Simulations of sphere-forming diblock copolymer melts

Molecular dynamics simulations are used to study melts of asymmetric sphere-forming diblock copolymers with two significantly different values of the invariant degree of polymerization, $\overline{N}=$ 3820 and 960. In both systems, changes in parameters that correspond to decreasing temperature lead to the appearance of micelles at a critical micelle temperature (CMT) and crystallization at a lower order-disorder transition temperature (ODT). The CMT is identifiable in simulations by the appearance of large clusters with a strongly segregated core region, but has no equally clear signature in scattering experiments on systems of modest $\overline{N}$. The value of the product $\chi N$ at the CMT (where $\chi$ is the Flory-Huggins parameter and $N$ is degree of polymerization) is close to that predicted by SCFT for the ODT, while the value at the actual ODT is larger and increases with decreasing $\overline{N}$. Micelles exhibit significant and comparable dispersity in aggregation number in the crystalline and liquid phases near the ODT. Both the liquid and crystal phases exhibit transient dimers consisting of pairs of neighboring spherical micelles with cores connected by a bridge of core-block material.

Figure 1

Visualization of the core of a micellar cluster composed of 83 chains. Minority block beads belonging to all molecules in the cluster are shown as as overlapping translucent red spheres. The conformation of one representative molecule is shown as sequence of points representing beads connected by line segments, with blue points for majority/corona block beads and red points for minority/core block beads.

Figure 2

Structure factor $S\left(q\right)$ vs nondimensionalized wavenumber $q{R}_{g0}$, for (a) $\overline{N}=960$ and (b) $\overline{N}=3820$ at several values of $\chi N$. $R_{g0}=b\sqrt{N/6}$ is the unperturbed polymer radius of gyration. Solid lines are fits of behavior near the peak to the functional form predicted by the random-phase approximation (RPA). Panel (b) reproduced with permission from Ref. [8]. 

Figure 3

Nondimensionalized inverse peak intensity $cN{S}^{-1}\left(q^{*}\right)/2$ vs $\chi N$ for $\overline{N}=960$ (solid-red circles) and $\overline{N}=3820$ (solid-blue diamonds). The diagonal dashed line is the RPA prediction. The vertical-dotted line marks location of ${\left(\chi N\right)}_{\mathrm{odt}}^{\mathrm{scf}}$. Error bars are smaller than the data points and provided in the SM [37]. 

Figure 4

Normalized peak wavenumber $q^{*}{R}_{g0}$ vs $\chi N$ for $\overline{N}=960$ (solid circles) and $\overline{N}=3820$ (solid-blue diamonds), where $R_{g0}=b\sqrt{N/6}$. The dotted-vertical line and the solid-vertical-gray line marks the disordered-FCC phase transition at ${\left(\chi N\right)}_{\mathrm{odt}}^{\mathrm{scf}}=36.6$ and the order-order transition from FCC to BCC at $\chi N=38.5$, respectively. The solid-horizontal-black line shows the RPA prediction for $q^{*}{R}_{g0}$ in the disordered phase, which is shown only for $\chi N<{\left(\chi N\right)}_{\mathrm{odt}}^{\mathrm{scf}}$. The dot-dashed and dashed curves plotted for $\chi N>{\left(\chi N\right)}_{\mathrm{odt}}^{\mathrm{scf}}$ show SCFT predictions for the value of $q^{*}{R}_{g0}$ corresponding to the primary Bragg peaks in the BCC and FCC phases, respectively. 

Figure 5

Symbols show the calculated intermicellar correlations $Z\left(q\right)$ at two values of $\chi N$ for (a) $\overline{N}=960$ and (b) $\overline{N}=3820$. The solid lines in the plot are fits to the Percus-Yevick theory, in which the sphere radius $R_{hs}$, and the effective volume fraction, $\eta$, and an overall constant of proportionality are adjusted to fit the data. The resulting fit parameters are in Table 2. Note that a different ordinate scale is used for each $\overline{N}$. 

Figure 6

Mole fraction $x_n$ of chains that belong to clusters of aggregation number $n$ for (a) $\overline{N}=960$ and (b) $\overline{N}=3820$. Note that the axis limits differ between panels. Panel (b) reproduced with permission from Ref. [8]. 

Figure 7

Fraction of free chains $x_{\mathrm{free}}$ as a function of $\chi N$ from simulations (symbols) and as predicted by SCFT. Diamonds and circles represent simulation results for $\overline{N}=3820$ and $\overline{N}=960$, respectively. Open and closed symbols represent results of simulations of disordered and ordered phases, respectively. 

Figure 8

Most probable micelle aggregation number $n^{*}$ vs $\chi N$ for (a) $\overline{N}=960$ and (b) $\overline{N}=3820$. Results are shown only for values of $\chi N$ for which $x_n$ exhibits a local maximum. Open and closed symbols represent results obtained from simulations of disordered and ordered states, respectively. Results obtained from ordered states are shown for all values of $\chi N$ for which the ordered state remained stable. Lines represent SCFT predictions for BCC and FCC crystals, which are visually indistinguishable on the scale of this plot. Note that the axis limits differ between panels. 

Figure 9

Average mole fraction of chains $x_n$ present in clusters of aggregation number $n$ for $\overline{N}=960$ at $\chi N=68.5$ for the BCC phase (solid line) and the disordered phase (dashed line). 

Figure 10

Local volume fraction of the minority B-block as a function of the distance from the micelle center for different values of $\chi N$ for (a) $\overline{N}=960$ and (b) $\overline{N}=3820$. The dotted vertical lines indicate the distances $r_c$ used to compute the core volume fractions ${\varphi }_B^{\left(c\right)}$. Note that the abscissa scales are different in the two panels. Panel (b) reproduced with permission from Ref. [8]. 

Figure 11

Probability density for the minority block volume fraction in the micelle core at different values of $\chi N$ for (a) $\overline{N}=960$ and (b) $\overline{N}=3820$. Note that the axis limits differ between the two panels. Panel (b) reproduced with permission from Ref. [8]. 

Figure 12

Statistical characteristics of the minority component core volume fraction ${\varphi }_B^{\left(c\right)}$. (a) Most probable value of ${\varphi }_B^{\left(\mathrm{c}\right)}$ for both models ($\overline{N}=960$ and $\overline{N}=3820$). (b) Mean value of ${\varphi }_B^{\left(\mathrm{c}\right)}$ for both models. In both plots, the vertical-dotted line shows the SCFT ODT value ${\left(\chi N\right)}_{\mathrm{odt}}^{\mathrm{scf}}$, and the dashed purple line shows the SCFT prediction for the the average B volume fraction at the center of a micelle within a BCC crystal for $\chi N>{\left(\chi N\right)}_{\mathrm{odt}}^{\mathrm{scf}}$. 

Figure 13

Semilogarithmic plot of the mole fraction $x_n$ of aggregation number $n$ for the disordered phase (solid-blue line) and ordered BCC phase (dashed-red line). Results for the BCC phase are vertically shifted for clarity by multiplying $x_n$ by ${10}^6$. Data for (a) $\overline{N}=$ 960 are for $\chi N=$ 68.5 and (b) data for $\overline{N}=$ 3820 are at $\chi N=$ 46.5. Note that the axis limits differ between panels. 

Figure 14

Heat map for the joint probability density of specified aggregation number $n$ and shape anisotropy $\mathrm{\Delta }$ in simulations of the BCC ordered phase for (a) $\overline{N}=960$ at $\chi N=68.5$ and (b) $\overline{N}=$ 3820 at $\chi N=46.5$. The red star marker in each plot is the theoretically calculated shape anisotropy for a dimer of two micelles on nearest-neighbor lattice positions. Straight lines in (b) are boundaries of the regions used to divide clusters into individual micelle and micelle multiplets of different multiplicity. Note that the abscissa scales differ between panels. 

Figure 15

Heat map for the frequency of observing shape anisotropy $\mathrm{\Delta }$ and aggregation number $n$, similar to Fig. 14, but now for the disordered phase for (a) $\overline{N}=960$ at $\chi N=68.5$ and (b) $\overline{N}=3820$ at $\chi N=46.5$. Note that the abscissa scales differ between panels. 

Figure 16

Short lived (top row) and long lived (bottom row) micelle dimer profiles showing the variation in an effective radius $r_{\mathrm{eff}}$ vs the distance $z$ from the center of the micelle dimer. $z$ is normalized by $r_0/2$ where $r_0$ is the distance between the nearest-neighboring micelles in a BCC crystal with a unit cell length of $L/3$ where $L$ is the simulation box length. $r_{\mathrm{eff}}$ is given in simulation units, in which the range $\sigma$ of the pair interaction is set to unity. Note that the axis limits differ between panels.