have seen that there are many more microphases than can occur than just lamellar spacing, but in general the same basic quantitative formalism should apply, i.e. minimizing interfacial area and maximizing conformational entropy.

Self-Assembly Isn’t Perfect: Defects and Experimental Techniques to Deal with Them

Now unfortunately for many applications like ligthographic masks, computer chips, optimal and electronic materials, batteries, etc. this self-assembly ordering transition is not perfect and long-ranged. In real systems perfect ordering over long range does not exist due to the presence of defects in the system, just like in we discussed in the Defect lecture in ENGR045. Moreover, block copolymers tend to have the easy onset of defects since there is only a minimal cost for forming them given the relatively low cost of forming interfaces, and as a result we typically see grain boundaries in BCP ordered phases. The origin of these grain boundaries comes from the growth of ordered phases in multiple parts of a sample at once that may not necessarily orient in the same direction, leading to defects when the different ordered regions grow and will eventually impinge upon each other. However hope isn’t lost. To minimize and

avoid the creation of defects we have some tools at our disposal to bias the growth of ordered phases in a particular direction. Some of these strategies include:

• Applying bias field - this involves using some sort of external biasing field, such as applied stress in some direction, an applied electric field, etc., to reduce the energy barrier for growth in some preferred direction.
• Substrate patterning - this involves functionalizing the growth substrate with regions of specific chemistry that guide the growth.
• Substrate constraints - using physical barriers, such as walls or trenches in the substrate, as guides to growth of ordered phases. To avoid defects it’s also important to ensure that the length scale of any physical constraint is compatible with the length scale of the growing BCP - that is, if the growing BCP hits a wall it should make contact at the boundary between phases in order to avoid compression or other effects that may induce defects.

These strategies all fall under the broad classification of applying a biasing field. One of experimental techniques at our disposal to apply a biasing field is via graphoepitaxy.

Graphoepitaxy

Epitaxy refers to the process of patterning some substrate via some lithographic technique. Generally, this pattern will be some chemical monolayer that has regions that interacts preferentially with one of the two blocks in a growing BCP structure. Putting a BCP mixture on this surface in a proportion that would normally form lamellae then would form new morphologies due to the preference for matching the pattern of the chemically modified surface. This process is called surface reconstruction and allows for the fine-tuned control of ordered BCP morphologies with minimal defect formation.

Growing a BCP phase under an applied stress can also lead to an anisotropic growth process, as well as providing a means of actually identifying a phase transition since the response to that stress should change dramatically when a transition occurs. The preferred direction of the stress will lead to microstructural growth in a specific direction by biasing the direction of diffusion and thermodynamially decreasing the equilibrium free energy when phases are preferentially aligned with the applied stress. This allows us to create morphologies that are well-aligned, avoiding defects due to the overlap of unaligned domains. The major issue with applying a stress versus epitaxy is that stress is typically applied on the macroscale and does not give the nanoscale control that epitaxy does.

A new technique, used heavily by Ross group at MIT and many many others, which combines both chemical and physical constraints is graphoepitaxy. This technique consists of coating a surface with both specific chemistries AND some topographical constraints, i.e. roughness, posts, channels, etc. The combination of these two constraints forces the growing microphase to be commensurate with the length scale of the physical constraints while also inducing anisotropy from preferential interactions with the chemistries present on the surface. One example is the creation of an array of ordered posts coated with a polymer that mixes preferentially with one of the two BCP blocks. When a BCP blend is added to the array of posts, microphase separation will occur with a long range order imposed by the posts since polymer can only grow in between them. Because the posts are coated with polymer, only one of the two species can interact with them, so the posts serve as nucleation sites for an ordered phase, which then sets the length scale for growing additional microphases between the posts. Thus, if the posts are spaced according to the same length scale that would occur if the BCP mixture separated without posts, then perfect ordering can be achieved! If the spacing is wrong, however, then there will be some strain energy that may lead to defect formation.

Graphoepitaxy also allows complex morphologies - e.g. aligning cylinders, very useful for nanowires. You can basically just change the spacing of posts to orient different cylindrical phases. A problem with this, however, is that there are some rotational directions that are exactly equivalent when assembly cylindrical phases, and there will thus be defects when these different rotationally identical directions overlap. To overcome this, you can elongate posts in one direction to increase surface energy, biasing toward one direction over another. To conclude, the basic idea is use some topology to impose order on the system, inducing the same type of ordering in the resulting BCP matrix, but with some preferential growth direction(s) to ensure perfect order!

BCP-Homopolymer Blends:

Key Definitions/Terms:

• Micelle - a spherical aggregate of diblocks (or other two-component molecules) which forms to segregate one component in a core region and the second component in a corona region. The corona region is exposed to surrounding solvent (or homopolymer) and thus forms to ”shield” the core from unfavorable interactions with the surrounding solvent.

• Critical micelle concentration (CMC) - the concentration of diblocks at which micelles first form; below this, the mixing entropy of the diblocks dominates and prevents aggregation. However, since diblocks tend to be large and thus have minimal mixing entropy, the CMC tends to occur for small concentrations.

So far we have been talking about block copolymers alone in solution, but we can also introduce additional phase behavior by mixing in homopolymer (of one of the two types present in the BCB i.e. A or B) with the BCPs. In this situation the homopolymer should preferentially mix with the corresponding block in the BCP, assuming that the chemical structure of the homopolymer is identical to the homopolymer of the block and hence their χ parameter is roughly 0. If this the case we will see some new phase behavior which includes the formation of micelles.

Micelle Formation:

A micelle is a a spherical assembly of molecules that have one region that prefers to mix with solvent and one region that tends to avoid mixing with solvent, leading to the formation of an inner core region shielded from solvent and an outer corona region that maximizes solvent exposure. On extremely important system that exhibits micelle formation is a collection lipids, which have a hydrophilic head region and hydrophobic tail region (amphilic).

At very low lipid concentrations, the translational entropy of the lipid mixture tends to be very high and any ordering transition is unfavorable due to the high entropic cost. At a specific critical concentration, the cricitical micelle concentration (CMC), ordering is favored in order to reduce the number of unfavorable water-tail contacts, leading to the formation of micelles with lipid heads facing outward to mix with water and hydrophobic tails forming an oil-like inner core. This morphology thus minimizes unfavorable hydrophobic interactions. At even higher concentrations of lipid, more ordered phases can form, including cylinders, bilayers, inverse micelles, etc. You can also add a third component that mixes well with lipid tails (some other hydrophobic component like oil) to yield an even more complicated ternary phase diagram.

In analogy to the small molecule lipid-water system, we can also think of the same ordering process in diblock copolymer - homopolymer mixtures, where we assume the homopolymer mixes perfectly with one of the two blocks (akin to water mixing well with the hydrophilic heads). In this system the homopolymer is thus the solvent and the diblock is the solute. In a very dilute system, the diblocks will tend to assume a disordered, homogeneous phase, identical to lipids. At the CMC the polymers will start interacting, leading to aggregation of the block that has an unfavorable enthalpic interaction with the surrounding homopolymer medium. As we increase the concentration of diblock even more, existing micelles will begin to order with each other to minimize the total amount of unfavorable interfacial interactions with solvent. Recall that since the diblock alone will tend to microphase separate as the amount of diblock gets larger and larger we will see phases appear that are similar to phases exhibited by block copolymer alone.

We can now use a very simple formalism to understand the concentration at which micelle formation is preferred. As always, here again the major competition is between enthalpy and entropy. Forming micelles costs entropy due to the ordering of diblocks, but saves in enthalpy by minimizing unfavorable interactions between the A and B block, in this case by sequestering B blocks in the core of the micelle. Hence the only enthalpic term is the interfacial energy, and the driving force for micelle formation is enthalpic in nature. However, compared to the case of diblocks alone, we must now also consider the influence of the free homopolymer in the system.

Determining Critical Micelle Concentration:

We will now develop a model to predict when micelle formation is preferred. We draw upon a lot of the same ideas that we already used to discuss microphase separation but now have to include the influence of the homopolymer component.

Let’s start by identifying variables in the system. Just like previously we will have a diblock copolymer consisting of an A block with degree of polymerization N_A and a B block with degree of polymerization N_B. The total degree of polymerization is thus N = N_A + N_B, and in this example we assume N_A = N_B = N/2 for simplicity. We characterize the enthalpic interactions between the two blocks with the FloryHuggins parameter χ_AB, which we assume to be greater than 0 and positive. We also assume to have a homopolymer with degree of polymerization equal to N, and also of type A, where χ_AA = 0.

Now what are the initial and final states of the system?

Initially we have a disordered mixture of copolymer and homopolymer which we can characterize with Flory-Huggins theory in the same way as in the case of diblocks alone.

The final state of micelle formation will occur at some certain diblock volume fraction φ_CMC. At this concentration we would expect the spontaneous formation of a micelle with core radius R_C, composed of B type blocks, and corona radius L_C, composed of A type blocks since the solvent is homopolymer of type A. One difference in this case is that we are comparing two systems in which the number of total molecules is not fixed. This is because the micelle will not occupy all of the block copolymer in the solution in order to reach equilibrium. The condition for equilibrium is that the chemical potential of diblocks in the disordered state and the micelle state must be equal - that is, the change in free energy for moving a single diblock from the micelle to disordered solution is 0. Therefore, we want to calculate the chemical potential of both the micelle and disordered state and set them equal to each other. For simplicity, we will set the chemical potential equal to the change in free energy per diblock (i.e. ∂G/∂n ≈ ∆G/∆n).

Free Energy of Disordered DB Solute and Homopolymer Solvent

Free Energy of Disordered State:

We previously described the free energy of the disordered state of diblocks using Flory-Huggins theory and we can do the same here to describe the free energy of mixing (same equation we had way back in Flory-Huggins)

$$\frac{\Delta G_{mix}}{N_0} = kT \lbrack \chi \Phi_1 \Phi_2 + \frac{\Phi _1}{x_1} \ln \Phi _1 + \frac{\Phi _2}{x_2} \ln \Phi _2 \rbrack \nonumber$$

Now we need to make a quick notation change since now we are working with BCPs. Here x₁ = N_A and x₂ = N_B. Since both of these degrees of polymerization are large for polymers, Φ₁/x₁ and Φ₂/x₂ are very small and we can safely ignore the two mixing terms and concentrate only on the enthalpic term. We also wrote N₀ as the total number of lattice sites in the imaginary lattice - in this case, if we instead let N₀ = N, where N is the length of a single diblock in our current formalism, the expression instead gives the free energy change for a SINGLE molecule. Multiplying through thus gives us the free energy change per polymer (approx. equal to the chemical potential)

$$\Delta G_{dis} \approx NkT\chi \Phi_A \Phi_B \nonumber$$

where we have now switched notation to the volume fractions of the two types, A and B.

Now, we have to put the two volume fractions of type A and B into terms of the volume fraction of diblock vs. homopolymer, since the volume fraction of diblock includes both A and B components but the homopolymer only has type A. We can thus write

$$\phi_A = \phi_A^{homo} + \phi_A^{diblock} \nonumber$$

$$\phi_B = \phi_B^{diblock} \nonumber$$

If we let φ = φ^diblock, then from the condition that N_A = N_B = N/2 we can say

that

$$\phi_A = (1-\phi) + \phi/2 \nonumber$$

$$\phi_B = \phi/2 \nonumber$$

$$\Delta G_{dis} \approx NkT\chi(1-\phi/2)(\phi/2) \nonumber$$

As a final approximation, we can assume that the volume fraction of the diblock solute is much smaller than that of the homopolymer solvent - that is, φ << 1. hence, we expect that 1 − φ/2 ≈ 1 and simplify to

$$\Delta G_{dis} \approx \frac{NkT\chi\phi}{2} \nonumber$$

This is our final expression for the free energy PER POLYMER in the disordered state.

Free Energy of the Micelle State:

Now we have to calculate the free energy per polymer in the micelle state. Here like there will be three contributions

1. Energy penalty due to stretching - stretching blocks in the corona and core of the micelle leads to entropic spring restoring force
2. Favorable increase in entropy due to mixing- gain in mixing entropy from mixing homopolymer with blocks in the corona
3. Energy penalty due to creating interface (IMDS) - the cost in forming an interface between the unfavorable A and B blocks

$$\Delta G_{micelle} = \Delta G_{stretch} + \Delta G_{mix} + \Delta G_{interface} \nonumber$$

Stretching Free Energy:

The stretching free energy is something that we have derived previously and we used it in the diblock case as $\Delta G \approx T \Delta S \approx \frac{3kT}{2}\frac{r^2}{r_0^2}$. Here, we recall that in our previous derivation we assume no enthalpy change upon stretching chains, and hence the free energy change is proportional to the final end-to-end distance of the chain squared over the initial squared end to end distance. In the case of the micelle, we assume the most general possible case - that both the blocks in the corona and core are capable of stretching or compressing, and that they assume a final length of R_C and L_C respectively. Because it is unclear whether they are elongated or compressed in the free energy minimized state, we will leave all 4 terms in the free energy expression

$$\Delta G_{stretch} \approx \frac{3kT}{2} \left [ \frac{R_C^2}{(N/2)b^2} + \frac{L_C^2}{(N/2)b^2} + \frac{(N/2)b^2}{R_C^2} + \frac{(N/2)b^2}{L_C^2} - 4 \right ] \nonumber$$

Here, the first two terms correspond to elongation, where the initial state is given by the unperturbed end-to-end distance of the chain in the core/corona respectively (where b is the monomer size corresponding to the Physicist’s Chain model). The second two terms correspond to compression of the same two regions. Also note that we have set unperturbed lengths based on each block having a number of monomers equal to N/2. In principle, we would find at equilibrium that for the core and corona only one of the two terms (i.e. compression or elongation) would apply, respectively, but without performing a full free energy minimization it is unclear whether stretching or compression is preferred. Finally, note that this term is already written PER POLYMER since there is no prefactor reflecting the number of polymers in the micelle.

Mixing Free Energy:

Next, we can consider the mixing free energy. Recall that the species being mixed are type A homopolymer in the type A corona, so no enthalpic term should be present since all enthalpic interactions are 0. The only term then is a mixing entropy, but again this is only between really a single component. Given this, let’s look at our old Flory Huggins Mixing term

$$\frac{\Delta G_{mix}}{N_0} = kT \lbrack \chi \Phi_1 \Phi_2 + \frac{\Phi _1}{x_1} \ln \Phi _1 + \frac{\Phi _2}{x_2} \ln \Phi _2 \rbrack \nonumber$$

But now we can ignore the enthalpy term completely since χ_AA = 0. Also here there is only one component so we can ignore the second mixing term. This leaves a single mixing term, which we can write as $\frac{\phi_H}{N_H}\ln \phi_H$, where φ_H is the volume fraction of homopolymer, which is the same as 1 − φ (recall that φ is the volume fraction of diblock). Similarly, N_H = N is the degree of polymerization of the homopolymer. Since the free energy is divided by N₀, which in Flory-Huggins theory is the number of lattice sites and hence an effective volume, we can make this a free energy PER POLYMER by dividing by the density of polymers in the corona. The density is approximated as the number of polymers in the micelle, n_M, divided by the total volume of the corona, ((R_C + L_C)³ − R_C³ ), or the volume of the sphere surrounding corona + core minus the volume of the core. We can now write the free energy change as

$$\Delta G_{mix} \approx \frac{kT}{n_M}\left ( ( R_C+L_C)^3 - R_C^3 \right ) \frac{\phi_H}{N_H} \ln \phi_H \nonumber$$

Free Energy of Interface:

The final free energy change is the interfacial energy, which we can derive directly from our treatment of microphase separation and the IMDS. We previously had that the interfacial energy per unit area, γ_AB, is given by

$$\gamma_{AB} = \frac{kT}{b^2}\sqrt{\frac{\chi_{AB}}{6}} \nonumber$$

We then wrote that the the free energy change was a simple function of the interfacial area per chain, Σ and γ_AB. The only distinction now is that the interfacial area per chain is different, and given from the surface area of the spherical core (because the unfavorable interface is between the core of type B and the corona of type A) rather than the area of a flat interface. This area is

$$\Sigma = \frac{4\pi R_C^2}{n_M} \nonumber$$

where again we have divided by n_M, the number of polymer chains in the micelle, to put this as the interfacial area per chain. Hence, we can write the total interfacial energy PER POLYMER as:

$$\Delta G_{interface} \approx \frac{kT}{b^2} \sqrt{\frac{\chi_{AB}}{6}}\frac{4 \pi R_C^2}{n_M} \nonumber$$

Final Total Free Energy:

Now the final total free energy change in the micelle state PER POLYMER is

$$\Delta G_{micelle} = \Delta G_{stretch} + \Delta G_{mix} + \Delta G_{interface} \nonumber$$

$$\Delta G_{stretch} \approx \frac{3kT}{2} \left [ \frac{R_C^2}{(N/2)b^2} + \frac{L_C^2}{(N/2)b^2} + \frac{(N/2)b^2}{R_C^2} + \frac{(N/2)b^2}{L_C^2} - 4 \right ] \nonumber$$

$$\Delta G_{mix} \approx \frac{kT}{n_M}\left ( ( R_C+L_C)^3 - R_C^3 \right ) \frac{\phi_H}{N_H} \ln \phi_H \nonumber$$

$$\Delta G_{interface} \approx \frac{kT}{b^2} \sqrt{\frac{\chi_{AB}}{6}}\frac{4 \pi R_C^2}{n_M} \nonumber$$

Simplify Total Free Energy Change in Micelle State:

The free energy per polymer chain expression for the micelle is very messy, so we can use a variety of simple approximations to reduce the expression considerably.

1. Assume diblock chains are ideal even in the micelle state- both blocks in the core and corona are in their unperturbed state. This is a significant assumption but we know that in the bulk this is true for polymers so that will give ∆G_stretch = 0
2. ∆G_mix = 0 - Because degree of polymerization N is very large and since degree of polymerization of homopolymer is also N the mixing term is close to 0 so ∆G_mix = 0. This should make sense as large homopolymers will have trouble penetrating the corona.

With these two assumptions, we have thus shown that the change in interfacial free energy is the dominant term, and is the term that drives aggregation of diblocks into micelles. We can further simplify this term by assuming an incompressibility condition - that is, the volume occupied by chains in the solution is equivalent to the volume occupied in the micelle. We can thus solve for n_M by assuming that the average volume of a B block is given as the number of monomers in a B block, N/2, times the average volume of a B type monomer, b³. The total volume occupied by the chains in the core is then

$$n_M \frac{N}{2} b^3 = \frac{4\pi}{3} R_C^3 \nonumber$$

where the right term is the volume of the core in terms of our variable R_C. Using this relation to simplify the interfacial energy gives

$$\Delta G_{interface} = \frac{3kT}{2} \sqrt{\frac{\chi_{AB}}{6}} \frac{Nb}{R_C} \nonumber$$

we have eliminated some constants on the order of 1 here. We can make one final simplification by again relating R_C to the unperturbed length of a B block, invoking the same assumption that the chains are in their ideal state. This gives R_C ≈ N^1/2b, and gives a final expression of

$$\Delta G_{interface} = \frac{3kT}{2} \sqrt{\frac{\chi_{AB}}{6}} N^{1/2} \nonumber$$

So we have the free energy change per chain in the disordered state as

$$\Delta G_{dis} \approx \frac{NkT\chi\phi}{2} \nonumber$$

The free energy change per chain in the micelle state is given by

$$\Delta G_{micelle} = \Delta G_{stretch} + \Delta G_{mix} + \Delta G_{interface} \nonumber$$

$$\approx \Delta G_{interface} \nonumber$$

$$\approx \frac{3kT}{2} \sqrt{\frac{\chi}{6}} N^{1/2} \nonumber$$

So the critical micelle concentration (CMC) will be defined as the concentration (or the volume fraction) for which these two chemical potentials are equivalent which is

$$\frac{NkT\chi\phi_{CMC}}{2} = \frac{3kT}{2} \sqrt{\frac{\chi}{6}} N^{1/2} \nonumber$$

$$\phi_{CMC} \approx \frac{1}{\sqrt{\chi N}} \nonumber$$

Hence we find via our simple model that the critical micelle concentration varies inversely with the square root of χN, the same parameters (χ and N) we used to describe the phase behavior of both polymer blends and diblock copolymers, so it would make sense that a blend of diblocks with homopolymer would depend on the same parameters. Also we see that this result makes qualitative sense - if χN is high, the critical concentration is low - that is, as the strength (and number) of unfavorable enthalpic interactions gets higher, it requires less diblock to initiate aggregation into micelles. This makes sense since the micelle phase minimizes unfavorable enthalpic interactions by clustering together same-type blocks. Finally, note that in the model we employ here, we have effectively ignored the thermodynamic contribution of the homopolymer by ignoring the mixing entropy in the corona - as a result, we see that the role of the homopolymer is strictly to modify the volume fractions of the two components A and B in the system; we might expect that this model is also then suitable for describing diblocks with compositional asymmetry.

Experimentally Observing the CMC

To determine experimentally when the CMC is reached, micelles can be observed visually via TEM measurements, where the number of micelles per nm² can be observed. At some point this goes to 0 - this indicates the onset of the critical micelle composition. It’s also possible to look at the fraction of free copolymer as a function of total copolymer fraction - at the CMC, micelles will begin to form and the concentration of free diblocks will begin to be used up in micelles, and hence the concentration of FREE copolymer stays constant above the CMC while the total amount of diblocks increases. Changing N as you graph this relation also shows how the onset of the CMC changes.

If you increase the amount of copolymer above the CMC, at some point there will be a large number of micelles in solution that are capable of interacting with each other to generate additional order. At high concentrations, ordered spheres and cylinders can be observed, as these morphologies can again be thought of as minimizing unfavorable interfacial energy interactions. Additional morphologies are also available - as in the case of just diblock copolymers, we again see a wide range of morphologies when mixing diblocks with homopolymers.

Strategies to Create More Complex Polymer Morphologies:

• Manipulate the IMDS: The IMDS will always form to minimize surface area and maximize conformation entropy/polymer deformation or stretching. So we can change the shape of the IMDS by blending polymers or adding assymetric BCP that may have branches or loops.
• Designer BCPs: Similar to the last point but you can change the structure of the copolymer to add branches or loops purposefully for applications i.e. use monomers that don’t coil but act as rods (very inflexible backbones) to create nanowires. Or disk polymers like graphene. These monomers will have lots of pi-pi bonds which will lead to strong packing and intermolecular interactions between monomers and can lead to interesting electrical/optical behavior
• Utilize triblock copolymers: Using polymers that contain 3 different blocks that can have different architectures (linear, ring, or miktoarm, similar to star polymers). By simply adding this 3rd block and assuming all 3 are incompatible there are 64 different morphologies that can be achieved depending on relative volume fractions. You might also see ordered structures at the interface of two other blocks, like cylinders at the interfaces of lamellae, which occurs in a very similar manner to a heterogeneous nucleation process (formation of one ordered phase lowers interfacial energy between two other phases and forms at the interface)

Summary of Phase Behavior of Different Systems:

• Phase separation for A-B polymer blend (macrophase separation) χN = 2
• Phase separation for A-B BCP (microphase separation) χN = 10.5
• Phase separation for homopolymer-BCP mixtures and composition for micelles formation$\phi_{CMC} \approx \frac{1}{\sqrt{\chi N}}$