Cell-Encapsulating Droplet Formation and Freezing
MATHEMETICAL MODEL
DROP FORMATION
The standard level set method, also known as the two-phase level set method has been
employed for incompressible two-phase viscous flows [1-3]. For a two-phase flow, the following
equations of motion within each fluid are expressed as
(1)
(2)
where the subscript i denotes the phase present at a given point in space, i.e., liquid or gas. Here, u, p,
D,
, and
denote the velocity vector, the pressure of the fluid, the rate of deformation tensor,
the density and the viscosity of the fluid, respectively. The density and viscosity are discontinuous
across the interface and are functions of time and space, which are defined by using the level set
function. The velocity across the interface is
(3)
where [ ] denotes the jump at the interface between gas and liquid. The interfacial boundary
conditions are applied at the interface as follows:
(4)
in which the normal n is the direction from the liquid to the gas and
denotes a gradient in the
local free surface coordinate. Here, the second term on the right-hand side represents the stress due to
the gradients of surface tension, which is usually important when large gradients of temperature are
present. The interface curvature
is obtained from the following equation:
(5)
The corresponding stress jump condition is given by
(6)
When the jump conditions are incorporated into Equations (1) and (2), the following equation is
obtained:
(7)
in where
is a distance function that has a negative value in the air, a positive value in the liquid,
and zero at the interface, and
is a smoothed delta function. Thus, the location of the interface is
given by the zero level set of the function
. Now that the free surface becomes a material interface
without interfacial mass transfer such as evaporation or condensation, the governing equation for the
level set function is given by
(8)
In addition, the normal vector is written ass
(9)
Please note that because the surface tension increases up to an infinite value in an infinitesimal
volume and the chemophysical properties change abruptly across the interface, it is difficult to obtain
a direct solution. As a result, the interface is smoothed across a finite thickness region. For example,
the density of fluid and the delta function are modified as
(10)
(11)
where g and l indicate the gas and liquid phase, respectively. Here, the smoothed Heaviside function
is given by
(12)
in which
denotes the half-thickness of the property transition region. The time dependent inlet
pressure was imposed as a boundary condition. For the level set,
was applied as the initial
condition. The equations stated above were calculated from finite element simulation using an
iterative solver for the unsteady terms.
PHASE CHANGE
Since the droplets addressed in this study are pico or nanoliter in volume, it is
assumed that the evaporation and the internal circulation of droplets are negligible. Therefore,
the primary mode of heat transfer is not convection but conduction in the droplet (the Peclet
number,
, where D is the diameter of droplets,
velocity of droplets, and the thermal diffusivity
the internal flow
). Moreover, it is presumed that the
polar conduction is not significant compared to the radial conduction in the droplet
(
) [4]. Liquid nitrogen vapor layer is thick enough to cover
the droplet completely. As the temperature of the droplets is reduced, no instabilities or
cellular motion appeared.
The resulting energy equation is given by [5]
(13)
in which  is a differential vector operator and x is the degree of crystallinity (0<x<1).
Chemophysical properties are functions of temperature and the amount of crystallization.
Axisymmetric condition is taken into account. The corresponding boundary conditions are
at r=0 and
at
. The initial condition is
at t=0.
Once the temperature of the liquid reaches around its melting temperature, the liquid
starts to solidify, by ice crystal nucleation and growth. Three main methods macroscopically
model the liquid solidification: uncoupled method, Stefan approach, and zone model [6].
Neglecting the heat generation of the liquid due to latent heat during the cooling process, the
energy equation is decoupled from the kinetics of the liquid.
Stefan’s approach uses
classical moving boundary solutions to model the solidification and sharply divides the entire
domain into the solid and the liquid domain. However, it was reported that the sharp interface
assumption is not always valid. Here, we employ the zone model. One can model the
crystallization process as a propagating zone, depending on the processing conditions and
materials parameters. Furthermore, zone models are based on the hypothesis that the total
heat content can be calculated by an enthalpy function [7]. In this study, the following nonisothermal kinetic equation proposed by Boutron is employed [8]:
(14)
During cooling, homogeneous nucleation first takes place between the homogeneous
nucleation temperature Th and the melting temperature Tm, and then heterogeneous
nucleation follows in the range of the glass transition temperature Tg to homogeneous
nucleation temperature [9]. In the nucleation regimes, the probability for a volume of ice
crystal (V) can be given by J(T,P)Vt, where J(T,P) is the nucleation rate as a function of
temperature T and pressure P. Rapid cooling can shorten the time duration in the nucleation
regime (t), thereby decreasing the probability of ice crystal nucleation. For example, a
cooling rate of ~108 °C/min makes it possible to vitrify even pure water [10]. It is noted that
there exists a critical droplet size beyond which incipient nuclei grow, leading to the
formation of ice crystals. Otherwise, incipient nuclei collapse and an ice crystal is not created.
The thermodynamic characteristic temperatures (Th, Tm, and Tg) change according to the
composition and concentration of solutes. In the current study, we took into account
cryoprotective agents (CPAs), which are chemicals used to minimize damage to cells or
tissues during freezing. As Th and Tm decrease, Tg increases, with increasing CPA
concentration.
EXPERIMENTS
All chemicals including bovine serum albumin (BSA), trehalose dihydrate, 1,2-propanediol,
and phosphate buffered saline (PBS) were purchased from Aldrich. To generate a droplet, an air line
was connected to the droplet ejector to provide the suitable pressure needed to overcome surface
tension at the orifice of the ejector. The solenoid ejector was controlled by a pulse generator. The
freezing medium consisted of 10% trehalose dihydrate (v/v), 4% BSA and 3M 1,2-propanediol in
PBS. After filtering the solution, the solution was kept at at 37 C. Similarly, the thawing medium
was prepared with 10% trehalose dihydrate (v/v), 4% BSA and 1.M 1,2-propanediol in PBS. The
images of the frozen droplets in the liquid nitrogen and the ejected droplets in the air were taken using
a polarized filter and a charge-coupled device (CCD) camera system, respectively. According to the
following protocol, we carried out the experiments.
For the preparation of the freezing medium, 10% trehalose dihydrate (v/v) and 4% BSA were
dissolved in PBS. After then, 1,2-propanediol was added to bring molarity to 3 M, and the solution
was filtered using a syringe filter. We kept the solution at 37 C. For the thawing medium, 10%
trehalose dihydrate (v/v) and 4% BSA were dissolved in PBS. 1,2-propanediol was added to bring
molarity to 1.5 M and then the solution was filtered using a syringe filter.
For the equipment setup, the droplet ejector was connected to the syringe using a needle and
proper tubing. Then the freezing media containing cells was loaded into the syringe. The air line was
connected to the syringe to provide suitable pressure that can overcome the surface tension at the
orifice of the ejector. The pressurized air is provided from a nitrogen gas tank. The solenoid ejector
was connected to the pulse generator.
For the preparation of cells for the experiments, cells were trypsinized to detach them from
the flask. After then, the trypsin solution was diluted with the cell culture media. All the contents of
the flask were transferred into a 15 ml centrifuge tube. The cells were spun down at 200 g for 5 min
using a centrifuge, and then the supernatant was aspirated off. The cell pellets were resuspended in
PBS, and a hemacytometer count was performed on a 50 l sample of the solution. The cell pellets
were resuspended in the first freezing medium for 5 min. The cells were spun down for 5 min with the
centrifuge, and the supernatant was then suck off. Then, the cells were resuspended at 1x106 cells ml-1
of the second freezing medium. 1 ml of cell-suspended cryoprotectant media was loaded into the
ejector.
Cell-encapsulating droplets were ejected into the liquid nitrogen directly. The cell strainer
full of vitrified droplets was removed from the liquid nitrogen bath and plunged into the first thawing
as quickly as possible. The cell strainer was taken out of the thawing medium after giving the cells
enough time to disperse into the medium.
REFERENCES
[1] J.A. Sethian, PNAS 20, 1591-1595 (1996).
[2] A. Iafrati, A. Di Mascio, and E.F. Campana, Int J Numer Meth Fluids 35, 281-297 (2001).
[3] M. Sussman, P. Smereka, and S. Osher, J Comput Phys 114, 146-159 (1994).
[4] S. Prakash and W.A. Sirignano, Int J Heat Mass Transfer 23, 253-268 (1980).
[5] A. Jiao, X. Han, J.K. Critser, and H. Ma, Cryobiology 52, 386-392 (2006).
[6] G. Astarita and J.M. Kenny, Chem Eng Commun 53, 69-84 (1987).
[7] A. Benard and S.G. Advani, Int J Heat Mass Trans 38, 819-832 (1995).
[8] P. Boutron and P. Mehl, Cryobiology 27, 359-377 (1990).
[9] P.G. Debenedetti and H.E. Stanley, Phys Today 56, 40-46 (2003).
[10] G.P. Johair, A. Hallbrucker, and E. Mayer, Nature 330, 552-553 (1987).