Bridging transitions for spheres and cylinders

We study bridging transitions between spherically and cylindrically shaped particles (colloids) of radius $R$ separated by a distance $H$ that are dissolved in a bulk fluid (solvent). Using macroscopics, microscopic density-functional theory, and finite-size scaling theory, we study the location and order of the bridging transition and also the stability of the liquid bridges, which determines spinodal lines. The location of the bridging transitions is similar for cylinders and spheres, so that at bulk coexistence, for example, the distance $H_b$ at which a transition between bridged and unbridged configurations occurs is proportional to the colloid radius $R$. However, all other aspects, particularly the stability of liquid bridges, are very different in the two systems. Thus, for cylinders the bridging transition is typically strongly first-order, while for spheres it may be first-order, critical, or rounded as determined by a critical radius $R_c$. The influence of thick wetting films and fluctuation effects beyond mean field are also discussed in depth.

Figure 1

Macroscopic phase diagram for first-order bridging transitions between completely wet cylinders immersed in a solvent at two-phase coexistence. The line represents the bridging coexistence curve $H_b=\pi R$. The bridged phase remains metastable for all $H>H_b$.

Figure 2

Macroscopic phase diagram for first-order bridging transitions between completely wet cylinders immersed in a solvent off coexistence. The solid line represents the bridging coexistence curve $H_b/L=2R/Lf(R/L)$, while the dashed line is the spinodal curve $H_{\mathrm{spin}}/L=2\sqrt{\frac{R^2}{L^2}+\frac{2R}{L}}$ at which the bridging film pinches.

Figure 3

Illustration of the length scales characterizing a macroscopic bridged configuration for spheres; see text for details.

Figure 4

Contact angle dependence of the parameter $\alpha \left(\theta \right)$ appearing in the macroscopic prediction $H_b=\alpha R$ for spheres (blue) and cylinders (red, uppermost), in which case the parameter $\alpha$ is given by Eq. (2). No bridging is possible for $\theta =\frac{\pi }{2}$. The dashed line represents the contact angle dependence of ${\alpha }_{\mathrm{spin}}\left(\theta \right)$ determining the spinodal $H_{\mathrm{spin}}={\alpha }_{\mathrm{spin}}\left(R\right)$ for spheres that lie close to the coexistence line.

Figure 5

Macroscopic phase diagram for first-order bridging transitions between completely wet spheres immersed in a solvent at bulk coexistence. The solid line represents the bridging coexistence curve $H_b\approx 2.32R$, while the dashed line is the spinodal curve $H_{\mathrm{spin}}/L\approx 2.38R$. The inset shows the dependence of the parameter $a$, characterizing the catenoid shape $acosh\frac{x}{a}$, as a function of $H$. The termination of the curve determines the spinodal at which the catenoid no longer touches the spheres.

Figure 6

Schematic behavior of the solvation force showing possible scenarios for the bridging transition between spheres or cylinders: (a) first-order bridging, (b) critical bridging, and (c) rounded bridging.

Figure 7

Phase diagram for cylindrical bridging transitions obtained for $T=0.89T_c$ at a two-phase coexistence. The symbols represent DFT results obtained from a numerical solution of Eq. (21). The red dashed line is a straight line with a slope of $\pi$ in accord with the macroscopic prediction given by Eq. (1). The black dotted line is a fit containing a log correction due to complete wetting by gas around the parametrized results of the cylinders as $H_b=\pi R+alnR/\sigma$. We find with $a=3.34\sigma$, in good agreement with the predicted value $a=\pi {\xi }_g\approx 3.52$.

Figure 8

Two-dimensional density profiles of the saturated liquid near two infinite cylinders each with a radius $R_c=70\sigma$ whose centers are separated by a distance $H=230\sigma$. The density profiles correspond to a first-order bridging transition at which the unbridged state (a) and the bridged state (b) coexist.

Figure 9

Phase diagram for cylindrical bridging transitions obtained for $T=0.89T_c$ off coexistence showing a comparison between DFT (symbols) and macroscopic theory (solid line) as given by Eqs. (5) and (6). Here, the Laplace radius $L\approx 5\sigma$. The bridging transition remains first-order down to microscopic radii, but it exhibits a critical point below the radius $R=3\sigma$ (see the text). Note that in contrast to the case of coexistence shown in Fig. 7, where the macroscopic theory was confirmed by analyzing the slope of the phase boundary for sufficiently large $R$, here the full comparison is made for the whole range of radii considered.

Figure 10

Two-dimensional density profiles of the supersaturated liquid near two cylindrical colloids each with a radius $R=\sigma$ (a) and $R=5\sigma$ (b) corresponding to the bridging transition. For $R=5\sigma$, the transition is first-order and the boundary of the bridging film is near circular with a radius of curvature $L\approx 5\sigma$ in line with the macroscopic considerations. For $R=\sigma$, the transition from a bridged to an unbridged state is continuous.

Figure 11

Phase diagram for spherical bridging transitions obtained for $T=0.89T_c$. The symbols refer to the DFT results: the filled squares denote points for which the bridging transition is first-order, while the empty squares correspond to a continuous bridging transition. The line represents a linear fit with slope $2.38$ close to the predicted macroscopic value $2.32$. The critical radius is estimated as $R_c\approx 10\sigma$.

Figure 12

Two-dimensional density profiles of the saturated liquid near two spherical colloids each with a radius $R=25\sigma$ whose centers are separated by a distance $H=64\sigma$. The density profiles correspond to a first-order bridging transition at which the unbridged state (a) and the bridged state (b) coexist. A fit to a catenoid shape is shown (black line).

Figure 13

DFT results for the solvation force $f_s\left(H\right)$ of a fluid with short-range forces adsorbed between two spheres as a function of separation for two different radii. (a) $R=25\sigma >R_c$, first-order bridging (blue line); (b) $R=7.5\sigma <R_c$, rounded bridging (red line, rightmost). The dotted vertical line represents the macroscopic prediction for $H_b/R=2.32$.

Figure 14

Schematic phase diagram for bridging transitions and typical configurations between completely wet (dry) cylinders, taking into account the effect of fluctuations. The dotted line represents the location of the macroscopic phase boundary $H_b=\pi R$, while the dashed line represents the crossover region of width $\sim e^{-2\pi \beta {\gamma }_{\text{lg}}{R}^2}$ centered around $H_b\left(r\right)$ over which the bridging transition is rounded. In the crossover region, we have highlighted the typical size of the domains of bridged and unbridged regions.