Concentration dependence of the sol-gel phase behavior of agarose-water system observed by the optical bubble pressure tensiometry

We have studied an expansion behavior of pressurized bubbles at the orifice of a capillary inserted in gelator-solvent (agarose-water) mixtures as a function of the gelator concentration in which the phase transition points are included. The pressure (P) -dependence of the radius of the curvature (R) of the bubbles monitored by laser beam has shown a discontinuous decrease in the exponent (m) of the experimental power law R = KΔP−m (K: constant) from 1 to 1/2 and a discontinuous increase in the average surface tension (γave) obtained from the work-area plots of the mixtures exceeding that of pure water (72.6 mN/m) at 0.02 < [agarose] < 0.03 wt%, which is attributed to the disappearance of the fluidity. The apparent surface tension (γapp = ΔP/2 R) of the system in the concentration range of 0.03–0.20 wt% has been analyzed by a modified Shuttleworth equation γapp = σ0 + τln(A/A0), where σ0 is an isotropic constant component and the second term is a surface area (A) -dependent elastic component, in which τ is the coefficient and A0 is the area of the orifice. The analysis has indicated that σ0 coincides with the γapp value of the mixture of 0.02 wt% and the second term at >0.02 wt% is the dominant component. From the appearance of the elastic component and concentration dependence of τ, the plateau of τ for the agarose-water mixtures at 0.03–0.10 wt% (Region II) has been explained to the phase separation giving two-phase mixtures of 0.02 wt% sol and 0.10 wt% gel and the upward inflection of τ at 0.10 wt% has been assigned to an increase in the elasticity of the gel with the increase of the agarose concentration in the range of >0.10 wt% (Region III). On considering the concentration dependence of the surface tension of agarose-water mixtures, the discontinuous and inflection points were assigned to the 1st- and 2nd-order phase transition concentrations of the agarose gel, respectively. Given the results with our tensiometry based on the optical bubble pressure method, distinct gelation points for other systems could be determined both mechanically and thermodynamically which will provide a diagnostic criterion of sol-gel transitions.

window at the bottom) to the depth of h = 10.0 mm by lifting a sample cell with a rack-and-pinion stage of a precision of 0.1 mm. The static pressure (Ps) operating to the meniscus is given by the depth h as Ps = Pout = gh, where  is the density of the sample liquid. A gas inlet was connected to a manometer using silicone tubing equipped with a T-joint and a syringe to pressurize the capillary. An aqueous solution of sodium dodecyl sulfate (SDS, [SDS] = 5 × 10 mol dm −3 , density: 0 = 0.99889 g cm −3 at 20 °C) was used as a gauge liquid of the manometer. The pressure applied to the meniscus (Pin) was measured by the manometer with a precision of 1 Pa. A laser beam from a semiconductor laser (650 nm, 0.2 mW, ≈0.9 mm diameter) was passed through an aperture (A1, ≈0.8 mm diameter) and the center of the capillary. The dispersed beam was focused by an external plano-convex lens (L, focal length of f2 = 100.0 mm, diameter of 50 mm) onto another aperture (A2) in the back where a silicon photodiode (Hamamatsu, S6675, bias voltage: 3.0 V) as a detector was placed as to maximize output voltage when the beam is focused onto the aperture. The diameter of the laser beam was also adjusted by means of an iris (I) in a range of 0.1-0.8 mm. The use of the whole meniscus as a lens was avoided to remove the spherical aberration caused by the peripheral part which may contain the effect of the interaction between the capillary wall and the sample, and the central part was used by adjusting the iris to select the diameter of the laser beam of ≈0.7 mm. The beam profile of the laser was not Gaussian and used without any modification by optical devices such as telescope. The output of the photodetector was monitored by a digital oscilloscope (Iwatsu DS-5320) at a rate of 10 Hz. When a gas pressure was applied to the capillary, a bubble grew and departed from the orifice. The gas pressure was gradually increased and noted, then, the position of the external lens was adjusted with a precision of 0.1 mm to maximize the output of the photodetector. The back focal length (BFL, f) of the combination of the meniscus and the lens can be determined from the positions of the meniscus and the external lens from the aperture (eq S1), where d is the distance between the meniscus and the lens L.
From the negative focal length f1, the radius of the meniscus (R) can be derived using a value of refractive index of the liquid (n) by eq S2.
The measured focal length was corrected with the thickness of the window of the optical cell (quartz, d' = 2 mm, nD 20 = n'= 1.458). The insertion of the window into the optical path shifts the imaginary focal point of the meniscus (eq S3). Figure S1. The set-up of the tensiometer using a pressurizing capillary and a laser for monitoring the radius of the curvature of the meniscus. The diverged laser beam by the meniscus is focused onto an aperture (A2) in front of the photodiode. Holders and posts of the optical components are omitted.

Measurement of surface tension of water and agarose-water mixtures
We considered the refractive indices of the samples and glasses used in this study for 650 nm light to be the same as those for 589 nm light (nD 20  The observed change in the radius of the curvature of the meniscus R can be considered to be a result of the change in the depth of the meniscus zc. Since the meniscus is a part of the sphere with a radius of R and its cross section has a radius of the capillary (R0), the depth zc can be estimated using eq S4 ( Figure S3).
Surface area (A) and volume (V) for a meniscus are given by eq S5 and S6 as functions of zc and R0. 19,32 The Gibbs energy change (G) for the increase of the volume of the meniscus (V) relative to the plane of the orifice upon the increase of the applied pressure difference (P) and was calculated by the pointto-point calculation of the area of the P-V curve (eqs S7 and S8) , where i is the number of the point.   Figure S4. A plot of the applied pressure difference (P) versus volume of the meniscus (V) calculated from the radius of the curvature of the meniscus with eq S6 for a mixture of agarose-water (agarose content 0.14 wt%) at 20℃.

Discontinuity of the surface tension at the phase boundary
As shown in Figure 5, discontinuity and inflection of the surface tension of agarose-water mixture were observed. We have to mention for the use of the terms, 1st-and 2nd-order transitions for the binary system and for the assignment of the discontinuity and inflection with respect to the concentration of agarose to the 1st-and 2nd-order transitions, although the original Ehrenfest's classification of phase transition deals with single component systems based on the first and second derivatives of the Gibbs energy with respect to temperature or pressure. A -transition in the heat capacity of Cu-Zn alloy (-brass) is exemplified for the 2nd-order phase transition of a binary mixture in a recent textbook of physical chemistry (ref. S1). We also mention for the parameter to describe the phase transition is the chemical potential () due to its intensive nature and not the Gibbs energy in the textbook (ref. S1).
The observed discontinuity of the surface tension at the concentration ≈0.025 wt% does not assure mathematically the discontinuity of the surface tension at 20℃ with respect to the temperature.
However, it is a cross point of the binodal curve with the T = 20℃ line and this point obviously indicates the transition concentration. Since the surface tension  has a negative gradient toward temperature (a') <  (a) in general and is discontinuous at the point (T = 20℃ and c = ≈0.025 wt%)  (a) <  (b), then, we will obtain similar discontinuity of  (b) <  (b') at T = 20℃ at the limit of T0 = Phase 1 and Phase 2 exhibits different concentration dependence of the surface tension. A concentration change process (a→b) can be replaced with a 3-step process (a→a'→ b'→b) because of the path-independent nature of thermodynamic state functions.