Critical Casimir effect in classical binary liquid mixtures

If a fluctuating medium is confined, the ensuing perturbation of its fluctuation spectrum generates Casimir-like effective forces acting on its confining surfaces. Near a continuous phase transition of such a medium the corresponding order parameter fluctuations occur on all length scales and therefore close to the critical point this effect acquires a universal character, i.e., to a large extent it is independent of the microscopic details of the actual system. Accordingly it can be calculated theoretically by studying suitable representative model systems. We report on the direct measurement of critical Casimir forces by total internal reflection microscopy with femtonewton resolution. The corresponding potentials are determined for individual colloidal particles floating above a substrate under the action of the critical thermal noise in the solvent medium, constituted by a binary liquid mixture of water and 2,6-lutidine near its lower consolute point. Depending on the relative adsorption preferences of the colloid and substrate surfaces with respect to the two components of the binary liquid mixture, we observe that, upon approaching the critical point of the solvent, attractive or repulsive forces emerge and supersede those prevailing away from it. Based on the knowledge of the critical Casimir forces acting in film geometries within the Ising universality class and with equal or opposing boundary conditions, we provide the corresponding theoretical predictions for the sphere—planar wall geometry of the experiment. The experimental data for the effective potential can be interpreted consistently in terms of these predictions and a remarkable quantitative agreement is observed.

Figure 1

(a) Geometry of the Derjaguin approximation for the plate-sphere geometry and (b) cross section through the center of the sphere and normal to the plate. In (a), the base of a cap of the sphere with radius $r\left(\theta \right)$ is shown as a thin line. The inner circle of the grey ring $dS\left(\theta \right)$ is this thin line shifted by $R\sin \theta d\theta$ toward the center of the sphere. In (b), $\delta =r(\theta +d\theta )-r\left(\theta \right)$ is the cross section of the gray ring $dS\left(\theta \right)$ shown in (a).

Figure 2

Scaling functions ${\Theta }_{(+,+)}$ and ${\Theta }_{(+,-)}$ of the Casimir potential ${\Phi }_C$ [see Eq. (12)] for (+,+) and (+,−) BCs, respectively, within the Derjaguin approximation and for the three-dimensional Ising universality class, as functions of $u=\tau {(z∕{\xi }_0^{+})}^{1∕\nu }$ with $\nu \simeq 0.630$. The thick solid and dashed lines have been obtained via Eq. (13) on the basis of the Monte Carlo estimates for ${\vartheta }_{(+,+),(+,-)}$ presented in Refs. 15, 16, indicated by (i) and (ii), respectively, in Figs. 9 and 10 of Ref. 16. [The thick solid lines agree with the estimates reported in Fig. 2(d) of Ref. 37.] ${\Theta }_{(+,+)}$ attains its minimum value ${\Theta }_{(+,+)}^{\left(\min \right)}\simeq -3.6$ (solid line) and $-2.8$ (dashed line) both for $u_{\min }\simeq 0.54$, whereas ${\Theta }_{(+,-)}$ attains (smoothly) its maximum value ${\Theta }_{(+,-)}^{\left(\max \right)}\simeq 19$ (solid line) and 14 (dashed line) both for $u_{\min }\simeq -0.03$. The first derivatives of ${\Theta }_{(+,-)}$ and ${\Theta }_{(+,+)}$ diverge logarithmically for $u\to 0$. The thin lines for $u>1$ indicate the asymptotic behaviors of $\Theta (u⪢1)$ given in Eq. (14) with the numerical values of the coefficients $A_{\pm }$ indicated from top to bottom for the corresponding curves. For (+,+) boundary conditions the asymptotic expressions are indistinguishable from ${\Theta }_{(+,+)}\left(u\right)$ for $u\gtrsim 1$.

Figure 3

Scaling functions ${\widehat{\vartheta }}_{(+,+)}$ and ${\widehat{\vartheta }}_{(+,-)}$ of the Casimir force $F_C$ [see Eq. (10)] for (+,+) and (+,−) BCs, respectively, within the Derjaguin approximation and for the three-dimensional Ising universality class, as functions of $u=\tau {(z∕{\xi }_0^{+})}^{1∕\nu }$ with $\nu \simeq 0.630$. The thick solid and dashed lines have been obtained via Eq. (11) on the basis of the Monte Carlo estimates for ${\vartheta }_{(+,+),(+,-)}$ presented in Refs. 15, 16, indicated by (i) and (ii), respectively, in Figs. 9 and 10 of Ref. 16. ${\widehat{\vartheta }}_{(+,+)}$ attains its minimum value ${\widehat{\vartheta }}_{(+,+)}^{\left(\min \right)}\simeq -4.9$ (solid line) and $-3.8$ (dashed line) both for $u_{\min }\simeq 2.6$, whereas ${\widehat{\vartheta }}_{(+,-)}$ attains its maximum value ${\widehat{\vartheta }}_{(+,-)}^{\left(\max \right)}\simeq 21$ (solid line) and 16 (dashed line) both for $u_{\min }\simeq -1.5$. The second derivatives of ${\widehat{\vartheta }}_{(+,+)}\left(u\right)$ and ${\widehat{\vartheta }}_{(+,-)}\left(u\right)$ diverge logarithmically for $u\to 0$. The thin lines for $u>1$ indicate the asymptotic behaviors $\widehat{\vartheta }(u⪢1)$ given in Eq. (14) with the numerical values of the coefficients $A_{\pm }$ indicated from top to bottom for the corresponding curves. For (+,+) boundary conditions the asymptotic expressions are indistinguishable from ${\widehat{\vartheta }}_{(+,+)}\left(u\right)$ for $u\gtrsim 5$. In the inset we compare the estimate for ${\widehat{\vartheta }}_{(+,+)}\left(x\right)∕{\widehat{\vartheta }}_{(+,+)}\left(0\right)$ as a function of $x=z∕\xi$ based on the Monte Carlo data of Refs. 15, 16 (solid line, MC) with the one presented in Ref. 19 and obtained by interpolating linearly and pointwise the exactly known film scaling functions in $d=2$ and $d=4$ in order to obtain an estimate for $d=3$ (dotted line, interpol.). The Monte Carlo estimate for this ratio is the same for both data sets (i) and (ii) in Ref. 16.

Figure 4

(Color online) Schematic phase diagram of a binary liquid mixture with a lower demixing transition point in terms of temperature $T$ and concentration $c_A$ of the $A$ species. The solid curve encloses the two-phase region separating via first-order phase transitions the $\alpha$ and $\beta$ phases rich in $A$ and $B$ species, respectively, terminating at the CP. The dashed line indicates the first-order bridging phase transition which occurs if a fluid mixture is confined between a planar wall and a sphere of radius $R$ and at distance $L$ possessing the same adsorption preference, here for the $\alpha$ phase. The bridging transition ends at the critical point $Q$ and separates a region in the bulk phase diagram in which a phase preferable by walls condenses and forms a bridge connecting the wall and the sphere, from the region in the bulk phase diagram where such a bridge is absent. Although the bridging transition is a (quasi-) first-order phase transition, in the bulk phase diagram it is described by a line instead of a coexistence region, because it is an interfacial phase transition.

Figure 5

(Color online) Data acquisition system (see main text for details). The green laser light generated by the optical tweezers is deflected by a double prism into the microscope objective (in the figure represented as a gray vertical cylinder above the spherical particle) and it provides an optical potential which confines the spherical colloid laterally. The blue light which is scattered by this particle out of the evanescent field of intensity $I_{\mathrm{ev}}\left(z\right)$ is collected by the same microscope objective, focused and then optically directed into the photon counter via a combination of prisms and mirrors (schematically represented in the upper part of the figure).

Figure 6

Data analysis. (a) Raw data for the total number of scattered photons $n_{\mathrm{sc}}\left(t\right)$ detected at a certain time $t$ within a time interval $\Delta t=1\mathrm{ms}$ from the single photon counter as a function of time, taken for a sampling time $t_{\mathrm{samp}}\simeq 15\min$. (b) Histogram calculated from this time series yielding the distribution function $p_{\mathrm{sc}}\left(n_{\mathrm{sc}}\right)$ of the number $n_{\mathrm{sc}}$ of scattered photons. (c) The knowledge of the relation [see Eq. (18)] between the scattered intensity $I_{\mathrm{sc}}\simeq n_{\mathrm{sc}}∕\Delta t$ and the position $z$ of the colloid in the evanescent field allows one to determine the probability distribution $p_z\left(z\right)=\zeta n_{\mathrm{sc}}\left(z\right)p_{\mathrm{sc}}\left(n_{\mathrm{sc}}\left(z\right)\right)$ for the particle-substrate distance $z$ from $p_{\mathrm{sc}}\left(n_{\mathrm{sc}}\right)$ and therewith by inversion of the Boltzmann factor the interaction potential. Further details are given in the main text.

Figure 7

Diffusion coefficient $D_{\perp }$ of a spherical particle of radius $R$ moving perpendicular to a wall at a distance $z$ [62, 71]. $D_{\infty }$ is the bulk diffusion coefficient. Note that for the present experimental conditions $z∕R<1$ and thus this distance dependence is pronounced.

Figure 8

(Color online) Bulk phase diagram of the binary liquid mixture of water and 2,6-lutidine (dimethylpyridine ${\mathrm{C}}_7{\mathrm{H}}_9\mathrm{N}$) [56, 73] at constant volume. The relevant thermodynamic variables are the temperature $T$ and the mass fraction $c_L$ of lutidine in the mixture. Open symbols in panel (a) refer to actual experimental data [73]. The schematic side view of a vertical sealed cell filled with the binary liquid mixture is shown by insets (i) and (ii) of panel (a) for thermodynamic states outside and inside of the coexistence loop, respectively. In (ii) W and L indicate the water- and the lutidine-rich phases, respectively. The mixture separates into the lutidine-rich and the water-rich phases within the two-phase coexistence area encircled by the solid first-order transition line. At the lowest and highest points (LCP and UCP, respectively) of this line the demixing transition is continuous. The detailed phase diagram in the vicinity of the lower critical point is shown in panel (b), together with the typical thermodynamic paths (dashed vertical lines) experimentally investigated here within the gray region as well as for $c_L=0.2$.

Figure 9

(Color online) Permittivities $\epsilon \left(i\omega \right)$ of the materials relevant for our experiment as functions of the frequency $\omega$ (rad/s) on a logarithmic scale. As on the left, from bottom to top, we report the curves corresponding to pure lutidine, polystyrene $\left({\epsilon }_{\mathrm{coll}}\right)$, silica $\left({\epsilon }_{\text{glass}}\right)$, a water-lutidine mixture with a volume concentration ${\varphi }_{\mathrm{L}}\simeq 0.25$ of lutidine $\left({\epsilon }_{\mathrm{liq}}\right)$, and pure water. These curves are based on the parametrizations and on the material properties reported in Refs. 72, 77. The four vertical lines indicate the frequencies ${\left\{{\omega }_n\right\}}_{n=1,\dots ,4}$ which within our approximation enter into the determination of the Hamaker constant in Eq. (29). [Note that $\mathrm{Im}\epsilon \left(\omega \right)=0$ if, as in the present case, $\mathrm{Re}\omega =0$ [72].]

Figure 10

(Color online) Effective interaction potential $\Phi \left(z\right)$ between a wall and a spherical particle of radius of $1.2\mu \mathrm{m}$ immersed in a water-lutidine mixture at the critical concentration $c_L^c$ as a function of the distance $z$ from the wall and for various values of the temperature in the one-phase region $(T<T_c)$ [37]. The gravitational and the offset contribution to the potential [see Eq. (26)] have been subtracted. The set of solid and dashed lines, which are barely distinguishable on this scale, correspond to the theoretical predictions (see the main text for details). The potentials reported here refer to the (−,−) boundary conditions (other cases are reported in Figs. 11, 12) and show that an increasingly attractive force contributes to the total potential upon approaching the critical temperature, i.e., upon decreasing $\Delta T=T_c-T$. Here $T_c$ is the nominal value of the critical temperature, corresponding to the anomaly in the background scattering, which signals the onset of critical opalescence in the sample. Only the data to the right of the vertical dotted line are considered for the comparison with the theoretical predictions (see the main text).

Figure 11

(Color online) Interaction potential $\Phi \left(z\right)$ as in Fig. 10 for the (+,−) boundary conditions and $R\simeq 1.85\mu \mathrm{m}$ [37]. An increasingly repulsive force contributes to the total potential upon approaching the critical temperature. The set of solid and dashed lines, which are barely distinguishable on this scale, correspond to the theoretical predictions (see the main text for details). Only the data to the right of the vertical dotted line are considered for the comparison with the theoretical predictions (see the main text).

Figure 12

(Color online) Interaction potential $\Phi \left(z\right)$ as in Figs. 10, 11 for the (+,+) boundary conditions [37]. As for (−,−) boundary conditions (see Fig. 10), upon approaching the critical temperature an increasingly attractive force contributes to the total potential.

Figure 13

(Color online) Interaction potential $\Phi \left(z\right)$ as in Figs. 10, 11, 12 for the (−,+) boundary conditions. As in the case of the (+,−) boundary conditions (see Fig. 11) an increasingly repulsive force contributes to the total potential upon approaching the critical temperature. The set of solid and dashed lines, which are barely distinguishable on this scale, correspond to the theoretical predictions (see the main text for details). Only the data to the right of the vertical dotted line are considered for the comparison with the theoretical predictions (see the main text).

Figure 14

(Color online) Correlation length $\xi$ as a function of the temperature deviation $\Delta T=T_c-T$ from the experimentally located critical temperature $T_c$ for the water-lutidine mixture at the critical concentration. The sets of data points are obtained by optimizing the agreement between the experimental data from Fig. 10 and the corresponding theoretical predictions for the Casimir potential for the (−,−) boundary conditions (Fig. 2, see the main text for details). The upper (lower) set of points has been determined by using as the scaling function ${\Theta }_{(-,-)}={\Theta }_{(+,+)}$ of the Casimir potential the one reported by the dashed (solid) line in Fig. 2. The dashed and solid curves are the best fit of the corresponding data sets based on Eq. (32) with the corresponding values $\Delta T_c^{\left(\mathrm{fit}\right)}=T_c-T_c^{\left(\mathrm{fit}\right)}$ indicated by the vertical dashed and solid lines, respectively, i.e., $\xi$ diverges at $T_c^{\left(\mathrm{fit}\right)}$ corresponding to $\Delta T(T=T_c^{\left(\mathrm{fit}\right)})=T_c-T_c^{\left(\mathrm{fit}\right)}=\Delta T_c^{\left(\mathrm{fit}\right)}$.

Figure 15

(Color online) Same as Fig. 14, obtained by optimizing the agreement between the experimental data from Fig. 11 and the corresponding theoretical predictions for the critical Casimir potential for the (+,−) boundary conditions (Fig. 2).

Figure 16

(Color online) Same as Figs. 14, 15, obtained by optimizing the agreement between the experimental data from Fig. 13 and the corresponding theoretical predictions for the critical Casimir potential for the (−,+) boundary conditions (Fig. 2).

Figure 17

(Color online) Effective interaction potential $\Phi \left(z\right)$ between a wall and a particle of diameter $2R=3.69\mu \mathrm{m}$, immersed in a water-lutidine mixture at mass fraction $c_L=0.2<c_L^c$, as a function of the distance $z$ and for various values of the temperature close to the demixing phase boundary [37]. This system corresponds to the (+,+) boundary conditions. The gravitational and offset contributions to the potential [see Eq. (26)] have been subtracted. The abrupt formation of a narrow minimum of the potential close to the wall upon increasing the temperature is interpreted as the formation of a liquid bridge between the particle and the wall (see the main text).