Discotic Dispersions Mediated by Depletion

Álvaro González García

doi:10.1007/978-3-030-33683-7_6

Polymer-Mediated Phase Stability of Colloids pp 85-109 | Cite as

Discotic Dispersions Mediated by Depletion

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Álvaro González García

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First Online: 29 October 2019

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Part of the Springer Theses book series (Springer Theses)

Abstract

In this Chapter we show how free volume theory (FVT) is an efficient and tractable thermodynamic framework capable of unravelling the complicated multi-phase behaviour of disc–polymer mixtures. Three principal reference phases for the hard platelets are considered: isotropic (I), nematic (N), and columnar (C). We derive analytical expressions that enable us to systematically trace the different types of phase coexistences revealed upon adding depletants, and confirm the predictive power of FVT by testing the calculated diagrams against phase stability scenarios from alternative approaches. A wide range of multi-phase equilibria is revealed, involving two-phase isostructural transitions of all phase symmetries (I, N, C) considered as well as the possible three-phase coexistences. Moreover, we identify the system parameters, relative disc shapes and colloid–polymer size ratios, at which four-phase equilibria are expected. These involve a remarkable coexistence of all three phase states commonly encountered in discotics including isostructural coexistences I$_1$–I$_2$–N–C, I–N$_1$–N$_2$–C, and I–N–C$_1$–C$_2$. The isostructural C$_1$–C$_2$ coexistence is analysed in detail and compared to direct-coexistence Monte Carlo computations. We improve FVT for disc–polymer mixtures, particularly accounting for a better depletant partitioning over the discotic columnar phases. From theory and simulations it is clear that in the C$_1$ phase depletants are present between the flat faces of the discs, as opposed to the denser (C$_2$) phase. Consequently, the C$_1$–C$_2$ coexistence is driven by the depletant partitioning in the intra-columnar direction. This study helps to understand the role of compartmentalisation in highly asymmetric, crowded systems.

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6.A Thermodynamics of Pure Platelet Suspensions

Various thermodynamic properties of pure platelet suspensions have been studied in detail previously [26, 49], and we solely report here the key ingredients required to calculate the final phase diagrams of model colloidal platelet-polymer mixtures. Entropy-driven phase transitions [50] as considered here depend on the excluded volume between two colloidal particles. This excluded volume is defined as the volume inaccessible to a second particle in the system as a consequence of the presence of a first particle. For two colloidal platelets, the excluded volume ($v_\text {exc}^\text {p-p}$) per particle volume ($v_\text {c}$) reads [51]:

$$\begin{aligned} \frac{v_\text {exc}^\text {p-p}}{v_c}&= 2 \left[ \left| \cos \gamma \right| +\frac{4 E\left( \sin \gamma \right) }{\pi }+1\right] +\frac{8 \Lambda \sin \gamma }{\pi }+\frac{2 \sin \gamma }{\Lambda } \text {,} \end{aligned}$$

(6.10)

where $\gamma $ defines the relative orientation between two colloidal discs, and E(x) is the complete elliptic integral of the second kind. Considering the symmetry of a platelet, its orientation can be defined via a unit vector ($\hat{u}$) in the axis of symmetry of the cylinder. Hence, $\cos \gamma = \hat{u}\cdot \hat{u}'$, where $\hat{u}'$ simple refers to a second cylinder. For the isotropic and nematic phases we consider Onsager–Parsons-Lee theory [26, 52]. The free energy of both the isotropic and nematic phases reads:

$$\begin{aligned} \frac{\widetilde{F}_k}{\phi _\text {c}} =\ln {\widetilde{v}_\text {c}}+\ln \phi _\text {c}-1+\sigma _k[f(\hat{u})]+\frac{2}{\pi }\frac{\phi _\text {c}}{\Lambda }G_\text {P}\langle \langle \Theta _\text {Exc}(\hat{u},\hat{u}') \rangle \rangle \text {.} \end{aligned}$$

(6.11)

The first two terms on the right-hand-side of Eq. (6.11) correspond to the finite-volume normalization of the energy ($\widetilde{v}_\text {c}$ being the dimensionless thermal volume of a platelet) and the ideal gas contribution to the free energy.

In order to calculate the free energy of the isotropic and nematic phases, an orientational distribution function [ODF, $f(\hat{u})$] needs to be accounted for. A system in which particle orientations are taken into account can be envisaged as a multi-component system in which each component corresponds to a possible particle orientation [51, 53]. Hence, the ODF is a measure of the probability of finding a particle with a given orientation $\hat{u}$. The rotational entropy term, $\sigma _k[f(\hat{u})]$ is defined as [52]:

$$\begin{aligned} \sigma _k[f(\hat{u})]&= \int {f(\hat{u})\ln [4\pi f(\hat{u})]\text {d}\hat{u}} \text {.} \end{aligned}$$

(6.12)

The dimensionless ensemble-averaged excluded volume follows from:

$$\begin{aligned} \langle \langle \Theta _k^\text {Exc}(\hat{u},\hat{u}') \rangle \rangle&= \frac{1}{\sigma ^3} \int {\int { f(\hat{u}) f(\hat{u}') v_\text {exc}^\text {p-p}(\hat{u}\cdot \hat{u}')\text {d}\hat{u} \text {d}\hat{u}'}} \text {.} \end{aligned}$$

(6.13)

Finally, effects beyond the second osmotic virial coefficient are accounted for in an approximate manner via the Parsons-Lee scaling factor [54, 55]:

$$\begin{aligned} G_\text {P}&= \frac{4-3 \phi _\text {c} }{4 (1-\phi _\text {c} )^2} \text {.} \end{aligned}$$

Formally, at each platelet concentration the free energy of the system must be minimized with respect to the ODF, $f(\hat{u})$. Analytical expressions for the ODF can be obtained for the isotropic state by considering equiprobability of orientations: $f(\hat{u}) = 1/(4\pi )$. In this case, $\sigma _\text {I}[f(\hat{u})] = 0$, and by applying the so-called isotropic averages ($\langle \langle \sin \gamma \rangle \rangle _\text {I} = \pi /4 $, $\langle \langle E\{\sin \gamma \} \rangle \rangle _\text {I} = \pi ^2/8 $, and $ \langle \langle \cos \gamma \rangle \rangle _\text {I} = 1/2 $), the free energy of an isotropic ensemble of discs can be written as [26]:

$$\begin{aligned} \frac{\widetilde{F}_\text {I}}{\phi _\text {c}} = \ln {\widetilde{v}_\text {c}} + \ln \phi _\text {c}-1 + \frac{2}{\pi }\frac{\phi _\text {c}}{\Lambda }G_\text {P}\Theta _\text {I}^\text {Exc} \text {,} \end{aligned}$$

(6.14)

with:

$$\begin{aligned} \Theta _\text {I}^\text {Exc} = \frac{\pi ^2}{8} + \left( \frac{3 \pi }{4}+\frac{\pi ^2}{4}\right) \Lambda + \frac{\pi \Lambda ^2}{2} \text {.} \end{aligned}$$

(6.15)

Furthermore, closed expressions for the free energy of the nematic phase can be obtained via a Gaussian approximation [56] for $f(\hat{u})$. Considering that all relative orientations can be defined as a Gaussian perturbation from the nematic director vector provides a closed-form expression for the free energy [26]. This Gaussian ODF reads

$$\begin{aligned} f_\text {N--G}(\theta )&= \frac{\kappa }{4\pi }\exp \left[ -\frac{1}{2}\kappa \theta ^2\right] \text {,} \end{aligned}$$

(6.16)

where $\theta $ is the polar angle between the nematic director and the orientation of the platelet. Minimizing the free energy with respect to the unknown parameter of the Gaussian ODF $\kappa $ provides a closed form for the free energy of the nematic phase [26]:

$$\begin{aligned} \frac{\widetilde{F}_\text {N--G}}{\phi _\text {c}} = \ln {\widetilde{v}_\text {c}} + \ln \phi _\text {c}-1 + \sigma _\text {N--G} + \frac{2}{\pi }\frac{\phi _\text {c}}{\Lambda }G_\text {P}\Theta _\text {N--G}^\text {Exc} \text {,} \end{aligned}$$

(6.17)

with:

$$\begin{aligned} \sigma _\text {N--G} = \ln \left[ \frac{\pi \phi _\text {c}^2 G_\text {P}^2}{\Lambda ^2}\right] -1\text {, } \end{aligned}$$

and

$$\begin{aligned} \Theta _\text {N--G}^\text {Exc}= \frac{1}{2}\pi ^{3/2}{\kappa (\phi _\text {c},\Lambda )}^{-1/2}+2 \pi \Lambda \text {.} \end{aligned}$$

For the columnar phase, a modified Lennard-Jones–Devonshire (LJD) cell-theory [57] approach provides a closed expression for the free energy [26, 43]:

$$\begin{aligned} \frac{\widetilde{F}_\text {C}}{\phi _\text {c}}&= \ln {\widetilde{v}_\text {c}} + \ln \phi _\text {c} -3 -2 \ln \left[ 1-\frac{1}{\widehat{\Delta }_\bot }\right] +2 \ln \left[ \frac{3\widehat{\Delta }_\bot ^2 \phi _\text {c}^\text {r}}{2 \Lambda \left( 1-\widehat{\Delta }_\bot ^2 \phi _\text {c}^\text {r}\right) }\right] -\ln \left[ \frac{1}{3} \left( 1-\widehat{\Delta }_\bot ^2 \phi _\text {c}^\text {r}\right) \right] \text {,} \end{aligned}$$

(6.18)

where the lateral spacing (inter-columnear direction) is:

$$\begin{aligned} \widehat{\Delta }_\bot \equiv {\Delta }_\bot /\sigma&= \frac{\root 3 \of {2} \bar{K}^{2/3}-\root 3 \of {3} 4 \phi _\text {c}^\text {r}}{6^{2/3} \root 3 \of {\bar{K}} \phi _\text {c}^\text {r}}\text {,} \end{aligned}$$

(6.19)

where

$$\begin{aligned} \bar{K}&= \sqrt{3(\phi _\text {c}^\text {r})^{3} (243 \phi _\text {c}^\text {r}+32)}+27 (\phi _\text {c}^\text {r})^{2} \text {,} \end{aligned}$$

(6.20)

and with

$$\begin{aligned} \phi _\text {c}^\text {r}&= {\phi _\text {c}}/{\phi _\text {c}^\text {cp}} \text {,} \end{aligned}$$

(6.21)

and

$$\begin{aligned} \phi _\text {c}^\text {cp} = {\pi }/{(2\sqrt{3})} \approx 0.907 \text {.} \end{aligned}$$

(6.22)

The approximated expression for the intra-columnar spacing [43] $\Delta _\parallel $ is:

$$\begin{aligned} \widehat{\Delta }_\parallel \equiv \Delta _\parallel /L&=\frac{1}{\phi _\text {c}^\text {r}\widehat{\Delta }_\bot ^2} \end{aligned}$$

(6.23)

The (dimensionless) osmotic pressures and chemical potentials follow the relations in Chap. 1:

$$\begin{aligned} \beta \mu \equiv \widetilde{\mu } = \left( \frac{\partial \widetilde{F}_\text {c}}{\partial \phi _\text {c}}\right) _{T,V} \quad \text {;}\quad \beta \Pi v_\text {c}\equiv \widetilde{\Pi } = \phi _\text {c}\widetilde{\mu }-\widetilde{F}_\text {c} \text {,} \end{aligned}$$

(6.24)

6.B Geometrical Free Volume Fraction in the Columnar State

We follow the ideas put forward in Chap. 3 to calculate a geometrical free volume fraction for PHS in a columnar state. Let $V_\text {UC}$ be the volume of the columnar unit cell, such that:

$$\begin{aligned} V_\text {UC}/v_\text {c}=\frac{3 \pi ^2}{16\phi _\text {c}}\text {.} \end{aligned}$$

(6.25)

If there is no overlap of depletion zones (colloidal concentrations far from the close packing), the free volume for depletants is simply the volume unoccupied by the depletion zones. Overlap of the depletion zones lead to an increase of the free volume fraction for depletants. In the case of the platelets, overlap of the depletion zones occurs either from the side or from the flat phases of the hard disc. These two contributions can be conveniently split. Due to the different scalings of the unit cell, [26] overlap in the intracolumnar direction occurs at lower colloidal concentrations than in the intercolumnar one. One must account for the total number of overlaps: nine in the intercolumnar direction and three in the intracolumnar one. This allows to cast $\alpha _\text {col}^\text {geo}$ in a generic form:

$$\begin{aligned} \alpha _\text {C}^\text {geo} = {\left\{ \begin{array}{ll} 1- \frac{3v_\text {excl}^{\text {HP-PHS}}}{V_\text {UC}} &{}\text { if } \phi _\text {c}<\phi _\text {c}^\parallel \text { (no overlap)}, \\ 1- \left( \frac{3v_\text {excl}^{\text {HP-PHS}}}{V_\text {UC}} - \frac{3v_\text {overl}^\parallel }{V_\text {UC}} \right) &{}\text { if } \phi _\text {c}^\parallel \le \phi _\text {c} < \phi _\text {c}^\bot \text { (overlap in}\, r_\parallel ) \\ 1- \left( \frac{3v_\text {excl}^{\text {HP-PHS}}}{V_\text {UC}} - \frac{3v_\text {overl}^\parallel }{V_\text {UC}} - \frac{9v_\text {overl}^\bot }{V_\text {UC}} \right) &{}\text { if } \phi _\text {c}^\bot \le \phi _\text {c} \text { (overlap in}\, r_\parallel \text { and }r_\bot ), \end{array}\right. } \end{aligned}$$

(6.26)

where $\phi _\text {c}^\parallel $ is the solution of the equation

$$\begin{aligned} \widehat{\Delta }_\parallel (\phi _\text {c}) = 1+q/\Lambda \text {,} \end{aligned}$$

(6.27)

and $\phi _\text {c}^\bot $ is the solution of the equation

$$\begin{aligned} \widehat{\Delta }_\bot (\phi _\text {c}) = 1+q. \end{aligned}$$

(6.28)

Note first that the term $v_\text {excl}^{\text {HP-PHS}}$ accounts for the depletion zone volume [equivalent to Eq. (6.2) minus the term corresponding to the sphere volume, $2q^3/(3\Lambda ^4)$]. Also note that with the approach described here we only account for two-body overlaps of the depletion zones, which is sufficient for small enough q-values. Further, in Eq. (6.26) the condition ‘no free volume for depletants’ is not shown for simplicity. Upon some algebra, these three different contributions read:

$$\begin{aligned} \begin{aligned} \frac{3v_\text {excl}^{\text {HP-PHS}}}{V_\text {UC}}&= \phi _\text {c}\left[ (1+q)^2+\frac{q}{6\Lambda }\left( 6+3\pi q +4 q^2\right) \right] \\ \frac{3v_\text {overl}^\parallel }{V_\text {UC}}&= \phi _\text {c} \left( 1+\frac{q}{\Lambda }\right) - \frac{\pi }{2\sqrt{3}\widehat{\Delta }_\bot } + \frac{2\phi _\text {c}}{3}\left[ 3\Lambda A_2(q/(2\Lambda ),\widehat{\Delta }_\parallel -1) + \Lambda ^2 \left( \widehat{\Delta }_\parallel -1\right) ^3 + \frac{q^3}{\Lambda } \right] \\ \frac{9v_\text {overl}^\bot }{V_\text {UC}}&= \frac{\phi _\text {c}}{\pi }{12 A_2\left( \frac{1+q}{2},\widehat{\Delta }_\bot \right) }\text {,} \end{aligned} \end{aligned}$$

(6.29)

where $A_2$ is the area of intersect between two discs with radius R at a distance r:

$$\begin{aligned} A_2(R,r)=2 R^2 \cos ^{-1}\left( \frac{r}{2 R}\right) -\frac{1}{2} r \sqrt{4 R^2-r^2} \end{aligned}$$

(6.30)

We shall finally note here that the algebraic complexity of Eq. (6.26) arises mostly due to the contribution to the depletion zone of the edges of the platelet.

6.C Direct Coexistence Monte Carlo Simulations

The (modified) direct Monte Carlo coexistence simulations consider hard colloidal platelets as hard oblate hard spherocylinders (OHSC) mixed with a non-adsorbing species which is modelled as penetrable hard spheres (PHSs). The direct colloid–colloid and depletant–colloid interactions are hard, whereas there are no depletant–depletant interactions. Simulations start with two non-equilibrated simulation boxes in contact. Each box contains either the C$_1$ or C$_2$ FVT-predicted coexistence volume fractions of colloids and polymers. The oblates are arranged in two columnar phases with the same column axes, whereas the depletants are distributed randomly without colloid–polymer overlaps. The whole simulation box contains N particles (discotics plus polymers), and a (MC) cycle is defined as N trials to displace and/or rotate a randomly chosen particle plus an attempt to change the aspect ratio of the simulation box (its volume is fixed). Two different equilibration steps are considered. Firstly, $1\times 10^6$ cycles are conducted restricting the depletants to the volumes that they occupied in the initial configuration (equilibration of the discotic phases). Secondly, $3\times 10^6$ cycles are carried out without restrictions (equilibration of the direct coexistence). Ensemble-averaged equilibrium colloid and polymer volume fractions are collected over the last $2\times 10^6$ cycles. The method used here is a modified direct coexistence Monte Carlo [58] approach. It is applied, as far as we are aware of, for the first time to directly study isostructural coexistence in highly dense discotic systems explicitly accounting for a binary mixture (OHSCs and PHSs). These equilibrium direct-coexistence MC colloid and polymer volume fractions are used for two independent sets of simulations ($1\times 10^6$ cycles to equilibrate, plus $2\times 10^6$ cycles for production), from which colloid–colloid and colloid–depletant distribution functions elucidate the structural details of the C$_1$ and C$_2$ phases. These direct coexistence simulations were blind-tested: two starting configurations at different colloid packing fractions in absence of depletants melt into a single one. Further details of the simulation method are beyond the scope of this thesis.

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Álvaro González García
- 1
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1.Van ’t Hoff Laboratory for Physical and Colloid Chemistry, Department of Chemistry and Debye InstituteUtrecht UniversityUtrechtThe Netherlands

Discotic Dispersions Mediated by Depletion

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