Diffusion through colloidal shells under stress

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Entry by Emily Redston, AP 225, Fall 2011

Work in Progress


Diffusion through colloidal shells under stress by J. Guery, J. Baudry, D. A. Weitz, P. M. Chaikin, and J. Bibette. Phys. Rev. E 79, 060402(R) (2009).


One area of great interest in soft matter is encapsulation. With applications in almost all types of industrial domains, from the oil industry to food packaging, efficient storage of gases and liquid in solid containers is of tremendous technological and economical importance. Furthermore, encapsulation of active ingredients such as drugs, proteins, nutrients, or vitamins is essential for a myriad of applications, such as drug delivery and agrichemicals.

The goal of encapsulation is to protect the delicate substances inside from a harsh environment, and to retain their activity until some required time. Long-term storage of liquids or gases often involve internal pressures, which increase the tensile stress of the container wall. At a colloidal scale, it is the osmotic pressure difference between the internal and external medium that is relevant, rather than the hydrostatic pressure. Eyring proposed that the permeability of solids is associated with a diffusive process involving an activation mechanism. Unfortunately, many of his ideas have not been tested quantitatively and thus many fundamental concepts are not fully understood.

In this paper, the authors propose a very cute little experiment that tests some of Eyring's ideas. By using core-shell (liquid core - solid shell) colloidal particles that are sensitive to osmotic pressure, they are able to follow the permeation of encapsulated probes at various stresses.

The Core-Shell Colloids

The authors create these colloidal shells using a double emulsion, which consists of droplets of water in larger drops of crystallizable oil. The oil is a mixture of crystallizable triglycerids, which allows the authors to make the emulsion at high temperature, around 70°C, where the oil is fluid, and then cool the sample to solidify the oil, creating a robust shell. The robust solid shell is only achieved at temperatures below the melting temperature of the oil (around 45°C). NaCl is added to the water to prevent ripening of the droplets, and glucose is also added to match the chemical potential of the inner water to the continuous phase water. The osmotic pressure difference is then deduced from the concentration of salt and glucose.

Experiments and Results

Figure 1. (a) Temporal evolution of the fractional release (X) for a shell composed of pure oil DM, at different temperatures: (<math>\circ</math>) 15 °C; (<math>\vartriangle</math>) 45 ° C; (<math>\diamond</math>) 60 ° C; and (<math>\Box</math>) 65 ° C. The solid lines correspond to the best adjustment of the data with the equation: X(t)=1−[(1−<math>X_{0}</math>)exp(−<math>\Beta_{0}</math>t)]. (b) Evolution of the characteristic relaxation rate <math>\Beta_{0}</math> in logarithmic scale with 1/<math>k_{B}</math><math>T_{r}</math>. The solid line corresponds to an exponential adjustment.
Figure 2. (a) Temporal evolution of the fractional release (X), for different dilution factors (T = 15°C); (<math>\circ</math>) d=0, <math>\Delta</math><math>\Pi</math>=0 atm; (<math>\times</math>)d=1.4, <math>\Delta</math><math>\Pi</math>=5 atm; (<math>\triangledown</math>) d=1.8, <math>\Delta</math><math>\Pi</math>=8 atm; (<math>\vartriangle</math>) d=4, <math>\Delta</math><math>\Pi</math>=13 atm; (<math>\diamond</math>) d=10, <math>\Delta</math><math>\Pi</math>=15 atm; (<math>\Box</math>) d=100, <math>\Delta</math><math>\Pi</math>=17 atm. The solid lines correspond to the best numerical adjustment of the data with Eq. (2). (b) Evolution of dX/dt (t=0) in logarithmic scale with 1−(1/d). The solid line corresponds to an exponential adjustment.

Potentiometric titration is used to measure the concentration of chloride ions as they are released into the continuous water phase. The relative ionic concentraion in the water ouside of the colloids, X, can then be found by normalizing the concentration at each time by the value obtained when all ions are released. Fig. 1(a) shows this fractional release X as a function of time at different temperatures (from 15 to 65°C for the oil). At 15°C the oil is essentially solid, while at 65°C it is entirely melted. The authors believe that the measured release results entirely from the diffusion of sodium and chloride ions from the inner droplets toward the external phase because there is no observed coalescence of the inner droplets with the globule interface.

This passive permeation must obey Fick’s law given by J = −d<math>N_{i}</math> / dt = PS(<math>C_{i}</math> - <math>C_{e}</math>), where J is the flux of salt from inside toward the external phase, <math>N_{i}</math> is the total number of salt ions inside the globule, P is the permeation coefficient of the shell, S is the total shell surface, and C is the concentration of salt inside (i) and in the external phase (e). The authors then assume that the limiting step is the permeation through the outer shell. The oil film between inner water droplets is much thinner than the shell thickness. Hence the salt concentration Ci is essentially homogeneous within the inner water droplets during the leakage process. If the globule volume fraction <math>\phi</math> is small, we have

dX/dt = <math>\Beta_{0}</math>(1 − X), (1)

where the ion concentration X = <math>C_{e}</math>(t) / <math>C_{e}</math> (t = infinity), and where <math>\Beta_{0}</math> = 3P/a is the characteristic relaxation rate, with a being the globule radius. Therefore X(t)=1−[(1−<math>X_{0}</math>)exp(−<math>\Beta_{0}</math>t)], where <math>X_{0}</math> is the initial burst fractional release which arises during the double emulsification process. For all temperatures studied the release mechanism is well described by a single exponential relaxation of time scale 1/<math>\Beta_{0}</math>, in agreement with the original Eyring assumption that the permeation coefficient P is proportional to exp(<math>E_{a}</math> / <math>k_{B}</math>T), where <math>E_{a}</math> is the activation energy and <math>k_{B}</math>T is the thermal energy. Fig. 1(b) shows ln<math>\Beta_{0}</math> as a function of 1 / <math>k_{B}</math>TT, from which the authors deduce that the activation energy <math>E_{A}</math> is ~ (5 – 7)<math>k_{B}</math><math>T_{r}</math>, <math>T_{r}</math> = 25° C. In the absence of osmotic mismatch the leakage of the colloidal system is strictly diffusion driven, with no discontinuity at the liquid-solid transition again in agreement with the Eyring picture.