Swollen lamellar phases confined in capillarylike pores

Structure and mechanical properties of swollen lamellar phases confined in square-base pipes are studied in a mesoscopic lattice model for oil-water-surfactant mixtures. Structure depends crucially on a thermodynamic state and is quite different far and close to the coexistence with a uniform phase. Lamellar domains with different orientations of lamellas are formed in most cases, and the mechanics is determined mainly by domain-wall energies. Shift of phase equilibria in square-base pipes compared to the bulk is just opposite to the shift in slits. We find capillary delamellarization for short-period and for swollen phases, for hydrophilic and for neutral external surfaces.

Figure 1

The cross section through the square-base capillary. The characteristic lengths and vectors are shown. See text for more details.

Figure 2

(a) Schematic of the frozen lamellar structure with $\widehat{\mathbf{n}}=(1,1,0)$ and $\lambda =12d$. (b) Schematic of the frozen lamellar structure with $\widehat{\mathbf{n}}=(1,0,0)$ and $\lambda =6a$. The filled and open circles denote water and oil clusters, respectively, and the arrow denotes the cluster of oriented surfactant.

Figure 3

(Color online) Water-covered walls, (a) $L∕d=82$, (b) $L∕d=84$, and (c) $L∕d=90$. The case (c) corresponds to the equilibrium width with no stress, $\Pi =0$. (d) Neutral walls $L∕d=82$. The thermodynamic variables $kT∕b=2.65$, $\mu ∕b=4.6$ and material constants $c∕b=2.5$, $g∕b=1$ correspond to a stable lamellar phase with the period $\lambda ∕d=12$.

Figure 4

(a) The excess pressure $\Pi =-(1∕2L)\partial {\Omega }_{\mathit{ex}}^z∕\partial L$; (b) $\Pi \left(L\right)+({\sigma }_{\Vert }+{\sigma }_{\perp })∕L$ [see Eqs. (7, 8)] for the square-base pore with side $L$ as functions of $L$. The walls are neutral (surface fields $h_i=0$). The solid lines in (b) are given by the first term on the rhs of Eq. (8) for $L$ such that $\mid L-L_N\mid ∕d⩽2$. The surface tensions are ${\sigma }_{\Vert }=0.190b∕d^2$ and ${\sigma }_{\perp }=0.188b∕d^2$. The optimal value of the fitting parameter $B_l$ is $B_l=0.0003kT∕d^3$. The thermodynamic variables $kT∕b=2.65,\mu ∕b=4.6$ and material constants $c∕b=2.5,g∕b=1$ correspond to a stable lamellar phase with the period $\lambda ∕d=12$. For any value of $L$ the square-base pore favors the lamellar structure Fig. 3d. Lengths are measured in $d$ units, and $b$ is the water-water interaction energy.

Figure 5

(a) $[{\Omega }_{\mathit{ex}}^z\left(L_N\right)-4{\sigma }_{\Vert }{L}_N]∕\left(2\sqrt{2}\right)$ at the values of the width of the pore $L=L_N$ corresponding to no stress. ${\sigma }_{\Vert }=-0.483b∕d^2$. From the slope of the straight solid line we can deduce the value of ${\sigma }_{\mathit{dw}}∕\left(2\sqrt{2}\right)$ in the absence of the stress for the phase with $\lambda =12d$. Note that ${\sigma }_{\mathit{dw}}∕\left(2\sqrt{2}\right)$ is analogous to ${\gamma }_o$ for the onion structure, and that for short-period phases the same value of ${\gamma }_o\approx 0.034b∕a^2$ was obtained in Ref. 5 for $kT∕b=2.1$ and $\mu ∕b=3.1$. (b) ${\Omega }_{\mathit{ex}}^z\left(L\right)-(4{\sigma }_{\Vert }+{\sigma }_{\mathit{dw}})L$ for the square base pore with side $L$, as a function of $L$. The walls are water covered. The thermodynamic variables are $kT∕b=2.65,\mu ∕b=4.6$ and the material constants are $c∕b=2.5$ and $g∕b=1$ (off phase coexistence).

Figure 6

(a) $\Pi \left(L\right)+2{\sigma }_{\Vert }∕L$, and (b) $\Pi \left(L\right)+(2{\sigma }_{\Vert }+{\sigma }_{\mathit{dw}}∕2)∕L$, for the square-base pore with side $L$, as functions of $L$. The walls are water covered. ${\sigma }_{\Vert }=-0.483b∕d^2$ and ${\sigma }_{\mathit{dw}}=0.068\sqrt{2}b∕d^2$. The solid lines are the best fits of the elastic term with $B_o=0.0002b∕d^3$. The thermodynamic variables and material constants are the same as in Fig. 5.

Figure 7

The difference between densities of water and oil, ${\rho }_w-{\rho }_o$, as a function of $x$ for $y=L∕2$. (a) $L∕d=L_4^t=106$, and (b) $L∕d=120=L_5^t-2$. There are four lamellar layers (oil and water-rich domains) absorbed at the walls, and the thickness of the central uniform oil-rich tube is $L_u\approx L-4\lambda$. The walls are water covered, the thermodynamic variables $kT∕b=2.65,\mu ={\mu }_{\mathit{coex}}(1-0.0016)$ and material constants $c∕b=2.5,g∕b=1$ are close to the bulk coexistence of the lamellar phase with the period $\lambda ∕d=16$, and the uniform, oil-, and water-rich phases.

Figure 8

(a) ${\Omega }_{\mathit{ex}}^z-4{\sigma }_{\Vert }L$ at the values of the width of the pore $L=L_N^t$ corresponding to the minimal size of the central water-, or oil-rich domain. ${\sigma }_{\Vert }=-0.480b∕d^2$. (b) ${\Omega }_{\mathit{ex}}^z\left(L\right)-(4{\sigma }_{\Vert }+{\sigma }_{\mathit{dw}})L$; (c) ${\Omega }_{\mathit{ex}}^z\left(L\right)-4{\sigma }_{\Vert }L-{\sigma }_{\mathit{dw}}N\lambda$ for the square-base pore with the side length $L$, as a function of $L$. The walls are water covered. The thermodynamic variables $kT∕b=2.65,\mu ∕b={\mu }_{\mathit{coex}}(1-0.0016)$, and material constants $c∕b=2.5,g∕b=1$ correspond to the stability of the lamellar phase with period $\lambda ∕d=16$.

Figure 9

The difference between densities of the water and oil, ${\rho }_w-{\rho }_o$, as a function of $x$ for $y=L∕2$ in the onion phase with (a) five and (b) six lamellar layers absorbed at the walls at the capillary delamellarization transition given by $kT∕b=2.65,{\mu }_{cc}={\mu }_{\mathit{coex}}(1-0.0045)$. The pore size $L∕d=106$ and the walls are water covered. The values of the thermodynamic variables and material constants $c∕b=2.5,g∕b=1$ correspond to the stability of the lamellar phase with the period $\lambda ∕d=16$.

Figure 10

Structural transitions between onion phases with different numbers of lamellar layers at the walls in the square-base capillary for $kT∕b=2.65$. The walls of the pore are water covered. The lines separate the stability region of the onion structures with $N$ layers (above) from the stability region the structures with $N+1$ layers (below in $\Delta \mu ∕{\mu }_{\mathit{coex}}$). The case of $1⩽N⩽5$ is shown. The horizontal line corresponds to the value of the chemical potential, at which the results shown in Fig. 8 have been obtained.