Kinetics of Micellization and Liquid–Liquid Phase Separation in Dilute Block Copolymer Solutions

Abstract

A lattice theory for block copolymer solutions near the boundary between the micellization and liquid–liquid phase separation regions proposes a new kinetic process of micellization where small concentrated-phase droplets are first formed and then transformed into micelles in the early stage of micellization. Moreover, the thermodynamically stable concentrated phase formed from metastable micelles by a unique ripening process in the late stage of phase separation, where the growing concentrated-phase droplet size is proportional to the square root of the time.

1. Introduction

An amphiphilic diblock copolymer forms a micelle in a selective solvent. If the block copolymer is thermosensitive, pH-sensitive, or ionic strength-sensitive, the kinetics of the micelle formation in dilute homogeneous block copolymer solutions can be studied experimentally by temperature-jump, pH-jump, or salt-jump experiments. Such kinetic studies have been carried out extensively for various block copolymer solutions thus far [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16]. The micellization process of block copolymers is much slower than that of small molecular surfactants, so the former kinetics is easier than the latter, but the micellization of the block copolymers is still fast to study experimentally.

The micellization kinetics of both block copolymers and small molecular surfactants have been analyzed on the basis of the stepwise aggregation mechanism of monomers [17,18,19,20,21]. This mechanism was first proposed by Aniansson and Wall in 1974 [17] to explain the micellization kinetics of small molecular surfactants, which resembles the kinetic theory of nucleation (the formation of a new phase within a metastable phase) [19,21], and was also utilized to analyze block copolymer micellization kinetics [3,4,5,6].

More recently, our group investigated the self-assemblies of several thermosensitive and ionic strength-sensitive block copolymers in dilute aqueous solutions, and found the competition between the micellization and liquid–liquid phase separation of those block copolymers [22,23,24,25,26]. The competition comes from weaker amphiphilicity of the block copolymers, which was explained semi-quantitatively by the lattice theory [27]. Furthermore, the lattice theory predicted that micellization prefers to liquid–liquid phase separation at a lower hydrophobic content of the copolymers, and the prediction was demonstrated experimentally using a thermosensitive block copolymer in aqueous solutions [26].

The present study deals with the kinetics of the micellization as well as of the liquid–liquid phase separation in dilute block copolymer solutions, where the micellization competes with the liquid–liquid phase separation. The liquid–liquid phase separation starts from the formation of small concentrated-phase droplets, which are thermodynamically less stable than the macroscopic concentrated phase due to the extra interfacial Gibbs energy. The preference of concentrated-phase droplets and micelles may be reversed during the growing process of micellization or phase separation. The judgement of the preference needs to quantitatively compare the Gibbs energy of formation of the concentrated-phase droplet with that of the micelle. Such a comparison proposes the unique kinetics of micellization and concentrated-phase droplet growth in the early and late stage, respectively, as explained in what follows.

2. Thermodynamics

2.1. Models and Mixing Gibbs Energy Densities [27]

Let us consider an aqueous solution of a block copolymer that consists of a hydrophilic A-chain of the degree of polymerization P_A and a hydrophobic B-chain of the degree of polymerization P_B. The hydrophilic and hydrophobic contents of the copolymer are given by x_A = P_A/(P_A + P_B) and x_B = P_B/(P_A + P_B), respectively. The monomer units of the A- and B-chains as well as the solvent water molecule are assumed to possess the same size a (the unit lattice size).

Figure 1a shows a schematic illustration of the spherical micelle. We assumed that both the hydrophobic core and hydrophilic shell regions of the micelle were uniform, and that the A- and B-block chains were completely excluded from the core and shell regions of the micelle, respectively, as shown in Figure 1, to calculate the thermodynamic quantities of the micellar phase. In this theory, the micelle is regarded as a thermodynamic phase. This simplified model of the spherical micelle resembles that used by Leibler et al. [28], although in the present model, the radii of the core R_core and of the whole micelle R_m are given by

$$R_{\mathrm{core}}/a=P_{\mathrm{B}}{}^{\alpha }, \left(R_{\mathrm{m}}-R_{\mathrm{core}}\right)/a=P_{\mathrm{A}}{}^{\alpha }$$

(1)

Here, the exponent α was assumed to be 0.5 for the Gaussian chain in this paper, but this α value does not essentially change the following semi-quantitative argument. Leibler et al. [28] considered a homopolymer as the solvent, and treated R_core and R_m as variables, determined from the free energy minimization condition.

The average volume fraction of the copolymer ϕ_P in the micellar phase is given by ϕ_P = 3(P_A + P_B)m/(4πR_m³), where m is the aggregation number of copolymer chains in the spherical micelle. The volume fractions of the A- and B-block chains in the shell and core regions (ϕ_A,s and ϕ_B,c), respectively, can be calculated from ϕ_P by

$${\varphi }_{\mathrm{A},\mathrm{s}}=\frac{x_{\mathrm{A}}{\varphi }_{\mathrm{P}}}{{\Phi }_s},{\varphi }_{\mathrm{B},\mathrm{c}}=\frac{x_{\mathrm{B}}{\varphi }_{\mathrm{P}}}{{\Phi }_c},{\Phi }_c=1-{\Phi }_s\equiv \frac{R_{\mathrm{core}}{}^3}{R_{\mathrm{m}}{}^3}$$

(2)

By extending the Flory–Huggins theory [29], we can formulate the mixing Gibbs energy density (per the unit lattice site) ∆g_m(ϕ_P) in the micellar phase as [27]

$$\begin{array}{c}\frac{\Delta g_{\mathrm{m}}({\varphi }_{\mathrm{P}})}{k_{\mathrm{B}}T}=\frac{{\varphi }_{\mathrm{P}}}{P}\ln \left(\kappa {\varphi }_{\mathrm{P}}\right)+{\Phi }_{\mathrm{s}}\left(1-{\varphi }_{\mathrm{A},\mathrm{s}}\right)\ln \left(1-{\varphi }_{\mathrm{A},\mathrm{s}}\right)+{\Phi }_{\mathrm{c}}\left(1-{\varphi }_{\mathrm{B},\mathrm{c}}\right)\ln \left(1-{\varphi }_{\mathrm{B},\mathrm{c}}\right)\\ +\left[x_{\mathrm{A}}\left(1-{\varphi }_{\mathrm{A},\mathrm{s}}\right){\chi }_{\mathrm{AS}}+x_{\mathrm{B}}\left(1-{\varphi }_{\mathrm{B},\mathrm{c}}\right){\chi }_{\mathrm{BS}}-x_{\mathrm{A}}{x}_{\mathrm{B}}{\chi }_{\mathrm{AB}}\right]{\varphi }_{\mathrm{P}}+\frac{3{\Phi }_c}{R_{\mathrm{core}}/a}\frac{a^2{\gamma }_{\mathrm{c}}}{k_{\mathrm{B}}T}\end{array}$$

(3)

where k_BT is the Boltzmann constant multiplied by the absolute temperature, and κ is a constant defined by

$$\kappa =\frac{\pi }{27}{\left(P_{\mathrm{A}}+P_{\mathrm{B}}\right)}^{2+\alpha }{x}_{\mathrm{A}}{x}_{\mathrm{B}}{}^{1-2\alpha }{\left(x_{\mathrm{A}}{}^{\alpha }+x_{\mathrm{B}}{}^{\alpha }\right)}^3$$

(4)

χ_AS, χ_BS, and χ_AB are the interaction parameters between the A-chain and the solvent, between the B-chain and the solvent, and between the A- and B-chains, respectively, and γ_c is the interfacial tension between the core and shell regions. Using the theory of Noolandi and Hong [30], as explained in Appendix A, the expression of γ_c as well as the thicknesses of the corresponding interfaces d_I,c are given by

$$\frac{a^2{\gamma }_{\mathrm{c}}}{k_{\mathrm{B}}T}=\sqrt{\frac{\left({\varphi }_{\mathrm{A},\mathrm{s}}+{\varphi }_{\mathrm{B},\mathrm{c}}\right)\Delta f_{\mathrm{c}}(0)}{3}},\frac{d_{\mathrm{I},\mathrm{c}}}{a}=\sqrt{\frac{{\varphi }_{\mathrm{A},\mathrm{s}}+{\varphi }_{\mathrm{B},\mathrm{c}}}{12\Delta f_{\mathrm{c}}(0)}}$$

(5)

where ∆f_c(0) is defined as

$$\begin{array}{c}4\Delta f_{\mathrm{c}}(0)\equiv {\chi }_{\mathrm{AB}}{\varphi }_{\mathrm{A},\mathrm{s}}{\varphi }_{\mathrm{B},\mathrm{c}}+\left({\chi }_{\mathrm{BS}}{\varphi }_{\mathrm{B},\mathrm{c}}-{\chi }_{\mathrm{AS}}{\varphi }_{\mathrm{A},\mathrm{s}}\right)\left({\varphi }_{\mathrm{B},\mathrm{c}}-{\varphi }_{\mathrm{A},\mathrm{s}}\right)\\ +2{\varphi }_{\mathrm{S},\mathrm{c}}\ln \left(\frac{{\varphi }_{\mathrm{S},\mathrm{c}}+{\varphi }_{\mathrm{S},\mathrm{s}}}{2{\varphi }_{\mathrm{S},\mathrm{c}}}\right)+2{\varphi }_{\mathrm{S},\mathrm{s}}\ln \left(\frac{{\varphi }_{\mathrm{S},\mathrm{c}}+{\varphi }_{\mathrm{S},\mathrm{s}}}{2{\varphi }_{\mathrm{S},\mathrm{s}}}\right)\end{array}$$

(6)

with ϕ_S,s = 1 − ϕ_A,s and ϕ_S,c = 1 − ϕ_B,s. Although not shown in Figure 1a, the interfacial region of the micelle has a sharp concentration gradient, as illustrated in Figure A1 in Appendix A. In Equation (3), the term of the interfacial tension between the shell region of the micelle and the coexisting dilute phase was neglected, because ϕ_A,s and χ_AS were considerably smaller than ϕ_P,d and $\overline{\chi }$ (cf. Equation (8)), respectively.

Figure 1b illustrates the schematic diagram of the spherical concentrated-phase droplet, where both volume fractions of the A- and B-chains are uniform. Through a simple extension of the Flory–Huggins theory [29], the mixing Gibbs energy density ∆g_d(ϕ_P) in the concentrated-phase droplet is written as

$$\frac{\Delta g_{\mathrm{d}}({\varphi }_{\mathrm{P}})}{k_{\mathrm{B}}T}={\varphi }_{\mathrm{S}}\ln {\varphi }_{\mathrm{S}}+\frac{{\varphi }_{\mathrm{P}}}{P}\ln {\varphi }_{\mathrm{P}}+\overline{\chi }{\varphi }_{\mathrm{S}}{\varphi }_{\mathrm{P}}+\frac{3a}{R_{\mathrm{d}}}\frac{a^2{\gamma }_{\mathrm{d}}}{k_{\mathrm{B}}T}$$

(7)

where ϕ_P and ϕ_S = 1 − ϕ_P are the volume fractions of the copolymer and the solvent in the droplet phase, respectively, and $\overline{\chi }$ is the average interaction parameter defined by

$$\overline{\chi }\equiv {\chi }_{\mathrm{AS}}{x}_{\mathrm{A}}+{\chi }_{\mathrm{BS}}{x}_{\mathrm{B}}-{\chi }_{\mathrm{AB}}{x}_{\mathrm{A}}{x}_{\mathrm{B}}$$

(8)

and γ_d is the interfacial tension between the concentrated-phase droplet and the homogeneous (dilute) solution phase. Not illustrated in Figure 1b, the interfacial region of the droplet must possess a sharp concentration gradient. There are two limiting cases: one is that only A-block chains make contact with the coexisting dilute phase at the interface, like the micelle, and the other is to assume that the composition ϕ_A(x)/ϕ_B(x) (cf. Figure A1 in Appendix A) is uniform, even at the interface. Here, we took the latter limiting case (i.e., the total copolymer volume fraction ϕ_P(x) changes under the constant composition). Then, the expression of γ_d and the thickness of this interface d_I,d are obtained from Equations (A4)–(A6) in Appendix A by

$$\frac{a^2{\gamma }_{\mathrm{d}}}{k_{\mathrm{B}}T}=\sqrt{\frac{\Delta f_{\mathrm{d}}(0)}{3\left[{\varphi }_{\mathrm{P}}+{\varphi }_{\mathrm{P},\mathrm{h}}(R_{\mathrm{d}})\right]}}\left[{\varphi }_{\mathrm{P}}-{\varphi }_{\mathrm{P},\mathrm{h}}(R_{\mathrm{d}})\right],\frac{d_{\mathrm{I},\mathrm{d}}}{a}=\frac{{\varphi }_{\mathrm{P}}-{\varphi }_{\mathrm{P},\mathrm{h}}(R_{\mathrm{d}})}{\sqrt{12\left[{\varphi }_{\mathrm{P}}+{\varphi }_{\mathrm{P},\mathrm{h}}(R_{\mathrm{d}})\right]\Delta f_{\mathrm{d}}(0)}}$$

(9)

with the copolymer volume fraction ϕ_P,h(R_d) in the dilute solution phase coexisting with the concentrated-phase droplet of the radius R_d and

$$\begin{array}{c}4\Delta f_{\mathrm{d}}(0)\equiv \overline{\chi }\left[{\varphi }_{\mathrm{P}}-{\varphi }_{\mathrm{P},\mathrm{h}}(R_{\mathrm{d}})\right]\left[{\varphi }_{\mathrm{S},\mathrm{h}}(R_{\mathrm{d}})-{\varphi }_{\mathrm{S}}\right]\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+2{\varphi }_{\mathrm{S}}\ln \left[\frac{{\varphi }_{\mathrm{S}}+{\varphi }_{\mathrm{S},\mathrm{h}}(R_{\mathrm{d}})}{2{\varphi }_{\mathrm{S}}}\right]+2{\varphi }_{\mathrm{S},\mathrm{h}}(R_{\mathrm{d}})\ln \left[\frac{{\varphi }_{\mathrm{S}}+{\varphi }_{\mathrm{S},\mathrm{h}}(R_{\mathrm{d}})}{2{\varphi }_{\mathrm{S},\mathrm{h}}(R_{\mathrm{d}})}\right]\end{array}$$

(10)

[ϕ_S = 1 − ϕ_P and ϕ_S,h(R_d) = 1 − ϕ_P,h(R_d)].

The excess chemical potentials (per molecule) of the solvent ∆μ_S,d and of the copolymer ∆μ_P,d in the droplet phase can be calculated from Equation (7) as

$$\begin{array}{c}\frac{\Delta {\mu }_{\mathrm{S},\mathrm{d}}}{k_{\mathrm{B}}T}=\ln {\varphi }_{\mathrm{S}}+\left(1-\frac{1}{P}\right){\varphi }_{\mathrm{P}}+\overline{\chi }{\varphi }_{\mathrm{P}}{}^2+\frac{2}{\left(R_{\mathrm{d}}/a\right)}\frac{a^2{\gamma }_{\mathrm{d}}}{k_{\mathrm{B}}T}-\frac{3{\varphi }_{\mathrm{P}}}{\left(R_{\mathrm{d}}/a\right)}\left(\frac{\mathrm{d}}{\mathrm{d}{\varphi }_{\mathrm{P}}}\frac{a^2{\gamma }_{\mathrm{d}}}{k_{\mathrm{B}}T}\right)\\ \frac{\Delta {\mu }_{\mathrm{P},\mathrm{d}}/P}{k_{\mathrm{B}}T}=\frac{1}{P}\ln {\varphi }_{\mathrm{P}}-\left(1-\frac{1}{P}\right){\varphi }_{\mathrm{S}}+\overline{\chi }{\varphi }_{\mathrm{S}}{}^2+\frac{2}{\left(R_{\mathrm{d}}/a\right)}\frac{a^2{\gamma }_{\mathrm{d}}}{k_{\mathrm{B}}T}+\frac{3{\varphi }_{\mathrm{S}}}{\left(R_{\mathrm{d}}/a\right)}\left(\frac{\mathrm{d}}{\mathrm{d}{\varphi }_{\mathrm{P}}}\frac{a^2{\gamma }_{\mathrm{d}}}{k_{\mathrm{B}}T}\right)\end{array}$$

(11)

Finally, the mixing Gibbs energy density ∆g_h(ϕ_P) as well as the excess chemical potentials of the solvent ∆μ_S,h and of the copolymer ∆μ_P,h in the homogeneous copolymer solution phase are given by [29]

$$\begin{array}{c}\frac{\Delta g_{\mathrm{h}}({\varphi }_{\mathrm{P}})}{k_{\mathrm{B}}T}={\varphi }_{\mathrm{S}}\ln {\varphi }_{\mathrm{S}}+\frac{{\varphi }_{\mathrm{P}}}{P}\ln {\varphi }_{\mathrm{P}}+\overline{\chi }{\varphi }_{\mathrm{S}}{\varphi }_{\mathrm{P}}\\ \frac{\Delta {\mu }_{\mathrm{S},\mathrm{h}}}{k_{\mathrm{B}}T}=\ln {\varphi }_{\mathrm{S}}+\left(1-\frac{1}{P}\right){\varphi }_{\mathrm{P}}+\overline{\chi }{\varphi }_{\mathrm{P}}{}^2,\frac{\Delta {\mu }_{\mathrm{P},\mathrm{h}}/P}{k_{\mathrm{B}}T}=\frac{1}{P}\ln {\varphi }_{\mathrm{P}}-\left(1-\frac{1}{P}\right){\varphi }_{\mathrm{S}}+\overline{\chi }{\varphi }_{\mathrm{S}}{}^2\end{array}$$

(12)

which correspond to Equations (7) and (11) in the limit of infinite R_d.

2.2. Phase Diagram

When $\overline{\chi }$ is sufficiently large, concentrated-phase droplets with the copolymer volume fraction ϕ_P,d(R_d) appear in the homogeneous copolymer solution, and if the amphiphilicity is strong enough (or χ_BS >> χ_AS), the micellar phase with the copolymer volume fraction ϕ_P,m is formed in the homogeneous copolymer solution. The copolymer volume fractions ϕ_P,d(R_d) and ϕ_P,m can be calculated from the following phase equilibrium conditions

$$\Delta {\mu }_{\mathrm{S},\mathrm{h}}({\varphi }_{\mathrm{P},\mathrm{h}}(R_{\mathrm{d}}))=\Delta {\mu }_{\mathrm{S},\mathrm{d}}({\varphi }_{\mathrm{P},\mathrm{d}}(R_{\mathrm{d}})),\Delta {\mu }_{\mathrm{P},\mathrm{h}}({\varphi }_{\mathrm{P},\mathrm{h}}(R_{\mathrm{d}}))=\Delta {\mu }_{\mathrm{P},\mathrm{d}}({\varphi }_{\mathrm{P},\mathrm{d}}(R_{\mathrm{d}}))$$

(13a)

$$\Delta {\mu }_{\mathrm{S},\mathrm{h}}({\varphi }_{\mathrm{P},\mathrm{h}}{}^{\left(\mathrm{m}\right)})=\Delta {\mu }_{\mathrm{S},\mathrm{m}}({\varphi }_{\mathrm{P},\mathrm{m}}),\Delta {\mu }_{\mathrm{P},\mathrm{h}}({\varphi }_{\mathrm{P},\mathrm{h}}{}^{\left(\mathrm{m}\right)})=\Delta {\mu }_{\mathrm{P},\mathrm{m}}({\varphi }_{\mathrm{P},\mathrm{m}})$$

(13b)

where ∆μ_S,I and ∆μ_P,i (i = h, d, m) are the excess chemical potentials (per molecule) of the solvent and of the copolymer, respectively, in the homogeneous solution (i = h), in the droplet phase (i = d), and in the micellar phase (i = m); ϕ_P,h(R_d) and ϕ_P,h^(m) are the copolymer volume fractions of the homogeneous (dilute) phase coexisting with the droplet and micellar phases, respectively.

As a case study, Figure 2 shows three mixing Gibbs energy densities, ∆g_h, ∆g_d, and ∆g_m as functions of ϕ_P for an AB diblock copolymer solution with P_A = P_B = 100 (x_A = x_B = 0.5), χ_AS = 0.4, χ_AB = 0, and χ_BS = 1.122. The blue curve (∆g_d) for the droplet phase shifts slightly upward from the black curve (∆g_h) for the homogeneous solution due to the interfacial tension term in Equation (7). The red curve (∆g_m) for the micellar phase has stronger curvature than the black and blue curves. By choosing χ_BS = 1.122, we can draw a common tangent to the black and red curves (although not clearly shown, the black curve was convex downward at ϕ_P~0), which makes contact with the black and red curves at ϕ_P,h(∞) (=ϕ_P,h^(m)), ϕ_P,m, and ϕ_P,d(∞). When χ_BS < 1.122, the red curve shifts upward, and when χ_BS > 1.122, the red curve shifts downward relative to the black curve, indicating that the homogeneous solution and the micellar phase are thermodynamically stabler at lower and higher χ_BS, respectively [27].

From the Gibbs–Duhem relation, the mixing Gibbs energy density is related to the excess chemical potential by

$$\Delta g_{\mathrm{h}}({\varphi }_{\mathrm{P}})=\left(1-{\varphi }_{\mathrm{P}}\right)\Delta {\mu }_{\mathrm{S},\mathrm{h}}+{\varphi }_{\mathrm{P}}\frac{\Delta {\mu }_{\mathrm{P},\mathrm{h}}}{P},\Delta g_{\mathrm{m}}({\varphi }_{\mathrm{P}})=\left(1-{\varphi }_{\mathrm{P}}\right)\Delta {\mu }_{\mathrm{S},\mathrm{m}}+{\varphi }_{\mathrm{P}}\frac{\Delta {\mu }_{\mathrm{P},\mathrm{m}}}{P}$$

(14)

and this equation verifies that ϕ_P,h^(m) and ϕ_P,m in Figure 2 fulfill the phase equilibrium conditions given by Equation (13b). On the other hand, the Gibbs–Duhem relation does not hold for the droplet phase due to the interfacial tension term in Equation (7), which is not proportional to the total number of molecules in the system. However, the coexisting phase volume fractions ϕ_P,d(R_d) and ϕ_P,h(R_d) can be calculated by solving the simultaneous Equation (13a).

Pairs of the coexisting phase volume fractions, ϕ_P,h(∞) and ϕ_P,d(∞), ϕ_P,d(R_d) and ϕ_P,h(R_d), and ϕ_P,h^(m) and ϕ_P,m, are obtained as functions of χ_BS, at constant χ_AS = 0.4 and χ_AB = 0 [and at R_d/a = 20 for the pair of ϕ_P,d(R_d) and ϕ_P,h(R_d)]. Figure 3 shows the phase diagram under this condition. Here, the black, blue, and red curves indicate coexisting homogeneous, droplet, and micellar phases, respectively, and solid and broken curves the stable and unstable (or metastable) states, respectively. The small black circle in the figure indicates the critical point for the liquid–liquid phase separation. It was noted that there is no critical point for the homogeneous-droplet phase equilibrium. For the coexistence of the micellar and homogeneous phases, we can draw two common tangents (the two tangents become identical at χ_BS = 1.122 in Figure 2), and the phase diagram has two coexistence regions of the micelle + dilute phase and micelle + concentrated phase, as shown in Figure 3. In what follows, we focused only on micellization and the concentrated-phase droplet formation in dilute copolymer solutions.

The micellar phase formation thermodynamically prefers the formation of the droplet phase with R_d/a = 20 at χ_BS > 1.048, as indicated by the blue thin dotted line in Figure 3. This phase boundary shifts to lower χ_BS than the boundary between the micellization and macroscopic liquid–liquid phase separation (cf. the red thin dotted line in Figure 3), because the interfacial tension term in ∆g_d(ϕ_P) given by Equation (7) destabilizes the droplet concentrated phase. With increasing R_d, the boundary between the micelle and droplet phases approaches the red thin dotted line, located at χ_BS = 1.122.

As already shown in a previous work [27], when χ_AS decreases, the biphasic region of the liquid–liquid phase separation goes up (Figure 3), while the binodal curves for the micelle–homogeneous phase equilibrium do not change as much. Thus, the stable liquid–liquid phase separation region becomes narrower and finally disappears in the phase diagram. A similar change in the phase diagram occurs when χ_AB increases.

In Figure 3, the black thin dot-dash curve indicates the spinodal, calculated by [29]

$${\varphi }_{\mathrm{P},\mathrm{sp}\pm }=\frac{2\overline{\chi }-1+P^{-1}\pm \sqrt{{\left(2\overline{\chi }-1+P^{-1}\right)}^2-8P^{-1}\overline{\chi }}}{4\overline{\chi }}$$

(15)

within the spinodal region, the homogeneous phase is not stable with respect to long-ranged concentration fluctuation induced by thermal agitation, and the spinodal decomposition can take place in the early stage of the liquid–liquid phase separation. On the other hand, the concentration fluctuation is unstable in the homogeneous solution, outside the spinodal region. Thus, when the homogeneous solution is jumped into the region between the spinodal and binodal curves in the dilute region, the concentrated-phase droplet and spherical micelle may be formed in the solution through the nucleation–growth mechanism.

3. Kinetics of Micellization and Liquid–Liquid Phase Separation

3.1. Growth of the Concentrated-Phase Droplet and Spherical Micelle in the Early Stage

Let us consider a metastable dilute homogeneous solution with a copolymer volume fraction ϕ_P,h between ϕ_P,h(R_d) and the lower spinodal volume fraction ϕ_P,sp− (cf. Equation (15)). When a spherical concentrated-phase droplet with the radius R_d and the copolymer volume fraction ϕ_P,d is formed in this homogeneous copolymer solution, the Gibbs energy of the droplet formation δG_d is given by

$$\delta G_{\mathrm{d}}=\frac{\Delta g_{\mathrm{d}}({\varphi }_{\mathrm{P},\mathrm{d}})-\Delta g_{\mathrm{h}}({\varphi }_{\mathrm{P},\mathrm{h}})}{{\varphi }_{\mathrm{P},\mathrm{d}}}\left(P_{\mathrm{A}}+P_{\mathrm{B}}\right)m$$

(16)

where ϕ_P,d is assumed to be equal to the equilibrium copolymer volume fraction ϕ_P,d(R_d) calculated by the phase equilibrium condition, Equations (13a) and (13b), as a function of the droplet radius R_d, and the aggregation number m is calculated by

$$m=\frac{4\pi }{3\left(P_{\mathrm{A}}+P_{\mathrm{B}}\right)}{R}_{\mathrm{d}}{}^3{\varphi }_{\mathrm{P},\mathrm{d}}$$

(17)

Because the droplet of the radius R_d is in equilibrium with the homogeneous solution with ϕ_P,h(R_d) (<ϕ_P,h), there was a concentration gradient in the vicinity of the droplet surface and the droplet grew through the diffusion flow of the copolymer from the homogeneous solution.

Similarly, when a spherical micelle with the aggregation number m and the copolymer volume fraction ϕ_P,m formed in the dilute homogeneous copolymer solution with the copolymer volume fraction ϕ_P,h (ϕ_P,h^(m) < ϕ_P,h < ϕ_P,sp−), the Gibbs energy of the micelle formation δG_m is given by

$$\delta G_{\mathrm{m}}=\frac{\Delta g_{\mathrm{m}}({\varphi }_{\mathrm{P},\mathrm{m}})-\Delta g_{\mathrm{h}}({\varphi }_{\mathrm{P},\mathrm{h}})}{{\varphi }_{\mathrm{P},\mathrm{m}}}\left(P_{\mathrm{A}}+P_{\mathrm{B}}\right)m$$

(18)

where ϕ_P,m is calculated from m by

$${\varphi }_{\mathrm{P},\mathrm{m}}=\frac{a^3\left(P_{\mathrm{A}}+P_{\mathrm{B}}\right)}{\left(4\pi /3\right)R_{\mathrm{m}}{}^3}m$$

(19)

where R_m is calculated by Equation (1).

In Figure 4, δG_d and δG_m are plotted against the aggregation number m in the copolymer solution at ϕ_P,h = 0.002. As seen in Figure 3, micellization is preferred to the liquid–liquid phase separation at χ_BS = 1.3, but δG_m for the micelle is higher than δG_d for the concentrated-phase droplet at m < 23. Thus, if χ_BS abruptly changes from a low value (<0.85, the one-phase condition) to 1.3, small concentrated-phase droplets form more easily than the micelles with the same m (<23) in the copolymer solution. However, when these concentrated-phase droplets grow, they become less stable than the micelle at m > 23. This indicates that the core–shell structure may be developed inside the growing droplets. Finally, the micelles with m = 34 (the red circle in Figure 4 for χ_BS = 1.3) form in the copolymer solution in the equilibrium state.

A similar relation between δG_d and δG_m also holds at χ_BS = 1.1. Concentrated-phase droplets first form in the copolymer solution, and they may be transformed into micelles with m = 31 (the red circle in Figure 4 for χ_BS = 1.1) during the growing process. Although the liquid–liquid phase separation is preferred to the micellization at χ_BS = 1.1 in the final equilibrium state, as seen in Figure 3, the micelle is a metastable state, and the formation of the concentrated phase of a large size must overcome the activation Gibbs energy barrier. The late stage of the concentrated-phase droplet growth in such a solution is discussed in the next Section 3.2.

At χ_BS = 0.9, δG_d is smaller than δG_m in the whole m range, and the concentrated-phase droplet keeps growing without transforming into the micelle. In the equilibrium state, the liquid–liquid phase separation is preferred to micellization (cf. Figure 3).

The kinetics of liquid–liquid phase separation and micellization in the early stage has been argued in the scenario of the nucleation–growth mechanism thus far. That is, the nuclei of the concentrated phase and the micelle are formed step-wise from the unimer (m = 1) [17,18,19,20,21]. In the concentrated-phase droplet, the copolymer chain was assumed to take the random coil conformation, and the concentration gradient near the droplet surface was neglected to derive Equation (7). Thus, Equation (7) for ∆g_d(ϕ_P) holds only at R_d larger than <R²>^1/2/2 + d_I,d, where <R²>^1/2 is the mean-square end-to-end distance of the copolymer chain [=(P_A + P_B)a²], and d_I,d is calculated by Equation (9). In Figure 4, m = 10 is close to the lower limit where Equation (7) holds. Moreover, in the micelle model used in this work, R_m and R_core are given by Equation (1), which may hold only when ϕ_B,c is sufficiently high. For small m, where ϕ_B,c is low, B-block chains forming the core may shrink to escape contact with the solvent, so our micelle model cannot be used for such a small m. Under the condition of Figure 4, ϕ_B,c ≈ 0.25 at m = 10, and our micelle model may not be applied at m considerably smaller than 10. Therefore, Equations (16) and (18) cannot be used to discuss the nucleation processes of liquid–liquid phase separation and micellization [21].

Although these nucleation processes were not discussed, the above argument proposes a new kinetics of the micellization process in the early stage (i.e., the transformation from the small concentrated-phase droplet to the spherical micelle). As above-mentioned, the stable liquid–liquid phase separation region becomes narrower and finally disappears in the phase diagram, when χ_AS decreases and/or χ_AB increases. Nevertheless, the same kinetics of the micellization process in the early stage can also be expected at lower χ_AS and higher χ_AB, because δG_d < δG_m at small m, even if the liquid–liquid phase separation region becomes unstable, as demonstrated in Figure 4 at χ_BS = 1.3.

3.2. Modification of the Lifshitz–Slyosov Theory in the Late Stage of the Phase Separation

As above-mentioned, the micellar phase is the metastable state at χ_BS = 1.1 (more generally at χ_BS from 1.03 to 1.122) within the biphasic region in Figure 3, and the thermodynamically equilibrium state is the liquid–liquid phase separation (where the concentrated-phase droplet size is macroscopic) in that region in Figure 3. The phase boundary (the blue horizontal dotted line in Figure 3 at R_d/a = 20) shifts to χ_BS = 1.1 at R_d/a = 50, so the concentrated-phase droplet with R_d/a > 50 is stabler than the micelle. If the concentrated-phase droplet of such a large size is formed occasionally by overcoming the activation Gibbs energy barrier, it keeps growing without developing the micellar structure. Since ϕ_P,h(R_d) < ϕ_P,h^(m) at R_d/a > 50, the growth of such large droplets in the solution is accompanied by the dissociation of micelles existing in the solution. This droplet ripening process in the micellar solution is different from the conventional Ostwald ripening in the usual liquid–liquid phase separation of the late stage.

The conventional Ostwald ripening process is quantitatively dealt with by the Lifshitz–Slyozov theory [31,32]. In the theory, the droplet with the radius R_d is in equilibrium with the homogeneous dilute solution (the mother solution) with the concentration ϕ_P,h(R_d) given by (in terms of our notations)

$${\varphi }_{\mathrm{P},\mathrm{h}}(R_{\mathrm{d}})={\varphi }_{\mathrm{P},\mathrm{h}}(\infty )+\frac{\sigma }{R_{\mathrm{d}}}$$

(20)

where σ is a constant parameter proportional to the interfacial tension. When the average concentration of the mother solution is denoted as ${\overline{\varphi }}_{\mathrm{P},\mathrm{h}}$, larger droplets with ϕ_P,h(R_d) < ${\overline{\varphi }}_{\mathrm{P},\mathrm{h}}$ can grow, while smaller droplets with ϕ_P,h(R_d) > ${\overline{\varphi }}_{\mathrm{P},\mathrm{h}}$ dissociate along with diffusing the solute into the mother solution. The critical radius R* for the growing droplet is given by

$$R^{\ast }=\frac{\sigma }{{\overline{\varphi }}_{\mathrm{P},\mathrm{h}}-{\varphi }_{\mathrm{P},\mathrm{h}}(\infty )}$$

(21)

Along with the droplet growth, the concentration of the mother solution is reduced, and R* increases with time t. Although the σ calculated from the results of ϕ_P,h(R_d) obtained by Equations (13a) and (13b) was slightly dependent on R_d, we neglected this dependence according to Lifshitz and Slyozov in what follows, of which the approximation is good only at sufficiently large R_d, or in the late stage of the phase separation

The concentration of the mother solution containing spherical micelles should maintain the critical micelle concentration ϕ_P,h^(m), even during the growth of concentrated-phase droplets. Thus, Equation (21) should be replaced by

$$R^{\ast }=\frac{\sigma }{{\varphi }_{\mathrm{P},\mathrm{h}}{}^{\left(\mathrm{m}\right)}-{\varphi }_{\mathrm{P},\mathrm{h}}(\infty )}$$

(22)

Thus, R* is independent of t until all micelles disappear in the solution, which is in contrast with the Lifshitz–Slyozov theory, and the theory should be modified as follows.

New parameters u and t_r are defined by

$$u\equiv \frac{R_{\mathrm{d}}}{R^{\ast }},t_{\mathrm{r}}\equiv \frac{\sigma D}{R^{\ast }{}^3}t$$

(23)

where D is the diffusion coefficient of the solute in the mother solution. Using these parameters, the droplet growth rate du/dt_r can be written as

$$\frac{\mathrm{d}u}{\mathrm{d}{t}_{\mathrm{r}}}=\frac{1}{u}\left(1-\frac{1}{u}\right)$$

(24)

By integrating this equation, we obtain

$$\frac{1}{2}\left(u+3\right)(u-1)+\ln (u-1)+C_0=t_{\mathrm{r}}$$

(25)

with the integration constant C₀. Figure 5 shows the time dependence of R_d for the growing droplet (R_d = R* at t = 0) in the micellar solution, calculated by Equation (25). When u and t_r are sufficiently large, Equation (25) can be approximated to

$$u\approx \sqrt{2t_{\mathrm{r}}}$$

(26)

This is in contrast with the Lifshitz–Slyozov theory, where the average radius of the droplet is proportional to t^1/3 in the late stage [31,32]. After all micelles disappear in the solution, the droplet growth should obey the Ostwald ripening (i.e., larger droplets grow by consuming the solute provided by dissociating smaller droplets). As the result, the average radius of the growing droplet is proportional to t^1/3.

4. Concluding Remarks

A discussion has been made on the phase separation kinetics in the block copolymer solution where the micellization competes with the liquid–liquid phase separation. When the block copolymer solution is jumped from the one-phase to biphasic region near the micelle–liquid–liquid phase separation boundary, small droplets of the concentrated phase are first formed, and afterward, the micellar structure may be developed in the concentrated-phase droplets. This is a kinetic mechanism of the micellization newly proposed in this study.

The kinetics of the micelle formation in block copolymer solutions has been investigated experimentally by many researchers by using the temperature-jump and stopped flow experiments [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16]. When homogeneous block copolymer solutions were brought into the micelle region, the scattering intensity and hydrodynamic radius in the copolymer solutions increased with time, and these experimental results were analyzed in terms of the step-by-step association model [17,18,19,20,21], but the detailed structure inside the scattering particles have not been investigated thus far. It is not enough to only measure the scattering intensity and hydrodynamic radius to distinguish the micelle from the concentrated-phase droplet experimentally. The newly proposed micellization kinetics must be checked in future work.

According to the above micellization kinetics, the micelles exist as a metastable state, even out of the micellization region in the thermodynamically stable phase diagram. In the late stage of the phase separation, the metastable micelle may be transformed into the stabler concentrated phase through the ripening process of the concentrated-phase droplet. This ripening process differs from the conventional Ostwald ripening occurring in the usual liquid–liquid phase separation systems [31,32], because the droplet growth in our block copolymer solution takes place by consuming the solute provided by metastable micelles in the solution. The growing droplet size is proportional to the square root of time t in the micellar solution, and the conventional t^1/3 dependence of the (average) droplet size in the Ostwald ripening is recovered after all micelles are consumed.

Recently, Narang and Sato [33] investigated the self-assembly of an amphiphilic amino acid derivative in aqueous dimethylsulfoxide, where concentrated-phase droplets coexist with micellar particles. Unfortunately, after the temperature jump of the solution, light scattering indicated the existence of sub-micron size droplets, but the droplets did not essentially grow any more during the light scattering and small-angle X-ray scattering measurements, maybe because of the fast ripening process in this small molecular system. To the best of the author’s knowledge, there have been no reports demonstrating the t^1/2 dependence of the droplet size in phase separating solutions.