Figure 1

Effect of confinement. Snapshots of the anomalous liquid (green, small spheres) confined in a fixed matrix of NPs (yellow, large spheres) in a DIST (a) and RND (b) configuration. Polynomial fits of simulated isochores of densities $0.89\le \rho /{\rho }_c^{\mathrm{bulk}}\le 1.59$ (bottom to top in the one-phase region) for DIST (c) and RND (d). Randomness reduces the temperature and pressure of the LLCP (circles), the separation between the HDL (lower filled triangles) and LDL spinodals (upper open triangles), and the separation between the TMD (diamonds) and the temperature of minimum densities (TminD, squares). Samples of error bars on $P$ are given in panel (c). Lines connecting symbols are guides for the eyes.Reuse & Permissions

Figure 2

The effect of different confinements on the parameters of the LLCP. Color coded circles (from dark blue for lower $\rho$ to bright red for higher$\rho$) represent the LLCP parameters in the $P-T-\rho$ phase space (a) for the liquid confined in the fixed matrix of NPs with CUBE, DIST, and RND configuration. Increasing disorder in the confinement, from CUBE to DIST to RND, shifts the LLCP down in $\rho$, $T$, and $P$. (b) Upon decreasing concentration $x_{\mathrm{NP}}$ (label near the symbols) for the CUBE confinement of the liquid (ii), the LLCP approaches the bulk case. Here, we use $N_{\mathrm{NP}}=64\mathrm{NPs}$ with $D_{\mathrm{NP}}/a=3$ in $V/a^3={20.6}^3$ at $x_{\mathrm{NP}}=24.5\%$ with $1452\le N\le 2508$ (with spontaneous crystallization below the LLCP), or in $V/a^3={24}^3$ at $x_{\mathrm{NP}}=15.5\%$ with $2570\le N\le 4439$, or in $V/a^3={28}^3$ at $x_{\mathrm{NP}}=9.8\%$ with $4358\le N\le 7528$. We find the same behavior for liquid (i). The models of liquid with two potentials (i) and (ii) are described in the text.Reuse & Permissions

Figure 3

The liquid adsorbs onto the NPs. The fluid density profile $g_{\mathrm{NP}–\mathrm{liq}}(r)$ at $T/T_c^{\mathrm{bulk}}=1.12$ for CUBE (leftmost), DIST (center), and RND (rightmost) confinements for the density $\rho /{\rho }_c^{\mathrm{bulk}}=1.59$ (solid lines) display large maxima (values in parenthesis) at the closest NP-liquid particle distance $r=r_0\equiv 2a$. Local compressibility $K_T$ (dashed lines) shows large peaks near the minimum of $g_{\mathrm{NP}–\mathrm{liq}}(r)$. The results for different confinements are shifted horizontally for clarity. Inset: Schematic representation of calculation of $g_{\mathrm{NP}–\mathrm{liq}}(r)$ and local $K_T$ inside equal-volume ($\Delta W=2.77a^3$) conical regions between two concentric spheres with different radii $R$ and $R+\Delta R$ centered at the NP (small yellow hemisphere in the middle), where $\Delta R=0.2a$ and $R=2.0a,2.2a,\dots ,8.0a$ (one such conical region is shown in dark red). The axis of the conical region is chosen at random 10 000 times for each NP. $g_{\mathrm{NP}–\mathrm{liq}}(r)$ is computed by counting the number of liquid particles and local $K_T=[(⟨n^2⟩/⟨n{⟩}^2)-1](\Delta W/k_{B}T)$ from fluctuations of number of liquid particles $n$.Reuse & Permissions

Figure 4

The distribution of local density $\mathcal{D}({\rho }_i/{\rho }_c^{\mathrm{bulk}})$ of the liquid inside the pockets for global liquid density $\rho /{\rho }_c^{\mathrm{bulk}}=0.94$ and $T/T_c^{\mathrm{bulk}}=0.88$. (a) In the DIST confinement, at $P/P_c^{\mathrm{bulk}}=1.3$, ${\mathcal{D}}^{\mathrm{DIST}}({\rho }_i/{\rho }_c^{\mathrm{bulk}})$ is a Gaussian centered at $\rho /{\rho }_c^{\mathrm{bulk}}=0.94$ with standard deviation ${\sigma }_D=0.055$. Inset: Cut through the simulation box. The liquid density (high to low color coded from blue to red) is computed inside spheres of radius $1.5a$ that do not intersect NPs. Areas for which we cannot evaluate liquid density with this method are in black. (b) In RND confinement, at $P/P_c^{\mathrm{bulk}}=1.25$, the broad ${\mathcal{D}}^{\mathrm{RND}}({\rho }_i/{\rho }_c^{\mathrm{bulk}})$ (red dashed line) is the result of two Gaussian components, both centered in $\rho /{\rho }_c^{\mathrm{bulk}}=0.94$, but with different standard deviations: one is due to the local density fluctuations (black line) with ${\sigma }_{R1}=0.052$, as in DIST, and the other with ${\sigma }_{R2}=0.159$ (shaded) due to the heterogeneity in pockets volumes. Inset: As in panel (a), but for RND. (c) Calculation of $P^{\mathrm{RND}}$ by taking into account the component of ${\mathcal{D}}^{\mathrm{RND}}({\rho }_i/{\rho }_c^{\mathrm{bulk}})$ due to the heterogeneity in pocket volumes. Polynomial fits of the isotherms, $P^{\mathrm{DIST}}(\rho )$ at constant $T$ (black dashed lines: from top to bottom $T/T_c^{\mathrm{bulk}}=0.77,0.69,0.61$), are used in Eq. (1) to get an estimate of $P^{\mathrm{RND}}(\rho )$ at the same $T$ (red solid lines), which compare well, within a range of densities close to the LLPT, with the simulation data for RND (symbols). Inset: 2D representation of the exclusion spheres (black, grey, white circles) with their radii $r_e$ (yellow arrows), which for $r_e=r_0$ (black circles) coincides with NPs. A cavity is shown as a blue diamond and a pocket is shown as a red square-like segment. For clarity, the liquid (colored regions) is shown inside cavities only.Reuse & Permissions