# Diffusion through colloidal shells under stress

Entry by Emily Redston, AP 225, Fall 2011

Work in Progress

## Contents

## Reference

*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).

## Introduction

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 harsh outer environments 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 of encapsulation 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 created 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

Potentiometric titration was used to measure the concentration of chloride ions as they are released into the continuous water phase. The relative ionic concentration in the water outside 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 should 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 the 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 in this process is the permeation through the outer shell. Since the oil film between inner water droplets is much thinner than the shell thickness, the salt concentration <math>C_{i}</math> is essentially homogeneous within the inner water droplets during the leakage process. If the globule volume fraction <math>\phi</math> is small, you can write

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>T. From this, the authors deduce that the activation energy <math>E_{A}</math> is ~ (5 – 7)<math>k_{B}</math><math>T_{r}</math>, where <math>T_{r}</math> = 25° C. In the absence of any osmotic mismatch, the leakage of the colloidal system is strictly diffusion driven, with no discontinuity at the liquid-solid transition. This agrees very well with the Eyring picture.

Dilution of the double emulsion by a factor of 100 using pure water leads to an osmotic pressure difference of <math>\Delta</math><math>\Pi</math> = 17 atm. Due to a slight permeability of water through the shell, the chemical potential of the water equilibrates rapidly. This causes an immediate increase in the the internal pressure. At 15°C, where the oil shell is solid, there is a rapid increase in the release of Cl- ions, followed by slower approach to the asymptotic limit, as shown by the open squares in Fig. 2(a). The authors vary the driving osmotic pressure, <math>\Delta</math><math>\Pi</math>, by using water with increasing concentrations of glucose, diluting the double emulsion by the same factor each time, thereby reducing the rate of release. For <math>\Delta</math><math>\Pi</math> = 0, there is a residual, very slow release of Cl− shown by the open circles in Fig. 2(a). For <math>\Delta</math><math>\Pi</math> <math>\ne</math> 0, the osmotic pressure mismatch causes a tensile stress on the shell, which should modify the activation energy.

To check this hypothesis, the authors looked at the relation between the osmotic pressure difference and the resulting tensile stress that acts on the colloidal shell. In the limit of a thin shell of thickness <math>\delta</math> , this relation can be simply derived from a force balance argument. The force exerted by the osmotic pressure mismatch on each half shell, <math>\pi</math> <math>a^2</math><math>\Delta</math><math>\Pi</math>, and the force due to the tensile stress,<math>\tau</math>, acting on the perimeter that holds the two half shells together, 2<math>\pi</math>a<math>\delta</math><math>\tau</math>, must be equal, <math>\tau</math> = <math>\Delta</math><math>\Pi</math>(a/2<math>\delta</math>). Part of the local activation energy <math>E_{A}</math> for an ion permeating a solid comes from the deformation energy of adding its additional volume to the solid matrix. The worklike energy for adding this volume increases under compressive stress and decreases under tensile stress in a complex manner, which nonetheless must be linear in applied stress and ionic volume, <math>\lambda^3</math>. The activation energy is expected to be linearly modified by the tensile stress as E(<math>\tau</math>)=<math>E_{A}</math>−<math>\tau</math><math>\lambda^3</math>.

The osmotic pressure difference <math>\Delta</math><math>\Pi</math> can be approximated as <math>\Delta</math><math>\Pi</math> = RT(2<math>C_{i}</math> − 2<math>C_{e}</math> − <math>C_{sugar}</math>) ~ RT(2<math>C_{i}</math> − <math>C_{sugar}</math>). <math>C_{e}</math> can be neglected as long as the globule volume fraction remains small. <math>C_{sugar}</math> is the final sugar concentration imposed by the dilution, and R is the ideal gas constant. <math>\Delta</math><math>\Pi</math> can now be expressed as <math>\Delta</math><math>\Pi</math> = 2RT<math>C_{i0}</math>(1 − X − 1/d), where <math>C_{i0}</math> is the initial internal salt concentration and d is the dilution factor of the external phase, which sets the initial osmotic pressure difference <math>\Delta</math><math>\Pi_{0}</math>. Thus, Eq. (1) becomes dX/dt = <math>\beta</math>(X)(1 − X) and the characteristic relaxation rate <math>\beta</math>(X)can be expressed as

<math>\beta</math>(X) = <math>\beta_{0}</math>exp[<math>\epsilon</math>(1 − 1/d)]exp(−<math>\epsilon</math>X), (2)

with <math>\epsilon</math>=(a/<math>\delta</math>)(<math>\lambda^3</math>/(<math>k_{B}</math>T)(RT<math>C_{i0}</math>). In kT units, <math>\epsilon</math>(1−1/d) is the initial drop of the activation energy induced by the initial tensile stress. Thus, at short time, when few ions have diffused through the shells and the osmotic shock is still at its maximum, the permeation rate <math>\beta</math> increases by the large factor exp[<math>\epsilon</math>(1 − 1/d)]. At longer time, the second term exp(−<math>\epsilon</math>X), which reflects the equilibration of the osmotic stress, slows down the relaxation rate. A plot of log[dX/dt (t=0)] as a function of 1−1/d is linear, with slope <math>\epsilon</math>, as seen in Fig. 2(b). From the slope, the authors found that <math>\epsilon</math>=2.5<math>\pm</math>0.3. The authors then solved equation (2) numerically and compared it to their data, as shown in Fig. 2(a). They also took the nonideal dependence of the osmotic pressure with the sugar and salt concentrations into account when making this fit. The fit allows for the determination of the two unknown parameters <math>\epsilon</math> and <math>\beta_{0}</math>.

## Conclusion

Encapsulation has the potential to be an extremely rewarding field. Unfortunately, there are still a lot of unknowns in regards to the fundamental physics of encapsulation. These authors present some of the first direct experimental evidence that an imposed tensile stress linearly reduces the local energy barrier for an activated process. This leads to an exponential increase in diffusion and permeability, as envisioned by Eyring almost 100 years ago. In other words, they found that the activation energy associated with the permeation process is decreased in proportion to an applied tensile stress, even though the diffusion species are not submitted to any external forces. Their rather simple experiment lead to findings that could provide some guidance in colloidal controlled drug delivery systems and encapsulation design.